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LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor M. Reid, Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom The titles below are available from booksellers, or from Cambridge University Press at http://www.cambridge.org/mathematics 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350

Topics on Riemann surfaces and Fuchsian groups, E. BUJALANCE, A.F. COSTA & E. MARTÍNEZ (eds) Surveys in combinatorics, 2001, J.W.P. HIRSCHFELD (ed) Aspects of Sobolev-type inequalities, L. SALOFF-COSTE Quantum groups and Lie theory, A. PRESSLEY (ed) Tits buildings and the model theory of groups, K. TENT (ed) A quantum groups primer, S. MAJID Second order partial differential equations in Hilbert spaces, G. DA PRATO & J. ZABCZYK Introduction to operator space theory, G. PISIER Geometry and integrability, L. MASON & Y. NUTKU (eds) Lectures on invariant theory, I. DOLGACHEV The homotopy category of simply connected 4-manifolds, H.-J. BAUES Higher operads, higher categories, T. LEINSTER (ed) Kleinian groups and hyperbolic 3-manifolds, Y. KOMORI, V. MARKOVIC & C. SERIES (eds) Introduction to Möbius differential geometry, U. HERTRICH-JEROMIN Stable modules and the D(2)-problem, F.E.A. JOHNSON Discrete and continuous nonlinear Schrödinger systems, M.J. ABLOWITZ, B. PRINARI & A.D. TRUBATCH Number theory and algebraic geometry, M. REID & A. SKOROBOGATOV (eds) Groups St Andrews 2001 in Oxford I, C.M. CAMPBELL, E.F. ROBERTSON & G.C. SMITH (eds) Groups St Andrews 2001 in Oxford II, C.M. CAMPBELL, E.F. ROBERTSON & G.C. SMITH (eds) Geometric mechanics and symmetry, J. MONTALDI & T. RATIU (eds) Surveys in combinatorics 2003, C.D. WENSLEY (ed.) Topology, geometry and quantum field theory, U.L. TILLMANN (ed) Corings and comodules, T. BRZEZINSKI & R. WISBAUER Topics in dynamics and ergodic theory, S. BEZUGLYI & S. KOLYADA (eds) Groups: topological, combinatorial and arithmetic aspects, T.W. MÜLLER (ed) Foundations of computational mathematics, Minneapolis 2002, F. CUCKER et al (eds) Transcendental aspects of algebraic cycles, S. MÜLLER-STACH & C. PETERS (eds) Spectral generalizations of line graphs, D. CVETKOVIC, P. ROWLINSON & S. SIMIC Structured ring spectra, A. BAKER & B. RICHTER (eds) Linear logic in computer science, T. EHRHARD, P. RUET, J.-Y. GIRARD & P. SCOTT (eds) Advances in elliptic curve cryptography, I.F. BLAKE, G. SEROUSSI & N.P. SMART (eds) Perturbation of the boundary in boundary-value problems of partial differential equations, D. HENRY Double affine Hecke algebras, I. CHEREDNIK ˇ (eds) L-functions and Galois representations, D. BURNS, K. BUZZARD & J. NEKOVÁR Surveys in modern mathematics, V. PRASOLOV & Y. ILYASHENKO (eds) Recent perspectives in random matrix theory and number theory, F. MEZZADRI & N.C. SNAITH (eds) Poisson geometry, deformation quantisation and group representations, S. GUTT et al (eds) Singularities and computer algebra, C. LOSSEN & G. PFISTER (eds) Lectures on the Ricci flow, P. TOPPING Modular representations of finite groups of Lie type, J.E. HUMPHREYS Surveys in combinatorics 2005, B.S. WEBB (ed) Fundamentals of hyperbolic manifolds, R. CANARY, D. EPSTEIN & A. MARDEN (eds) Spaces of Kleinian groups, Y. MINSKY, M. SAKUMA & C. SERIES (eds) Noncommutative localization in algebra and topology, A. RANICKI (ed) Foundations of computational mathematics, Santander 2005, L.M PARDO, A. PINKUS, E. SÜLI & M.J. TODD (eds) Handbook of tilting theory, L. ANGELERI HÜGEL, D. HAPPEL & H. KRAUSE (eds) Synthetic differential geometry (2nd Edition), A. KOCK The Navier–Stokes equations, N. RILEY & P. DRAZIN Lectures on the combinatorics of free probability, A. NICA & R. SPEICHER Integral closure of ideals, rings, and modules, I. SWANSON & C. HUNEKE Methods in Banach space theory, J.M.F. CASTILLO & W.B. JOHNSON (eds) Surveys in geometry and number theory, N. YOUNG (ed) Groups St Andrews 2005 I, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) Groups St Andrews 2005 II, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) Ranks of elliptic curves and random matrix theory, J.B. CONREY, D.W. FARMER, F. MEZZADRI & N.C. SNAITH (eds) Elliptic cohomology, H.R. MILLER & D.C. RAVENEL (eds) Algebraic cycles and motives I, J. NAGEL & C. PETERS (eds) Algebraic cycles and motives II, J. NAGEL & C. PETERS (eds) Algebraic and analytic geometry, A. NEEMAN Surveys in combinatorics 2007, A. HILTON & J. TALBOT (eds) Surveys in contemporary mathematics, N. YOUNG & Y. CHOI (eds) Transcendental dynamics and complex analysis, P.J. RIPPON & G.M. STALLARD (eds) Model theory with applications to algebra and analysis I, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) Model theory with applications to algebra and analysis II, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds)

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Finite von Neumann algebras and masas, A.M. SINCLAIR & R.R. SMITH Number theory and polynomials, J. MCKEE & C. SMYTH (eds) Trends in stochastic analysis, J. BLATH, P. MÖRTERS & M. SCHEUTZOW (eds) Groups and analysis, K. TENT (ed) Non-equilibrium statistical mechanics and turbulence, J. CARDY, G. FALKOVICH & K. GAWEDZKI Elliptic curves and big Galois representations, D. DELBOURGO Algebraic theory of differential equations, M.A.H. MACCALLUM & A.V. MIKHAILOV (eds) Geometric and cohomological methods in group theory, M.R. BRIDSON, P.H. KROPHOLLER & I.J. LEARY (eds) Moduli spaces and vector bundles, L. BRAMBILA-PAZ, S.B. BRADLOW, O. GARCÍA-PRADA & S. RAMANAN (eds) Zariski geometries, B. ZILBER Words: Notes on verbal width in groups, D. SEGAL Differential tensor algebras and their module categories, R. BAUTISTA, L. SALMERÓN & R. ZUAZUA Foundations of computational mathematics, Hong Kong 2008, F. CUCKER, A. PINKUS & M.J. TODD (eds) Partial differential equations and fluid mechanics, J.C. ROBINSON & J.L. RODRIGO (eds) Surveys in combinatorics 2009, S. HUCZYNSKA, J.D. MITCHELL & C.M. RONEY-DOUGAL (eds) Highly oscillatory problems, B. ENGQUIST, A. FOKAS, E. HAIRER & A. ISERLES (eds) Random matrices: High dimensional phenomena, G. BLOWER Geometry of Riemann surfaces, F.P. GARDINER, G. GONZÁLEZ-DIEZ & C. KOUROUNIOTIS (eds) Epidemics and rumours in complex networks, M. DRAIEF & L. MASSOULIÉ Theory of p-adic distributions, S. ALBEVERIO, A.YU. KHRENNIKOV & V.M. SHELKOVICH ´ Conformal fractals, F. PRZYTYCKI & M. URBANSKI Moonshine: The first quarter century and beyond, J. LEPOWSKY, J. MCKAY & M.P. TUITE (eds) Smoothness, regularity and complete intersection, J. MAJADAS & A. G. RODICIO Geometric analysis of hyperbolic differential equations: An introduction, S. ALINHAC Triangulated categories, T. HOLM, P. JØRGENSEN & R. ROUQUIER (eds) Permutation patterns, S. LINTON, N. RUŠKUC & V. VATTER (eds) An introduction to Galois cohomology and its applications, G. BERHUY Probability and mathematical genetics, N. H. BINGHAM & C. M. GOLDIE (eds) Finite and algorithmic model theory, J. ESPARZA, C. MICHAUX & C. STEINHORN (eds) Real and complex singularities, M. MANOEL, M.C. ROMERO FUSTER & C.T.C WALL (eds) Symmetries and integrability of difference equations, D. LEVI, P. OLVER, Z. THOMOVA & P. WINTERNITZ (eds) ˇ Forcing with random variables and proof complexity, J. KRAJÍCEK Motivic integration and its interactions with model theory and non-Archimedean geometry I, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) Motivic integration and its interactions with model theory and non-Archimedean geometry II, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) Entropy of hidden Markov processes and connections to dynamical systems, B. MARCUS, K. PETERSEN & T. WEISSMAN (eds) Independence-friendly logic, A.L. MANN, G. SANDU & M. SEVENSTER Groups St Andrews 2009 in Bath I, C.M. CAMPBELL et al (eds) Groups St Andrews 2009 in Bath II, C.M. CAMPBELL et al (eds) Random fields on the sphere, D. MARINUCCI & G. PECCATI Localization in periodic potentials, D.E. PELINOVSKY Fusion systems in algebra and topology, M. ASCHBACHER, R. KESSAR & B. OLIVER Surveys in combinatorics 2011, R. CHAPMAN (ed) Non-abelian fundamental groups and Iwasawa theory, J. COATES et al (eds) Variational problems in differential geometry, R. BIELAWSKI, K. HOUSTON & M. SPEIGHT (eds) How groups grow, A. MANN Arithmetic dfferential operators over the p-adic Integers, C.C. RALPH & S.R. SIMANCA Hyperbolic geometry and applications in quantum chaos and cosmology, J. BOLTE & F. STEINER (eds) Mathematical models in contact mechanics, M. SOFONEA & A. MATEI Circuit double cover of graphs, C.-Q. ZHANG Dense sphere packings: a blueprint for formal proofs, T. HALES A double Hall algebra approach to affine quantum Schur-Weyl theory, B. DENG, J. DU & Q. FU Mathematical aspects of fluid mechanics, J. ROBINSON, J.L. RODRIGO & W. SADOWSKI (eds) Foundations of computational mathematics: Budapest 2011, F. CUCKER, T. KRICK, A. SZANTO & A. PINKUS (eds) Operator methods for boundary value problems, S. HASSI, H.S.V. DE SNOO & F.H. SZAFRANIEC (eds) Torsors, étale homotopy and applications to rational points, A.N. SKOROBOGATOV (ed) Appalachian set theory, J. CUMMINGS & E. SCHIMMERLING (eds) The maximal subgroups of the low-dimensional finite classical groups, J.N. BRAY, D.F. HOLT & C.M. RONEY-DOUGAL Complexity science: the Warwick master’s course, R. BALL, R.S. MACKAY & V. KOLOKOLTSOV (eds) Surveys in combinatorics 2013, S. BLACKBURN, S. GERKE & M. WILDON (eds) Representation theory and harmonic analysis of wreath products of finite groups, T. CECCHERINI SILBERSTEIN, F. SCARABOTTI & F. TOLLI Moduli spaces, L. BRAMBILA-PAZ, O. GARCIA-PRADA, P. NEWSTEAD & R. THOMAS (eds) Automorphisms and equivalence relations in topological dynamics, D.B. ELLIS & R. ELLIS Optimal transportation: theory and applications, Y. OLLIVIER, H. PAJOT & C. VILLANI (eds)

London Mathematical Society Lecture Note Series: 415

Automorphic Forms and Galois Representations Volume 2

Edited by

FRED DIAMOND King’s College London PAY MA N L . KASSAEI McGill University, Montréal M I N H YO N G K I M University of Oxford

University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107693630 c Cambridge University Press 2014 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 Printed in the United Kingdom by CPI Group Ltd, Croydon CR0 4YY A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Automorphic forms and galois representations / edited by Fred Diamond, King’s College London, Payman L. Kassaei, McGill University, Montréal, Minhyong Kim, University of Oxford. volumes cm. – (London Mathematical Society lecture note series ; 414, 415) Papers presented at the London Mathematical Society, and EPSRC (Great Britain Engineering and Physical Sciences Research Council), Symposium on Galois Representations and Automorphic Forms, held at the University of Durham from July 18–28, 2011. ISBN 978-1-107-69192-6 (v. 1) – ISBN 978-1-107-69363-0 (v. 2) 1. Automorphic forms–Congresses. 2. Automorphic functions–Congresses. 3. Forms (Mathematics)–Congresses. 4. Galois theory–Congresses. I. Diamond, Fred, editor of compilation. II. Kassaei, Payman L., 1973– editor of compilation. III. Kim, Minhyong, editor of compilation. IV. Symposium on Galois Representations and Automorphic Forms (2011 : Durham, England) QA353.A9A925 2014 515 .9–dc23 2014001841 ISBN 978-1-107-69363-0 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

List of contributors Preface

page vi vii

1 On the local structure of ordinary Hecke algebras at classical weight one points Mladen Dimitrov 2 Vector bundles on curves and p-adic Hodge theory Laurent Fargues and Jean-Marc Fontaine

1 17

3 Around associators Hidekazu Furusho

105

4 The stable Bernstein center and test functions for Shimura varieties Thomas J. Haines

118

5 Conditional results on the birational section conjecture over small number fields Yuichiro Hoshi

187

6 Blocks for mod p representations of GL2 (Q p ) Vytautas Pašk¯unas

231

7 From étale P+ -representations to G-equivariant sheaves on G/P Peter Schneider, Marie-France Vigneras, and Gergely Zabradi

248

8 Intertwining of ramified and unramified zeros of Iwasawa modules Chandrashekhar Khare and Jean-Pierre Wintenberger

367

v

Contributors

Mladen Dimitrov, UFR Mathématiques, Université Lille 1, Lille, 59655 Villeneuve d’Ascq Cedex, France. Laurent Fargues, CNRS, Institut de Mathématiques de Jussieu, Paris, 75252, France. Jean-Marc Fontaine, Laboratoire de Mathématiques d’Orsay, Université Paris Sud, Paris 91405 Orsay, France. Hidekazu Furusho, Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan. Thomas J. Haines, Department of Mathematics, Univeristy of Maryland, College Park, MD 20742-4015, USA. Yuichiro Hoshi, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan. Chandrashekhar Khare, Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA. Vytautas Pašk¯unas, Fakultät für Mathematik, Universität Duisburg–Essen, Essen 45127, Germany. Peter Schneider, Mathematischen Instituts, Westfälische Wilhelms-Universität Münster, 48149 Germany. Marie-France Vignéras, Institut de Mathématiques de Jussieu, Université Paris Diderot (Paris 7), 775205 Paris Cedex 13, France. Jean-Pierre Wintenberger, Département de Mathématiques, Université de Strasbourg, 67084 Strasbourg Cedex, France. Gergely Zabradi, Institute of Mathematics, Eötvös Loránd University, H-1518 Budapest, Pf. 120, Hungary. vi

Preface

The London Mathematical Society Symposium – EPSRC Symposium on Galois Representations and Automorphic Forms was held at the University of Durham from 18th July until 28th July 2011. These topics have been playing an important role in present-day number theory, especially via the Langlands program and the connections it entails. The meeting brought together researchers from around the world on these and related topics, with lectures on a variety of recent major developments in the area. Roughly half of these talks were individual lectures, while the rest constituted series on the following themes: • • • • •

p-adic local Langlands Curves and vector bundles in p-adic Hodge theory The fundamental lemma and trace formula Anabelian geometry Potential automorphy

These Proceedings present much of the progress described in those lectures. The organizers are very grateful to all the speakers and to others who contributed articles. We also wish to thank the London Mathematical Society and EPSRC for the financial support that made the meeting possible. We warmly appreciate the assistance and hospitality provided by the University of Durham’s Department of Mathematics and Grey College. These institutions have helped to make the Symposia such a well established and highly valued event in the number theory community. Fred Diamond Payman Kassaei Minhyong Kim

vii

1 On the local structure of ordinary Hecke algebras at classical weight one points Mladen Dimitrov

Abstract The aim of this chapter is to explain how one can obtain information regarding the membership of a classical weight one eigenform in a Hida family from the geometry of the Eigencurve at the corresponding point. We show, in passing, that all classical members of a Hida family, including those of weight one, share the same local type at all primes dividing the level.

1. Introduction Classical weight one eigenforms occupy a special place in the correspondence between Automorphic Forms and Galois Representations since they yield two dimensional Artin representations with odd determinant. The construction of those representations by Deligne and Serre [5] uses congruences with modular forms of higher weight. The systematic study of congruences between modular forms has culminated in the construction of the p-adic Eigencurve by Coleman and Mazur [4]. A p-stabilized classical weight one eigenform corresponds then to a point on the ordinary component of the Eigencurve, which is closely related to Hida theory. An important result of Hida [11] states that an ordinary cuspform of weight at least two is a specialization of a unique, up to Galois conjugacy, primitive Hida family. Geometrically this translates into the smoothness of the Eigencurve at that point (in fact, Hida proves more, namely that the map The author is partially supported by Agence Nationale de la Recherche grants ANR-10-BLAN-0114 and ANR-11-LABX-0007-01. Automorphic Forms and Galois Representations, ed. Fred Diamond, Payman L. Kassaei and c Cambridge University Press 2014. Minhyong Kim. Published by Cambridge University Press.

1

2

Mladen Dimitrov

to the weight space is etale at that point). Whereas Hida’s result continues to hold at all non-critical classical points of weight two or more [13], there are examples where this fails in weight one [6]. The purely quantitive question of how many Hida families specialize to a given classical p-stabilized weight one eigenform, can be reformulated geometrically as to describe the local structure of the Eigencurve at the corresponding point. An advantage of the new formulation is that it provides group theoretic and homological tools for the study of the original question thanks to Mazur’s theory of deformations of Galois representations. Moreover, this method gives more qualitative answers, since the local structure of the Eigencurve at a given point contains more information than the collection of all Hida families passing through that point. The local structure at weight one forms with RM was first investigated by Cho and Vatsal [3] in the context of studying universal deformation rings, who showed that in many cases the Eigencurve is smooth, but not etale over the weight space, at those points. The main result of a joint work with Joël Bellaïche [1] states that the p-adic Eigencurve is smooth at all classical weight one points which are regular at p and gives a precise criterion for etalness over the weight space at those points. The author has learned recently that the work [10] of Greenberg and Vatsal contains a slightly weaker version of this result. It would be interesting to describe the local structure at irregular points, to which we hope to come back in a future work. The chapter is organized as follows. Section 2 describes some p-adic aspects in the theory of weight one eigenforms. Sections 3 and 4 introduce, respectively, the ordinary Hecke algebras and primitive Hida families, which are central objects in Hida theory [12]. In Section 5 various Galois representations are studied with emphasis on stable lattices, leading to the construction of a representation (1.10) which is a bridge between a primitive Hida family and its classical members. This is used in Section 6 to establish the rigidity of the local type in a Hida family, including in weight one (see Proposition 1.8). Section 7 quotes the main results of [1] and describes their consequences in classical Hida theory (see Corollary 1.15). The latter would have been rather straightforward, should the Eigencurve have been primitive, in the sense that the irreducible component of its ordinary locus would have corresponded (after inverting p) to primitive Hida families. Lacking a reference for the construction of such an Eigencurve, we establish a local isomorphism, at the points of interest, between the reduced Hecke algebra, used in the definition of the Eigencurve, and the new quotient of the full Hecke algebra, used in the definition of primitive Hida families (see Corollary 1.14).

Ordinary Hecke algebras at classical weight one points

3

Acknowledgements. The author would like to thank Joël Bellaïche for many helpful discussions, as well as the referee for his careful reading of the manuscript and for pointing out some useful references.

2. Artin modular forms and the Eigencurve ¯ ⊂ C be the field of algebraic numbers, and denote by Gal(Q/ ¯ Q) the We let Q absolute Galois group of Q. For a prime we fix a decomposition subgroup G ¯ Q) and denote by I its inertia subgroup and by Frob the arithmetic of Gal(Q/ Frobenius in G /I . ¯ → Q ¯ p. We fix a prime number p and an embedding Q Let f (z) = n≥1 an q n be a newform of weight one, level M and central character . Thus a1 = 1 and for every prime M (resp. | M) f is an eigenvector with eigenvalue a for the Hecke operator T (resp. U ). By a theorem of Deligne and Serre [5] there exists a unique continuous irreducible representation: ¯ Q) → GL2 (C), ρ f : Gal(Q/

(1.1)

such that its Artin L-function L(ρ f , s) equals L( f, s) =

an n

ns

=

M

(1 − a −s + ()−2s )−1

(1 − a −s )−1 .

|M

It follows that if a = 0 for | M, then a is the eigenvalue of ρ f (Frob ) acting on the unique line fixed by I . Since ρ f has finite image, a is an eigenvalue of a finite order matrix, hence it is a root of unity. Similarly, for M the characteristic polynomial X 2 − a X + () of ρ f (Frob ) has two (possibly equal) roots α and β which are both roots of unity. In order to deform f p-adically, one should first choose a p-stabilization of f with finite slope, that is an eigenform of level 1 (M) ∩ 0 ( p) sharing the same eigenvalues as f away from p and having a non-zero U p -eigenvalue. By the above discussion if such a stabilization exists, then it should necessarily be ordinary. We distinguish two cases: If p does not divide M, then f has two p-stabilizations f α (z) = f (z) − β p f ( pz) and f β (z) = f (z) − α p f ( pz) with U p -eigenvalue α p and β p , respectively. If p divides M and a p = 0, then f is already p-stabilized. We let then α p = a p and f α = f . Denote by N the prime to p-part of M.

4

Mladen Dimitrov

Definition 1.1. We say that f α is regular at p if either p divides M and a p = 0, or p does not divide M and α p = β p . The Eigencurve C of tame level 1 (N ) is a rigid analytic curve over Q p parametrizing systems of eigenvalues for the Hecke operators T ( N p) and U p appearing in the space of finite slope overconvergent modular forms of tame level dividing N . We refer to the original article of Coleman and Mazur [4] for the case N = 1 and p > 2, and to Buzzard [2] for the general case. Recall that C is reduced and endowed with a flat and locally finite weight map κ : C → W, where W is the rigid space over Q p representing homomorphisms × Z× p ×(Z /N Z) → Gm . The p-stabilized newform f α defines a point on the ordinary component of C, whose image by κ is a character of finite order. Theorem 1.2. [1] Let f be a classical weight one cuspidal eigenform form which is regular at p. Then the Eigencurve C is smooth at the point defined by f α , so there is a unique irreducible component of C containing that point. In particular, if f has CM by a quadratic field in which p splits, then all classical points of that component also have CM by the same field. Moreover, C is etale over the weight space W at the point defined by f α , unless f has RM by a quadratic field in which p splits. In Section 7 we will revisit this theorem from the perspective of Hida families.

3. Ordinary Hecke algebras The results in this and the following two sections are due to Hida [11, 12] when p is odd and have been completed for p = 2 by Wiles [18] and Ghate–Kumar [8]. Let = Z p [[Gal(Q∞ / Q)]] Z p [[1 + p ν Z p ]] be the Iwasawa algebra of the cyclotomic Z p extension Q∞ of Q, where ν = 2 if p = 2 and ν = 1 otherwise. It is a complete local Z p -algebra which is an integral domain of Krull dimension 2. Let χcyc be the universal -adic cyclotomic character obtained by ¯ Q) Gal(Q∞ / Q) with the canonical continuous group composing Gal(Q/ homomorphism from Gal(Q∞ / Q) to the units of its completed group ring . We say that a height one prime ideal p of a finite -algebra T is of weight k (an integer ≥ 1) if P = p ∩ is the kernel of a Z p -algebra homomor¯ p whose restriction to a finite index subgroup of 1 + p ν Z p phism → Q is given by x → x k−1 . Such an ideal p induces a Galois orbit of Z p -algebra ¯ p called specializations in weight k. homomorphisms T → T / p → Q By definition a -adic ordinary cuspform of level N (a positive integer not divisible by p) is a formal q-expansion with coefficients in the integral closure

Ordinary Hecke algebras at classical weight one points

5

of in some finite extension of its fraction field, whose specialization in any weight k ≥ 2 yield the q-expansion of a p-stabilized, ordinary, normalized cuspform of tame level N and weight k. However, specializations in weight one are not always classical. The ordinary Hecke algebra T N of tame level N is defined as the -algebra generated by the Hecke operators U (resp. T , ) for primes dividing N p (resp. not dividing N p) acting on the space of -adic ordinary cuspforms of tame level N . Hida proved that T N is free of finite rank over and its height one primes of weight k ≥ 2 are in bijection with the (Galois orbits of) classical ordinary eigenforms of weight k and tame level dividing N . A -adic ordinary cuspform of level N is said to be N -new if all specializations in weights ≥ 2 are p-stabilized, ordinary cuspforms of tame level N which are N -new. Define Tnew N as the quotient of T N acting faithfully on the space of -adic ordinary cuspforms of level N , which are N -new. A result of Hida (see [12, Corollaries 3.3 and 3.7]) states that Tnew N is a finite, reduced, torsion free algebra, whose height one primes of weight k ≥ 2 are in bijection with the Galois orbits of classical ordinary eigenforms of weight k and tame level N which are N -new.

4. Primitive Hida families A primitive Hida family F = n≥1 An q n of tame level N is by definition a adic ordinary cuspform, new of level N and which is a normalized eigenform for all the Hecke operators, i.e., a common eigenvector of the operators U , T and as above. The relations between coefficients and eigenvalues for the Hecke operators are the usual ones for newforms. One can see from [12, p. 265] that primitive Hida families can be used to write down a basis of the space of

-adic ordinary cuspforms in the same fashion as classically newforms can be used to write down a basis of the space of cuspforms. The central character ψ F : (Z /N pν )× → C× of the family is defined by ψ F () = eigenvalue of . Galois orbits of primitive Hida families of level N are in bijection with the minimal primes of T = Tnew N . More precisely, a primitive Hida family determines and is uniquely determined by a -algebra homomorphism T → Frac( ), sending each Hecke operator to its eigenvalue on F, whose kernel is a minimal prime a ⊂ T. Since T is a finite and reduced -algebra, its localization Ta is a finite field extension of Frac( ). Hence, we obtain the following homomorphisms of -algebras: ∼ T T / a → T / a → Ta → K F ⊂ Frac( ),

(1.2)

6

Mladen Dimitrov

where T / a denotes the integral closure of the domain T / a in its field of fractions Ta . In particular, the image K F of Ta in Frac( ) is a finite extension of Frac( ) generated by the coefficients of F. By definition all specializations of F in weight k ≥ 2 yield p-stabilized, ordinary newforms of tame level N and weight k. In weight one, there are only finitely many classical specializations, unless F has CM by a quadratic field in which p splits (see [9] and [6]). Nevertheless, a theorem of Wiles [18] asserts that any p-stabilized newform of weight one occurs as a specialization of a primitive Hida family. n Given a primitive Hida family F = n≥1 An q of level N , Hida constructed in [11, Theorem 2.1] an absolutely irreducible continuous representation: ¯ Q) → GL2 (K F ), ρ F : Gal(Q/

(1.3)

unramified outside N p, such that for all not dividing N p the trace of the image of Frob equals A . Moreover det ρ F = ψ F χcyc . Finally by Wiles [18, Theorem 2.2.2] the space of I p -coinvariants is a line on which Frob p acts by A p .

5. Galois representations 5.1. Minimal primes

The total quotient field of T is given by T ⊗ Frac( ) a Ta where the product is taken over all minimal primes of T. The representation (1.3) can be rewritten as ¯ Q) → GL2 (Ta ) ρa : Gal(Q/

(1.4)

and by putting those together we obtain a continuous representation ¯ Q) → GL2 (T ⊗ Frac( )) ρT : Gal(Q/

(1.5)

unramified outside N p, such that for all not dividing N p the trace of the image of Frob equals T . Moreover the space of I p -coinvariants is free of rank one and Frob p acts on it as U p .

5.2. Maximal primes Since T is a finite -algebra, it is semi-local, and is isomorphic to the direct product m Tm where the product is taken over all maximal primes. By composing (1.5) with the canonical projection, one obtains:

Ordinary Hecke algebras at classical weight one points ¯ Q) → GL2 (Tm ⊗ Frac( )). ρm : Gal(Q/

7

(1.6)

The composition: Tr(ρm )

¯ Q) −→ Tm → T / m Gal(Q/

(1.7)

is a pseudo-character taking values in a field and sending the complex conjugation to 0. By a result of Wiles [18, §2.2] it is the trace of a unique semi-simple representation: ¯ Q) → GL2 (T / m). ρ¯m : Gal(Q/

(1.8)

Note that whereas each minimal prime a ⊂ T is contained in a unique maximal prime, there may be several minimal primes contained in a given maximal prime m, those corresponding to primitive Hida families sharing the same residual Galois representation ρ¯m .

5.3. Galois stable lattices A lattice over a noetherian domain R (or R-lattice) is a finitely generated R-submodule of a finite dimensional Frac(R)-vector space which spans the latter. This definition extends to a noetherian reduced ring R and its total quotient field a Ra , where a runs over the (finitely many) minimal primes of R. The continuity of ρa implies the existence of a Galois stable T / a-lattice 2 in Ta , and similar statements hold for ρ F , ρT and ρm . It is worth mentioning that ρT cannot necessarily be defined over the normalization of T in a Ta . In other words ρa does not necessarily stabilize a free T / a-lattice. There is an exception: if K F = Frac( ) and p > 2 the regularity of implies that ρ F always admits a Galois stable free -lattice (see [11, §2]). If m is a maximal prime such that the residual Galois representation ρ¯m is absolutely irreducible, then by a result of Nyssen [15] and Rouquier [16] ρm stabilizes a free Tm -lattice. It follows that for every minimal prime a ⊂ m, the representation ρa stabilizes a free lattice over Tm / a = T / a.

5.4. Height one primes Let f be a p-stabilized, ordinary, newform of tame level N and weight k. It determines uniquely a height one prime p ⊂ T and an embedding of Tp / p ¯ p , although not every height one prime of T of weight one is obtained into Q in this way. Our main interest is in the structure of the P -algebra Tp , where P = p ∩ . The ring Tp is local, noetherian, reduced of Krull dimension 1, but is not necessarily integrally closed. It might even not be a domain, since f

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Mladen Dimitrov

could be a specialization of several, non Galois conjugate, Hida families (see [6, §7.4]), hence there may be several minimal primes a of T contained in p. ¯ Q) → GL2 (Q ¯ p ) be the continuous irreducible representaLet ρ f : Gal(Q/ tion attached to f by Deligne when k ≥ 2 and by (1.1) when k = 1 via the ¯ → Q ¯ p . Since ρ f is odd, it can be defined over the fixed embeddings C ⊃ Q ¯ p generated by its coefficients, and hence ring of integers of the subfield of Q defines an isomorphic representation: ¯ Q) → GL2 (Tp / p), ρ¯p : Gal(Q/

(1.9)

admitting a model over the integral closure of T / p in its field of fractions Tp / p. The normalization of Tp in its total quotient field a⊂p Ta is given by a⊂p Tp / a, where Tp / a (T / a)p is the integral closure of Tp / a

(T / a)p in Ta . Denote by T p / a the completion of the discrete valuation ring T p / a. Note

that they share the same residue field which is a finite extension of Tp / p and that there is a natural bijection between the set of T p / a-lattices in a given Ta -vector space V and the set of T / a-lattices in V ⊗ p Ta Frac(Tp / a). Since ρ¯p is absolutely irreducible and Tp / a is local and complete, by a result of Nyssen [15] and Rouquier [16] the representation ρa ⊗Ta Frac(T p / a) sta bilizes a free Tp / a-lattice. The latter lattice yields (by intersection) a free T p / a-lattice stable by ρa . In other terms there exists a unique, up to conjugacy, continuous representation: ¯ Q) → GL2 (T ρpa : Gal(Q/ p / a),

(1.10)

such that ρpa ⊗T Ta ρa and ρpa mod p ρ¯p . p /a This representation is a bridge between a form and a family and will be used in Section 6 to transfer properties in both directions. The exact control theorem for ordinary Hecke algebras, proved by Hida for p > 2 and by Ghate–Kumar [8] for p = 2, has the following consequence: Theorem 1.3. [11, Corollary 1.4] Assume that k ≥ 2. Then the local algebra Tp is etale over the discrete valuation ring P . In particular, f is a specialization of a unique, up to Galois conjugacy, Hida family corresponding to a minimal prime a. Assume for the rest of this section that Tp is a domain. Then the field of fractions of Tp is isomorphic to Ta , where a is the unique minimal prime of T contained in p. Since normalization and localization commute, we have

Ordinary Hecke algebras at classical weight one points

9

p . Therefore, the collection of representations (1.10) / a)p (T / a)p T (T are replaced by a unique, up to conjugacy, continuous representation: ¯ Q) → GL2 (T p ), ρp : Gal(Q/

(1.11)

such that ρp ⊗Tp Ta ρa and ρp mod p ρ¯p . If we further assume that Tp is etale over P , then Tp is itself a discrete p . valuation ring, hence Tp T

6. Rigidity of the automorphic type in a Hida family By definition, all specializations in weight at least two of a primitive Hida family F of level N share the same tame level. Also, by [7, Proposition 2.2.4], the tame conductor of ρ F equals N . The aim of this section is to show that the tame level of all classical weight one specializations of F is also N , and to show that all classical specializations of F (including those of weight one) share the same automorphic type at all primes dividing N .

6.1. Minimally ramified Hida families Recall that a newform f is said to be minimally ramified if it has minimal level amongst the underlying newforms of all its twists by Dirichlet characters. Lemma 1.4. Let F be a primitive Hida family and let χ be a Dirichlet character of conductor prime to p. There exists a unique primitive Hida family Fχ underlying F ⊗ χ , in the sense that the p-stabilized, ordinary newform underlying a given specialization of F ⊗χ can be obtained by specializing Fχ . Proof. By [12, p. 250] one can write any -adic ordinary cuspform as a linear combination of translates of primitive Hida families of lower or equal level. Since F ⊗χ is an eigenform for all but finitely many Hecke operators, it is necessarily a linear combination of translates of the same primitive Hida family, denoted Fχ . It follows that any specialization of Fχ in weight at least two is the p-stabilized, ordinary newform underlying the corresponding specialization of F ⊗ χ. Definition 1.5. We say that a primitive Hida family F of level N is minimally ramified if for every Dirichlet character χ of conductor prime to p, the level of Fχ is a multiple of N . As for newforms, it is clear that any primitive Hida family admits a unique twist which is minimally ramified.

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Lemma 1.4 implies that being minimally ramified is pure with respect to specializations in weight at least two, that is to say, all specializations of a minimally ramified primitive Hida family are minimally ramified, and a primitive Hida family admitting a minimally ramified specialization is minimally ramified. This observation together with the classification of the admissible representations of GL2 (Q ), easily implies: Lemma 1.6. Let F = n≥1 An q n be a minimally ramified, primitive Hida family of level N and let be a prime dividing N . Denote by unr(C) the unramified character of G sending Frob to C. (i) If ψ F is unramified at and 2 does not divide N , then every specialization in weight at least two corresponds to an automorphic form which is special at . In particular A = 0 and the restriction of ρ F to G is an unramified twist of an extension of 1 by unr(). (ii) If the conductor of ψ F and N share the same -part, then every specialization in weight at least two corresponds to an automorphic form which is a ramified principal series at . In particular A = 0 and the restriction of ρ F to G equals unr(A ) ⊕ unr(B )ψ F , for some B ∈ K F . (iii) In all other cases, every specialization in weight at least two corresponds to an automorphic form which is supercuspidal at . In particular A = 0 and the restriction of ρ F to G is irreducible.

6.2. General case Definition 1.7. Let F be a primitive Hida family of level N and let be a prime dividing N . We say that F is special (resp. ramified principal series or supercuspidal) at , if a minimally ramified twist of F falls in case (i) (resp. (ii) or (iii)) of Lemma 1.6. It follows from Lemma 1.6, that being special, principal series or supercuspidal is pure with respect to specializations, that is to say, all specializations in weight at least two are of the same type. We will now describe the local automorphy type in greater detail and deduce information about classical weight one specializations. Proposition 1.8. Let F be a primitive Hida family of level N and let be a prime dividing N . If F is special at , so are all its specializations in weight at least two and F does not admit any classical weight one specialization. Otherwise, ρ F (I ) is a finite group invariant under any classical specialization, including in weight one. More precisely

Ordinary Hecke algebras at classical weight one points

11

(i) If F is a ramified principal series at , then the restriction of ρ F to G is isomorphic to ϕ ⊕ ϕ , where ϕ and ϕ are characters whose restrictions to inertia have finite order. (ii) If F is supercuspidal at , then either the restriction of ρ F to G is induced from a character of an index two subgroup of G whose restriction to inertia has finite order, or = 2 and all classical specializations of F are extraordinary supercuspidal representations at 2. In particular, all classical weight one specializations of F have tame level N . Proof. Although parts of the proposition seem to be well-known to experts, for the benefit of the reader, we will give a complete proof. If F is special at , then the claim about specializations in weight at least two follows directly from Lemma 1.6(i). Moreover in this case ρ F |G is by definition reducible and the quotient of the two characters occurring in its semisimplification equals unr(). Since is not a root of unity, F does not admit any classical weight one specializations. Suppose now that F is not special at . Since = p and ρ F is continuous, Grothendieck’s -adic monodromy theorem implies that ρ F (I ) is finite. Let p be a height one prime of T corresponding to a classical cusp form f of weight k, containing the minimal prime a defined by F. Denote by L the free rank two a a T p / a-lattice on which ρp acts (see (1.10)). Recall that ρp mod p ρ¯p and consider the natural projection: ρ F (I ) ρpa (I ) ρ¯p (I ) ρ f (I ),

(1.12)

which we claim is an isomorphism. In fact, an eigenvalue ζ of an element of the kernel has to be a root of unity since the latter is a finite group, in particular ¯ p . Since by assumption (ζ −1)2 ∈ p, the product of all its G p -conjugates ζ ∈Q belongs to p ∩ Q p , which is {0} because Tp is a Q p -algebra. Hence ζ = 1 which implies that the kernel is trivial. The claim (i) follows directly from 1.6(ii), so we can assume for the rest of the proof that F is supercuspidal at . Denote by W the wild inertia subgroup of I . Suppose first that ρa |W is reducible, isomorphic to ⊕ with = . Since W is normal in G , it follows easily that extends to an index two subgroup of G , as claimed. If ρa |W is reducible and isotropic, then by taking an eigenvector for a topological generator of I / W one sees that ρa | I is reducible too, which allows us to conclude as in the previous case. Suppose finally that ρa |W is irreducible. Then by a classical result on hyper-solvable groups its image is a dihedral group, hence = 2. Assume further ρa |G is not dihedral, since this case can be handled as above. Then, any specialization in weight at least two of F yields an eigenform f which

12

Mladen Dimitrov

is an extraordinary supercuspidal representation at = 2. The isomorphism (1.12) implies that all other classical specializations of F are also extraordinary supercuspidal representations at = 2 (we refer to [17] and [14, §5.1] for a detailed analysis of this case).

7. Local structure of the ordinary Hecke algebras at classical weight one points 7.1. A deformation problem Let f α be a weight one p-stabilized newform of tame level N as in Section 2. Assume that f is regular at p. By ordinarity the restriction of ρ f to G p is a sum of two characters ψ1 and ψ2 , and by regularity exactly one of those characters, say ψ1 , is the unramified character sending Frob p to α p . By (1.9) the Galois representation ρ f is defined over a finite extension E = Tp / p of Q p , where p denotes the height one prime of T determined by f . Consider the functor D sending a local Artinian ring A with maximal ideal m A and residue field A/ m A = E to the set of strict equivalence classes of ¯ Q) → GL2 (A) such that ρ representations ρ : Gal(Q/ mod m A ρ f and fitting in an exact sequence 2 → ρ 1 → 0 0→ψ → ψ 1 is an unramified character of A[G p ]-modules, free over A, and such that ψ whose reduction modulo m A equals ψ1 . We define D as the subfunctor of D consisting of deformation with constant determinant. Finally, define Dmin ) as the subfunctor of D (resp. D ) of deformations ρ (resp. Dmin such that for all dividing N such that a = 0, the I -invariants in ρ are a free A-module of rank one. The functors D and Dmin are pro-representable by local noetherian complete E-algebras R and Rmin , while D and Dmin are representable by local Artinian E-algebras R and Rmin . Denote by tD the tangent space of D, etc. Using the interpretation of tD and tD in terms of Galois cohomology groups, the main technical result of [1] states: Theorem 1.9. If f is regular at p, then dim tD = 1. If we further assume that f does not have RM by a quadratic field in which p splits, then tD = 0.

7.2. Modular deformations p the completion of Tp The composition: Denote by T T) ¯ Q) Tr(ρ p Gal(Q/ −→ T → Tp → T

(1.13)

Ordinary Hecke algebras at classical weight one points

13

is a two dimensional pseudo-character taking values in a complete local ring and whose reduction modulo the maximal ideal is the trace of the absolutely irreducible representation ρ¯p . By a result of Nyssen [15] and Rouquier [16] the pseudo-character (1.13) is the trace of a two dimensional irreducible representation: ¯ Q) → GL2 (T p ). ρ

p : Gal(Q/ (1.14) This representation contains more information than the collection (ρpa )a⊂p and plays a central role in the analysis of the p-adic deformations of f . Note that

P is formal power series over its residue field

P /P P /P which is a finite extension of Q p . Consider the local Artinian Q p -algebra p ⊗

T =T P /P.

P

(1.15)

Reducing (1.14) modulo P yields a continuous representation: ¯ Q) → GL2 (T ), ρT : Gal(Q/

(1.16)

such that det(ρT ) = det(ρ f ). In fact, for all ∈ (Z /N p)× , the image of in T is given by the henselian lift of its image in T / p, hence is fixed. As in [1], one can describe the local behavior of ρ

p at bad primes. Proposition 1.10. (i) For all dividing N such that a = 0, the space of I -invariants in ρ

p is free of rank one and Frob acts on it as U . (ii) Assume that f is regular at p. Then ρ

p is ordinary, in the sense that the space of I p -coinvariants is free of rank one and Frob p acts on it as U p . By Proposition 1.10 the Galois representation (1.14) (resp. (1.16)) defines a ). One deduces the following surjective homomorpoint of Dmin (resp. of Dmin phisms of local reduced

P -algebras (resp. local Artinian E-algebras): p , and R Rmin T

(1.17)

R Rmin T .

7.3. Smoothness and etaleness Lemma 1.11. The P -algebra Tp is etale if, and only if, T is a field. Proof. Since T is flat over , so is T ⊗ P over P . The algebra T ⊗ P is unramified over P if, and only if, T ⊗ P /P is unramified over P /P, that is to say is a product of fields. Since T ⊗ P = Tp , we have

p ∩ =P

Tp ⊗ P P /P = T ⊗ P /P T ⊗

P /P =

p ∩ =P

p ∩ =P

p ⊗

T

P P /P.

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Mladen Dimitrov

p is unramified One deduces that Tp is unramified over P if, and only if, T over

if, and only if, T = T ⊗

/P is a field. P p

P P Proposition 1.12. Suppose that f is regular at p. Then Tp is a discrete valuation ring and the homomorphisms in (1.17) are isomorphisms. Moreover, if f does not have RM by a quadratic field in which p splits, then Tp is etale over P , and otherwise, under the additional assumption that dim E R ≤ 2, the ramification index of Tp over P equals 2. Proof. Since f is regular at p, Theorem 1.9 implies that dim tD = 1. Since p > 0, one deduces that the natural surjective homomorphism R T p dim T

is an isomorphism of discrete valuation rings, hence Tp is a discrete valuation p modulo P we obtain the ring too. By reducing the isomorphism R T isomorphism R T . If tD = 0, by Nakayama’s lemma the structural homomorphisms E → R and

P → R are isomorphisms. By Lemma 1.11, it follows that Tp is etale over P , as claimed. Assume now that dim tD = 1, in which case Theorem 1.9 implies that f has RM by a quadratic field in which p splits. Since dim E R ≤ 2 by assumption and dim E T ≥ 2 by [6, Proposition 2.2.4], we deduce that the ramification index is 2. Remark 1.13. Cho and Vatsal [3] have proved that dim E R ≤ 2 under some additional assumptions, and their method is expected to continue to work under the only assumption of regularity at p.

7.4. Reduced Hecke algebras Define the reduced ordinary Hecke algebra T = TN of tame level N as the subalgebra of T N generated over by the Hecke operators U p , T and for primes not dividing N p. By the theory of newforms, the natural composition: TN → T N → Tnew (1.18) N , N |N

is injective, in particular TN is reduced. A classical result from Hida theory says that the localization of (1.18) at any height one prime of weight at least two yields an isomorphism. Let p be a height one prime T corresponding to a p-stabilized classical weight one eigenform f α and denote by p the corresponding height one prime of T . Corollary 1.14. Suppose that f is regular at p. Then the localization of (1.18) yields an isomorphism Tp Tp .

Ordinary Hecke algebras at classical weight one points

15

Proof. Proposition 1.8 implies Tnew N ,p = {0} for all N < N , hence localizing (1.18) at p yields an injective homomomorphism Tp → Tp . Since Tp and Tp are finite over the Zariski ring P , it is enough to check the surjectivity after completion. This follows from Proposition 1.12, since the surjective p factors through T homomorphism R T p .

We will conclude this chapter, by giving a partial answer to the original question that motivated this research. Corollary 1.15. Let f be a classical weight one cuspidal eigenform form which is regular at p. Then there exists a unique Hida family specializing to f α . In particular, if f has CM by a quadratic field in which p splits, then the family has also CM by the same field.

References [1] J. B ELLAÏCHE AND M. D IMITROV . On the Eigencurve at classical weight one points, arXiv:1301.0712, submitted. [2] K. B UZZARD, Eigenvarieties, in L-functions and Galois Representations, vol. 320 of London Math. Soc. Lecture Note Ser., Cambridge University Press, Cambridge, 2007, pp. 59–120. [3] S. C HO AND V. VATSAL, Deformations of induced Galois representations, J. Reine Angew. Math., 556 (2003), 79–98. [4] R. C OLEMAN AND B. M AZUR, The eigencurve, in Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996), vol. 254 of London Math. Soc. Lecture Note Ser., Cambridge University Press, Cambridge, 1998, pp. 1–113. [5] P. D ELIGNE AND J.-P. S ERRE, Formes modulaires de poids 1, Ann. Sci. École Norm. Sup. 7 (4), (1974), 507–530. [6] M. D IMITROV AND E. G HATE, On classical weight one forms in Hida families, J. Théor. Nombres Bordeaux, 24, 3 (2012), 669–690 [7] M. E MERTON , R. P OLLACK AND T. W ESTON, Variation of Iwasawa invariants in Hida families, Invent. Math., 163 (2006), 523–580. [8] E. G HATE AND N. K UMAR, Control theorems for ordinary 2-adic families of modular forms, to appear in the Proceedings of the International Colloquium on Automorphic Representations and L-functions, TIFR, 2012. [9] E. G HATE AND V. VATSAL, On the local behaviour of ordinary -adic representations, Ann. Inst. Fourier, Grenoble, 54, 7 (2004), 2143–2162. [10] R. G REENBERG AND V. VATSAL, Iwasawa theory for Artin representations, in preparation. [11] H. H IDA, Galois representations into GL2 (Z p [[X ]]) attached to ordinary cusp forms, Invent. Math., 85 (1986), 545–613. [12] , Iwasawa modules attached to congruences of cusp forms, Ann. Sci. Ecole Norm. Sup. 19 (4), (1986), 231–273. [13] M. K ISIN, Overconvergent modular forms and the Fontaine-Mazur conjecture, Invent. Math., 153 (2003), 373–454.

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[14] P. K UTZKO, The Langlands conjecture for Gl2 of a local field, Ann. of Math. 112 (2), (1980), pp. 381–412. [15] L. N YSSEN, Pseudo-représentations, Math. Ann., 306 (1996), 257–283. [16] R. ROUQUIER, Caractérisation des caractères et pseudo-caractères, J. Algebra, 180 (1996), 571–586. [17] A. W EIL, Exercices dyadiques, Invent. Math., 27 (1974), 1–22. [18] A. W ILES, On ordinary λ-adic representations associated to modular forms, Invent. Math., 94 (1988), 529–573.

2 Vector bundles on curves and p-adic Hodge theory Laurent Fargues and Jean-Marc Fontaine

Contents Introduction 1 Holomorphic functions of the variable π 2 The space |Y | 3 Divisors on Y 4 Divisors on Y/ϕ Z 5 The curve 6 Vector bundles 7 Vector bundles and ϕ-modules References

page 17 18 30 44 48 54 62 83 103

Introduction This text is an introduction to our work [12] on curves and vector bundles in p-adic Hodge theory. This is a more elaborate version of the reports [13] and [14]. We give a detailed construction of the “fundamental curve of p-adic Hodge theory” together with sketches of proofs of the main properties of the objects showing up in the theory. Moreover, we explain thoroughly the classification theorem for vector bundles on the curve, giving a complete proof for rank two vector bundles. The applications to p-adic Hodge theory, the theorem “weakly admissible implies admissible” and the p-adic monodromy theorem, are not given here but can be found in [13] and [14]. We would like to thank the organizers of the EPSRC Durham Symposium “Automorphic forms and Galois representations” for giving us the opportunity to talk about this subject. Automorphic Forms and Galois Representations, ed. Fred Diamond, Payman L. Kassaei and c Cambridge University Press 2014. Minhyong Kim. Published by Cambridge University Press.

17

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Laurent Fargues and Jean-Marc Fontaine

1. Holomorphic functions of the variable π 1.1. Background on holomorphic functions in a p-adic punctured disk after Lazard ([25]) 1.1.1. The Frechet algebra B Let F be a complete non-archimedean field for a non trivial valuation v : F −→ R ∪ {+∞}, with characteristic p residue field. We note |·| = p −v(·) the associated absolute value. Consider the open punctured disk D∗ = {0 < |z| < 1} ⊂ A1F as a rigid analytic space over F, where z is the coordinate on the affine line. If I ⊂]0, 1[ is a compact interval set D I = {|z| ∈ I } ⊂ D∗ , ∗ an annulus that is an affinoïd domain in D if I = [ρ1 , ρ2 ] with ρ1 , ρ2 ∈ |F × |. There is an admissible affinoïd covering

D∗ = DI I ⊂]0,1[

where I goes through the preceding type of compact intervals. Set now B = O(D∗ ) = an z n | an ∈ F, ∀ρ ∈]0, 1[ n∈Z

lim |an |ρ n = 0

|n|→+∞

the ring of holomorphic functions on D∗ . In the preceding description of B, one checks the infinite set of convergence conditions associated to each ρ ∈]0, 1[ can be rephrased in the following two conditions ⎧ ⎨lim inf v(an n ) ≥ 0 n→+∞ (2.1) ⎩ lim v(a−n ) = +∞. n For ρ ∈]0, 1[ and f =

n→+∞

n an z

n

∈ B set

| f |ρ = sup{|an |ρ n }. n∈Z

If ρ = p −r with r > 0 one has | f |ρ = p −vr ( f ) with vr ( f ) = inf {v(an ) + nr }. n∈Z

Vector bundles on curves and p-adic Hodge theory

19

Then | · |ρ is the Gauss supremum norm on the annulus {|z| = ρ}. It is in fact a multiplicative norm, that is to say vr is a valuation. Equipped with the set of norms (| · |ρ )ρ∈]0,1[ , B is a Frechet algebra. The induced topology is the one of uniform convergence on compact subsets of the Berkovich space associated to D∗ . If I ⊂]0, 1[ is a compact interval then B I = O(D I ) equipped with the set of norms (| · |ρ )ρ∈I is a Banach algebra. In fact if I = [ρ1 , ρ2 ] then by the maximum modulus principle, for f ∈ B I sup | f ρ | = sup{| f |ρ1 , | f |ρ2 }. ρ∈I

One then has B = lim B I ←− I ⊂]0,1[

as a Frechet algebra written as a projective limit of Banach algebras. For f = n an z n ∈ B one has | f |1 = lim | f |ρ = sup |an | ∈ [0, +∞]. ρ→1

n∈Z

We will later consider the following closed sub-O F -algebra of B B+ = { f ∈ B | f 1 ≤ 1} = an z n ∈ B | an ∈ O F n∈Z

=

an z n | an ∈ O F ,

n∈Z

v(a−n ) = +∞ . n→+∞ n lim

Set now Bb = { f ∈ B | ∃N ∈ N, z N f ∈ O(D) and is bounded on D} = an z n | an ∈ F, ∃C ∀n |an | ≤ C . n−∞

This is a dense sub-algebra of B. In particular one can find back B from Bb via completion with respect to the norms (| · |ρ )ρ∈]0,1[ . In the same way b Bb,+ = B ∩ B+ = an z n | an ∈ O F

= is dense in

B+

n−∞ O F z[ 1z ]

and thus one can find back B+ from Bb,+ via completion.

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Laurent Fargues and Jean-Marc Fontaine

1.1.2. Zeros and growth of holomorphic functions Recall the following properties of holomorphic functions overs C. Let f be holomorphic on the open disk of radius 1 (one could consider the punctured disk but we restrict to this case to simplify the exposition). For ρ ∈ [0, 1[ set M(ρ) = sup | f (z)|. |z|=ρ

The following properties are verified: • the function ρ → − log M(ρ) is a concave function of log ρ (Hadamard), • if f (0) = 0, f has no zeros on the circle of radius ρ and (a1 , . . . , an ) are its zeros counted with multiplicity in the disk of radius ρ, then as a consequence of Jensen’s formula − log | f (0)| ≥

n

(− log |ai |) − nρ − log M(ρ).

i=1

In the non-archimedean setting we have an exact formula linking the growth of an holomorphic function and its zeros. For this, recall the formalism of the Legendre transform. Let ϕ : R −→] − ∞, +∞] be a convex decreasing function. Define the Legendre transform of ϕ as the concave function (see Figure 1) L (ϕ) : ]0, +∞[ −→ [−∞, +∞[ λ −→ inf {ϕ(x) + λx}. x∈R

L(ϕ)(λ) ϕ(x) x

λ

Figure 1 The Legendre transform of ϕ evaluated at the slope λ where by definition the slope is the opposite of the derivative (we want the slopes to be the valuations of the roots for Newton polygons).

Vector bundles on curves and p-adic Hodge theory

21

If ϕ1 , ϕ2 are convex decreasing functions as before define ϕ1 ∗ ϕ2 as the convex decreasing function defined by (ϕ1 ∗ ϕ2 )(x) = inf {ϕ1 (a) + ϕ2 (b)}. a+b=x

We have the formula L (ϕ1 ∗ ϕ2 ) = L (ϕ1 ) + L (ϕ2 ). One can think of the Legendre transform as being a “tropicalization” of the Laplace transform: tropicalization

(R, +, ×) −−−−−−−−→ (R, inf, +) Laplace transform /o /o /o / Legendre transform usual convolution ∗ /o /o o/ / tropical ∗ just defined. The function ϕ is a polygon, that is to say piecewise linear, if and only if L (ϕ) is a polygon. Moreover in this case: • the slopes of L (ϕ) are the x-coordinates of the breakpoints of ϕ, • the x-coordinates of the breakpoints of L (ϕ) are the slopes of ϕ. Thus L and its inverse give a duality slopes ←→ x-coordinates of breakpoints. From these considerations one deduces that if ϕ1 and ϕ2 are convex decreasing polygons such that ∀i = 1, 2, ∀λ ∈]0, +∞[, L (ϕi )(λ) = −∞ then the slopes of ϕ1 ∗ ϕ2 are obtained by concatenation from the slopes of ϕ1 and the ones of ϕ2 . For f = n∈Z an z n ∈ B set now Newt(f) = decreasing convex hull of {(n, v(an ))}n∈Z . This is a polygon with integral x-coordinate breakpoints. Moreover the function r → vr ( f ) defined on ]0, +∞[ is the Legendre transform of Newt(f). Then, the statement of the “ p-adic Jensen formula” is the following: the slopes of Newt(f) are the valuations of the zeros of f (with multiplicity). Example 2.1. Take f ∈ O(D), f (0) = 0. Let ρ = p −r ∈]0, 1[ and (a1 , . . . , an ) be the zeros of f in the ball {|z| ≤ ρ} counted with multiplicity. Then, as a consequence of the fact that r → vr ( f ) is the Legendre transform

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Laurent Fargues and Jean-Marc Fontaine

v(f(0)) v(a1) v(a2) vr(f)

r

1

v(an)

2

n

Figure 2 An illustration of the “ p-adic Jensen formula”. The numbers over each line are their slopes.

of a polygon whose slopes are the valuations of the roots of f , one has the formula n v( f (0)) = vr ( f ) − nr + v(ai ) i =1

(see Figure 2). Finally, remark that B+ is characterized in terms of Newton polygons: B+ = { f ∈ B | Newt(f) ⊂ upper half plane y ≥ 0}. 1.1.3. Weierstrass products For a compact interval I = [ρ1 , ρ2 ] ⊂]0, 1[ with ρ1 , ρ2 ∈ |F × | the ring B I is a P.I.D. with Spm(B I ) = |D I |. In particular Pic(D I ) = 0. Now let’s look at Pic(D∗ ). In fact in the following we will only be interested in the submonoid of effective line bundles Pic+ (D∗ ) = {[L ] ∈ Pic(D∗ ) | H 0 (D∗ , L ) = 0}. Set

Div+ (D∗ ) = D = m x [x] | m x ∈ N, ∀I ⊂]0, 1[ x∈|D∗ |

compact supp(D) ∩ D I is finite

the monoid of effective divisors on D∗ . There is an exact sequence div

0 → B \ {0}/B× −−→ Div+ (D∗ ) −→ Pic+ (D∗ ) −→ 0. We are thus led to the question: for D ∈ Div+ (D∗ ), does there exist f ∈ B\{0} such that div( f ) = D ?

Vector bundles on curves and p-adic Hodge theory

23

This is of course the case if supp(D) is finite. Suppose thus it is infinite. We will suppose moreover F is algebraically closed (the discrete valuation case is easier but this is not the case we are interested in) and thus |D∗ | = m F \ {0} where m F is the maximal ideal of O F . Suppose first there exists ρ0 ∈]0, 1[ such that supp(D) ⊂ {0 < |z| ≤ ρ0 }. Then we can write D= [ai ], ai ∈ m F \ {0}, lim |ai | = 0. i→+∞

i ≥0

The infinite product

+∞

1−

i =0

ai z

,

converges in the Frechet algebra B and its divisor is D. We are thus reduced to the case supp(D) ⊂ {ρ0 < |z| < 1} for some ρ0 ∈]0, 1[. But if we write D= [ai ], lim |ai | = 1 i≥0

i→+∞

then neither of the infinite products “ i≥0 1 − azi ” or “ i ≥0 1 − azi ” converges. Recall that over C this type of problem is solved by introducing renormalization factors. Typically, if we are looking for a holomorphic func tion f on C such that div( f ) = n∈N [−n] then the product “z n∈N 1 + nz ” z does not converge but z n∈N 1 + nz e− n = eγ z 1(z) does. In our nonarchimedean setting this problem has been solved by Lazard. Theorem 2.2 (Lazard [25]). If F is spherically complete then there exists a sequence (h i )i≥0 of elements of B× such that the product i≥0 [(z − ai ).h i ] converges. Thus, if F is spherically complete Pic+ (D∗ ) = 0 (and in fact Pic(D∗ ) = 0). In the preceding problem, it is easy to verify that for any F there always exist renormalization factors h i ∈ B\{0} such that i≥0 [(z − ai ).h i ] converges and thus an f ∈ B \ {0} such that div( f ) ≥ D. The difficulty is thus to introduce renormalization factors that do not add any new zero. Let’s conclude this section with a trick that sometimes allows us not to introduce any renormalization factors. Over C this is the following. Suppose we are looking for a holomorphic on C whose divisor is n∈Z [n]. The infi function nite product “z n∈Z\{0} 1 − nz ” does not converge. Nevertheless, regrouping the terms, the infinite product z n≥1 1 − nz 1 + nz = sinππ z converges.

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Laurent Fargues and Jean-Marc Fontaine

In the non-archimedean setting there is a case where this trick works. This is the following. Suppose E|Q p is a finite extension and E is an algebraic closure of E. Let LT be a Lubin–Tate group law over O E . Its logarithm logLT is a rigid analytic function on the open disk D with zeros the torsion points of the Lubin–Tate group law, LT [π ∞ ] = {x ∈ m E | ∃n ≥ 1, [π n ]LT (x) = 0}. The infinite product z

1−

ζ ∈LT [π ∞ ]\{0}

z ζ

does not converge since in this formula |ζ | → 1. Nevertheless, the infinite product z z 1− ζ n n−1 n≥1

ζ ∈LT [π ]\LT [π

]

converges in the Frechet algebra of holomorphic functions on D and equals logLT . This is just a reformulation of the classical formula logLT = lim π −n [π n ]LT . n→+∞

1.2. Analytic functions in mixed characteristic 1.2.1. The rings B and B+ Let E be a local field with uniformizing element π and finite residue field Fq . Thus, either E is a finite extension of Q p or E = Fq ((π )). Let F|Fq be a valued complete extension for a non trivial valuation v : F → R ∪ {+∞}. Suppose moreover F is perfect (in particular v is not discrete). Let E |E be the unique complete unramified extension of E inducing the extension F|Fq on the residue fields, OE /π OE = F. There is a Teichmüller lift [·] : F → OE and E = [xn ]π n | xn ∈ F (unique writing). n−∞

If charE = p then [·] is additive, E |F, and E = F((π )). If E|Q p then E = WO E (F)[ π1 ] = W (F) ⊗W (Fq ) E the ramified Witt vectors of F. There is a Frobenius ϕ acting on E , q ϕ [xn ]π n = [xn ]π n . n

n

Vector bundles on curves and p-adic Hodge theory

25

f

If E|Q p then on W (F) ⊗W (Fq ) E one has ϕ = ϕQ p ⊗ Id where in this formula

q = p f and ϕQ p is the usual Frobenius of the Witt vectors. In this case the addition law of WO E (F) is given by [xn ]π n + [yn ]π n = [Pn (x0 , . . . , xn , y0 , . . . , yn )]π n (2.2) n≥0

n≥0

n≥0

q i−n q j −n where Pn ∈ Fq X i , Y j are generalized polynomials. The 0≤i, j ≤n multiplication law is given in the same way by such kind of generalized polynomials. Definition 2.3. (1) Define Bb = [xn ]π n ∈ E | ∃C, ∀n |xn | ≤ C n−∞

b,+

B

=

[xn ]π n ∈ E | xn ∈ O F

n−∞

= WO E (O F ) π1 if E|Q p (2) For x =

= O F π [ π1 ] if E = Fq ((π )). n [x n ]π

n

∈ Bb and r ≥ 0 set vr (x) = inf {v(xn ) + nr }. n∈Z

If ρ = q −r ∈]0, 1] set |x|ρ = q −vr (x) . (3) For x = n [xn ]π n ∈ Bb set Newt(x) = decreasing convex hull of {(n, v(xn ))}n∈Z . In the preceding definition one can check that the function vr does not depend on the choice of a uniformizing element π . In the equal characteristic case, that is to say E = Fq ((π )), setting z = π one finds back the rings defined in Section 1.1. For x ∈ Bb the function r → vr (x) defined on ]0, +∞[ is the Legendre transform of Newt(x). One has v0 (x) = lim vr (x). The Newton polygon of r→0

x is +∞ exactly on ] − ∞, vπ (x)[ and moreover lim Newt(x) = v0 (x). One +∞

has to be careful that since the valuation of F is not discrete, this limit is not always reached, that is to say Newt(x) may have an infinite number of strictly positive slopes going to zero. One key point is the following proposition whose proof is not very difficult but needs some work. Proposition 2.4. For r ≥ 0, vr is a valuation on Bb .

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Laurent Fargues and Jean-Marc Fontaine

Thus, for all ρ ∈]0, 1], | · |ρ is a multiplicative norm. One deduces from this that for all x, y ∈ Bb , Newt(xy) = Newt(x) ∗ Newt(y) (see 1.1.2). In particular the slopes of Newt(xy) are obtained by concatenation from the slopes of Newt(x) and the ones of Newt(y). For example, as a consequence, if a1 , . . . , an ∈ m F \ {0}, then Newt (π − [a1 ]) . . . (π − [an ]) is +∞ on ] − ∞, 0[, 0 on [n, +∞[ and has non-zero slopes v(a1 ), . . . , v(an ). Definition 2.5. Define • B = completion of Bb with respect to (| · |ρ )ρ∈]0,1[ , • B+ = completion of Bb,+ with respect to (| · |ρ )ρ∈]0,1[ , • for I ⊂]0, 1[ a compact interval B I = completion of Bb with respect to (| · |ρ )ρ∈I . The rings B and B+ are E-Frechet algebras and B+ is the closure of Bb,+ in B. Moreover, if I = [ρ1 , ρ2 ] ⊂]0, 1[, for all f ∈ B sup | f |ρ = sup{| f |ρ1 , | f |ρ2 } ρ∈I

because the function r → vr ( f ) is concave. Thus, B I is an E-Banach algebra. As a consequence, the formula B = lim B I ←− I ⊂]0,1[

expresses the Frechet algebra B as a projective limit of Banach algebras. Remark 2.6. Of course, the preceding rings are not new and appeared for example under different names in the work of Berger ([2]) and Kedlaya ([21]). The new point of view here is to see them as rings of holomorphic functions of the variable π . In particular, the fact that vr is a valuation (Proposition 2.4) had never been noticed before. The Frobenius ϕ extends by continuity to automorphisms of B and B+ , and ∼ for [ρ1 , ρ2 ] ⊂]0, 1[ to an isomorphism ϕ : B[ρ1 ,ρ2 ] − → B[ρ q ,ρ q ] . 1

2

Remark 2.7. In the case E = Fq ((π )), setting z = π as in Section 1.1, the q n n Frobenius ϕ just defined is given by ϕ n x n z = n x n z. This is thus an arithmetic Frobenius, the geometric one being n xn z n → n xn z qn .

Vector bundles on curves and p-adic Hodge theory

27

The ring B+ satisfies a particular property. In fact, if x ∈ Bb,+ and r ≥ r > 0 then vr (x) ≥

r vr (x). r

(2.3)

Thus, if 0 < ρ ≤ ρ < 1 we have + + B+ [ρ,ρ ] = Bρ ⊂ Bρ b,+ where for a compact interval I ⊂]0, 1[ we note B+ I for the completion of B + + with respect to the (| · |ρ )ρ∈I , and Bρ := B{ρ} . One deduces that for any ρ0 ∈]0, 1[, B+ ρ0 is stable under ϕ and B+ = ϕ n B+ ρ0 n≥0

the biggest sub-algebra of B+ ρ0 on which ϕ is bijective. Suppose E = Q p and choose ρ ∈ |F × |∩]0, 1[. Let a ∈ F such that |a| = ρ. Define B+ cris,ρ = p-adic completion of the P.D. hull of the ideal W (O F )[a] of W (O F ) ⊗Z p Q p . This depends only on ρ since W (O F )[a] = {x ∈ W (O F ) | |x|0 ≥ ρ}. We thus have n 1 [a ] B+ = W (O ) . F cris,ρ n! n≥1 p One has + + B+ ρ p ⊂ Bcris,ρ ⊂ Bρ p−1 .

From this one deduces B+ =

ϕ n B+ cris,ρ ,

n≥0 + the ring usually denoted “Brig ” in p-adic Hodge theory. The ring B+ cris,ρ appears naturally in comparison theorems where it has a natural interpretation in terms of crystalline cohomology. But the structure of the ring B+ is simpler as we will see. Moreover, if k ⊂ O F is a perfect sub-field, K 0 = W (k)Q and (D, ϕ) a k-isocrystal, then ϕ=Id ϕ=Id D ⊗ K 0 B+ = D ⊗ K 0 B+ cris,ρ + because ϕ is bijective on D. Replacing B+ cris,ρ by B is thus harmless most of the time.

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Laurent Fargues and Jean-Marc Fontaine

Remark 2.8. Suppose E|Q p and let (xn )n∈Z be a sequence of O F such that lim v(xn−n ) = +∞. Then, the series n→+∞

[xn ]π n

n∈Z

converges in

B+ .

But:

• we don’t know if each element of B+ is of this form, • for such an element of B+ we don’t know if such a writing is unique, • we don’t know if the sum or product of two element of this form is again of this form. The same remark applies to B. Nevertheless, there is a sub E-vector space of B+ where the preceding remark does not apply. One can define for any O E -algebra R the group of (ramified) Witt bivectors BWO E (R). Elements of BWO E (R) have a Teichmüller expansion that is infinite on the left and on the right. One has BWO E (O F ) :=

BWO E (O F /a)

lim ←−

a⊂O F non zero ideal

=

n∈Z

=

Vπn [xn ] | xn ∈ O F , liminf v(xn ) > 0 n→−∞

[yn ]π n | yn ∈ O F , liminf q n v(x n ) > 0 ⊂ B+ . n→−∞

n∈Z

The point is that in BWO E (O F ) [xn ]π n + [yn ]π n = lim Pk (xn−k , . . . , xn , yn−k , . . . , yn ) π n n∈Z

n∈Z

n∈Z

k→+∞

where the generalized polynomials q i −k q j −k Pk ∈ Fq X i , Y j 1≤i, j ≤k give the addition law of the Witt vectors as in formula (2.2) and the limits in the preceding formulas exist thanks to [15], prop.1.1 chap. II, and the convergence condition appearing in the definition of the bivectors. In fact, periods of π -divisible O E -modules (that is to say p-divisible groups when E = Q p ) lie in BWO E (O F ), and BWO E (O F ) contains all periods whose Dieudonné–Manin slopes lie in [0, 1]. In equal characteristic, when E = Fq ((π )), there is no restriction on Dieudonné–Manin slopes of formal O E -modules (what we call here a formal O E -module is a Drinfeld module in dimension 1). This gives a meta-explanation to the fact that in equal

Vector bundles on curves and p-adic Hodge theory

29

characteristic all elements of B have a unique power series expansion and the fact that this may not be the case in unequal characteristic. 1.2.2. Newton polygons Since the elements of B may not be written uniquely as a power series n n∈Z [x n ]π , we need a trick to define the Newton polygon of such elements. The following proposition is an easy consequence of the following Dini type theorem: if a sequence of concave functions on ]0, +∞[ converges point-wise then the convergence is uniform on all compact subsets of ]0, +∞[ (but not on all ]0, +∞[ in general). Proposition 2.9. If (xn )n≥0 is a sequence of Bb that converges to x ∈ B \ {0} then for all I ⊂]0, 1[ compact, ∃N, n ≥ N and q −r ∈ I =⇒ vr (xn ) = vr (x). One deduces immediately: Corollary 2.10. For x ∈ B the function r → vr (x) is a concave polygon with integral slopes. This leads us to the following definition. Definition 2.11. For x ∈ B, define Newt(x) as the inverse Legendre transform of the function r → vr (x). Thus, Newt(x) is a polygon with integral x-coordinate breakpoints. Moreover, if (λi )i ∈Z are its slopes, where λi is the slope on the segment [i, i + 1] (we set λi = +∞ if Newt(x) is +∞ on this segment), then lim λi = 0 and lim λi = +∞.

i →+∞

i→−∞

In particular lim Newt(x) = +∞. Those properties of Newt(x) are the only −∞

restrictions on such polygons. Remark 2.12. If xn −→ x in B with xn ∈ Bb and x = 0 then one checks n→+∞

using Proposition 2.9 that in fact for any compact subset K of R, there exists an integer N such that for n ≥ N , Newt(xn )|K = Newt(x)|K . The advantage of Definition 2.11 is that it makes it clear that Newt(x) does not depend on the choice of a sequence of Bb going to x. Example 2.13. (1) If (xn )n∈Z is a sequence of F satisfying the two conditions of formula n is the decreasing convex hull (2.1) then the polygon Newt [x ]π n∈Z n of {(n, v(xn ))}n∈N .

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Laurent Fargues and Jean-Marc Fontaine

(2) If (an )n≥0 is a sequence of F × going to zero then the infinite product [an ] converges in B and its Newton polygon is zero on [0, +∞[ n≥0 1− π and has slopes the (v(an ))n≥0 on ] − ∞, 0]. Of course the Newton polygon of x does not give more information than the polygon r → vr (x). But it is much easier to visualize and its interest lies in the fact that we can appeal to our geometric intuition from the usual case of holomorphic functions recalled in Section 1.1 to guess and prove results. Here is a typical application: the proof of the following proposition is not very difficult once you have convinced yourself it has to be true by analogy with the usual case of holomorphic functions. Proposition 2.14. We have the following characterizations: (1) B+ = {x ∈ B | Newt(x) ≥ 0}. (2) Bb = {x ∈ B | Newt(x) is bounded below and ∃A, Newt(x)|]−∞,A] = +∞}. (3) The algebra {x ∈ B | ∃ A, Newt(x)|]−∞,A] = +∞} is a subalgebra of ! v(x n ) n WO E (F)[ π1 ] equal to n−∞ [x n ]π | liminf n ≥ 0 . n→+∞

This has powerful applications that would be difficult to obtain without Newton polygons. For example one obtains the following. Corollary 2.15. × ! (1) B× = Bb = x ∈ Bb | Newt(x) has 0 as its only non infinite slope . d (2) One has Bϕ=π = 0 for d < 0, Bϕ=Id = E and for d ≥ 0, ϕ=π d d Bϕ=π = B+ . Typically, the second point is obtained in the following way. If x ∈ B satisfies ϕ(x) = π d x then Newt(ϕ(x)) = Newt(π d x) that is to say Newt(x) satisfies the functional equation qNewt(x)(t) = Newt(x)(t − d). By solving this functional equation and applying Proposition 2.14 one finds the results.

2. The space |Y | 2.1. Primitive elements We would like to see the Frechet algebra B defined in the preceding section as an algebra of holomorphic functions on a “rigid analytic space” Y . This is of course the case if E = Fq ((π )) since we can take Y = D∗ a punctured disk as in Section 1.1. This is not the case anymore when E|Q p , at least as a Tate rigid space. But nevertheless we can still define a topological space |Y | that embeds

Vector bundles on curves and p-adic Hodge theory

31

in the Berkovich space M(B) of rank 1 continuous valuations on B. It should be thought of as the set of classical points of this “space” Y that would remain to construct. To simplify the exposition, in the following we always assume E|Q p , that is to say we concentrate on the most difficult case. When E = Fq ((π )), all stated results are easy to obtain by elementary manipulation and are more or less already contained in the backgrounds of Section 1.1. Definition 2.16. (1) An element x = n≥0 [xn ]π n ∈ WO E (O F ) is primitive if x0 = 0 and there exists an integer n such that xn ∈ O× F . For such a primitive element x we define deg(x) as the smallest such integer n. (2) A primitive element of strictly positive degree is irreducible if it can not be written as a product of two primitive elements of strictly lower degree. If k F is the residue field of O F there is a projection WO E (O F ) WO E (k F ). Then, x is primitive if and only if x ∈ / π WO E (O F ) and its projection x˜ ∈ WO E (k F ) is non-zero. For such an x, deg(x) = vπ (x). ˜ We deduce from this that the product of a degree d by a degree d primitive element is a degree d +d primitive element. Degree 0 primitive elements are the units of WO E (O F ). Any primitive degree 1 element is irreducible. In terms of Newton polygons, x ∈ WO E (O F ) is primitive if and only if Newt(x)(0) = +∞ and Newt(x)(t) = 0 for t 0. Definition 2.17. Define |Y | to be the set of primitive irreducible elements modulo multiplication by an element of WO E (O F )× . There is a degree function deg : |Y | → N≥1 given by the degree of any representative of a class in |Y |. If x is primitive note x¯ ∈ O F for its reduction modulo π. We have |x| ¯ = | y¯ | if y ∈ WO E (O F )× .x. There is thus a function · : |Y | −→ ]0, 1[ WO E (O F )× .x −→ |x| ¯ 1/ deg(x) . A primitive element x ∈ WO E (O F ) of strictly positive degree is irreducible if and only if the ideal generated by x is prime. In fact, if x = yz with y, z ∈ WO E (O F ) and x primitive then projecting to W O E (k F ) and O F the

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Laurent Fargues and Jean-Marc Fontaine

preceding equality one obtains that y and z are primitive. There is thus an embedding |Y | ⊂ Spec(WO E (O F )). The Frobenius ϕ induces a bijection ∼

ϕ : |Y | − → |Y | that leaves invariant the degree and satisfies ϕ(y) = yq . Remark 2.18. When E = Fq ((π )), replacing WO E (O F ) by O F z in the preceding definitions (we set z = π ) there is an identification |Y | = |D∗ |. In fact, according to Weierstrass, any irreducible primitive f ∈ O F z has a unique irreducible unitary polynomial P ∈ O F [z] in its O F z× -orbit satisfying: P(0) = 0 and the roots of P have absolute value < 1. Then for y ∈ |D∗ |, deg(y) = [k(y) : F] and y is the distance from y to the origin of the disk D.

2.2. Background on the ring R For an O E -algebra A set q R(A) = x (n) n≥0 | x (n) ∈ A, x (n+1) = x (n) . If A is π -adic, I is a closed ideal of A such that A is I + (π )-adic, then the reduction map induces a bijection ∼

R(A) −→ R(A/I ) with inverse given by

x (n)

n≥0

−→

lim

k→+∞

x (n+k)

q k n≥0

,

where x (n+k) ∈ A is any lift of x (n+k) ∈ A/I , and the preceding limit is for the I + (π )-adic topology. In particular, applying this for I = (π ), we deduce that the set valued functor R factorizes canonically as a functor R : π -adic O E -algebras −→ perfect Fq -algebras. If WO E stands for the (ramified) Witt vectors there is then a couple of adjoint functors π -adic O E -algebras o

R WO E

/

perfect Fq -algebras

Vector bundles on curves and p-adic Hodge theory

33

where WO E is left adjoint to R and the adjunction morphisms are given by: ∼ A −→ R WO E (A) −n a −→ a q n≥0

and θ : WO E (R(A)) −→ A [xn ]π n −→ x n(0) π n . n≥0

n≥0

If L|Q p is a complete valued extension for a valuation w : L → R ∪ {+∞} extending a multiple of the p-adic valuation, then R(L) equipped with the valuation x −→ w(x (0) ) is a characteristic p perfect complete valued field with ring of integers R(O L ) (one has to be careful that the valuation on R(L) may be trivial). It is not very difficult to prove that if L is algebraically closed then R(L) is too. A reciprocal to this statement will be stated in the following sections.

2.3. The case when is F algebraically closed Theorem 2.19. Suppose F is algebraically closed. Let p ∈ Spec(WO E (O F )) generated by a degree one primitive element. Set A = WO E (O F )/p and θ : WO E (O F ) A the projection. The following properties are satisfied: (1) There is an isomorphism ∼

O F −→ R(A) −n x −→ θ x q

n≥0

.

(2) The map O F −→ A x −→ θ ([x]) is surjective. (3) There is a unique valuation w on A such that for all x ∈ O F , w(θ ([x])) = v(x).

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Laurent Fargues and Jean-Marc Fontaine

Moreover, A[ π1 ] equipped with the valuation w is a complete algebraically closed extension of E with ring of integers A. There is an identification of valued fields F = R A[ π1 ] . (4) If π ∈ R(A) is such that π (0) = π then p = ([π] − π ). Remark 2.20. One can reinterpret points (2) and (4) of the preceding theorem in the following way. Let x be primitive of degree 1. Then: • we have a Weierstrass division in WO E (O F ): given y ∈ WO E (O F ), there exists z ∈ WO E (O F ) and a ∈ O F such that y = zx + [a], • we have a Weierstrass factorization of x: there exists u ∈ WO E (O F )× and b ∈ O F such that x = u.(π − [b]). One has to be careful that, contrary to the classical situation, the remainder term a is not unique in such a Weierstrass division. Similarly, b is not uniquely determined by x in the Weierstrass factorization. Indications on the proof of Theorem 2.19. Statement (1) is an easy consequence of general facts about the ring R recalled in Section 2.2: if p = (x) ∼

R(A) −→ R(A/π A) = R(O F /x) ¯ = OF since O F is perfect. According to point (1), point (2) is reduced to proving that any element of A has a q-th root. If E 0 = W (Fq )Q , the norm map N E/E 0 induces a norm map WO E (O F ) → W (O F ) sending a primitive element of degree 1 to a primitive element of degree 1. Using this one can reduce the problem to the case E = Q p . Suppose p = 2 to simplify. Then any element of 1 + p 2 W (O F ) has a p-th root. Using this fact plus some elementary manipulations one is reduced to solving some explicit equations in the truncated Witt vectors of length 2, W2 (O F ). One checks this is possible, using the fact that F is algebraically closed (here this hypothesis is essential, the hypothesis F perfect is not sufficient). In fact, the preceding proof gives that for any integer n, any element of A has an n-th root, the case when n is prime to p being easier than the case n = p we just explained (since then any element of 1 + pWO E (O F ) has an n-th root).

Vector bundles on curves and p-adic Hodge theory

35

Point (4) is an easy consequence of the following classical characterization (0) n of ker θ: an element y = ∈ π A× is a n≥0 [yn ]π ∈ ker θ such that y0 generator of ker θ . In fact, if y is such an element then ker θ = (y) + π ker θ and one concludes ker θ = (y) by applying the π -adic Nakayama lemma (ker θ is π -adically closed). In point (3), the difficulty is to prove that the complete valued field L = A[ π1 ] is algebraically closed; other points following easily from point (2). Using the fields of norms theory one verifies that L contains an algebraic closure of Q p . More precisely, one can suppose thanks to point (4) that p = (π − [π ]). There is then an embedding Fq ((π )) ⊂ F that induces F := Fq ((π )) ⊂ F. This induces a morphism L = WO E (O F ) π1 /(π −[π]) → L. But thanks to the fields of norms theory, L is the completion of an algebraic closure of E. In particular, L contains all roots of unity. Let us notice that since O L /π O L = O F /πO F , the residue field of L is the same as the one of F and is thus algebraically closed. Now, we use the following proposition that is well known in the discrete valuation case thanks to the theory of ramification groups (those ramification groups do not exist in the non discrete valuation case, but one can define some ad hoc one to obtain the proposition). Proposition 2.21. Let K be a complete valued field for a rank 1 valuation and K |K a finite degree Galois extension inducing a trivial extension on the residue fields. Then the group Gal(K |K ) is solvable. Since for any integer n any element of L has an n-th root, one concludes L is algebraically closed using Kümmer theory. Note now OC = WO E (O F )/p with fraction field C. One has to be careful that the valuation w on C extends only a multiple of the π-adic valuation of E: q −w(π ) = p. The quotient morphism Bb,+ −→ C extends in fact by continuity to surjective morphisms / Bb / B /B Bb,+E EE nn I ~ n ~ n EE ~~ nnn EE EE" ~ ~~n~nnnn " ~~ wnw n C where I ⊂]0, 1[ is such that p ∈ I . This is a consequence of the inequality q −w( f ) ≤ | f |ρ for f ∈ Bb and ρ = p. If p = (x) then all kernels of those surjections are the principal ideals generated by x in those rings.

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Laurent Fargues and Jean-Marc Fontaine

Convention: we will now see |Y |deg=1 as a subset of Spm(B). For m ∈ |Y |deg=1 , we note Cm = B/m, θm : B Cm and vm the valuation such that vm (θm ([x])) = v(x).

Theorem 2.22. If F is algebraically closed then the primitive irreducible elements are of degree one. Remark 2.23. One can reinterpret the preceding theorem as a factorization statement: if x ∈ WO E (O F ) is primitive of degree d then x = u.(π − [a1 ]) . . . (π − [ad ]), u ∈ WO E (O F )× , a1 , . . . , ad ∈ m F \ {0}. Indications on the proof of theorem 2.22. Let f ∈ WO E (O F ) be primitive. The theorem is equivalent to saying that f “has a zero in |Y |deg=1 ” that is to say there exists m ∈ |Y |deg=1 such that θm ( f ) = 0. The method to construct such a zero is a Newton type method by successive approximations. To make it work we need to know it converges in a sense that has to be specified. We begin by proving the following. Proposition 2.24. For m1 , m2 ∈ |Y |deg=1 set d(m1 , m2 ) = q −vm1 (a) if θm1 (m2 ) = OCm1 a. Then: deg=1 . (1) This defines an ultrametric distance on |Y | (2) For any ρ ∈]0, 1[, |Y |deg=1,·≥ρ , d is a complete metric space.

Remark 2.25. In equal characteristic, if E = Fq ((π )), then |Y | = |D∗ | = m F \{0} and this distance is the usual one induced by the absolute value |·| of F. The approximation algorithm then works like this. We define a sequence (mn )n≥1 of |Y |deg=1 such that: • (mn )n≥1 is constant, • it is a Cauchy sequence, • lim vmn ( f ) = +∞. n→+∞

Write f = k≥0 [xk ]π k . The Newton polygon of f as defined in Section 1.2.2 k is the same as the Newton polygon of g(T ) = k≥0 x k T ∈ O F T . Let z ∈ m F be a root of g(T ) with valuation the smallest one among the valuations

Vector bundles on curves and p-adic Hodge theory

37

of the roots of g(T ) (that is to say the smallest non-zero slope of Newt(f)). Start with m1 = (π − [z]) ∈ |Y |deg=1 . If mn is defined, mn = (ξ ) with ξ primitive of degree 1, we can write f = [ak ]ξ k k≥0

in WO E (O F ) (this is a consequence of point (2) of Theorem 2.19 and the fact that WO E (O F ) is ξ -adic). We check the power series h(T ) = k≥0 ak T k ∈ O F T is primitive of degree d. Let z be a root of h(T ) of maximal valuation. Then ξ − [z] is primitive of degree 1 and we set mn+1 = (ξ − [z]). We then prove the sequence (mn )n≥1 satisfies the required properties.

2.4. Parametrization of |Y | when F is algebraically closed Suppose F is algebraically closed. We see |Y | as a subset of Spm(B). As we saw, for any m ∈ |Y | there exists a ∈ m F \ {0} such that m = (π − [a]). The problem is that such an a is not unique. Moreover, given a, b ∈ m F \ {0}, there is no simple rule to decide whether (π − [a]) = (π − [b]) or not. Here is a solution to this problem. Let LT be a Lubin–Tate group law over O E . We note Q = [π]LT ∈ O E T and G the associated formal group on Spf(O E ). We have G(O F ) = m F , + . LT

The topology induced by the norms (| · |ρ )ρ∈]0,1[ on WO E (O F ) is the “weak topology” on the coefficients of the Teichmüller expansion, that is to say the product topology via the bijection ∼

ON F −→ WO E (O F ) (xn )n≥0 −→ [xn ]π n . n≥0

If a ∈ m F \ {0}, this coincides with the ([a], π )-adic topology. Moreover WO E (O F ) is complete, that is to say closed in B. If WO E (O F )00 = [xn ]π n | x0 ∈ m F = {x ∈ WO E (O F ) | x mod π ∈ m F }, n≥0

then

G WO E (O F ) = WO E (O F )00 , + . LT

One verifies the following proposition.

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Laurent Fargues and Jean-Marc Fontaine

Proposition 2.26. For x ∈ m F , the limit [x] Q := lim [π n ]LT

n→+∞

xq

−n

exists in WO E (O F ), reduces to x modulo π , and defines a “lift” [·] Q : G(O F ) → G(WO E (O F )). The usual Teichmüller lift [·] is well adapted to the multiplicative group law: [x y] = [x].[y]. The advantage of the twisted Teichmüller lift [·] Q is that it is more adapted to the Lubin–Tate one: [x] Q + [y] Q = x + y Q . LT

LT

When E = Q p and LT = Gm one has [x] Q = [1 + x] − 1. Definition 2.27. For ∈ m F \ {0} define u =

[] Q ∈ WO E (O F ). [ 1/q ] Q

This is a primitive degree one element since it is equal to the power series For example, if E = Q p and LT = Gm , setting

Q(T ) evaluated at [ 1/q ] Q . T = 1 + one has

1 p−1 u = 1 + p + · · · + p .

Proposition 2.28. There is a bijection ∼ G(O F ) \ {0} /O× E −→ |Y |

O× E . −→ (u ).

The inverse of this bijection is given by the following rule. For m ∈ |Y |, define ! X (G)(OCm ) = (xn )n≥0 | xn ∈ G(OCm ), π.xn+1 = xn . More generally, X (G) will stand for the projective limit “lim G” where the ←− n≥0

transition mappings are multiplication by π (one can give a precise geometric meaning to this but this is not our task here, see [12] for more details). The reduction modulo π map induces a bijection ∼

X (G)(OCm ) −→ X (G)(OCm /πOCm ) with inverse given by (xn )n →

lim π k xˆn+k

k→+∞

n

Vector bundles on curves and p-adic Hodge theory

39

where xˆn+k is any lift of xn+k . But [π]LT modulo π is the Frobenius Frobq . We thus have X (G)(OCm /πOCm ) = G R(OCm /π OCm ) = G(O F ) since ∼

∼

O F −−→ R(OCm ) −−→ R(OCm /π OCm ). θm ◦[·]

The Tate module Tπ (G) = {(xn )n≥0 ∈ X (G)(OCm ) | x0 = 1} ⊂ X (G)(OCm ) embeds thus in G(O F ). This is a rank 1 sub-O E -module. The inverse of the bijection of Proposition 2.28 sends m to this O E -line in G(O F ).

2.5. The general case: Galois descent Let now F be general (but still perfect), that is to say not necessarily algebraically closed. Let F be an algebraic closure of F with Galois group G F = Gal(F |F). Since the field F will vary we now " put " a subscript in the " is equipped with an preceding notation to indicate this variation. The set "Y

F action of G F . Theorem 2.29. For p ∈ |Y F | ⊂ Spec(WO E (O F )) set L = WO E (O F )/p π1 and θ : Bb,+ F L. Then: (1) There is a unique valuation w on L such that for x ∈ O F , w(θ ([x])) = v(x). (2) (L , w) is a complete valued extension of E. (3) (L , w) is perfectoid in the sense that the Frobenius of O L /π O L is surjective. (4) Via the embedding O F −→ R(L) −n a −→ θ a q

n≥0

one has R(L)|F and this extension is of finite degree [R(L) : F] = deg p.

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Laurent Fargues and Jean-Marc Fontaine

Remark 2.30. What we call here perfectoid is what we called “strictly p-perfect” in [13] and [14]. The authors decided to change their terminology because meanwhile the work [29] appeared. Indications on the proof. The "proof " is based on Theorems 2.19 and 2.22 via " " to |Y F |. For this we need the following. a Galois descent argument from Y

F Theorem 2.31. One has m F .H 1 G F , O

= 0. F Using Tate’s method ([31]) this theorem is a consequence of the following “almost etalness” statement whose proof is much easier than in characteristic 0. Proposition 2.32. If L|F is a finite degree extension then m F ⊂ tr L/F (O L ). Sketch of proof. The trace tr L/F commutes with the Frobenius ϕ = Frobq . Choosing x ∈ O L such that tr L/F (x) = 0 one deduces that lim |tr L/F (ϕ −n (x))| = 1.

n→+∞

From Theorem 2.31 one deduces that H 1 G F , m

= 0 which implies F H 1 G F , 1 + WO E m

=0 (2.4) F where

× × 1 + WO E m

= ker W O −→ W kF .

O O E E F F

Finally, let us notice that thanks to Ax’s theorem (that can be easily deduced from 2.32) one has G F WO E O

= WO E (O F ). F Proposition 2.33. Let x ∈ WO E O

be a primitive element such that ∀σ ∈ F G F , (σ (x)) = (x) as an ideal of WO E O

. Then, there exists y ∈ WO E (O F ) F such that (x) = (y). Proof. If x˜ ∈ WO E (k F ) is the projection of x via WO E (O

) → WO E (k F ), up F deg x to multiplying x by a unit, one can suppose x˜ = π . Looking at the cocycle σ → σ (x) , the proposition is then a consequence of the vanishing (2.4). x Let now x ∈ WO E (O F ) be primitive irreducible of degree d. There exist y1 , . . . , yr ∈ WO E O

primitive of degree 1 satisfying (yi ) = (y j ) for i = j, F a1 , . . . , ar ∈ N≥1 and u ∈ WO E (O

)× such that F x = u.

r i=1

yiai .

Vector bundles on curves and p-adic Hodge theory

The finite subset (yi )

! 1≤i ≤r

41

⊂ |Y

| F

is stable under G F . Using Proposition 2.33 and the irreducibility of x one verifies that this action is transitive and a1 = · · · = ar = 1. In particular one has r = d. Note mi = (yi ) and C mi the associated algebraically closed residue field. Let K |F be the degree d extension of F in F such that G K = StabG F (m1 ). One has Bb,+

/(x) = F

and thus

d

Cmi

i =1

G F GK Bb,+ = Cm .

/(x) 1 F

Now, one verifies using that x is primitive that if a ∈ m F \ {0} then 1 WO E (O F )/(x) π1 = WO E (O F )/(x) [a] . Theorem 2.31 implies that for such an a, [a].H 1 G F , WO E O

= 0. F One deduces from this that

G F b,+ L = Bb,+ /(x) = B /(x)

F F

that is thus a complete valued field. Moreover GK GK R(L) = R Cm = R(Cm1 )G K =

F = K. 1 Other statements of Theorem 2.29 are easily deduced in the same way. In the preceding theorem the quotient morphism θ : Bb,+ F L extends by continuity to a surjection B F L with kernel the principal ideal B F p. From now on, we will see |Y F | as a subset of Spm(B). If m ∈ |Y F | we note L m = B F /m, θm : B L m . The preceding arguments give the following. " "G F −fin " " " " whose G F -orbit is Theorem 2.34. Let "Y

be the elements of "Y

F F finite. There is a surjection " "G F −fin " β : "Y

−→ |Y F | F

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Laurent Fargues and Jean-Marc Fontaine

whose fibers are the G F -orbits: " "G F −fin ∼ "Y " /G F −→ |Y F |. F Moreover: • for m ∈ |Y F |, one has #β −1 (m) = [R(L m ) : F] = deg m, " "G F −fin " • for n ∈ "Y

, if m = β(n), one has an extension Cn |L m that identifies F Cn with the completion of the algebraic closure L m of L m and ∼ Gal F |R(L m ) −→ Gal L m |L m . Remark 2.35. (1) One has to be careful that θm (WO E (O F )) ⊂ O L m is only an order. It is equal to O L m if and only if deg m = 1. Using Theorem 2.31 one can verify that this order contains the maximal ideal of O L m . (2) Contrary to the case when F is algebraically closed, in general an m ∈ |Y F | of degree 1 is not generated by an element of the form π −[a], a ∈ m F \{0}.

2.6. Application to perfectoid fields Reciprocally, given a complete valued field L|E for a rank 1 valuation, it is perfectoid if and only if the morphism θ : WO E (R(O L )) −→ O L is surjective. In this case one can check that the kernel of θ is generated by a primitive degree 1 element. The preceding considerations thus give the following. Theorem 2.36. (1) There is an equivalence of categories between perfectoid fields L|E and the category of couples (F, m) where F|Fq is perfectoid and m ∈ |Y F | is of degree 1. (2) In the preceding equivalence, if L corresponds to (F, m), the functor R induces an equivalence between finite étale L-algebras and finite étale F-algebras. The inverse equivalence sends the finite extension F |F to B F /B F m.

Vector bundles on curves and p-adic Hodge theory

43

Example 2.37. (1) The perfectoid field L = Q p (ζ p∞ ) corresponds to F = F p ((T ))perf and p−1 m = 1 + [T 1/ p + 1] + · · · + [T p + 1] .

with (2) Choose p ∈ R(Q p ) such that p (0) = p. The perfectoid field L = M (n) perf M = ∪n≥0 Q p ( p ) corresponds to F = F p ((T )) and m = ([T ] − p). Remark 2.38. In the preceding correspondence, F is maximally complete if and only if L is. In particular, one finds back the formula given 1 in [26] for p-adic maximally complete fields: they are of the form W (O F ) p /([x] − p) where F is maximally complete of characteristic p and x ∈ F × satisfies v(x) > 0. Remark 2.39. Suppose F = F p ((T )). One can ask what are the algebraically closed residue fields Cm up to isomorphism when m goes through |Y F |. Let us note C p the completion of an algebraic closure of Q p . Thanks to the fields of norms theory it appears as a residue field Cm for some m ∈ |Y F |. The question is: is it true that for all m ∈ |Y F |, Cm C p ? The authors do not know the answer to this question. They know that for each integer n ≥ 1, OCm / p n OCm OC p / p n OC p but in a non canonical way. As a consequence of the preceding theorem we deduce almost étalness for characteristic 0 perfectoïd fields. Corollary 2.40. For L|E a perfectoïd field and L |L a finite extension we have m L ⊂ tr L |L (O L ). Proof. Set F = R(L ) and F = R(L). If L corresponds to m ∈ |Y F |, a = {x ∈ O F | |x| ≤ m} and a = {x ∈ O F | |x| ≤ m} we have identifications O L /πO L = O F /a O L /π O L = O F /a . According to point (2) of the preceding theorem, with respect to those identifications the map tr L |L modulo π is induced by the map tr F |F . The result is thus a consequence of Proposition 2.32. Remark 2.41. (1) Let K be a complete valued extension of Q p with discrete valuation and perfect residue field and M|K be an algebraic infinite degree arithmetically profinite extension. Then, by the fields of norms theory ([32]),

is perfectoid and point (2) of the preceding theorem is already L = M contained in [32].

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Laurent Fargues and Jean-Marc Fontaine

(2) In [29] Scholze has obtained a different proof of point (2) of Theorem 2.36 and of Corollary 2.40. Using Sen’s method ([30]), Corollary 2.40 implies the following. Theorem 2.42. Let L|E be a perfectoid field with algebraic closure L. Then the functor V → V ⊗ L

L induces an equivalence of categories between finite dimensional L-vector spaces and finite dimensional

L-vector spaces equipped with a continuous semi-linear action of Gal(L|L). An inverse is given by the functor W → W Gal(L|L) . Using this theorem one deduces by dévissage the following that we will use later. Theorem 2.43. Let m ∈ |Y F | and note B+ = B+

F ,d R,m

m ∈|Y | F β(m )=m

F ,d R,m

the B

m-adic completion of B

(see Definition 2.44). Then the functor F F M −→ M ⊗B+

F,d R,m

B+

F ,d R,m

induces an equivalence of categories between finite type B+ F,d R,m -modules and + finite type B

-modules equipped with a continuous semi-linear action of F ,d R,m

Gal(F |F). An inverse is given by the functor W → W Gal( F |F ) .

3. Divisors on Y 3.1. Zeros of elements of B We see |Y | as a subset of Spm(B). For m ∈ |Y |, we set L m = B/m and θm : B L m . We note vm the valuation on L m such that vm (θm ([a])) = v(a). One has m = q −vm (π )/ deg m where . was defined after Definition 2.17. Definition 2.44. For m ∈ |Y | define Bd+R,m as the m-adic completion of B.

Vector bundles on curves and p-adic Hodge theory

45

The ring B+ d R,m is a discrete valuation ring with residue field L m and the + natural map B → B+ d R,m is injective. We note again θm : Bd R,m → L m . We note ordm : B+ d R,m −→ N ∪ {+∞} its normalized valuation. Example 2.45. If E = Fq ((π )) then |Y | = |D∗ | and if m ∈ |Y | corresponds ∗ to x ∈ |D∗ | then B+ d R,m = OD ,x . Theorem 2.46. For f ∈ B, the non-zero finite slopes of Newt(f) are the − logq m with multiplicity ordm ( f ) deg(m) where m goes through the elements of |Y | such that θm ( f ) = 0. Indications on the proof. It suffices to prove that for any finite non-zero slope λ of Newt(f) there exists m ∈ |Y | such that q −λ = m and θm ( f ) = 0. As in Proposition 2.24 there is a metric d on |Y | such that for all ρ ∈]0, 1[, {m ≥ ρ} is complete. For m1 , m2 ∈ |Y |, if θm1 (m2 ) = O L m1 x then d(m1 , m2 ) = q −vm1 (x)/ deg m2 . We begin with the case when f = [xn ]π n ∈ WO E (O F ). n≥0

If d ≥ 0 set fd =

d

[xn ]π n .

n=0

For d 0, λ appears as a slope of Newt(fd ) with the same multiplicity as in Newt(f). For each d, f d = [ad ].gd for some ad ∈ O F and gd ∈ WO E (O F ) primitive. Thanks to the preceding results, we already know the result for each gd . Thus, setting X d = {m ∈ |Y | | m = q −λ and θm ( f d ) = 0}, we know that for d 0, X d = ∅. Moreover #X d is bounded when d varies. Now, if m ∈ X d , looking at vm (θm ( f d+1 )), one verifies that there exists m ∈ X d+1 such that d(m, m ) ≤ q

−

(d+1)λ−v(x0 ) #X d+1

.

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Laurent Fargues and Jean-Marc Fontaine

From this one deduces there exists a Cauchy sequence (md )d0 where md ∈ X d . If m = lim md then θm ( f ) = 0. This proves the theorem when d→+∞

f ∈ WO E (O F ) and thus when f ∈ Bb . The general case is now obtained in the same way by approximating f ∈ B by a converging sequence of elements of Bb . Example 2.47. As a corollary of the preceding theorem and Proposition 2.14, for f ∈ B \ {0} one has f ∈ B × if and only if for all m ∈ |Y |, θm ( f ) = 0.

3.2. A factorization of elements of B when F is algebraically closed Set B[0,1[ = { f ∈ B | ∃A, Newt(f)|]−∞,A] = +∞} "" v(x n ) = [xn ]π n ∈ WO E (F) π1 " liminf ≥0 n→+∞ n n−∞ (see Proposition 2.14 for this equality). Suppose F is algebraically closed. Given f ∈ B, applying Theorem 2.46, if λ1 , . . . , λr > 0 are some slopes of Newt(f) one can write f = g.

r

1−

i=1

[ai ] π

where v(ai ) = λi . Using this one proves the following. Theorem 2.48. Suppose F is algebraically closed. For f ∈ B there exists a sequence (ai )i≥0 of elements of m F going to zero and g ∈ B[0,1[ such that f = g.

+∞ i=0

1−

[ai ] . π

If moreover f ∈ B+ there exists such a factorization with g ∈ WO E (O F ).

3.3. Divisors and closed ideals of B Definition 2.49. Define " Div+ (Y ) = am [m] " ∀I ⊂]0, 1[ m∈|Y |

compact {m | am = 0 and m ∈ I } is finite .

Vector bundles on curves and p-adic Hodge theory For f ∈ B \ {0} set div( f ) =

47

ordm ( f )[m] ∈ Div+ (Y ).

m∈|Y |

Definition 2.50. For D ∈ Div+ (Y ) set a−D = { f ∈ B | div( f ) ≥ D}, an ideal of B. For each m ∈ |Y |, the function ordm : B → N ∪ {+∞} is upper semicontinuous. From this one deduces that the ideal a−D is closed in B. Theorem 2.51. The map D → a−D induces an isomorphism of monoids between Div+ (Y ) and the monoid of closed non-zero ideals of B. Moreover, D ≤ D if and only if a−D ⊂ a−D . If a is a closed ideal of B then ∼

a −→ lim B I a. ←− I ⊂]0,1[

The result is thus a consequence of the following. Theorem 2.52. For a compact non-empty interval I ⊂]0, 1[: • if I = {ρ} with ρ ∈ / |F × | then B I is a field • if not then the ring B I is a principal ideal domain with maximal ideals {B I m | m ∈ |Y |, m ∈ I }. Sketch of proof. The proof of this theorem goes as follows. First, given f ∈ B I , one can define a bounded Newton polygon NewtI (f). If f ∈ Bb then NewtI (f) is obtained from Newt(f) by removing the slopes λ such that q −λ ∈ / I (if there is no such slope we define NewtI (f) as the empty polygon). Now, if f ∈ B the method used in Definition 2.11 to define the Newton polygon does not apply immediately. For example, if I = {ρ} and f ∈ Bb then | f |ρ does not determine NewtI (f) (more generally, if I = [q −λ1 , q −λ2 ] one has the same problem with the definition of the pieces of NewtI (f) where the slopes are λ1 and λ2 ). But one verifies that if f n −→ f in B I with n→+∞

f n ∈ Bb then for n 0, NewtI (fn ) is constant and does not depend on the sequence of Bb going to f . Then, if f ∈ B I we prove a theorem that is analogous to Theorem 2.46: the slopes of NewtI (f) are the − logq m with multiplicity ordm ( f ) deg m where

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Laurent Fargues and Jean-Marc Fontaine

m ∈ I and θm ( f ) = 0. This gives a factorization of any f ∈ B I as a product r f = g. ξi i =1

where the ξi are irreducible primitive, ξi ∈ I , and g ∈ B I satisfies NewtI (g) = ∅. Finally, we prove that if f ∈ B I satisfies NewtI (f) = ∅ then f ∈ B× I . For f ∈ Bb this is verified by elementary manipulations. Then if f n −→ f in n→+∞

B I with f n ∈ Bb , since NewtI (f) = ∅ for n 0 one has NewtI (fn ) = ∅. But then for n 0 and ρ ∈ I , −1 −1 | f n+1 − f n−1 |ρ = | f n+1 |−1 ρ .| f n |ρ .| f n+1 − f n |ρ −→ 0. n→+∞

Thus the sequence

( f n−1 )n0

of B I converges towards an inverse of f .

Example 2.53. For f, g ∈ B \ {0}, f is a multiple of g if and only if div( f ) ≥ div(g). In particular there is an injection of monoids div : B \ {0}/B× −→ Div+ (Y ). Corollary 2.54. The set |Y | is the set of closed maximal ideals of B. Remark 2.55. Even when F is spherically complete we do not know whether div : B× → Div+ (Y ) is surjective or not (see 2.2).

4. Divisors on Y/ϕ Z 4.1. Motivation Suppose we want to classify ϕ-modules over B, that is to say free B-modules equipped with a ϕ-semi-linear automorphism. This should be the same as vector bundles on Y /ϕ Z where Y is this “rigid” space we did not really define but that should satisfy • (Y, OY ) = B • |Y | is the set of “classical points” of Y . Whatever this space Y is, since ϕ(m) = mq , ϕ acts in a proper discontinuous way without fixed point on it. Thus, Y /ϕ Z should have a sense as a “rigid” space. Let’s look in more details at what this space Y /ϕ Z should be.

Vector bundles on curves and p-adic Hodge theory

49

It is easy to classify rank 1 ϕ-modules over B. They are parametrized by Z: to n ∈ Z one associates the ϕ-module with basis e such that ϕ(e) = π n e. We thus should have ∼

Z −→ Pic(Y/ϕ Z ) n −→ L ⊗n where L is a line bundle such that for all d ∈ Z, H 0 (Y/ϕ Z , L ⊗d ) = Bϕ=π . d

If E = Fq ((π )) and F is algebraically closed Hartl and Pink classified in [20] the ϕ-modules over B, that is to say ϕ-equivariant vector bundles on D∗ . The first step in the proof of their classification ([20] theo.4.3) is that if (M, ϕ) is such a ϕ-module then for d 0 M ϕ=π = 0. d

The same type of result appears in the context of ϕ-modules over the Robba ring in the work of Kedlaya (see for example [22] prop. 2.1.5). From this one deduces that the line bundle L should be ample. We are thus led to study the scheme # d Proj Bϕ=π d≥0

for which one hopes it is “uniformized by Y ” and allows us to study ϕ-modules over B. In fact if (M, ϕ) is such a ϕ-module, we hope the quasi-coherent sheaf # d M ϕ=π d≥0

is the vector bundle associated to the ϕ-equivariant vector bundle on Y attached to (M, ϕ).

4.2. Multiplicative structure of the graded algebra P Definition 2.56. Define P=

#

Bϕ=π

d

d≥0

as a graded E-algebra. We note Pd = Bϕ=π the degree d homogeneous elements. d

In fact we could replace B by B+ in the preceding definition since Bϕ=π = (B+ )ϕ=π d

(Corollary 2.15). One has P0 = E.

d

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Laurent Fargues and Jean-Marc Fontaine

Definition 2.57. Define Div+ (Y/ϕ Z ) = {D ∈ Div+ (Y ) | ϕ ∗ D = D}. There is an injection |Y |/ϕ Z −→ Div+ (Y /ϕ Z ) m −→ [ϕ n (m)] n∈Z

that makes Div+ (Y/ϕ Z ) a free abelian monoid on |Y |/ϕ Z . If x ∈ Bϕ=π \ {0} then div(x) ∈ Div+ (Y /ϕ Z ). d

Theorem 2.58. If F is algebraically closed the morphism of monoids div : Pd \ {0} /E × −→ Div+ (Y/ϕ Z ) d≥0

is an isomorphism. Let us note the following important corollary. Corollary 2.59. If F is algebraically closed the graded algebra P is graded factorial with irreducible elements of degree 1. In the preceding theorem, the injectivity is an easy application of Theorem 2.51. In fact, if x ∈ Pd and y ∈ Pd are non-zero elements such that div(x) = div(y) then x = uy with u ∈ B× . But B× = (Bb )× (see the comment after Proposition 2.14). Thus, $ 0 if d = d b ϕ=π d−d u ∈ (B ) = E if d = d . The surjectivity uses Weierstrass products. For this, let x ∈ WO E (O F ) be a primitive degree d element and D = div(x) its divisor. We are looking for f ∈ Pd \ {0} satisfying div( f ) = ϕ n (D). n∈Z

Up to multiplying x by a unit we can suppose x ∈ π d + WO E (m F ). Then the infinite product + (x) =

ϕ n (x) n≥0

πd

Vector bundles on curves and p-adic Hodge theory

51

converges. For example, if x = π − [a], + (π − [a]) =

n [a q ] 1− . π

n≥0

One has div(+ (x)) =

ϕ n (D).

n≥0

We then would like to define “− (x) =

ϕ n (x)”

n0

n≥0

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for any ρ0 . Moreover, for any x ∈ B[ρ,1[ there is defined a Newton polygon Newt(x) and ! B+ ρ = x ∈ B[ρ,1[ | Newt(x) ≥ 0 ! B+ = x ∈ B | Newt(x) ≥ 0 . In fact, by concavity of the Gauss valuation r → vr (x), for any x ∈ B[ρ,1[ the limit |x|1 := lim |x|ρ ρ→1

exists in [0, +∞] and equals q −v0 (x) for x ∈ Bb and ! B+ ρ = x ∈ B[0,ρ[ | |x|1 ≤ 1 . Let us note |x|1 := q −v0 (x) . One has v0 (x) = lim Newt(x) +∞

B+ ρ , v0

and on is a valuation extending the valuation previously defined on Bb . One has to be careful that, contrary to B, the Frechet algebra B+ is not Frechet–Stein since the rings B+ ρ are not noetherian and it is not clear whether + + + ϕ : Bρ → Bρ (that is to say the inclusion B+ ρ q ⊂ Bρ ) is flat or not. Note ϕ-ModB+ and ϕ-ModB+ρ for the associated categories of finite rank free modules equipped with a semi-linear isomorphism. The category ϕ-ModB+ρ does not depend on ρ. There is a scalar extension functor ϕ-ModB+ −→ ϕ-ModB and, using Proposition 2.123, a functor ϕ-ModB+ρ −→ ϕ-ModB (M, ϕ) −→ ϕ n M ⊗B+ρ B[ρ,1[ . n≥0

Proposition 2.129. The functors ϕ-ModB+ → ϕ-ModB and ϕ-ModB+ρ → ϕ-ModB are fully faithful. Proof. Let’s treat the case of ϕ-ModB+ , the case of ϕ-ModB+ρ being identical. Using internal Hom’s this is reduced to proving that for (M, ϕ) ∈ ϕ-ModB+ one has ϕ=I d ∼ M ϕ=I d −→ M ⊗B+ B . Let us fix a basis of M and for x = (x1 , . . . , xn ) ∈ Bn M ⊗ B and r > 0 set Wr (x) = inf vr (xi ). 1≤i ≤n

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99

An element a ∈ B+ satisfies vr (a) ≥ 0 for r 0. Let us fix r0 0 such that all the coefficients (ai j )i, j of the matrix of ϕ in the fixed basis of M satisfy vr0 (ai, j ) ≥ 0. Then if x ∈ M ⊗ B satisfies ϕ(x) = x one has qW r0 (x) = Wr0 (ϕ(x)) ≥ Wr0 (x) q

and thus for k ≥ 1 W r0n (x) ≥ q

1 Wr (x). qn 0

Taking the limit when n → +∞ one obtains W0 (x) ≥ 0 that is to say x ∈ (B+ )n M. The preceding proposition together with Theorem 2.126 then gives the following. Theorem 2.130. Suppose F is algebraically closed. For A ∈ {B+ , B+ ρ } there is a bijection ∼

{λ1 ≥ · · · ≥ λn | n ∈ N, λi ∈ Q} −→ ϕ-Mod A / ∼ n # (λ1 , . . . , λn ) −→ A(−λi ) . i =1

One deduces there are equivalences of categories ∼

∼

∼

ϕ-ModB+ −→ ϕ-ModB+ρ −→ ϕ-ModB −→ ϕ-ModR F where an inverse of the first equivalence is given by M → ∩n≥0 ϕ n (M).

7.8. Another proof of the classification of ϕ-modules over B+ and B+ ρ We explain how to give a direct proof of Theorem 2.130 without using Kedlaya’s Theorem 2.121. This proof is much simpler and in fact applies even if the field F is not algebraically closed: Theorem 2.131. Theorem 2.130 remains true for any F with algebraically closed residue field. This relies on the introduction of a new ring called B. Set ! P = x ∈ Bb,+ | v0 (x) > 0 ! = [xn ]π n |x n ∈ O F , ∃C > 0, ∀n, v(xn ) ≥ C n−∞

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and B = Bb,+ /P. If k stands for the residue field of O F , there is a reduction morphism Bb,+ −→ WO E (k)Q . Then, B is a local ring with residue field WO E (k)Q . Let’s begin by classifying ϕ-modules over B. Theorem 2.132. Any ϕ-module over B is isomorphic to a direct sum of B(λ), λ ∈ Q. Sketch of proof. One first proves that for λ, μ ∈ Q, Ext1 (B(λ), B(μ)) = 0. For two ϕ-modules M and M one has Ext1 (M, M ) = H 1 (M ∨ ⊗ M ) where for a ϕ-module M , H 1 (M ) = coker(I d − ϕ M ). Up to replacing ϕ by a power of itself, that is to say replacing E by an unramified extension, we are thus reduced to proving that for any d ∈ Z, I d − π d ϕ : B −→ B is surjective. For d > 0, this is a consequence of the fact that I d − π d ϕ : WO E (O F ) −→ WO E (O F ) is surjective since π d ϕ is topologically nilpotent on WO E (O F ) for the π -adic topology. For d < 0 this is deduced in the same way using I d − π −d ϕ −1 . For d = 0, this is a consequence of the fact that I d − ϕ is bijective on WO E (m F ) since ϕ is topologically nilpotent on WO E (m F ) for the ([a], π )-adic topology for any a ∈ m F \ {0} (the topology induced by the Gauss norms (|.|ρ )ρ∈]0,1[ ) and since the residue field k of F is algebraically closed. Let (M, ϕ) ∈ ϕ-ModB and Mk = M ⊗ WO E (k)Q be the associated isocrystal. Let λ be the smallest slope of (Mk , ϕ). It suffices now to prove that M has a sub ϕ-module isomorphic to B(λ) whose underlying B-module is a direct factor. Up to raising ϕ to a power and twisting one is reduced to the case λ = 0. Then Mk has a sub-lattice stable under ϕ. Let us remark that any element of Bb,+ whose image in WO E (k)Q lies in WO E (k) is congruent modulo p to an element of WO E (O F ). Lifting the basis of to a basis of M (recall B is a local ring), one then checks there is a free ϕ-module N over WO E (O F ) together with ∼ a morphism N → M inducing an isomorphism N ⊗WO E (O F ) B − → M and such ∼

that N ⊗ WO E (k) − → . For such an N , there is an isomorphism ∼

N ϕ=I d −→ ϕ=I d

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101

such that N ϕ=I d ⊗ E WO E (O F ) is a direct factor in N . In fact, after fixing a basis of N , N WO E (O F )n is complete with respect to the family of norms (.ρ )ρ∈]0,1[ where (x1 , . . . , xn )ρ = sup1≤i≤n |xi |ρ . Moreover the Frobenius of N is topologically nilpotent on WO E (m F )n for this set of norms. The result is deduced (for the direct factor assertion, one has to use that WO E (m F ) is contained in the Jacobson radical of WO E (O F ) together with Nakayama lemma). To make the link between B+ , B+ ρ and B we need the following. Lemma 2.133. For any a ∈ m F \ {0}, B+ = [a]B+ + Bb,+ and B+ ρ = b,+ . [a]B+ + B ρ In fact, for any x = n−∞ [x n ]π n let us note x + = ]π n and n≥0 [x n − n + + x = n 0} + + and the same for B+ ρ . From this we deduce surjections B → B and Bρ → B. Now, Theorem 2.130 is a consequence of Theorem 2.132 and the following.

Proposition 2.134. The reduction functor ϕ-ModB+ → ϕ-ModB is fully faithful. The same holds for B+ ρ. Sketch of proof. We treat the case of B+ , the case of B+ ρ being identical. This is reduced to proving that for (M, ϕ) ∈ ϕ-ModB+ , if M is the associated mod∼ ϕ=I d ule over B, then M ϕ=I d − →M . If = pM this is reduced to proving that ∼

I d − ϕ : −→ . Fix a basis of M (B+ )n and note A = (ai j )i, j ∈ GLn (B+ ) the matrix of ϕ in this base. For each r ≥ 0 and m = (x1 , . . . , xn ) ∈ M set mr = inf1≤i≤n vr (xi ). Let r0 > 0 be fixed. Then = pn is complete with respect to the set of additive norms (.)r>0 . Note Ar = infi, j vr (ai, j ). Fix an r > 0. One first checks that for any m ∈ M and k ≥ 1 ϕ k (m)r ≥ q k m

r qk

+

k−1 i=0

q i A ri . q

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Laurent Fargues and Jean-Marc Fontaine

Now, according to inequality (2.3) of Section 1.2.1 (the inequality is stated for Bb,+ but extends by continuity to B+ ), for any r ≤ r one has Ar ≥

r Ar r

and thus k−1

q i A

i =0

r qk

≥ αk + β

for some constants α, β ∈ R. But now, if m ∈ , lim mr = m0 > 0 and r →0

thus

lim ϕ k (m)r = +∞.

k→+∞

We deduce that ϕ is topologically nilpotent on and thus I d − ϕ is bijective on it. Remark 2.135. The preceding proof does not use the fact that F is algebraically closed and thus Theorem 2.130 remains true when F is any perfectoid field with algebraically closed residue field. On this point, there is a big difference between ϕ-modules over B+ and the ones over B. In fact, one can prove that ϕ-modules over B satisfy Galois descent like vector bundles (Theorem 2.103) for any perfectoid field F and thus for a general F Theorem 2.126 is false. Remark 2.136. As a consequence of the classification theorem and the first part of the proof of Theorem 2.132, for any M, M ∈ ϕ-ModB+ , Ext1 (M, M ) = 0. This is not the case for ϕ-modules over B. The equivalence 2.127 is an equivalence of exact categories and for example Ext1ϕ -ModB (B, B(1)) = 0. In fact, although the scalar extension functor ∼

ϕ-ModB+ − → ϕ-ModB is exact, its inverse is not. Let us conclude with a geometric interpretation of the preceding result. Set = Spec(Z p ) and for an F p -scheme S note F -IsocS/ for the category of F-isocrystals. If S → S is a thickening then F-Isoc S/ F-Isoc S / . Let now a ∈ m F \ {0}, ρ = |a| and Sρ = Spec(O F /O F a). The category F-Isoc Sρ / does not depend on the choice of ρ ∈]0, 1[. The crystalline site Cris(Sρ /) has an initial object Acris,ρ such that Acris,ρ 1p = Bcris,ρ (see Section 1.2.1). We thus have an equivalence F-Isoc Sρ / ϕ-ModB+

cris,ρ

.

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103

+ + But since B+ ρ p ⊂ Bcris,ρ ⊂ Bρ p−1 we have an equivalence

ϕ-ModB+

cris,ρ

ϕ-ModB+ρ .

Moreover, one can think of ϕ-ModB+ as being the category of “convergent F-isocrystals on Sρ ”. We thus have proved the following. Theorem 2.137. Suppose F is a perfectoid field with algebraically closed residue field. Then any F-isocrystal, resp. convergent F-isocrystal, on Sρ is isotrivial that is to say comes from the residue field of k after the choice of a s / / k. splitting O F

References [1] Y. André. Slope filtrations. Confluentes Mathematici, 1:1–85, 2009. [2] L. Berger. Représentations p-adiques et équations différentielles. Invent. Math., 148(2):219–284, 2002. [3] Laurent Berger. Construction de (φ, )-modules: représentations p-adiques et B-paires. Algebra Number Theory, 2(1):91–120, 2008. [4] F. Cherbonnier and P. Colmez. Représentations p-adiques surconvergentes. Invent. Math., 133(3):581–611, 1998. [5] J.-F. Dat, S. Orlik, and M. Rapoport. Period domains over finite and p-adic fields, volume 183 of Cambridge Tracts in Mathematics. Cambridge University Press, 2010. [6] Vladimir G. Drinfeld. Coverings of p-adic symmetric domains. Functional Analysis and its Applications, 10(2):29–40, 1976. [7] G. Faltings. Group schemes with strict O-action. Mosc. Math. J., 2(2):249–279, 2002. [8] G. Faltings. Coverings of p-adic period domains. J. Reine Angew. Math., 643:111–139, 2010. [9] L. Fargues. L’isomorphisme entre les tours de Lubin-Tate et de Drinfeld et applications cohomologiques. In L’isomorphisme entre les tours de Lubin-Tate et de Drinfeld, Progress in math., 262, pages 1–325. Birkhäuser, 2008. [10] L. Fargues. La filtration de Harder-Narasimhan des schémas en groupes finis et plats. J. Reine Angew. Math., 645:1–39, 2010. [11] L. Fargues. La filtration canonique des points de torsion des groupes p-divisibles. Annales scientifiques de l’ENS, 44(6):905–961, 2011. [12] L. Fargues and J.-M. Fontaine. Courbes et fibrés vectoriels en théorie de Hodge p-adique. Prépublication. [13] L. Fargues and J.-M. Fontaine. Factorization of analytic functions in mixed characteristic. To appear in the proceedings of a conference in Sanya. [14] L. Fargues and J.-M. Fontaine. Vector bundles and p-adic Galois representations. Stud. Adv. Math. 51, 2011. [15] J.-M. Fontaine. Groupes p-divisibles sur les corps locaux. Société Mathématique de France, Paris, 1977. Astérisque, No. 47-48.

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[16] J.-M. Fontaine. Le corps des périodes p-adiques. Astérisque, (223):59–111, 1994. With an appendix by Pierre Colmez, Périodes p-adiques (Bures-sur-Yvette, 1988). [17] B. Gross and M. Hopkins. Equivariant vector bundles on the Lubin-Tate moduli space. In Topology and representation theory (Evanston, IL, 1992), volume 158 of Contemp. Math., pages 23–88. Amer. Math. Soc., 1994. [18] A. Grothendieck. Sur la classification des fibrés holomorphes sur la sphère de Riemann. Amer. J. Math., 79:121–138, 1957. [19] A. Grothendieck. Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I. Inst. Hautes Études Sci. Publ. Math., (11):167, 1961. [20] U. Hartl and R. Pink. Vector bundles with a Frobenius structure on the punctured unit disc. Compos. Math., 140(3):689–716, 2004. [21] K. Kedlaya. Slope filtrations revisited. Doc. Math., 10:447–525, 2005. [22] K. Kedlaya. Slope filtrations for relative Frobenius. Astérisque, (319):259–301, 2008. Représentations p-adiques de groupes p-adiques. I. Représentations galoisiennes et (φ, )-modules. [23] K. Kedlaya, J. Pottharst, and L. Xiao. Cohomology of arithmetic families of (phi,gamma)-modules. arXiv:1203.5718v1. [24] G. Laffaille. Groupes p-divisibles et corps gauches. Compositio Math., 56(2):221–232, 1985. [25] M. Lazard. Les zéros des fonctions analytiques d’une variable sur un corps valué complet. Inst. Hautes Études Sci. Publ. Math., (14):47–75, 1962. [26] B. Poonen. Maximally complete fields. Enseign. Math. (2), 39(1-2):87–106, 1993. [27] M. Rapoport and T. Zink. Period spaces for p-divisible groups. Number 141 in Annals of Mathematics Studies. Princeton University Press, 1996. [28] P. Schneider and J. Teitelbaum. Algebras of p-adic distributions and admissible representations. Invent. Math., 153(1):145–196, 2003. [29] P. Scholze. Perfectoid spaces. Preprint. [30] S. Sen. Continuous cohomology and p-adic Galois representations. Invent. Math., 62(1):89–116, 1980/81. [31] J. T. Tate. p-divisible groups. In Proc. Conf. Local Fields (Driebergen, 1966), pages 158–183. Springer, 1967. [32] J.-P. Wintenberger. Le corps des normes de certaines extensions infinies de corps locaux; applications. Ann. Sci. École Norm. Sup. (4), 16(1):59–89, 1983.

3 Around associators Hidekazu Furusho

Abstract This is a concise exposition of recent developments around the study of associators. It is based on the author’s talk at the Mathematische Arbeitstagung in Bonn, June 2011 (cf. [F11b]) and at the Automorphic Forms and Galois Representations Symposium in Durham, July 2011. The first section is a review of Drinfeld’s definition [Dr] of associators and the results [F10, F11a] concerning the definition. The second section explains the four pro-unipotent algebraic groups related to associators; the motivic Galois group, the Grothendieck– Teichmüller group, the double shuffle group and the Kashiwara–Vergne group. Relationships, actually inclusions, between them are also discussed.

1. Associators We recall the definition of associators [Dr] and explain our main results in [F10, F11a] concerning the defining equations of associators. The notion of associators was introduced by Drinfeld in [Dr]. They describe monodromies of the KZ (Knizhnik–Zamolodchikov) equations. They are essential for the construction of quasi-triangular quasi-Hopf quantized universal enveloping algebras ([Dr]), for the quantization of Lie-bialgebras (Etingof–Kazhdan quantization [EtK]), for the proof of formality of chain operad of little discs by Tamarkin [Ta] (see also Ševera and Willwacher [SW]) and also for the combinatorial reconstruction of the universal Vassiliev knot invariant (the Kontsevich invariant [Kon, Ba95]) by Bar-Natan [Ba97], Cartier [C], Kassel and Turaev [KssT], Le and Murakami [LM96a] and Piunikhin [P]. Automorphic Forms and Galois Representations, ed. Fred Diamond, Payman L. Kassaei and c Cambridge University Press 2014. Minhyong Kim. Published by Cambridge University Press.

105

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Hidekazu Furusho

Notation 3.1. Let k be a field of characteristic 0 and k¯ be its algebraic closure. Denote by U F2 = k X 0 , X 1

the non-commutative formal power series ring defined as the universal enveloping algebra of the completed free Lie algebra F2 with two variables X 0 and X 1 . An element ϕ = ϕ(X 0 , X 1 ) of U F2 is called group-like1 if it satisfies (ϕ) = ϕ ⊗ ϕ and ϕ(0, 0) = 1

(3.1)

ˆ F2 is given by (X 0 ) = X 0 ⊗ 1 + 1 ⊗ X 0 and where : U F2 → U F2 ⊗U (X 1 ) = X 1 ⊗ 1 + 1 ⊗ X 1 . For any k-algebra homomorphism ι : U F2 → S, the image ι(ϕ) ∈ S is denoted by ϕ(ι(X 0 ), ι(X 1 )). Denote by U a3 (resp. U a4 ) the universal enveloping algebra of the completed pure braid Lie algebra a3 (resp. a4 ) over k with 3 (resp. 4) strings, which is generated by ti j (1 i, j 3 (resp. 4)) with defining relations tii = 0, ti j = t ji , [ti j , tik + t jk ] = 0 (i, j ,k: all distinct) and [ti j , tkl ] = 0 (i, j ,k,l: all distinct). Note that X 0 → t12 and X 1 → t23 give an isomorphism U F2 U a3 . Definition 3.2 ([Dr]). A pair (μ, ϕ) with a non-zero element μ in k and a group-like series ϕ = ϕ(X 0 , X 1 ) ∈ U F2 is called an associator if it satisfies one pentagon equation ϕ(t12 , t23 + t24 )ϕ(t13 + t23 , t34 ) = ϕ(t23 , t34 )ϕ(t12 + t13 , t24 + t34 )ϕ(t12 , t23 ) (3.2) in U a4 and two hexagon equations μ(t13 + t23 ) μt13 μt23 } = ϕ(t13 , t12 ) exp{ }ϕ(t13 , t23 )−1 exp{ }ϕ(t12 , t23 ), 2 2 2 (3.3) μ(t12 + t13 ) μt μt 13 12 exp{ } = ϕ(t23 , t13 )−1 exp{ }ϕ(t12 , t13 )exp{ }ϕ(t12 , t23 )−1 2 2 2 (3.4) in U a3 . exp{

Remark 3.3. (i) Drinfeld [Dr] proved that such a pair always exists for any field k of characteristic 0. (ii) The equations (3.2)–(3.4) reflect the three axioms of braided monoidal categories [JS]. We note that for any k-linear infinitesimal tensor category C, each associator gives a structure of a braided monoidal category on C[[h]] (cf. [C, Dr, KssT]). Here C[[h]] denotes the category whose set of objects is equal to that of C and whose set of morphisms MorC [[h]] (X, Y ) is MorC (X, Y ) ⊗ k[[h]] (h: a formal parameter). 1 It is equivalent to ϕ ∈ exp F . 2

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107

Actually, the two hexagon equations are a consequence of the one pentagon equation: Theorem 3.4 ([F10]). Let ϕ = ϕ(X 0 , X 1 ) be a group-like element of U F2 . Suppose that ϕ satisfies the pentagon equation (3.2). Then there always exists μ ∈ k¯ (unique up to signature) such that the pair (μ, ϕ) satisfies two hexagon equations (3.3) and (3.4). Recently several different proofs of the above theorem were obtained (see [AlT, BaD, Wi]). One of the nicest examples of associators is the Drinfeld associator: Example 3.5. The Drinfeld associator Φ K Z = Φ K Z (X 0 , X 1 ) ∈ C X 0 , X 1

is defined to be the quotient Φ K Z = G 1 (z)−1 G 0 (z) where G 0 and G 1 are the solutions of the formal KZ-equation, which is the following differential equation for multi-valued functions G(z) : C\{0, 1} → C X 0 , X 1

X0 d X1 G(z) = + G(z), dz z z−1 such that G 0 (z) ≈ z X 0 when z → 0 and G 1 (z) ≈ (1 − z) X 1 when z → 1 (cf. √ [Dr]). It is shown in [Dr] (see also [Wo]) that the pair (2π −1, Φ K Z ) √ forms an associator for k = C. Namely Φ K Z satisfies (3.1)∼(3.4) with μ = 2π −1. Remark 3.6. (i) The Drinfeld associator is expressed as follows: k −1 Φ K Z (X 0 , X 1 ) = 1 + (−1)m ζ (k1 , · · · , km )X 0km −1 X 1 · · · X 01 X 1 m,k1 ,...,km ∈N km >1

+ (regularized terms). Here ζ (k1 , · · · , km ) is the multiple zeta value (MZV in short), the real number defined by the following power series ζ (k1 , · · · , km ) :=

0 1, ζ (a)ζ (b) =

a−1 b−1+i i=0

+

i

ζ (a − i, b + i)

b−1 a−1+ j ζ (b − j, a + j ), j j=0

ζ (a)ζ (b) = ζ (a, b) + ζ (a + b) + ζ (b, a). The former follows from (3.1) and the latter follows from (3.6).

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109

The regularized double shuffle relations are also a consequence of the pentagon equation: Theorem 3.10 ([F11a]). Let ϕ = ϕ(X 0 , X 1 ) be a group-like element of U F2 . Suppose that ϕ satisfies the pentagon equation (3.2). Then it also satisfies the regularized double shuffle relations (3.6). This result attains the final goal of the project posed by Deligne– Terasoma [Te]. Their idea is to use some convolutions of perverse sheaves, whereas our proof is to use Chen’s bar construction calculus. It would be our next project to complete their idea and to get another proof of Theorem 3.10. Remark 3.11. Our Theorem 3.10 was extended cyclotomically in [F12]. The following Zagier’s relation which is essential for Brown’s proof of Theorem 3.17 might be also one of the most fascinating ones. Theorem 3.12 ([Z]). For a, b 0 ζ (2{a} , 3, 2{b} ) = 2 with

Ara,b

=

2r 2a+2

a+b+1

r (−1)r (Ara,b − Ba,b )ζ (2r + 1)ζ (2{a+b+1−r} )

r=1

r = (1 − 2−2r ) 2r . and Ba,b 2b+1

2. Four groups We explain recent developments on the four pro-unipotent algebraic groups related to associators; the motivic Galois group, the Grothendieck–Teichmüller group, the double shuffle group and the Kashiwara–Vergne group, all of which are regarded as subgroups of Aut exp F2 . In the end of this section we discuss natural inclusions between them.

2.1. Motivic Galois group We review the formulations of the motivic Galois groups (consult also [An] as a nice exposition). Notation 3.13. We work in the triangulated category D M(Q)Q of mixed motives over Q (a part of an idea of mixed motives is explained in [De] §1) constructed by Hanamura, Levine and Voevodsky. Tate motives Q(n) (n ∈ Z) are (Tate) objects of the category. Let D M T (Q)Q be the triangulated subcategory of D M(Q)Q generated by Tate motives Q(n) (n ∈ Z). By the work

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of Levine a neutral tannakian Q-category M T (Q) = M T (Q)Q of mixed Tate motives over Q is extracted by taking the heart with respect to a t-structure of D M T (Q)Q . Deligne and Goncharov [DeG] introduced the full subcategory M T (Z) = M T (Z)Q of unramified mixed Tate motives inside of M T (Q)Q , All objects there are mixed Tate motives M (i.e. an object of M T (Q)) such that for each subquotient E of M which is an extension of Q(n) by Q(n + 1) for n ∈ Z, the extension class of E in Ext1M T (Q) (Q(n), Q(n + 1)) = Ext1M T (Q) (Q(0), Q(1)) = Q× ⊗ Q is equal to in Z× ⊗ Q = {0}. In the category M T (Z) of unramified mixed Tate motives, the following holds: $ 1 (m = 3, 5, 7, . . . ), 1 dimQ Ext M T (Z) (Q(0), Q(m)) = (3.7) 0 (m : others), dimQ Ext2M T (Z) (Q(0), Q(m)) = 0.

(3.8)

The category M T (Z) forms a neutral tannakian Q-category (consult [DeM]) with the fiber functor ωcan : M T (Z) → VectQ (VectQ : the category of Q-vector spaces) sending each motive M to W ⊕n Hom(Q(n), Gr−2n M). Definition 3.14. The motivic Galois group here is defined to be the Galois group of M T (Z), which is the pro-Q-algebraic group defined by GalM (Z) := Aut⊗ (M T (Z) : ωcan ). By the fundamental theorem of tannakian category theory, ωcan induces an equivalence of categories M T (Z) Rep GalM (Z)

(3.9)

where the right-hand side of the isomorphism denotes the category of finite dimensional Q-vector spaces with GalM (Z)-action. Remark 3.15. The action of GalM (Z) on ωcan (Q(1)) = Q defines a surjection GalM (Z) → Gm and its kernel GalM (Z)1 is the unipotent radical of GalM (Z). There is a canonical splitting τ : Gm → GalM (Z) which gives a negative grading on its associated Lie algebra LieGalM (Z)1 . From (3.7) and (3.8) it follows that the Lie algebra is the graded free Lie algebra generated by one element in each degree −3, −5, −7, . . . . (consult [De] §8 for the full story).

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→ − The motivic fundamental group π1M (P1 \{0, 1, ∞} : 01) constructed in [DeG] §4 is a (pro-) object of M T (Z). The Drinfeld associator (cf. Example 3.5) is essential in describing the Hodge realization of the motive (cf. [An, DeG, F07]). By our tannakian equivalence (3.9), it gives a (pro-) object of the right-hand side of (3.9), which induces a (graded) action : GalM (Z)1 → Aut exp F2 .

(3.10)

Remark 3.16. For each σ ∈ GalM (Z)1 (k), its action on exp F2 is described by e X 0 → e X 0 and e X 1 → ϕσ−1 e X 1 ϕσ for some ϕσ ∈ exp F2 . The following has been conjectured (Deligne–Ihara conjecture) for a long time and finally proved by Brown by using Zagier’s relation (Theorem 3.12). Theorem 3.17 ([Br]). The map " is injective. It is a pro-unipotent analogue of the so-called Bely˘ı’s theorem [Bel] in the pro-finite group setting. The theorem says that all unramified mixed Tate motives are associated with MZVs.

2.2. Grothendieck–Teichmüller group The Grothendieck–Teichmüller group was introduced by Drinfeld [Dr] in his study of deformations of quasi-triangular quasi-Hopf quantized universal enveloping algebras. It was defined to be the set of “degenerated” associators. The construction of the group was also stimulated by the previous idea of Grothendieck, un jeu de Lego–Teichmüller, which was posed in his article Esquisse d’un programme [G]. Definition 3.18 ([Dr]). The Grothendieck–Teichmüller group G RT1 is defined to be the pro-algebraic variety whose set of k-valued points consists of degenerated associators, which are group-like series ϕ ∈ U F2 satisfying the defining equations (3.2)–(3.4) of associators with μ = 0. Remark 3.19. (i) By Theorem 3.4, G RT1 is reformulated to be the set of group-like series satisfying (3.2) without quadratic terms. (ii) It forms a group [Dr] by the multiplication below ϕ2 ◦ ϕ1 := ϕ1 (ϕ2 X 0 ϕ2−1 , X 1 ) · ϕ2 = ϕ2 · ϕ1 (X 0 , ϕ2−1 X 1 ϕ2 ).

(3.11)

By the map X 0 → X 0 and X 1 → ϕ −1 X 1 ϕ, the group G RT1 is regarded as a subgroup of Aut exp F2 . (iii) Ihara came to the Lie algebra of G RT1 independently of Drinfeld’s work in his arithmetic study of Galois action on fundamental groups (cf. [Iy90]).

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(iv) The cyclotomic analogues of associators and that of the Grothendieck– Teichmüller group were introduced by Enriquez [En]. Some elimination results on their defining equations in special case were obtained in [EnF]. Geometric interpretation (cf. [Dr, Iy90, Iy94]) of equations (3.2)–(3.4) implies the following (for a proof, see also [An, F07]) Theorem 3.20. Im ⊂ G RT1 . Related to the questions posed in [De, Dr, Iy90], it is expected that they are isomorphic. Remark 3.21. (i) The Drinfeld associator Φ K Z is an associator (cf. Example 3.5) but is not a degenerated associator, i.e. Φ K Z ∈ G RT1 (C). p (ii) The p-adic Drinfeld associator Φ K Z introduced in [F04] is not an p associator but a degenerated associator, i.e. Φ K Z ∈ G RT1 (Q p ) (cf. [F07]).

2.3. Double shuffle group The double shuffle group was introduced by Racinet as the set of solutions of the regularized double shuffle relations with “degeneration” condition (no quadratic terms condition). Definition 3.22 ([R]). The double shuffle group D M R0 is the pro-algebraic variety whose set of k-valued points consists of the group-like series ϕ ∈ U F2 satisfying the regularized double shuffle relations (3.6) without linear terms and quadratic terms. Remark 3.23. (i) We note that D M R stands for double mélange regularisé ([R]). (ii) It was shown in [R] that it forms a group by the operation (3.11). (iii) In the same way as in Remark 3.19 (ii), the group D M R0 is regarded as a subgroup of Aut exp F2 . It is also shown that Im" is contained in D M R0 (cf. [F07])). Actually it is expected that they are isomorphic. Theorem 3.10 follows the inclusion between G RT1 and D M R0 : Theorem 3.24 ([F11a]). G RT1 ⊂ D M R0 . It is also expected that they are isomorphic.

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Remark 3.25. (i) The Drinfeld associator Φ K Z satisfies the regularized double shuffle relations (cf. Remark 3.8) but it is not an element of the double shuffle group, i.e. Φ K Z ∈ D M R0 (C), because its quadratic term is non-zero, actually is equal to ζ (2)X 1 X 0 − ζ (2)X 0 X 1 . p (ii) The p-adic Drinfeld associator Φ K Z satisfies the regularized double shuffle relations (cf. [BeF, FJ]) and it is an element of the double shuffle group, p i.e. Φ K Z ∈ D M R0 (Q p ), which also follows from Remark 3.21.(ii) and Theorem 3.24.

2.4. Kashiwara–Vergne group In [KswV] Kashiwara and Vergne proposed a conjecture related to the Campbell–Baker–Hausdorff series which generalizes Duflo’s theorem (Duflo isomorphism) to some extent. The conjecture was settled generally by Alekseev and Meinrenken [AlM]. The Kashiwara–Vergne group was introduced as a “degeneration” of the set of solutions of the conjecture by Alekseev and Torossian in [AlT], where they gave another proof of the conjecture by using Drinfeld’s [Dr] theory of associators. The following is one of the formulations of the conjecture stated in [AlET]. Generalized Kashiwara–Vergne problem: Find a group automorphism P : exp F2 → exp F2 such that P belongs to T Aut exp F2 (that is, P ∈ Aut exp F2 such that P(e X 0 ) = p1 e X 0 p1−1 and P(e X 1 ) = p2 e X 1 p2−1 for some p1 , p2 ∈ exp F2 ) and P satisfies P(e X 0 e X 1 ) = e(X 0 +X 1 ) and the coboundary Jacobian condition δ ◦ J (P) = 0. Here J stands for the Jacobian cocycle J : T Aut exp F2 → tr2 and δ denotes the differential map δ : trn → trn+1 for n = 1, 2, . . . (for their precise definitions see [AlT]). We note that P is uniquely determined by the pair ( p1 , p2 ). The following is essential for the proof of the conjecture. Theorem 3.26 ([AlT, AlET]). Let (μ, ϕ) be an associator. Then the pair ( p1 , p2 ) = ϕ(X 0 /μ, X ∞ /μ), e X ∞ /2 ϕ(X 1 /μ, X ∞ /μ) with X ∞ = −X 0 − X 1 gives a solution to the above problem.

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The Kashiwara–Vergne group is defined to be the set of solutions of the problem with “degeneration condition” (“the condition μ = 0”): Definition 3.27 ([AlT, AlET]). The Kashiwara–Vergne group K RV is defined to be the set of P ∈ Aut exp F2 which satisfies P ∈ T Aut exp F2 , P(e(X 0 +X 1 ) ) = e(X 0 +X 1 ) and the coboundary Jacobian condition δ ◦ J (P) = 0. The above K RV forms a subgroup of Aut exp F2 . We denote by K RV0 the subgroup of K RV consisting of P without linear terms in both p1 and p2 . Theorem 3.26 yields the inclusion below. Theorem 3.28 ([AlT, AlET]). G RT1 ⊂ K RV0 . Actually it is expected that they are isomorphic (cf. [AlT]). A recent result of Schneps in [S] also leads to Theorem 3.29 ([S]). D M R0 ⊂ K RV0 .

2.5. Comparison By Theorem 3.17, 3.20, 3.24, 3.28 and 3.29, we obtain Theorem 3.30. GalM (Z)1 ⊆ G RT1 ⊆ D M R0 ⊆ K RV0 . We finish our exposition by posing the following question: Question 3.31. Are they all equal? Namely, GalM (Z)1 = G RT1 = D M R0 = K RV0 ? These four groups were constructed independently and there are no philosophical reasonings why we expect that they are all equal. Though it might be not so good mathematically to believe such equalities without any strong conceptual support, the author believes that it might be good at least spiritually to imagine a hidden theory to relate them.

References [An] André, Y.; Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses, 17, Société Mathématique de France, Paris, 2004. [AlET] Alekseev, A., Enriquez, B. and Torossian, C.; Drinfeld associators, braid groups and explicit solutions of the Kashiwara-Vergne equations, Publ. Math. Inst. Hautes Études Sci. 112 (2010), 143–189.

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[AlM] [AlT] [Ba95] [Ba97]

[BaD] [Bel] [BeF]

[Br] [C]

[De]

[DeG] [DeM]

[Dr] [En] [EnF] [EtK] [Eu]

[F03] [F04] [F07]

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and Meinrenken, E.; On the Kashiwara-Vergne conjecture, Invent. Math. 164 (2006), no. 3, 615–634. and Torossian, C.; The Kashiwara-Vergne conjecture and Drinfeld’s associators, Ann. of Math. (2) 175 (2012), no. 2, 415–463. Bar-Natan, D.; On the Vassiliev knot invariants, Topology 34 (1995), no. 2, 423–472. ; Non-associative tangles, Geometric topology (Athens, GA, 1993), 139–183, AMAPIP Stud. Adv. Math., 2.1, Amer. Math. Soc., Providence, RI, 1997. and Dancso, Z.; Pentagon and hexagon equations following Furusho, Proc. Amer. Math. Soc. 140 (2012), no. 4, 1243–1250. Bely˘ı, G. V., Galois extensions of a maximal cyclotomic field, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 2, 267–276, 479. Besser, A. and Furusho, H.; The double shuffle relations for p-adic multiple zeta values, Primes and Knots, AMS Contemporary Math, Vol 416, 2006, 9–29. Brown, F.; Mixed Tate Motives over Spec(Z ), Annals of Math., 175, (2012) no. 2, 949–976. Cartier, P.; Construction combinatoire des invariants de Vassiliev-Kontsevich des nœuds, C. R. Acad. Sci. Paris Ser. I Math. 316 (1993), no. 11, 1205–1210. Deligne, P.; Le groupe fondamental de la droite projective moins trois points, Galois groups over Q (Berkeley, CA, 1987), 79–297, Math. S. Res. Inst. Publ., 16, Springer, New York-Berlin, 1989. and Goncharov, A.; Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. Ecole Norm. Sup. (4) 38 (2005), no. 1, 1–56. and Milne, J.; Tannakian categories, in Hodge cycles, motives, and Shimura varieties (P.Deligne, J.Milne, A.Ogus, K.-Y.Shih editors), Lecture Notes in Mathematics 900, Springer-Verlag, 1982. Drinfel’d, V. G.; On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(Q/Q), Leningrad Math. J. 2 (1991), no. 4, 829–860. Enriquez, B.; Quasi-reflection algebras and cyclotomic associators, Selecta Math. (N.S.) 13 (2007), no. 3, 391–463. and Furusho, H.; Mixed Pentagon, octagon and Broadhurst duality equation, J. Pure Appl. Algebra, 216, (2012), no. 4, 982–995. no. 4, Etingof, P. and Kazhdan, D.; Quantization of Lie bialgebras. II, Selecta Math. 4 (1998), no. 2, 213–231. Euler, L., Meditationes circa singulare serierum genus, Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 140–186 and Opera Omnia: Series 1, Volume 15, pp. 217–267 (also available from www.math.dartmouth.edu/euler/). Furusho, H.; The multiple zeta value algebra and the stable derivation algebra, Publ. Res. Inst. Math. Sci. 39, (2003), no. 4, 695–720. ; p-adic multiple zeta values I – p-adic multiple polylogarithms and the p-adic K Z equation, Invent. Math. 155, (2004), no. 2, 253–286. ; p-adic multiple zeta values II – tannakian interpretations, Amer. J. Math. 129, (2007), no 4, 1105–1144.

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[F10] [F11a] [F11b] [F12]

[FJ]

[G]

[IkKZ] [Iy90]

[Iy94]

[JS] [KswV] [KssT] [Kon] [LM96a] [LM96b] [P] [R] [S] [SW] [Ta] [Te]

; Pentagon and hexagon equations, Annals of Math. 171 (2010), no. 1, 545–556. ; Double shuffle relation for associators, Annals of Math. 174 (2011), no. 1, 341–360. ; Four groups related to associators, report of the Mathematische Arbeitstagung in Bonn, June 2011, arXiv:1108.3389. ; Geometric interpretation of double shuffle relation for multiple L-values, Galois-Teichmüller theory and Arithmetic Geometry, Advanced Studies in Pure Math 63 (2012), 163–187. and Jafari, A.; Regularization and generalized double shuffle relations for p-adic multiple zeta values, Compositio Math. 143, (2007), 1089–1107. Grothendieck, A.; Esquisse d’un programme, 1983, available on pp. 243–283. London Math. Soc. LNS 242, Geometric Galois actions, 1, 5–48, Cambridge University Press. Ihara, K., Kaneko, M. and Zagier, D.; Derivation and double shuffle relations for multiple zeta values, Compos. Math. 142 (2006), no. 2, 307–338. Ihara, Y.; Braids, Galois groups, and some arithmetic functions, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 99–120, Math. Soc. Japan, Tokyo, 1991. London Math. Soc. LNS ; On the embedding of Gal(Q/Q) into GT, 200, The Grothendieck theory of dessins d’enfants (Luminy, 1993), 289–321, Cambridge University Press, Cambridge, 1994. Joyal, A. and Street, R.; Braided tensor categories, Adv. Math. 102 (1993), no. 1, 20–78. Kashiwara, M. and Vergne, M.; The Campbell-Hausdorff formula and invariant hyperfunctions, Invent. Math. 47 (1978), no. 3, 249–272. Kassel, C. and Turaev, V.; Chord diagram invariants of tangles and graphs, Duke Math. J. 92 (1998), no. 3, 497–552. Kontsevich, M.; Vassiliev’s knot invariants, I. M. Gelfand Seminar, 137–150, Adv. Soviet Math., 16, Part 2, Amer. Math. Soc., Providence, RI, 1993. Le, T.Q.T. and Murakami, J.; The universal Vassiliev-Kontsevich invariant for framed oriented links, Compositio Math. 102 (1996), no. 1, 41–64. ; Kontsevich’s integral for the Kauffman polynomial, Nagoya Math. J. 142 (1996), 39–65. Piunikhin, S.; Combinatorial expression for universal Vassiliev link invariant, Comm. Math. Phys. 168 (1995), no. 1, 1–22. Racinet, G.; Doubles melangés des polylogarithmes multiples aux racines de l’unité, Publ. Math. Inst. Hautes Etudes Sci. 95 (2002), 185–231. Schneps, L.; Double shuffle and Kashiwara-Vergne Lie algebra, J. Algebra 367 (2012), 54–74. Ševera, P and Willwacher, T.; Equivalence of formalities of the little discs operad, Duke Math. J. 160 (2011), no. 1, 175–206. Tamarkin, D. E.; Formality of chain operad of little discs, Lett. Math. Phys. 66 (2003), no. 1–2, 65–72. Terasoma, T.; Geometry of multiple zeta values, International Congress of Mathematicians. Vol. II, 627–635, Eur. Math. Soc., Zürich, 2006.

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[Wi] Willwacher, T.; M. Kontsevich’s graph complex and the GrothendieckTeichmueller Lie algebra, arXiv:1009.1654, preprint (2010). [Wo] Wojtkowiak, Z.; Monodromy of iterated integrals and non-abelian unipotent periods, Geometric Galois actions, 2, 219–289, London Math. Soc. LNS 243, Cambridge University Press, Cambridge, 1997. [Z] Zagier, D.; Evaluation of the multiple zeta values ζ (2, ..., 2, 3, 2, ..., 2), Annals of Math., 175, (2012), no. 2, 977–1000.

4 The stable Bernstein center and test functions for Shimura varieties Thomas J. Haines

Abstract We elaborate the theory of the stable Bernstein center of a p-adic group G, and apply it to state a general conjecture on test functions for Shimura varieties due to the author and R. Kottwitz. This conjecture provides a framework by which one might pursue the Langlands–Kottwitz method in a very general situation: not necessarily PEL Shimura varieties with arbitrary level structure at p. We give a concrete reinterpretation of the test function conjecture in the context of parahoric level structure. We also use the stable Bernstein center to formulate some of the transfer conjectures (the “fundamental lemmas”) that would be needed if one attempts to use the test function conjecture to express the local Hasse–Weil zeta function of a Shimura variety in terms of automorphic L-functions.

Contents 1 2 3 4 5 6 7 8

Introduction page 119 Notation 122 Review of the Bernstein center 123 The local Langlands correspondence 129 The stable Bernstein center 131 The Langlands–Kottwitz approach for arbitrary level structure 142 Test functions in the parahoric case 153 Overview of evidence for the test function conjecture 159

Research partially supported by NSF DMS-0901723. Automorphic Forms and Galois Representations, ed. Fred Diamond, Payman L. Kassaei and c Cambridge University Press 2014. Minhyong Kim. Published by Cambridge University Press.

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9 Evidence for conjectures on transfer of the Bernstein center 10 Explicit computation of the test functions 11 Appendix: Bernstein isomorphisms via types References

119

161 163 166 183

1. Introduction The main purpose of this chapter is to give precise statements of some conjectures on test functions for Shimura varieties with bad reduction. In the Langlands–Kottwitz approach to studying the cohomology of a Shimura variety, one of the main steps is to identify a suitable test function that is “plugged into” the counting points formula that resembles the geometric side of the Arthur–Selberg trace formula. To be more precise, let (G, h −1 , K p K p ) denote Shimura data where p is a fixed rational prime such p that the level-structure group factorizes as K p K p ⊂ G(A f )G(Q p ). This data gives rise to a quasi-projective variety Sh K p := Sh(G, h −1 , K p K p ) over a number field E ⊂ C. Let p ∈ Gal(Q p /Q p ) denote a geometric Frobenius element. Then one seeks to prove a formula for the semi-simple Lefschetz number Lefss (rp , Sh K p ) trss (rp , H•c (Sh K p ⊗E Q p , Q )) = c(γ0 ; γ , δ) Oγ (1 K p ) TOδθ (φr ), (γ0 ;γ ,δ)

(4.1) (see §6.1 for more details). The test function φr appearing here is the most interesting part of the formula. Experience has shown that we may often find a test function belonging to the center Z(G(Q pr ), K pr ) of the Hecke algebra H(G(Q pr ), K pr ), in a way that is explicitly determined by the E-rational conjugacy class {μ} of 1-parameter subgroups of G associated to the Shimura data. In most PEL cases with good reduction, where K p ⊂ G(Q p ) is a hyperspecial maximal compact subgroup, this was done by Kottwitz (cf. e.g. [Ko92a]). When K p is a parahoric subgroup of G(Q p ) and when GQ p is unramified, the Kottwitz conjecture preK

p dicts that we can take φr to be a power of p times the Bernstein function z −μ, j arising from the Bernstein isomorphism for the center Z(G(Q pr ), K pr ) of the parahoric Hecke algebra H(G(Q pr ), K pr ) (see Conjecture 4.43 and §11). In fact Kottwitz formulated (again, for unramified groups coming from Shimura data) a closely related conjecture concerning nearby cycles on Rapoport–Zink local models of Shimura varieties, which subsequently played an important role in the study of local models (Conjecture 4.44). It also inspired important developments in the geometric Langlands program,

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e.g. [Ga]. Both versions of Kottwitz’ conjectures were later proved in several parahoric cases attached to linear or symplectic groups (see [HN02a, H05]). In a recent breakthrough, Pappas and Zhu [PZ] defined group-theoretic versions of Rapoport–Zink local models for quite general groups, and proved in the unramified situations the analogue of Kottwitz’ nearby cycles conjecture for them. These matters are discussed in more detail in §7 and §8. Until around 2009 it was still not clear how one could describe the test functions φr in all deeper level situations. In the spring of 2009 the author and Kottwitz formulated a conjecture predicting test functions φr for general level structure K p . This is the test function conjecture, Conjecture 4.30. It postulates that we may express φr in terms of a distribution Z E j0 in the Bernstein center Z(G(Q pr )) associated to a certain represenV−μ, j

E

j0 L tation V−μ, j (defined in (4.16)) of the Langlands L-group (G Q pr ). Let d = dim(Sh K p ). Then Conjecture 4.30 asserts that we may take

φr = pr d/2 Z

E j0 V−μ, j

∗ 1 K pr ∈ Z(G(Q pr ), K pr )

the convolution of the distribution Z

E j0

V−μ, j

with the characteristic function 1 K pr

of the subgroup K pr . As shown in §7, this specializes to the Kottwitz conjecture for parahoric subgroups in unramified groups. Conjecture 4.30 was subsequently proved for Drinfeld case Shimura varieties with 1 ( p)-level structure by the author and Rapoport [HRa1], and for modular curves and for Drinfeld case Shimura varieties with arbitrary level structure by Scholze [Sch1, Sch2]. The distributions in Conjecture 4.30 are best seen as examples of a construction V ; Z V which attaches to any algebraic representation V of the Langlands dual group L G (for G any connected reductive group over any p-adic field F), an element Z V in the stable Bernstein center of G/F. This chapter elaborates the theory of the stable Bernstein center, following the lead of Vogan [Vo]. The set of all infinitesimal characters,

i.e. the set of all G-conjugacy classes of admissible homomorphisms λ : W F → L G (where W F is the Weil group of the local field F), is given the structure of an affine algebraic variety over C, and the stable Bernstein center Zst (G/F) is defined to be the ring of regular functions on this variety.1 In order to describe the precise conjectural relation between the Bernstein and stable Bernstein centers of a p-adic group, it was necessary to formulate an 1 The difference between our treatment and Vogan’s is in the definition of the variety structure

on the set of infinitesimal characters.

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enhancement LLC+ of the usual conjectural local Langlands correspondence LLC for that group. Having this relation in hand, the construction V ; Z V provides a supply of elements in the usual Bernstein center of G/F, which we call the geometric Bernstein center. It is for such distributions that one can formulate natural candidates for (Frobenius-twisted) endoscopic transfer, which we illustrate for standard endoscopy in Conjecture 4.35 and for stable base change in Conjecture 4.36. These form part of the cadre of “fundamental lemmas” that one would need to pursue the “pseudostabilization” of (4.1) and thereby express the cohomology of Sh K p in terms of automorphic representations along the lines envisioned by Kottwitz [Ko90] but for places with arbitrary bad reduction. In the compact and non-endoscopic situations, we prove in Theorem 4.40 that the various Conjectures we have made yield an expression of the semisimple local Hasse–Weil zeta function in terms of semi-simple automorphic L-functions. Earlier unconditional results in this direction, for nice PEL situations, were established in [H05], [HRa1], [Sch1, Sch2]. We stress that the framework here should not be limited to PEL Shimura varieties, but should work more generally. In recent work of Scholze and Shin [SS], the connection of the stable Bernstein center with Shimura varieties helped them to give nearly complete descriptions of the cohomology of many compact unitary Shimura varieties with bad reduction at p; they consider the “EL cases” where GQ p is a product of Weil restrictions of general linear groups. It would be interesting to extend the connection to further examples. Returning to the original Kottwitz conjecture for parahoric level structure, Conjecture 4.30 in some sense subsumes it, since it makes sense for arbitrary level structure and without the hypothesis that GQ pr be unramified. However, Conjecture 4.30 has the drawback that it assumes LLC+ for GQ pr . Further, it is still of interest to formulate the Kottwitz conjecture in the parahoric cases for arbitrary groups in a concrete way that can be checked (for example) by explicit comparison of test functions with nearby cycles. In §7 we formulate the Kottwitz conjecture for general groups, making use of the transfer homomorphisms of the Appendix §11 to determine test functions on arbitrary groups from test functions on their quasi-split inner forms. The definition of transfer homomorphisms requires a theory of Bernstein isomorphisms more general than what was heretofore available. Therefore, in the Appendix we establish these isomorphisms in complete generality in a nearly self-contained way, and also provide some related structure theory results that should be of independent interest.

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Here is an outline of the contents of this chapter. In §3 we review the Bernstein center of a p-adic group, including the algebraic structure on the Bernstein variety of all supercuspidal supports. In §4 we recall the conjectural local Langlands correspondence (LLC), and discuss additional desiderata we need in our elaboration of the stable Bernstein center in §5. In particular in §5.2 we describe the enhancement (LLC+) which plays a significant role throughout the chapter, and explain why it holds for general linear groups in Remark 4.8 and Corollary 4.11. The distributions Z V are defined in §5.7, and are used to formulate the test function conjecture, Conjecture 4.30, in §6.1. In the rest of §6, we describe the nearby cycles variant Conjecture 4.31 along with some of the endoscopic transfer conjectures needed for the “pseudostabilization”, and assuming these conjectures we prove in Theorem 4.40 the expected form of the semi-simple local Hasse–Weil zeta functions, in the compact and nonendoscopic cases. In §7 we give a concrete reformulation of the test function conjecture in parahoric cases, recovering the Kottwitz conjecture and generalizing it to all groups using the material from the Appendix. The purpose of §8 and §9 is to list some of the available evidence for Conjectures 4.30 and 4.36. In §10 certain test functions are described very explicitly. Finally, the Appendix gives the treatment of Bernstein isomorphisms and the transfer homomorphisms, alluded to above. Acknowledgments. I am very grateful to Guy Henniart for supplying the proof of Proposition 4.10 and for allowing me to include his proof in this chapter. I warmly thank Timo Richarz for sending me his unpublished article [Ri] and for letting me quote a few of his results in Lemma 4.56. I am indebted to Brooks Roberts for proving Conjecture 4.7 for GSp(4) (see Remark 4.8). I thank my colleagues Jeffrey Adams and Niranjan Ramachandran for useful conversations. I also thank Robert Kottwitz for his influence on the ideas in this chapter and for his comments on a preliminary version. I thank Michael Rapoport for many stimulating conversations about test functions over the years. I am grateful to the referee for helpful suggestions and remarks.

2. Notation If G is a connected reductive group over a p-adic field F, then R(G) will denote the category of smooth representations of G(F) on C-vector spaces. We will write π ∈ R(G)irred or π ∈ (G/F) if π is an irreducible object in R(G). If G as above contains an F-rational parabolic subgroup P with F-Levi factor M and unipotent radical N , define the modulus function δ P : M(F) →

Stable Bernstein center and test functions R>0 by

123

δ P (m) = |det(Ad(m) ; Lie(N (F)))| F 1/2

where | · | F is the normalized absolute value on F. By δ P (m) we mean the positive square-root of the positive real number δ P (m). For σ ∈ R(M), we frequently consider the normalized induced representation G(F)

1/2

i PG (σ ) = Ind P(F) (δ P σ ). We let 1 S denote the characteristic function of a subset S of some ambient space. If S ⊂ G, let g S = gSg −1 . If f is a function on S, define the function g f on g S by g f (·) = f (g −1 · g). Throughout the chapter we use the Weil form of the local or global Langlands L-group L G.

3. Review of the Bernstein center We shall give a brief synopsis of [BD] that is suitable for our purposes. Other useful references are [Be92], [Ren], and [Roc]. The Bernstein center Z(G) of a p-adic group G is defined as the ring of endomorphisms of the identity functor on the category of smooth representations R(G). It can also be realized as an algebra of certain distributions, as the projective limit of the centers of the finite-level Hecke algebras, and as the ring of regular functions on a certain algebraic variety. We describe these in turn.

3.1. Distributions We start by defining the convolution algebra of distributions. We write G for the rational points of a connected reductive group over a p-adic field. Thus G is a totally disconnected locally compact Hausdorff topological group. Further G is unimodular; fix a Haar measure d x. Let Cc∞ (G) denote the set of C-valued compactly supported and locally constant functions on G. Let H(G, d x) = (Cc∞ (G), ∗d x ), the convolution product ∗d x being defined using the Haar measure d x. A distribution is a C-linear map D : Cc∞ (G) → C. For each f ∈ C ∞ (G) we define f˘ ∈ C ∞ (G) by f˘(x) = f (x −1 ) for x ∈ G. We set ˘ f ) := D( f˘). D( We can convolve a distribution D with a function f ∈ Cc∞ (G) and get a new function D ∗ f ∈ C ∞ (G), by setting ˘ · f ), (D ∗ f )(g) = D(g

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where (g · f )(x) := f (xg). The function D ∗ f does not automatically have compact support. We say D is essentially compact provided that D ∗ f ∈ Cc∞ (G) for every f ∈ Cc∞ (G). We define g f by g f (x) := f (g −1 xg) for x, g ∈ G. We say that D is G of G-invariant G-invariant if D( g f ) = D( f ) for all g, f . The set D(G)ec ∞ essentially compact distributions on Cc (G) turns out to have the structure of a commutative C-algebra. We describe next the convolution product and its properties. Given distributions D1 , D2 with D2 essentially compact, we define another distribution D1 ∗ D2 by (D1 ∗ D2 )( f ) = D˘ 1 (D2 ∗ f˘). Lemma 4.1. The convolution products D ∗ f and D1 ∗ D2 have the following properties: (a) For φ ∈ Cc∞ (G) let Dφ d x (sometimes abbreviated φ d x) denote the . essentially compact distribution given by f → G f (x)φ(x) d x. Then D φ d x ∗ f = φ ∗d x f . (b) If f ∈ Cc∞ (G), then D ∗ ( f d x) = (D ∗ f ) d x. In particular, Dφ1 d x ∗ Dφ2 d x = Dφ1 ∗d x φ2 d x . (c) If D2 is essentially compact, then (D1 ∗ D2 ) ∗ f = D1 ∗ (D2 ∗ f ). If D1 and D2 are each essentially compact, so is D1 ∗ D2 . (d) If D2 and D3 are essentially compact, then (D1 ∗D2 )∗D3 = D1 ∗(D2 ∗D3 ). (e) An essentially compact distribution D is G-invariant if and only if D ∗ (1U g d x) = (1U g d x) ∗ D for all compact open subgroups U ⊂ G and g ∈ G. Here 1U g is the characteristic function of the set U g. (f) If D is essentially compact and f 1 , f 2 ∈ C c∞ (G), then D ∗ ( f 1 ∗d x f 2 ) = (D ∗ f 1 ) ∗d x f 2 . G , ∗) is a commutative and associative Corollary 4.2. The pair (D(G)ec C-algebra.

3.2. The projective limit Let J ⊂ G range over the set of all compact open subgroups of G. Let H(G) denote the convolution algebra of compactly-supported measures on G, and let H J (G) ⊂ H(G) denote the ring of J -bi-invariant compactly-supported measures, with center Z J (G). The ring H J (G) has as unit e J = 1 J d x J , where 1 J is the characteristic function of J and d x J is the Haar measure with vold x J (J ) = 1. Note that if J ⊂ J , then d x J = [J : J ] d x J .

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Let Z(G, J ) denote the center of the algebra H(G, J ) consisting of compactly-supported J -bi-invariant functions on G with product ∗d x J . There is an isomorphism Z(G, J ) → Z J (G) by z J → z J d x J . For J ⊂ J there is an algebra map Z(G, J ) → Z(G, J ), given by z J → z J ∗d x J 1 J . Equivalently, we have Z J (G) → Z J (G) given by z J d x J → z J d x J ∗ (1 J d x J ). We can view R(G) as the category of non-degenerate smooth H(G)modules, and any element of lim Z(G, J ) acts on objects in R(G) in a way that ← − commutes with the action of H(G). Hence there is a canonical homomorphism lim Z(G, J ) → Z(G). ← − There is also a canonical homomorphism G lim Z (G, J ) → D(G)ec ← − since Z = (z J ) J ∈ lim Z(G, J ) gives a distribution on f ∈ Cc∞ (G) as ← − follows: choose J ⊂ G sufficiently small that f is right-J -invariant, and set / Z( f ) = z J (x) f (x) d x J . (4.2) G

This is independent of the choice of J . Note that for f ∈ H(G, J ) we have Z ∗ f = z J ∗d x J f , and in particular Z ∗ 1 J = z J , for all J . To see Z = (z J ) J as a distribution is really G-invariant, note that for f ∈ H(G, J ), the identities Z ∗ f = z J ∗d x J f = f ∗d x J z J imply that Z ∗ ( f d x) = ( f d x) ∗ Z . This in turn shows that Z is G-invariant by Lemma 4.1(e). Now §1.4–1.7 of [BD] show that the above maps yield isomorphisms G Z(G) ← lim Z(G, J ) → D(G)ec . (4.3) ← − Corollary 4.3. Let Z ∈ Z(G), and suppose a finite-length representation π ∈ R(G) has the property that Z acts on π by a scalar Z (π ).

(a) For every compact open subgroup J ⊂ G, Z ∗ 1 J acts on the left on π J by the scalar Z (π ). (b) For every f ∈ H(G), tr(Z ∗ f | π ) = Z (π ) tr( f | π ).

3.3. Regular functions on the variety of supercuspidal supports 3.3.1. Variety structure on set of supercuspidal supports We describe the variety of supercuspidal supports in some detail. Also we will describe it in a slightly unconventional way, in that we use the Kottwitz homomorphism to parametrize the (weakly) unramified characters on G(F). This will be useful later on, when we compare the Bernstein center with the stable Bernstein center.

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Let us recall the basic facts on the Kottwitz homomorphism [Ko97]. Let

un of the maximal unramified extension F un in some L be the completion F algebraic closure of F, and let L¯ ⊃ F¯ denote an algebraic closure of L. Let ∼ ¯ ¯ un ) denote the inertia group. Let ∈ Aut(L/F) be I = Gal( L/L) = Gal( F/F the inverse of the Frobenius automorphism σ . In [Ko97] is defined a functorial surjective homomorphism for any connected reductive F-group H

)) I , κ H : H (L) X ∗ (Z ( H

(4.4)

= H

(C) denotes the Langlands dual group of H . By [Ko97, §7], it where H remains surjective on taking -fixed points:

)) κ H : H (F) X ∗ (Z( H I . We define H (L)1 := ker(κ H ) H (F)1 := ker(κ H ) ∩ H (F). We also define H (F)1 ⊇ H (F)1 to be the kernel of the map H (F) →

)) /tor s derived from κ H . X ∗ (Z ( H I If H is anisotropic modulo center, then H (F)1 is the unique maximal compact subgroup of H (F) and H (F)1 is the unique parahoric subgroup of H (F) (see e.g. [HRo]). Sometimes the two subgroups coincide: for example if H is any unramified F-torus, then H (F)1 = H (F)1 . We define X (H ) := Homgrp (H (F)/H (F)1 , C× ), the group of unramified characters on H (F). This definition of X (H ) agrees with the usual one as in [BD]. We define X w (H ) := Homgrp (H (F)/H (F)1 , C× ) and call it the group of weakly unramified characters on H (F). We follow the notation of [BK] in discussing supercuspidal supports and inertial equivalence classes. As indicated earlier in §3.1, for convenience we will sometimes write G when we mean the group G(F) of F-points of an F-group G. A cuspidal pair (M, σ ) consists of an F-Levi subgroup M ⊆ G and a supercuspidal representation σ on M. The G-conjugacy class of the cuspidal pair (M, σ ) will be denoted (M, σ )G . We define the inertial equivalence classes: we write (M, σ ) ∼ (L , τ ) if there exists g ∈ G such that g Mg −1 = L and g σ = τ ⊗ χ for some χ ∈ X (L). Let [M, σ ] denote the equivalence class of G (M, σ )G .

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If π ∈ R(G)irred , then the supercuspidal support of π is the unique element (M, σ )G such that π is a subquotient of the induced representation i PG (σ ), where P is any F-parabolic subgroup having M as a Levi subgroup. Let XG denote the set of all supercuspidal supports (M, σ )G . Denote by the symbol s = [M, σ ]G a typical inertial class. For an inertial class s = [M, σ ]G , define the set Xs = {(L , τ )G | (L , τ ) ∼ (M, σ )}. We have * XG = Xs . s

We shall see below that XG has a natural structure of an algebraic variety, and the Bernstein components Xs form the connected components of that variety. First we need to recall the variety structure on X (M). As is well-known, X (M) has the structure of a complex torus. To describe this, we first consider the weakly unramified character group X w (M). This is a diagonalizable group over C. In fact, by Kottwitz we have an isomorphism

I ) = X ∗ ((Z ( M)

I ) ). M(F)/M(F)1 ∼ = X ∗ (Z ( M) This means that

I ) )], C), X w (M)(C) = Homgrp (M(F)/M(F)1 , C× ) = Homalg (C[X ∗ ((Z( M)

in other words,

I ) . X w (M) = (Z ( M)

(4.5)

Another way to see (4.5) is to use Langlands’ duality for quasicharacters, which is an isomorphism

Homcont (M(F), C× ) ∼ = H 1 (W F , Z ( M)). (Here W F is the Weil group of F; see §4.) Under this isomorphism, X w (M)

I) → is identified with the image of the inflation map H 1 (W F /I, Z ( M)

that is, with H 1 ( , Z ( M)

I ) = (Z ( M)

I ) . This last idenH 1 (W F , Z ( M)), cocyc

I ) to tification is given by the map sending a cocycle ϕ ∈ Z 1 ( , Z( M) cocyc I w

) . The two ways of identifying X (M) with (Z ( M)

I ) , ϕ () ∈ (Z ( M) that via the Kottwitz isomorphism and that via Langlands duality, agree.2 For a more general result which implies this agreement, see [Kal], Prop. 4.5.2. Now we can apply the same argument to identify the torus X (M). We first get X (M)(C) = Homgrp (M(F)/M(F)1 , C× ). 2 We normalize the Kottwitz homomorphism as in [Ko97], so that κ × Gm : L → Z is the

valuation map sending a uniformizer $ to 1. Then the claimed agreement holds provided we normalize the Langlands duality for tori as in (4.7).

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Since M(F)/M(F)1 is the quotient of M(F)/M(F)1 by its torsion, it follows that

I ) )◦ =: (Z ( M)

I )◦ , X (M) = ((Z ( M) the neutral component of X w (M). Now we turn to the variety structure on Xs . We fix a cuspidal pair (M, σ ) representing s M = [M, σ ] M and s = [M, σ ]G . Let the corresponding Bernstein components be denoted Xs and Xs M . As sets, we have Xs = {(M, σ χ )G } and Xs M = {(M, σ χ ) M }, where χ ∈ X (M). The torus X (M) acts on Xs M by χ → (M, σ χ ) M . The isotropy group is stabσ := {χ | σ ∼ = σ χ }. Let Z (M)◦ denote the neutral component of the center of M. Then stabσ belongs to the kernel of the map X (M) → X (Z(M)◦ ), χ → χ | Z (M)◦ (F) , hence stabσ is a finite subgroup of X (M). Thus Xs M is a torsor under the torus X (M)/stabσ , and thus has the structure of an affine variety over C. There is a surjective map Xs M → Xs (M, σ χ ) M → (M, σ χ )G , χ ∈ X (M)/stabσ . ∼ σ χ for some χ ∈ X (M)}. Then Let N G ([M, σ ] M ) := {n ∈ N G (M) | n σ = G the fibers of Xs M → Xs are precisely the orbits the finite group W[M,σ ] M := N G ([M, σ ]G )/M on Xs M . Via G Xs = W[M,σ ] M \Xs M

the set Xs acquires the structure of an irreducible affine variety over C. Up to isomorphism, this structure does not depend on the choice of the cuspidal pair (M, σ ). 3.3.2. The center as regular functions on X G An element z ∈ Z(G) determines a regular function X: for a point (M, σ )G ∈ Xs , z acts on i G P (σ ) by a scalar z(σ ) and the function (M, σ )G → z(σ ) is a regular function on XG . This is the content of [BD, Prop. 2.11]. In fact we have by [BD, Thm. 2.13] an isomorphism Z(G) → C[XG ].

(4.6)

Together with (4.3) this gives all the equivalent ways of realizing the Bernstein center of G.

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4. The local Langlands correspondence We need to recall the general form of the conjectural local Langlands correspondence (LLC) for a connected reductive group G over a p-adic field F. Let ¯ F¯ denote an algebraic closure of F. Let W F ⊂ Gal( F/F) =: F be the Weil group of F. It fits into an exact sequence of topological groups 1

/ IF

/ WF

val

/Z

/ 1,

where I F is the inertia subgroup of F and where, if ∈ W F is a geometric Frobenius element (the inverse of an arithmetic Frobenius element), then val() = −1. Here I F has its profinite topology and Z has the discrete topology. Sometimes we write I for I F in what follows. Recall the Weil–Deligne group is W F := W F C, where wzw−1 = |w|z val(w) for w ∈ W F and z ∈ C, with |w| := q F for q F = #(O F /($ F )), the cardinality of the residue field of F. A Langlands parameter is an admissible homomorphism ϕ : W F → L G,

W F . This means: where L G := G • ϕ is compatible with the projections W F → W F and ν : L G → W F ; • ϕ is continuous and respects Jordan decompositions of elements in W F and L G (cf. [Bo79, §8], for the definition of Jordan decomposition in the group W F C and what it means to respect Jordan decompositions here); • if ϕ(W F ) is contained in a Levi subgroup of a parabolic subgroup of L G, then that parabolic subgroup is relevant in the sense of [Bo79, §3.3]. (This condition is automatic if G/F is quasi-split.)

Let (G/F) denote the set of G-conjugacy classes of admissible homomor L phisms ϕ : W F → G and let (G/F ) = R(G(F))irred the set of irreducible smooth (or admissible) representations of G(F) up to isomorphism. Conjecture 4.4 (LLC). There is a finite-to-one surjective map (G/F) → (G/F), which satisfies the desiderata of [Bo79, §10]. The fiber ϕ over ϕ ∈ (G/F) is called the L-packet for ϕ. We mention a few desiderata of the LLC that will come up in what follows. First, LLC for Gm is nothing other than Langlands duality for Gm , which we

, normalize as follows: for T any split torus torus over F, with dual torus T

) Homconts (T (F), C× ) = Homconts (W F , T ξ ↔ ϕξ

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) and w ∈ W F , satisfies, for every ν ∈ X ∗ (T ) = X ∗ (T ν(ϕξ (w)) = ξ(ν(Art−1 F (w))).

(4.7)

ab × Here Art−1 F : W F → F is the reciprocity map of local class field theory which sends any geometric Frobenius element ∈ W F to a uniformizer in F. Next, we think of Langlands parameters in two ways, either as continuous L-homomorphisms

ϕ : W F →

L

G

modulo G-conjugation, or as continuous 1-cocycles

ϕ cocyc : W F → G

via the projection W → W F ). modulo 1-coboundaries (where W F acts on G F The dictionary between these is

WF ϕ(w) = (ϕ cocyc (w), w) ¯ ∈G for w ∈ W F and w¯ the image of w under W F → W F . The desideratum we will use explicitly is the following (a special case of [Bo79, 10.3(2)]: given any Levi pair (M, σ ) (where σ ∈ (M/F)) with repcocyc

and any unramified 1-cocycle z χcocyc : resenting 1-cocycle ϕσ : W F → M,

I representing χ ∈ X (M) via the Langlands correspondence for W F → Z ( M) quasi-characters on M(F), we have ϕσcocyc = ϕσcocyc · z χcocyc χ

We may view z cocyc modulo 1-coboundaries with values in M. as a 1-cocycle χ

the righton W F which is trivial on C; since it takes values in the center of M, hand side is a 1-cocycle whose cohomology class is independent of the choices cocyc cocyc of 1-cocycles ϕσ and z χ in their respective cohomology classes. Hence the condition just stated makes sense. Concretely, if χ ∈ X (M) lifts to an

I , then up to M-conjugacy

element z ∈ Z( M) we have

WF . ϕσ χ () = (z, 1)ϕσ () ∈ M

(4.8)

Remark 4.5. There is a well-known dictionary between equivalence classes of admissible homomorphisms ϕ : W F C → L G and equivalence classes of admissible homomorphisms W F × SL2 (C) → L G. For a complete explanation, see [GR, Prop. 2.2]. Because of this equivalence, it is common in the literature for the Weil–Deligne group W F to sometimes be defined as W F C, and sometimes as W F × SL2 (C).

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5. The stable Bernstein center 5.1. Infinitesimal characters

Following Vogan [Vo], we term a G-conjugacy class of an admissible3 homomorphism λ : WF →

L

G

an infinitesimal character. Denote the G-conjugacy class of λ by (λ)G . In this section we give a geometric structure to the set of all infinitesimal characters for a group G. It should be noted that the variety structure we define here differs from that put forth by Vogan in [Vo, §7]. If ϕ : W F → L G is an admissible homomorphism, then its restriction ϕ|W F represents an infinitesimal character. Here it is essential to consider restriction along the proper embedding W F → W F : if W F is thought of as W F C, then this inclusion is w → (w, 0); if W F is thought of as W F × SL2 (C), then the inclusion is w → (w, diag(|w|1/2 , |w|−1/2 )). If ϕπ ∈ (G/F) is attached by LLC to π ∈ (G/F), then following Vogan [Vo] we shall call the

G-conjugacy class (ϕπ |W F )G

the infinitesimal character of π . If G is quasi-split over F, then conjecturally every infinitesimal character λ is represented by a restriction ϕπ |W F : W F → L G for some π ∈ (G/F). Assume LLC holds for G/F. Let λ be an infinitesimal character for G. Define the infinitesimal class to be the following finite union of L-packets * λ := ϕ . ϕ ;λ

Here ϕ ranges over G-conjugacy classes of admissible homomorphisms W F → L G such that (ϕ|W F )G = (λ)

, and ϕ is the corresponding L-packet G of smooth irreducible representations of G(F).

5.2. LLC+ In order to relate the Bernstein variety X with the variety Y of infinitesimal characters, we will assume the Local Langlands Correspondence (LLC) for G and all of its F-Levi subgroups. We assume all the desiderata listed by Borel in [Bo79]. There are two additional desiderata of LLC we need. 3 “Admissible” is defined as for the parameters ϕ : W → L G (e.g. λ(W ) consists of F F semisimple elements of L G) except that we omit the “relevance condition”. This is because

the restriction ϕ|W F of a Langlands parameter could conceivably factor through a non-relevant Levi subgroup of L G (even though ϕ does not) and we want to include such restrictions in what we call infinitesimal characters.

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Definition 4.6. We will declare that G satisfies LLC+ if the LLC holds for G and its F-Levi subgroups, and these correspondences are compatible with normalized parabolic induction in the sense of the Conjecture 4.7 below, and invariant under certain isomorphisms in the sense of Conjecture 4.12 below. Let M ⊂ G denote an F-Levi subgroup. Then the inclusion M → G

induces an embedding L M → L G which is well-defined up to G-conjugacy (cf. [Bo79, §3]). Conjecture 4.7. (Compatibility of LLC with parabolic induction) Let σ ∈ (M/F) and π ∈ (G/F) and assume π is an irreducible subquotient of i PG (σ ), where P = M N is any F-parabolic subgroup of G with F-Levi factor M. Then the infinitesimal characters ϕπ | W F : W F →

L

G

and ϕσ | W F : W F →

L

M →

L

G

are G-conjugate. Remark 4.8. (1) The conjecture implies that the restriction ϕπ |W F depends only on the supercuspidal support of π. This latter statement is a formal consequence of Vogan’s Conjecture 7.18 in [Vo], but the Conjecture 4.7 is slightly more precise. In Proposition 4.23 we will give a construction of the map f in Vogan’s Conjecture 7.18, by sending a supercuspidal support (M, σ )G (a “classical infinitesimal character” in [Vo]) to the infinitesimal character (ϕσ |W F )G . With this formulation, the condition on f imposed in Vogan’s Conjecture 7.18 is exactly the compatibility in the conjecture above. (2) The conjecture holds for GLn , and is implicit in the way the local Langlands correspondence for GLn is extended from supercuspidals to all representations (see Remark 13.1.1 of [HRa1]). It was a point of departure in Scholze’s new characterization of LLC for GLn [Sch3], and that paper also provides another proof of the conjecture in that case. (3) I was informed by Brooks Roberts (private communication), that the conjecture holds for GSp(4). (4) Given a parameter ϕ : W F → L G, there exists a certain P = M N and a certain tempered parameter ϕ M : W F → L M and a certain real-valued unramified character χ M on M(F) whose parameter is in the interior of the Weyl chamber determined by P, such that the L-packet φ consists of Langlands quotients J (π M ⊗ χ M ), for π M ranging over the packet

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ϕ M . The parameter ϕ is the twist of ϕ M by the parameter associated to the character χ M . This reduces the conjecture to the case of tempered representations. One can further reduce to the case of discrete series representations. The following is a very natural kind of functoriality which should be satisfied for all groups. Conjecture 4.9. (Invariance of LLC under isomorphisms) Suppose φ : (G, π ) → (G , π ) is an isomorphism of connected reductive F-groups together with irreducible smooth representations on them. Then the induced isomorphism L φ : L G → L G (well-defined up to an inner automorphism of

G), takes the G -conjugacy class of ϕπ : W F → L G to the G-conjugacy L class of ϕπ : W F → G. Proposition 4.10. Conjecture 4.9 holds when G = GLn . Proof. (Guy Henniart). It is enough to consider the case where G = GLn and φ is an F-automorphism of GLn . The functorial properties in the Langlands correspondence for GLn are: (i) Compatibility with class field theory, that is, with the case where n = 1. (ii) The determinant of the Weil–Deligne group representation corresponds to the central character: this is Langlands functoriality for the homomorphism det : GLn (C) → GL1 (C). (iii) Compatibility with twists by characters, i.e., Langlands functoriality for the obvious homomorphism of dual groups GL1 (C) × GLn (C) → GLn (C). (iv) Compatibility with taking contragredients: this is Langlands functoriality with respect to the automorphism g → t g −1 (transpose inverse), since it is known that for GLn (F) this sends an irreducible representation to a representation isomorphic to its contragredient. These properties are enough to imply the desired functoriality for F-automorphisms of GLn . When n = 1, the functoriality is obvious for any F-endomorphism of GL1 . When n is at least 2, an F-automorphism of GLn induces an automorphism of SLn hence an automorphism of the Dynkin diagram which must be the identity or, (when n ≥ 3) the opposition automorphism. Hence up to conjugation by GLn (F), the F-automorphism is the identity on SLn , or possibly (when n ≥ 3) transpose inverse. Consequently the F-automorphism can be reduced (by composing with an inner automorphism or possibly with transpose inverse) to one which is the identity on SLn , hence is of the form g → g · c(det(g)) where c ∈ X ∗ (Z (GLn )). But this

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implies that it is the identity unless n = 2, in which case it could also be g → g · det(g)−1 . In that exceptional case, the map induced on the dual group GL2 (C) is also g → g · det(g)−1 , and the desired result holds by invoking (ii) and (iii) above. Corollary 4.11. Let M = GLn 1 × · · · × GLnr ⊂ GLn be a standard Levi subgroup. Let g ∈ GLn (F). Then Conjecture 4.9 holds for the isomorphism cg : M → g M given by conjugation by g. Proof. It is enough to consider the case where g belongs to the normalizer of M in GLn . Let T ⊂ M be the standard diagonal torus in GLn . Then g ∈ N G (T )M. Thus composing g with a permutation matrix which normalizes M we may assume that cg preserves each diagonal factor GLn i . The desired functoriality follows by applying Proposition 4.10 to each GLn i . For the purposes of comparing the Bernstein center and the stable Bernstein center as in Proposition 4.23, we need only this weaker variant of Conjecture 4.9. Conjecture 4.12. (Weak invariance of LLC) Let M ⊆ G be any F-Levi subgroup and let g ∈ G(F). Then Conjecture 4.9 holds for the isomorphism cg : M → g M.

5.3. Variety structure on the set of infinitesimal characters

⊃

It is helpful to rigidify things on the dual side by choosing the data G B⊃T of a Borel subgroup and maximal torus which are stable under the action of

and which form part of the data of a F -invariant splitting for G

F on G (cf. [Ko84a, §1]). The variety structure we will define will be independent

of this choice, up to isomorphism, since different choices such that

B ⊃ T L F

([Ko84a, Cor. 1.7]). Let B :=

are conjugate under G B W F and LT = T

WF . Following [Bo79, §3.3], we say a parabolic subgroup P ⊆ L G is standard

is a W F -stable standard if P ⊇ L B. Then its neutral component P ◦ := P ∩ G

parabolic subgroup of G (containing B), and P = P ◦ W F . Every parabolic

subgroup in L G is G-conjugate to a unique standard parabolic subgroup. Assume P is standard and let M◦ ⊂ P ◦ be the unique Levi factor with

; it is W F -stable. Then M := NP (M◦ ) is a Levi subgroup of P in M◦ ⊇ T the sense of [Bo79, §3.3], and M = M◦ W F . The Levi subgroups M ⊂ L G which arise this way are called standard. Every Levi subgroup in L G is

G-conjugate to at least one standard Levi subgroup; two different standard Levi

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135

Denote by {M} the set of standard Levi subgroups may be conjugate under G.

subgroups in L G which are G-conjugate to a fixed standard Levi subgroup M. L Now suppose λ : W F → G is an admissible homomorphism. Then there exists a minimal Levi subgroup of L G containing λ(W F ). Any two such are

commuting conjugate by an element of Cλ◦ , where Cλ is the subgroup of G with λ(W F ), by (the proof of) [Bo79, Prop. 3.6].

Suppose λ1 , λ2 : W F → L G are G-conjugate. Then there exists a + +

G-conjugate λ1 (resp. λ2 ) of λ1 (resp. λ2 ) and a standard Levi subgroup M1 + + −1 (resp. M2 ) containing λ+ = 1 (W F ) (resp. λ2 (W F )) minimally. Write gλ1 g + −1

Then the Levi subgroups gM1 g and M2 contain λ2 for some g ∈ G. λ+ 2 (W F ) minimally, hence by [Bo79, Prop. 3.6] are conjugate by an element s ∈ C ◦+ . Then sg(M1 )(sg)−1 = M2 , and thus {M1 } = {M2 }. λ2

Hence any G-conjugacy class (λ)G gives rise to a unique class of standard Levi subgroups {Mλ }, with the property that the image of some element λ+ ∈ (λ)G

is contained minimally by Mλ for some Mλ in this class. A similar argument shows the following lemma. + + Lemma 4.13. Let λ+

= 1 and λ2 be admissible homomorphisms with (λ1 )G + + + (λ2 )G , and suppose λ (W ) and λ (W ) are contained minimally by a

F F 1 2 standard Levi subgroup M. Then there exists n ∈ NG (M) such that n λ + = λ+ . 1 2

The following lemma is left to the reader. Lemma 4.14. If M ⊆

LG

is a standard Levi subgroup, then

◦ ◦ NG

(M) = {n ∈ N G

(M ) | nM is W F -stable}.

Consequently, conjugation by n ∈ NG (M) preserves the set (Z (M◦ ) I )◦ . More generally, if M1 and M2 are standard Levi subgroups of L G and if we define the transporter subset by

| gM1 g −1 = M2 }, TransG (M1 , M2 ) := {g ∈ G then ◦ ◦ ◦ ◦ TransG

(M1 , M2 ) = {g ∈ TransG

(M1 , M2 ) | gM1 = M2 g is W F -stable}.

Consequently, conjugation by g ∈ TransG (M1 , M2 ) sends (Z (M◦1 ) I )◦ into (Z (M◦2 ) I )◦ . We can now define the notion of inertial equivalence (λ1 )G

∼ (λ2 )G

of infinitesimal characters.

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Definition 4.15. We say (λ1 )G and (λ2 )G

are inertially equivalent if • {Mλ1 } = {Mλ2 }; + • there exists M ∈ {Mλ1 }, and λ+

and λ2 ∈ (λ2 )G

whose images 1 ∈ (λ1 )G are minimally contained by M, and an element z ∈ (Z (M◦ ) I )◦ , such that + (zλ+ 1 )M◦ = (λ2 )M◦ .

We write [λ]G for the inertial equivalence class of (λ)G

. Note that M automatically contains (zλ+ 1 )(W F ) minimally if it contains minimally.

λ+ 1 (W F )

Lemma 4.16. The relation ∼ is an equivalence relation on the set of infinitesimal characters. Proof. Use Lemmas 4.13 and 4.14.

⊃ B

⊃T

Remark 4.17. To define (λ1 )G ∼ (λ2 )G we used the choice of G

(which was assumed to form part of a F -invariant splitting for G) in order to define the notion of standard Levi subgroup of L G. However, the equivalence relation ∼ is independent of this choice, since as remarked above, any two

are conjugate under G

F , by [Ko84a, Cor. 1.7]. F -invariant splittings for G Remark 4.18. The property we need of standard Levi subgroups M ⊆ L G

is W F -stable, and is that they are decomposable, that is, M◦ := M ∩ G ◦ M = M W F . Any standard Levi subgroup is decomposable. In our discussion, we could have avoided choosing a notion of standard Levi, by associating to each (λ)G

a unique class of decomposable Levi subgroups {M},

all of which are G-conjugate, such that λ factors minimally through some M ∈ {M}. Now fix a standard Levi subgroup M ⊆ L G. We write tM◦ for an inertial equivalence class of admissible homomorphisms W F → M. We write YtM◦ for the set of M◦ -conjugacy classes contained in this inertial class. We want to give this set the structure of an affine algebraic variety over C. Define the torus Y (M◦ ) := (Z(M◦ ) I )◦ . Then Y (M◦ ) acts transitively on YtM◦ . Fix a representative λ : WF → M for this inertial class, so that tM◦ = [λ]M◦ .

(4.9)

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Lemma 4.19. The Y (M◦ )-stabilizer stabλ := {z ∈ Y (M◦ ) | (zλ)M◦ = (λ)M◦ } is finite. Proof. There exists an integer r ≥ 1 such that r acts trivially on M◦ . The group stabλ is contained in the preimage of the finite group Z (M◦ ) F ∩ (M◦ )der under the norm homomorphism Nr : (Z (M◦ ) I ) → Z (M◦ ) F , z → z(z) · · · r−1 (z) and the kernel of this homomorphism is finite. Then YtM◦ is a torsor under the quotient torus YtM◦ ∼ = Y (M◦ )/stabλ . In this way the left-hand side acquires the structure of an affine algebraic variety. Up to isomorphism, this structure is independent of the choice of λ representing tM◦ . Now let t denote an inertial class of infinitesimal characters for G, and let Yt denote the set of infinitesimal characters in t. Recall t gives rise to a unique class of standard Levi subgroups {M}, having the property that some representative λ for t factors minimally through some M ∈ {M}. Fix such a representative λ : W F → M → L G for t, so that t = [λ]G and tM◦ = [λ]M◦ . By our previous work, there is a surjective map YtM◦ → Yt (zλ)M◦ → (zλ)G . where z ∈ Y (M◦ )/stabλ . Let n ◦ NG

(M, [λ]M◦ ) = {n ∈ N G

(M) | ( λ)M◦ = (zλ)M◦ , for some z ∈ Y (M )}.

From the above discussion we see the following. Lemma 4.20. The fibers of YtM◦ → Yt are precisely the orbits of the finite

G ◦ group W[λ] := N G

(M, [λ]M◦ )/M on YtM◦ . M◦

Hence Yt = WtGM◦ \YtM◦ acquires the structure of an affine variety over + C. Thus Y = t Yt is an affine variety over C and each Yt is a connected component. Let Zst (G) denote the ring of regular functions on the affine variety Y. We call this ring the stable Bernstein center of G/F.

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5.4. Base change homomorphism of the stable Bernstein center Let E/F be a finite extension in F/F with ramification index e and residue field extension k E /k F of degree f . Then W E ⊂ W F and I E ⊆ I F . Further, we f can take E := F as a geometric Frobenius element in W E . Let YG/F resp. G/E Y denote the variety of infinitesimal characters associated to G resp. G E . Proposition 4.21. The map (λ)G → (λ|W E )G determines a morphism of algebraic varieties YG/F → YG/E . Definition 4.22. We call the corresponding map b E/F : Zst (G E ) → Zst (G) the base change homomorphism for the stable Bernstein center.

W F factors minimally through the standard Proof. Suppose λ : W F → G

WE Levi subgroup M ⊂ G W F and that its restriction λ|W E : W E → G

factors minimally through the standard Levi subgroup M E ⊂ G W E . We may assume M◦E ⊆ M◦ and thus Z(M◦ ) ⊂ Z(M◦E ). There is a homomorphism of tori Y (M◦ ) = (Z(M◦ ) I F )◦ F −→ (Z(M◦E ) I E )◦

f

F

= Y (M◦E )

(4.10) f −1

z −→ z f := N f (z) := z · F (z) · · · F

(z).

Recall that z ∈ (Z (M◦ ) I F ) F is identified with the image of the element z( F ) ∈ Z(M◦ ) I F , where z is viewed as a cohomology class z ∈ H 1 ( F , Z (M◦ ) I F ). Using the same fact for E in place of F, it follows that (zλ)|W E = z f λ|W E , where z f is defined as above. Thus the map (zλ)G

→ ((zλ)|W E )G lifts to the map (zλ)M◦ → (z f λ|W E )M◦ → (z f λ|W E )G

, and being induced by (4.10), the latter is an algebraic morphism.

5.5. Relation between the Bernstein center and the stable Bernstein center The varieties X and Y are defined unconditionally. In order to relate them, we need to assume LLC+ holds. Proposition 4.23. Assume LLC+ holds for the group G. Then the map (M, σ )G → (ϕσ |W F )G defines a quasi-finite morphism of affine algebraic varieties f : X → Y. It is surjective if G/F is quasi-split.

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The reader should compare this with Conjecture 7.18 in [Vo]. Our variety structure on the set Y is different from that put forth by Vogan, and our f is given by a simple and explicit rule. In view of LLC+ our f automatically satisfies the condition which Vogan imposed on the map in his Conjecture 7.18: if π has supercuspidal support (M, σ )G , then the infinitesimal character of π is f ((M, σ )G ). Proof. It is easy to see that the map (M, σ )G → (ϕσ |W F )G is well-defined. We need to show that an isomorphism cg : (M, σ ) → ( g M, g σ ) given by conjugation by g ∈ G(F) gives rise to parameters ϕσ : W F → L M → L G and ϕ g σ : W F → L ( g M) → L G which differ by an inner automorphism of

In view of Conjecture 4.12 applied to M, the isomorphism L ( g M) → G. LM

takes ϕ g σ to an M-conjugate of ϕσ . On the other hand the embeddings L M → L G and L ( g M) → L G are defined using based root systems in such a way

that it is obvious that they are G-conjugate. To examine the local structure of this map, we first fix a λ and a standard Mλ through which λ factors minimally. Let t = [λ]G . Then over Yt the map f takes the form * Xs M → Yt . (4.11) s M ;t

Here s M ranges over the inertial classes [M, σ ]G such that (ϕσ |W F )G is inertially equivalent to (λ)G . We now fix a representative (M, σ ) for s M . Given

such a ϕσ , its restriction ϕσ |W F factors through a G-conjugate of L M. But

ϕσ |W F factors mini(ϕσ |W F )G

∼ (λ)G

implies that (up to conjugation by G) L mally through Mλ . Thus we may assume that Mλ ⊆ M. The corresponding

→ Z (M◦ ) induces a morphism of algebraic tori inclusion Z ( M) λ

= (Z ( M)

I )◦ → (Z (M◦λ ) I )◦ = Y (M◦λ ). Y ( M)

by the Kottwitz isomorphism (or the Langlands Further, recall X (M) ∼ = Y ( M) cocyc duality for quasi-characters), by the rule χ → z χ (). Taking (4.8) into account, we see that (4.11) on Xs M , given by (M, σ χ )G → (ϕσ χ |W F )G

for χ ∈ X (M)/stabσ , lifts to the map X (M)/stabσ → Y (M◦λ )/stabλ ,

(4.12)

→ Y (M◦ ), up to transwhich is the obvious map induced by X (M) → Y ( M) λ ◦ lation by an element in Y (Mλ ) measuring the difference between (ϕσ |W F )M◦λ and (λ)M◦λ . The map (4.12) is clearly a morphism of algebraic varieties. Hence the map f is a morphism of algebraic varieties.

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The fibers of f are finite by a property of LLC. Finally, if G/F is quasi-split, the morphism f is surjective by another property of LLC. Corollary 4.24. Assume G/F satisfies LLC+, so that the map f in Proposition 4.23 exists. Then f induces a C-algebra homomorphism Zst (G) → Z(G). It is injective if G/F is quasi-split. Remark 4.25. For the group GLn the constructions above are unconditional because the local Langlands correspondence and its enhancement LLC+ are known (cf. Remark 4.8(2) and Corollary 4.11). One can see that XGLn → YGLn is an isomorphism and hence Zst (GLn ) = Z(GLn ). Remark 4.26. As remarked by Scholze and Shin [SS, §6], one may conjecturally characterize the image of Zst (G) → Z(G) in a way that avoids direct mention of L-parameters. According to them it should consist of the distriG such that, for any function f ∈ C ∞ (G(F) whose stable butions D ∈ D(G)ec c orbital integrals vanish at semi-simple elements, the function D∗ f also has this property. See [SS, §6] for further discussion of this. From conjectured relations between stable characters and stable orbital integrals, one can conjecturally rephrase the condition on D in terms of stable characters, as SOϕ (D ∗ f ) = 0, ∀ϕ, if SOϕ ( f ) = 0, ∀ϕ.

(4.13)

An element of Zst (G) acts by the same scalar on all π ∈ ϕ , and so the above condition holds if D ∈ f (Zst (G)). The converse direction is much less clear, and implies non-trivial statements about the relation between supercuspidal supports, L-packets, and infinitesimal classes. Indeed, suppose we are given D ∈ Z(G) that satisfies (4.13). This should mean that it acts by the same scalar on all π ∈ ϕ . On the other hand, saying D comes from Zst (G) would mean that D acts by the same scalar on all π ∈ λ and those scalars vary algebraically as λ ranges over Y. So if for some λ the infinitesimal class * λ = ϕ ϕ ;λ

contains an L-packet ϕ0 λ such that the set of supercuspidal supports coming from ϕ0 does not meet the set of those coming from any ϕ with ϕ not conjugate to ϕ0 , then one could construct a regular function D ∈ Z(G) which is constant on the L-packets ϕ but not constant on λ , and thus not in f (Zst (G)). In that case (4.13) would not be sufficient to force D ∈ f (Zst (G)), and the conjecture of Scholze–Shin would be false. In that case, one could define the subring Zst∗ (G) ⊆ Z(G) of regular functions on the Bernstein variety which take the same value on all supercuspidal supports of representations in the same L-packet. This would then perhaps better deserve the title “stable

Stable Bernstein center and test functions

141

Bernstein center” and it would be strictly larger than f (Zst (G)) at least in some cases. To illustrate this in a more specific setting, suppose G/F is quasi-split and λ does not factor through any proper Levi subgroup of L G. Then by Proposition 4.27 below, we expect λ to consist entirely of supercuspidal representations. If λ contains at least two L-packets ϕ , then there would exist a D ∈ Z(G) which is constant on the ϕ ’s yet not constant on λ , and the Scholze–Shin conjecture should be false. Put another way, if the Scholze–Shin conjecture is true, we expect that whenever λ does not factor through a proper Levi in L G, the infinitesimal class λ consists of at most one L-packet.4

5.6. Aside: when does an infinitesimal class consist only of supercuspidal representations? Proposition 4.27. Assume G/F is quasi-split and LLC+ holds for G. Then λ consists entirely of supercuspidal representations if and only if λ does not factor through any proper Levi subgroup L M L G. Proof. If λ contains a nonsupercuspidal representation π with supercuspidal support (M, σ )G for M G, then by LLC+, we may assume ϕπ |W F , and hence λ, factors through the proper Levi subgroup L M L G. Conversely, if λ factors minimally through a standard Levi subgroup Mλ L G, then we must show that contains a nonsupercuspidal representation λ of G. Since G/F is quasi-split, we may identify Mλ = L Mλ for an F-Levi subgroup Mλ G. Now for t = [λ]G , the map (4.11) is surjective. For any F-Levi subgroup

I )◦ < M Mλ , a component of the form X[M,σ ]G has dimension dim (Z ( M) I ◦ dim (Z ( Mλ ) ) = dim Yt . Thus the union of the components of the form X[M,σ ]G with M Mλ cannot surject onto Yt . Thus there must be a component of the form X[Mλ ,σλ ] appearing in the left-hand side of (4.11). We may cocyc assume ϕσλ factors through L Mλ along with λ. Writing (λ) M ϕσλ ) M

λ = (z χ

λ for some χ ∈ X (Mλ ), it follows that λ contains the nonsupercuspidal representations with supercuspidal support (Mλ , σλ χ )G .

5.7. Construction of the distributions ZV Let (r, V ) be a finite-dimensional algebraic representation of L G on a complex vector space. Given a geometric Frobenius element ∈ W F and an admissible homomorphism λ : W F → L G, we may define the semi-simple trace 4 In fact this statement holds: if λ does not factor through a proper Levi subgroup of L G, then there is at most one way to extend it to an admissible homomorphism ϕ: W F → L G.

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Thomas J. Haines trss (λ(), V )) := tr(r λ(), V rλ(I F ) ).

Note this is independent of the choice of . This notion was introduced by Rapoport [Ra90] in order to define semisimple local L-functions L(s, π p , r ), and is parallel to the notion for -adic Galois representations used in [Ra90] to define semi-simple local zeta functions ζpss (X, s); see also [HN02a], [H05]. The following result is an easy consequence of the material in §5.3. Proposition 4.28. The map λ → trss (λ(), V ) defines a regular function on the variety Y hence defines an element Z V ∈ Zst (G) by ss Z V ((λ)G

) = tr (λ(), V ).

We use the same symbol Z V to denote the corresponding element in Z(G) given via Zst (G) → Z(G). The latter has the property Z V (π ) = trss (ϕπ (), V )

(4.14)

for every π ∈ (G/F), where Z V (π ) stands for Z V ((M, σ )G ) if (M, σ )G is the supercuspidal support of π . Remark 4.29. One does not really need the full geometric structure on the set Y in order to construct Z V ∈ Z(G): one may show directly, assuming that LLC and Conjecture 4.7 hold, that π → trss (ϕπ (), V ) descends to give a regular function on X and hence (4.14) defines an element Z V ∈ Z(G). Using the map f simply makes the construction more transparent (but has the drawback that we also need to assume Conjecture 4.12).

6. The Langlands–Kottwitz approach for arbitrary level structure 6.1. The test functions Let (G, X ) be a Shimura datum, where X is the G(R)-conjugacy class of an R-group homomorphism h : RC/R Gm → GR . This gives rise to the reflex field E ⊂ C and a G(C)-conjugacy class {μ} ⊂ X ∗ (GC ) which is defined over E. Choose a quasi-split group G∗ over Q and an inner twisting ψ : G∗ → G of Q-groups. In particular we get an inner twisting G∗E → GE as well as an isomorphism of L-groups L (GE ) → L (G∗E ). Let Q ⊂ C denote the algebraic numbers, so that we have an inclusion E ⊂ Q and we can regard {μ} as a G(Q)-conjugacy class in X ∗ (GQ ) which is

Stable Bernstein center and test functions

143

defined over E (cf. [Ko84b, Lemma 1.1.3]). Using ψ regard {μ} as a G∗ (Q)conjugacy class in X ∗ (G∗ ), defined over E. By Kottwitz’ lemma ([Ko84b, Q 1.1.3]), {μ} is represented by an E-rational cocharacter μ : Gm → G∗E . Following Kottwitz’ argument in [Ko84b, 2.1.2], it is easy to show that there exists a unique representation (r−μ , V−μ ) of L (GE ) such that as a represen ∗ it is an irreducible representation with extreme weight −μ and the tation of G Weil group WE acts trivially on the highest-weight space corresponding to any

∗ . E -fixed splitting for G E Using ψ we can regard (r−μ , V−μ ) as a representation of L (GE ). The isomorphism class of this representation depends only on the equivalence class of the inner twisting ψ, thus only on G and {μ}. Now we fix a rational prime p and set G := GQ p . Choose a prime ideal p ⊂ E lying above p, and set E := Ep . Choose an algebraic closure Q p of Q p and fix henceforth an isomorphism of fields C ∼ = Q p such that the embedding E → C ∼ = Q p corresponds to the prime ideal p. This gives rise to an embedding Q → Q p extending E → Q p , and thus to an embedding W E → WE . We get from this an embedding L (G E ) → L (GE ). Via this embedding we can regard (r−μ , V−μ ) as a representation (r−μ , V−μ ) of L (G E ). Associated to (r−μ , V−μ ) ∈ Rep( L (G E )) we have an element Z V−μ in the Bernstein center Z(G E ). Of course here and in what follows, we are assuming LLC+ holds for G E . Now we review briefly the Langlands–Kottwitz approach to studying the local Hasse–Weil zeta functions of Shimura varieties. Let Sh K p = Sh(G, h −1 , K p K p ) denote the canonical model5 over E for the Shimura variety attached to the data (G, h −1 , K p K p ) for some sufficiently small compact p open subgroup K p ⊂ G(A f ) and some compact open subgroup K p ⊂ G(Q p ). We limit ourselves to constant coefficients Q in the generic fiber of Sh K p (here = p is a fixed rational prime). Let p denote any geometric Frobenius element in Gal(Q p /Q p ). Then in the Langlands–Kottwitz approach to the semi-simple local zeta function ζpss (s, Sh K p ), one needs to prove an identity of the form trss (rp , H•c (Sh K p ⊗E Q p , Q )) =

c(γ0 ; γ , δ) Oγ (1 K p ) TOδθ (φr ).

(γ0 ;γ ,δ)

(4.15) Here the semi-simple Lefschetz number Lefss (rp , Sh K p ) on the left-hand side is the alternating semi-simple trace of Frobenius on the compactly-supported 5 We use this term in the same sense as Kottwitz [Ko92a], comp. Milne [Mil, §1, esp. 1.10].

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-adic cohomology groups6 of Sh K p (see [Ra90] and [HN02a] for the notion of semi-simple trace). The expression on the right has precisely the same form as the counting points formula proved by Kottwitz in certain good reduction cases (PEL type A or C, K p hyperspecial; cf. [Ko92a, (19.6)]). The integer r ≥ 1 ranges over integers of the form j · [k E0 : F p ], j ≥ 1, where E 0 /Q p is the maximal unramified subextension of E/Q p and k E0 is its residue field. Thus j rp = p where p is a geometric Frobenius element in Gal(Q p /E). Finally, φr is an element in the Hecke algebra H(G(Q pr ), K pr ) with values in Q , where Q pr is the unique degree r unramified extension in Q p /Q p , and where K pr ⊂ G(Q pr ) is a suitable compact open subgroup which is assumed to be a natural analogue of K p ⊂ G(Q p ). To be more precise about K pr , in practice there is a smooth connected Z p -model G for G, such that K p = G(Z p ). In that case, we always take K pr = G(Z pr ), where Z pr is the ring of integers in Q pr . In forming TOδσ (φr ), the Haar measure on G(Q pr ) is normalized to give K pr measure 1. Let E j /E be the unique unramified extension of degree j in Q p /E. Let E j0 /Q p be the maximal unramified subextension of E j /Q p . So E/E 0 and E j /E j0 are totally ramified of the same degree, and E j 0 = Q pr . √ 1/2 We make the choice of p ∈ Q , and use it to define δ P as a function √ with values in Q( p) ⊂ Q . We can now specify the test function φr ∈ Z(G(E j0 ), K j 0 ), which will take values in Q . In the construction of the elements Z V ∈ Zst (G), everything works the same

) W F on a Q -vector space. way for (r, V ) a representation of L G := G(Q We henceforth take this point of view. Let (r−μ, j , V−μ, j ) ∈ Rep( L (G E j )) denote the restriction of (r−μ , V−μ ) ∈ Rep( L (G E )) via L (G E j ) → L (G E ). E

E

j0 j0 L We can then induce to get a representation (r−μ, j , V−μ, j ) of (G E j0 ). By Mackey theory, we get the same representation if we first induce to L (G E0 ) and then restrict to L (G E j0 ), that is, we have

E

E

GW E j0

j0 j0 (r−μ,

j , V−μ, j ) := IndGW

Ej

GW E0

E ResGW r−μ = ResGW

GW Ej

GW E

E j0

0 IndGW r−μ .

E

(4.16) This gives rise to Z

E j0

V−μ, j

∈

Zst (G

E j0 ). By abuse of notation, we use the same

symbol to denote the image of this in the Bernstein center: Z

E j0

V−μ, j

∈ Z(G E j 0 ).

6 The Langlands–Kottwitz method really applies to the middle intersection cohomology groups

of the Baily–Borel compactification and not just to the cohomology groups with compact supports; see [Ko90] and [Mor] for some general conjectures and results in this context, at primes of good reduction. The identity (4.15) corresponds to the contribution of the interior, at primes of arbitrary reduction, and is a first step toward understanding the intersection cohomology groups.

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Of course here we are viewing Z(G E j 0 ) as Q -valued regular functions on the Bernstein variety, or equivalently as Q -valued invariant essentially compact distributions: the topology on C playing no role, it is harmless to identify it with Q . The following is the conjecture formulated jointly with R. Kottwitz. Conjecture 4.30. (Test function conjecture) Let d = dim(Sh K p ). The test r d/2 function φr in (4.15) may be taken to be p Z E j 0 ∗ 1 K pr . In particular, V−μ, j

φr may be taken in the center Z(G(Q pr ), K pr ) of H(G(Q pr ), K pr ) and these test functions vary compatibly with change in the level K p in an obvious sense. The same test functions should be used when one incorporates arbitrary Hecke operators away from p into (4.15). Following Rapoport’s strategy (cf. [Ra90], [Ra05], [H05]), one seeks to find a natural integral model M K p over O E for Sh K p , and then rephrase the above conjecture using the method of nearby cycles R" := R" M K p (Q ). Conjecture 4.31. There exists a natural integral model M K p /O E for Sh K p , such that trss (rp , R"x ) = c(γ0 ; γ , δ) O(1 K p ) TOδθ (φr ), (4.17) x∈M K p (k E j0 )

where φr = pr d/2 Z

(γ0 ;γ ,δ)

E j0 V−μ, j

∗ 1 K pr

as in Conjecture 4.30.

Remark 4.32. Implicit in this conjecture is that the method of nearby cycles can be used for compactly-supported cohomology. In fact we could conjecture there exists a suitably nice compactification of M K p /O E so that the natural map Hic (M K p ⊗O E F p , R"(Q )) → Hic (Sh K p ⊗ E Q p , Q ) is a Galois-equivariant isomorphism. For G = GSp2g and where M K p is the natural integral model for Sh K p for K p an Iwahori subgroup, this was proved by Benoit Stroh. Of course, one is really interested in intersection cohomology groups of the Baily–Borel compactification (see footnote 5), and in fact Stroh [Str] computed the nearby cycles and verified the analogue of the Kottwitz conjecture on nearby cycles (see Conjecture 4.44 below) for these compactifications. Remark 4.33. Some unconditional versions of Conjectures 4.30 and 4.31 have been proved. See §8.

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6.2. Endoscopic transfer of the stable Bernstein center Part of the Langlands–Kottwitz approach is to perform a “pseudostabilization” of (4.15), and in particular prove the “fundamental lemmas” that are required for this. The stabilization expresses (4.15) in the form H i(G, H) STe∗ (h), the sum over global Q-elliptic endoscopic groups H for G of the (G, H)regular Q-elliptic part of the geometric side of the stable trace formula for (H, h) (cf. notation of [Ko90]), for a certain transfer function h ∈ Cc∞ (H(A)). (By contrast in “pseudostabilization” which is used in certain situations, one instead writes (4.15) in terms of the trace formula for G and not its quasisplit inner form, and this is sometimes enough, as in e.g. Theorem 4.40 below.) For stabilization one needs to produce elements h p ∈ Cc∞ (H(Q p )) which are Frobenius-twisted endoscopic transfers of φr . The existence of such transfers h p is due mainly to the work of Ngô [Ngo] and Waldspurger [Wal97, Wal04, Wal08]. But we hope to have a priori spectral information about the transferred functions h p . A guiding principle is that the nearby cycles on an appropriate “local model” for Sh K p should naturally produce a central element as a test function φr , which should coincide with that given by the test function conjecture (cf. Conjecture 4.31); then its spectral behavior is known by construction. In that case one can formulate a conjectural endoscopic transfer h p of φr with known spectral behavior. General Frobenius-twisted endoscopic transfer homomorphisms Zst (G Q pr ) → Zst (HQ p ) will be described elsewhere. Here for simplicity we content ourselves to describe two special cases: standard (untwisted) endoscopic transfer of the geometric Bernstein center, and the base change transfer for the stable Bernstein center. 6.2.1. Endoscopic transfer of the geometric Bernstein center Let us fix an endoscopic triple (H, s, η0 ) for G over a p-adic field F (cf. [Ko84a, §7]), and suppose we have fixed an extension η : L H → L G of

→ G

(we suppose we are in a situation, e.g. G der = G sc , where such η0 : H extensions always exist). We could hope the natural map Y H/F −→ YG/F (λ) H −→ (η ◦ λ)G

would be algebraic and hence would induce an endoscopic transfer homomorphism Zst (G) → Zst (H ). By invoking further expectations about endoscopic lifting, one would then formulate a map on the level of Bernstein centers, Z(G) → Z(H ), which we could write as Z → Z |η . But these assertions

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are not obvious. Fortunately, in practice we need this construction rather on the geometric Bernstein center. Definition 4.34. Assume LLC+ holds for G/F. We define the geometric Bernstein center Zgeom (G) to be the subalgebra of Zst (G) generated by the elements Z V as V ranges over Rep( L G). The terminology geometric Bernstein center is motivated by § 6.4 below. Let V |η ∈ Rep( L H ) denote the restriction of V ∈ Rep( L G) along η. Further assume LLC+ also holds for H/F. Then Z V → Z V |η determines a map Zgeom (G) → Zgeom (H ). Write Z VG (resp. Z VH|η ) for the image of Z V (resp. Z V |η ) in Z(G) (resp. Z(H )). Conjecture 4.35. Assume LLC+ holds for both G and H . Then in the above situation the distribution Z VH|η ∈ Z(H ) is the endoscopic transfer of Z VG ∈

Z(G) in the following sense: whenever a function φ H ∈ Cc∞ (H (F)) is a transfer of a function φ ∈ Cc∞ (G(F)), then Z VH|η ∗ φ H is a transfer of Z VG ∗ φ. This conjecture and its Frobenius-twisted analogue were announced by the author in April 2011 at Princeton [H11]. A very similar statement subsequently appeared as Conjecture 7.2 in [SS]. Considering the untwisted case for simplicity, the difference is that in [SS], the authors take in place of Z V an element in the stable Bernstein center essentially of the form (λ)G → tr(λ( F ), V−μ ), where here the usual trace, not the semi-simple trace, is used. That conjecture is proved in [SS] in all EL or quasi-EL cases, by invoking special features of general linear groups such as the existence of base change representations. Formally, Conjecture 4.35 contains as a special case the “fundamental lemma implies spherical transfer” result of Hales [Hal] (see also Waldspurger [Wal97]). Indeed if K , K H are hyperspecial maximal compact subgroups in G(F), H (F), then 1 K H is a transfer of 1 K by the fundamental lemma, and hence Z VH|η ∗ 1 K H is a transfer of Z VG ∗ 1 K . But by the Satake isomorphism, every K -spherical function on G(F) is a linear combination of functions of the form Z VG ∗ 1 K for some representation V (comp. §6.4). Even in more general situations, Conjecture 4.35 is most useful when applied to a pair φ, φ H of unit elements in appropriate Hecke algebras. At least when G splits over F un , Kazhdan–Varshavsky proved in [KV] that for some explicit scalar c, the Iwahori unit c1 I H is a transfer of the Iwahori unit 1 I . As another example, if K nG ⊂ G(F) is the n-th principal congruence subgroup in G(F), then for some explicit scalar c the function c1 K nH is a transfer

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of 1 K nG (proved by Ferrari [Fer] under some mild restrictions on the residue characteristic of F), and thus c(Z VH|η ∗ 1 K nH ) should be an explicit transfer of Z VG ∗ 1 K nG . A Frobenius-twisted analogue of Ferrari’s theorem together with the Frobenius-twisted analogue of Conjecture 4.35 would give an explicit Frobenius-twisted transfer of the test function φr from Conjecture 4.30, if K p is a principal congruence subgroup.

6.2.2. Base change of the stable Bernstein center We return to the situation of Proposition 4.21, but we specialize it to cyclic Galois extensions of F and furthermore we assume G/F is quasi-split. Let E/F be any finite cyclic Galois subextension of F/F with Galois group θ , and with corresponding inclusion of Weil groups W E → W F . If φ ∈ H(G(E)) and f ∈ H(G(F)) are functions in the corresponding Hecke algebras of locally constant compactly-supported functions, then we say φ, f are associated (or f is a base-change transfer of φ), if the following result holds for the stable (twisted) orbital integrals: for every semisimple element γ ∈ G(F), we have SOγ ( f ) = (γ , δ) SOδθ (φ) (4.18) δ

where the sum is over stable θ -conjugacy classes δ ∈ G(E) with semisimple norm N δ, and where (γ , δ) = 1 if N δ = γ and (γ , δ) = 0 otherwise. See e.g. [Ko86], [Ko88], [Cl90], or [H09] for further discussion. Conjecture 4.36. In the above situation, consider Z ∈ Zst (G E ), and consider its image, also denoted by Z , in Z(G E ). Consider b E/F (Z) ∈ Zst (G) (cf. Definition 4.22) and also denote by b E /F (Z ) its image in Z(G). Then b E/F (Z) is the base-change transfer of Z ∈ Z(G E ), in the following sense: whenever a function f ∈ Cc∞ (G(F)) is a base-change transfer of φ ∈ Cc∞ (G(E)), then b E/F (Z ) ∗ f is a base-change transfer of Z ∗ φ. Proposition 4.37. Conjecture 4.36 holds for GLn . Proof. The most efficient proof follows Scholze’s proof of Theorem C in [Sch2] which makes essential use of the existence of cyclic base change lifts for GLn . Let π ∈ (GLn /F) be a tempered irreducible representation with base change lift ∈ (GLn /E), a tempered representation which is characterized by the character identity % ((g, θ )) = %π (N g) for all elements g ∈ GLn (E) with regular semisimple norm N g ([AC, Thm. 6.2, p. 51]). Here (g, θ ) ∈ GLn (E) Gal(E/F) and θ acts on by the normalized intertwiner Iθ : → of [AC, p. 11].

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Suppose f is a base-change transfer of φ. Using the Weyl character formula and its twisted analogue (cf. [AC, p. 36]), we see that tr((φ, θ ) | ) = tr( f | π ). Multiplying by the constant Z () = b E/F (Z )(π ), we get tr((Z ∗ φ, θ ) | ) = tr(b E/F (Z) ∗ f | π ). (Use Corollary 4.3 and its twisted analogue.) There exists a base-change transfer h ∈ Cc∞ (GLn (F)) of Z ∗ φ ([AC, Prop. 3.1]). Using the same argument as above for the pair Z ∗φ and h, we conclude that tr(b E/F (Z)∗ f −h | π ) = 0 for every tempered irreducible π ∈ (GLn /F). By Kazhdan’s density theorem (Theorem 1 in [Kaz]) the regular semi-simple orbital integrals of b E/F (Z ) ∗ f and h agree. Thus the (twisted) orbitals integrals of b E /F (Z ) ∗ f and φ match at all regular semi-simple elements, and hence at all semi-simple elements by Clozel’s Shalika germ argument [Cl90, Prop. 7.2]. Remark 4.38. Unconditional versions of Conjecture 4.36 are available for parahoric and pro-p Iwahori–Hecke algebras, when G/F is unramified.7 See §9.

6.3. Application: local Hasse–Weil zeta functions By Kottwitz’ base change fundamental lemma for units [Ko86], we know 1 K p is a base-change transfer of 1 K pr whenever K p = G(Z p ) and K pr = G(Z pr ) for a smooth connected Z p -model G for G. Then Conjectures 4.30 and 4.36 together say that ( j)

fp

:= pr d/2 b E j 0 /Q p (Z

E j0

V−μ, j

) ∗ 1K p

(4.19)

is a base-change transfer of a test function φr that satisfies (4.15). Setting E

GW E

E

0 (r−μ0 , V−μ0 ) := IndGW r−μ

(4.20)

E

we have for any admissible parameter ϕ : WQ p → ϕ (G/Q p ) the identity ( j)

K

L (G

Qp )

E

and any π p ∈

tr( f p | π p ) = pr d/2 dim(π p p ) trss (ϕ(rp ) , V−μ0 ),

(4.21)

7 The pro-p Sylow subgroup of an Iwahori subgroup I ⊂ G(F) coincides with its pro-unipotent radical I + , and it has become conventional to term the Hecke algebra Cc∞ (I + \G(F)/I + ) the

pro- p Iwahori–Hecke algebra.

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where r = j[E 0 : Q p ]. In the compact and non-endoscopic cases, the above discussion allows us to express ζpss (s, Sh K p ) in terms of semi-simple automorphic L-functions. To explain this we need a detour on the point of view taken in [L1, L2] (comp. [Ko84b, §2.2]). Recall (r−μ , V−μ ) ∈ Rep( L (G E )). Consider the Langlands representation

GW

r := IndG WQ r−μ , E

and for each prime p of E dividing p, consider

GW Qp

rp := IndGW

Ep

GW Q

GW

p E ResGW r−μ = IndGW r−μ .

Ep

Mackey theory gives

W G

Q ResGW r=

Qp

Ep

#

rp .

p| p

If p is understood, let Ep0 /Q p denote the maximal unramified subextension of Ep /Q p , and set E = Ep and E 0 := Ep0 . Then we have

WQ G

WQ G

E

p p rp = IndGW r−μ = IndGW r−μ0 .

E

E0

(4.22)

Lemma 4.39. Suppose π p ∈ ϕ (G/Q p ). Then $ j E trss (ϕ(p ), r−μ0 ) if r = j [E 0 : Q p ] −1 ss r [E 0 : Q p ] tr (ϕ( p ), rp ) = 0, if [E 0 : Q p ] |r . (4.23)

WQ p -modules Proof. There is an isomorphism of G E0

WQ p ] ⊗C[G W ] r−μ rp ∼ , = C[G E 0

and

ϕ(IQ ) rp p

has a C-basis of the form {ϕ(ip ) ⊗ wk } where 0 ≤ i ≤ [E 0 :

Q p ]−1 and {wk } comprises a C-basis for (r−μ0 )ϕ(IQ p ) . The lemma follows. E

The following result shows the potential utility of Conjectures 4.30 and 4.36. It applies not just to PEL Shimura varieties, but to any Shimura variety where these conjectures are known. Similar results will hold when incorporating Hecke operators away from p. Theorem 4.40. Suppose Gder is anisotropic over Q, so that the associated Shimura variety Sh K p = Sh(G, h −1 , K p K p ) is proper over E. Suppose G has “no endoscopy”, in the sense that the group K(Gγ0 /Q) is trivial for every semisimple element γ0 ∈ G(Q), as in e.g. [Ko92b]. Let p be a prime ideal of E

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dividing p. Assume (LLC+) (cf. §5.2), and Conjectures 4.30 and 4.36 hold for all groups GQ pr . Then in the notation above, we have K d ζpss (s, Sh K p ) = L ss (s − , π p , rp )a(π f ) dim(π f ) , (4.24) 2 π f

πp

where π f = ⊗ π p runs over irreducible admissible representations of G(A f ) and the integer a(π f ) is given by a(π f ) = m(π f ⊗ π∞ ) tr( f ∞ |π∞ ), π∞ ∈∞

where m(π f ⊗π∞ ) is the multiplicity of π f ⊗ π∞ in L2 (G(Q)AG (R)◦ \G(A)). Here AG is the Q-split component of the center of G (which we assume is also its R-split component). Further ∞ is the set of irreducible admissible representations of G(R) which have trivial infinitesimal and central characters, and f ∞ is defined as in [Ko92b] to be (−1)dim(Sh K ) times a pseudo-coefficent 0 ∈ . of an essentially discrete series member π∞ ∞ Proof. The method follows closely the argument of Kottwitz in [Ko92b] (comp. [HRa1, §13.4]), so we just give an outline. We will use freely the notation of Kottwitz and [HRa1]. Set f = [Ep0 : Q p ]. By definition we have log ζpss (s, Sh K p ) =

∞

j

Lefss (p , Sh K p )

j =1

p− j f s . j

(4.25)

By using (4.15) together with Conjectures 4.30 and 4.36, the arguments of Kottwitz [Ko92b] show that for each j ≥ 1 j ( j) Lefss (p , Sh K p ) = τ (G) SOγ0 ( f p f p f ∞ ), (4.26) γ0

( j)

where f p is defined as in (4.19) and f p is the characteristic function of K p ⊂ p G(A f ). Here γ0 ranges over all stable conjugacy classes in G(Q). Since Gder is anisotropic over Q, the trace formula for any f ∈ Cc∞ (AG (R)◦ \G(A)) takes the simple form τ (Gγ )Oγ ( f ) = m(π ) tr( f |π ), (4.27) γ

π

where γ ranges over conjugacy classes in G(Q) and π ranges over irreducible representations in L2 (G(Q)AG (R)◦ \G(A)). By [Ko92b, Lemma 4.1], the vanishing of all K(Gγ0 /Q) means that

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τ (Gγ ) Oγ ( f ) = τ (G)

γ

SOγ0 ( f ).

γ0

It follows that j

Lefss (p , Sh K p ) ( j) = m(π ) tr( f p f p f ∞ |π ) π

=

π f π∞ ∈∞

=

πf

( j)

p

m(π f ⊗ π∞ ) · tr( f p |π f ) · tr( f p |π p ) · tr( f ∞ |π∞ ) j

E

a(π f ) dim(π Kf ) p j f d/2 trss (ϕπ p (p ), V−μ0 ),

the last equality by (4.21). By definition we have log L ss (s, π p , rp ) =

∞

trss (ϕπ p (rp ), rp )

r =1

p −r s . r

Now (4.24) follows by invoking (4.23). Remark 4.41. Unconditional versions of Theorem 4.40 are available for some parahoric or pro-p-Iwahori level cases, or for certain compact “Drinfeld case” Shimura varieties with arbitrary level; these cases are alluded to in §8.

6.4. Relation with geometric Langlands For simplicity, assume G is split over a p-adic or local function field F. Assume G satisfies LLC+. From the construction of Z V in Proposition 4.28, we have a map

→ Z(G, J ) K 0 RepC (G)

(4.28)

V → Z V ∗ 1 J for any compact open subgroup J ⊂ G(F), which gives rise to a commutative diagram Z(G, J) 7 −∗ J 1 I

Z(G, I) 5 k k k Bernkkkk −∗ I 1 K kkk∼ kkk Sat / H(G, K )

K 0 Rep(G) ∼

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whenever J ⊆ I ⊂ K where I resp. K is an Iwahori resp. special maximal compact subgroup, and where the bottom two arrows are the Bernstein resp. Satake isomorphisms. We warn the reader that the oblique arrow

→ Z(G, J ) is injective but not surjective in general, and also it is K 0 Rep(G) additive but not an algebra homomorphism in general. Gaitsgory [Ga] constructed the two arrows Sat and Bern geometrically when F is a function field, using nearby cycles for a degeneration of the affine Grassmannian GrG to the affine flag variety Fl for G. One can hope that, as in the

→ Z(G, J ) catIwahori case [Ga], one can construct the arrow K 0 Rep(G) egorically using nearby cycles for a similar degeneration of GrG to a “partial affine flag variety”, namely an fpqc-quotient LJ/L + J where J is a smooth connected group scheme over F p [[t]] with generic fiber JF p ((t)) = G F p ((t)) and J(F p [[t]]) = J . Here LJ (resp. L + J) is the ind-scheme (resp. scheme) over F p representing the sheaf of groups for the fpqc-topology whose sections for an F p -algebra R are given by LJ(Spec R) = J(R[[t]][ 1t ]) (resp. L + J(Spec R) = J(R[[t]])). At least for J = I + , the pro-p Iwahori subgroup, this will be realized in forthcoming joint work of the author and Benoit Stroh. In a related vein, the geometric Satake equivalence of Mirkovic–Vilonen [MV] is a categorical version of the Satake isomorphism Sat, and this is usually stated when G is a split group over F = F p ((t)). One can ask for a version of this when G is nonsplit, possibly not even quasi-split, over such a field F. The correct Satake isomorphism to “categorify” appears to be the one described in [HRo]. In many cases where G is quasi-split and split over a tamely ramified extension of F, this has been carried out in recent work of X. Zhu [Zhu].

7. Test functions in the parahoric case We fix r = j[E 0 : Q p ] for some j ∈ N. We assume K p is a parahoric subgroup of G(Q p ), and we let K pr denote the corresponding parahoric subgroup of G(Q pr ). Assuming LLC+ holds for G Q pr , we can speak of the test function φr = pr d/2 Z E j 0 ∗ 1 K pr ∈ Z(G(Q pr ), K pr ). (4.29) V−μ, j

We wish to give a more concrete description of this function, making use of Bernstein’s isomorphism for Z(G(Q pr ), K pr ) which is detailed in the Appendix, §11. In the next two subsections, we are concerned with the case where G Q pr is quasi-split. We write F := Q pr . Choose a maximal F-split torus A in G, and let T denote its centralizer in G. Fix an F-rational Borel subgroup B containing T . Let K F ⊂ G(F) denote the parahoric subgroup corresponding to K p .

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By Kottwitz [Ko84b, Lemma (1.1.3)], the G(Q p )-conjugacy class {μ} is represented by an F-rational cocharacter μ ∈ X ∗ (T ) F = X ∗ (A). It is clear that E, the field of definition of {μ}, is contained in any subfield of Q p which splits G. Given π ∈ (G/F) with π K F = 0, to understand (4.29) we need to compute the scalar tr(ϕπ ( F ), (V−μ0 )ϕπ (I F ) ). E

(4.30)

There is an unramified character χ of T (F) such that π is a subquotient of i BG (χ ), and we may assume ϕπ |W F = ϕχ |W F . Since χ is unramified, ϕχ (I F ) =

W F . Regarding χ as an element of T

, (4.7) implies that we may 1 IF ⊂ T

write ϕχ ( F ) = χ F ∈ T W F . Then we need to compute E

tr(χ F , (V−μ0 )1I F ).

(4.31)

7.1. Unramified groups and the Kottwitz conjecture Let us consider the case where G Q pr is unramified. Since we are assuming G splits over an unramified extension of Q p , it follows that E/Q p is unramified, E i.e. E = E 0 and V−μ0 = V−μ . Moreover F = E j0 contains E with degree j.

1I F Further, since G splits over F un , we have V−μ = V−μ . So we are reduced to computing tr(χ F , V−μ ). Exactly as in Kottwitz’ calculation of the Satake transform in [Ko84b, p. 295], we see that (4.30) is tr(χ F , V−μ ) = (−λ)(χ ). (4.32) λ∈W (F)·μ

Here W (F) = W (G, A) is the relative Weyl group for G/F, and we view

. This proves the following result. λ ∈ X ∗ (A) = X ∗ (T ) F as a character on T Lemma 4.42. In the above situation, Z

E j0

V−μ, j

∗ 1 K pr = z −μ, j ,

(4.33)

where the Bernstein function z −μ, j (cf. Definition 4.72) is the unique element of Z(G(F), K F ) which acts (on the left) on the normalized induced representation i BG (χ ) K F by the scalar λ∈W (F)·μ (−λ)(χ ), for any unramified character χ : T (F) → C× . Of course the advantage of z −μ, j is that unlike the left-hand side of (4.33), it is defined unconditionally. A relatively self-contained, elementary, and efficient approach to Bernstein functions is given in §11. Thus Conjecture 4.30 in this situation is equivalent to the Kottwitz Conjecture.

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Conjecture 4.43. (Kottwitz conjecture) In the situation where GQ pr is unramified and K p is a parahoric subgroup, the function φr in (4.15) may be taken to be pr d/2 z −μ, j . Conjecture 4.43 was formulated by Kottwitz in 1998, about 11 years earlier than Conjecture 4.30. There is a closely related conjecture of Kottwitz concerning nearby cycles on Rapoport–Zink local models Mloc K p for Sh K p . We refer to [RZ, Ra05] for definitions of local models, and to [H05, HN02a] for further details about the following conjecture in various special cases. Conjecture 4.44. (Kottwitz Conjecture for Nearby Cycles) Write G for the Bruhat–Tits parahoric group scheme over Z pr with generic fiber G Q pr and with G(Z pr ) = K pr . Let Gt denote the analogous parahoric group scheme over F pr [[t]] with the “same” special fiber as G. Then there is an L + Gt,F pr -equivariant embedding of Mloc K p ,F r into the affine flag variety p

LGt,F pr /L + Gt,F pr , via which we can identify the semisimple trace of FrobeMloc Kp

nius function x → trss (Fr pr , R"x

r ) on x ∈ Mloc K p (F p ) with the function

p dr/2 z −μ, j ∈ Z(Gt (F pr ((t))) , Gt (F pr [[t]])).

7.2. The quasi-split case

I F is a possibly disconnected reductive group, with maximal torus The group G I ◦

F ) (see the proof of Theorem 8.2 of [St]). Now we may restrict the rep(T E

IF W F ⊂ G

W F . Let χ be a weakly resentation V−μ0 to the subgroup G

I F ) F . The only unramified character of T (F); by (4.5) we can view χ ∈ (T E 0 1I F I F

T -weight spaces of (V−μ ) which contribute to (4.31) are indexed by the

I F ) F . (It is important to note that it is F -fixed weights, i.e. by those in X ∗ (T

I F , and not for the maximal the weight spaces for the diagonalizable group T I ◦ F

torus (T ) , which come in here.) This is consistent with Theorem 4.71 of the Appendix, and may be expressed as follows. Proposition 4.45. In the general quasi-split situation, Z

E j0

V−μ, j

∗ 1 K pr is the

unique function in Z(G(Q pr ), K pr ) which acts on the left on each weakly unramified principal series representation i BG (χ ) K F by the scalar (4.31), and thus is a certain linear combination of Bernstein functions z −λ, j where −λ ∈ E0

I F ) F ranges over the W (G, A)-orbits of F -fixed T

I F -weights in V−μ X ∗ (T . It is an interesting exercise to write out the linear combinations of Bernstein functions explicitly in each given case. Once this is done, the result can be used to find explicit descriptions of test functions for inner forms of quasi-split groups. This is the subject of the next subsection.

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7.3. Passing from quasi-split to general cases via transfer homomorphisms 7.3.1. Test function conjecture via transfer homomorphisms We use freely the notation and set-up explained in the Appendix §11.12. Let G ∗ be a quasi-split F-group with an inner twisting ψ : G → G ∗ . Let J ∗ ⊂ G ∗ (F) resp. J ⊂ G(F) be parahoric subgroups and consider the normalized transfer homomorphism t˜ : Z(G ∗ (F), J ∗ ) → Z(G(F), J ) from Definition 4.78. The following conjecture indicates that test functions for the quasi-split group G ∗ should determine test functions for G. This is compatible with the global considerations which led to Theorem 4.40. Conjecture 4.46. Let K pr resp. K ∗pr be parahoric subgroups of G(Q pr ) resp. G ∗ (Q pr ), with corresponding normalized transfer homomorphism t˜ : Z(G ∗ (Q pr ), K ∗pr ) → Z(G(Q pr ), K pr ). If φr∗ ∈ Z(G ∗ (Q pr ), K ∗pr ) is the ∗ function pr d/2 (Z G E j 0 ∗ 1 K ∗r ) described in Proposition 4.45 for the data V−μ, j (G ∗Q r , {−μ}, K ∗pr ), then p

p

φr := t˜(φr∗ ) ∈ Z(G(Q pr ), K pr ) is a test function satisfying (4.15) for the original data (G Q pr , {−μ}, K pr ). Assuming Conjecture 4.30 holds, another way to formulate this is that the ∗ normalized transfer homomorphism t˜ takes the function Z G E j 0 ∗ 1 K ∗r ∈ Z(G ∗ (Q

pr ),

K ∗pr )

to the function Z

V−μ , j

G E j0

V−μ, j

p

∗ 1 K pr ∈ Z(G(Q pr ), K pr ). But

the point of Conjecture 4.46 is to provide an explicit test function for the non-quasi-split data (G Q pr , {−μ}, K pr ) which can be compared with direct geometric calculations of the nearby cycles attached to this data, and thus to provide a method to prove an unconditional analogue of Conjecture 4.30 for such data. This is illustrated in §7.3.3 below. The next two paragraphs show that Conjecture 4.46 is indeed reasonable. 7.3.2. A calculation for GL2 Take = GL2,F and G = D × , where D is the central simple division algebra over F of dimension 4. Here we will explicitly calculate and compare the test functions associated to (GL2,F , {−μ}, I F ) and (D × , {−μ}, O× D ), where μ = (1, 0), and where I F ⊂ × × GL2 (F) and O D ⊂ D are the standard Iwahori subgroups. This calculation will show that the normalized transfer homomorphism takes one test function to the other. This is required in order for both Conjectures 4.30 and 4.46 to hold true. G∗

Stable Bernstein center and test functions

157 ×

∗ Write z −μ = Z V−μ2 ∗ 1 I F ∈ Z(GL2 (F), I F ) and z −μ = Z VD−μ ∗ 1O× ∈ D H(D × , O× ) ∼ = C[Z]. The last isomorphism is induced by the Kottwitz homoGL

D

morphism, which in this case is the normalized valuation valF ◦ Nrd D : D × Z, where val F is the normalized valuation for F and Nrd D : D → F is the reduced norm. Write μ¯ = (0, 1) and let B ∗ denote the Borel subgroup of lower triangular ∗ acts on the left on i GL2 (χ ) I F by the scalar matrices in GL2 . Then z −μ B∗ tr(χ F , V−μ ) = (−μ − μ)(χ ¯ ), for any unramified character χ ∈ Hom(T ∗ (F)/T ∗ (F)1 , C× ). We may view χ as a diagonal 2 × 2 complex matrix χ = diag(χ1 , χ2 ). To calculate z −μ we need a few preliminary remarks. First we parametrize × unramified characters η ∈ Hom(D × /O× D , C ) by writing η = η0 ◦ Nrd D , where η0 ∈ C× is viewed as the unramified character on F × which sends $ F → η0 . The map Nrd D : D × → F × is Langlands dual to the diagonal embedding Gm (C) → GL2 (C), and it follows that the cocycles z η and z η0 attached by Langlands duality to the quasicharacters η and η0 satisfy z η0 0 η0 0 zη = , and thus, z η ( F ) = . 0 z η0 0 η0 On the other hand, if 1 denotes the trivial 1-dimensional representation of D × , × × then its Langlands parameter ϕ1D satisfies ϕ1D ( F ) = diag(q −1/2 , q 1/2 ) (see, e.g., [PrRa, Thm. 4.4]). So using (4.8), we obtain ×

tr(ϕηD ( F ), V−μ ) η q −1/2 0 0 = tr , V −μ 0 η0 q 1/2 =

−1/2 (δ B ∗ ($ −μ ) · −μ|

Z

+

−1/2 δ B ∗ ($ −μ¯ ) · −μ| ¯

Z)

η

0

0

0 . η0

∗ . Using the definition of t˜ we deduce the following Here

Z is the center of G result. Proposition 4.47. The normalized transfer homomorphism t˜ : Z(GL2 (F), ∗ I F ) → H(D × , O× D ) sends z −μ to z −μ . 7.3.3. Compatibility with nearby cycles in some anisotropic cases Suppose we are in a situation where E = Q p . As before, write F = Q pr . Suppose G F = (D ⊗ F)× × Gm , where D is a central division algebra over E of degree n 2 , for n > 2. This situation arises in the setting of “fake unitary”

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simple Shimura varieties (see, e.g. [H01, §5]). Let G ∗ = GLn × Gm , a split inner form of G over Q p . ∗ = Suppose that V−μ = ∧m (Cn ) for 0 < m < n, i.e. the representation of G × GLn (C) × C where the first factor acts via the irreducible representation with highest weight (1m , 0n−m ) and the second factor acts via scalars. Consider the local models M∗loc = Mloc (G ∗ , {−μ}, K ∗p ) and Mloc = Mloc (G, {−μ}, K p ), where K ∗p ⊂ G ∗ (F) and K p ⊂ G(F) are Iwahori subgroups. We can choose the inner twist G → G ∗ and the subgroups K ∗p and K p so that M∗loc (F p ) = Mloc (F p ) and where the action of geometric Frobenius p on the right-hand side is given by p = Ad(c p ) · ∗p where ∗p is the usual Frobenius action (on the lefthand side) and where τ → Ad(cτ ) represents the class in H 1 (Q p , PGLn ) corresponding to the inner twist G → G ∗ . Assume (r, n) = 1 and set q = pr . Then Mloc (G, {−μ}, K p )(Fq ) consists of a single point. To understand the corresponding test function we may ignore the Gm -factor and pretend that G = D × and G ∗ = GLn . Then the Kottwitz homomorphism κG : G(F) → Z induces an isomorphism H(G(Q pr ), K pr ) ∼ = C[Z]. The test function for the Shimura variety giving rise to the local Shimura data (G, {−μ}, K p ) should be calculated by understanding the function trace of Frobenius on nearby cycles on Mloc , similarly to Conjecture 4.44 in the unramified case. The test function should be of the form Cq · 1m ∈ C[Z] = H(G(Q p ), K pr ) for some scalar Cq . Proposition 4.48. In the above situation, Conjecture 4.46 predicts that the coefficient Cq is given by Cq = #Gr(m, n)(Fq ), the number of Fq -rational points on the Grassmannian variety Gr(m, n) parametrizing m-planes in n-space. This is compatible with calculations of Rapoport of the trace of Frobenius on nearby cycles of the local models for such situations, see [Ra90]. Thus the normalized transfer homomorphism gives a group-theoretic framework with which we could make further predictions about nearby cycles on the local models attached to non-quasiplit groups G, assuming we know explicitly the corresponding test function for a quasi-split inner form of G. Proof. By the final sentence of Proposition 4.79, we simply need to inte∗ grate the function pr d/2 z −μ ∈ Z(GLn (Q pr ), K ∗pr ) over the fiber of the

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159

Kottwitz homomorphism val ◦ det over 1m ∈ C[Z]. This is a combinatorial ∗ explicitly. However, it problem which could be solved since we know pr d/2 z −μ is easier to use geometry. Translating “integration over the fiber of the Kottwitz homomorphism” in terms of local models gives us the equality ∗loc Cq = Tr(rp , R" M (Q )x ). x∈M∗loc (Fq )

(Here is a rational prime with = p.) But the special fiber of M∗loc embeds into the affine flag variety FlGLn for GLn /F p , and under the projection p : FlGLn → GrGLn to the affine Grassmannian, M∗loc maps onto Gr(m, n) and ∗loc Rp∗ (R" M (Q )) = Q , the constant -adic sheaf on Gr(m, n) in degree 0. Thus we obtain Cq = Tr(rp , (Q )x ) = #Gr(m, n)(Fq ) x∈Gr(m,n)(Fq )

as desired. (The reader should note the similarity with Prop. 3.17 in [Ra90], which is justified in a slightly different way.)

8. Overview of evidence for the test function conjecture 8.1. Good reduction cases In case GQ p is unramified and K p is a hyperspecial maximal compact subgroup of G(Q p ), we expect Sh(G, h −1 , K p K p ) to have good reduction over OEp . In PEL cases this was proved by Kottwitz [Ko92a]. In the same paper for PEL cases of type A or C, it is proved that the function φr = 1 K pr μ( p−1 )K pr satisfies (4.15), which can be viewed as verifying Conjecture 4.30 for these cases.

8.2. Parahoric cases Assume K p is a parahoric subgroup. We will discuss only PEL Shimura varieties. Here the approach is via the Rapoport–Zink local model Mloc K p for a suitable integral model M K p for Sh K p and the main ideas are due to Rapoport. We refer to the survey articles [Ra90], [Ra05], and [H05] for more about how local models fit in with the Langlands–Kottwitz approach. For much more about the geometry of local models, we refer the reader to the survey article of Pappas– Rapoport–Smithing [PRS] and the references therein.

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Using local models, the first step to proving Conjecture 4.31 is to prove Conjecture 4.44. The first evidence was purely computational: in [H01], z −μ, j was computed explicitly in the Drinfeld case and the result was compared with Rapoport’s computation of the nearby cycles in that setting, proving Conjecture 7.1.3 directly. This result motivated Kottwitz’ more general conjecture and also inspired Beilinson and Gaitsgory to construct the center of an affine Hecke algebra via a nearby cycles construction, a feat carried out in [Ga]. Then in [HN02a] Gaitsgory’s method was adapted to prove Conjecture 4.44 for the split groups GLn and GSp2n . This in turn was used to demonstrate Conjecture 4.43 for certain special Shimura varieties in [H05], and then to prove the analogue of Theorem 4.40 for those special Shimura varieties with parahoric level structure at p. The harmonic analysis ingredient needed for the latter was provided by [H09]. In his 2011 PhD thesis, Sean Rostami proved Conjecture 4.44 when G is an unramified unitary group. In a recent breakthrough, Pappas and Zhu defined group-theoretic local models Mloc K p whenever G splits over a tamely ramified extension, and for unramified groups G proved Conjecture 4.44, see [PZ], esp. Theorem 10.16.

8.3. Deeper level cases We again limit our discussion to PEL situations, where progress to date has occurred. It is again natural to study directly the nearby cycles relative to a suitable integral model for the Shimura variety and hope that it gives rise to a test distribution in the Bernstein center. For Shimura varieties in the “Drinfeld case” with K p a pro-p Iwahori subgroup of G(Q p ) = GLn (Q p ) × Q× p (“1 ( p)-level structure at p”), one may use Oort–Tate theory to define suitable integral models and prove Conjectures 4.31 and 4.30 for them. This was done by the author and Rapoport [HRa1] (and [H12] provided the harmonic analysis ingredient needed to go further and prove Theorem 4.40 in this case). Around the same time as [HRa1], Scholze studied in [Sch1] nearby cycles on suitable integral models for the modular curves with arbitrarily deep full level structure at p. In this way he proved Conjectures 4.31 and 4.30 in these cases, taking the compactifications also into account, and thereby proved the analogue of Theorem 4.40 for the compactified modular curves at nearly all primes of bad reduction. The nearby cycles on his integral models naturally gave rise to some remarkable distributions in the Bernstein center, for which he gave explicit formulas (see § 10).

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161

Then in [Sch2] Scholze generalized the approach of [Sch1] to compact Shimura varieties in the Drinfeld case, again finding an explicit description of nearby cycles. In this case, he was still able to produce a test function to plug into (4.15), or rather, simultaneously incorporating the base-change transfer results he needed in precisely this case, he found a test function that goes directly into the pseudostabilization of (4.15). This allowed him to prove Theorem 4.40. In contrast to the modular curve situation, in higher rank the nearby cycles on Scholze’s integral models do not directly produce distributions in the Bernstein center, and an explicit description of his test functions seems hopeless. But nevertheless Scholze was able to prove by indirect means Conjecture 4.30 in this case. The description of the nearby cycles in [Sch2] provided one ingredient for Scholze’s subsequent paper [Sch3] which gave a new and streamlined proof of the local Langlands conjecture for general linear groups. In later work Scholze [Sch4] formalized his method of producing test functions in many cases, using deformation spaces of p-divisible groups, and this is used to give a nearly complete description of the cohomology groups of many compact unitary Shimura varieties in his joint work [SS] with S. W. Shin; their main assumption at p is that GQ p is a product of Weil restrictions of general linear groups. The advantage of what we could call the Langlands–Kottwitz– Scholze approach in this situation is that it yields in [SS] a new construction of the Galois representations constructed earlier by Shin [Sh], in a shorter way that avoids Igusa varieties. In these later developments, Conjecture 4.30 does not play a central part, but the stable Bernstein center does nevertheless still play a clarifying role in the pseudostabilization process (e.g. in [SS]). It seems that only certain integral models, such as those we see in many parahoric or pro-p Iwahori level cases, have the favorable property that their nearby cycles naturally give rise to distributions in the Bernstein center. It remains an interesting problem to find such integral models in more cases, and to better understand the role of the Bernstein center in the study of Shimura varieties.

9. Evidence for conjectures on transfer of the Bernstein center Here we present some evidence for the general principle that the (stable/geometric) Bernstein center is particularly well-behaved with respect to (twisted) endoscopic transfer. The primary evidence thus far consists of some unconditional analogues of Conjecture 4.36.

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Let G/F be an unramified group, and let Fr /F be the degree r unramified extension of F in some algebraic closure of F. In [H09, H12], the author defined base change homomorphisms br : Z(G(Fr ), Jr ) → Z(G(F), J ), where J ⊂ G(F) is either a parahoric subgroup or a pro-p Iwahori subgroup, and where Jr is the corresponding subgroup of G(Fr ). Then we have “basechange fundamental lemmas” of the following form.8 Theorem 4.49. For any φr ∈ Z(G(Fr ), Jr ), the function br (φr ) is a basechange transfer of φr in the sense of (4.18). By Kottwitz [Ko86], the function 1 J is a base-change transfer of 1 Jr . Hence for any Vr ∈ Rep( L (G Fr )), Conjecture 4.36 predicts that b Fr /F (Z Vr ) ∗ 1 J is a base-change transfer of Z Vr ∗ 1 Jr . This is a consequence of Theorem 4.49, because of the following compatibility between the base-change operations in [H09, H12] and in the context of stable Bernstein centers (cf. Prop. 4.21). Lemma 4.50. In the above situations, br (Z Vr ∗ 1 Jr ) = b Fr /F (Z Vr ) ∗ 1 J . Proof. First assume J is a parahoric subgroup. Let χ be any unramified character of T (F). It is enough to show that the two functions act on the left by the same scalar on every unramified principal series representation i BG (χ ) J . Let Nr : T (Fr ) → T (F) be the norm homomorphism. By the definition of br in [H09], br (Z Vr ∗ 1 Jr ) acts by the scalar by which Z Vr ∗ 1 Jr acts on i BGrr (χ ◦ Nr ) Jr . This is the scalar by which Z Vr acts on i BGrr (χ ◦ Nr ), which in view of LLC+ is Tr trss (ϕχ◦N (rF ), Vr ) = trss (ϕχT (rF ), Vr ). r

(4.34)

But the right-hand side is the scalar by which b Fr /F (Z Vr ) ∗ 1 J acts on i BG (χ ) J . Tr The equality ϕχ◦N (rF ) = ϕχT (rF ) we used in (4.34) follows from the r commutativity of the diagram of Langlands dualities for tori Homconts (T (F), C× ) Nr

Homconts (T (Fr ), C× )

/ H 1 (W F , L T ) conts Res

/ H 1 (W F , L Tr ) conts r

which was proved in [KV, Lemma 8.1.3]. 8 Relating to pro-p Iwahori level, a much stronger result is proved in [H12] concerning the

base-change transfer of Bernstein centers of Bernstein blocks for depth-zero principal series representations.

Stable Bernstein center and test functions

163

Now suppose J = I + is a pro-p Iwahori subgroup. Then the same argument works given the following fact: for any depth-zero character χ : T (F)1 → C× and any extension of it to a character χ˜ on T (F), and any zr ∈ Z(G(Fr ), Ir+ ), + the function br (z r ) acts on i BG (χ) ˜ I by the same scalar by which zr acts on + i BGrr (χ˜ ◦ Nr ) Ir . This follows from the definition of br given in Definition 10.0.3 of [H12], using [H12, Lemma 4.2.1]. Let us also mention again Scholze’s Theorem C in [Sch2], which essentially proves Conjecture 4.36 for GLn (see Proposition 4.37).

10. Explicit computation of the test functions 10.1. Parahoric cases Conjecture 4.30 implies that test functions are compatible with change of level. Therefore for the purposes of computing them for parahoric level, the key case is where K p is an Iwahori subgroup. Thus, for the rest of this subsection we consider only Iwahori level structure. Since test functions attached to quasisplit groups should determine, in an computable way, those for inner forms (by Conjecture 4.46 and Proposition 4.79), it is enough to understand quasi-split groups. Via Proposition 4.45 this boils down to giving explicit descriptions of the Bernstein functions z −λ, j , assuming we have already expressed the test function explicitly in term of these – this is automatic for unramified groups using the Kottwitz Conjecture (Conjecture 4.43). Let us therefore consider the problem of explicitly computing Bernstein functions z μ attached to any group G/F and an Iwahori subgroup I ⊂ G(F) (F being any local nonarchimedean field). For simplicity consider the case where G/F is unramified, and regard μ as a dominant coweight in X ∗ (A). The μ which arise in Conjecture 4.43 are minuscule; however, we consider μ denote the extended affine which are not necessarily minuscule here. Let W Weyl group of G over F (cf. §11). Attached to μ is an the μ-admissible set , defined by Adm(μ) ⊂ W | x ≤ tλ , for some λ ∈ W (G, A) · μ}, Adm(μ) = {x ∈ W determined by the Iwahori subwhere ≤ denotes the Bruhat order on W corresponding group I and where tλ denotes the translation element in W to λ ∈ X ∗ (A). The μ-admissible set has been studied for its relation to the stratification by Iwahori-orbits in the local model Mloc K p ; for much information see [KR], [HN02b], [Ra05]. The strongest combinatorial results relating local models and Adm(μ) are due to Brian Smithling, see e.g. [Sm1, Sm2, Sm3].

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For our purposes, the set Adm(μ) enters because it is the set indexing the double cosets in the support of z μ . 0 Proposition 4.51. The support of z μ is the union x∈Adm(μ) I x I . Proof. This was proved using the theory of alcove walks as elaborated by Görtz [G], in the Appendix to [HRa1]. It applies to affine Hecke algebras with arbitrary parameters, hence the corresponding result holds for arbitrary groups, not just unramified groups. The following explicit formula was proved in [H01] and in [HP]. Let Tx = . In the formulas here and below, q = pr is the cardinality of 1 I x I for x ∈ W the residue field of F. Proposition 4.52. Assume μ is minuscule. Assume the parameters for the Iwahori Hecke algebra are all equal. Then q (tμ )/2 z μ = (−1)(tμ ) (−1)(x) R x,tλ(x) (q) Tx , x∈Adm(μ)

and where Rx,y (q) where x decomposes as x = tλ(x) w ∈ X ∗ (A) W = W denotes the R-polynomial of Kazhdan–Lusztig [KL], and the length function, . A similar formula holds in the context of affine for the quasi-Coxeter group W Hecke algebras with arbitrary parameters. In the Drinfeld case G = GLn and μ = (1, 0n−1 ), the coefficient of Tx is (1 − q)(tμ )−(x) . There are also explicit formulas for Bernstein functions z μ when μ is not minuscule, but they tend to be much more complicated. For related computations see [HP] and [GH].

10.2. A Pro-p Iwahori level case In the Drinfeld case where GQ p = GLn × Gm and μ = (1, 0n−1 ) × 1, and where K p is a pro-p Iwahori subgroup, an explicit formula for the test function φr for Sh(G, h −1 , K p K p ) was found by the author and Rapoport. We shall rephrase this slightly by ignoring the Gm factor and giving the formula for G = GLn . Proposition 4.53 ([HRa1], Prop. 12.2.1). Let q = pr . Let Ir+ denote the standard pro-p Iwahori subgroup of G r := GLn (Q pr ). Let T denote the standard diagonal torus in GLn . In terms of natural embeddings T (Fq ) → G r , and → G r giving elements tw−1 ∈ G r representing Ir+ \G r /Ir+ , we have w∈W

Stable Bernstein center and test functions

φr (Ir+ tw−1 Ir+ ) =

⎧ ⎪ ⎪ ⎨0,

165

if w ∈ / Adm(μ)

if w ∈ Adm(μ) but Nr (t) ∈ / TS(w) (F p ) ⎪ ⎪ ⎩(−1)n ( p − 1)n−|S(w)| (1 − q)|S(w)|−n−1 , otherwise. 0,

(4.35) Here S(w) is the set of critical indices for w, equivalently S(w) is the set of standard basis vectors e j ∈ Zn such that w ≤ te j in the Bruhat order on W determined by the standard Iwahori subgroup of GLn .

10.3. Deeper level structures Here the known explicit descriptions pertain only to G = GL2 and first were proved by Scholze [Sch1]. It remains an interesting question whether one can find explicit descriptions of test functions in higher rank groups with arbitrary level structure: even the Drinfeld case G = GLn , μ = (1, 0n−1 ) looks difficult, cf. [Sch2]. To state Scholze’s result, we need some notation. As usual let F be a nonarchimedean local field with ring of integers O, uniformizer $ , and residue field cardinality q. Let B denote the O-subalgebra M2 (O) of M2 (F). For any j ∈ Z set B j = $ j B. Let K = B × , the standard maximal compact subgroup of G = GL2 (F). For n ≥ 1, let K n = 1 + Bn ; so K n is a principal congruence subgroup and is a normal subgroup of K . Scholze defines a (compatible) family of functions φn ∈ H(G, K n ) for n ≥ 1. His definition uses two functions, : G → Z ∪ {∞} and k : G → Z. Let (g) = val ◦ det(1 − g). Let k(g) be the unique integer k such that g ∈ Bk and g ∈ / Bk+1 . By definition φn is 0 unless val ◦ det(g) = 1, tr(g) ∈ O, and g ∈ B1−n . Assume these conditions, in which case one can check that 1 − n ≤ k(g) ≤ 0 and (g) ≥ 0. Now define ⎧ ⎪ if tr(g) ∈ $ O, ⎪ ⎨−1 − q, φn (g) := 1 − q 2(g) , if tr(g) ∈ O× and (g) < n + k(g), ⎪ ⎪ ⎩1 + q 2(n+k(g))−1 , if tr(g) ∈ O× and (g) ≥ n + k(g). Proposition 4.54. (Scholze [Sch1]) For each n ≥ 1, the function z n := q−1 [K :K n ] · φn belongs to the center Z(GL2 (F), K n ), and the family (z n )n is compatible with change of level and thus defines a distribution in the sense of (4.2). This distribution is q 1/2 Z V where V is the standard representation C2 of the Langlands dual group GL2 (C). In an unpublished work, Kottwitz gave another proof of this proposition and also described the same distribution in terms of a family (n )n of functions

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n ∈ Z(GL2 (F), In ) where In ranges over the “barycentric” Moy–Prasad filtration in the standard Iwahori subgroup I ⊂ GL2 (F). By a completely different technique, in [Var] Sandeep Varma extended both the results of Scholze and Kottwitz stated above, by describing the distributions attached to V = Symr (C2 ) where r is any odd natural number less than p, the residual characteristic of F.

11. Appendix: Bernstein isomorphisms via types 11.1. Statement of Purpose Alan Roche proved the following beautiful result in [Roc, Theorem 1.10.3.1]. Theorem 4.55 (Roche). Let e be an idempotent in the Hecke algebra H = H(G(F)). View H as a smooth G(F)-module via the left regular repre sentation, and write e = s∈S es according to the Bernstein decomposition ) H = Se = {s ∈ S | es = 0}, and consider the cates∈S Hs . Let Se = gory RSe (G(F)) = s∈Se Rs (G(F)) and its categorical center Z s∈Se Zs . Let Z(eHe) denote the center of the algebra eHe. Then the map z → z(e) defines an algebra isomorphism ZSe → Z(eHe). Roche’s proof is decidedly non-elementary: besides the material developed in [Roc], it relies on some deep results of Bernstein cited there, most importantly Bernstein’s Second Adjointness Theorem and the construction of an explicit progenerator for each Bernstein block Rs (G(F)). In this chapter we use only the very special case of Roche’s theorem where e = e J for a parahoric subgroup J ⊂ G(F). We will explain a more elementary approach to this special case. It will rely only on the part of Bernstein’s theory embodied in Proposition 4.68 below. Formally, the inputs needed are, first, the existence of Bernstein’s categorical decomposition R(G) = s Rs (G), which is proved for instance in [Roc, Theorem 1.7.3.1], in an elementary way, and, second, the internal structure of the Bernstein block Rs (G) associated to a cuspidal pair s = [(M(F), χ˜ ]G where M is a minimal F-Levi subgroup of G and χ˜ is a character on M(F) which is trivial on its unique parahoric subgroup. For such components, progenerators can be constructed in an elementary way, without using Bernstein’s Second Adjointness Theorem. In fact in what follows we describe this internal structure using a few straightforward elements of the theory of Bushnell–Kutzko types, all of which are contained in [BK]. For e = e J Roche’s theorem gives the identification of the center of the parahoric Hecke algebra, in other words a Bernstein isomorphism for the most

Stable Bernstein center and test functions

167

general case, where G/F is arbitrary and J ⊂ G(F) is an arbitrary parahoric subgroup. However we will provide a proof only for the crucial case of J = I , an Iwahori subgroup of G(F). The general parahoric case should follow formally from the Iwahori case, following the method of Theorem 3.1.1 of [H09], provided one is willing to rely on some basic properties of intertwiners for principal series representations (a purely algebraic theory of such intertwiners was detailed for split resp. unramified groups in [HKP] resp. [H07], and the extension to arbitrary groups should be similar to [H07]). The Iwahori case is approached in a different way by S. Rostami [Ro]. Rostami’s proof yields more information, describing the Iwahori–Matsumoto and Bernstein presentations for the Iwahori–Hecke algebra and deducing the description of its center from its Bernstein presentation.

11.2. Some notation un and let σ ∈ The notation will largely come from [HRo]. Recall L = F Aut(L/F) the Frobenius automorphism, which has fixed field F. We use the

σ . Moreover, if S denotes a maxisymbol G as an abbreviation for X ∗ (Z(G)) IF mal L-split torus in G which is defined over F, with centralizer T = CentG (S), un = N G (S)(L)/T (L)1 , the extended then G will denote the subgroup of W affine Weyl group for G/L, which preserves the alcove a in the apartment S corresponding to S, in the building B(G, L) of G over L. ◦& As always, we let I be the Iwahori subgroup I = Ga (O F ), which we recall we have chosen to be in good position relative to A: the corresponding alcove aσ ⊂ B(G, F) is required to belong in the apartment A corresponding to A. Let v F ∈ aσ be a special vertex with corresponding special maximal ◦& parahoric subgroup K = Gv F (O F ). Thus K ⊃ I . Recall M is a minimal F-Levi subgroup of G. Further, if I is an Iwahori subgroup of G(F), then I M := M(F) ∩ I = M(F)1 is the corresponding Iwahori subgroup of M(F) (cf. [HRo, Lemma 4.1.1]). Use the symbol 1 to denote the trivial 1-dimensional representation of any group.

11.3. Preliminary structure theory results Several of the results discussed here were proved independently by S. Rostami and will appear with somewhat different proofs in [Ro]. 11.3.1. Iwahori–Weyl group over F The following lemma concerns variations on well-known results, and were first proved by Timo Richarz [Ri].

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Let G 1 denote the subgroup of G(F) generated by all parahoric subgroups of G(F). By [HRa1, Lemma 17] and [Ri], we have G 1 = G(F)1 . Let N1 = N G (A)(F) ∩ G 1 , and let S denote the set of reflections through the walls of a. Then by [BT2, Prop. 5.2.12], the quadruple (G 1 , I, N1 , S) is a (double) Tits system with affine Weyl group Waff = N1 /I ∩ N1 , and the inclusion G 1 → G(F) is B N -adapted9 of connected type. Lemma 4.56 (T. Richarz [Ri]). (a) Let M1 = M(F)1 . Define the Iwahori– := NG (A)(F)/M1 . Then there is an isomorphism W = Weyl group as W Waff G , which is canonical given the choice of base alcove a. This gives the structure of a quasi-Coxeter group. W (b) If S ⊂ G is a maximal L-split torus which is F-rational and contains un := N G (S)(L)/T (L)1 , then A, and if we set T := CentG (S) and W σ the natural map NG (S)(L) → N G (A)(F) induces an isomorphism un )σ → . (W W Thus, in light of (b) we may reformulate the Bruhat–Tits decomposition of [HRa1, Prop. 8 and Rem. 9], as follows. Lemma 4.57. The map N G (A)(F) → G(F) induces a bijection ∼ W = I \G(F)/I.

(4.36)

Further, the Bruhat order ≤ and length function on Waff extend in the usual , and we have for w ∈ W and s ∈ Waff representing a simple affine way to W reflection, the usual BN-pair relations $ I swI, if w < sw IsIwI = (4.37) I wI ∪ I sw I, if sw < w. 11.3.2. Iwahori factorization Let P = M N be an F-rational parabolic subgroup with Levi factor M, unipotent radical N and opposite unipotent radical N . LetI H = I ∩ H for H = N , N , or M. Lemma 4.58. In the above situation, we have the Iwahori factorization I = IN · IM · IN .

(4.38)

Proof. We use the notation of [BT2]. By [BT2, 5.2.4] with := a, we have +&

−&

◦&

G◦a (O& ) = Ua Ua Na , 9 In [BT2] the symbol B is used in place of I .

Stable Bernstein center and test functions ◦&

&

169

&

where Na := N & ∩ Z◦ (O& )Ua . Since Z◦ (O & )Ua ⊂ G◦a (O& ), we have ◦&

Na = N & ∩ G◦a (O& ) = Z◦ (O& ). The key inclusion here, N & ∩ G◦a (O& ) ⊆ Z◦ (O& ), translates in our notation to N G (A)(F) ∩ I ⊆ M(F)1 , which can be deduced from Lemma 4.56(a). Translating again back to our notation we get I = I N · I N · I M which is the desired equality since I M normalizes I N . 11.3.3. On M(F)1 /M(F)1 Lemma 4.59. The following basic structure theory results hold: (a) In the notation of [HRo], we have M(F)1 /M(F)1 = K˜ /K , which injects into G(F)1 /G(F)1 . Thus M(F)1 = M(F)1 ∩ G(F)1 . (b) The Weyl group W (G, A) acts trivially on M(F)1 /M(F)1 . (c) Let a ⊂ B(G, L) denote the alcove invariant under the group Aut(L/F) ⊃ σ which corresponds to the Iwahori I ⊂ G(F). We assume I ⊂ K . Then the naive Iwahori I˜ := G(F)1 ∩ Fix(aσ ) has the following properties • M(F)1 /M(F)1 = I˜/I = K˜ /K . • I˜ = I · M(F)1 . Proof. Part (a): in the notation of [HRo], we know that M,tors = K˜ /K ([HRo, Prop. 11.1.4]). Applying this to G = M we get M,tors = M(F)1 /M(F)1 . So M(F)1 /M(F)1 = K˜ /K . By (8.0.1) and Lemma 8.0.1 in [HRo], the latter injects into G(F)1 /G(F)1 . The final statement follows. Part (b): By [HRo, Lemma 5.0.1], W (G, A) has representatives in K ∩ NG (A)(F). Thus it is enough to show that if n ∈ K ∩ N G (A)(F) and m ∈ M(F)1 , then nmn −1 m −1 ∈ M(F)1 . This follows from (a), since we clearly have nmn −1 m −1 ∈ M(F)1 ∩ G(F)1 . Part (c): First note that M(F)1 ⊂ I˜ and M(F)1 ⊂ I . Thus there is a commutative diagram / K˜ /K 8 q q q q q qq qqq M(F)1 /M(F)1 I˜/I O

The oblique arrow is bijective by (a). We claim the horizontal arrow is injective, that is, I˜ ∩ K = I . But I˜ ∩ K = G(F)1 ∩ Fix(aσ ) ∩ G(F)1 ∩ Fix(v F ), where v F is the special vertex in B(G ad , F) corresponding to K (cf. [HRo, Lem. 8.0.1]). Thus I˜ ∩ K = G(F)1 ∩ Fix(aσ ) = I by Remark 8.0.2 of [HRo].

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It now follows that all arrows in the diagram are bijective. This implies both statements in (c). Remark 4.60. Let P = M N be as above. We deduce from (c) and (4.38) the Iwahori factorization for I˜ I˜ = I N · M(F)1 · I N ,

(4.39)

using the fact that M(F)1 normalizes I N and I N . 11.3.4. Iwasawa decomposition Next we need to establish a suitable form of the Iwasawa decomposition. Let P = M N be as above. Lemma 4.61. The inclusion N G ( A)(F) → G(F) induces bijections := NG (A)(F)/M(F)1 → W M(F)1 N (F)\G(F)/I W (G, A) = NG (A)(F)/M(F) → P(F)\G(F)/I.

(4.40) (4.41)

= σ W (G, A) (cf. Lemma Proof. In view of the decomposition W M 3.0.1(III) of [HRo] plus Lemma 4.56(b)) and the Kottwitz isomorphism σM → M(F)/M(F)1 (cf. Lemma 3.0.1 [HRo]), it suffices to prove (4.40). For x ∈ B(G, F), let Px ⊂ G(F) denote the subgroup fixing x. By [Land, Prop. 12.9], we have G(F) = N (F) · N G ( A)(F) · Px . For sufficiently generic points x ∈ aσ , we have Px = I˜, which is M(F)1 I by Lemma 4.59(c). Since M(F)1 ⊂ NG (A)(F), we have G(F) = N (F) · N G (A)(F) · I and the map (4.40) is surjective. To prove injectivity, assume n 1 = um 0 · n 2 · j for u ∈ N (F), m 0 ∈ M(F)1 , n 1 , n 2 ∈ N G ( A)(F), and j ∈ I . There exists z ∈ Z (M)(F) such that zuz −1 ∈ I N (cf. e.g. [BK, Lem. 6.14]). Then zn 2 = (zuz −1 )m 0 · zn 2 · j ∈ I zn 2 I, and so by (4.36), zn 2 ≡ zn 2 modulo M(F)1 . and M(F)1 N (F)x I ∩ I y I = ∅, then x ≤ y in the Lemma 4.62. If x, y ∈ W determined by I . Bruhat order on W Proof. This follows from the BN-pair relations (4.37) as in the proof of the Claim in Lemma 1.6.1 of [HKP].

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11.3.5. The universal unramified principal series module M Define M = Cc (M(F)1 N (F)\G(F)/I ). The subscript “c” means we are considering functions supported on finitely many double cosets. Some basic facts about M were given in [HKP] for the special case where G is split, and here we need to state those facts in the current general situation. Abbreviate by setting H = H(G(F), I ) and R = C[ M ]. Then f ∈ H acts on the left on M by right convolutions by f˘, which is defined by f˘(g) = f (g −1 ). The same goes for the normalized induced representation 1/2 i PG (χ) ˜ I = IndGP (δ P χ) ˜ I , where χ˜ is a character on M(F)/M(F)1 . Moreover, R acts on the left on M by normalized left convolutions: for r ∈ R and φ ∈ M, m ∈ M(F), / 1/2 (r · φ)(m) = r (y)δ P (y)φ(y −1 m) dy M(F)

where voldy (M(F)1 ) = 1. The actions of R and H commute, so M is an (R, H )-bimodule. Lemma 4.63. The following statements hold. (a) Any character χ˜ −1 : M(F)/M(F)1 → C× extends to an algebra homomorphism χ˜ −1 : R → C, and there is an isomorphism of left H -modules C ⊗ R,χ˜ −1 M = i PG (χ˜ ) I . (b) For w ∈ W (G, A) =: W , set vw = 1 M(F)1 N (F)w I ∈ M. Then M is free of rank 1 over H with canonical generator v1 . (c) M is free as an R-module, with basis {vw }w∈W . Proof. The proofs for (a–b) are nearly identical to their analogues for split groups in [HKP]. Part (a) is formal. Part (b) relies on the Bruhat–Tits decomposition (4.36), the Iwasawa decomposition (4.40), and Lemma 4.62. Part (c) was not explicitly mentioned in [HKP]. But it can be proved using (4.40) along with the relations analogous to [HKP, (1.6.1–1.6.2)], for which the Iwahori factorization (4.38) is the main ingredient.

11.4. Why (M(F)1 , 1) is an S M -type We let χ range over the characters of M(F)1 /M(F)1 . Let χ˜ denote any extension to a character of the finitely generated abelian group M(F)/M(F)1 . Fix one such extension χ˜ 0 . Note that the inertial class [M(F), χ˜ 0 ] M consists of all

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pairs (M(F), χ), ˜ since M(F)-conjugation does not introduce any new characters on M(F). Therefore we may abuse notation and denote this inertial class by [M(F), χ] ˜ M =: sχM . Let S M := {[M(F), χ˜ ] M | χ ranges as above}. This is a finite set of inertial classes, in bijective correspondence with M(F)1 /M(F)1 . Proposition 4.64. The pair (M(F)1 , 1) is a Bushnell–Kutzko type for S M . Note: This proposition simply makes precise the last paragraph of [BK, 9.2]. Proof. Let σ be an irreducible smooth representation of M(F). We must show that σ = χ˜ for some χ˜ iff σ | M(F )1 ⊃ 1. (⇒): Obvious. (⇐): We see that σ . v = 0 on which M(F)1 acts trivially. Since σ is irreducible, it coincides with the smallest M(F)-subrepresentation containing v, and then since M(F)1 / M(F), we see that M(F)1 acts trivially on all of σ ; further, σ is necessarily finite-dimensional over C. Since M(F)/M(F)1 is abelian, σ contains an M(F)-stable line, since a commuting set of matrices can be simultaneously triangularized. This line is all of σ since σ is irreducible. Thus σ is 1-dimensional, and so σ = χ˜ for some χ˜ .

11.5. Why ( I, 1) is an S G -type We define SG = {[t]G | [t] M ∈ S M }. The map [M, χ] ˜ M → [M, χ˜ ]G is injective: if [M, χ˜1 ]G = [M, χ˜2 ]G , then there exists n ∈ NG (A)(F) such that n (χ˜ ) = χ˜ η for some character η on M(F)/M(F)1 . Restricting to M(F)1 1 2 and using n (χ1 ) = χ1 (Lemma 4.59(b)), we see χ1 = χ2 . So S M ∼ = SG via [t] M → [t]G . We saw above that (M(F)1 , 1) is an S M -type. The fact that (I, 1) is an SG -type follows from [BK], Theorem 8.3, once we check the following proposition. Proposition 4.65. The pair (I, 1) is a G-cover for (M(F)1 , 1) in the sense of [BK, Definition 8.1]. Proof. We need to check the three conditions (i–iii) of Definition 8.1. First (i), the fact that (I, 1) is decomposed with respect to (M, P) in the sense of [BK, (6.1)], follows from the Iwahori factorization I = I N · I M · I N discussed in Remark 4.60. The equality I ∩ M(F) = M(F)1 gives condition (ii). Finally we must prove (iii): for every F-parabolic P with Levi factor M, there exists an invertible element of H(G(F), I ) supported on I z P I , where z P belongs to Z (M)(F) and is strongly (P, I )-positive. The existence of elements z P ∈ Z (M)(F) which are strongly (P, I )-positive is proved in [BK, Lemma 6.14]. Any corresponding characteristic function 1 I z P I is invertible

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in H(G(F), I ), as follows from the Iwahori–Matsumoto presentation of H(G(F), I ). (This presentation itself is easy to prove using (4.37).)

11.6. Structure of the Bernstein varieties Let R(G) denote the category of smooth representations of G(F), and let Rχ (G) denote the full subcategory corresponding to the inertial class [M, χ˜ ]G . That is, a representation (π, V ) ∈ R(G) is an object of Rχ (G) if and only if for each irreducible subquotient π of π , there exists an extension χ˜ of χ such 1/2 that π is a subquotient of IndGP (δ P χ˜ ). We review the structure of the Bernstein varieties XχG and XχM . In this discussion, for each χ we fix an extension χ˜ of χ once and for all – the structures we define will be independent of the choice of χ, ˜ i.e. uniquely determined by (M, χ ) up to a unique isomorphism. As a set XχG (resp. XχM ) consists of the elements (M, χ˜ η)G (resp. (M, χ˜ η) M ) belonging to the inertial equivalence class [M, χ] ˜ G (resp. [M, χ] ˜ M ) as η × ranges over the set X (M) of unramified C -valued characters on M(F) (unramified means it factors through M(F)/M(F)1 ). The map X (M) → XχM , η → (M, χ˜ η) M , is a bijection. Since X (M) is a complex torus, this gives XχM the structure of a complex torus. More canonically, XχM is just the variety of all extensions χ˜ of χ , and it is a torsor under the torus X (M). Now fix χ˜ again. There is a surjective map XχM → XχG (M, χ˜ η) M → (M, χ˜ η)G . Since W := W (G, A) acts trivially on M(F)1 /M(F)1 (Lemma 4.59), one can prove that the fibers of this map are precisely the W -orbits on XχM . Thus as sets W \XχM = XχG , and this gives XχG the structure of an affine variety over C. Having chosen the isomorphism X (M) ∼ = XχM as above, we can transport the W -action on XχM over to an action on X (M). This action depends on the choice of χ˜ and is not the usual action unless χ˜ is W -invariant. We obtain XχG = W \χ˜ X (M), where the latter denotes the quotient with respect to this unusual action on X (M). Let C[XχG ] denote the ring of regular functions on the variety XχG . The algebraic morphism XχM → XχG induces an isomorphism of algebras C[XχG ] → C[XχM ]W .

(4.42)

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11.7. Consequences of the theory of types Let us define convolution in H(G(F), I ) using the Haar measure d x on G(F) which gives I volume 1. Let Z(G(F), I ) denote the center of H(G(F), I ). We define for each χ ∈ (M(F)1 /M(F)1 )∨ a function eχ ∈ H(G(F), I ) by requiring it to be supported on I˜, and by setting eχ (y) = vold x ( I˜)−1 χ( y¯ )−1 if y ∈ I˜. Here we regard χ as a character on I˜/I (cf. Lemma 4.59) and let y¯ ∈ I˜/I denote the image of y. If y = n + · m 1 · n − ∈ I N · M(F)1 · I N , then eχ (y) = vold x ( I˜)−1 χ (m 1 )−1 . Lemma 4.66. The functions {eχ }χ give a complete set of central orthogonal idempotents for H(G(F), I ): (a) eχ ∈ Z(G(F ), I ); (b) eχ eχ = δχ ,χ eχ , there δχ ,χ is the Kronecker delta function; (c) 1 I = χ eχ . Proof. The proof is a straightforward exercise for the reader. For parts (a–b), use the fact that M(F)1 normalizes I , that G(F) = I · NG (A)(F) · I , and that W (G, A) acts trivially on M(F)1 /M(F)1 (cf. Lemma 4.59). Proposition 4.67. The idempotents eχ are the elements in the Bernstein center which project the category R(G) onto the various Bernstein components Rχ (G). That is, there is a canonical isomorphism of algebras H(G(F), I ) = eχ H(G(F), I )eχ χ

and, for any smooth representation (π, V ) ∈ R(G), the G(F)-module spanned by the χ -isotypical vectors V χ = π(eχ )V is the component of V lying in the subcategory Rχ (G). Finally, eχ H(G(F), I )eχ = H(G(F), I˜, χ ), the right-hand side being the algebra of I -bi-invariant C-valued functions f ∈ Cc (G(F)) such that f (i˜1 g i˜2 ) = χ (i˜1 )−1 f (g)χ (i˜2 )−1 for all g ∈ G(F) and i˜1 , i˜2 ∈ I˜. The following records the standard consequences of the fact that (I, 1) is an SG -type (see [BK, Theorem 4.3]). Let R I (G) denote the full subcategory of R(G) whose objects are generated as G-modules by their I -invariant vectors. Proposition 4.68. As subcategories of R(G), we have the equality R I (G) = χ Rχ (G). In particular, an irreducible representation (π, V ) ∈ R(G)

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belongs to R I (G) if and only if π ∈ Rχ (G) for some χ . Furthermore, there is an equivalence of categories R I (G) → H(G(F), I )-Mod (π, V ) → V I . Finally, Z(G(F), I ) is isomorphic with the center of the category χ Rχ (G), which according to Bernstein’s theory is the ring χ C[XχG ]. Concretely, the map Z(G(F), I ) → χ C[XχG ], z → zˆ , is characterized as follows: for every χ and every (M, χ˜ )G ∈ XχG , z ∈ Z(G(F), I ) acts on 1/2

IndGP (δ P χ˜ ) I by the scalar zˆ (χ). ˜ Let us single out what happens in the special case of G = M. We can identify H(M(F), M(F)1 ) = C[ M ]. Let eχM denote the idempotent in H(M(F), M(F)1 ) analogous to eχ , for the case G = M. By Propositions 4.67 and 4.68 for G = M, we have H(M(F), M(F)1 ) = eχM H(M(F), M(F)1 )eχM = C[XχM ], (4.43) χ

χ

the last equality holding since H(M(F), M(F)1 ) is already commutative. Thus, the ring eχM H(M F), M(F)1 )eχM can be regarded as the ring of regular functions on the variety XχM of all extensions χ˜ of χ .

11.8. The embedding of C[ M ]W into Z(G(F), I) We make use of the following special case of a general construction of Bushnell–Kutzko [BK]: for any F-parabolic P with Levi factor M, there is an injective algebra homomorphism t P : H(M(F), M(F)1 ) → H(G(F), I ) which is uniquely characterized by the property that for each (π, V ) ∈ R I (G), v ∈ V I , and h ∈ H(M(F), M(F)1 ), we have the identity qπ (t P (h) · v) = h · qπ (v).

(4.44)

Here qπ : V I → VNM(F)1 is an isomorphism, which is induced by the canonical projection V → V N to the (unnormalized) Jacquet module. See [BK, Thm. 7.9].

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It turns out that it is better to work with a different normalization. We define another injective algebra homomorphism θ P : H(M(F), M(F)1 ) → H(G(F), I ) −1/2

h → t P (δ P

h).

Then using (4.44) θ P satisfies −1/2

qπ (θ P (h) · v) = (δ P

h) · qπ (v).

(4.45)

We view χ˜ as a varying element of the Zariski-dense subset S of the variety of all characters on the finitely generated abelian group M(F)/M(F)1 , conG 1/2 sisting of those regular characters χ˜ such that V (χ˜ ) := i G ˜ P (χ˜ ) := Ind P (δ P χ) is irreducible as an object of R(G). We apply the above discussion to the representations V := V (χ) ˜ with χ˜ ∈ S. By a result of Casselman [Cas], we know that as M(F)-modules # 1/2 VN = δ P ( w χ˜ ) w∈W

and that M(F)1 acts trivially on this module. Now suppose h ∈ C[ M ]W . −1/2 M(F)1 Then δ P h acts on VN = V N by the scalar h(χ) ˜ (viewing h as a regular M function on Xχ ). It follows from (4.45) that θ P (h) acts by the scalar h(χ) ˜ on i PG (χ) ˜ I , for χ˜ ∈ S. Now let f ∈ H be arbitrary, and set := f ∗ θ P (h) − θ P (h) ∗ f ∈ H . We see that acts by zero on i PG (χ˜ ) I for every χ˜ ∈ S.

(4.46)

We claim that = 0. Recall that ∈ H gives an R-linear endomorphism of M, hence by Lemma 4.63(c) may be represented by an |W | × |W | matrix E with entries in R. Now Spec(R) = Spec(C[ M ]) is a diagonalizable group scheme over C with character group M . Hence R is a reduced finite-type C-algebra and its maximal ideals are precisely the kernels of the C-algebra homomorphisms χ˜ −1 : R → C coming into Lemma 4.63(a). By that Lemma and (4.46), we see that E ≡ 0 (mod m) for a Zariski-dense set of maximal ideals m in Spec(R). Since R is reduced and finite-type over C, this implies that the entries of E are identically zero. This proves the claim because M is free of rank 1 over H (Lemma 4.63(b)). Since f was arbitrary, we get θ P (h) ∈ Z(G(F), I ), as desired. We have proved the following result. Lemma 4.69. The map θ P : C[ M ] → H(G(F), I ) restricts to give an embedding C[ M ]W → Z(G(F), I ).

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11.9. The center of the Iwahori–Hecke algebra Theorem 4.70. The map θ P gives an algebra isomorphism C[ M ]W → Z(G(F), I ). Further, this isomorphism is independent of the choice of parabolic P containing M as a Levi factor. Proof. The description of θ P above, and the preceding discussion, show that we have a commutative diagram ∼ / M W C[ M ]W χ C[Xχ ] O θP

Z(G(F), I )

∼

/

χ

C[XχG ] .

The left vertical arrow is bijective because the right vertical arrow is, by (4.42).

11.10. Bernstein isomorphisms and functions Putting together Roche’s theorem 4.55 with Theorem 4.70, we deduce a more general result that holds for any parahoric subgroup J ⊇ I . Theorem 4.71. The composition C[ M ]W

θP

/ Z(G(F), I )

−∗ I 1 J

/ Z(G(F), J )

(4.47)

is an isomorphism. We call this map the Bernstein isomorphism. Definition 4.72. Given μ ∈ M , we define the Bernstein function z μ ∈ Z(G(F), J ) to be the image of the symmetric monomial function λ∈W ·μ λ ∈ C[ M ]W under the Bernstein isomorphism (4.47).

11.11. Compatibility with constant terms Recall M = CentG (A) is a minimal F-Levi subgroup of G and P = M N is a minimal F-parabolic subgroup with Levi factor M and unipotent radical N . Let Q = L R be another F-parabolic subgroup with F-Levi factor L and unipotent radical R. Assume Q ⊇ P; then L ⊇ M and R ⊆ N . Further L contains a minimal F-parabolic subgroup L ∩ P = M · (L ∩ N ), and N = L ∩ N · R. If J ⊂ G(F) is a parahoric subgroup corresponding to a facet in the apartment of the Bruhat–Tits building of G(F) corresponding to A, then JL := J ∩ L is a parahoric subgroup of L(F) (by [HRo, Lem. 4.1.1]).

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Given f ∈ H(G(F), J ), define f (Q) ∈ H(L(F), JL ) by / / 1/2 −1/2 f (Q) (l) = δ Q (l) f (lr ) dr = δ Q (l) f (rl) dr, R

R

where voldr (J ∩ R) = 1. An argument similar to Lemma 4.7.2 of [H09] shows that f → f (Q) sends Z(G(F), J ) into Z(L(F), JL ), and determines a map cG L making the following diagram commute: (G, A) C[ M ]W _

C[ M ]W (L ,A)

∼

/ Z(G(F), J )

(4.48)

cG L

∼

/ Z(L(F), JL ).

The diagram shows that c G L is indeed an (injective) algebra homomorphism and, as the notation suggests, is independent of the choice of parabolic subgroup Q having L as a Levi factor. We call cG L the constant term homomorphism. By its very construction, the map θ M : C[ M ] → H(M(F), M(F)1 ) has its inverse induced by the Kottwitz isomorphism κ M (F) : M(F)/M(F)1 → M . By taking L = M in (4.48), this remark allows us to write down the inverse of θ P in general. W (G, A) Corollary 4.73. The composition κ M (F) ◦ c G M takes values in C[ M ] and is the inverse of the Bernstein isomorphism θ P .

11.12. Transfer homomorphisms 11.12.1. Construction Transfer homomorphisms were defined for special maximal parahoric Hecke algebras in [HRo]. By virtue of the Bernstein isomorphisms (4.47), we can now define these homomorphisms on the level of centers of arbitrary parahoric Hecke algebras. Let us recall the general set-up from [HRo, §11.2]. Let G ∗ be a quasi-split group over F. Let F s denote a separable closure of F, and set = Gal(F s /F). Recall that an inner form of G ∗ is a pair (G, ") consisting of a connected reductive F-group G and a -stable G ∗ad (F s )-orbit " of F s -isomorphisms ψ : G → G ∗ . The set of isomorphism classes of pairs (G, ") corresponds bijectively to H 1 (F, G ∗ad ). Fix once and for all parahoric subgroups J ⊂ G(F) and J ∗ ⊂ G ∗ (F). Choose any maximal F-split tori A ⊂ G and A∗ ⊂ G ∗ such that the facet fixed by J (resp. J ∗ ) is contained in the apartment of the building B(G, F)

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(resp. B(G ∗ , F)) corresponding to the torus A (resp. A∗ ). Let M = CentG (A) and T ∗ = CentG ∗ (A∗ ), a maximal F-torus in G ∗ . Now choose an F-parabolic subgroup P ⊂ G having M as Levi factor, and an F-rational Borel subgroup B ∗ ⊂ G ∗ having T ∗ as Levi factor. Then there exists a unique parabolic subgroup P ∗ ⊂ G ∗ such that P ∗ ⊇ B ∗ and P ∗ is G ∗ (F s )-conjugate to ψ(P) for every ψ ∈ ". Let M ∗ be the unique Levi factor of P ∗ containing T ∗ . Then define " M = {ψ ∈ " | ψ(P) = P ∗ , ψ(M) = M ∗ }. ∗ (F s )-orbit of F s -isomorphisms M → The set " M is a nonempty -stable Mad ∗ M , and so (M, " M ) is an inner form of M ∗ . Choose any ψ0 ∈ " M . Then since ψ0 |A is F-rational, ψ0 (A) is an F-split torus in Z(M ∗ ) and hence ψ0 (A) ⊆ A∗ .

→ ∗ and hence a Now ψ0 induces a -equivariant map Z ( M) Z( M ∗ ) → T homomorphism ∗

∗ ) ∗F ]W (G t A∗ ,A : C[X ∗ (T I

∗ , A∗ )

F

F ]W (G,A) , −→ C[X ∗ (Z( M)) IF

where (·)∗ designates the Galois action on G ∗ (for Weyl-group equivariance ∗ this homomorphism does not see [HRo, §12.2]). Since " M is a torsor for Mad depend on the choice of ψ0 ∈ " M . Further, it depends only on the choice of A∗ and A, and not on the choice of the parabolic subgroups P ⊃ M and B ∗ ⊃ T ∗ we made in constructing it. Definition 4.74. Let J ⊂ G(F) and J ∗ ⊂ G ∗ (F ) be any parahoric subgroups and choose compatible maximal F-split tori A resp. A∗ as above. Then we define the transfer homomorphism t : Z(G ∗ (F), J ∗ ) → Z(G(F), J ) to be the unique homomorphism making the following diagram commute t

Z(G ∗ (F ), J ∗ ) 0

/ Z(G(F), J ) 0

∗

∗ )∗F ]W (G ∗ , A∗ ) C[X ∗ (T I

t A∗ , A

F

/ C[X ∗ (Z ( M))

F ]W (G, A) , IF

where the vertical arrows are the Bernstein isomorphisms. By [BT2, 4.6.28], any two choices for A (resp. A∗ ) are J -(resp. J ∗ -) conjugate. Using Corollary 4.73 it follows that t is independent of the choice of A and A∗ and is a completely canonical homomorphism. Remark 4.75. The map ∗

∗ )∗F → X ∗ (Z ( M))

F t A∗ ,A : X ∗ (T IF I F

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is surjective. Via the Kottwitz homomorphism we may view this as the composition / M ∗ (F)/M ∗ (F)1

T ∗ (F)/T ∗ (F)1

ψ0−1 ∼

/ M(F)/M(F)1

(4.49)

where the first arrow is induced by the inclusion T ∗ → M ∗ . It is enough to observe that M ∗ (F) = T ∗ (F) · M ∗ (F)1 , which in turn follows from the Iwasawa decomposition (cf. (4.41)) for M ∗ (F), which states that M ∗ (F) = ∗ (F) · K ∗ for an F-rational Borel subgroup B ∗ = T ∗ · U ∗ and T ∗ (F) · U M ∗ M M∗ M∗ a special maximal parahoric subgroup K M ∗ in M ∗ , and from the vanishing of ∗ (F) · K ∗ . the Kottwitz homomorphism on U M ∗ M 11.12.2. Normalized transfer homomorphism The transfer homomorphism is slightly too naive, and it is necessary to normalize it in order to get a homomorphism which has the required properties. We need to define normalized homomorphisms t A∗ , A on Weyl-group invariants, for which the following lemma is needed. Lemma 4.76. Recall that T ∗ = CentG ∗ (A∗ ) is a maximal torus in G ∗ defined over F; let S ∗ be the F un -split component of T ∗ , a maximal F un -split torus in G ∗ defined over F and containing A∗ . We have T ∗ = CentG ∗ (A∗ ) = CentG ∗ (S ∗ ). Choose a maximal F un -split torus S ⊂ G which is defined over F and which contains A, and set T = CentG (S). Choose ψ0 ∈ " M such that ψ0 is defined over F un and satisfies ψ0 (S) = S ∗ and hence ψ0 (T ) = T ∗ . Then the diagram ψ

&

0 W (G, A) _ _ _ _ _ ∼_ _/ W (G ∗ , A∗ )/ W (M ∗ , A∗ )

0

W (G, S)/W (M, S) F

0

ψ0 ∼

∗ / W (G ∗ , S ∗ )/ W (M ∗ , S ∗ ) F

&

defines a bijective map ψ0 . It depends on the choice of the data P, B ∗ used to define " M and M ∗ , but it is independent of the choices of S and ψ0 ∈ " M with the stated properties. Proof. The left vertical arrow is [HRo, Lem. 6.1.2]. The right vertical arrow is described in [HRo, Prop. 12.1.1]. The proof of the latter justifies the lower horizontal arrow. Indeed, given w ∈ W (G, A) we may choose a representative n ∈ NG (S)(L) F (cf. [HRo]). We have ψ0−1 ◦ ∗F ◦ ψ0 ◦ −1 F = Int(m ) for s some m ∈ N M (S)(F ). Since n is F -fixed, we get ∗F (ψ0 (n)) = ψ0 (n) · ψ0 (n)−1 ψ0 (m )ψ0 (n)ψ0 (m )−1 .

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181

As n normalizes M and hence ψ0 (n) normalizes M ∗ , this shows that ψ0 (n)W (M ∗ , S ∗ ) is ∗F -fixed. There exists m ∗n ∈ N M ∗ (S ∗ )(L) such that ψ0 (n)m ∗n ∈ N G ∗ (A∗ )(F). Then & ψ0 (w) is the image of ψ0 (n)m ∗n in W (G ∗ , A∗ )/W (M ∗ , A∗ ). The independence statement is proved using this description. Via the Kottwitz homomorphism we can view t A∗ , A as induced by the composition (4.49). We now alter this slightly. Lemma 4.77. Given the choices of P ⊃ M and B ∗ ⊃ T ∗ needed to define " M and given any F un -rational ψ0 ∈ " M , we define an algebra homomorphism C[T ∗ (F)/T ∗ (F)1 ] −→ C[M(F)/M(F)1 ] (4.50) −1/2 1/2 at ∗ t ∗ −→ at ∗ δ B ∗ (t ∗ )δ P (m) · m, t∗

m

t ∗ →m

where t ∗ ranges over T ∗ (F)/ T ∗ (F)1 and m ranges over M(F)/M(F)1 and t ∗ → m means that ψ0−1 (t ∗ ) ∈ m M(F)1 , (cf. (4.49)). Then (4.50) takes W (G ∗ , A∗ )-invariants to W (G, A)-invariants, and the resulting homomorphism t˜A∗ , A : C[T ∗ (F)/T ∗ (F)1 ]W (G

∗ ,A∗ )

→ C[M(F)/M(F)1 ]W (G, A)

is independent of the choices of P, B ∗ , and F un -rational ψ0 ∈ " M . Proof. To check the Weyl-group invariance, we may fix P and B ∗ , and choose S and ψ0 as in Lemma 4.76. Suppose t ∗ at ∗ t ∗ is W (G ∗ , A∗ )-invariant. We need to show that the function on M(F)/M(F)1 −1/2 1/2 m → at ∗ δ B ∗ (t ∗ )δ P (m) (4.51) t ∗ →m

is W (G, A)-invariant, and independent of the choice of P and B ∗ . For w ∈ W (G, A) choose n and m ∗n as in the proof of Lemma 4.76, and define n by ψ0 (n ) = ψ0 (n)m ∗n . Thus ψ0 (n ) ∈ NG ∗ (A∗ )(F) and hence it normalizes T ∗ (F). We claim that (4.51) takes the same values on m M(F)1 and on n m M(F)1 . First we observe that n m M(F)1 = n m M(F)1 . Setting m n := ψ0−1 (m ∗n ), it is enough to prove m n m M(F)1 = m M(F)1 . But as ψ0 is L-rational we have m n ∈ M(L) and so conjugation by m n induces the identity map on M(L)/M(L)1 and hence on its subset M(F)/M(F)1 as well. Now we write the value of (4.51) on n m M(F)1 as −1/2 1/2 at ∗∗ δ B ∗ (t ∗∗ )δ P ( n m).

t ∗∗ → n m M(F)1

182

Thomas J. Haines −1

Setting t ∗ = ψ0 (n ) t ∗∗ and using at ∗ = at ∗∗ (which follows from W (G ∗ , A∗ )invariance), we write the above as −1/2 1/2 at ∗ δ B ∗ ( ψ0 (n ) t ∗ ) δ P ( n m). t ∗ →m

The index set is stable under the W (M ∗ , A∗ )-action on T ∗ (F)/T ∗ (F)1 . If we look at the sum over the W (M ∗ , A∗ )-orbit of a single element t0∗ , with stabilizer group Stab(t0∗ ), we get −1/2 1 · at0∗ · δ B ∗ ( ψ0 (n )y t0∗ ), ∗ M∗ |Stab(t0 )| y

(4.52)

where y ranges over W (M ∗ , A∗ ). Now n ∈ NG (S)(L) F ⊆ NG (A)(F) = N G (M)(F). Hence ψ0 (n)m ∗n = ψ0 (n ) normalizes M ∗ as well as T ∗ , and thus ∗ to another F-rational Borel subgroup of M ∗ conjugation by ψ0 (n ) takes B M ∗ ∗ containing T . Using this it is clear that (4.52) is unchanged if the superscript ψ0 (n ) is omitted, and this proves our claim. For the same reason (4.51) is independent of the choice of P and B ∗ , and similarly t˜A∗ , A is independent of the choice of P and B ∗ , and of the choice of F un -rational ψ0 ∈ " M . Now we give a normalized version of Definition 4.74. Definition 4.78. We define the normalized transfer homomorphism t˜ : Z(G ∗ (F), J ∗ ) → Z(G(F), J ) to be the unique homomorphism making the following diagram commute Z(G ∗ (F), J ∗ ) 0

∗ ) C[X ∗ (T

/ Z(G(F), J ) 0

∗F I F∗

t˜

]W (G

∗ ,A∗ )

t˜A∗ , A

/ C[X ∗ (Z ( M))

F ]W (G, A) , IF

where the vertical arrows are the Bernstein isomorphisms. As was the case for t, the homomorphism t˜ is independent of the choice of A and A∗ , and it is a completely canonical homomorphism. The following shows it is compatible with constant term homomorphisms. 11.12.3. Normalized transfer homomorphisms and constant terms We use the notation of §11.11. Write L = CentG (A L ) for some torus A L ⊆ A. Let L ∗ = CentG ∗ (A∗L ∗ ) for a subtorus A∗L ∗ ⊆ A∗ . Without loss of generality, we may assume that our inner twist G → G ∗ restricts to give an inner twist L → L ∗ .

Stable Bernstein center and test functions

183

Proposition 4.79. In the above situation, the following diagram commutes: Z(G ∗ (F), J ∗ )

t˜

/ Z(G(F), J )

∗

cG L

cG L∗

Z(L ∗ (F), JL∗∗ )

(4.53)

t˜

/ Z(L(F), JL ).

Taking L = M, the diagram shows in order to compute t˜ it is enough to compute it in the case where G ad is anisotropic. In that case if z ∈ Z(G ∗ (F), J ∗ ), the function t˜(z) is given by integrating z over the fibers of the Kottwitz homomorphism κG ∗ (F). Proof. The commutativity boils down to the fact that the quantities (4.52) do not depend on the ambient group G. Remark 4.80. The final sentence in Proposition 4.79 is the key to explicit computation of t˜(z) given z, and is illustrated in §7.3.3. This final sentence was incorrectly asserted to hold for the unnormalized transfer homomorphisms (for special maximal parahoric Hecke algebras) in Prop. 12.3.1 of [HRo].

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5 Conditional results on the birational section conjecture over small number fields Yuichiro Hoshi

Abstract In the present chapter, we give necessary and sufficient conditions for a birational Galois section of a projective smooth curve over either the field of rational numbers or an imaginary quadratic field to be geometric. As a consequence, we prove that, over such a small number field, to prove the birational section conjecture for projective smooth curves, it suffices to verify that, roughly speaking, for any birational Galois section of the projective line, the local points associated to the birational Galois section avoid three distinct rational points, and, moreover, a certain Galois representation determined by the birational Galois section is unramified at all but finitely many primes.

Contents Introduction 0 Notations and conventions 1 Birational Galois sections and their geometricity 2 Local geometricity of birational Galois sections 3 Galois sections of tori that locally arise from points 4 Conditional results on the birational section conjecture Appendix A Ramification of Galois sections References

page 188 193 195 199 205 210 221 229

2000 Mathematics Subject Classification. 14H30. Automorphic Forms and Galois Representations, ed. Fred Diamond, Payman L. Kassaei and c Cambridge University Press 2014. Minhyong Kim. Published by Cambridge University Press.

187

188

Yuichiro Hoshi

Introduction Let k be a field of characteristic 0, k an algebraic closure of k, and X a projecdef

tive smooth geometrically connected curve over k. Write G k = Gal(k/k) for the absolute Galois group of k determined by the fixed algebraic closure k of k. Now we have a natural surjection π1 (X ) G k from the étale fundamental group π1 (X ) of X to G k induced by the structure morphism of X . Then Grothendieck’s section conjecture may be stated as follows: If k is finitely generated over the field of rational numbers, and X is of genus ≥ 2, then any section of this surjection π1 (X ) G k arises from a k-rational point of X , i.e., the image of any section of this surjection coincides with, or, equivalently, is contained in, a decomposition subgroup of π1 (X ) associated to a k-rational point of X . In the present chapter, we discuss the birational version of this conjecture, i.e., the birational section conjecture. Denote by k(X ) the function field of X . Fix an algebraic closure k(X ) of k(X ) def

containing k and write G k(X ) = Gal(k(X )/k(X )). Then the natural inclusions k → k(X ), k → k(X ) determine a surjection G k(X ) G k , which factors through the above surjection π1 (X ) G k . We shall refer to a section of this surjection G k(X ) G k as a [pro-Primes] birational Galois section of X/k [cf. Definition 5.2]. In the present chapter, we discuss the geometricity of birational Galois sections. Let x be a closed point of X and D x ⊆ G k(X ) a decomposition subgroup of G k(X ) associated to x. Then, as is well-known, the image of the composite Dx → G k(X ) G k coincides with the open subgroup G k(x) ⊆ G k of G k corresponding to the residue field k(x) of X at x, and, moreover, the resulting surjection Dx G k(x) admits a [not necessarily unique] section. In particular, if, moreover, k(x) = x, i.e., x ∈ X (k), then the closed subgroup Dx ⊆ G k(X ) of G k(X ) contains the image of a [not necessarily unique] birational Galois section of X/k. We shall say that a birational Galois section of X/k is geometric if its image is contained in a decomposition subgroup of G k(X ) associated to a [necessarily k-rational] closed point of X [cf. Definition 5.3]. The birational section conjecture over local fields has been solved affirmatively. In [8], Koenigsmann proved that if k is either a p-adic local field for some prime number p [i.e., a finite extension of the p-adic completion of the field of rational numbers] or the field of real numbers, then any birational Galois section of X/k is geometric [cf. [8] Proposition 2.4, (2)]. Moreover,

Conditional results on birational section conjecture

189

in [12], Pop obtained, by a refined discussion of Koenigsmann’s discussion, a result concerning birational Galois sections over p-adic local fields [cf. [12], Theorem A], which leads naturally to a proof of the geometrically pro- p version of Koenigsmann’s result over p-adic local fields [cf. Proposition 5.10]. In [18], Wickelgren proved a strong version of the birational section conjecture over the field of real numbers [cf. [18], Corollary 1.2]. In the remainder of the Introduction, we discuss the geometricity of birational Galois sections over number fields; suppose that k is a number field [i.e., a finite extension of the field of rational numbers]. First, let us recall that, in [1], Esnault and Wittenberg proved that if the Shafarevich–Tate group of the Jacobian variety of X over k is finite, then the existence of a birational Galois section of X/k implies the existence of a divisor of degree 1 on X ; more precisely, the existence of a section of the natural surjection G k(X ) /[G k·k(X ) , G k·k(X ) ] G k , where we write def

G k·k(X ) = Gal(k(X )/k · k(X )) and [G k·k(X ) , G k·k(X ) ] for the closure of the commutator subgroup of G k·k(X) , is equivalent to the existence of a divisor of degree 1 on X [cf. [1], Theorem 2.1]. Next, let us recall that, in [4], Harari and Stix proved, as a consequence of results obtained by Stoll in [16], that if there exist an abelian variety A over k and a nonconstant morphism X → A over k such that both the Mordell–Weil group and the Shafarevich–Tate group of A/k are finite, then any birational Galois section of X/k is geometric [cf. [4], Theorem 17]. This result of Harari and Stix gives us some examples of X/k for which any birational Galois section is geometric [cf. [4], Remark 18, (1)]. To state our main results, let us discuss local points associated to a biraf tional Galois section. Write Pk for the set of nonarchimedean primes of k. For f each p ∈ Pk , fix an algebraic closure k p of the p-adic completion kp of k con def f taining k and write G p = Gal(k p /kp ) ⊆ G k . Finally, write Ak ⊆ p∈P f kp k for the finite part of the adele ring of k, i.e., the subring of p∈P f kp conk sisting of elements (ap )p∈P f ∈ p∈P f kp such that ap is contained in the k

k

f

ring of integers of kp for all but finitely many p ∈ Pk . Then it follows from f a result obtained in [8], as well as [12], that, for each p ∈ Pk , a birational Galois section s of X/k uniquely determines a kp -valued point xp of X such that, for any open subscheme U ⊆ X of X , the image of the homomorphism ∼ G p → π1 (U ⊗k kp ) naturally determined by the isomorphism π1 (U ⊗k kp ) → s π1 (U ) ×G k G p and the composite G p → G k → G k(X ) π1 (U ) is contained in a decomposition subgroup of π1 (U ⊗k kp ) associated to xp [cf. Proposition 5.24]; we shall refer to the kp -valued point xp as the kp -valued point of X associated to s [cf. Definition 5.20]. In particular, [since X is projective f over k] the birational Galois section s uniquely determines an Ak -valued point

190

Yuichiro Hoshi def

f

xA = (x p )p∈P f ∈ X (Ak ) ⊆ f Ak -valued

k

f X (kp ) of X ; we shall p∈Pk f Ak -valued point of X associated to

refer to the

point xA as the s [cf. Definition 5.20]. Note that if the birational Galois section s is geometric, then there exists a [necessarily unique] k-rational point x ∈ X (k) of X such that, for each f p ∈ Pk , the kp -valued point of X determined by x is the kp -valued point of X associated to s [cf. Remark 5.21]. The following result is the main result, which gives necessary and sufficient conditions for a birational Galois section of a projective smooth geometrically connected curve over a small number field, i.e., either the field of rational numbers or an imaginary quadratic field, to be geometric [cf. Theorem 5.40 in the case where C consists of all finite groups]. Theorem A. Let k be either the field of rational numbers or an imaginary quadratic field, X a projective smooth geometrically connected curve over k, and s a [pro-Primes] birational Galois section of X/k [cf. Definition 5.2]. Then the following conditions are equivalent: (1) s is geometric [cf. Definition 5.3]. (2) The following two conditions are satisfied: (2-i) There exists a finite morphism φ : X → P1k over k such that, for f each p ∈ Pk , the composite φ

xp

Spec kp −→ X −→ P1k determines a kp -valued point of P1k \ {0, 1, ∞} ⊆ P1k . (2-ii) For each open subscheme U ⊆ X of X which is a hyperbolic curve over k [cf. §0], there exists a prime number lU such that the pro-lU Galois section of U/k [cf. Definition 5.2] naturally determined by s is either cuspidal [cf. Definition 5.37, (i)] or unramified almost everywhere [cf. Definition 5.37, (ii)]. (3) There exists a finite morphism φ : X → P1k over k such that the composite f

xA

φ

Spec Ak −→ X −→ P1k f

determines an Ak -valued point of P1k \ {0, 1, ∞} ⊆ P1k . f f (4) There exist a finite subset T ⊆ Pk of Pk and a closed subscheme Z ⊆ X f of X which is finite over k such that, for each p ∈ Pk \ T , [the image of] the kp -valued point xp of X is contained in Z ⊆ X . The outline of the proof of Theorem A is as follows: The implications (1) ⇒ (2) ⇒ (3) follow immediately from the various definitions involved, together with some results that are proved in Appendix and derived from the discussion

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given in [7]. Next, to verify the implications (3) ⇒ (4) ⇒ (1), let us observe that since k is either the field of rational numbers or an imaginary quadratic field, the following assertion (†) holds [cf. Lemma 5.32]: f

(†): for every [pro-Primes] birational Galois section of P1k /k, if the associated Ak def f valued point of P1k determines an Ak -valued point of Gm,k = P1k \ {0, ∞}, then the induced [pro-Primes] Galois section of Gm,k /k arises from a k-rational point of Gm,k .

Then the implication (3) ⇒ (4) follows immediately from (†). Thus, it remains to verify the implication (4) ⇒ (1). Since X is projective, we have a closed immersion X → PkN for some positive integer N . Now by condition (4), we may assume without loss of generality that, for every nonarchimedean prime p of k, the kp -valued point of PkN determined by this closed immersion X → PkN and the kp -valued point of X associated to the given birational Galois section s lies on the open subscheme Gm,k ×k · · · ×k Gm,k ⊆ AkN ⊆ PkN of PkN ; in particular, we have a kp -valued point of Gm,k ×k · · · ×k Gm,k . Moreover, again by condition (4), together with the above assertion (†), one verifies easily that, for each p, any one of the N factors of the coordinate of the kp valued point of Gm,k ×k · · · ×k Gm,k is k-rational. In particular, it follows that X admits a k-rational point. Thus, by applying this observation to the various open subgroups of G k(X ) that contain the image of s, we conclude that s is geometric. This completes the explanation of the outline of the proof of Theorem A. Note that Theorem A is a result without any assumption on the finiteness of a Shafarevich–Tate group. Next, let us observe that the equivalence (1) ⇔ (3) of Theorem A may be regarded as a tripod analogue of the result due to Harari and Stix discussed above, i.e., [4], Theorem 17. The condition that k is either the field of rational numbers or an imaginary quadratic field [i.e., the assumption that the group of units of the ring of integers of k is finite] in the statement of Theorem A may be regarded as an analogue of the finiteness condition on the Mordell–Weil group in the statement of [4], Theorem 17; on the other hand, since any abelian variety is proper, in the case of [4], Theorem 17, the condition corresponding that the birational Galois sec to our condition f tion determines [not only a f k p -valued point but also] an Ak -valued p∈P k

point of the tripod P1k \ {0, 1, ∞} in Theorem A is automatically satisfied. Let us also observe that our main result leads naturally to some examples – which, however, were already essentially obtained by Stoll – of projective smooth curves for which any prosolvable birational Galois section is geometric [cf. Remarks 5.35; 5.41, (iv)].

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As a corollary of Theorem A, we prove the following result [cf. Corollary 5.42]. Theorem B. Let k be either the field of rational numbers or an imaginary quadratic field. Then the following assertions are equivalent: (1) Any [pro-Primes] birational Galois section [cf. Definition 5.2] of any projective smooth geometrically connected curve over k is geometric [cf. Definition 5.3]. (2) Any [pro-Primes] birational Galois section of P1k /k is geometric. (3) Any [pro-Primes] birational Galois section s of P1k /k satisfies the following two conditions: (3-i) There exist three distinct elements a, b, c ∈ P1k (k) of P1k (k) such that, for any nonarchimedean prime p of k, the kp -valued point of P1k associated to s is ∈ {a, b, c} ⊆ (P1k (k) ⊆) P1k (kp ). (3-ii) There exists a prime number l such that the pro-l Galois section of P1k \ {0, 1, ∞} [cf. Definition 5.2] naturally determined by s is either cuspidal [cf. Definition 5.37, (i)] or unramified almost everywhere [cf. Definition 5.37, (ii)]. (4) Any [pro-Primes] birational Galois section s of P1k /k satisfies the following two conditions: (4-i) There exist three distinct elements a, b, c ∈ P1k (k) of P1k (k) such that, for any nonarchimedean prime p of k, the kp -valued point of P1k associated to s is ∈ {a, b, c} ⊆ (P1k (k) ⊆) P1k (kp ). (4-ii) Write s P for the pro-Primes Galois section of P1k \ {0, 1, ∞} [cf. Definition 5.2] naturally determined by s. Then it holds either that s P is cuspidal [cf. Definition 5.37, (i)], or that there exists a prime number l such that the l-adic Galois representation sP

G k −→ π1 (P1k \ {0, 1, ∞}) −→ GL2 (Zl ) – where the second arrow π1 (P1k \ {0, 1, ∞}) → GL2 (Zl ) is the l-adic representation of π1 (P1k \ {0, 1, ∞}) determined by the Legendre family of elliptic curves over P1k \ {0, 1, ∞}, i.e., the elliptic curve over P1k \ {0, 1, ∞} = Spec k[u ±1 , (1 − u)−1 ] determined by the equation “y 2 = x(x − 1)(x − u)” – is unramified at all but finitely many primes of k. As a consequence [cf. the equivalences (1) ⇔ (3) and (1) ⇔ (4) of Theorem B], for a number field k which is either the field of rational numbers or an imaginary quadratic field, to prove the birational section conjecture over k [i.e., assertion (1) of Theorem B], it suffices to verify that, roughly

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speaking, for any birational Galois section of the projective line over k, the local points associated to the birational Galois section avoid three distinct rational points [cf. conditions (3-i), (4-i)], and, moreover, a certain Galois representation determined by the birational Galois section is unramified at all but finitely many primes [cf. conditions (3-ii), (4-ii)]. However, it is not clear to the author at the time of writing whether or not these are always satisfied. Finally, the author should mention that the referee pointed out that, after the present paper, Stix presented, in [15], results that are similar to and stronger than some results of the present paper. In fact, for instance, a similar result to the equivalence (1) ⇔ (2) of Theorem A (respectively, the equivalence (1) ⇔ (4) of Theorem B; Lemma 5.32; Theorem 5.33; Corollary 5.34) of the present paper may be found as [15], Theorem A (respectively, [15], Theorem C; [15], Proposition 4; [15], Corollary 9; [15], Corollary 11).

Acknowledgements The author would like to thank Akio Tamagawa, Takahiro Tsushima, and Seidai Yasuda for their helpful comments and, especially, discussions concerning §3. The author also would like to thank the referee for some comments and suggestions. This research was supported by Grant-in-Aid for Young Scientists (B), No. 22740012, Japan Society for the Promotion of Science.

0. Notations and conventions Numbers: The notation Primes will be used to denote the set of all prime numbers. The notation Z will be used to denote the ring of rational integers. If ⊆ Primes, then we shall refer to a nonzero integer whose prime divisors are contained in as a -integer, and we shall write

Z for the pro- completion def of Z, i.e.,

Z = lim Z/nZ, where the projective limit is over all positive ← − integers n. We shall refer to a finite (respectively, finitely generated) extension of the field of rational numbers as a number field (respectively, finitely generated field of characteristic 0). If p ∈ Primes, then the notation Z p will be used to denote the p-adic completion of Z, and we shall refer to a finite extension of the p-adic completion of the field of rational numbers as a p-adic local field. Profinite groups: Let G be a profinite group and H ⊆ G a closed subloc (H ) the centralizer, group of G. Then we shall denote by Z G (H ), NG (H ), Z G normalizer, local centralizer of H in G, respectively, i.e.,

194

Yuichiro Hoshi Z G (H ) = { g ∈ G | ghg −1 = h for any h ∈ H } , def

NG (H ) = { g ∈ G | g · H · g −1 = H } , def

def

loc ZG (H ) = lim Z G (U ) − → U

– where the injective limit is over all open subgroups U ⊆ H of H . We shall def

def

loc (G) as the center, local center refer to Z (G) = Z G (G), Z loc (G) = Z G of G, respectively. We shall say that G is center-free, slim if Z (G) = {1}, Z loc (G) = {1}, respectively. Let ⊆ Primes be a nonempty subset of Primes [where we refer to the discussion entitled “Numbers” concerning the set Primes]. Then we shall say that a finite group G is a -group if the cardinality of G is a -integer [where we refer to the discussion entitled “Numbers” concerning the term “-integer”]. Let C be a full formation [i.e., a family of finite groups that is closed under taking quotients, subgroups, and extensions]. We shall say that a finite group is a C-group if [a finite group which is isomorphic to] the finite group is contained in C. We shall say that a profinite group is a pro-C group if every finite quotient of the profinite group is a C-group. We shall write (C) ⊆ Primes for the set of prime numbers p ∈ Primes such that Z/ pZ is a C-group. Here, we note that one verifies easily that (C) = Primes if and only if C contains all finite solvable groups. If C consists of all -groups for some nonempty subset ⊆ Primes, then we shall refer to a pro-C group as a pro- group. Let G be a profinite group. Then we shall write Aut(G) for the group of [continuous] automorphisms of G, Inn(G) ⊆ Aut(G) for the group of inner automorphisms of G, and def

Out(G) = Aut(G)/Inn(G) . If, moreover, G is topologically finitely generated, then one verifies easily that the topology of G admits a basis of characteristic open subgroups, which thus induces a profinite topology on the group Aut(G), hence also a profinite topology on the group Out(G). Curves: Let S be a scheme and X a scheme over S. Then we shall say that X is a smooth curve over S if there exist a scheme X cpt which is smooth, proper, geometrically connected, and of relative dimension 1 over S and a closed subscheme D ⊆ X cpt of X cpt which is finite and étale over S such that the complement X cpt \ D of D in X cpt is isomorphic to X over S. Note

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that, as is well-known, if X is a smooth curve over [the spectrum of] a field k, then the pair “(X cpt , D)” is uniquely determined up to canonical isomorphism over k; we shall refer to X cpt as the smooth compactification of X over k and to a geometric point of X cpt whose image lies on D as a cusp of X . Let S be a scheme. Then we shall say that a smooth curve X over S is a hyperbolic curve (respectively, tripod) over S if there exist a pair (X cpt , D) satisfying the condition in the above definition of the term “smooth curve” and a pair (g, r ) of nonnegative integers such that 2g − 2 + r > 0 (respectively, (g, r ) = (0, 3)), any geometric fiber of X cpt → S is [a necessarily smooth, proper, and connected curve] of genus g, and the degree of D ⊆ X cpt over S is r. Let S be a scheme, U ⊆ S an open subscheme of S, and X a hyperbolic curve over U . Then we shall say that X admits good reduction over S if there exists a hyperbolic curve X S over S such that X S × S U is isomorphic to X over U .

1. Birational Galois sections and their geometricity In the present §1, we discuss the notion of a birational Galois section. In the present §1, let C be a full formation, k a field of characteristic 0, and k an algebraic closure of k. For a finite extension k (⊆ k) of k, write def

G k = Gal(k/k ). Definition 5.1. Let X be a quasi-compact scheme which is geometrically integral over k. (i) We shall write k(X ) for the function field of X . (ii) We shall write CX/k for the pro-C geometric fundamental group of X , i.e., the maximal pro-C quotient of π1 (X ⊗k k), and CX/k for the geometrically pro-C fundamental group of X , i.e., the quotient of π1 (X) by the kernel of the natural surjection π1 (X ⊗k k) CX/k . If X is the spectrum of a k-algebra R, then we shall write

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Yuichiro Hoshi CR/k = CX/k ; CR/k = CX/k . def

def

Thus, we have a commutative diagram of profinite groups 1 −−−−→ Ck(X )/k −−−−→ Ck(X )/k −−−−→ ⏐ ⏐ ⏐ ⏐ 3 3 1 −−−−→ CX/k

−−−−→ CX/k

G k −−−−→ 1 4 4 4

−−−−→ G k −−−−→ 1

[cf. (i)] – where the horizontal sequences are exact [cf. [3], Exposé IX, Théorème 6.1]. If C consists of all -groups [cf. §0] for some nonempty subset ⊆ Primes [cf. §0], then we shall write C C X/k = X/k ; X/k = X/k . def

def

Definition 5.2. Let X be a quasi-compact scheme which is geometrically integral over k. Then we shall refer to a section of the upper (respectively, lower) exact sequence of the commutative diagram of Definition 5.1, (ii), as a pro-C birational Galois section (respectively, pro-C Galois section) of X/k. The Ck(X )/k -conjugacy (respectively, CX/k -conjugacy) class of a pro-C birational Galois section (respectively, pro-C Galois section) of X/k as the conjugacy class of the pro-C birational Galois section (respectively, pro-C Galois section). If C consists of all -groups for some nonempty subset ⊆ Primes, then we shall refer to a pro-C birational Galois section (respectively, pro-C Galois section) of X/k as a pro- birational Galois section (respectively, pro- Galois section) of X/k. Definition 5.3. Let X be a smooth curve over k [cf. §0] and s a pro-C birational Galois section (respectively, pro-C Galois section) of X/k [cf. Definition 5.2]. Then we shall say that s is geometric if the image of s is contained in a decomposition subgroup of Ck(X )/k (respectively, CX/k ) associated to a [necessarily k-rational] closed point of the [uniquely determined] smooth compactification of X over k. Remark 5.4. Let X be a smooth curve over k. Then it follows immediately from the various definitions involved that the geometricity of a pro-C birational Galois section (respectively, pro-C Galois section) of X/k depends only on its conjugacy class [cf. Definition 5.2]. Remark 5.5. Let X , Y be smooth curves over k and Y → X a dominant morphism over k, which thus determines a finite extension k(X ) → k(Y ) over k. If a pro-C birational Galois section (respectively, pro-C Galois section) s

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of Y/k is geometric, then it follows immediately from the various definitions involved that the pro-C birational Galois section (respectively, pro-C Galois section) of X/k determined by s and the morphism Y → X [i.e., the pro-C birational Galois section (respectively, pro-C Galois section) of X/k obtained as the composite of s and the natural open homomorphism Ck(Y )/k → Ck(X )/k (respectively, CY/k → CX/k ) induced by Y → X ] is geometric.

Remark 5.6. Let X be a projective smooth curve over k, U ⊆ X an open subscheme of X , and s a pro-C birational Galois section of X/k. Then it follows immediately from the various definitions involved that if s is geometric, then the pro-C Galois section of U/k naturally determined by s [i.e., the pro-C Galois section of U/k obtained as the composite of s and the natural surjection C ] is geometric. Ck(X )/k U/k Lemma 5.7. Let X be a hyperbolic curve over k [cf. §0] and x, y closed points of the [uniquely determined] smooth compactification of X . Suppose that k is generalized sub- p-adic [i.e., k is isomorphic to a subfield of a finitely generated extension of the p-adic completion of the maximal unramified extension of the p-adic completion of the field of rational numbers – cf. [11], Definition 4.11] for some p ∈ (C) [cf. §0]. Then the following conditions are equivalent: (1) x = y. (2) There exist respective decomposition subgroups Dx , D y ⊆ CX/k of CX/k associated to x, y such that the image of the composite Dx ∩ D y → CX/k G k is open. Proof. The implication (1) ⇒ (2) is immediate. Next, we verify the implication (2) ⇒ (1). Suppose that condition (2) is satisfied. Then it is immediate that, to verify the implication (2) ⇒ (1), by replacing CX/k by an open subgroup of CX/k , we may assume without loss of generality that X is of genus ≥ 2, and, moreover, the displayed composite of condition (2) is surjective, hence also that x and y are k-rational. Thus, to verify the implication (2) ⇒ (1), by replacing X by its smooth compactification, we may assume without loss of generality that x, y ∈ X (k). Then, by considering the quo{ p} tient CX/k X/k of CX/k , the implication (2) ⇒ (1) follows immediately from [11], Theorem 4.12 [cf. also [11], Remark following Theorem 4.12], together with a similar argument to the argument used in the proof of [10], Theorem C. This completes the proof of the implication (2) ⇒ (1), hence also of Lemma 5.7.

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Lemma 5.8. Let X be a hyperbolic curve over k, s a pro-C Galois section of X/k [cf. Definition 5.2], and k (⊆ k) a finite extension of k. Suppose that k is either (a) a finitely generated field of characteristic 0 [cf. §0] or (b) a p-adic local field for some p ∈ (C) [cf. §0]. Then the following conditions are equivalent: (1) s is geometric [cf. Definition 5.3]. (2) The pro-C Galois section s|G k of X ⊗k k /k determined by s is geometric. (3) For any open subgroup H ⊆ CX/k of CX/k containing the image of s, the [uniquely determined] smooth compactification of the finite étale covering of X corresponding to H ⊆ CX/k admits a k -valued point. Proof. The equivalence (1) ⇔ (2) follows immediately from a similar argument to the argument used in the proof of [7], Lemma 54. Here, we note that if k satisfies the condition (b), then, in order to apply a similar argument to the argument used in the proof of [7], Lemma 54, we have to replace “[Moc99, Theorem C]” (respectively, the finiteness of the set “(X n )cpt (k)” obtained by Mordell–Faltings’s theorem) in the proof of [7], Lemma 54, by Lemma 5.7 (respectively, the compactness of the set “(X n )cpt (k)” obtained by the consideration of a suitable model of “(X n )cpt ” over the ring of integers of k). The equivalence (2) ⇔ (3) follows immediately from a similar argument to the argument applied in the proof of [17], Proposition 2.8, (iv). This completes the proof of Lemma 5.8. Lemma 5.9. Let X be a smooth curve over k, s a pro-C birational Galois section of X/k [cf. Definition 5.2], and k (⊆ k) a finite extension of k. Suppose that k is either (a) a finitely generated field of characteristic 0 [cf. §0] or (b) a p-adic local field [cf. §0] for some p ∈ (C) [cf. §0]. Then the following conditions are equivalent: (1) s is geometric [cf. Definition 5.3]. (2) The pro-C birational Galois section s|G k of X ⊗k k /k determined by s is geometric. (3) For any open subgroup H ⊆ Ck(X )/k of Ck(X)/k containing the image of s, the [uniquely determined] smooth compactification of the normalization of X in the finite extension of k(X ) corresponding to H ⊆ Ck(X )/k admits a k -valued point.

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Proof. This follows immediately from a similar argument to the argument applied in the proof of Lemma 5.8. The following result was essentially proved in [12] by a refined discussion of the discussion given in [8]. Proposition 5.10. Let p be a prime number and X a smooth curve over k. Suppose that p ∈ (C) [cf. §0], and that k is a p-adic local field. Then any pro-C birational Galois section of X/k [cf. Definition 5.2] is geometric [cf. Definition 5.3]. Proof. It follows from the equivalence (1) ⇔ (3) of Lemma 5.9 that, to verify Proposition 5.10, it suffices to verify that, for any open subgroup H ⊆ Ck(X )/k

of Ck(X )/k containing the image of s, the [uniquely determined] smooth compactification of the normalization of X in the finite extension of k(X ) corresponding to H ⊆ Ck(X )/k admits a k(ζ p )-valued point, where we use the notation ζ p ∈ k to denote a primitive p-th root of unity. On the other hand, this follows immediately from [12], Theorem A, (2). This completes the proof of Proposition 5.10.

2. Local geometricity of birational Galois sections In the present §2, we discuss the notion of the local geometricity of birational Galois sections of smooth curves over number fields. In the present §2, let C be a full formation, k a number field [cf. §0], k an algebraic closure of k, and X a smooth curve over k [cf. §0]. Write ok ⊆ k for the ring of integers of k, f

Pk

for the set of all nonarchimedean primes of k, and X cpt for the [uniquely determined] smooth compactification of X over k. Moreover, f for each p ∈ Pk , write kp for the p-adic completion of k and op ⊆ kp

200

Yuichiro Hoshi f

for the ring of integers of kp . For each p ∈ Pk , let us fix an algebraic closure k p of kp containing k and write def

def

G p = Gal(k p /kp ) ⊆ G k = Gal(k/k) . Definition 5.11. Let s be a pro-C Galois section of X/k [cf. Definition 5.2]. f For a nonarchimedean prime p ∈ Pk of k, we shall say that s is geometric at p if the pro-C Galois section of X ⊗k kp /kp naturally determined by s [i.e., the pro-C Galois section of X ⊗k kp /kp determined by the natural isomorphism ∼

CX ⊗k kp /kp −→ CX/k ×G k G p and the composite G p → G k → CX/k ] s

f

f

is geometric [cf. Definition 5.3]. For a subset S ⊆ Pk of Pk , we shall say that s is geometric at S if, for each p ∈ S, s is geometric at p. Finally, we shall say f that s is locally geometric if s is geometric at Pk . Remark 5.12. In the notation of Definition 5.11, it is immediate that if s is geometric [cf. Definition 5.3], then s is locally geometric. f

f

Definition 5.13. Let S ⊆ Pk be a subset of Pk . Then we shall write def f Ak | S = kp ; p∈S f

def

Ak | S =

" " f (ap )p∈S ∈ Ak | S " ap ∈ op for all but finitely many p ∈ S ; f def f f def f Ak = Ak | P f ; Ak = Ak | P f . k

k

f

f

Remark 5.14. Since X cpt is proper over k, for any subset S ⊆ Pk of Pk , the f f natural injection X cpt (Ak | S ) → X cpt ( Ak | S ) is bijective. Definition 5.15. Let s be a pro-C Galois section of X/k [cf. Definition 5.2]. If f s is geometric at a nonarchimedean prime p ∈ Pk of k [cf. Definition 5.11], cpt i.e., there exists a kp -valued point xp ∈ X (kp ) = (X cpt ⊗k kp )(kp ) of X cpt such that the image of the pro-C Galois section of X ⊗k kp /kp naturally determined by s is contained in a decomposition subgroup of CX⊗k kp /kp associated to xp , then we shall refer to such a kp -valued point “xp ” of X cpt as a kp -valued f f point of X cpt associated to s. If s is geometric at a subset S ⊆ Pk of Pk , f then we shall refer to an Ak | S -valued point, or, equivalently [cf. Remark 5.14],

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f

an Ak | S -valued point, of X cpt determined by kp -valued points of X cpt associf ated to s – where p ranges over elements of S – as an Ak | S -valued point, or, f equivalently, an Ak | S -valued point, of X cpt associated to s. Remark 5.16. In the notation of Definition 5.15, suppose that s is geometric [cf. Definition 5.3], hence also locally geometric [cf. Definition 5.11; Remark 5.12]. Then it is immediate that there exists a k-rational point x ∈ f X cpt (k) of X cpt such that, for each p ∈ Pk , the kp -valued point of X cpt cpt determined by x is a kp -valued point of X associated to s. In particular, f f the Ak -valued point of X cpt determined by x is an Ak -valued point of X cpt associated to s. Note that if C contains all finite solvable groups, and X is a hyperbolic curve over k [cf. §0], then it follows from Theorem 5.33 below that the converse holds, i.e., if s is locally geometric, and there exists a k-rational point f x ∈ X cpt (k) of X cpt such that, for each p ∈ Pk , the kp -valued point of cpt cpt X determined by x is a kp -valued point of X associated to s, then s is geometric. Lemma 5.17. Let s be a pro-C birational Galois section of X/k [cf. Deff inition 5.2] and p ∈ Pk . For an open subscheme U ⊆ X cpt of X cpt , write s[U ] for the pro-C Galois section of U/k [cf. Definition 5.2] naturally determined by s [i.e., the pro-C Galois section of U/k obtained as the composite of s and C ]; the natural surjection Ck(X )/k U/k s[U, p] for the pro-C Galois section of U ⊗k kp /kp naturally determined by s [i.e., the pro-C Galois section of U ⊗k kp /kp determined by the natural isomorphism ∼

C C U ⊗k kp /kp −→ U/k ×G k G p

and the composite C G p → G k → Ck(X )/k U/k ]. s

Then the following conditions are equivalent: (1) There exists a kp -valued point xp ∈ X cpt (kp ) = (X cpt ⊗k kp )(kp ) of X cpt such that, for any open subscheme U ⊆ X cpt of X cpt , the image of the proC Galois section s[U, p] of U ⊗k kp /kp is contained in a decomposition C subgroup of U ⊗k kp /kp associated to x p .

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(2) For any open subscheme U ⊆ X cpt of X cpt , the pro-C Galois section s[U ] of U/k is geometric at p [cf. Definition 5.11], i.e., the pro-C Galois section s[U, p] of U ⊗k kp /kp is geometric [or, equivalently, the image of s[U, p] C is contained in a decomposition subgroup of U ⊗k k p /kp associated to a kp -rational point of X cpt ⊗k kp ]. (3) For any open subgroup H ⊆ Ck(X )/k of Ck(X)/k containing the image of s, the [uniquely determined] smooth compactification of the normalization of X in the finite extension of k(X ) corresponding to H ⊆ Ck(X )/k admits a kp -valued point. (4) The image of the homomorphism G p → Ck(X )/k ×G k G p induced by s is contained in the image of a decomposition subgroup of Ckp (X ⊗k kp )/kp associated to a [necessarily kp -rational] closed point of X cpt ⊗k kp by the natural surjection Ckp (X ⊗k kp )/kp Ck(X )/k ×G k G p .

Proof. The implications (4) ⇒ (1) ⇒ (2) ⇒ (3) are immediate. Finally, the implication (3) ⇒ (4) follows immediately from a similar argument to the argument applied in the proof of [17], Proposition 2.8, (iv). This completes the proof of Lemma 5.17. Definition 5.18. Let s be a pro-C birational Galois section of X/k [cf. Deff inition 5.2]. For a nonarchimedean prime p ∈ Pk of k, we shall say that s is geometric at p if the pair (s, p) satisfies equivalent conditions (1), (2), (3), f f and (4) of Lemma 5.17. For a subset S ⊆ Pk of Pk , we shall say that s is geometric at S if, for each p ∈ S, s is geometric at p. Finally, we shall say that f s is locally geometric if s is geometric at Pk . Remark 5.19. In the notation of Definition 5.18, it is immediate that if s is geometric [cf. Definition 5.3], then s is locally geometric. Definition 5.20. Let s be a pro-C birational Galois section of X/k [cf. Deff inition 5.2]. If s is geometric at a nonarchimedean prime p ∈ Pk of k [cf. Definition 5.18], i.e., the pair (s, p) satisfies condition (1) of Lemma 5.17, then we shall refer to a kp -valued point “xp ” of X cpt appearing in condition (1) of Lemma 5.17 as a kp -valued point of X cpt associated to s. If s is geometric f f f at a subset S ⊆ Pk of Pk , then we shall refer to an Ak | S -valued point, or, f equivalently [cf. Remark 5.14], an Ak | S -valued point, of X cpt determined by kp -valued points of X cpt associated to s – where p ranges over elements of f f S – as an Ak | S -valued point, or, equivalently, an Ak | S -valued point, of X cpt associated to s. Remark 5.21. In the notation of Definition 5.20, suppose that s is geometric [cf. Definition 5.3], hence also locally geometric [cf. Definition 5.18;

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Remark 5.19]. Then it is immediate that there exists a k-rational point x ∈ f X cpt (k) of X cpt such that, for each p ∈ Pk , the kp -valued point of X cpt determined by x is a kp -valued point of X cpt associated to s. In particular, f f the Ak -valued point of X cpt determined by x is an Ak -valued point of X cpt associated to s. Note that if C contains all finite solvable groups, then it follows from Theorem 5.33 below that the converse holds, i.e., if s is locally geometric, and there f exists a k-rational point x ∈ X cpt (k) of X cpt such that, for each p ∈ Pk , the cpt cpt kp -valued point of X determined by x is a kp -valued point of X associated to s, then s is geometric. Lemma 5.22. Let s be a pro-C birational Galois section (respectively, prof f C Galois section) of X/k [cf. Definition 5.2] and S ⊆ Pk a subset of Pk . Suppose that s is geometric at S [cf. Definition 5.18 (respectively, Definition 5.11)], and that, for each p ∈ S, the residue characteristic of p is ∈ (C) [cf. §0]. Suppose, moreover, that if s is a pro-C Galois section of X/k, then f X is a hyperbolic curve over k [cf. §0]. Then an Ak | S -valued point of X cpt associated to s [cf. Definition 5.20 (respectively, Definition 5.15)] is uniquely determined by s. Proof. Observe that, to verify Lemma 5.22, by replacing S by a subset of S of cardinality 1, we may assume without loss of generality that S = {p} for f some p ∈ Pk . Then the uniqueness in question follows immediately from Lemma 5.7. This completes the proof of Lemma 5.22. Lemma 5.23. Let s be a pro-C birational Galois section of X/k [cf. Definif f f f tion 5.2], S ⊆ Pk a subset of Pk , and xA ∈ X cpt (Ak | S ) an Ak | S -valued point cpt of X . Suppose that s is geometric at S [cf. Definition 5.18]. Write s[X ] for the pro-C Galois section of X/k [cf. Definition 5.2] naturally determined by s. Then the following hold: (i) s[X ] is geometric at S [cf. Definition 5.11]. f f (ii) If xA ∈ X cpt (Ak | S ) is an Ak | S -valued point of X cpt associated to s [cf. f f Definition 5.20], then xA ∈ X cpt (Ak | S ) is an Ak | S -valued point of X cpt associated to s[X ] [cf. Definition 5.15; assertion (i)]. (iii) Suppose, moreover, that, for each p ∈ S, the residue characteristic of p is ∈ (C) [cf. §0], and that X is a hyperbolic curve over k. Then it holds f f that xA ∈ X cpt (Ak | S ) is an Ak | S -valued point of X cpt associated to s if f f and only if xA ∈ X cpt (Ak | S ) is an Ak | S -valued point of X cpt associated to s[X ] [cf. assertion (i)].

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Proof. Assertions (i), (ii) follow immediately from the various definitions involved. Assertion (iii) follows immediately from Lemma 5.22, together with assertion (ii). This completes the proof of Lemma 5.23. The following result was essentially proved in [12] by a refined discussion of the discussion given in [8]. Proposition 5.24. Let s be a pro-C birational Galois section of X/k [cf. Defif f nition 5.2] and S ⊆ Pk a subset of Pk such that, for each p ∈ S, the residue characteristic of p is ∈ (C) [cf. §0]. Then s is geometric at S [cf. Definif tion 5.18]. In particular, s determines a unique Ak | S -valued point of X cpt [cf. Definition 5.20]. f

Proof. If s is geometric at S, then the uniqueness of an Ak | S -valued point of X cpt associated to s follows from Lemma 5.22. Thus, to verify Proposition 5.24, it suffices to verify that s is geometric at S. Moreover, it follows immediately from the various definitions involved that, to verify that s is geometric at S, by replacing S by a subset of S of cardinality 1, we may assume f without loss of generality that S = {p} for some p ∈ Pk , whose residue characteristic we denote by p. Thus, it follows from Lemma 5.25 below [cf. condition (5) of Lemma 5.25 below] that, to verify Proposition 5.24, it suffices to verify that, for any open subgroup H ⊆ Ck(X)/k of Ck(X )/k containing the image of s, the [uniquely determined] smooth compactification Y of the normalization of X in the finite extension of k(X ) corresponding to H ⊆ Ck(X )/k admits a kp (ζ p )-valued point, where we use the notation ζ p ∈ k to denote a primitive p-th root of unity. On the other hand, by considering the restriction of the pro-C birational Galois section of Y/k naturally determined by s to the closed subgroup Gal(k/k(ζ p )h ) of G k , where we write k(ζ p )h ⊆ k for the algebraic closure of k(ζ p ) in kp (ζ p ), we conclude from [12], Theorem B, (2), that Y (k(ζ p )h ) = ∅, hence also that Y (kp (ζ p )) = ∅. This completes the proof of Proposition 5.24. Lemma 5.25. In the notation of Lemma 5.17, suppose, moreover, that the residue characteristic of p is ∈ (C) [cf. §0]. Let kp (⊆ k p ) be a finite extension of kp . Then equivalent conditions (1), (2), (3), and (4) of Lemma 5.17 are equivalent to the following conditions: (5) For any open subgroup H ⊆ Ck(X )/k of Ck(X)/k containing the image of s, the [uniquely determined] smooth compactification of the normalization of X in the finite extension of k(X ) corresponding to H ⊆ Ck(X )/k admits a kp -valued point.

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(6) The image of the composite of the natural inclusion Gal(k p /kp ) → G p and the homomorphism G p → Ck(X )/k ×G k G p induced by s is contained in the image of a decomposition subgroup of Ckp (X ⊗k kp )/kp associated to a closed point of X cpt ⊗k kp [necessarily defined over a subfield of kp ] by the natural surjection Ckp (X⊗k kp )/kp Ck(X )/k ×G k G p .

Proof. The implication (3) ⇒ (5) is immediate. Moreover, by applying the implication (3) ⇒ (4) of Lemma 5.17 to the restriction of s to a suitable open subgroup of G k , we conclude that the implication (5) ⇒ (6) holds. Finally, the implication (6) ⇒ (4) follows immediately from a similar argument to the argument applied in the proof of [7], Lemma 54, by replacing “[Moc99, Theorem C]” (respectively, the finiteness of the set “(X n )cpt (k)” obtained by Mordell–Faltings’s theorem) in the proof of [7], Lemma 54, by Lemma 5.7 (respectively, the compactness of the set “(X n )cpt (k)” obtained by the consideration of a suitable model of “(X n )cpt ” over the ring of integers of k). This completes the proof of Lemma 5.25. Proposition 5.26. Suppose that X is a hyperbolic curve over k. Let s be a f pro-C Galois section of X/k [cf. Definition 5.2] and S ⊆ Pk a subset of f Pk such that, for each p ∈ S, the residue characteristic of p is ∈ (C) [cf. §0]. Suppose that s arises from a pro-C birational Galois section of X/k [cf. Definition 5.2]. Then s is geometric at S [cf. Definition 5.11]. In particular, s f determines a unique Ak | S -valued point of X cpt [cf. Definition 5.15]. Proof. The fact that s is geometric at S follows immediately from Proposition 5.24, together with Lemma 5.23, (i). The fact that s determines a f unique Ak | S -valued point of X cpt follows immediately from Lemma 5.22. This completes the proof of Proposition 5.26.

3. Galois sections of tori that locally arise from points In the present §3, we discuss Galois sections of tori that locally arise from points. We maintain the notation of the preceding §2. Let ⊆ Primes be a nonempty subset of Primes [cf. §0]. Write def # def Div(ok ) = Z Pic(ok ) = Pic(Spec ok ) ; f

p∈Pk

dk for the minimal positive integer such that dk · Pic(ok ) = {0}; Gm,Z = P1Z \ {0, ∞} = Spec Z[u ±1 ] ; def

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for each p ∈ Pk , vp : k × −→ Z for the p-adic valuation which induces a surjection kp× Z; divk : k × a

−→ Div(ok ) → f vp (a) · p . p∈P k

[Thus, we have an exact sequence of modules divk

× 0 −→ o× k −→ k −→ Div(ok ) −→ Pic(ok ) −→ 0 .]

Write, moreover, for a ring R, def

Gm,R = Gm,Z ⊗Z R . Let us identify Gm,R (R) with R × by the invertible function u ∈ R[u ±1 ]× : Gm,R (R) R × . Definition 5.27. Let M be a module. Then we shall write def

M[] = lim M/n M ← − – where n ranges over positive -integers [cf. §0]. Lemma 5.28. (i) For every [not necessarily algebraic] extension k of k, if we write GS (Gm,k /k ) for the set of conjugacy classes of pro- Galois sections of Gm,k /k [cf. Definition 5.2], then the map GS (Gm,k /k ) −→ H 1 (k , Gm,k /k ) determined by the natural isomorphism G

m,k /k

∼

−→ Gm,k /k

induced by k → k and the map GS (Gm,k /k ) −→ H 1 (k , G

m,k /k

)

given by mapping an element s ∈ GS (Gm,k /k ) to the element of H 1 (k , G /k ) obtained by considering the difference of s and the elem,k

ment of GS (Gm,k /k ) arising from the k-rational point 1 ∈ (k )×

Gm,k (k ) is bijective.

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∼

(ii) There exists a natural isomorphism Gm,k /k → Z (1) [where “(1)” denotes a Tate twist] such that, for every [not necessarily algebraic] extension k of k, the following diagram of sets commutes: ∼

Gm,k (k ) −−−−→ GS (Gm,k /k ) −−−−→ H 1 (k , G /k ) ⏐ ⏐ m,k ⏐ ⏐ 03 03 (k )×

−−−−→

∼

−−−−→ H 1 (k ,

Z (1)) .

(k )× []

Here, the left-hand upper horizontal arrow is the natural map given by mapping a k -rational point of Gm,k to the conjugacy class of a pro- Galois section of Gm,k /k associated to the k -rational point, the righthand upper horizontal arrow is the bijection of (i), the left-hand vertical arrow is the natural identification by the fixed invertible function u, the right-hand vertical arrow is the isomorphism induced by the isomorphism ∼

in question Gm,k /k → Z (1), the left-hand lower horizontal arrow is the natural homomorphism [cf. Definition 5.27], and the right-hand lower horizontal arrow is the natural isomorphism given by the Kummer theory. f f (iii) Let S ⊆ Pk be a subset of Pk . Then there exists a natural isomorphism between the commutative diagram of sets Gm,k (k) ⏐ ⏐ 3

Gm,k (k)

Gm,k (Ak | S ) −−−−→ Gm,k ( Ak | S ) −−−−→ f

GS (Gm,k /k) ⏐ ⏐ 3

−−−−→

f

p∈S

GS (Gm,kp /kp )

and the commutative diagram of modules k× ⏐ ⏐ 3

k×

−−−−→

f f (Ak | S )× −−−−→ ( Ak | S )× −−−−→

k × [] ⏐ ⏐ 3 × p∈S (k p []) .

Proof. Assertion (i) follows immediately from the various definitions involved. Next, we verify assertion (ii). For a positive integer n, write μn ⊆ k for the group of n-th roots of unity. Then, as is well-known, for a positive -integer n, there exist natural isomorphisms ∼

∼

1 ±1 × ±1 × n H 1 ( Gm,k /k , μn ) −→ H (Gm,k , μn ) ←− k[u ] /(k[u ] ) .

Thus, the invertible function u ∈ k[u ±1 ]× determines an element of ∼

H 1 ( Gm,k /k , Z (1)) −→ Hom(Gm,k /k , Z (1)) .

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On the other hand, one verifies easily that the resulting homomorphism

Gm,k /k → Z (1) is an isomorphism and satisfies the condition in the statement of assertion (ii). This completes the proof of assertion (ii). Assertion (iii) follows immediately from assertion (ii). This completes the proof of Lemma 5.28. Lemma 5.29. The following hold: (i) The exact sequence of modules divk

× 1 −→ o× k −→ k −→ Div(ok )

determines an exact sequence of modules divk []

× 1 −→ o× k [] −→ k [] −→ Div(ok )[] .

(ii) There is no nontrivial element of the cokernel of the natural homomorphism k × → k × [] which is annihilated by a -integer. def

Proof. First, we verify assertion (i). Write M = Im(divk ) ⊆ Div(ok ) for the image of divk . Then since M is a free Z-module, there exists a section of the natural surjection k × M; thus, we obtain a noncanonical isomorphism ∼ × × × o× k × M → k . In particular, the natural homomorphism ok [] → k [] is injective. Thus, to verify assertion (i), it suffices to verify that the kernel of × divk [] is contained in o× k [] ⊆ k [], or, equivalently [by the existence of ∼

× the noncanonical isomorphism o× k × M → k ], the natural homomorphism M[] → Div(ok )[] is injective. f For an element ai · pi ∈ Div(ok ), where ai ∈ Z and pi ∈ Pk , write [ ai · pi ] ∈ Pic(ok ) = Div(ok )/M for the element of the cokernel of divk determined by ai · pi ∈ Div(ok ). Now, for an element a ∈ Pic(ok ), let us f fix a nonarchimedean prime qa ∈ Pk of k such that a = [1 · qa ]. [Note that it follows immediately from Chebotarev’s density theorem that the subset of f f Pk consisting of p ∈ Pk such that a = [1 · p] is of density 1/'Pic(ok ); in def

f

particular, such a qa always exists.] Write T = { qa ∈ Pk | a ∈ Pic(ok ) }. f

def

Moreover, for each p ∈ Pk \ T , write xp = 1 · p − 1 · q[1·p] ∈ Div(ok ). Then one verifies easily that the Z-submodule N ⊆ Div(ok ) generated by f { xp | p ∈ Pk \ T } is contained in M and determines a section of the natural ) projection Div(ok ) Z. In particular, we obtain a commutative f p∈Pk \T diagram of free Z-modules

Conditional results on birational section conjecture 0 −−−−→ N −−−−→ 4 4 4

−−−−→

M ⏐ ⏐ 3

M/N ⏐ ⏐ 3

209

−−−−→ 0

0 −−−−→ N −−−−→ Div(ok ) −−−−→ Div(ok )/N −−−−→ 0 – where the horizontal sequences are exact, and the vertical arrows are injective. On the other hand, since Pic(ok ), hence also T , is finite, M/N and Div(ok )/N are finitely generated free Z-modules. In particular, one verifies easily that the natural homomorphism (M/N )[] → (Div(ok )/N )[] is injective. Thus, it follows immediately that the natural homomorphism in question M[] → Div(ok )[] is injective. This completes the proof of assertion (i). Next, we verify assertion (ii). Now let us observe that one verifies easily that there is no nontrivial element of the cokernel of the natural homomorphism Z →

Z which is annihilated by a -integer. Thus, assertion (ii) follows immediately from the existence of the [noncanonical] isomorphism o× k × ∼ M → k × obtained in the proof of assertion (i), together with the well-known fact that o× k is finitely generated. This completes the proof of assertion (ii). Remark 5.30. The observation given in the proof of Lemma 5.29 was related to the author by A. Tamagawa and S. Yasuda. Lemma 5.31. By applying Lemma 5.28, (iii), let us identify Gm,k (k) f (respectively, Gm,k (Ak ); GS (Gm,k /k); p∈P f GS (Gm,kp /kp )) with k × k f (respectively, (Ak )× ; k × []; p∈P f (kp× [])). Suppose that k is either the k field of rational numbers or an imaginary quadratic field. Let (ap )p∈P f ∈ (Ak )× Gm,k (Ak ) f

f

k

a ∈ k × [] GS (Gm,k /k) be such that their images in p∈P f (kp× []) p∈P f GS (Gm,kp /kp ) [cf. k k the diagrams of Lemma 5.28, (iii)] coincide. Then the following hold: (i) a dk ∈ k × [] GS (Gm,k /k) is contained in the image of the natural homomorphism k × Gm,k (k) → k × [] GS (Gm,k /k). (ii) If dk is a -integer, then a ∈ k × [] GS (Gm,k /k) is contained in the image of the natural homomorphism k × Gm,k (k) → k × []

GS (Gm,k /k). (iii) If dk is a -integer, and we fix an element a ∈ k × Gm,k (k) whose × image in k [] GS (Gm,k /k) coincides with a [cf. (ii)], then, for each f p ∈ Pk whose residue characteristic is ∈ , the difference ap · a −1 ∈ kp× is a root of unity whose order is a (Primes \ )-integer.

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Proof. First, we verify assertion (i). Since the image of a ∈ k × []

GS (Gm,k /k) in p∈P f (kp× [])

f GS (Gm,kp /k p ) is contained p∈P k

k

in the image of the natural homomorphism (Ak )× Gm,k (Ak ) → × f (k p [])

f GS (Gm,kp /kp ), one verifies easily that the image p∈P p∈P k

f

f

k

of a ∈ k × [] by the homomorphism divk [] : k × [] → Div(ok )[] is contained in the Z-submodule Div(ok ) ⊆ Div(ok )[]. Thus, it follows immediately from the definition of dk that there exists b ∈ k × such that the images d k b and a in Div(ok )[] coincide. On the other hand, since k is either the field of rational numbers or an imaginary quadratic field, it holds that o× k is finite, × which thus implies that o× → o [] is surjective. Thus, it follows immek k diately from Lemma 5.29, (i), that, by replacing b by a suitable element of k × , we conclude that a dk coincides with the image of b ∈ k × in k × []. This completes the proof of assertion (i). Assertion (ii) follows immediately from Lemma 5.29, (ii), together with assertion (i); our assumption that dk is a -integer. Finally, we verify assertion f (iii). One verifies easily that, for each p ∈ Pk whose residue characteristic is ∈ , the kernel of the natural homomorphism kp× → kp× [] consists of roots of unity in kp whose orders are (Primes \ )-integers. Thus, assertion (iii) follows immediately from assertion (ii). This completes the proof of assertion (iii). Lemma 5.32. In the notation of Lemma 5.31, if = Primes, then the commutative diagram of sets Gm,k (k) −−−−→ ⏐ ⏐ 3 f

Gm,k (Ak ) −−−−→

GS (Gm,k /k) ⏐ ⏐ 3 f

p∈Pk

GS (Gm,kp /kp )

is cartesian. Proof. This follows immediately from Lemma 5.31, (ii), (iii).

4. Conditional results on the birational section conjecture In the present §4, we prove conditional results on the birational section conjecture for projective smooth curves over number fields. We maintain the notation of the preceding §3. First, let us recall the following result that was essentially proved in [16]. It seems to the author that [at least, a similar result to] the following result

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211

is likely to be well-known to experts. Since, however, the result could not be found in the literature, the author decided to give a proof. Theorem 5.33. Let C be a full formation that contains all finite solvable groups, k a number field [cf. §0], X a projective smooth curve (respectively, hyperbolic curve) over k [cf. §0], and s a pro-C birational Galois section (respectively, locally geometric pro-C Galois section) of X/k [cf. Defif nition 5.2 (respectively, Definitions 5.2; 5.11)]. Write Pk for the set of nonarchimedean primes of k and X cpt for the [uniquely determined] smooth f compactification of X over k. For each p ∈ Pk , write kp for the p-adic completion of k. Then the following conditions are equivalent: (1) s is geometric [cf. Definition 5.3]. f f (2) There exist a subset T ⊆ Pk of Pk of density 0 and a closed subscheme f Z ⊆ X cpt of X cpt which is finite over k such that, for each p ∈ Pk \ T , the [image of the uniquely determined – cf. Lemma 5.22] kp -valued point of X cpt associated to s [cf. Definition 5.20; Proposition 5.24 (respectively, Definition 5.15)] is contained in Z ⊆ X cpt . Proof. First, we verify Theorem 5.33 in the case where s is a locally geometric pro-C Galois section. The implication (1) ⇒ (2) is immediate [cf. also Remark 5.16]. Next, we verify the implication (2) ⇒ (1). Now observe that it follows from the equivalence (1) ⇔ (2) of Lemma 5.8 that, to verify the implication (2) ⇒ (1), by replacing k by a suitable finite extension of k, we may assume without loss of generality that k is totally imaginary. Next, observe that, for each open subgroup H ⊆ CX/k of CX/k containing the image of s, if we write Y for the connected finite étale covering of X corresponding to H ⊆ CX/k [thus, CY /k = H ⊆ CX/k ], then since the morphism Y → X is finite, one verifies easily that the pro-C Galois section of Y/k naturally determined by s [which is necessarily locally geometric by the various definitions involved] satisfies condition (2). Thus, to verify the implication (2) ⇒ (1), by replacing X by such a suitable Y , we may assume without loss of generality that X is of genus ≥ 2; moreover, it follows from the equivalence (1) ⇔ (3) of Lemma 5.8 that, to verify the implication (2) ⇒ (1), by applying the conclusion to various open subgroups of CX/k containing the image of s, it suffices to verify that X cpt (k) = ∅. In particular, since X is of genus ≥ 2, and [one verifies easily that] the pro-C Galois section of X cpt /k naturally determined by s is locally geometric and satisfies condition (2), to verify that X cpt (k) = ∅, by replacing X by X cpt , we may assume without loss of generality that X cpt = X . Now since s is locally geometric, and k is totally imaginary, it follows immediately from the definition of “X (Ak )f-ab • ” [cf. [16], Definition 5.4, (3)] that the

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[uniquely determined] kp -valued points of X associated to s – where p ranges over nonarchimedean primes of k – form a part of an element of X (Ak )f-ab • . Thus, it follows immediately from [16], Theorem 8.2, together with condition (2), that Z (k) = ∅, hence also X (k) = ∅. This completes the proof of the implication (2) ⇒ (1), hence also of Theorem 5.33 in the case where s is a locally geometric pro-C Galois section. Next, we verify Theorem 5.33 in the case where s is a pro-C birational Galois section. The implication (1) ⇒ (2) is immediate [cf. also Remark 5.21]. Next, we verify the implication (2) ⇒ (1). First, observe that it follows immediately from a similar argument to the argument applied in the proof of Theorem 5.33 in the case where s is a locally geometric pro-C Galois section that, to verify Theorem 5.33 in the case where s is a pro-C birational Galois section, by replacing Ck(X )/k by an open subgroup of Ck(X )/k containing the image of s, we may assume without loss of generality that X is of genus ≥ 2; moreover, it follows from the equivalence (1) ⇔ (3) of Lemma 5.9 that, to verify the implication (2) ⇒ (1), by applying the conclusion to various open subgroups of Ck(X )/k containg the image of s, it suffices to verify that X (k) = ∅. On the other hand, since X is of genus ≥ 2, in light of Proposition 5.26, by applying Theorem 5.33 in the case where s is a locally geometric pro-C Galois section to the pro-C Galois section of X/k naturally determined by s, we conclude that X (k) = ∅. This completes the proof of the implication (2) ⇒ (1), hence also of Theorem 5.33 in the case where s is a pro-C birational Galois section. Theorem 5.33 naturally leads to the following corollary that was essentially proved by Stoll [cf., e.g., [16], Theorem 8.6]. Corollary 5.34. Let C be a full formation that contains all finite solvable groups, k a number field [cf. §0], and X a projective smooth curve (respectively, hyperbolic curve) over k [cf. §0]. Suppose that there exist an abelian variety A over k and a nonconstant morphism X → A over k such that both the Mordell–Weil group and the Shafarevich–Tate group of A/k are finite. Then any pro-C birational Galois section (respectively, any locally geometric pro-C Galois section) of X/k [cf. Definition 5.2 (respectively, Definitions 5.2; 5.11)] is geometric [cf. Definition 5.3]. Proof. Write f (A/k) def Sel = lim Ker H 1 (k, A(k)[n]) → H 1 (kp , A(k)) ← − n

f

p∈Pk

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– where the projective limit is over all positive integers n, and A(k)[n] is the subgroup of A(k) consisting of elements of A(k) that are annihilated by n. Then the well-known natural G k -equivariant isomorphism ∼ Primes Primes f (A/k) → A(k)[n] → A/k /n A/k induces a natural injection Sel Primes

H 1 (k, A/k ); moreover, it follows immediately from the various definitions involved that the pro-Primes Kummer homomorphism A(k) → Primes H 1 (k, A/k ) associated to A [cf., e.g., [6], Remark 1.1.4, (iii)] factors f Primes through Sel (A/k) ⊆ H 1 (k, ), which thus implies that we have a A/k

f (A/k). [Here, this injectivity is a formal consenatural injection A(k) → Sel quence of the well-known fact that there is no nontrivial divisible element of A(k).] On the other hand, since the Shafarevich–Tate group of A/k is finite, in light of the fact that the absolute Galois group of the completion of k at an archimedean prime is either Z/2Z or {1}, one verifies easily that, for each positive integer n, the cokernel of the natural homomorphism A(k)/n A(k) −→ Ker H 1 (k, A(k)[n]) → H 1 (kp , A(k)) f

p∈Pk

is annihilated by a positive integer which does not depend on n. Thus, since the Mordell–Weil group of A/k is finite, it follows immediately that the resulting f (A/k) is an isomorphism. injection A(k) → Sel Let s be a pro-C birational Galois section (respectively, locally geometric pro-C Galois section) of X/k. Write s A for the pro-Primes Galois section of A/k obtained as the composite Primes

G k −→ Ck(X )/k −→ CX/k −→ CA/k = A/k s

Primes

(respectively, G k −→ CX/k −→ CA/k = A/k s

)

– where the third (respectively, second) arrow is the homomorphism over G k induced by the nonconstant morphism X → A over k. Then s A natuPrimes rally determines an element of H 1 (k, A/k ) [cf., e.g., [6], Remark 1.1.4, (ii)]; moreover, it follows immediately from Proposition 5.24 (respectively, our assumption that s is locally geometric), together with the various defi∼ f (A/k) ⊆ nitions involved, that this element is contained in A(k) → Sel Primes 1 H (k, A/k ). In particular, since X → A is nonconstant, and the Mordell–Weil group A(k) is finite, it follows immediately from the injectivity of the pro-Primes Kummer homomorphism associated to A ⊗k kp [that is a formal consequence of the well-known fact that there is no nontrivial divisible element of A(kp )], together with [6], Remark 1.1.4, (iii), that s satisfies condition (2) of Theorem 5.33. Thus, it follows from the implication

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(2) ⇒ (1) of Theorem 5.33 that s is geometric. This completes the proof of Corollary 5.34. Remark 5.35. As in the cases of [4], Theorem 17; [16], Theorem 8.6, one may apply Corollary 5.34 to obtain some examples of projective smooth curves over number fields for which any prosolvable birational Galois section [i.e., any pro-C birational Galois section in the case where C consists of all finite solvable groups] is geometric [cf., e.g., the discussions in [4], Remark 18, (1); [16], Example 8.7; [16], Corollary 8.8]. Remark 5.36. The observation given in the proof of Corollary 5.34 was related to the author by A. Tamagawa and S. Yasuda. Definition 5.37. Suppose that X is a hyperbolic curve over [the number field] k. Let s be a pro-C Galois section of X/k [cf. Definition 5.2]. (i) We shall say that s is cuspidal if the image of s is contained in a decomposition subgroup of CX/k associated to a cusp of X/k. (ii) We shall say that s is unramified almost everywhere if the composite G k −→ CX/k −→ Aut(CX/k ) s

– where the second arrow is the action of CX/k on CX/k obtained by f

conjugation – is unramified for all but finitely many p ∈ Pk . Remark 5.38. In the notation of Definition 5.37, it is immediate that if s is cuspidal [cf. Definition 5.37, (i)], then s is geometric [cf. Definition 5.3]. Proposition 5.39. Suppose that is finite. Then any geometric [cf. Definition 5.3] pro- Galois section [cf. Definition 5.2] of a hyperbolic curve over a number field is either cuspidal [cf. Definition 5.37, (i)] or unramified almost everywhere [cf. Definition 5.37, (ii)]. Proof. This follows immediately from Proposition 5.A.50. Next, we prove the main result of the chapter. Theorem 5.40. Let C be a full formation, k either the field of rational numbers or an imaginary quadratic field, X a projective smooth curve over k [cf. §0], and s a pro-C birational Galois section of X/k [cf. Definition 5.2]. f Write ok for the ring of integers of k and Pk for the set of nonarchimedean f primes of k. For each p ∈ Pk , write kp for the p-adic completion of k and f op for the ring of integers of kp . Write, moreover, Ak for the finite part of the adele ring of k, i.e.,

Conditional results on birational section conjecture f def

Ak =

(ap )p∈P f ∈ k

215

" " kp " ap ∈ op for all but finitely many p .

f

p∈Pk

Suppose that the following three conditions are satisfied: (a) The pro-C birational Galois section s is locally geometric [cf. Definition 5.18]. (b) (C) [cf. §0] is cofinite, i.e., Primes \ (C) [cf. §0] is finite. def

(c) Pic(ok ) = Pic(Spec ok ) is annihilated by a (C)-integer [cf. §0]. [Note that it follows from Proposition 5.24 that if (C) = Primes, or, equivalently [cf. §0], C contains all finite solvable groups, then the above three conditions are satisfied.] Then the following conditions are equivalent: (1) The pro-C birational Galois section s is geometric [cf. Definition 5.3]. (2) The following two conditions are satisfied: (2-i) There exist a finite morphism φ : X → P1k over k and, for each f p ∈ Pk , a kp -valued point xp of X associated to s [cf. Definition 5.20; condition (a)] [note that if the residue characteristic of p is ∈ (C), then the kp -valued point x p of X associated to s is uniquely determined – cf. Lemma 5.22] such that the composite φ

xp

Spec kp −→ X −→ P1k determines a kp -valued point of P1k \ {0, 1, ∞} ⊆ P1k . (2-ii) For each open subscheme U ⊆ X of X which is a hyperbolic curve over k [cf. §0], there exists a prime number lU ∈ (C) contained in (C) such that the pro-lU Galois section of U/k [cf. Definition 5.2] naturally determined by s is either cuspidal [cf. Definition 5.37, (i)] or unramified almost everywhere [cf. Definition 5.37, (ii)]. f (3) There exist a finite morphism φ : X → P1k over k and an Ak -valued point xA of X associated to s [cf. Definition 5.20; condition (a)] [note that if f (C) = Primes, then the Ak -valued point x A of X associated to s is uniquely determined – cf. Lemma 5.22] such that the composite f

xA

φ

Spec Ak −→ X −→ P1k f

determines an Ak -valued point of P1k \ {0, 1, ∞} ⊆ P1k . f f (4) There exist a finite subset T ⊆ Pk of Pk and a closed subscheme Z ⊆ X f of X which is finite over k such that, for each p ∈ Pk \ T whose residue characteristic is ∈ (C), the [image of the uniquely determined – cf. Lemma 5.22] kp -valued point x p of X associated to s [cf. Definition 5.20; condition (a)] is contained in Z ⊆ X .

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Proof. The implication (1) ⇒ (2) follows immediately from Proposition 5.39, together with Remark 5.21. Next, we verify the implication (2) ⇒ (3). Supf pose that condition (2) is satisfied. Then, by condition (2-i), for each p ∈ Pk , the composite φ

xp

Spec kp −→ X −→ P1k determines a kp -valued point of P1k \ {0, 1, ∞}. Thus, to verify the implication (2) ⇒ (3), it suffices to verify that the above kp -valued point of P1k \ {0, 1, ∞} obtained as the composite φ ◦ xp determines an op -valued point f of P1op \ {0, 1, ∞} for all but finitely many p ∈ Pk . Write U ⊆ X for the open subscheme of X obtained as the inverse image of P1k \ {0, 1, ∞} ⊆ P1k by φ. Then, by condition (2-ii), there exists a prime number lU ∈ (C) contained in (C) such that the pro-lU Galois section s U of U/k obtained as the composite {l }

U G k −→ Ck(X )/k −→ U/k

s

is either cuspidal or unramified almost everywhere. Write s P for the pro-lU Galois section of P1k \ {0, 1, ∞} obtained as the composite {l }

U G k −→ Ck(X )/k −→ U/k −→

s

{lU } (P1k \{0,1,∞})/k

– where the third arrow is the homomorphism over G k induced by φ. Then since the morphism U → P1k \ {0, 1, ∞} induced by φ is finite, one verifies {lU } {l } easily that the homomorphism U/k → U1 maps injectively any {l }

(Pk \{0,1,∞})/k

U cuspidal decomposition subgroup of U/k associated to a cusp of U/k to a

cuspidal decomposition subgroup of

{lU } (P1k \{0,1,∞})/k

associated to a cusp of

P1k \ {0, 1, ∞}. Thus, it follows immediately that if s U is cuspidal, then s P is f cuspidal. On the other hand, by applying Lemma 5.7 [to “φ ◦ xp ” for p ∈ Pk whose residue characteristic is = lU ], it follows immediately from condition (2-i) that s P is not cuspidal. Thus, we conclude that s U is not cuspidal, hence also [by condition (2-ii)] unramified almost everywhere. In particular, it follows Proposition 5.A.55, (ii), that s P is unramified almost everywhere. Therefore, it follows immediately from Proposition 5.A.50, together with condition (2-i), that the kp -valued point of P1k obtained as the composite φ ◦ xp determines an op -valued point of P1op \ {0, 1, ∞} for all but finitely many f

p ∈ Pk . This completes the proof of the implication (2) ⇒ (3). Next, we verify the implication (3) ⇒ (4). Suppose that condition (3) is def

satisfied. Write s G for the pro-(C) Galois section of Gm,k = P1k \ {0, ∞} over k obtained as the composite

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217

(C )

G k → Ck(X )/k → Ck(P1 )/k CGm,k /k = Gm,k /k s

k

– where the second arrow is the homomorphism over G k induced by φ – and t G for the pro-(C) Galois section of Gm,k over k obtained as the composite ∼

(C )

G k → Ck(X )/k → Ck(P1 )/k → Ck(P1 )/k CGm,k /k = Gm,k /k s

k

k

– where the second arrow is the homomorphism over G k induced by φ, and the third arrow is the automorphism over G k induced by the automorphism of P1k over k given by “u → 1 − u”. Then it follows immediately f from condition (3) that there exists an element (ap )p∈P f ∈ ( Ak )× such that k

(ap )p∈P f , (1 − ap )p∈P f ∈ (Ak )× Gm,k (Ak ), and, moreover, the respecf

k

f

k

tive images of the pro-(C) Galois sections s G , t G ∈ GS(C ) (Gm,k /k)

k × [(C)] [cf. Lemma 5.28, (i), (ii)] in the set p∈P f GS(C ) (Gm,kp /kp )

k × f (k p [(C)]) [cf. the diagrams of Lemma 5.28, (iii)] coincide with the p∈P k

respective images of the elements (ap )p∈P f , (1 − ap )p∈P f ∈ (Ak )×

k k f Gm,k (Ak ) in the set p∈P f (kp× [(C)]) p∈P f GS(C ) (Gm,kp /kp ). Thus, k k it follows from Lemma 5.31, (ii), (iii); together with condition (c), that f

(∗): there exist as , at ∈ k × such that, for p ∈ Pk , if we write u p = ap · as−1 , def

f

vp = (1 − ap ) · at−1 ∈ kp× , and the residue characteristic of p is ∈ (C), then u p , vp are roots of unity of kp whose orders are (Primes \ (C))-integers. def

f

Now let us observe that, for p ∈ Pk , the pair (u p , vp ) satisfies the equation 1 = as · u p + at · v p . Thus, it follows immediately from [2], Theorem 1.1, together with condition (b), that the set {(u p , vp )}p∈P f , hence also the set {u p }p∈P f , is finite. In partick k ular, since ap = as ·u p [cf. (∗)], it follows immediately that the pro-C birational Galois section of P1k /k obtained as the composite G k −→ Ck(X )/k −→ Ck(P1 )/k s

k

– where the second arrow is the homomorphism over G k induced by φ – satisfies condition (4). Therefore, since φ is finite, one verifies easily that the pro-C birational Galois section s satisfies condition (4). This completes the proof of the implication (3) ⇒ (4). Finally, we verify the implication (4) ⇒ (1). Suppose that condition (4) is f f satisfied. Let us fix an element p0 ∈ Pk \ T of Pk \ T such that the residue

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Yuichiro Hoshi

characteristic of p0 is ∈ (C) [note that, by condition (b), such a p0 always exists] and write r (p0 ) for the cardinality of the set of roots of unity of kp0 . Now observe that, for any open subgroup H ⊆ Ck(X )/k of Ck(X )/k containing the image of s, if we write Y for the normalization of X in the finite extension of k(X ) corresponding to H ⊆ Ck(X)/k [thus, Ck(Y )/k = H ⊆ Ck(X )/k ], then since the morphism Y → X is finite, the pro-C birational Galois section of Y/k f determined by s satisfies condition (4) relative to the finite subset “T ” ⊆ Pk appearing in condition (4). Thus, it follows from the equivalence (1) ⇔ (3) of Lemma 5.9 that, to verify condition (1), by applying the conclusion to various such H ’s, it suffices to verify that X admits a k(ζr(p0 ) )-valued point – where we use the notation ζr(p0 ) ∈ k to denote a primitive r(p0 )-th root of unity. f For each p ∈ Pk , let us fix a kp -valued point xp of X associated to s [cf. condition (a)]. Now since X is projective, there exists a closed immersion X → PkN over k for some positive integer N . Then it follows immediately from condition (4) that there exists a hyperplane H ⊆ PkN defined over k such f that, for any p ∈ Pk , [the image of] the fixed kp -valued point xp of X is contained in X \ (X ∩ H ) ⊆ PkN \ H AkN . Moreover, again by condition (4) – by considering a suitable automorphism of A1k over k – we may assume without f loss of generality that, for each i ∈ {1, · · · , N } and p ∈ Pk , the kp -valued point of A1k obtained as the composite xp

pri

Spec kp → X \ (X ∩ H ) → PkN \ H AkN → A1k def

factors through Gm,k = A1k \{0} ⊆ A1k . Therefore, we conclude that there exist an open subscheme U ⊆ X of X and a closed immersion U → Gm,k ×k · · ·×k def f Gm,k over k such that the A -valued point xA = (xp ) f of X determined by k

p∈Pk

the fixed kp -valued points xp lies on U . On the other hand, again by condition f (4), one verifies easily that, for each i ∈ {1, · · · , N }, the Ak -valued point of Gm,k obtained as the composite f xA

pri

Spec Ak → U → Gm,k ×k · · · ×k Gm,k → Gm,k f

determines an Ak -valued point of Gm,k . Thus, it follows immediately from Lemma 5.31, (ii), (iii); condition (c), that, for each i ∈ {1, · · · , N }, the kp0 valued point of Gm,k obtained as the composite x p0

pri

Spec kp0 → U → Gm,k ×k · · · ×k Gm,k → Gm,k determines a k(ζr(p0 ) )-valued point of Gm,k . In particular, since U → Gm,k ×k · · · ×k Gm,k is a closed immersion, one verifies easily that the kp0 -valued

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219

point xp0 of U , hence also X , determines a k(ζr(p0 ) )-valued point. This completes the proof of the implication (4) ⇒ (1), hence also of Theorem 5.40. Remark 5.41. (i) Theorem 5.40 is a result without any assumption on the finiteness of a Shafarevich–Tate group. (ii) The equivalence (1) ⇔ (3) of Theorem 5.40 may be regarded as a tripod analogue of [4], Theorem 17. The condition that k is either the field of rational numbers or an imaginary quadratic field [i.e., the assumption that o× k is finite] in the statement of Theorem 5.40 may be regarded as an analogue of the finiteness condition on the Mordell–Weil group in the statement of [4], Theorem 17; the condition that Pic(ok ) is annihilated by a (C)-integer in the statement of Theorem 5.40 may be regarded as an analogue of the finiteness condition on the Shafarevich–Tate group in the statement of [4], Theorem 17. On the other hand, since any abelian variety is proper, in the case of [4], Theorem 17, the condition corresponding to our condition that the birational Galois section determines [not only an f f Ak -valued point but also] an Ak -valued point of the tripod P1k \ {0, 1, ∞} in Theorem 5.40 is automatically satisfied. (iii) If C contains all finite solvable groups, then Theorem 5.33 implies the equivalence (1) ⇔ (4) of Theorem 5.40. (iv) One verifies easily that the proof of the equivalence (1) ⇔ (4) of Theorem 5.40 gives us an alternative proof of Corollary 5.34 in the case where s is a pro-C birational Galois section, and k is either the field of rational numbers or an imaginary quadratic field. Indeed, in the notation of Corollary 5.34, it follows from the argument given in the proof of Corollary 5.34 that every pro-C birational Galois section s of X/k satisfies condition (4) of Theorem 5.40. Thus, it follows from the equivalence (1) ⇔ (4) of Theorem 5.40 that s is geometric. Corollary 5.42. Let k be either the field of rational numbers or an imagidef

nary quadratic field and k an algebraic closure of k. Write G k = Gal(k/k) f f and Pk for the set of nonarchimedean primes of k. For each p ∈ Pk , write kp for the p-adic completion of k. Then the following assertions are equivalent: (1) Any pro-Primes birational Galois section [cf. Definition 5.2] of any projective smooth curve over k [cf. §0] is geometric [cf. Definition 5.3]. (2) Any pro-Primes birational Galois section of P1k /k is geometric. (3) Any pro-Primes birational Galois section s of P1k /k satisfies the following two conditions:

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Yuichiro Hoshi

(3-i) There exist three distinct elements a, b, c ∈ P1k (k) of P1k (k) such that, for any nonarchimedean prime p of k, the [uniquely determined – cf. Lemma 5.22] kp -valued point of P1k associated to s [cf. Definition 5.20; Proposition 5.24] is ∈ {a, b, c} ⊆ (P1k (k) ⊆) P1k (kp ). (3-ii) There exists a prime number l such that the pro-l Galois section of P1k \ {0, 1, ∞} [cf. Definition 5.2] naturally determined by s is either cuspidal [cf. Definition 5.37, (i)] or unramified almost everywhere [cf. Definition 5.37, (ii)]. (4) Any pro-Primes birational Galois section s of P1k /k satisfies the following two conditions: (4-i) There exist three distinct elements a, b, c ∈ P1k (k) of P1k (k) such that, for any nonarchimedean prime p of k, the [uniquely determined – cf. Lemma 5.22] kp -valued point of P1k associated to s [cf. Definition 5.20; Proposition 5.24] is ∈ {a, b, c} ⊆ (P1k (k) ⊆) P1k (kp ). (4-ii) Write s P for the pro-Primes Galois section of P1k \ {0, 1, ∞} naturally determined by s. Then it holds either that s P is cuspidal, or that there exists a prime number l such that the l-adic Galois representation sP

Primes (P1k \{0,1,∞})/k

G k −→

−→ GL2 (Zl )

– where we refer to Definition 5.1, (ii), concerning the profiPrimes Primes nite group 1 ; the second arrow 1 → (Pk \{0,1,∞})/k

(Pk \{0,1,∞})/k

GL2 (Zl ) is the l-adic representation determined by the Legendre family of elliptic curves over P1k \ {0, 1, ∞}, i.e., the elliptic curve over P1k \ {0, 1, ∞} = Spec k[u ±1 , (1 − u)−1 ] determined by the equation “y 2 = x(x − 1)(x −u)” – is unramified at all but finitely f many p ∈ Pk . Proof. The implications (1) ⇒ (2) ⇒ (4) are immediate [cf. also Remark 5.21]. On the other hand, the implication (2) ⇒ (1) follows immediately from the fact that any projective smooth curve over k may be obtained as the normalization of P1k in the finite extension of k(P1k ) corresponding to an Primes open subgroup of 1 . Moreover, let us observe that it follows immedik(Pk )/k

ately from Proposition 5.39, together with Remark 5.21, that the implications (2) ⇒ (3) holds. Finally, we verify the implication (3) ⇒ (2) (respectively, (4) ⇒ (2)). Suppose that assertion (3) (respectively, assertion (4)) holds. Let s be a pro-Primes birational Galois section of P1k /k. For each nonarchimedean prime p of k, write xp for the [uniquely determined] kp -valued point of P1k associated to s. Then it

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221

follows from condition (3-i) (respectively, condition (4-i)) that, by considering a suitable automorphism of P1k over k, we may assume without loss of generalf ity that, for any p ∈ Pk , xp ∈ P1k (kp ) is ∈ {0, 1, ∞} ⊆ P1k (kp ). Thus, for any f prime number l, by applying Lemma 5.7 [to “x p ” for p ∈ Pk whose residue characteristic is = l], it follows immediately that the pro-l Galois section s P,{l} of P1k \ {0, 1, ∞} obtained as the composite s

G k −→

Primes k(P1k )/k

−→

{l} (P1k \{0,1,∞})/k

,

hence also the pro-Primes Galois section s P of P1k \ {0, 1, ∞} obtained as the composite s

G k −→

Primes k(P1k )/k

−→

Primes (P1k \{0,1,∞})/k

,

is not cuspidal. Thus, by condition (3-ii) (respectively, condition (4-ii)), we conclude that there exists a prime number l0 such that s P,{l0 } is unramified almost everywhere (respectively, the l0 -adic Galois representation obtained as the displayed composite of condition (4-ii) is unramified at all but finitely many f f p ∈ Pk ). Thus, since [we have assumed that] for any p ∈ Pk , xp ∈ P1k (kp ) 1 is ∈ {0, 1, ∞} ⊆ Pk (kp ), it follows immediately from Proposition 5.A.50 (respectively, [14], Theorem 1) that the birational pro-Primes Galois section s of P1k /k satisfies condition (3) of Theorem 5.40, hence also [by the equivalence (1) ⇔ (3) of Theorem 5.40] that s is geometric. This completes the proof of the implication (3) ⇒ (2) (respectively, (4) ⇒ (2)), hence also of Corollary 5.42.

Appendix A. Ramification of Galois sections In the present §A, we discuss the ramification of Galois sections of hyperbolic curves over p-adic local fields. In the present §A, let ⊆ Primes be a nonempty subset of Primes [cf. §0], k a p-adic local field for some prime number p [cf. §0], k an algebraic closure of k, and X a hyperbolic curve over k [cf. §0]. For a finite extension k (⊆ k) of k, write G k = Gal(k/k ) , def

Ik ⊆ G k for the inertia subgroup of G k , and ok ⊆ k

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Yuichiro Hoshi

for the ring of integers of k . Write, moreover, X cpt for the [uniquely determined] smooth compactification of X over k; X/k for the pro- geometric fundamental group of X , i.e., the maximal pro- quotient of π1 (X ⊗k k); X/k for the geometrically pro- fundamental group of X , i.e., the quotient of π1 (X ) by the kernel of the natural surjection π1 (X ⊗k k) X/k . Thus, we have an exact sequence of profinite groups [cf. [3], Exposé IX, Théorème 6.1] 1 −→ X/k −→ X/k −→ G k −→ 1 .

Let s be a pro- Galois section of X/k [cf. [6], Definition 1.1, (i)], i.e., a section of the above exact sequence of profinite groups. Definition 5.A.43. We shall say that s is unramified (respectively, potentially unramified) if the image of the composite s

Ik −→ G k −→ X/k −→ Aut( X/k ) – where the third arrow is the action of X/k on X/k obtained by conjugation – is trivial (respectively, finite).

Proposition 5.A.44. The following hold: (i) If p ∈ , then any pro- Galois section of X/k [cf. [6], Definition 1.1, (i)] is not potentially unramified, hence also not unramified [cf. Definition 5.A.43]. (ii) If X does not admit good reduction over ok [cf. §0], then any pro- Galois section of X/k is not unramified. Proof. Let s be a pro- Galois section of X/k. First, we verify assertion (i). Now one verifies easily that there exists a characteristic open subgroup H ⊆ X/k of X/k such that the connected finite étale covering of X corresponding to the open subgroup H · Im(s) of X/k topologically generated by H and Im(s) is of genus ≥ 1. On the other hand, since H ⊆ X/k is characteristic, and [as is well-known] X/k is slim [cf. §0], it follows from [7], Lemma 5, that we have a natural injection Aut( X/k ) → Aut(H ). Thus, to verify assertion (i), by replacing by the open subgroup H ·Im(s), we may assume without X/k

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223

loss of generality that X cpt is of genus ≥ 1. Next, let us observe that, as is well-known, since p ∈ , and X cpt is of genus ≥ 1, there exist G k -equivariant isomorphisms 2 cpt H 2 ( ⊗k k, Z p ) Z p (1) X cpt /k , Z p ) H (X

– where “(1)” denotes a Tate twist. In particular, [the restriction to Ik of] the p-adic cyclotomic representation χ p : Ik → Aut(Z p (1)) factors through the displayed composite of Definition 5.A.43. On the other hand, one may verify easily that the image of χ p is infinite. Thus, s is not potentially unramified. This completes the proof of assertion (i). Next, we verify assertion (ii). Suppose that X does not admit good reduction over ok . Now it follows from assertion (i) that, to verify assertion (ii), we may assume without loss of generality that p ∈ . Then it follows immediately from [17], Theorem 0.8, that the image of the composite Ik −→ Aut( X/k ) −→ Out( X/k )

– where the first arrow is the displayed composite of Definition 5.A.43 – is nontrivial, hence that s is not unramified. This completes the proof of assertion (ii). Definition 5.A.45. If X admits good reduction X over ok [cf. §0], then we shall write (π (X ) ) -ét 1

X/k

X/k

for the quotient of π1 (X ) by the normal closed subgroup topologically normally generated by the kernels of the natural surjections π1 (X ) X/k , π1 (X ) π1 (X ). Thus, the natural surjection X/k G k determines a -ét surjection X/k G k /Ik . We shall write -ét X/k -ét for the kernel of the surjection X/k G k /Ik . Thus, we have a commutative diagram of profinite groups 1 −−−−→ → −−−→ X/k −−−− X/k − ⏐ ⏐ ⏐ ⏐ 3 3

Gk ⏐ ⏐ 3

−−−−→ 1

-ét -ét −−−→ G /I −−−−→ 1 1 −−−−→ → k k X/k −−−− X/k − – where the horizontal sequences are exact.

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Remark 5.A.46. In the notation of Definition 5.A.45, as is well-known, if -ét p ∈ , then the left-hand vertical arrow X/k → X/k of the commutative diagram of Definition 5.A.45 is an isomorphism. In particular, the right-hand upper horizontal arrow X/k → G k induces an isomorphism ∼ -ét Ker( ) → I , and the right-hand square of the commutative X/k

k

X/k

diagram of Definition 5.A.45 is cartesian. Proposition 5.A.47. The following conditions are equivalent: (1) s is unramified [cf. Definition 5.A.43]. (2) p ∈ , X admits good reduction over ok [cf. §0], and the image of the composite s I → G → -ét k

k

X/k

X/k

[cf. Definition 5.A.45] is trivial. (3) p ∈ , X admits good reduction over ok , and the composite s

Ik → G k → X/k determines an isomorphism ∼ -ét Ik −→ Ker( X/k X/k ) . (4) p ∈ , and, for any open subgroup H ⊆ X/k of X/k containing the image of s, the connected finite étale covering of X corresponding to H ⊆ X/k admits good reduction over ok .

Proof. First, we verify the equivalence (1) ⇔ (2). It follows immediately from Proposition 5.A.44 that both (1) and (2) imply that p ∈ , and that X admits good reduction over ok . Thus, suppose that these conditions are satisfied. Write J for the image of the displayed composite of condition (2). Then it follows immediately from the existence of the commutative diagram -ét -ét of Definition 5.A.45 that J ⊆ X/k ⊆ X/k . Thus, it follows immediately from Remark 5.A.46 that the displayed composite Ik → Aut( X/k ) of Definition 5.A.43 factors as -ét -ét ∼ Ik J → X/k → Aut( X/k ) ← Aut( X/k ) -ét -ét – where the third arrow is the action of X/k on X/k obtained by conju∼ gation. Now since, as is well-known, → -ét is center-free, the third X/k

X/k

arrow of this composite is injective. Therefore, it follows immediately that the condition that s is unramified is equivalent to the condition that J = {1}. This completes the proof of the equivalence (1) ⇔ (2).

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The equivalence (2) ⇔ (3) follows immediately from Remark 5.A.46. Next, we verify the implication (3) ⇒ (4). Suppose that condition (3) is satisfied. Then it is immediate that if an open subgroup of X/k contains the image of -ét ; thus, it follows immediately s, then it arises from an open subgroup of X/k from the various definitions involved that the corresponding connected finite étale covering of X admits good reduction over ok . This completes the proof of the implication (3) ⇒ (4). Finally, we verify the implication (4) ⇒ (3). Suppose that condition (4) is satisfied. Let H ⊆ X/k be an open subgroup of X/k containing the image of s. Write Y → X for the connected finite étale covering of X corresponding to H ⊆ X/k ; thus, Y /k = H ⊆ X/k . Then it follows from condition (4) that Y admits good reduction over ok . Thus, it follows from [9], Lemma 8.3, that the morphism Y → X extends to a morphism between their [uniquely determined] smooth models. In particular, it follows immediately -ét -ét from the definitions of X/k and Y /k that the inclusion Y/k ⊆ X/k -ét -ét determines an inclusion Ker( Y/k Y /k ) ⊆ Ker( X/k X/k ). Thus, -ét it follows immediately from Remark 5.A.46 that Ker(Y /k Y /k ) = -ét -ét Ker( X/k X/k ), hence that Ker( X/k X/k ) ⊆ Y/k = H . Therefore, by considering the intersection of such H ’s, we obtain that Ker( X/k -ét X/k ) ⊆ Im(s). Thus, again by Remark 5.A.46, we conclude that condition (3) holds. This completes the proof of the implication (4) ⇒ (3), hence also of Proposition 5.A.47. Lemma 5.A.48. Suppose that p ∈ , and that X admits good reduction -ét -ét over ok [cf. §0]. Let ⊆ X/k be an open subgroup of X/k . Write so s -ét for the composite G k → X/k X/k , k (⊆ k) for the [necessarily unramified] finite extension of k corresponding to the image of the compos-ét ite → X/k G k /Ik , Y → X for the connected finite étale covering of -ét cpt for the [uniquely X corresponding to the open subgroup ⊆ X/k , and Y determined] smooth compactification of Y over k . [Here, it follows immediately from the various definitions involved that Y is a hyperbolic curve over -ét -ét k ; Y , hence also Y cpt , admits good reduction over ok ; Y /k = ⊆ X/k .] Suppose, moreover, that Y is of genus ≥ 2. Then the image of the composite s (I ) ∩ -ét → -ét -ét ( -ét )ab o

k

Y/k

Y /k

Y cpt /k

Y cpt /k

– where the second arrow is the surjection induced by the open immersion Y → Y cpt – is trivial. Proof. This follows immediately from a similar argument to the argument used in the proof of assertion (ii) in the proof of [6], Lemma 3.3.

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Proposition 5.A.49. Suppose that p ∈ , and that X admits good reduction over ok [cf. §0]. Then the following conditions are equivalent: (1) s is ramified, i.e., not unramified [cf. Definition 5.A.43]. (2) The image of the composite s -ét Ik → G k → X/k X/k

-ét is a nontrivial closed subgroup of a cuspidal inertia subgroup of X/k associated to a cusp of X/k. (3) The image of the composite s

Ik → G k → X/k X/k – where X/k is the quotient of X/k defined in [7], Definition 1, (iv), i.e., the quotient of X/k by the kernel Z ( X/k ) of the homomorphism X/k

X/k → Aut( X/k ) obtained by conjugation – is a nontrivial closed subgroup of a cuspidal inertia subgroup of X/k associated to a cusp of X/k. (4) There exists an element l ∈ of such that the pro-l Galois section of X/k [cf. [6], Definition 1.1, (i)] naturally determined by s is ramified.

Proof. First, we verify the equivalence (1) ⇔ (2). It follows from the equivalence (1) ⇔ (2) of Lemma 5.A.47 that s is ramified if and only if the image of the composite of condition (2) is nontrivial. On the other hand, it follows immediately from the existence of the commutative diagram of Definition 5.A.45, together with Remark 5.A.46, that the composite of condition (2) factors through the maximal pro- quotient of Ik , which is, as is wellknown, procyclic. Thus, it follows immediately from Lemma 5.A.48, together with [5], Lemma 1.6, that the equivalence (1) ⇔ (2) holds. This completes the proof of the equivalence (1) ⇔ (2). Next, let us observe that the implication (3) ⇒ (1) follows immediately from the various definitions involved [cf. also the definition of the quotient X/k ]. Next, we verify the implication (2) ⇒ (3). Since, as is well-known, X/k is slim [cf. §0], it follows from [7], Proposition 6, (ii), together with Remark 5.A.46, that we have a -ét sequence of natural surjections X/k X/k X/k , which induces an ∼ injection → -ét → . Thus, one verifies easily that the implicaX/k

X/k

X/k

tion (2) ⇒ (3) holds. This completes the proof of the implication (2) ⇒ (3). Finally, we verify the equivalence (1) ⇔ (4). For each nonempty subset ⊆ of , write J for the image of the composite

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227

-ét Ik → G k → X/k X/k X/k .

Then it follows immediately from the verified equivalence (1) ⇔ (2), together with the well-known structure of the maximal pro- quotient of the fundamental group of a smooth curve over an algebraically closed field of characteristic ∈ , that, for each nonempty subset ⊆ of , the image J is procyclic, and, moreover, J is the maximal pro- quotient of J [relative to the natural surjection J J ]. In particular, we conclude that J = {1} if and only if J{l} = {1} for any l ∈ , i.e., the equivalence (1) ⇔ (4) holds. This completes the proof of the equivalence (1) ⇔ (4), hence also of Proposition 5.A.49. Proposition 5.A.50. Suppose that p ∈ , and that s is geometric [cf. [6], Definition 1.1, (iii)]. Let x ∈ X cpt (k) be a k-rational point of X cpt such that a decomposition subgroup of X/k associated to x contains the image of s. Consider the following three conditions: (1) s is unramified [cf. Definition 5.A.43]. (2) x ∈ X (k), and the hyperbolic curve X \ {x}, hence also the hyperbolic curve X , over k admits good reduction over ok [cf. §0]. (3) x ∈ X (k). Then we have implications (2) =⇒ (1) =⇒ either (2) or (3) . In particular, if x ∈ X (k), then we have an equivalence (1) ⇐⇒ (2) . Proof. To verify Proposition 5.A.50, it is immediate that it suffices to verify that if x ∈ X (k), then condition (1) is equivalent to condition (2). Thus, suppose that x ∈ X (k). Now let us observe that it follows immediately from Proposition 5.A.44, (ii), that both (1) and (2) imply that X admits good reduction over ok . Thus, we may assume without loss of generality that X admits good reduction over ok . Moreover, observe that it follows immediately from the equivalence (1) ⇔ (4) of Proposition 5.A.49 that, to verify the equivalence (1) ⇔ (2), by considering the pro-l Galois section of X/k naturally determined by s – where l ranges over elements of – we may assume without loss of generality that is of cardinality 1. On the other hand, since is of cardinality 1, it follows immediately from [7], Proposition 19, (ii), that s the kernel of the composite G k → X/k → Aut( X/k ) coincides with the kernel of the pro- outer Galois representation associated to the hyperbolic curve X \ {x} over k. Thus, the equivalence (1) ⇔ (2) follows immediately from [17], Theorem 0.8. This completes the proof of Proposition 5.A.50.

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Yuichiro Hoshi

Proposition 5.A.51. Suppose that p ∈ , that X admits good reduction over ok [cf. §0], and that X is proper over k. Then any pro- Galois section of X/k [cf. [6], Definition 1.1, (i)] is unramified [cf. Definition 5.A.43]. Proof. This follows immediately from the equivalence (1) ⇔ (2) of Proposition 5.A.49. Remark 5.A.52. Proposition 5.A.51 may be regarded as a Galois section version of the valuative criterion for properness of morphisms of schemes. Remark 5.A.53. In [13], Saïdi proved the existence of a nongeometric pro Galois section in the situation of Proposition 5.A.51 [cf. [13], Proposition 4.2.1]. Proposition 5.A.54. Suppose that p ∈ , and that X admits good reduction over ok [cf. §0]. Then s is unramified if and only if s is potentially unramified [cf. Definition 5.A.43]. Proof. This follows immediately from the equivalence (1) ⇔ (2) of Proposition 5.A.49, together with the well-known fact that any cuspidal inertia -ét

subgroup of X/k associated to a cusp of X/k is isomorphic to Z as an abstract profinite group. Proposition 5.A.55. Let Y be a hyperbolic curve over k and X → Y a dominant morphism over k. Write sY for the pro- Galois section of Y /k [cf. [6], s Definition 1.1, (i)] determined by s, i.e., the composite G k → X/k → Y/k . Then the following hold: (i) Write X/k , Y /k for the respective quotients of X/k , Y /k defined in [7], Definition 1, (iv) [cf. also the statement of condition (3) of Proposi tion 5.A.49]. Then the natural homomorphism X/k → Y/k induces a homomorphism X/k → Y/k . (ii) If s is unramified (respectively, potentially unramified) [cf. Definition 5.A.43], then sY is unramified (respectively, potentially unramified). (iii) Suppose that X → Y is finite, and that X and Y admit good reduction over ok [cf. §0]. Then s is unramified if and only if sY is unramified.

Proof. First, we verify assertion (i). Now since, as is well-known, the profi nite group Y/k is slim [cf. §0], for any open subgroup H ⊆ Y /k of Y/k , it follows immediately from [7], Lemma 5, that N (H ) ∩ Z (Y /k ) = Y/k Y/k Z (H ). Thus, it follows immediately from the fact that the natural homoY/k

morphism X/k → Y /k is open that the natural homomorphism X/k →

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229

Y/k induces a homomorphism X/k → Y /k . This completes the proof of assertion (i). Assertion (ii) follows immediately from the various definitions involved, together with assertion (i) [cf. also the definitions of the quotients X/k , Y /k ]. Finally, we verify assertion (iii). It follows from Proposition 5.A.44, (i), that both the condition that s is unramified and the condition that sY is unramified imply that p ∈ . Thus, suppose that p ∈ . On the other hand, since X → Y is finite, one verifies easily that the restriction of the natural homomorphism X/k → Y /k to any cuspidal inertia sub group of X/k associated to a cusp of X/k is injective. Thus, assertion (iii) follows immediately from assertions (i), (ii); the equivalence (1) ⇔ (3) of Proposition 5.A.49, together with the fact that the sequences of natural surjec -ét -ét tions X/k X/k X/k , Y/k Y/k Y /k induce injections ∼ ∼ → -ét → , → -ét → , respectively [cf. the X/k

X/k

X/k

Y/k

Y /k

Y /k

proof of the implication (2) ⇒ (3) of Proposition 5.A.49]. This completes the proof of assertion (iii).

References [1] H. Esnault and O. Wittenberg, On abelian birational sections, J. Amer. Math. Soc. 23 (2010), no. 3, 713–724. [2] J.-H. Evertse, H. P. Schlickewei, and W. M. Schmidt, Linear equations in variables which lie in a multiplicative group, Ann. of Math. (2) 155 (2002), no. 3, 807–836. [3] A. Grothendieck et al., Revêtements Étales et Groupe Fondamental (SGA 1), Séminaire de géométrie algébrique du Bois Marie 1960–61, Documents Mathématiques, 3. Société Mathématique de France, Paris, 2003. [4] D. Harari and J. Stix, Descent obstruction and fundamental exact sequence, The Arithmetic of Fundamental Groups - PIA 2010, 147–166, Contributions in Mathematical and Computational Sciences, vol. 2, Springer-Verlag, Berlin Heidelberg, 2012. [5] Y. Hoshi and S. Mochizuki, On the combinatorial anabelian geometry of nodally nondegenerate outer representations, Hiroshima Math. J. 41 (2011), no. 3, 275– 342. [6] Y. Hoshi, Existence of nongeometric pro- p Galois sections of hyperbolic curves, Publ. Res. Inst. Math. Sci. 46 (2010), no. 4, 829–848. [7] Y. Hoshi, On monodromically full points of configuration spaces of hyperbolic curves, The Arithmetic of Fundamental Groups - PIA 2010, 167–207, Contributions in Mathematical and Computational Sciences, vol. 2, Springer-Verlag, Berlin Heidelberg, 2012. [8] J. Koenigsmann, On the ‘section conjecture’ in anabelian geometry, J. Reine Angew. Math. 588 (2005), 221–235.

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[9] S. Mochizuki, The profinite Grothendieck conjecture for closed hyperbolic curves over number fields, J. Math. Sci. Univ. Tokyo 3 (1996), no. 3, 571–627. [10] S. Mochizuki, The local pro- p anabelian geometry of curves, Invent. Math. 138 (1999), no. 2, 319–423. [11] S. Mochizuki, Topics surrounding the anabelian geometry of hyperbolic curves, Galois groups and fundamental groups, 119–165, Math. Sci. Res. Inst. Publ. 41, Cambridge University Press, Cambridge, 2003. [12] F. Pop, On the birational p-adic section conjecture, Compos. Math. 146 (2010), no. 3, 621–637. [13] M. Saïdi, Good sections of arithmetic fundamental groups, arXiv:1010.1313, 2010. [14] J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. [15] J. Stix, On the birational section conjecture with local conditions, arXiv:1203.3236, 2012. [16] M. Stoll, Finite descent obstructions and rational points on curves, Algebra Number Theory 1 (2007), no. 4, 349–391. [17] A. Tamagawa, The Grothendieck conjecture for affine curves, Compositio Math. 109 (1997), no. 2, 135–194. [18] K. Wickelgren, 2-nilpotent real section conjecture, arXiv:1006.0265, 2010.

6 Blocks for mod p representations of GL2 (Q p ) Vytautas Pašk¯unas

Abstract Let π1 and π2 be absolutely irreducible smooth representations of G = GL2 (Q p ) with a central character, defined over a finite extension of F p . We show that if there exists a non-split extension between π1 and π2 then they both appear as subquotients of the reduction modulo p of a unit ball in a crystalline Banach space representation of G. The results of Berger–Breuil describe such reductions and allow us to organize the irreducible representation into blocks. The result is new for p = 2; the proof, which works for all p, is new.

1. Introduction Let L be a finite extension of Q p , with the ring of integers O, a uniformizer $ , and residue field k, and let G = GL2 (Q p ) and let B be the subgroup of upper-triangular matrices in G. Theorem 6.1. Let π1 , π2 be smooth, absolutely irreducible k-representations of G with a central character. Suppose that Ext1G (π2 , π1 ) = 0 then after replacing L by a finite extension, we may find integers (l, k) ∈ Z × N and × unramified characters χ1 , χ2 : Q× p → L with χ2 = χ1 | |, such that π1 and ss ss π2 are subquotients of , where is the semi-simplification of the reduction modulo $ of an open bounded G-invariant lattice in , where is the universal unitary completion of −1 l k−1 2 (IndG L . B χ1 ⊗ χ2 | | )sm ⊗ det ⊗ Sym

Automorphic Forms and Galois Representations, ed. Fred Diamond, Payman L. Kassaei and c Cambridge University Press 2014. Minhyong Kim. Published by Cambridge University Press.

231

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Vytautas Pašk¯unas

The results of Berger–Breuil [3], Berger [2], Breuil–Emerton [6] and [22] ss describe explicitly the possibilities for , see Proposition 6.13. These results and the Theorem imply that Ext1G (π2 , π1 ) vanishes in many cases. Let us make this more precise. Let Modsm G (O) be the category of smooth G-representation on O-torsion modules. It contains Modsm G (k), the category of smooth G-representations on k-vector spaces, as a full subcategory. Every irreducible object π of Modsm G (O) is killed by $ , and hence is an object of Modsm (k). Barthel–Livné [1] and G Breuil [4] have classified the absolutely irreducible smooth representations π admitting a central character. They fall into four disjoint classes: (i) (ii) (iii) (iv)

characters δ ◦ det; special series Sp ⊗δ ◦ det; principal series (IndG B δ1 ⊗ δ2 )sm , δ1 = δ2 ; supersingular representations;

where Sp is the Steinberg representation, that is the locally constant func× tions from P1 (Q p ) to k modulo the constant functions; δ, δ1 , δ2 : Q× p → k are smooth characters and we consider δ1 ⊗ δ2 as a character of B, which sends a0 db to δ1 (a)δ2 (d). Using their results and some easy arguments, see [25, §5.3], one may show that for an irreducible smooth representation π the following are equivalent: (1) π is admissible, which means that π H is finite dimensional for all open subgroups H of G; (2) EndG (π ) is finite dimensional over k; (3) there exists a finite extension k of k, such that π ⊗k k is isomorphic to a finite direct sum of distinct absolutely irreducible k -representations with a central character. Let Modl.adm (O) be the full subcategory of Modsm G G (O), consisting of representations, which are equal to the union of their admissible subrepresentations. l.adm The categories Modsm (O) are abelian, see [15, Prop.2.2.18]. G (O) and ModG l.adm We define ModG (k) in exactly the same way with O replaced by k. Let l.adm (O), then Irradm Irradm G be the set of irreducible representations in ModG G is sm the set of irreducible representations in ModG (O) satisfying the equivalent conditions described above. We define an equivalence relation ∼ on Irradm G : π ∼ τ , if there exists a sequence of irreducible admissible representations π = π1 , π2 , . . . , πn = τ , such that for each i one of the following holds: (1) πi ∼ = πi +1 ; (2) Ext1G (πi , πi+1 ) = 0; (3) Ext1G (πi +1 , πi ) = 0. We note that it does not matter for the definition of ∼, whether we compute Ext1G sm l.adm in Modsm (O) or Modl.adm (k), since we only care G (O), ModG (k), ModG G 1 about vanishing or non-vanishing of ExtG (πi , πi +1 ) for distinct irreducible representations. A block is an equivalence class of ∼.

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233

Corollary 6.2. The blocks containing an absolutely irreducible representation are given by the following: (i) B = {π } with π supersingular; −1 G −1 −1 (ii) B = {(IndG B δ1 ⊗ δ2 ω )sm , (Ind B δ2 ⊗ δ1 ω )sm } with δ2 δ1 ±1 ω , 1; −1 (iii) p > 2 and B = {(IndG B δ ⊗ δω )sm }; (iv) p = 2 and B = {1, Sp} ⊗ δ ◦ det; (v) p ≥ 5 and B = {1, Sp, (IndGB ω ⊗ ω−1 )sm } ⊗ δ ◦ det; (vi) p = 3 and B = {1, Sp, ω ◦ det, Sp ⊗ω ◦ det} ⊗ δ ◦ det;

=

× × × where δ, δ1 , δ2 : Q× p → k are smooth characters and where ω : Q p → k is the character ω(x) = x|x| (mod $ ).

One may view the cases (iii) to (vi) as degenerations of case (ii). A finitely generated smooth admissible representation of G is of finite length, [15, Thm.2.3.8]. This makes Modl.adm (O) into a locally finite category. It follows G from [17] that every locally finite category decomposes into blocks. In our situation we obtain: ∼ Modl.adm (O) Modl.adm (O)[B], (6.1) = G G B∈Irradm G /∼

where Modl.adm (O)[B] is the full subcategory of Modl.adm (O) consisting of G G representations, with all irreducible subquotients in B. One can deduce a similar result for the category of admissible unitary L-Banach space representations of G, see [25, Prop.5.32]. The result has been previously known for p > 2. Breuil and the author [7, §8], Colmez [8, §VII], Emerton [16, §4] and the author [23] have computed Ext1G (π2 , π1 ) by different characteristic p methods, which do not work in the exceptional cases, when p = 2. In this paper, we go via characteristic 0 and make use of a deep Theorem of Berger–Breuil. The proof is less involved, but it does not give any information about the extensions between irreducible representations lying in the same block. The motivation for these calculations comes from the p-adic Langlands correspondence for GL2 (Q p ). Colmez in [8] to a 2-dimensional absolutely irreducible L-representation of the absolute Galois group of Q p has associated an admissible unitary absolutely irreducible non-ordinary L-Banach space representation of G. He showed that his construction induces an injection on the isomorphism classes and asked whether it is a bijection, see [8, §0.13]. This has been answered affirmatively in [25] for p ≥ 5, where the knowledge of

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Vytautas Pašk¯unas

blocks has been used in an essential way. The results of this chapter should be useful in dealing with the remaining cases. Let us give a rough sketch of the argument. Let 0 → π1 → π → π2 → 0 be a non-split extension. The method of [7] allows us to embed π into , such that | K is admissible and an injective object in Modsm K (k), where K = GL2 (Z p ). Using the results of [24] we may lift to an admissible unitary L-Banach space representation E of G, in the sense that we may find a G-invariant unit ball E 0 in E, such that E 0 /$ E 0 ∼ = . Moreover, E| K is isomorphic to a direct summand of C(K , L)⊕r , where C(K , L) is the space of continuous function with the supremum norm. This implies, using an argument of Emerton, that the K -algebraic vectors are dense in E. As a consequence we find a closed G-invariant subspace of E, such that the reduction of ∩ E 0 modulo $ contains π as a subrepresentation, and con˜ c-IndG K Z 1i (T −ai )ni

⊗ detli ⊗ Symki −1 L 2 as a dense subrepresentation, where Z is the centre of G, 1˜ i : K Z → L × is a character, trivial on K , ai ∈ L, and ˜ c-IndG K Z 1i T is a certain Hecke operator in EndG (c-IndG 1˜ i ), such that is an

m tains ⊕i=1

KZ

(T −ai )

unramified principal series representation. Once we have this we are in a good shape to prove Theorem 6.1. Acknowledgements. I thank the anonymous referee for the comments, which led to an improvement of the exposition, and Jochen Heinloth for a stimulating discussion.

2. Notation Let L be a finite extension of Q p with the ring of integers O, uniformizer $ and residue field k. We normalize the valuation val on L so that val( p) = 1, and the norm | |, so that |x| = p − val(x) , for all x ∈ L. Let G = GL2 (Q p ); Z the centre of G; B the of upper triangular matrices; subgroup K = GL2 (Z p ); I = {g ∈ K : g ≡ ∗0 ∗∗ (mod p)}; I1 = {g ∈ K : g ≡ 10 ∗1 (mod p)}; let 0 × K be the G-normalizer of I ; let H = { [λ] 0 [μ] : λ, μ ∈ F p }, where [λ] is the Teichmüller lift of λ; let G be the subgroup of G generated by matrices 0p 0p , 0 1 + p 0 and H . Let G = {g ∈ G : val(det(g)) ≡ 0 (mod 2)}. Since we are working with representations of locally pro- p groups in characteristic p, these representations will not be semi-simple in general; socle is the maximal semisimple subobject. So for example, socG τ means the maximal semi-simple G-subrepresentation of τ . Let Banadm G (L) be the category of admissible unitary L-Banach space representations of G, studied in [26]. This category is abelian. Let be an admissible unitary L-Banach space representation of

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G, and let % be an open bounded G-invariant lattice in , then %/$ % is a smooth admissible k-representation of G. If %/$ % is of finite length as a ss G-representation, then we let be the semi-simplification of %/$ %. Since ss any two such %’s are commensurable, is independent of the choice of %. Universal unitary completions are discussed in [11, §1].

3. Main Let π1 , π2 be distinct smooth absolutely irreducible k-representations of G with a central character. It follows from [1] and [4] that π1 and π2 are admissible. We suppose that there exists a non-split extension in Modsm G (O): 0 → π1 → π → π2 → 0.

(6.2)

Since π1 and π2 are distinct and irreducible, by examining the long exact sequence induced by multiplication with $ , we deduce that π is killed by $ . A similar argument shows that the existence of a non-split extension implies that the central character of π1 is equal to the central character of π2 . Moreover, π also has a central character, which is then equal to the central character of π1 , see [23, Prop.8.1]. We denote this central character by ζ : Z → k × . After replacing L by a quadratic extension and twisting by a character we may assume that ζ ( 0p 0p ) = 1. Lemma 6.3. If π1I1 = π I1 then Theorem 6.1 holds for π1 and π2 . Proof. Since ζ is continuous, it is trivial on the pro- p group Z ∩ I1 . We thus may extend ζ to Z I1 , by letting ζ (zu) = ζ (z) for all z ∈ Z , u ∈ I1 . If τ is a smooth k-representation of G with a central character ζ then I1 is naturally an τ I1 ∼ = Hom I1 Z (ζ, τ ) ∼ = HomG (c-IndG K Z ζ, τ ). Thus τ G H := EndG (c-Ind I1 Z ζ ) module. Taking I1 -invariants of (6.2) we get an exact sequence of H-modules: 0 → π1I1 → π I1 → π2I1 .

(6.3)

Since π2 is irreducible, π2I1 is an irreducible H-module by [27]. Hence, if π1I1 = π I1 , then the last arrow is surjective. It is shown in [20], that if τ is a smooth k-representation of G, with a central character ζ , generated as a G-representation by its I1 -invariants, then the natural map τ I1 ⊗H c-IndG K Z ζ → τ is an isomorphism. This implies that the sequence 0 → π1I1 → π I1 → π2I1 → 0 is non-split, and hence defines a non-zero element of I Ext1H (π2I1 , π1I1 ). Since πi ∼ = πi 1 ⊗H c-IndG K Z ζ for i = 1, 2, the H-modules I1 I1 π1 and π2 are non-isomorphic. Non-vanishing of Ext1H (π2I1 , π1I1 ) implies

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that there exists a smooth character η : G → k × such that either (π1 ∼ = η and π2 ∼ = Sp ⊗η) or (π2 ∼ = η and π1 ∼ = Sp ⊗η), [25, Lem.5.24], where Sp is the Steinberg representation. In both cases the universal unitary completion of −1 (IndG ˜ where η˜ : G → O× is any smooth character lifting B | | ⊗ | | )sm ⊗ η, η, will satisfy the conditions of Theorem 6.1 by [13, 5.3.18]. Lemma 6.3 allows to assume that π I1 = π1I1 . We note that this implies that soc K π1 ∼ = soc K π , and, since I1 is contained in G + , the restriction of (6.2) to + G is a non-split extension of G + -representations. Now we perform a renaming trick, the purpose of which is to get around some technical issues, when p = 2. If either p > 2 or p = 2 and π1 is neither a special series nor a character then we let τ1 = π1 , τ = π and τ2 = π2 . If p = 2 and π1 is either a special series representation or a character, then we let 0 → τ1 → τ → τ2 → 0 be the exact sequence obtained by tensoring (6.2) with IndG 1. In particular, τ ∼ 1, which implies = π ⊗ IndG G+ G+ ∼ ∼ that τ |G + = π |G + ⊕ π |G + and τ1 |G + = π1 |G + ⊕ π1 |G + . Hence, τ I1 = τ1I1 and soc K τ ∼ = soc K τ1 ∼ = soc K π1 ⊕ soc K π1 . This implies that socG τ ∼ = socG τ1 . Lemma 6.4. socG τ ∼ = socG τ1 ∼ = π1 . Proof. We already know that socG τ ∼ = socG τ1 and we only need to consider the case p = 2 and π1 is either a special series or a character. The assumption on π1 implies that π1I1 is one dimensional. Let K be the normalizer of I1 in G, then I1 Z is a subgroup of K of index 2. We note that I = I1 as p = 2. Thus K acts on π1I1 by a character χ , such that the restriction of χ to I1 Z is equal to ζ . Since p = 2, we have an exact non-split sequence of G-representations 0 → 1 → IndG 1 → 1 → 0. We note that G + and hence Z I1 act trivially on all G+ the terms in this sequence. By tensoring with π1 we obtain an exact sequence 0 → π1 → τ1 → π1 → 0 of G-representations. Taking I1 -invariants, gives us I I an isomorphism of K-representations τ1 1 ∼ = π1 1 ⊗IndK Z I1 1. This representation is a non-split extension of χ by itself. Thus τ1 is a non-split extension of π1 by itself. Hence, socG τ1 ∼ = π1 . If p = 2 then τ1I1 is 2-dimensional and has a basis of the form {v, 0p 01 v}: if π1 is either a character or special series, this follows from the isomorphism I τ1I1 ∼ = π1 1 ⊗IndK Z I1 1, otherwise τ1 = π1 and the assertion follows from [7, Cor. 6.4 (i)] noting that the work of Bartel–Livné [1] and Breuil [4] on classification of irreducible representations of G implies that π I1 is isomorphic as a module of the pro- p Iwahori–Hecke algebra to M(r, λ, η) defined in [7, Def.6.2]. Since τ I1 = τ1I1 , [7, Prop.9.2] implies that the inclusion τ I1 → τ has a G-equivariant section.

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Proposition 6.5. There exists a G-equivariant injection τ → , where is a smooth k-representation of G, such that | K is an injective envelope of soc K τ p 0 K I1 ∼ in Modsm K (k), 0 p acts trivially on and |K = IndG . Proof. The existence of satisfying the first two conditions follows from [7, Cor.9.11]. The 0last condition is satisfied as a byproduct of the construction of 1 the action of p 0 in [7, Lem.9.6]. Corollary 6.6. Let be as above then soc K ∼ = soc K τ1 and socG ∼ = π1 . Proof. Since τ is a subrepresentation of , soc K τ is contained in soc K . Since | K is an injective envelope of soc K τ , every non-zero K -invariant subspace of intersects soc K τ non-trivially. This implies that soc K τ ∼ = soc K . ∼ This implies the first assertion, as soc K τ = soc K τ1 . Moreover, every G-invariant non-zero subspace of intersects τ non-trivially, since those are also K -invariant. This implies socG ∼ = socG τ ∼ = π1 , where the last isomorphism follows from Lemma 6.4. Lemma 6.7. Let κ be a finite dimensional k-representation of G on which p 0 0 p acts trivially. There exists an admissible unitary L-Banach space repre sentation (E, ) of K, such that E ⊂ |L|, 0p 0p acts trivially on E, and the reduction modulo $ of the unit ball in E is isomorphic to (IndK G κ)sm as a K-representation.

Proof. It is enough to prove the statement, when κ is indecomposable, which we now assume. Let p Z be the subgroup of G generated by 0p 0p . Since the order of H is prime to p, and H has index 2 in G/ pZ , κ is either a character or an induction of a character from H to G/ p Z . In both cases we may lift κ to a representation κ˜ 0 of G/ p Z on a free O-module of rank 1 or rank 2 respectively. Let κ˜ = κ˜ 0 ⊗O L and let be the gauge of κ˜ 0 . Then is G-invariant K/ pZ

and κ˜ 0 is the unit ball with respect to . Then (IndG / p Z κ) ˜ cont with the norm K/ p Z

f 1 := supg∈K/ p Z f (g) is a lift of (IndG / pZ κ)sm , where the subscript cont indicates continuous induction: the space of continuous functions with the right transformation property. Theorem 6.8. Let be any representation given by Proposition 6.5. Then there exists an admissibleunitary L-Banach space representation (E, ) of G, such that E ⊂ |L|, 0p 0p acts trivially on E, and the reduction modulo $ of the unit ball in E is isomorphic to as a G-representation. Proof. If p = 2 this is shown in [24, Thm.6.1]. We will observe that the renaming trick allows us to carry out essentially the same proof when p = 2. We make no assumption on p.

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We first lift | K to characteristic 0. Let σ be the K -socle of . Pontryagin duality induces an anti-equivalence of categories between Modsm K (k) and the pro.aug category of pseudocompact kK -modules, which we denote by Mod K (k). ∨ is a Since is an injective envelope of σ in Modsm (k), its Pontryagin dual K pro.aug σ ∨ be a projective envelope projective envelope of σ ∨ in Mod K (k). Let P σ ∨ /$ P σ ∨ is of σ ∨ in the category of pseudocompact OK -modules. Then P pro.aug ∨ a projective envelope of σ in Mod K (k). Since projective envelopes are σ ∨ /$ P σ ∨ . Since τ1 is admissiunique up to isomorphism, we obtain ∨ ∼ =P ∼ ble and σ = soc K τ1 by Corollary 6.6, σ is a finite dimensional k-vector space. In particular, σ ∨ is a finitely generated OK -module, and so there exists a surjection of OK -modules OK ⊕r σ ∨ . Since OK ⊕r is projective, and σ ∨ σ ∨ is essential, the surjection factors through OK ⊕r P σ ∨ , and so P Pσ ∨ is a finitely generated OK -module. Since Pσ ∨ is projective, we deduce that it is a direct summand of OK ⊕r , and hence it is O-torsion free. σ ∨ is an O-torsion free, finitely generated OK -module, and its Thus P pro.aug reduction modulo $ is isomorphic to ∨ in Mod K (k). Let E 0 = σ ∨ , L), and let 0 be the supremum norm. It follows from [26] Homcont ( P O that E 0 is an admissible unitary L-Banach space representation of K . More∨ over, the unit ball E 00 in E 0 is Homcont O ( Pσ , O) and cont cont ∼ ∼ Homcont O ( Pσ ∨ , O) ⊗O k = HomO ( Pσ ∨ , k) = Homk (Pσ ∨ , k) ∨ ∨ ∼ = ( ) ∼ = ,

see [24, §5] for details. We extend the action of K on E 0 to the action of K Z by letting 0p 0p act trivially. Since σ is finite dimensional, it follows from [21, Lem.6.2.4] that I1 is a I1 finite dimensional k-vector space. Since |K ∼ = (IndK G )sm by Proposition 6.5, Lemma 6.7 implies that there exists a unitary L-Banach space representation (E 1 , 1 ) of K, such that E 1 ⊆ |L|, 0p 0p acts trivially on E 1 and the reduction of the unit ball E 10 in E 1 modulo $ is isomorphic to |K . We claim that there exists an isometric, I Z -equivariant isomorphism ϕ : E 1 → E 0 such that the following diagram of I Z -representations: E 10 /$ E 10

ϕ

∼ =

/ E 0 /$ E 0 0 0

mod $

(6.4)

∼ =

id

/

commutes, where the left vertical arrow is the given K-equivariant isomorphism E 10 /$ E 10 ∼ = |K and the right vertical arrow is the given K Z equivariant isomorphism E 00 /$ E 00 ∼ = | K Z . Granting the claim, we may

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transport the action of K on E 0 by using ϕ to obtain a unitary action of K Z and K on E 0 , such that the two actions agree on K Z ∩ K, which is equal to I Z . The resulting action glues to the unitary action of G on E 0 , see [21, Cor.5.5.5], which is stated for smooth representations, but the proof of which works for any representation. The commutativity of the above diagram implies that E 00 ⊗O k ∼ = as a G-representation. 0 We will prove the claim now. Let M = Homcont O (E 1 , O) equipped with the pro.aug topology of pointwise convergence. Then M is an object of Mod I (O), pro.aug and M ⊗O k ∼ (k), see [24, Lem.5.4]. Since | K is injective = ∨ in Mod I sm in Modsm K (k), | I is injective in Mod I (k). Since I1 is a pro- p group, every non-zero I -invariant subspace of intersects I1 non-trivially. Thus | I is ∨ an injective envelope of I1 in Modsm I (k). Hence, is a projective envelope pro.aug I ∨ 1 of ( ) in Mod I (k). Since M is O-torsion free, and M ⊗O k is a propro.aug jective envelope of ( I1 )∨ in Mod I (k), [24, Prop.4.6] implies that M is pro.aug σ ∨ . a projective envelope of ( I1 )∨ in Mod I (O). The same holds for P Since projective envelopes are unique up to isomorphism, there exists an iso∼ = pro.aug σ ∨ → morphism ψ : P M in Mod I (O). It follows from [24, Cor.4.7] that σ ∨ ) → Autk I ( P σ ∨ /$ P σ ∨ ) is surjective. Using the natural map AutO I ( P pro.aug

this we may choose ψ so that the following diagram in Mod K σ ∨ /$ P σ ∨ P

ψ

∼ =

∨

/ M/$ M

mod $

(k):

(6.5)

∼ =

id

/ ∨

commutes. Dually we obtain an isometric I -equivariant isomorphism of unitary L-Banach space representations of I , ψ d : Homcont O (M, L) → σ ∨ , L). It follows from [26, Thm.1.2] that (E 1 , 1 ) is natHomcont ( P O urally and isometrically isomorphic to Homcont O (M, L) with the supremum norm. This gives our ϕ. The commutativity of (6.5) implies the commutativity of (6.4). Corollary 6.9. The Banach space representation (E, ) constructed in Theorem 6.8 is isometrically, K -equivariantly isomorphic to a direct summand of C(K , L)⊕r , where C(K , L) is the space of continuous functions from K to L, equipped with the supremum norm, and r is a positive integer. Proof. It follows from the construction of E, that (E, ) is isometrically, ∨ K -equivariantly isomorphic to Homcont O ( Pσ , L) with the supremum norm. σ ∨ is a direct sumMoreover, it follows from the proof of Theorem 6.8 that P mand of OK ⊕r . It is shown in [26, Lem.2.1, Cor.2.2] that the natural map

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K → OK , g → g induces an isometrical, K -equivariant isomorphism between C(K , L) and Homcont O (OK , L). If F is a finite extension of Q p then exactly the same proof works. We note that [7, Thm.9.8] is proved for GL2 (F). We record this as a corollary below. Let O F be the ring of integers of F, $ F a uniformizer, k F the let residue field, G F be the subgroup of GL2 (F) generated by the matrices $0F $0F , $0F 01 0 × and [λ] 0 [μ] , for λ, μ ∈ k F , where [λ] is the Teichmüller lift of λ. Let I1 be the standard pro- p Iwahori subgroup of G. Corollary 6.10. Let τ be an admissible smooth k-representation of GL2 (F), such that $0F $0F acts trivially on τ and if p = 2 assume that the inclusion τ I1 → τ has a G F -equivariant section. Then there exists a GL2 (F)-equivariant embedding τ → , such that |GL2 (O F ) is an injective envelope of GL2 (O F )-socle of τ in the category of smooth k-representations of GL2 (O F ) and $0F $0F acts trivially on . Moreover, we may lift to an admissible unitary L-Banach space representation of GL2 (F). Remark 6.11. We also note that one could work with a fixed central character throughout. Let Vl,k = detl ⊗ Symk−1 L 2 , for k ∈ N and l ∈ Z. Rather unfortunately k also denotes the residue field of L, we hope that this will not cause any confusion. Proposition 6.12. Let (E, ) be a unitary L-Banach space representation of K isomorphic in the category of unitary admissible L-Banach space representations of K to a direct summand of C(K , L)⊕r . The evaluation map # Hom K (Vl,k , E) ⊗ Vl,k → E (6.6) (l,k)∈Z×N

is injective and the image is a dense subspace of E. Moreover, the subspaces Hom K (Vl,k , E) are finite dimensional. Proof. The argument is the same as given in the proof of [14, Prop.5.4.1]. We have provided some details in the Appendix at the request of the referee. It is enough to prove the statement for C(K , L), since then it is true for C(K , L)⊕r and by applying the idempotent, which cuts out E, we may deduce the same statement for E. In the case E = C(K , L), the assertion follows from Proposition 6.A.17 applied to G = GL2 . We note that every rational irreducible representation of GL2 /L is isomorphic to Vl,k for a unique pair (l, k) ∈ Z×N. The last assertion follows from (6.A.13) below.

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−1 Proposition 6.13. Let ρ = (IndG B χ1 ⊗ χ2 | | )sm be a smooth principal × series representation of G, where χ1 , χ2 : Q p → L × smooth characters with χ1 || = χ2 . Let be the universal unitary completion of ρ ⊗Vl,k . Then is an admissible, finite length L-Banach space representation of G. Moreover, if ss is non-zero and we let be the semi-simplification of the reduction modulo ss $ of an open bounded G-invariant lattice in , then either is irreducible supersingular, or

ss

−1 ss G −1 ss ⊆ (IndG B δ1 ⊗ δ2 ω )sm ⊕ (Ind B δ2 ⊗ δ1 ω )sm ,

(6.7)

× for some smooth characters δ1 , δ2 : Q× p → k , where the superscript ss indicates the semi-simplification.

Proof. If = 0 then −(k + l) ≤ val(χ1 ( p)) ≤ −l, −(k + l) ≤ val(χ2 ( p)) ≤ −l and val(χ1 ( p)) + val(χ2 ( p)) = −(k + 2l), [24, Lem.7.9], [11, Lem.2.1]. If both inequalities are strict and χ1 = χ2 then it is shown in [3, §5.3] that is ss non-zero, admissible and absolutely irreducible. The assertion about then follows from [2]. If both inequalities are strict, χ1 = χ2 and is non-zero it is shown in [22, Prop.4.2] that there exist O-lattices M in ρ ⊗ Vl,k and M in ρ ⊗ Vl,k , −1 where ρ = (IndG B χ1 ⊗ χ2 | | )sm for some distinct smooth characters, × × χ1 , χ2 : Q p → L congruent to χ1 , χ2 modulo 1 + ($ ), such that both lattices are finitely generated O[G]-modules and their reductions modulo $ are isomorphic. Since M is O-torsion free, the completion of ρ ⊗ Vl,k with respect to the gauge of M is non-zero, and since M is a finitely generated O[G]module, the completion is the universal unitary completion, [11, Prop.1.17], thus is isomorphic to . Let 0 be the unit ball in with respect to the gauge of M. Then 0 /$ 0 ∼ = M/$ M ∼ = M /$ M . Now by the same argument the completion of ρ ⊗ Vl,k with respect to the gauge of M is the universal unitary completion of of ρ ⊗ Vl,k . Since χ1 = χ2 we may apply the results of Berger–Breuil [3] to conclude that the semi-simplification of M /$ M has the desired form. Suppose that either val(χ1 ( p)) = −l or val(χ2 ( p)) = −l. If χ1 = χ2 | | then this forces k = 1, so that Vl,k is a character and ρ ⊗ Vl,k ∼ = −1 ) × is a unitary character. It fol(IndG | | ⊗ | | ⊗ η, where η : G → L sm B lows from [13, Lem.5.3.18] that the universal unitary completion of ρ ⊗ Vl,k ss ss is admissible and of length 2. Moreover, ∼ = η ⊕ Sp ⊗η ∼ = (IndG B η ⊗ η)sm . If χ1 = χ2 | | then it follows from [6, Lem.2.2.1] that the universal unitary completion of ρ ⊗ Vl,k is isomorphic to a continuous induction of a unitary ss character. Hence is isomorphic to the semi-simplification of a principal series representation.

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Proof of Theorem 6.1. Let (E, ) be the unitary L-Banach space representation of G constructed in the proof of Theorem 6.8. Let E 0 be the unit ball in E, then by construction we have E 0 /$ E 0 ∼ = , where is a smooth k-representation of G, satisfying the conditions of Proposition 6.5. Let V = ⊕ Hom K (Vl,k , E)⊗Vl,k , where the sum is taken over all (l, k) ∈ Z×N. It follows from Corollary 6.9 and Proposition 6.12 that the natural map V → E is injective and the image is dense. Let {V i }i≥0 be any increasing, exhaustive filtration of V by finite dimensional K -invariant subspaces. Then V i ∩ E 0 i is a K -invariant O-lattice in V i , and we denote by V its reduction modulo $ . It follows from [24, Lem.5.5] that the reduction modulo $ induces a i i K -equivariant injection V → . The density of V in E implies that {V }i≥0 is an increasing, exhaustive filtration of by finite dimensional, K -invariant subspaces. Recall that contains τ as a subrepresentation, see Proposition 6.5. Now τ is finitely generated as a G-representation, since it is of finite length. Thus we may conclude, that there exists a finite dimensional K -invariant subspace W of V , such that τ is contained in the G-subrepresentation of generated by W . Let ϕ : Vl,k → E be a non-zero K -equivariant, L-linear homomorphism. Let R(ϕ) be the G-subrepresentation of E in the category of (abstract) G-representations on L-vector spaces, generated by the image of ϕ. Frobe˜ nius reciprocity gives us a surjection c-IndG R(ϕ), where KZ 1 ⊗ V l,k ˜1 : K Z → L × is an unramified character, such that p 0 acts trivially on 0 p ˜ Now EndG (c-IndG 1) ˜ is isomorphic to the ring of polynomials over L Vl,k ⊗1. KZ

in one variable T . It follows from the proof of [24, Cor.7.4] that the surjection c-IndG 1˜

factors through P(TK )Z ⊗ Vl,k R(ϕ), for some non-zero P(T ) ∈ L[T ]. Let R be the (abstract) G-subrepresentation of E generated by W , and let be the closure of R in E. Since W is isomorphic to a finite direct sum of Vl,k ’s, we deduce that if we replace L by a finite extension there exists a surjection: m ˜ # c-IndG K Z 1i ⊗ Vli ,ki R, n (T − ai ) i

(6.8)

i=1

c-IndG 1˜

KZ i for some ai ∈ L, n i ∈ N and (li , ki ) ∈ Z × N. Let ρi = T −a , then i using (6.8) we may construct a finite, increasing, exhaustive filtration {R j } j≥0 of R by G-invariant subspaces, such that for each j there exists a surjection ρi ⊗ Vli ,ki R j /R j −1 , for some 1 ≤ i ≤ m. Moreover, by choosing n i and m in (6.8) to be minimal, we may assume that HomG (ρi ⊗ Vli ,ki , R) is non-zero for all 1 ≤ i ≤ m. Let j be the closure of R j in E. We note that since E is admissible, j is an admissible unitary L-Banach space representation of

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j j G, moreover the category Banadm G (L) is abelian. Since R is dense in , its j j −1 image is dense in / . Hence, for each j there exists a G-equivariant map ϕ j : ρi ⊗ Vli ,ki → j / j −1 with a dense image. Let i be the universal unitary completion of ρi ⊗Vli ,ki . Since the target of ϕ j is unitary, we can extend it to a continuous G-equivariant map ϕ˜ j : i → j / j −1 . Moreover, since the target of ϕ j is admissible and the image is dense, ϕ˜ j is surjective. For each closed subspace U of E, we let U be the reduction of (U ∩ E 0 ) modulo $ . It follows from [24, Lem.5.5] that the reduction modulo $ induces an injection U → . Since contains W , will contain W . Since j is G-invariant, it will contain τ . Now { } j≥0 defines a finite, increasing, exhaustive filtration of by G-invariant subspaces. Since π2 is an irreducible subquotient of τ , there exists j , such that π2 is an irreducible subquotient of j j−1 / . Each representation ρi is an unramified principal series representation, considered in Proposition 6.13, see [5, Prop.3.2.1]. Hence, i is an admissible, ss finite length L-Banach space representation of G, moreover i is of finite length as described in Proposition 6.13. The surjection ϕ˜ j : i j / j−1 ss ss induces a surjection i ( j / j −1 ) . It follows from [24, Lem.5.5] that ss j j −1 the semi-simplification of / is isomorphic to ( j / j −1 ) . Thus π2 ss is a subquotient of i . Since HomG (ρi ⊗ Vli ,ki , ) is non-zero, there exists a non-zero continuous G-invariant homomorphism ϕ : i → . Let be the image of ϕ. Since i and are admissible, we have a surjection i and an injection → ss in the abelian category Banadm G (L). The surjection induces a surjection i ss . The injection induces an injection → → . Since socG ∼ = π1 ∼ by Corollary 6.6 and is non-zero, we deduce that π1 = socG . Hence, π1 ss is a subquotient of i .

Lemma 6.14. Let κ and λ be smooth k-representations of G and let l be a finite extension of k. Then ExtiG (κ, λ) ⊗k l ∼ = ExtiG (κ ⊗k l, λ ⊗k l), for all i ≥ 0, sm where the Ext groups are computed in Modsm G (k) and ModG (l), respectively. Proof. The assertion for i = 0 follows from [25, Lem.5.1]. Hence, it is enough to find an injective resolution of λ in Modsm G (k), which remains injective after G tensoring with l. Such resolution may be obtained by considering (Ind{1} V )sm , where {1} is the trivial subgroup of G and V is a k-vector space. We note that G G (Ind{1} V )sm ⊗k l ∼ V ⊗k l)sm , since l is finite over k. = (Ind{1} Proof of Corollary 6.2. Lemma 6.14 implies that replacing L by a finite extension does not change the blocks. It follows from Proposition 6.13 and Theorem 6.1 that an irreducible supersingular representation is in a block on

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its own. Let π{δ1 , δ2 } be the semi-simple representation defined by (6.7), × are smooth characters. We have to show that where δ1 , δ2 : Q× p → k all irreducible subquotients of π {δ1 , δ2 } lie in the same block. We adopt an argument used in [8]. It follows from [5, 5.3.3.1, 5.3.3.2, 5.3.4.1] that there exists an irreducible unitary L-Banach space representation of G, such that ss ∼ = π{δ1 , δ2 }, then [8, Prop.VII.4.5(i)] asserts that we may choose an open bounded G-invariant lattice % in such that %/$ % is indecomposable. It follows from (6.1) that all the irreducible subquotients of %/$ % lie in the same block. We will list explicitly the irreducible subquotients of π {δ1 , δ2 }. It is shown −1 in [1] that if δ2 δ1−1 = ω then (IndG B δ1 ⊗ δ2 ω )sm is absolutely irreducible, and there exists a non-split exact sequence 0 → δ1 ◦ det → (IndGB δ1 ⊗ δ2 ω−1 )sm → Sp ⊗δ1 ◦ det → 0

(6.9)

δ2 δ1−1

if = ω. Taking this into account there are the following possibilities for decomposing π {δ1 , δ2 } into irreducible direct summands depending on δ1 , δ2 and p: (i) If δ2 δ1−1 = ω±1 , 1 then −1 G −1 π {δ1 , δ2 } ∼ = (IndG B δ1 ⊗ δ2 ω )sm ⊕ (Ind B δ2 ⊗ δ1 ω )sm ;

(ii) if δ2 = δ1 (= δ) then (a) if p > 2 then π {δ, δ} ∼ = (IndGB δ ⊗ δω−1 )⊕2 sm ; (b) if p = 2 then π {δ, δ} ∼ = (Sp⊕2 ⊕1⊕2 ) ⊗ δ ◦ det. (iii) if δ2 δ1−1 = ω±1 then −1 (a) if p ≥ 5 then π{δ1 , δ2 } ∼ = (1 ⊕ Sp ⊕(IndG B ω ⊗ ω )sm ) ⊗ δ ◦ det; (b) if p = 3 then π {δ1 , δ2 } ∼ = (1 ⊕ Sp ⊕ω ◦ det ⊕ Sp ⊗ω ◦ det) ⊗ δ ◦ det; (c) if p = 2 then we are in the case (ii)(b), where δ is either δ1 or δ2 . Finally, we note that in the case (ii)(b) instead of using [5, 5.3.3.2], which is stated without proof, we could have observed that since (6.9) is non-split, Sp ⊗δ1 ◦ det and δ1 ◦ det lie in the same block.

Appendix A. Density of algebraic vectors Let X be an affine scheme of finite type over Z p and let A = (X, O X ). By choosing an isomorphism A ∼ = Z p [x1 , . . . , xn ]/( f 1 , . . . , f m ) we may identify the X (Z p ) with a closed subset of Znp . The induced topology on X (Z p ) is independent of a choice of the isomorphism, see [9, Prop.2.1]. Let C(X (Z p ), L) be the space of continuous functions from X (Z p ) to L. Since X (Z p ) is compact, C(X (Z p ), L) equipped with the supremum norm

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is an L-Banach space. Recall that X (Z p ) = HomZ p −alg (A, Z p ). We denote by C alg (X (Z p ), L) the functions f : X (Z p ) → L, which are obtained by evaluating elements of A ⊗Z p L at Z p -valued points of X . Lemma 6.A.15. C alg (X (Z p ), L) is a dense subspace of C(X (Z p ), L). Proof. We first look at the special case, when X = An , so that A = Z p [x1 , . . . , xn ] and X (Z p ) = Znp . Since addition and multiplication in Z p are continuous functions, we deduce that C alg (A(Z p ), L) is a subspace of C(A(Z p ), L). The density follows from the theory of Mahler expansions, see for example [19, III.1.2.4]. In the general case, we choose an isomorphism A ∼ = Z p [x1 , . . . , xn ]/( f 1 , . . . , f m ) and identify X (Z p ) with a closed subset of An (Z p ) = Znp . The restriction of functions to X (Z p ) induces a surjective map r : C(An (Z p ), L) → C(X (Z p ), L), see for example [10, Thm.3.1(1)]. Since C alg (An (Z p ), L) is dense in C(An (Z p ), L) and supx∈X (Z p ) |r ( f )(x)| ≤ supx∈An (Z p ) | f (x)| for all f ∈ C(An (Z p ), L) we deduce that r (C alg (An (Z p ), L)) is a dense subspace. Since it is equal to C alg (X (Z p ), L) we are done. Remark 6.A.16. If X is an affine scheme of finite type over O F , where O F is a ring of integers in a finite field extension F over Q p , then there are two ways to topologize X (O F ): as O F points of X and as Z p -points of the Weil restriction of X to Z p . However, they coincide, see [9, Ex.2.4]. Proposition 6.A.17. Let G be an affine group scheme of finite type over Z p such that G L is a split connected reductive group over L. Then the evaluation map # HomG(Z p ) (V , C(G(Z p ), L)) ⊗ V → C(G(Z p ), L), (6.A.10) [V ]

where the sum is taken over all the isomorphism classes of irreducible rational representations of G L , is injective and the image is equal to C alg (G(Z p ), L). In particular, the image of (6.A.10) is a dense subspace of C(G(Z p ), L). Proof. The category of rational representations of G L is semi-simple as L is of characteristic 0, see [18, II.5.6 (6)]. Hence, the regular representation O(G L ) decomposes into a direct sum of irreducible representations. Since we have assumed that G L is split, every irreducible rational representation V of G L is absolutely irreducible [18, II.2.9]. This implies that EndG L (V ) = L for every irreducible representation V . It follows from Frobenius reciprocity [18, I.3.7 (3)] and the semi-simplicity of O(G L ) that we have an isomorphism of G L -representations: # V∗ ⊗ V (6.A.11) O(G L ) ∼ = [V ]

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where the G L -action on V ∗ is trivial. The isomorphism (6.A.11) is G(L)equivariant, and hence G(Z p )-equivariant, which gives us an isomorphism of G(Z p )-representations: # C alg (G(Z p ), L) ∼ V ∗ ⊗ V. (6.A.12) = [V ]

The map ϕ → [v → ϕ(v)(1)] induces an isomorphism HomG(Z p ) (V, C(G(Z p ), L)) ∼ = V ∗,

(6.A.13)

with the inverse map given by → [v → [g → (gv)]]. Since every V is a finite dimensional L-vector space, we conclude from (6.A.12), (6.A.13) that the injection HomG(Z p ) (V , C alg (G(Z p ), L)) → HomG(Z p ) (V, C(G(Z p ), L)) (6.A.14) is an isomorphism. Moreover, as a byproduct we obtain that EndG(Z p ) (V ) = L and HomG(Z p ) (V , W ) = 0, if V and W are non-isomorphic irreducible representations of G L . We conclude that the evaluation map is injective, and the image is equal to C alg (G(Z p ), L). Lemma 6.A.15 implies the last assertion.

References [1] L. BARTHEL AND R. L IVNÉ, ‘Irreducible modular representations of GL2 of a local field’, Duke Math. J. 75, (1994) 261–292. [2] L. B ERGER , ‘Représentations modulaires de GL2 (Q p ) et représentations galoisiennes de dimension 2’, Astérisque 330 (2010), 263–279. [3] L. B ERGER AND C. B REUIL , ‘Sur quelques représentations potentiellement cristallines de GL2 (Q p )’, Astérisque 330 (2010) 155–211. [4] C. B REUIL, ‘Sur quelques représentations modulaires et p-adiques de GL2 (Q p ). I’, Compositio 138, (2003), 165–188. [5] C. B REUIL, ‘Sur quelques représentations modulaires et p-adiques de GL2 (Q p ). II’, J. Inst. Math. Jussieu 2, (2003), 1–36. [6] C. B REUIL AND M. E MERTON, ‘Représentations p-adiques ordinaires de GL2 (Q p ) et compatibilité local-global’, Astérisque 331 (2010), 255–315. ¯ [7] C. B REUIL AND V. PAŠK UNAS , ‘Towards a modulo p Langlands correspondence for GL2 ’, Memoirs of AMS, 216, 2012. [8] P. C OLMEZ, ‘Représentations de GL2 (Q p ) et (ϕ, )-modules’, Astérisque 330 (2010) 281–509. [9] B. C ONRAD, ‘Weil and Grothendieck approaches to adelic points’, Enseign. Math. (2) 58 (2012), no. 1–2, 61–97. [10] R.L. E LLIS , ‘Extending continuous functions on zero-dimensional spaces’, Math. Ann., 186, (1970), 114–122.

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[11] M. E MERTON, ‘ p-adic L-functions and unitary completions of representations of p-adic reductive groups’, Duke Math. J. 130 (2005), no. 2, 353–392. [12] M. E MERTON, ‘Locally analytic vectors in representations of locally p-adic analytic groups’, to appear in Memoirs of the AMS. [13] M. E MERTON, ‘Local-global compatibility conjecture in the p-adic Langlands programme for G L 2 /Q’, Pure and Applied Math. Quarterly 2 (2006), no. 2, 279–393. [14] M. E MERTON, ‘Local-global compatibility in the p-adic Langlands programme for G L 2 /Q’, Preprint 2011, available at www.math.uchicago.edu/~emerton/ preprints.html. [15] M. E MERTON, ‘Ordinary parts of admissible representations of p-adic reductive groups I. Definition and first properties’, Astérisque 331 (2010), 335–381. [16] M. E MERTON, ‘Ordinary parts of admissible representations of p-adic reductive groups II. Derived functors’, Astérisque 331 (2010), 383–438. [17] P. G ABRIEL, ‘Des catégories abéliennes’, Bull. Soc. Math. France 90 (1962) 323–448. [18] J.C. JANTZEN, Representations of algebraic groups, 2nd edn, Mathematical Surveys and Momographs, Vol. 107, AMS, 2003. [19] M. L AZARD, ‘Groupes analytiques p-adiques’, Publ. Math. IHES 26 (1965). [20] R. O LLIVIER, ‘Le foncteur des invariants sous l’action du pro-p-Iwahori de GL(2, F)’, J. für die Reine und Angewandte Mathematik 635 (2009) 149–185. ¯ [21] V. PAŠK UNAS , ‘Coefficient systems and supersingular representations of GL2 (F)’, Mémoires de la SMF, 99, (2004). ¯ [22] V. PAŠK UNAS ‘On some crystalline representations of GL2 (Q p )’, Algebra Number Theory 3 (2009), no. 4, 411–421. ¯ [23] V. PAŠK UNAS , ‘Extensions for supersingular representations of GL2 (Q p )’, Astérisque 331 (2010) 317–353. ¯ [24] V. PAŠK UNAS , ‘Admissible unitary completions of locally Q p -rational representations of GL2 (F)’, Represent. Theory 14 (2010), 324–354. ¯ [25] V. PAŠK UNAS , ‘The image of Colmez’s Montreal functor’, to appear in Publ. Math. IHES. DOI: 10.1007/s10240-013-0049-y. [26] P. S CHNEIDER AND J. T EITELBAUM, ‘Banach space representations and Iwasawa theory’, Israel J. Math. 127, (2002) 359–380. [27] M.-F. V IGNÉRAS, ‘Representations modulo p of the p-adic group G L(2, F)’, Compositio Math. 140 (2004) 333–358.

7 From étale P+-representations to G-equivariant sheaves on G/P Peter Schneider, Marie-France Vigneras, and Gergely Zabradi∗

Abstract Let K /Q p be a finite extension with ring of integers o, let G be a connected reductive split Q p -group of Borel subgroup P = T N and let α be a simple root of T in N . We associate to a finitely generated module D over the Fontaine ring over o endowed with a semilinear étale action of the monoid T+ (acting on the Fontaine ring via α), a G(Q p )-equivariant sheaf of o-modules on the compact space G(Q p )/P(Q p ). Our construction generalizes the representation D P1 of G L(2, Q p ) associated by Colmez to a (ϕ, )-module D endowed with a character of Q∗p .

Contents 1

2

Introduction 1.1 Notation 1.2 General overview 1.3 The rings α (N0 ) and OE ,α 1.4 Equivalence of categories 1.5 P-equivariant sheaves on C 1.6 Generalities on G-equivariant sheaves on G/P 1.7 (s, res, C)-integrals Hg 1.8 Main theorem 1.9 Structure of the chapter Induction IndG H for monoids H ⊂ G 2.1 Definition and remarks 2.2 From N to Z 2.3 (ϕ, ψ)-modules

page 250 250 251 252 252 253 254 255 257 258 261 261 262 264

∗ The third author was partially supported by OTKA Research grant no. K-101291.

Automorphic Forms and Galois Representations, ed. Fred Diamond, Payman L. Kassaei and c Cambridge University Press 2014. Minhyong Kim. Published by Cambridge University Press.

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3

Etale P+ -modules 3.1 Etale module M 3.2 Induced representation M P 3.3 Compactly induced representation McP 3.4 P-equivariant map Res : Cc∞ (N , A) → End A (M P ) 3.5 P-equivariant sheaf on N 3.6 Independence of N0 3.7 Etale A[P+ ]-module and P-equivariant sheaf on N 4 Topology 4.1 Topologically étale A[P+ ]-modules P 4.2 Integration on N with value in Endcont A (M ) 5 G-equivariant sheaf on G/P 5.1 Topological G-space G/P and the map α 5.2 Equivariant sheaves and modules over skew group rings 5.3 Integrating α when M is compact 5.4 G-equivariant sheaf on G/P 6 Integrating α when M is non compact 6.1 (s, res, C)-integrals 6.2 Integrability criterion for α 6.3 Extension of Res 6.4 Proof of the product formula 6.5 Reduction modulo pn 7 Classical (ϕ, )-modules on OE 7.1 The Fontaine ring OE 7.2 The group G L(2, Q p ) 7.3 Classical étale (ϕ, )-modules 8 A generalization of (ϕ, )-modules 8.1 The microlocalized ring (N0 ) et 8.2 The categories Met

(N0 ) (L ∗ ) and MOE , (L ∗ ) 8.3 Base change functors 8.4 Equivalence of categories 8.5 Continuity 9 Convergence in L + -modules on (N0 ) 9.1 Bounded sets 9.2 The module Msbd 9.3 The special family Cs when M is killed by a power of p 9.4 Functoriality and dependence on s 10 Connected reductive split group References

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266 267 271 274 275 279 281 282 283 283 287 290 292 295 301 302 304 305 307 311 312 316 320 320 320 321 328 328 330 334 336 340 346 346 347 350 353 361 366

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1. Introduction 1.1. Notation We fix a finite extension K /Q p of ring of integers o and an algebraic closure Q p of K . We denote by G p = Gal(Q p /Q p ) the absolute Galois group of Q p , by (Z p ) = o[[Z p ]] the Iwasawa o-algebra of maximal ideal M(Z p ), and by OE the Fontaine ring which is the p-adic completion of the localization of (Z p ) with respect to the elements not in p (Z p ). We put on OE the weak topology inducing the M(Z p )-adic topology on

(Z p ), a fundamental system of neighbourhoods of 0 being ( p n OE + M(Z p )n )n∈N . The action of Z p − {0} by multiplication on Z p extends to an action on OE . We fix an arbitrary split reductive connected Q p -group G and a Borel Q p -subgroup P = T N with maximal Q p -subtorus T and unipotent radical N . We denote by w0 the longest element of the Weyl group of T in G, by + the set of roots of T in N , and by u α : Ga → Nα , for α ∈ + , a Q p -homomorphism onto the root subgroup Nα of N such that tu α (x)t −1 = u α (α(t)x) for x ∈ Q p and t ∈ T (Q p ), and N0 = α∈+ u α (Z p ) is a subgroup of N (Q p ). We denote by T+ the monoid of dominant elements t in T (Q p ) such that val p (α(t)) ≥ 0 for all α ∈ + , by T0 ⊂ T+ the maximal subgroup, by T++ the subset of strictly dominant elements, i.e. val p (α(t)) > 0 for all α ∈ + , and we put P+ = N0 T+ , P0 = N0 T0 . The natural action of T+ on N0 extends to an action on the Iwasawa o-algebra

(N0 ) = o[[N0 ]]. The compact set G(Q p )/P(Q p ) contains the open dense subset C = N (Q p )w0 P(Q p )/P(Q p ) homeomorphic to N (Q p ) and the compact subset C0 = N0 w0 P(Q p )/P(Q p ) homeomorphic to N0 . We put P(Q p ) = w0 P(Q p )w0−1 . Each simple root α gives a Q p -homomorphism xα : N → Ga with section u α . We denote by α : N0 → Z p , resp. ια : Z p → N0 , the restriction of xα , resp. u α , to N0 , resp. Z p . For example, G = G L(n), P is the subgroup of upper triangular matrices, N consists of the strictly upper triangular matrices (1 on the diagonal), T is the diagonal subgroup, N0 = N (Z p ), the simple roots are α1 , . . . , αn−1 where αi (diag(t1 , . . . , tn )) = ti ti−1 +1 , x αi sends a matrix to its (i, i + 1)-coefficient, u αi (.) is the strictly upper triangular matrix, with (i, i + 1)-coefficient . and 0 everywhere else. We denote by C ∞ (X, o) the o-module of locally constant functions on a locally profinite space X with values in o, and by Cc∞ (X, o) the subspace of compactly supported functions.

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1.2. General overview Colmez established a correspondence V → (V ) from the absolutely irreducible K -representations V of dimension 2 of the Galois group G p to the unitary admissible absolutely irreducible K -representations of G L(2, Q p ) admitting a central character [6]. This correspondence relies on the construction of a representation D(V ) P1 of G L(2, Q p ) for any representation V (not necessarily of dimension 2) of G p and any unitary character δ : Q∗p → o∗ . When the dimension of V is 2 and when δ = (x|x|)−1 δV , where δV is the character of Q∗p corresponding to the representation det V by local class field theory, then D(V ) P1 is an extension of (V ) by its dual twisted by δ ◦ det. It is a general belief that the correspondence V → (V ) should extend to a correspondence from representations V of dimension d to representations of G L(d, Q p ). We generalize here Colmez’s construction of the representation D P1 of G L(2, Q p ), replacing G L(2) by the arbitrary split reductive connected Q p group G. More precisely, we denote by OE ,α the ring OE with the action of T+ via a simple root α ∈ (if the rank of G is 1, α is unique and we omit α). For any finitely generated OE ,α -module D with an étale semilinear action of T+ , we construct a representation of G(Q p ). It is realized as the space of global sections of a G(Q p )-equivariant sheaf on the compact quotient G(Q p )/P(Q p ). When the rank of G is 1, the compact space G(Q p )/P(Q p ) is isomorphic to P1 (Q p ) and when G = G L(2) we recover Colmez’s sheaf. We review briefly the main steps of our construction. 1. We show that the category of étale T+ -modules finitely generated over OE ,α is equivalent to the category of étale T+ -modules finitely generated over

α (N0 ), for a topological ring α (N0 ) generalizing the Fontaine ring OE , which is better adapted to the group G, and depends on the simple root α. 2. We show that the sections over C0 N0 of a P(Q p )-equivariant sheaf S of o-modules over C N is an étale o[P+ ]-module S(C0 ) and that the functor S → S(C0 ) is an equivalence of categories. 3. When S(C0 ) is an étale T+ -module finitely generated over α (N0 ), and the root system of G is irreducible, we show that the P(Q p )-equivariant sheaf S on C extends to a G(Q p )-equivariant sheaf over G(Q p )/P(Q p ) if and only if the rank of G is 1. 4. For any strictly dominant element s ∈ T++ , we associate functorially to an étale T+ -module M finitely generated over α (N0 ), a G(Q p )-equivariant sheaf Ys of o-modules over G(Q p )/P(Q p ) with sections over C0 a dense étale (N0 )[T+ ]-submodule Msbd of M. When the rank of G is 1, the sheaf

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Ys does not depend on the choice of s ∈ T++ , and Msbd = M; when G = G L(2) we recover the construction of Colmez. For a general G, the sheaf Ys depends on the choice of s ∈ T++ , the system (Ys )s∈T++ of sheaves is compatible, and we associate functorially to M the G(Q p )-equivariant sheaves Y∪ and Y∩ of o-modules over G(Q p )/P(Q p ) with sections over C0 equal to ∪s∈T++ Msbd and ∩s∈T++ Msbd , respectively.

1.3. The rings α (N0 ) and OE ,α Fixing a simple root α ∈ , the topological local ring α (N0 ), generalizing the Fontaine ring OE , is defined as in [11] with the surjective homomorphism α : N0 → Z p . We denote by M(Nα ) the maximal ideal of the Iwasawa o-algebra

(Nα ) = o[[Nα ]] of the kernel Nα of α . The ring α (N0 ) is the M(Nα )adic completion of the localization of (N0 ) with respect to the Ore subset of elements which are not in M(Nα ) (N0 ). This is a noetherian local ring with maximal ideal Mα (N0 ) generated by M(Nα ). We put on α (N0 ) the weak topology with fundamental system of neighbourhoods of 0 equal to (Mα (N0 )n + M(N0 )n )n∈N . The action of T+ on N0 extends to an action on

α (N0 ). We denote by OE ,α the ring OE with the action of T+ induced by (t, x) → α(t)x : T+ × Z p → Z p . The homomorphism α and its section ια induce T+ -equivariant ring homomorphisms α : α (N0 ) → OE ,α , ια : OE ,α → α (N0 ), such that α ◦ ια = id .

1.4. Equivalence of categories An étale T+ -module over α (N0 ) is a finitely generated α (N0 )-module M with a semi-linear action T+ × M → M of T+ which is étale, i.e. the action ϕt on M of each t ∈ T+ is injective and M = ⊕u∈J (N0 /t N0 t −1 ) uϕt (M), t −1 )

if J (N0 /t N0 ⊂ N0 is a system of representatives of the cosets N0 /t N0 t −1 ; in particular, the action of each element of the maximal subgroup T0 of T+ is invertible. We denote by ψt the left inverse of ϕt vanishing on uϕt (M) for u ∈ N0 not in t N0 t −1 . These modules form an abelian category Met

α (N0 ) (T+ ). We define analogously the abelian category Met (T ) of finitely generOE,α + ated OE ,α -modules with an étale semilinear action of T+ . The action ϕt of each element t ∈ T+ such that α(t) ∈ Z∗p is invertible. We show that the action T+ × D → D of T+ on D ∈ Met OE,α (T+ ) is continuous for the weak topology on D; the canonical action of the inverse T− of T is also continuous.

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Theorem 7.1. The base change functors OE ⊗α − and α (N0 )⊗ια − induce quasi-inverse isomorphisms D : Met

α (N0 )

et et (T+ ) → Met OE,α (T+ ), M : MOE,α (T+ ) → M

α (N0 )

(T+ ).

Using this theorem, we show that the action of T+ and of the inverse monoid T− (given by the operators ψ) on an étale T+ -module over α (N0 ) is continuous for the weak topology.

1.5. P-equivariant sheaves on C The o-algebra C ∞ (N0 , o) is naturally an étale o[P+ ]−module, and the monoid P+ acts on the o-algebra Endo M by (b, F) → ϕb ◦ F ◦ ψb . We show that there exists a unique o[P+ ]-linear map res : C ∞ (N0 , o) → Endo M sending the characteristic function 1 N0 of N0 to the identity id M ; moreover res is an algebra homomorphism which sends 1b.N0 to ϕb ◦ ψb for all b ∈ P+ acting on x ∈ N0 by (b, x) → b.x. For the sake of simplicity, we denote now by the same letter a group defined over Q p and the group of its Q p -rational points. Let M P be the o[P]-module induced by the canonical action of the inverse monoid P− of P+ on M; as a representation of N , it is isomorphic to the representation induced by the action of N0 on M. The value at 1, denoted by ev0 : M P → M, is P− -equivariant, and admits a P+ -equivariant splitting σ0 : M → M P sending m ∈ M to the function equal to n → nm on N0 and vanishing on N − N0 . The o[P]-submodule McP of M P generated by σ0 (M) is naturally isomorphic to A[P] ⊗ A[P+ ] M. When M = C ∞ (N0 , o) then McP = Cc∞ (N , o) and M P = C ∞ (N , o) with the natural o[P]-module structure. We have the natural o-algebra embedding F → σ0 ◦ F ◦ ev0 : Endo M → Endo M P , sending id M to the idempotent R0 = σ0 ◦ ev0 in Endo M P . Proposition 7.2. There exists a unique o[P]-linear map Res : Cc∞ (N , o) → Endo M P sending 1 N0 to R0 ; moreover Res is an algebra homomorphism. The topology of N is totally disconnected and by a general argument, the functor of compact global sections is an equivalence of categories from the

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P-equivariant sheaves on N C to the nondegenerate modules on the skew group ring Cc∞ (N , o)#P = ⊕b∈P bC c∞ (N, o) in which the multiplication is determined by the rule (b1 f 1 )(b2 f 2 ) = b b b1 b2 f 1 2 f 2 for bi ∈ P, f i ∈ Cc∞ (N, o) and f 1 2 (.) = f 1 (b2 .). Theorem 7.3. The functor of sections over N0 C0 from the P-equivariant sheaves on N C to the étale o[P+ ]-modules is an equivalence of categories. The space of global sections of a P-equivariant sheaf S on C is S(C) = S(C0 ) P .

1.6. Generalities on G-equivariant sheaves on G/ P The functor of global sections from the G-equivariant sheaves on G/P to the modules on the skew group ring AG/P = C ∞ (G/P, o)#G is an equivalence of categories. We have the intermediate ring A AC = C c∞ (C, o)#P ⊂ A = ⊕g∈G gCc∞ (g −1 C ∩ C, o) ⊂ AG/P , and the o-module Z = ⊕g∈G gC c∞ (C, o) which is a left ideal of AG/P and a right A-submodule. Proposition 7.4. The functor Z → Y (Z ) = Z ⊗A Z from the nondegenerate A-modules to the AG/P -modules is an equivalence of categories; moreover the G-sheaf on G/P corresponding to Y (Z ) extends the P-equivariant sheaf on C corresponding to Z |AC . Given an étale o[P+ ]-module M, we consider the problem of extending to A the o-algebra homomorphism Res : AC → Endo (McP ), b f b → b ◦ Res( f b ). b∈P

We introduce the subrings A0 = 1C0 A1C0 = ⊕g∈G gC ∞ (g −1 C0 ∩ C0 , o) ⊂ A, AC 0 = 1C0 AC 1C0 = ⊕b∈P bC ∞ (b−1 C0 ∩ C0 , o) ⊂ AC .

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The skew monoid ring AC0 = C ∞ (C0 , o)#P+ = ⊕b∈P+ bC ∞ (C0 , o) is contained in AC 0 . The intersection g −1 C0 ∩ C0 is not 0 if and only if g ∈ N0 P N0 . The subring Res(AC 0 ) of Endo (M P ) necessarily lies in the image of Endo (M). The group P acts on A by (b, y) → (b1G/P )y(b1G/P )−1 for b ∈ P, and the map b ⊗ y → (b1G/P )y(b1G/P )−1 gives o[P] isomorphisms o[P] ⊗o[P+ ] A0 → A and o[P] ⊗o[P+ ] AC 0 → AC . Proposition 7.5. Let M be an étale o[P+ ]-module. We suppose given, for any g ∈ N0 P N0 , an element Hg ∈ Endo (M). The map R0 : A0 → Endo (M), g f g → Hg ◦ res( f g ) g∈N0 P N0

g∈N0 P N0

is a P+ -equivariant o-algebra homomorphism which extends Res |AC0 if and only if, for all g, h ∈ N0 P N0 , b ∈ P ∩ N0 P N0 , and all compact open subsets V ⊂ C0 , the relations H1. res(1V ) ◦ Hg = Hg ◦ res(1g −1 V ∩C0 ), H2. Hg ◦ Hh = Hgh ◦ res(1h −1 C0 ∩C0 ), H3. Hb = b ◦ res(1b−1 C0 ∩C0 ) hold true. In this case, the unique o[P]-equivariant map R : A → End A (McP ) extending R0 is multiplicative. When these conditions are satisfied, we obtain a G-equivariant sheaf on G/P with sections on C0 equal to M.

1.7. (s, res, C)-integrals H g Let M be an étale T+ -module M over α (N0 ) with the weak topology. We denote by Endcont o (M) the o-module of continuous o-linear endomorphisms of M, and for g in N0 P N0 , by U g ⊆ N0 the compact open subset such that Ug w0 P/P = g −1 C0 ∩ C0 . For u ∈ Ug , we have a unique element α(g, u) ∈ N0 T such that guw0 N = α(g, u)uw0 N . We consider the map αg,0 : N0 → Endcont o (M) αg,0 (u) = Res(1C0 ) ◦ α(g, u) ◦ Res(1C0 ) for u ∈ Ug and αg,0 (u) = 0 otherwise.

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The module M is Hausdorff complete but not compact; also we introduce a notion of integrability with respect to a special family C of compact subsets C ⊂ M, i.e. satisfying: C(1) C(2) C(3) C(4)

Any compact subset of a compact set in C also lies in C. 0 If C1 , C2 , . . . , Cn ∈ C then in=1 Ci is in C, as well. For all C ∈ C we have N0 C ∈ C. 0 M(C) := C∈C C is an étale o[P+ ]-submodule of M.

A map from M(C) to M is called C-continuous if its restriction to any C ∈ C is continuous. The o-module HomCont (M(C), M) of C-continuous o-linear o homomorphisms from M(C) to M with the C-open topology, is a topological complete o-module. For s ∈ T++ , the open compact subgroups Nk = s k N0 s −k ⊂ N for k ∈ Z, form a decreasing sequence of union N and intersection {1}. A map F : N0 → HomCont A (M(C), M) is called (s, res, C)-integrable if the limit / Fd res := lim F(u) ◦ res(1u Nk ), N0

k→∞

u∈J (N0 /Nk )

where J (N0 /Nk ) ⊆ N0 , for any k ∈ N, is a set of representatives for the cosets in N0 /Nk , exists in HomCont A (M(C), M) and is independent of the choice of the sets J (N0 /Nk ). We denote by Hg,J (N0 /Nk ) the sum in the right-hand side when F = αg,0 (.)| M(C) . Proposition 7.6. For all g ∈ N0 P N0 , the map αg,0 (.)| M(C) : N0 → HomCont A (M(C), M) is (s, res, C)-integrable when C(5) For any C ∈ C the compact subset ψs (C) ⊆ M also lies in C. T(1) For any C ∈ C such that C = N0 C, any open A[N0 ]-submodule M of M, and any compact subset C+ ⊆ L + there exists a compact open subgroup P1 = P1 (C, M, C+ ) ⊆ P0 and an integer k(C, M, C+ ) ≥ 0 such that s k (1 − P1 )C+ ψsk ⊆ E(C, M)

for any k ≥ k(C, M, C+ ).

The integrals Hg of αg,0 (.)| M(C) satisfy the relations H1, H2, H3, when they belong to End A (M(C)), and when C(6) For any C ∈ C the compact subset ϕs (C) ⊆ M also lies in C. T(2) Given a set J (N0 /Nk ) ⊂ N0 of representatives for cosets in N0 /Nk , for k ≥ 1, for any x ∈ M(C) and g ∈ N0 P N0 there exists a compact A-submodule C x,g ∈ C and a positive integer k x,g such that Hg,J (N0 /Nk ) (x) ⊆ C x,g for any k ≥ k x,g .

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When C satisfies C(1), . . . , C(6) and the technical properties T(1), T(2) are true, we obtain a G-equivariant sheaf on G/P with sections on C0 equal to M(C).

1.8. Main theorem Let M be an étale T+ -module M over α (N0 ) with the weak topology and let s ∈ T++ . We have the natural T+ -equivariant quotient map M : M → D = OE ,α ⊗α M

,

m → 1 ⊗ m

from M to D = D(M) ∈ MOE,α (T+ ), of T+ -equivariant section ι D : D → M = α (N0 ) ⊗ια D,

d → 1 ⊗ d.

We note that o[N0 ]ι D (D) is dense in M. A lattice D0 in D is a (Z p )submodule generated by a finite set of generators of D over OE . When D is killed by a power of p, the o-module Msbd (D0 ) := {m ∈ M | M (ψsk (u −1 m)) ∈ D0 for all u ∈ N0 and k ∈ N} of M is compact and is a (N0 )-module. Let Cs be the family of compact subsets of M contained in Msbd (D0 ) for some lattice D0 of D, and let Msbd = ∪ D0 Msbd (D0 ) the union being taken over all lattices D0 in D. In general, M is p-adically complete, M/ p n M is an étale T+ -module over α (N0 ), and D/ p n D = D(M/ p n M). We denote by pn : M → M/ p n M the reduction modulo p n , and by Cs,n the family of compact subsets constructed above for M/ p n M. We define the family Cs of compact subsets C ⊂ M such that pn (C) ∈ Cs,n for all n ≥ 1, and the o-module Msbd of m ∈ M such that the set of M (ψsk (u −1 m)) for k ∈ N, u ∈ N0 is bounded in D for the weak topology. By reduction to the easier case where M is killed by a power of p, we show that Cs satisfies C(1), . . . , C(6) and that the technical properties T(1), T(2) are true. Proposition 7.7. Let M be an étale T+ -module M over α (N0 ) and let s ∈ T++ . (i) Msbd is a dense (N0 )[T+ ]-étale submodule of M containing ι D (D). (ii) For g ∈ N0 P N0 , the (s, res, Cs )-integrals Hg,s of αg,0 | Msbd exist, lie in Endo (Msbd ), and satisfy the relations H1, H2, H3. (iii) For s1 , s2 ∈ T++ , there exists s3 ∈ T++ such that Msbd contains Msbd ∪ 3 1 bd ∩ M bd . Msbd and H = H on M g,s1 g,s2 s1 s2 2

258

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The endomorphisms Hg,s ∈ Endo (Msbd ) induce endomorphisms of bd satisfying the relations ∩s∈T++ Msbd and of ∪s∈T++ Msbd = s∈T++ Ms H1, H2, H3. Moreover ∪s∈T++ Msbd and ∩s∈T++ Msbd are (N0 )[T+ ]-étale submodules of M containing ι D (D). Our main theorem is the following: Theorem 7.8. There are faithful functors Y∩ , (Ys )s∈T++ , Y∪ : Met OE,α (T+ ) −→ G-equivariant sheaves on G/P, sending D = D(M) to a sheaf with sections on C0 equal to the dense

(N0 )[T+ ]-submodules of M

Msbd , (Msbd )s∈T++ , and Msbd , s∈T++

s∈T++

respectively. When G = G L(2, Q p ), the sheaves Ys (D) are all equal to the G-equivariant sheaf on G/P P1 (Q p ) of global sections D P1 constructed by Colmez. When the root system of G is irreducible of rank > 1, we check that ∪s∈T++ Msbd is never equal to M.

1.9. Structure of the chapter In Section 2, we consider a general commutative (unital) ring A and A-modules M with two endomorphisms ψ, ϕ such that ψ ◦ ϕ = id. We show that the induction functor IndZ is exact and that the module A[Z] ⊗N,ϕ M N,ψ = lim ← −ψ is isomorphic to the subrepresentation of IndZ M generated by N,ψ (M) = lim ← −ψ k the elements of the form (ϕ (m))k∈N . In Section 3, we consider a general monoid P+ = N0 L + contained in a group P with the property that N0 is a group such that t N0 t −1 ⊂ N0 has a finite index for all t ∈ L + and we study the étale A[P+ ]-modules M. We show that the inverse monoid P− = L − N0 ⊂ P acts on M, the inverse of t ∈ L + acting by the left inverse ψt of the action ϕt of t with kernel uϕt (M) for u ∈ N0 not in t N0 t −1 . We add the hypothesis that L + contains a central element s such that the sequence (s k N0 s −k )k∈Z is decreasing of trivial intersection, of union a group N , and that P = N L is the semi-direct product of N and of L = ∪k∈N L − s k . An A[P+ ]-submodule of M is étale if and only if it is stable by ψs . The representation M P of P induced by M| P− , restricted to N is the representation induced from M| N0 , and restricted to s Z is the representation limψ M induced from M|s −N . The natural ← − s

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A[P+ ]-embedding M → M P generates a subrepresentation McP of M P isomorphic to A[P] ⊗ A[P+ ] M. When N is a locally profinite group and N0 an open compact subgroup, we show the existence and the uniqueness of a unitpreserving A[P+ ]-map res : C ∞ (N0 , A) → End A (M), we extend it uniquely to an A[P]-map Res : C ∞ (N , A) → End A (M P ), and we prove our first theorem: the equivalence between the P-equivariant sheaves of A-modules on N and the étale A[P+ ]-modules on N0 . In Section 4, we suppose that A is a linearly topological commutative ring, that P is a locally profinite group and that M is a complete linearly topological A-module with a continuous étale action of P+ such that the action of P− is also continuous, or equivalently ψs is continuous (we say that M is a topologically étale module). Then M P is complete for the compact-open topology and Res is a measure on N with values in the algebra E cont of continuous endomorphisms of M P . We show that E cont is a complete topocont logical ring for the topology defined by the ideals E L of endomorphisms P with image in an open A-submodule L ⊂ M , and that any continuous map N → E cont with compact support can be integrated with respect to Res. In Section 5, we introduce a locally profinite group G containing P as a closed subgroup with compact quotient set G/P, such that the double cosets P\G/P admit a finite system W of representatives normalizing L, of image in NG (L)/L equal to a group, and the image C = Pw0 P/P in G/P of a double coset (with w0 ∈ W ) is open dense and homeomorphic to N by the map n → nw0 P/P. We show that any compact open subset of G/P is a finite disjoint union of g −1 U w0 P/P for g ∈ G and U ⊂ N a compact open subgroup. We prove the basic result that the G-equivariant sheaves of A-modules on G/P identify with modules over the skew group ring C ∞ (G/P, A)#G, or with nondegenerate modules over a (non unital) subring A, and that an étale A[P+ ]-module M endowed with endomorphisms Hg ∈ End A (M), for g ∈ N0 P N0 , satisfying certain relations H1, H2, H3, gives rise to a nondegenerate A-module. For g ∈ G we denote Ng ⊂ N such that N g w0 P/P = g −1 C ∩ C. We study the map α from the set of (g, u) with g ∈ G and u ∈ N g to P defined by guw0 N = α(g, u)uw0 N . In particular, we show the cocycle relation α(gh, u) = α(g, h.u)α(h, u) when each term makes sense. When M is compact, then M P is compact and the action of P on M P induces a continuous map P → E cont . We show that the A-linear map A → E cont given by the integrals of α(g, .) f (.) with respect to Res, for f ∈ Cc∞ (Ng , A), is multiplicative. As explained above, we obtain a G-equivariant sheaf of A-modules on G/P with sections M on C0 .

260

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In Section 6, we do not suppose that M is compact and we introduce the notion of (s, res, C)-integrability for a special family C of compact subsets of M. We give an (s, res, C)-integrability criterion for the function αg,0 (u) = Res(1 N0 )α(gh, u) Res(1 N0 ) on the open subset Ug ⊂ N0 such that Ug w0 P/P = g −1 C0 ∩C0 , for g ∈ N0 w0 Pw0 N0 , a criterion which ensures that the integrals Hg of αg,0 satisfy the relations H1, H2, H3, as well as a method of reduction to the case where M is killed by a power of p. When these criteria are satisfied, as explained in Section 5, one gets a G-equivariant sheaf of A-modules on G/P with sections M on C0 . (2) Section 7 concerns classical (ϕ, )-modules over OE , seen as étale o[P+ ](2) module D, where the upper exponent indicates that P+ is the upper triangular monoid P+ of G L(2, Q p ). Using the properties of treillis we apply the method explained in Section 6 to this case and we obtain the sheaf constructed by Colmez. In Section 8 we consider the case where N0 is a compact p-adic Lie group (2) endowed with a continuous non-trivial homomorphism : N0 → N0 with a (2) section ι, that L ∗ ⊂ L is a monoid acting by conjugation on N0 and ι(N0 ), (2) that extends to a continuous homomorphism : P∗ = N0 L ∗ → P+ (2) sending L ∗ to L + and that ι is L ∗ equivariant. We consider the abelian categories of étale L ∗ -modules finitely generated over the microlocalized ring

(N0 ) resp. over OE (with the action of L ∗ induced by ). Between these categories we have the base change functors given by the natural L ∗ -equivariant algebra homomorphisms : (N0 ) → OE and ι : OE → (N0 ). We show our second theorem: the base change functors are quasi-inverse equivalences of categories. When L ∗ contains an open topologically finitely generated pro- p-subgroup, we show that an étale L ∗ -module over OE is automatically topologically étale for the weak topology; the result extends to étale L ∗ -modules over (N0 ), with the help of this last theorem. In Section 9, we suppose that : P → P (2) (Q p ) is a continuous homomorphism with (L) ⊂ L (2) (Q p ), and that ι : N (2) (Q p ) → N is a L-equivariant section of | N (as L acts on N (2) (Q p ) via ) sending (N0 ) in N0 . The assumptions of Section 8 are satisfied for L ∗ = L + . Given an étale L + -module M over

(N0 ), we exhibit a special family Cs of compact subsets in M which satisfies the criteria of Section 6 with M(Cs ) equal to a dense (N0 )[L + ]-submodule Msbd ⊂ M. We obtain our third theorem: there exists a faithful functor from the étale L + -modules over (N0 ) to the G-equivariant sheaves on G/P sending M to the sheaf with sections Msbd on C0 . In Section 10, we check that our theory applies to the group G(Q p ) of rational points of a split reductive group of Q p , to a Borel subgroup P(Q p )

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261

of maximal split torus T (Q p ) = L and to a natural homomorphism α : P(Q p ) → P (2) (Q p ) associated to a simple root α. We obtain our main theorem: there are compatible faithful functors from the étale T (Q p )+ -modules D over OE (where T (Q p )+ acts via α) to the G(Q p )-equivariant sheaves on G(Q p )/P(Q p ) sheaves with sections M(D)bd s on C0 , for all strictly dominant s ∈ T (Q p ). When the root system of G is irreducible of rank > 1, we show that ∪s Msbd = M = M(D). Acknowledgements: A part of the work on this article was done when the first and third authors visited the Institut Mathématique de Jussieu at the Universities of Paris 6 and Paris 7, and the second author visited the Mathematische Institut at the Universität Münster. We express our gratitude to these institutions for their hospitality. We thank heartily CIRM, IAS, the Fields Institute, as well as Durham, Cordoba and Caen Universities, for their invitations giving us the opportunity to present this work. Finally, we would like to thank the anonymous referee for a very careful reading of the manuscript and his suggestions for improving the presentation.

2. Induction IndG H for monoids H ⊂ G A monoid is supposed to have a unit.

2.1. Definition and remarks Let A be a commutative ring, let G be a monoid and let H be a submonoid of G. We denote by A[G] the monoid A-algebra of G and by M A (G) the category of left A[G]-modules, which has no reason to be equivalent to the category of right A[G]-modules. One can construct A[G]-modules starting from A[H ]-modules in two natural ways, by taking the two adjoints of the restriction functor ResG H : M A (G) → M A (H ) from G to H . For M ∈ M A (H ) and V ∈ M A (G) we have the isomorphism

Hom A[G] (A[G] ⊗ A[H ] M, V ) −−→ Hom A[H ] (M, V ) and the isomorphism

Hom A[G] (V, Hom A[H ] ( A[G], M)) −−→ Hom A[H ] (V, M).

(7.1)

For monoid algebras, restriction of homomorphisms induces the identification Hom A[H ] (A[G], M) = IndGH (M)

262

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where IndG H (M) is formed by the functions f : G → M such that f (hg) = h f (g) for any h ∈ H, g ∈ G; the group G acts by right translations, g f (x) = f (xg) for g, x ∈ G. The isomorphism (7.1) pairs φ of the left side and of the right side satisfying ([14] I.5.7) φ(v)(g) = (gv) for (v, g) ∈ V × G. It is well known that the left and right adjoint functors of ResG H are transitive (for monoids H ⊂ K ⊂ G), the left adjoint is right exact, the right adjoint is left exact. We observe important differences between monoids and groups: (a) The binary relation g ∼ g if g ∈ H g is not symmetric, there is no “quotient space” H \G, no notion of function with finite support modulo H in IndG H (M). (b) When h M = 0 for some h ∈ H such that hG = G, then IndG H (M) = 0. Indeed f (hg) = h f (g) implies f (hg) = 0 for any g ∈ G. (c) When G is a group generated, as a monoid, by H and the inverse monoid H −1 := {h ∈ G | h −1 ∈ H }, and when M in an A[H ]-module such that the action of any element h ∈ H on M is invertible, then f (g) = g f (1) for all g ∈ G and f ∈ IndG H (M). This can be seen by induction on the minimal number m ∈ N such that g = g1 . . . gm with gi ∈ H ∪ H −1 . Then g1 ∈ H implies f (g) = g1 f (g2 . . . gm ), and g1 ∈ H −1 implies f (g2 . . . gm ) = f (g1−1 g1 g2 . . . gm ) = g1−1 f (g). The representation IndGH (M) is isomorphic by f → f (1) to the natural representation of G on M.

2.2. From N to Z An A-module with an endomorphism ϕ is equivalent to an A[N]-module, ϕ being the action of 1 ∈ N, and an A-module with an automorphism ϕ is equivalent to an A[Z]-module. When ϕ is bijective, A[Z] ⊗ A[N] M and IndZ N (M) are isomorphic to M. In general, A[Z] ⊗ A[N] M is the limit of an inductive system and IndZ N (M) is the limit of a projective system. The first one is interesting when ϕ is injective, the second one when ϕ is surjective. For r ∈ N let Mr = M. The general element of Mr is written xr with x ∈ M. Let lim (M, ϕ) be the quotient of 2r∈N Mr by the − → equivalence relation generated by ϕ(x)r+1 ≡ xr , with the isomorphism

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263

induced by the maps xr → ϕ(x)r Mr → Mr of inverse induced by the maps xr → xr +1 Mr → Mr+1 . Let x → [x] : Z → A[Z] be the canonical map. The maps xr → [−r ] ⊗ x Mr → A[Z] ⊗ A[N] M for r ∈ N induce an isomorphism of A[Z]-modules →

lim M − →

A[Z] ⊗ A[N] M.

Let lim M := {x = (xm )m∈N ∈ ← −

M : ϕ(xm+1 ) = xm for any m ∈ N} (7.2)

m∈N

seen as an A[Z]-module via the automorphism x → (ϕ(x0 ), x0 , x1 , . . .) = (ϕ(x 0 ), ϕ(x1 ), ϕ(x2 ) . . .) of inverse x → (x1 , x2 , . . .). The map f → ( f (−m))m∈N is an isomorphism of A[Z]-modules IndZ N (M)

→

lim M. ← −

The submodules of M Mϕ

∞ =0

= ∪k∈N M ϕ

k =0

ϕ ∞ (M) = ∩n∈N ϕ n (M)

,

are stable by ϕ. The inductive limit sees only the quotient M/M ϕ projective limit sees only the submodule ϕ ∞ (M), lim M = lim (M/M ϕ − → − →

∞ =0

),

∞ =0

and the

lim M = lim (ϕ ∞ (M)). ← − ← −

Lemma 7.9. Let 0 → M1 → M2 → M3 → 0 be an exact sequence of A-modules with an endomorphism ϕ. (a) The sequence 0 → lim M1 → lim M2 → lim M3 → 0 − → − → − → is exact. (b) When ϕ is surjective on M1 , the sequence 0 → lim M1 → lim M2 → lim M3 → 0 ← − ← − ← − is exact. Proof. This is a standard fact on inductive and projective limits.

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2.3. (ϕ, ψ)-modules Let M be an A-module with two endomorphisms ψ, ϕ such that ψ ◦ ϕ = 1. Then ψ is surjective, ϕ is injective, the endomorphism ϕ ◦ ψ is a projector of M giving the direct decomposition M = ϕ(M) ⊕ M ψ=0 , and m ψ=0

m = (ϕ ◦ ψ)(m) + m ψ=0

(7.3)

M ψ=0

for m ∈ M ∈ the kernel of ψ. We consider the representation of Z induced by (M, ψ) as in Subsection 2.2, IndZ M. N,ψ (M) lim ← − ψ

On the induced representation ψ is an isomorphism and we introduce ϕ := ψ −1 . As ψ is surjective on M, the map ev0 IndZ N,ψ (M) → M, corresponding to the map lim M → M, (xm )m∈N → x0 ← − ψ

is surjective. A splitting is the map σ0 M → IndZ N,ψ (M) corresponding to M → lim M, ← −

x → (ϕ m (x))m∈N .

(7.4)

ψ

Obviously ev0 is ψ-equivariant, σ0 is ϕ-equivariant, ev0 ◦ σ0 = id M , and :=

R0

σ0 ◦ ev0 ∈ End A (IndZ N,ψ (M))

is an idempotent of image σ0 (M). Definition 7.10. The representation of Z compactly induced from (M, ψ) Z is the subrepresentation c-IndZ N,ψ (M) of IndN,ψ (M) generated by the image of σ0 (M). We note that, for any k ≥ 1, the endomorphisms ψ k , ϕ k satisfy the same properties as ψ, ϕ because ψ k ◦ ϕ k = 1. For any integer k ≥ 0, the value at k is a surjective map evk IndZ N,ψ (M) → M, corresponding to the map lim M → M, ← −

(xm )m∈N → xk

(7.5)

ψ

of splitting σk M → IndZ N,ψ (M) corresponding to the map M → lim M, ← −

x → (ψ k (x), . . . , ψ(x), x, ϕ(x), ϕ 2 (x), . . .) .

ψ

The following relations are immediate: evk = ev0 ◦ ϕ k = ψ ◦ evk+1 = evk+1 ◦ ψ, σk = ψ k ◦ σ0 = σk+1 ◦ ϕ = ϕ ◦ σk+1 .

(7.6)

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We deduce that σk (M) ⊂ σk+1 (M). Since σk (M) is ϕ-invariant we have

c-IndZ ψ k (σ0 (M)) = σk (M) = σk (M). (7.7) N,ψ (M) = k∈N

k∈N

k∈N

In lim (M) the subspace of (xm )m∈N such that xk+r = ϕ k (xr ) for all k ∈ N ← − ψ

Z and for some r ∈ N, is equal to c-IndZ N,ψ (M). The definition of c-IndN,ψ (M) is functorial. We get a functor c-IndZ N,ψ from the category of A-modules with two endomorphisms ψ, ϕ such that ψ ◦ ϕ = 1 (a morphism commutes with ψ and with ϕ) to the category of A[Z]-modules.

Proposition 7.11. The map A[Z] ⊗ A[N],ϕ M → Hom A[N],ψ (A[Z], M) = IndZ N,ψ (M) [k] ⊗ m → (ϕ k ◦ σ0 )(m) induces an isomorphism from the tensor product A[Z] ⊗ A[N],ϕ M to the compactly induced representation c-IndZ N,ψ (M) (note that ψ and ϕ appear). Proof. From (7.3) and the relations between the σk we have for m ∈ M, k ∈ N, k ≥ 1, σk (m) = σk−1 (ψ(m)) + σk (m ψ=0 ). By induction k∈N σk (M) = σ0 (M) + k≥1 σk (M ψ=0 ). Using (7.6) one checks that the sum is direct, hence by (7.7), ψ=0 c-IndZ )). N,ψ (M) = σ0 (M) ⊕ (⊕k≥1 σk (M

On the other hand, one deduces from (7.3) that A[Z] ⊗ A[N],ϕ M

=

([0] ⊗ M) ⊕ (⊕k≥1 ([−k] ⊗ M ψ=0 )).

With Lemma 7.9 we deduce: Corollary 7.12. The functor c-IndZ N,ψ is exact. We have two kinds of idempotents in End A (IndZ N,ψ (M)), for k ∈ N, defined by Rk = σ0 ◦ ϕ k ◦ ψ k ◦ ev0 ,

R−k := ψ k ◦ R0 ◦ ϕ k = σk ◦ evk .

(7.8)

The first ones are the images of the idempotents rk := ϕ k ◦ ψ k ∈ End A (M) via the ring homomorphism End A (M) → End A IndZ N,ψ (M),

f → σ0 ◦ f ◦ ev0 .

(7.9)

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The second ones give an isomorphism from IndZ N,ψ (M) to the limit of the projective system (σk (M), R−k : σk+1 (M) → σk (M)). Lemma 7.13. The map f IndZ N,ψ (M) to lim(σk (M)) ← −

:=

→ (R−k ( f ))k∈N is an isomorphism from

{( f k )k∈N | f k ∈ σk (M), f k = R−k ( f k+1 )

for k ∈ N }

R−k

of inverse ( f k )k∈N → f with evk ( f ) = evk ( f k ). Remark 7.14. As ϕ is injective, its restriction to ∩n∈N ϕ n (M) is an isomorphism and the following A[Z]-modules are isomorphic (Section 2.2): IndZ N,ϕ (M)

lim M ← − ϕ

∩n∈N ϕ n (M). ∞ =0

As ψ is surjective, its action on the quotient M/M ψ following A[Z]-modules are isomorphic (Section 2.2): A[Z] ⊗ A[N],ψ M

lim M − →

is bijective and the

M/M ψ

∞ =0

.

ψ

Remark 7.15. When the A-module M is noetherian, a ψ-stable submodule of M which generates M as a ϕ-module is equal to M. Proof. Let N be a submodule of M. As M is noetherian there exists k ∈ N such that the ϕ-stable submodule of M generated by N is the submodule Nk ⊂ M generated by N, ϕ(N ), . . . , ϕ k (N ). When N is ψ-stable we have ψ k (Nk ) = N and when N generates M as a ϕ-module we have M = Nk . In this case, M = ψ k (M) = ψ k (Nk ) = N .

3. Etale P+ -modules Let P = N L be a semi-direct product of an invariant subgroup N and of a group L and let N0 ⊂ N be a subgroup of N . For any subgroups V ⊂ U ⊂ N , the symbol J (U/V ) ⊂ U denotes a set of representatives for the cosets in U/V . The group P acts on N by (b = nt, x) → b.x = nt xt −1 for n, x ∈ N and t ∈ L. The P-stabilizer {b ∈ P | b.N0 ⊂ N0 } of N0 is a monoid P+ = N0 L +

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267

where L + ⊂ L is the L-stabilizer of N0 . Its maximal subgroup {b ∈ P | b.N0 = N0 } is the intersection P0 = N0 L 0 of P+ with the inverse monoid P− = L − N0 where L − is the inverse monoid of L + and L 0 is the maximal subgroup of L + . We suppose that the subgroup t.N0 = t N0 t −1 ⊂ N0 has a finite index, for all t ∈ L + . Let A be a commutative ring and let M be an A[P+ ]-module, equivalently an A[N0 ]-module with a semilinear action of L + . The action of b ∈ P+ on M is denoted by ϕb . If b ∈ P0 then ϕb is invertible and we also write ϕb (m) = bm, ϕb−1 (m) = b−1 m for m ∈ M. The action ϕt ∈ End A (M) of t ∈ L + is A[N0 ]-semilinear: ϕt (xm) = ϕt (x)ϕt (m)

for

x ∈ A[N0 ], m ∈ M.

(7.10)

3.1. Etale module M The group algebra A[N0 ] is naturally an A[P+ ]-module. For t ∈ L + the map ϕt is injective of image A[t N0 t −1 ], and A[N0 ] = ⊕u∈J (N0 /t N0 t −1 ) u A[t N0 t −1 ]. Definition 7.16. We say that M is étale if, for any t ∈ L + , the map ϕt is injective and M = ⊕u∈J (N0 /t N0 t −1 ) u ϕt (M).

(7.11)

An equivalent formulation is that, for any t ∈ L + , the linear map A[N0 ] ⊗ A[N0 ],ϕt M → M, x ⊗ m → xϕt (m) is bijective. For M étale and t ∈ L + , let ψt ∈ End A (M) be the unique canonical left inverse of ϕt of kernel M ψt =0 = uϕt (M). u∈(N0 −t N0 t −1 )

The trivial action of P+ on M is not étale, and obviously the restriction to P+ of a representation of P is not always étale. Lemma 7.17. Let M be an étale A[P+ ]-module. For t ∈ L + , the kernel M ψt =0 is an A[t N0 t −1 ]-module, the idempotents in End A M (u ◦ ϕt ◦ ψt ◦ u −1 )u∈J (N0 /t N0 t −1 ) are orthogonal of sum the identity. Any m ∈ M can be written uϕt (m u,t ) m= u∈J (N0 /t N0

t −1 )

for unique elements m u,t ∈ M, equal to m u,t = ψt (u −1 m).

(7.12)

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Proof. The kernel M ψt =0 is an A[t N0 t −1 ]-module because N0 − t N0 t −1 is stable by left multiplication by t N0 t −1 . The endomorphism ϕt ◦ ψt is an idempotent because ψt ◦ ϕt = id M . Then apply (7.11) and notice that m ∈ M is equal to m= (u ◦ ϕt ◦ ψt ◦ u −1 )(m). u∈J (N0 /t N0 t −1 )

Remark 7.18. (1) An A[P+ ]-module M is étale when, for any t ∈ L + , the action ϕt of t admits a left inverse f t ∈ End A M such that the idempotents (u ◦ ϕt ◦ f t ◦ u −1 )u∈J (N0 /t N0 t −1 ) are orthogonal of sum the identity. The endomorphism f t is the canonical left inverse ψt . (2) The A[P+ ]-module A[N0 ] is étale. As A[N0 ] is a left and right free A[t N0 t −1 ]-module of rank [N0 : t N0 t −1 ] we have for x ∈ A[N0 ], x= uϕt (xu,t ) = ϕt (xu,t )u −1 u∈J (N0 /t N0 t −1 )

u∈J (N0 /t N0 t −1 )

= ψ (xu) and ψ is the left inverse of ϕ of where xu,t = ψt (u −1 x), xu,t t t t kernel u A[t N0 t −1 ] = A[t N0 t −1 ]u −1 . u∈N0 −t N0 t −1

u∈N0 −t N0 t −1

Let M be an étale A[P+ ]-module and t ∈ L + . We denote m → m ψt =0 : M → M ψt =0 the projector id M −ϕt ◦ψt along the decomposition M = ϕt (M)⊕ M ψt =0 . Lemma 7.19. Let x ∈ A[N0 ] and m ∈ M. We have ψt (ϕt (x)m) = xψt (m), (ϕt (x)m)ψt =0 = ϕt (x)(m ψt =0 ),

ψt (xϕt (m)) = ψt (x)m, (xϕt (m))ψt =0 = x ψt =0 ϕt (m).

Proof. We multiply m = (ϕt ◦ ψt )(m) + m ψt =0 on the left by ϕt (x). By the A[N0 ]-semilinearity of ϕt we have ϕt (x)m = ϕt (xψt (m)) + ϕt (x)(m ψt =0 ). As M ψt =0 is an A[t N0 t −1 ]-module, the uniqueness of the decomposition implies ψt (ϕt (x)m) = xψt (m) and (ϕt (x)m)ψt =0 = ϕt (x)(m ψt =0 ). We multiply x = (ϕt ◦ ψt )(x) + x ψt =0 on the right by ϕt (m). By the semilinearity of ϕt we have xϕt (m) = ϕt (ψt (x)m) + x ψt =0 ϕt (m). As A[N0 ]ψt =0 ϕt (M) = M ψt =0 the uniqueness of the decomposition implies ψt (xϕt (m)) = ψt (x)m, (xϕt (m))ψt =0 = x ψt =0 ϕt (m).

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Lemma 7.20. Let x ∈ A[N0 ] and m ∈ M. We have ψt (xm) = ψt (xu)ψt (u −1 m). u∈J (N0 /t N0 t −1 )

Proof. Using (7.12), replace m by u∈J (N0 /t N0 t −1 ) uϕt (m u,t ) in ψt (xm). We get ψt (xm) = ψt xuϕt (m u,t ) = ψt (xu)m u,t u∈J (N0 /t N0 t −1 )

=

u∈J (N0 /t N0 t −1 )

ψt (xu)ψt (u −1 m)

u∈J (N0 /t N0 t −1 )

using the first line of Lemma 7.19.

Proposition 7.21. Let M be an étale A[P+ ]-module. The map b−1 = (ut)−1 → ψb := ψt ◦ u −1 P− → End A (M) for t ∈ L + , u ∈ N0 , defines a canonical action of P− on M. Proof. We check that ψb1 b2 = ψb2 ◦ ψb1 for b1 = u 1 t1 , b2 = u 2 t2 ∈ P+ . We −1 have ψb1 b2 = ψt1 t2 ◦ (u 1 t1 u 2 t1−1 )−1 and ψb2 ◦ ψb1 = ψt2 ◦ u −1 2 ◦ ψt1 ◦ u 1 . −1 −1 −1 As u 2 ◦ ψt1 = ψt1 ◦ t1 u 2 t1 , it remains only to show ψt2 ψt1 = ψt1 t2 . For the sake of simplicity, we note ϕi = ϕti , ψi = ψti . For m ∈ M we have m = ϕ1 (ϕ2 ◦ ψ2 (ψ1 (m)) + ψ1 (m)ψ2 =0 ) + m ψ1 =0 . This is also m = (ϕt1 t2 ◦ ψ2 ◦ ψ1 )(m) + ϕ1 (ψ1 (m)ψ2 =0 ) + m ψ1 =0 because ϕ1 ◦ ϕ2 = ϕt1 t2 . By the uniqueness of the decomposition m = (ϕt1 t2 ◦ ψt1 t2 )(m) + m ψt1 t2 =0 we are reduced to show that M ψt1 t2 =0

=

ϕ1 (M ψ2 =0 ) + M ψ1 =0 .

It is enough to prove the inclusion M ψt1 t2 =0 ⊂ ϕ1 (M ψ2 =0 ) + M ψ1 =0 to get the equality because M = ϕt1 t2 (M) ⊕ V with V equal to any of them. Hence we want to show uϕt1 t2 (M) ⊂ ϕ1 uϕ2 (M) u∈N0 −t1 t2 N0 (t1 t2 )−1

+

u∈N0 −t2 N0 t2−1

uϕ1 (M).

(7.13)

u∈N0 −t1 N0 t1−1

As ϕ1 ◦ u ◦ ϕ2 = t1 ut1−1 ◦ ϕt1 t2 the right side of (7.13) is uϕt1 t2 (M) + uϕ1 (M). u∈t1 N0 t1−1 −t1 t2 N0 (t1 t2 )−1

u∈N0 −t1 N0 t1−1

As ϕt1 t2 = ϕ1 ◦ ϕ2 we have ϕt1 t2 (M) ⊂ ϕ1 (M). Hence (7.13) is true.

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Lemma 7.22. Let f : M → M be an A-morphism between two étale A[P+ ]-modules M and M . Then f is P+ -equivariant if and only if f is P− -equivariant (for the canonical action of P− ). Proof. Let t ∈ L + . We suppose that f is N0 -equivariant and we show that f ◦ ϕt = ϕt ◦ f is equivalent to f ◦ ψt = ψt ◦ f . Our arguments follow the proof of ([5] Prop. II.3.4). (a) We suppose f ◦ ϕt = ϕt ◦ f . Then f (ϕt (M)) = ϕt ( f (M)) is contained in ϕt (M ) and f (M ψt =0 ) = u∈N0 −t N0 t −1 uϕt ( f (M)) is contained in M ψt =0 . By Lemma 7.17, this implies f ◦ ϕt ◦ ψt = ϕt ◦ ψt ◦ f . As f ◦ ϕt = ϕt ◦ f and ϕt is injective this is equivalent to f ◦ ψt = ψt ◦ f . (b) We suppose f ◦ ψt = ψt ◦ f . Let m ∈ M. Then f (ϕt (m)) belongs to ϕt (M) because ϕt (M) is the subset of x ∈ M such that ψt (u −1 x) = 0 for all u ∈ N0 − t N0 t −1 and we have ψt (u −1 f (ϕt (m))) = f (ψt (u −1 (ϕt (m)))). Let x(m) ∈ M be the element such that f (ϕt (m)) = ϕt (x(m)). We have x(m) = ψt ϕt (x(m)) = ψt ( f (ϕt (m))) = f (ψt ϕt (m)) = f (m). Therefore f (ϕt (m)) = ϕt ( f (m)).

Proposition 7.23. The category M A (P+ )et of étale A[P+ ]-modules is abelian and has a natural fully faithful functor into the abelian category M A (P− ) of A[P− ]-modules. Proof. From Proposition 7.21 and Lemma 7.22, it suffices to show that the kernel and the image of a morphism f : M → M between two étale modules M, M , are étale. Since the ring homomorphism ϕt is flat, for t ∈ L + , the functor t := A[N0 ] ⊗ A[N0 ],ϕt − sends the exact sequence (E)

0 → Ker f → M → M → Coker f → 0

(7.14)

to an exact sequence 0 → t (Ker f ) → t (M) → t (M ) → t (Coker f ) → 0, (7.15) and the natural maps j− : t (−) → − define a map t (E) → (E). The maps j M and j M are isomorphisms because M and M are étale, hence the maps jKer f and jCoker f are isomorphisms, i.e. Ker f and Coker f are étale. (t (E))

Note that a subrepresentation of an étale representation of P+ is not necessarily étale nor stable by P− .

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Remark 7.24. An arbitrary direct product or a projective limit of étale A[P+ ]modules is étale. Proof. Since the A[t N0 t −1 ]-module A[N0 ] is free of finite rank, for t ∈ L + , the tensor product A[N0 ] ⊗ A[t N0 t −1 ] − commutes with arbitrary projective limits.

3.2. Induced representation M P Let P be a locally profinite group, semi-direct product P = N L of closed subgroups N , L, let N0 ⊂ N be an open profinite subgroup, and let s be an element of the centre Z(L) of L such that L = L − s Z (notation of Section 3) and (Nk := s k N0 s −k )k∈Z is a decreasing sequence of union N and trivial intersection. As the conjugation action L × N → N of L on N is continuous and N0 is compact open in N , the subgroups L 0 ⊂ L , P0 ⊂ P are open and the monoids P+ , P− are open in P. We have P = P− s Z

=

s Z P+

because, for n ∈ N and t ∈ L, there exists k ∈ N and n 0 ∈ N0 such that n = s −k n 0 s k and ts −k ∈ L − . Thus tn = ts −k n 0 s k ∈ P− s k and (tn)−1 ∈ s −k P+ . In particular P is generated by P+ and by its inverse P− . Let M be an étale left A[P+ ]-module. We denote by ϕ the action of s on M and by ψ the canonical left inverse of ϕ, by M P = Ind PP− (M) the A[P]-module induced from the canonical action of P− on M (Section 2.1). When f : P → M is an element of M P , the values of f on s N determine the values of f on N and reciprocally because, for any u ∈ N0 , k ∈ N, f (s −k us k ) = (ψ k ◦ u)( f (s k )), f (s k ) = (v ◦ ϕ k )( f (s −k v −1 s k )).

(7.16)

v∈J (N0 /Nk )

The first equality is obvious from the definition of Ind PP− , the second equality is obvious by the first equality as the idempotents (v ◦ ϕ k ◦ ψ k ◦ v −1 )v∈J (N0 /Nk ) are orthogonal of sum the identity, by Lemma 7.17. Proposition 7.25. (a) The restriction to s Z is an A[s Z ]-equivariant isomorphism MP

→

Z

Indss −N (M).

(b) The restriction to N is an N -equivariant bijection from M P to Ind N N0 (M).

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Proof. (a) As P = P− s Z and s −N ⊂ P− ∩ s Z (it is an equality if N is not trivial), the restriction to s Z is a s Z -equivariant injective map M P → Z Z Indss −N (M). To show that the map is surjective, let φ ∈ Indss −N (M) and b ∈ P. Then, for b = b− s r with b− ∈ P− , r ∈ Z, f (b)

:=

b− φ(s r )

is well defined because the right side depends only on b, and not on the s r with b , b ∈ choice of (b− , r ). Indeed for two choices b = b− s r = b− − − P− , r ≥ r in Z, we have

r −r b− φ(s r ) = b− s φ(s r ) = b− φ(s r ).

The well defined function b → f (b) on P belongs obviously to M P and its restriction to s Z is equal to φ. (b) As P− ∩ N = N0 the restriction to N is an N -equivariant map M P → P Ind N N0 (M). The map is injective because the restriction to N of f ∈ M N determines the restriction of f to s by (7.16) which determines f by (a). We have the natural injective map f → φ f

Z

Indss −N (M) → M P → Ind N N0 (M)

:

φ f (s −k us k ) = (ψ k ◦ u)( f (s k ))

(7.17)

for

k ∈ N, u ∈ N0 ,

→

Indss −N (M)

and we have the map φ → f φ

:

Ind N N0 (M)

Z

defined by

f φ (s k ) =

(v ◦ ϕ k )(φ(s −k v −1 s k ))

for

k ∈ N.

v∈J (N0 /Nk )

Indeed the function f φ satisfies ψ( f φ (s k+1 )) = f φ (s k ): since ψ ◦ u ◦ ϕ k+1 = s −1 us ◦ ϕ k when u ∈ N1 and is 0 otherwise, we have ψ( f φ (s k+1 )) = ψ( (v ◦ ϕ k+1 )(φ(s −k−1 v −1 s k+1 )) v∈J (N0 /Nk+1 )

=

(s −1 vs ◦ ϕ k )(φ(s −k−1 v −1 s k+1 )).

v∈N1 ∩J (N0 /Nk+1 )

The last term is

(v ◦ ϕ k )(φ(s −k v −1 s k )) = f φ (s k )

v∈J (N0 /Nk )

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273

because s −1 (N1 ∩ J (N0 /Nk+1 ))s is a system of representatives of N0 /Nk and each term of the sum does not depend on the representative. Indeed for u ∈ N0 , (vs k us −k ◦ ϕ k )(φ(s −k (vs k us −k )−1 s k ) = (v ◦ ϕ k ◦ u)(φ(u −1 s −k v −1 s k )) = (v ◦ ϕ k )(φ(s −k v −1 s k )). For u ∈ N0 , k ∈ N, we have φ fφ (s −k us k ) = (ψ k ◦ u) f φ (s k ) = (ψ k ◦ uv ◦ ϕ k )(φ(s −k v −1 s k )) = φ(s −k us k ) v∈J (N0 /Nk )

k where the last equality comes from Ker ψ k = u∈N0 −Nk uϕ (M). Moreover, we have f φ f = f as a consequence of Lemma 7.17. Proposition 7.26. The induction functor Ind PP−

:

M A (P+ )et → M A (P− ) → M A (P)

is exact. Proof. The canonical action of any element of P− on an étale A[P+ ]-module is surjective. Apply Lemma 7.9. Proposition 7.27. Let f ∈ M P . Let n, n ∈ N and t ∈ L + and denote by k(n) the smallest integer k ∈ N such that n ∈ N−k . We have: (n f )(s m ) = (s m ns −m )( f (s m )) (t −1 f )(s m ) = ψt ( f (s m )) (s k f )(n ) =

for all m ≥ k(n),

(s f )(s m ) = f (s m+1 )

and k

vϕ ( f (s

−k −1 k

v

n s ))

for all m ∈ Z,

for all k ≥ 1,

v∈J (N0 /Nk )

(t −1 f )(n ) = ψt ( f (tn t −1 ))

and

(n f )(n ) = f (n n).

Proof. The formulas (s f )(s m ) = f (s m+1 ), (n f )(n ) = f (n n) are obvious. It is clear that (t −1 f )(s m ) = f (s m t −1 ) = f (t −1 s m ) = t −1 ( f (s m )) = ψt ( f (s m )), (t −1 f )(n ) = f (nt −1 ) = f (t −1 tn t −1 ) = t −1 ( f (tn t −1 )) = ψt ( f (tn t −1 )). n f (s m ) = f (s m n) = f (s m ns −m s m ) = (s m ns −m ) f (s m ).

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Using Lemma 7.17, we write (s k f )(n ) =

vϕ k (ψ k (v −1 ((s k f )(n )))),

v∈J (N0 /Nk )

ψ (v k

−1

((s f )(n ))) = ψ k (v −1 ( f (n s k ))) = ψ k ( f (v −1 n s k )) k

We obtain (s k f )(n ) =

= f (s −k v −1 n s k ).

v∈J (N0 /Nk ) vϕ( f (s

−k v −1 n s k )).

Definition 7.28. The s-model and the N -model of M P are the spaces Z Indss −N (M) lim M and Ind N N0 (M), respectively, with the action of P ← − ψ

described in Proposition 7.27.

3.3. Compactly induced representation McP The map ev0 M P → M,

f → f (1),

admits a splitting σ0 : M → M P . For m ∈ M, σ0 (m) vanishes on N − N0 and is equal to nm on n ∈ N0 and to ϕ k (m) on s k for k ∈ N. In particular, by Proposition 7.25.b, σ0 is independent of the choice of s. Lemma 7.29. The map ev0 is P− -equivariant, the map σ0 is P+ -equivariant, the A[P+ ]-modules σ0 (M) and M are isomorphic. Proof. It is clear on the definition of M P that ev0 is P− -equivariant. We show that σ0 is L + -equivariant using the s-model. Let t ∈ L + . We choose t ∈ L + , r ∈ N with t t = s r . Then ϕt ϕt = ϕ r and ϕt = ψt ϕ r . We obtain for tσ0 (m)(s k ) = σ0 (m)(s k t) the following expression σ0 (m)(t −1 s k+r ) = ψt (σ0 (m)(s k+r )) = ψt ϕ r+k (m) = ϕt ϕ k (m) = ϕ k ϕt (m) = σ0 (tm)(s k ). Hence tσ0 (m) = σ0 (tm). We show that σ0 is N0 -equivariant using the N -model. Let n 0 ∈ N0 and m ∈ M. Then n 0 σ0 (m) = σ0 (n 0 m), because for k ∈ N, u ∈ N0 , n 0 σ0 (m)(s −k us k ) = σ0 (m)(s −k us k n 0 ) = σ0 (m)(s −k us k n 0 s −k s k ) = (ψ k ◦ us k n 0 s −k ◦ ϕ k )(m) = (ψ k ◦ u ◦ ϕ k )(n 0 m) = σ0 (n 0 m)(s −k us k ).

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275

The compact induction of M from P− to P is defined to be the A[P]submodule c-Ind PP− (M) := McP of M P generated by σ0 (M). The space McP is the subspace of functions f ∈ M P with compact restriction to N , equivalently such that f (s k+r ) = ϕ k ( f (s r )) for all k ∈ N and some r ∈ N. The restriction to s Z is an s Z -isomorphism (Proposition 7.25) McP

→

Z

c-Indss −N ,ψ (M).

By Proposition 7.11, the map A[P] ⊗ A[P+ ] M → c-Ind PP− (M) [s −k ] ⊗ m → (ϕ −k ◦ σ0 )(m) is an isomorphism. Lemma 7.30. The compact induction functor from P− to P is isomorphic to c-Ind PP− A[P] ⊗ A[P+ ] M A (P+ )et → M A (P),

(7.18)

and is exact. Proof. For the exactness see Corollary 7.12.

3.4. P-equivariant map Res : C c∞ (N, A) → End A (M P ) Let Cc∞ (N , A) be the A-module of locally constant compactly supported functions on N with values in A, with the usual product of functions and with the natural action of P, P × Cc∞ (N, A) → Cc∞ (N , A),

(b, f ) → (b f )(x) = f (b−1 .x).

For any open compact subgroup U ⊂ N , the subring C ∞ (U, A) ⊂ Cc∞ (N , A) of functions f supported in U , has a unit equal to the characteristic function 1U of U , and is stable by the P-stabilizer PU of U . We have b1U = 1b.U . The A[PU ]-module C ∞ (U, A) and the A[P]-module C c∞ (N, A) are cyclic generated by 1U . The monoid P+ = N0 L + acts on End A (M) by P+ × End A (M) → End A (M) (b, F) → ϕb ◦ F ◦ ψb . Note that we have ψut = ψt ◦ u −1 .

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Proposition 7.31. There exists a unique P+ -equivariant A-linear map res C ∞ (N0 , A)

→

End A (M)

respecting the unit. It is a homomorphism of A-algebras. Proof. If the map res exists, it is unique because the A[P+ ]-module C ∞ (N0 , A) is generated by the unit 1 N0 . The existence of res is equivalent to Lemma 7.17 as we will show below. For b ∈ P+ we have the idempotent res(1b.N0 ) := ϕb ◦ ψb ∈ End A (M). (7.19) + We claim that for any finite disjoint union b.N0 = i∈I bi .N0 with bi ∈ P+ , the idempotents res(1bi .N0 ) are orthogonal of sum res(1b.N0 ). We may assume that b = 1, since the inclusion bi .N0 ⊂ b.N0 yields b−1 bi ∈ P+ . Write bi = u i ti with u i ∈ N0 and ti ∈ L + , and choose t ∈ L + such that t ∈ ti L + , say t = ti li (with li ∈ L + ). Let (n i j ) j be a system of representatives for N0 /li .N0 . Since M is étale, Lemma 7.17 shows that, for each i, the idempotents (ϕn i j li ◦ ψn i j li ) j are orthogonal, with sum id M . Note that (vi j := u i ti n i j ti−1 )(i, j) form a system of representatives for N0 /t .N0 , so again by Lemma 7.17 the idempotents (ϕvi j t ◦ ψvi j t )(i, j) are orthogonal with sum id M . The claim follows, since vi j t = bi (n i j li ), so ϕvi j t ◦ ψvi j t = ϕbi ◦ ϕn i j li ◦ ψn i j li ◦ ψbi . The claim being proved, we get an A-linear map res : C ∞ (N0 , A) → End A (M) which is clearly P+ -equivariant and respects the unit. It respects the product because, for f 1 , f 2 ∈ C ∞ (N0 , A), there exists t ∈ L + such that f 1 and f 2 are constant on each coset ut N0 t −1 ⊂ N0 . Hence res( f 1 f 2 ) = v∈J (N0 /t N0 t −1 ) f 1 (v) f 2 (v) res(1vt.N0 ) = res( f 1 ) ◦ res( f 2 ). The group P = N L acts on End A (M P ) by conjugation. We have the canonical injective algebra map F → σ0 ◦ F ◦ ev0

:

End A M → End A (M P ).

(7.20)

It is P+ -equivariant since, by Lemma 7.29 for b ∈ P+ , we have b ◦ σ0 ◦ F ◦ ev0 ◦b−1 = σ0 ◦ ϕb ◦ F ◦ ψb ◦ ev0 .

(7.21)

We consider the composite P+ -equivariant algebra homomorphism res

C ∞ (N0 , A) −−→ End A (M) −→ End A (M P ), sending 1 N0 to R0 := σ0 ◦ ev0 and, more generally, 1b.N0 to b ◦ R0 ◦ b−1 for b ∈ P+ .

G-equivariant sheaves on G/P For f ∈ M P , R0 ( f ) ∈ M P vanishes on N − N0 and R0 ( f )(s k ) In the N -model, R0 is the restriction to N0 . We show now that the composite morphism extends to Cc∞ (N , A).

ϕ k ( f (1)).

277 =

Proposition 7.32. There exists a unique P-equivariant A-linear map Res Cc∞ (N , A) → End A (M P ) such that Res(1 N0 ) = R0 . The map Res is an algebra homomorphism. Proof. If the map Res exists, it is unique because the A[P]-module Cc∞ (N , A) is generated by 1 N0 . For b ∈ P we define Res(1b.N0 ) := b ◦ R0 ◦ b−1 . We prove that b ◦ R0 ◦ b−1 depends only on the subset b.N0 ⊂ N , and that for any finite disjoint decomposition of b.N0 = 2i∈I bi .N0 with bi ∈ P, the idempotents bi ◦ R0 ◦ bi−1 are orthogonal of sum b ◦ R0 ◦ b−1 . The equivalence relation b.N0 = b .N0 for b, b ∈ P is equivalent to b P0 = b P0 because the normalizer of N0 in P is P0 . We have b ◦ R0 ◦ b−1 = R0 when b ∈ P0 because res(1b.N0 ) = res(1 N0 ) = id (Proposition 7.31). Hence b ◦ R0 ◦ b−1 depends only on b.N0 . By conjugation by b−1 , we reduce to prove that the idempotents bi ◦ R0 ◦ bi−1 are orthogonal of sum R0 for any disjoint decomposition of N0 = 2i ∈I bi .N0 and bi ∈ P. The bi belong to P+ , and Proposition 7.31 implies the equality. To prove that the A-linear map Res respects the product it suffices to check that, for any t ∈ L + , k ∈ N, the endomorphisms Res(1vt N0 t −1 ) ∈ End A (M P ) are orthogonal idempotents, for v ∈ J (N−k /t N0 t −1 ). We already proved this for k = 0 and for all t ∈ L + , and s k J (N−k /t N0 t −1 )s −k = J (N0 /s k t N0 t −1 s −k ). Hence we know that (s k ◦ Res(1vt N0 t −1 ) ◦ s −k )v∈J (N−k /t N0 t −1 ) are orthogonal idempotents. This implies that (Res(1vt N0 t −1 ))v∈J (N−k /t N0 t −1 ) are orthogonal idempotents. Remark 7.33. (i) The map Res is the restriction of an algebra homomorphism C ∞ (N, A) → End A (M P ), where C ∞ (N , A) is the algebra of all locally constant functions on N . For this we observe

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1. The A[P+ ]-module C ∞ (N0 , A) is étale. For t ∈ L + , the corresponding ψt satisfies (ψt f )(x) = f (t xt −1 ). 2. The map ( f, m) → res( f )(m) C ∞ (N0 , A) × M → M is ψt equivariant, hence induces to a pairing C ∞ (N0 , A) P × M P → M P . 3. The A[P]-module C ∞ (N0 , A) P is canonically isomorphic to C ∞ (N, A). (ii) The monoid P+ × P+ acts on End A (M) by ϕ(b1 ,b2 ) F := ϕb1 ◦ F ◦ ψb2 . For this action End A (M) is an étale A[P+ × P+ ]-module, and we have ψ(b1 ,b2 ) F = ψb1 ◦ F ◦ ϕb2 . Definition 7.34. For any compact open subsets V ⊂ U ⊂ N0 and m ∈ M, we denote resU := res(1U ), MU := resU (M), m U := resU (m), resU V := resV | MU : MU → M V . For any compact open subsets V ⊂ U ⊂ N and f ∈ M P ResU := Res(1U ), MU := ResU (M P ), fU := ResU ( f ), ResU V := ResV | MU : MU → M V . Remark 7.35. The notation is coherent for U ⊂ N0 , as follows from the following properties. For b ∈ P+ we have – resb.U = ϕb ◦ resU ◦ψb (Proposition 7.31); – b ◦ ResU = σ0 ◦ ϕb ◦ resU ◦ ev0 and ResU ◦b−1 = σ0 ◦ resU ◦ψb ◦ ev0 ; – (ResU f )(1) = resU ( f (1)). We note also that Proposition 7.32 implies: Corollary 7.36. For any compact open subset U ⊂ N equal to a finite disjoint union U = 2i∈I Ui of compact open subsets Ui ⊂ N , the idempotents ResUi are orthogonal of sum ResU . Corollary 7.37. For u ∈ N , the projector Resu N0 is the restriction to N0 u −1 in the N -model. Proof. We have Resu N0 = u ◦ Res N0 ◦u −1 and Res N0 is the restriction to N0 in the N -model. Hence for x ∈ N , (Resu N0 f )(x) = (Res N0 u −1 f )(xu) vanishes for x ∈ N − N0 u −1 and for v ∈ N0 , (Resu N0 f )(vu −1 ) = (u −1 f )(v) = f (vu −1 ). The constructions are functorial. A morphism f : M → M of A[P+ ]modules, being also A[P− ]-equivariant induces a morphism Ind PP− ( f ) :

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M P → M P of A[P]-modules. On the other hand, M P is a module over the non unital ring Cc∞ (N , A) through the map Res. The morphism Ind PP− ( f ) is Cc∞ (N , A)-equivariant. Since Res is P-equivariant , it suffices to prove that Ind PP− ( f ) respects R0 = σ0 ◦ ev0 which is obvious.

3.5. P-equivariant sheaf on N We formulate now Proposition 7.32 in the language of sheaves. Theorem 7.38. One can associate to an étale A[P+ ]-module M, a Pequivariant sheaf S M of A-modules on the compact open subsets U ⊂ N , with – sections MU on U , – restrictions ResU V for any open compact subset V ⊂ U , – action f → b f = Resb.U (b f ) MU → Mb.U of b ∈ P. Proof. (a) ResU U is the identity on MU = ResU (M) because ResU is an idempotent. V U (b) ResW ◦ ResU V = Res W for compact open subsets W ⊂ V ⊂ U ⊂ N . Indeed, we have ResW ◦ ResV = ResW on MU . (c) If U is the union of compact open subsets Ui ⊂ U for i ∈ I , and U U f i ∈ MUi satisfying ResUii ∩U j ( f i ) = ResUij∩U j ( f j ) for i, j ∈ I , there exists a unique f ∈ MU such that ResU Ui ( f ) = f i for all i ∈ I . (c1) True when (Ui )i∈I is a partition of U because I is finite and ResU is the sum of the orthogonal idempotents ResUi . (c2) True when I is finite because the finite covering defines a finite partition of U by open compact subsets V j for j ∈ J , such that V j ∩ Ui is empty or equal to V j for all i ∈ I, j ∈ J . By hypothesis on the f i , if V j ⊂ Ui , then the restriction of f i to V j does not depend on the choice of i, and is denoted by φ j . Applying (c1), there is a unique f ∈ MU such that ResV j ( f ) = φ j for all j ∈ J . Note also that the V j contained in Ui form a finite partition of Ui and that fi is the unique element of MUi such that ResV j ( f i ) = φ j for those j. We deduce that f is the unique element of MU such that ResUi ( f ) = f i for all i ∈ I . (c3) In general, U being compact, there exists a finite subset I ⊂ I such that U is covered by Ui for i ∈ I . By (c2), there exists a unique f I ∈ MU such that f i = ResUi ( f I ) for all i ∈ I . Let i ∈ I not belonging to I . Then the nonempty intersections Ui ∩U j for j ∈ I form a finite covering of Ui by compact open subsets. By (c2), f i is the unique element of MUi such that ResUi ∩U j ( f j ) = ResUi ∩U j ( f i ) for all nonempty Ui ∩ U j . The

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element ResUi ( f I ) has the same property, we deduce by uniqueness that f i = ResUi ( f I ). (d) Let f ∈ MU . When b = 1 we have clearly 1( f ) = f . For b, b ∈ P, we have (bb )( f ) = Res(bb ).U ((bb ) f ) = Resb.(b .U ) (b(b f )) = b(b f ). For a compact open subset V ⊂ U , we have b ◦ ResV ◦ ResU = ResbV ◦b ◦ ResU in End A M P hence b ResU V = Resb.V b. Proposition 7.39. Let H be a topological group acting continuously on a locally compact totally disconnected space X . Any H -equivariant sheaf F (of A-modules) on the compact open subsets of X extends uniquely to a H -equivariant sheaf on the open subsets of X . Proof. This is well known. See [4] §9.2.3 Prop. 1. Remark 7.40. The space of sections on an open subset U ⊂ X is the projective limit of the sections F (V ) on the compact open subsets V of U for the restriction maps F(V ) → F (V ) for V ⊂ V . By this general result, the P-equivariant sheaf defined by M on the compact open subsets of N (Theorem 7.38), extends uniquely to a P-equivariant sheaf S M on (arbitrary open subsets of) N . We extend the definitions 7.34 to arbitrary open subsets U ⊂ N . We denote by ResU V the restriction maps for open subsets V ⊂ U of N , by ResU = ResUN and by MU = ResU (M P ). In this way we obtain an exact functor M → (MU )U from M A (P+ )et to the category of P-equivariant sheaves of A-modules on N . Note that for a compact open subset U even the functor M → MU is exact. Proposition 7.41. The representation of P on the global sections of the sheaf S M is canonically isomorphic to M P . Proof. We have the obvious P-equivariant homomorphism (ResU )U

M P −−−−−→ M N = lim MU . ← − U

The group N is the union of s −k .N0 = s −k N0 s k for k ∈ N. Hence M N = limk M N−k . In the s-model of M P we have Ress −k .N0 = R−k and by the lemma ← − 7.13 the morphism f → (Ress −k .N0 ( f ))k∈N M P is bijective.

→

MN

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Corollary 7.42. The restriction ResUN : M N → MU from the global sections to the sections on an open compact subset U ⊂ N is surjective with a natural splitting. Proof. It corresponds to an idempotent ResU = Res(1U ) ∈ End A (M P ).

3.6. Independence of N0 Let U ⊂ N be a compact open subgroup. For n ∈ N and t ∈ L, the inclusion ntU t −1 ⊂ U is obviously equivalent to n ∈ U and tU t −1 ⊂ U . Hence the P-stabilizer PU := {b ∈ P | b.U ⊂ U } of U is the semi-direct product of U by the L-stabilizer L U of U . As the decreasing sequence (Nk = s k N0 s −k )k∈N forms a basis of neighbourhoods of 1 in N and N = ∪r∈Z N−r , the compact open subgroup U ⊂ N contains some Nk and is contained in some N−r . This implies that the intersection L U ∩ s N is not empty hence is equal to sUN where sU = s kU for some kU ≥ 1. The monoid PU = U L U and the central element sU of L satisfy the same conditions as (P+ = N0 L + , s), given at the beginning of Section 3.2. Our theory associates to each étale A[PU ]-module a P-equivariant sheaf on N . The subspace MU ⊂ M P (Definition 7.34) is stable by PU because b ◦ ResU = Resb.U ◦b for b ∈ P and Mb.U = Resb.U (M) ⊂ ResU (M) = MU . As MU = ⊕u∈J (U/t.U ) u Mt.U for t ∈ L U the A[PU ]-module MU is étale. Proposition 7.43. The P-equivariant sheaf S M on N associated to the étale A[P+ ]-module M is equal to the P-equivariant sheaf on N associated to the étale A[PU ]-module MU . Proof. For b ∈ PU we denote by ϕU,b the action of b on MU and by ψU,b the left inverse of ϕU,b with kernel MU −b.U . We have MU = Mb.U ⊕ MU −b.U and for fU ∈ MU , ϕU,b ( f U ) = b fU , ψU,b ( fU ) = b−1 Resb.U ( fU ), (ϕU,b ◦ ψU,b )( f U ) = Resb.U ( f U ).

(7.22)

By the last formula and Remark 7.35, the sections on b.U and the restriction maps from MU to Mb.U in the two sheaves are the same for any b ∈ PU . This implies that the two sheaves are equal on (the open subsets of) U . By symmetry they are also equal on (the open subsets of) N0 . The same arguments for arbitrary compact open subgroups U, U ⊂ N imply that the P-equivariant sheaves on N associated to the étale A[PU ]-module MU and to the étale A[PU ]-module MU are equal on (the open subsets of) U and on (the open

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subsets of) U . Hence all these sheaves are equal on (the open subsets of) the compact open subsets of N and also on (the open subsets of) N .

3.7. Etale A[ P+ ]-module and P-equivariant sheaf on N Proposition 7.44. Let M be an A[P+ ]-module such that the action ϕ of s on M is étale. Then M is an étale A[P+ ]-module. Proof. Let t ∈ L + . We have to show that the action ϕt of t on M is étale. As L = L + s −N with s central in L, there exists k ∈ N such that s k t −1 ∈ L + . This implies ϕ k = ϕs k t −1 ◦ ϕt in End A (M) and s k N0 s −k ⊂ t N0 t −1 . As ϕ is injective, ϕt is also injective. For any representative system J (t N0 t −1 /s k N0 s −k ) of t N0 t −1 /s k N0 s −k and any representative system J (N0 /t N0 t −1 ) of N0 /t N0 t −1 , the set of uv for u ∈ J (N0 /t N0 t −1 ) and v ∈ J (t N0 t −1 /s k N0 s −k ) is a representative system J (N0 /s k N0 s −k ) of N0 /s k N0 s −k . Let ψ be the canonical left inverse of ϕ. We have id = u◦ v ◦ ϕ k ◦ ψ k ◦ v −1 ◦ u −1 u∈J (N0 /t N0 t −1 )

=

u∈J (N0 /t N0 t −1 )

u◦

u∈J (N0 /t N0 t −1 )

=

v∈J (t N0 t −1 /s k N0 s −k )

v ◦ ϕt ◦ ϕt −1 s k ◦ ψ k ◦ v −1 ◦ u −1

v∈J (t N0 t −1 /s k N0 s −k )

u ◦ ϕt ◦ (

v ◦ ϕt −1 s k ◦ ψ k ◦ v −1 ) ◦ u −1 .

v∈J (N0 /t −1 s k N0 s −k t)

We deduce that ϕt is étale of canonical left inverse ψt the expression between parentheses. Corollary 7.45. An A[P+ ]-submodule M ⊂ M of an étale A[P+ ]-module M is étale if and only if it is stable by the canonical inverse ψ of ϕ. Proof. If M is ψ-stable, for m ∈ M every m u,s belongs to M in (7.12). Hence the action of s on M is étale, and M is étale by Proposition 7.44. Corollary 7.46. The space S(N0 ) of global sections of a P+ -equivariant sheaf S on N0 is an étale representation of P+ , when the action ϕ of s on S(N0 ) is injective. Proof. By Proposition 7.44 it suffices to show that S(N0 ) = ⊕u∈J (N0 /s N0 s −1 ) us(S(N0 )). But this equality is true because N0 is the disjoint sum of the open subsets us N0 s −1 = us.N0 and S(us.N0 ) = us(S(N0 )).

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The canonical left inverse ψ of the action ϕ of s on S(N0 ) vanishes on S(us N0 s −1 ) for u = 1 and on S(s N0 s −1 ) is equal to the isomorphism S(s N0 s −1 ) → S(N0 ) induced by s −1 . Theorem 7.47. The functor M → S M is an equivalence of categories from the abelian category of étale A[P+ ]-modules to the abelian category of Pequivariant sheaves of A-modules on N , of inverse the functor S → S(N0 ) of sections over N0 . Proof. Let S be a P-equivariant sheaf on N . By Corollary 7.46, the space S(N0 ) of sections on N0 is an étale representation of P+ because the action ϕ of s on S(N0 ) is injective. We show now that the representation of P on the space S(N )c of compact sections on N depends uniquely the representation of P+ on S(N0 ). The representation of N on S(N )c is defined by the representation of N0 on S(N0 ), because S(N )c = ⊕u∈J (N/N0 ) S(u N0 ) and S(u N0 ) = uS(N0 ) for u ∈ N . The group P is generated by N and L + . For t ∈ L + , the action of t on S(N )c is defined by the action of N on S(N )c and by the action of t on S(N0 ), because tS(u N0 ) = tut −1 tS(N0 ) with tut −1 ∈ N for u ∈ N . We deduce that the A[P]-module S(N )c is equal to the compact induced representation S(N0 )cP , and that the sheaves S and SS (N0 ) are equal. Conversely, let M be an étale A[P+ ]-module. The A[P+ ]-module S M (N0 ) of sections on N0 of the sheaf S M is equal to M (Theorem 7.38).

4. Topology 4.1. Topologically étale A[ P+ ]-modules We add to the hypothesis of Section 3.2 the following (a) A is a linearly topological commutative ring (the open ideals form a basis of neighbourhoods of 0). (b) M is a linearly topological A-module (the open A-submodules form a basis of neighbourhoods of 0), with a continuous action of P+ P+ × M → M (b, x) → ϕb (x). We call such an M a continuous A[P+ ]-module. If M is also étale in the algebraic sense (Definition 7.16) and the maps ψt , for t ∈ L + , are continuous we call M a topologically étale A[P+ ]-module.

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Lemma 7.48. Let M be a continuous A[P+ ]-module which is algebraically étale, then: (i) The maps ψt for t ∈ L + are open. (ii) If ψ = ψs is continuous then M is topologically étale. Proof. (i) The projection of M = M0 ⊕ M1 onto the algebraic direct summand M0 (with the submodule topology) is open. Indeed let V ⊂ M be an open subset, then M0 ∩ (V + M1 ) is open in M0 and is equal to the projection of V . We apply this to M = ϕt (M) ⊕ Ker ψt and to the projection ϕt ◦ ψt . Then we note that ψt (V ) = ϕt−1 ((ϕt ◦ ψt )(V )). (ii) Given any t ∈ L + we find t ∈ L + and n ∈ N such that t t = s n . Hence ψt t = ψt ◦ ψt = ψ n is continuous by assumption. As ψt is surjective and open, for any open subset V ⊂ M we have ψt−1 (V ) = ψt ((ψt ◦ ψt )−1 (V )) which is open. Lemma 7.49. (i) A compact algebraically étale A[P+ ]-module is topologically étale. (ii) Let M be a topologically étale A[P+ ]-module. The P− -action (b−1 , m) → ψb (m) P− × M → M on M is continuous. Proof. (i) The compactness of M implies that #

M = ϕt (M) ⊕

uϕt (M)

u∈(N0 −t N0 t −1 )

is a topological decomposition of M as the direct sum of finitely many closed submodules. It suffices to check that the restriction of ψt to each summand is continuous. On all summands except the first one ψt is zero. By compactness of M the map ϕt is a homeomorphism between M and the closed submodule ϕt (M). We see that ψt |ϕt (M) is the inverse of this homeomorphism and hence is continuous. (ii) Since P0 is open in P− = L −1 + P0 we only need to show that the restriction of the P− -action to t −1 P0 × M → M, for any t ∈ L + , is continuous. We contemplate the commutative diagram t −1 P0 × M t·× id

P0 × M

/M O ψt

/M

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where the horizontal arrows are given by the P− -action. The P0 -action on M induced by P− coincides with the one induced by the P+ -action. Therefore the bottom horizontal arrow is continuous. The left vertical arrow is trivially continuous, and ψt is continuous by assumption. Lemma 7.50. For any compact subgroup C ⊂ P+ , the open C-stable Asubmodules of M form a basis of neighbourhoods of 0. Proof. We have to show that any open A-submodule M of M contains an open C-stable A-submodule. By continuity of the action of P+ on M, there exists for each c ∈ C, an open A-submodule Mc of M and an open neighbourhood Hc ⊂ P+ of c such that ϕx (Mc ) ⊂ M for all x ∈ Hc . By the compactness of C, there exists a finite subset I ⊂ C such that C = ∪c∈I (Hc ∩C). By finiteness of I , the intersection M := ∩c∈I Mc ⊂ M is an open A-submodule such that M := c∈C ϕc (M ) ⊂ M. The A-submodule M is C-stable and, since M ⊂ M ⊂ M, also open. Let M be a topologically étale A[P+ ]-module. Since P0 is open in P the A-module M P is a submodule of the A-module C(P, M) of all continuous maps from P to M. We equip C(P, M) with the compact-open topology which makes it a linear-topological A-module. A basis of neighbourhoods of zero is given by the submodules C(C, M) := { f ∈ C(P, M) | f (C) ⊂ M} with C and M running over all compact subsets in P and over all open submodules in M, respectively. With M also C(P, M) is Hausdorff. It is well known that the regular action of P on C(P, M) is continuous (see for instance Proposition 7.52(ii) for a proof). Therefore M P is characterized inside C(P, M) by Z closed conditions and hence is a closed submodule. Similarly, Indss −N (M) and Z Ind N N0 (M) are closed submodules of C(s , M) and C(N , M), respectively, for the compact-open topologies. Clearly the homomorphisms of restricting maps Z (Proposition 7.25) M P → Indss −N (M) and M P → Ind N N0 (M) are continuous. Z

Lemma 7.51. The restriction maps M P → Indss −N (M) and M P → Ind N N0 (M) are topological isomorphisms. Proof. The topology on M P induced by the compact-open topology on the Z s-model Indss −N M is the topology with basis of neighbourhoods of zero Bk,M = { f ∈ M P | f (s m ) ∈ M for all − k ≤ m ≤ k}, for all k ∈ N and all open A-submodules M of M. One can replace Bk,M by Ck,M = { f ∈ M P | f (s k ) ∈ M},

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because Bk,M ⊂ Ck,M and conversely given (k, M) there exists an open Asubmodule M ⊂ M such that ψ m (M ) ⊂ M for all 0 ≤ m ≤ 2k as ψ is continuous (Lemma 7.49), hence C k,M ⊂ Bk,M . The topology on M P induced by the compact-open topology on the N -model Ind N N0 M is the topology with basis of neighbourhoods of zero Dk,M = { f ∈ M P | f (N−k ) ⊂ M}, for all (k, M) as above. We fix an auxiliary compact open subgroup P0 ⊂ P0 . It then suffices, by Lemma 7.50, to let M run, in the above families, over the open A[P0 ]-submodules M of M. Let C ⊂ P be any compact subset and let M be an open A[P0 ]-submodule of M. We choose k ∈ N large enough so that Cs −k ⊂ P− . Since Cs −k is compact and P0 is an open subgroup of P we find finitely many b1 , . . . , bm ∈ P+ −1 P . The continuity of the maps ψ implies such that Cs −k ⊂ b1−1 P0 ∪. . .∪bm bi 0 the existence of an open A[P0 ]-submodule M of M such that ψbi (M ) ⊂ M for any 1 ≤ i ≤ m. We deduce that Ck,M ⊂ C bi−1 P0 s k , M ⊂ C(C, M). i

Furthermore, by the continuity of the action of P+ on M, there exists an open submodule M such that v∈J (N0 /Nk ) vϕ k (M ) ⊂ M . The second part of the formula (7.16) then implies that Dk,M ⊂ Ck,M .

The maps ev0 : M P → M and σ0 : M → M P are continuous (Section 3.3). cont ⊂ E := End (M P ) the We denote by Endcont A A (M) ⊂ End A (M) and E subalgebra of continuous endomorphisms. We have the canonical injective algebra map (7.20) f → σ0 ◦ f ◦ ev0

:

cont Endcont . A (M) → E

Proposition 7.52. Let M be a topologically étale A[P+ ]-module. (i) If M is complete, resp. compact, the A-module M P is complete, resp. compact. (ii) The natural map P × M P → M P is continuous. (iii) Res( f ) ∈ E cont for each f ∈ Cc∞ (N , A) (Proposition 7.32).

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Proof. (i) If M is complete, by [3] TG X.9 Cor. 3 and TG X.25 Th. 2, the compact-open topology on C(P, M) is complete because P is locally compact. Hence, M P as a closed submodule is complete as well. If M is compact, the s-model of M P is compact as a closed subset of the compact space M N . Hence by Lemma 7.51, M P is compact. (ii) It suffices to show that the right translation action of P on C(P, M) is continuous. This is well known: the map in question is the composite of the following three continuous maps P × C(P, M) −→ P × C(P × P, M) (b, f ) −→ (b, (x, y) → f (yx)), P × C(P × P, M) −→ P × C(P, C(P, M)) (b, F) −→ (b, x → [y → F(x, y)]), and P × C(P, C(P, M)) −→ C(P, M) (b, ) −→ (b), where the continuity of the latter relies on the fact that P is locally compact. (iii) It suffices to consider functions of the form f = 1b.N0 for some b ∈ P. But then Res( f ) = b ◦ σ0 ◦ ev0 ◦b−1 is the composite of continuous endomorphisms.

P 4.2. Integration on N with value in Endcont A (M )

We suppose that M is a complete topologically étale A[P+ ]-module. We denote by E cont the ring of continuous A-endomorphisms of the complete A-module M P with the topology defined by the right ideals cont P EL = Homcont A (M , L)

for all open A-submodules L ⊂ M P . Lemma 7.53. E cont is a complete topological ring. Proof. It is clear that the maps (x, y) → x − y and (x, y) → x ◦ y from E cont × E cont to E cont are continuous, i.e. that E cont is a topological ring. The composite of the natural morphisms cont P P E cont → lim E cont /E L → lim Homcont A (M , M /L) ← − ← −

L

L

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is an isomorphism (the natural map M P → limL M P /L is an isomor← − phism), hence the two morphisms are isomorphisms since the kernel of the cont P P cont is map E cont → Homcont A (M , M /L) is E L . We deduce that E complete. Definition 7.54. An A-linear map Cc∞ (N, A) → E cont is called a measure on N with values in E cont . The map Res is a measure on N with values in E cont (Proposition 7.52). Let Cc (N, E cont ) be the space of compactly supported continuous maps from N to E cont . We will prove that one can “integrate” a function in Cc (N , E cont ) with respect to a measure on N with values in E cont . Proposition 7.55. There is a natural bilinear map Cc (N , E cont ) × Hom A (Cc∞ (N , A), E cont ) → E cont / ( f, λ) → f dλ. N

Proof. (a) Every compact subset of N is contained in a compact open subset. It follows that Cc (N , E cont ) is the union of its subspaces C(U, E cont ) of functions with support contained in U , for all compact open subsets U ⊂ N . cont (b) For any open A-submodule L of M P , a function in C(U, E cont /E L ) is cont cont locally constant because E /E L is discrete. An upper index ∞ means that we consider locally constant functions hence cont cont cont C(U, E cont /E L ) = C ∞ (U, E cont /E L ) = C ∞ (U, A)⊗ A E cont /E L .

There is a natural linear pairing cont cont (C ∞ (U, A) ⊗ A E cont /E L )× Hom A (C ∞ (U, A), E cont ) → E cont /E L

( f ⊗ x, λ) → xλ( f ). cont Note that E cont /E L is a right E cont -module. cont (c) Let f ∈ Cc (N, E ) and let λ ∈ Hom A (Cc∞ (N , A), E cont ). Let U ⊂ N be an open compact subset containing the support of f . For any open Acont submodule L of M P let f L ∈ Cc∞ (U, E cont /E L ) be the map induced by f . Let / cont f L dλ ∈ E cont /E L U

be of (b). The elements . the image of ( f L , λ) by the natural pairing cont = lim E cont /E cont to f dλ combine in the projective limit E U L L ← −L

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. . give an element U f dλ ∈ E cont . One checks easily that U f dλ does not depend on the choice of U . We define / / f dλ := f dλ. N

U

We recall that J (N /V ) is a system of representatives of N / V when V ⊂ N is a compact open subgroup. Corollary 7.56. Let f ∈ Cc (N , E cont ) and let λ be a measure on N with values in E cont . Then / lim f (v) λ(1vV ) = f dλ V →{1}

N

v∈J (N /V )

with the limit over compact open subgroups V ⊂ N shrinking to {1}. Proof. We choose an open compact subset U ⊂ N containing the support of f . Let L be an open o-submodule of M P and a compact open subgroup V ⊂ N such that uV .⊂ U and f L (proof of Proposition 7.55) is constant on uV for all u ∈ U . Then U f L dλ is the image of f (v) λ(1vV ) v∈J (N/V ) cont by the quotient map E cont → E cont /E L .

Lemma 7.57. Let f ∈ Cc (N , E cont ) be a continuous map with support in the compact open subset U ⊂ N , let λ be a measure on N with values in E cont , and let L ⊂ M P be any open A-submodule. There is a compact open subgroup VL ⊂ N such that U VL = U and / cont f 1uV dλ − f (u)λ(1uV ) ∈ E L N

for any open subgroup V ⊂ VL and any u ∈ U . Proof. The integral in question is the limit (with respect to open subgroups V ⊂ V ) of the net ( f (uv) − f (u))λ(1uvV ). v∈J (V /V ) cont is a right ideal it therefore suffices to find a compact open subgroup Since E L VL ⊂ N such that U VL = U and cont f (uv) − f (u) ∈ E L

for any u ∈ U and v ∈ VL .

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We certainly find a compact open subgroup V˜ ⊂ N such that U V˜ = U . The map U × V˜ → E cont (u, v) → f (uv) − f (u) is continuous and maps any (u, 1) to zero. Hence, for any u ∈ U , there is an open neighbourhood Uu ⊂ U of u and a compact open subgroup Vu ⊂ V˜ cont . Since U is compact we have U = such that Uu × Vu is mapped to E L Uu 1 ∪ . . . ∪ Uu s for finitely many appropriate u i ∈ U . The compact open subgroup VL := Vu 1 ∩ . . . ∩ Vu s then is such that U × VL is mapped to cont EL . Let C(N , E cont ) be the space of continuous functions from N to E cont . For any continuous function f ∈ C(N , E cont ), for any compact open subset U ⊂ N and for any measure λ on N with values in E cont we denote / / f dλ = f 1U dλ U

N

where 1U ∈ C ∞ (U, A) is the characteristic function of U hence f 1U ∈ Cc (N , E cont ) is the restriction of f to U . The “integral of f on U ” (with respect to the measure λ) is equal to the “integral of the restriction of f to U ”. Remark 7.58. For f ∈ Cc (N , E cont ) and φ ∈ Cc∞ (N , A) we have / / / f φd Res = φ f d Res = f d Res ◦ Res(φ). N

N

N

Proof. This is immediate from the construction of the integral and the multiplicativity of Res.

5. G-equivariant sheaf on G/ P Let G be a locally profinite group containing P = N L as a closed subgroup satisfying the assumptions of Section 3.2 such that (a) G/P is compact. (b) There is a subset W in the G-normalizer NG (L) of L such that – the image of W in NG (L)/L is a subgroup, – G is the disjoint union of Pw P for w ∈ W . We note that Pw P = N w P = PwN .

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(c) There exists w0 ∈ W such that N w0 P is an open dense subset of G. We call C := N w0 P/P the open cell of G/P. (d) The map (n, b) → nw0 b from N × P onto N w0 P is a homeomorphism. Remark 7.59. These conditions imply that G = P P P = C(w0 )C(w0−1 ) where P := w0 Pw0−1 and C(g) = Pg P for g ∈ G. Proof. The intersection of the two dense open subsets gC and C in G/P is open and not empty, for any g ∈ G. The group G acts continuously on the topological space G/P, G × G/P → G/P (g, x P) → gx P. For n, x ∈ N and t ∈ L we have nt xw0 P = nt xt −1 w0 P = (nt.x)w0 P hence the action of P on the open cell corresponds to the action of P on N introduced before Proposition 7.32, i.e. the homeomorphism N → C,

u → xu := uw0 P

is P-equivariant. When M is an étale A[P+ ]-module, this allows us to systematically view the map Res in the following as a P-equivariant homomorphism of A-algebras Res : Cc∞ (C, A) → End A (M P ) and the corresponding sheaf (Theorem 7.38) as a sheaf on C. Our purpose is to show that this sheaf extends naturally to a G-equivariant sheaf on G/P for certain étale A[P+ ]-modules. When M is a complete topologically étale A[P+ ]-module we note that also integration with respect to the measure Res (Proposition 7.55) will be viewed in the following as a map Cc (C, E cont ) → E cont / f → f d Res C

on the space Cc (C, to E cont .

E cont )

of compactly supported continuous maps from C

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5.1. Topological G-space G/ P and the map α Definition 7.60. An open subset U of G/P is called standard if there is a g ∈ G such that gU is contained in the open cell C. The inclusion gU ⊂ N w0 P/P is equivalent to U = g −1 U w0 P/P for a unique open subset U ⊂ N . An open subset of a standard open subset is standard. The translates by G of N0 w0 P/P form a basis of the topology of G/P. Proposition 7.61. A compact open subset U ⊂ G/P is a disjoint union 5 U= g −1 V w0 P/P g∈I

where V ⊂ N is a compact open subgroup and I ⊂ G a finite subset. Proof. We first observe that any open covering of U can be refined into a disjoint open covering. In our case, this implies that U has a finite disjoint covering by standard compact open subsets. Let g −1 U w0 P/P ⊂ G/P be a standard compact open subset. Then U = 2u∈J uV (disjoint union) with a finite set J ⊂ U and V ⊂ N is a compact open subgroup. Then g −1 U w0 P/P = 2h∈I h −1 V w0 P/P (disjoint union) where I ={u −1 g | u∈J }. For g ∈ G and x in the nonempty open subset g −1 C ∩ C of G/P (Remark 7.59), there is a unique element α(g, x) ∈ P such that, if x = uw0 P/P with u ∈ N , then guw0 N = α(g, x)uw0 N . We give some properties of the map α. Lemma 7.62. Let g ∈ G. Then (i) g −1 C ∩ C = C if and only if g ∈ P. (ii) The map α(g, .) : g −1 C ∩ C → P is continuous. (iii) We have gx = α(g, x)x for x ∈ g −1 C ∩ C and we have α(b, x) = b for b ∈ P and x ∈ C. Proof. (i) We have g −1 C ∩ C = C if and only if g N w0 P ⊂ N w0 P if and only if g ∈ P. Indeed, the condition h Pw0 P ⊂ Pw0 P on h ∈ G depends only on Ph P and for w ∈ W , the condition w Pw0 P ⊂ Pw0 P implies ww0 ∈ Pw0 P hence ww0 ∈ w0 L by the hypothesis (b) hence w ∈ L.

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(ii) Let N g ⊂ N be such that N g w0 P/P = g −1 C ∩ C. It suffices to show that the map u → α(g, uw0 P)u : Ng → P is continuous. This follows from the continuity of the maps u → guw0 N : N g → Pw0 P/N = Pw0 N/N and bw0 N → b : Pw0 N /N → P. (iii) Obvious. Lemma 7.63. Let g, h ∈ G and x ∈ (gh)−1 C ∩h −1 C ∩C. Then hx ∈ g −1 C ∩C and we have α(gh, x) = α(g, hx)α(h, x). Proof. The first part of the assertion is obvious. Let x = uw0 P and hx = vw0 P with u, v ∈ N . We have huw0 N = α(h, x)uw0 N , gvw0 N = α(g, hx)vw0 N, and α(gh, x)uw0 N = ghuw0 N . The first identity implies α(h, x)uw0 P = vw0 P, hence v −1 α(h, x)u ∈ P ∩ w0 Pw0−1 . Hypothesis (d) easily yields P ∩ w0 Pw0−1 = L, hence α(h, x)u = vt for some t ∈ L. Multiplying the second identity on the right by w0−1 tw0 we obtain gvtw0 N = α(g, hx)vtw0 N = α(g, hx)α(h, x)uw0 N . Finally, by inserting the first identity into the right-hand side of the third identity we get α(gh, x)uw0 N = gα(h, x)uw0 N = gvtw0 N = α(g, hx)α(h, x)uw0 N which is the assertion. It will be technically convenient later to work on N instead of C. For g ∈ G let therefore N g be the open subset of N such that C ∩ g −1 C = Ng w0 P/P. We have N g = N if and only if g ∈ P (Lemma 7.62 (i)). We have the ∼ homeomorphism u → xu := uw0 P/P : N − → C and the continuous map (Lemma 7.62 (ii)) N g −→ P u −→ α(g, xu ) such that gu = α(g, xu )u n(g, ¯ u) α(g, xu )u = n(g, u)t (g, u)

for some n(g, ¯ u) ∈ N := w0 N w0−1 , for some n(g, u) ∈ N, t (g, u) ∈ L .

(7.23)

Lemma 7.64. Fix g ∈ G and let V ⊂ g −1 C ∩ C be any compact open subset. ˙ . . . ∪˙ Vm by compact open subsets There exists a disjoint covering V = V1 ∪ Vi and points xi ∈ Vi such that

294

Peter Schneider, Marie-France Vigneras, and Gergely Zabradi α(g, xi )Vi ⊂ gV

for any 1 ≤ i ≤ m. ∼

Proof. We denote the inverse of the homeomorphism u → xu : N − → C by x → u x . The image C ⊂ P of V under the continuous map x → α(g, x)u x V → P is compact. As (Lemma 7.62 (iii)) α(g, x)x = gx ∈ gV for any x ∈ V , under the continuous action of P on C, every element in the compact set C maps the point w0 P into gV . It follows that there is an open neighbourhood V0 ⊂ C of w0 P such that C V0 ⊂ gV . This means that α(g, x)u x V0 ⊂ gV

for any x ∈ V .

Using Proposition 7.61 we find, by appropriately shrinking V0 , a disjoint covering of V of the form V = u 1 V0 ∪˙ . . . ∪˙ u m V0 with u i ∈ N . We put xi := u i w0 P. We denote by G X := {x ∈ G | x X ⊂ X } the G-stabilizer of a subset X ⊂ G/P and by G †X := {g ∈ G | g ∈ G X , g −1 ∈ G X } = {x ∈ G | x X = X } the subgroup of invertible elements of G X . If G X is open then its inverse monoid is open hence G †X is open (and conversely). Lemma 7.65. The G-stabilizers G U and G †U are open in G, for any compact open subset U ⊂ G/P. Proof. By Proposition 7.61 it suffices to consider the case where U = U w0 P/P for some compact open subgroup U ⊂ N . As U w0 P ⊂ G is an open subset containing w0 there exists an open subgroup K ⊂ G such that K w0 ⊂ U w0 P. The set U/(K ∩ U ) is finite because U is compact and (K ∩ U ) ⊂ U is an open subgroup. The finite intersection K := 6 6 −1 = −1 is an open subgroup of K which is u∈U/(U ∩K ) u K u u∈U u K u normalized by U . But K U = U K implies that K U w0 P = U K w0 P ⊂ U (U w0 P)P = U w0 P, and hence that K ⊂ G U . We deduce that G U is open. Hence G †U is open. Remark 7.66. The G-stabilizer of the open cell C is the group P. Proof. Lemma 7.62 (i). For U ⊂ C the map GU × U → P

,

(g, x) → α(g, x)

(7.24)

is continuous because, if U = U w0 P/P with U open in N , then the map (g, u) → guw0 N : G U × U → Pw0 P/N = Pw0 N /N is continuous (cf. the proof of Lemma 7.62 (ii)).

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5.2. Equivariant sheaves and modules over skew group rings Our construction of the sheaf on G/P will proceed through a module theoretic interpretation of equivariant sheaves. The ring Cc∞ (C, A) has no unit element. But it has sufficiently many idempotents (the characteristic functions 1V of the compact open subsets V ⊂ C). A (left) module Z over Cc∞ (C, A) is called nondegenerate if for any z ∈ Z there is an idempotent e ∈ Cc∞ (C, A) such that ez = z. It is well known that the functor sheaves of A-modules on C → nondegenerate Cc∞ (C, A)-modules which sends a sheaf S to the A-module of global sections with compact support 0 Sc (C) := V S(V ), with V running over all compact open subsets in C, is an equivalence of categories. In fact, as we have discussed in the proof of Theorem 7.38 a quasi-inverse functor is given by sending the module Z to the sheaf whose sections on the compact open subset V ⊂ C are equal to 1V Z . In order to extend this equivalence to equivariant sheaves we note that the group P acts, by left translations, from the right on Cc∞ (C, A) which we write as ( f, b) → f b (.) := f (b.). This allows to introduce the skew group ring AC := Cc∞ (C, A)#P = ⊕b∈P bCc∞ (C, A) in which the multiplication is determined by the rule (b1 f 1 )(b2 f 2 ) = b1 b2 f 1b2 f 2

for bi ∈ P and f i ∈ Cc∞ (C, A).

It is easy to see that the above functor extends to an equivalence of categories

P-equivariant sheaves of A-modules on C − → nondegenerate AC -modules. We have the completely analogous formalism for the G-space G/P. The only small difference is that, since G/P is assumed to be compact, the ring C ∞ (G/P, A) of locally constant A-valued functions on G/P is unital. The skew group ring AG/P := C ∞ (G/P, A)#G = ⊕g∈G gC ∞ (G/P, A) therefore is unital as well, and the equivalence of categories reads

G-equivariant sheaves of A-modules on G/P − → unital AG/P -modules. For any open subset U ⊂ G/P the A-algebra Cc∞ (U, A) of A-valued locally constant and compactly supported functions on U is, by extending functions by zero, a subalgebra of C ∞ (G/P, A). It follows in particular that AC is a subring

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of AG/P . There is for our purposes a very important ring in between these two rings which is defined to be A := AC ⊂G/P := ⊕g∈G gCc∞ (g −1 C ∩ C, A). That A indeed is multiplicatively closed is immediate from the following observation. If supp( f ) denotes the support of a function f ∈ C ∞ (G/P, A) then we have the formula supp( f 1 f 2 ) = g −1 supp( f 1 ) ∩ supp( f 2 ) g

for g ∈ G and f 1 , f 2 ∈ C ∞ (G/P, A).

(7.25)

In particular, if f i ∈ Cc∞ (gi−1 C ∩ C, A) then supp( f 1 2 f 2 ) ⊂ g2−1 (g1−1 C ∩ C) ∩ (g2−1 C ∩ C) ⊂ (g1 g2 )−1 C ∩ C. g

We also have the A-submodule Z := ⊕g∈G gCc∞ (C, A) of AG/P . Using (7.25) again one sees that Z actually is a left ideal in AG/P which at the same time is a right A-submodule. This means that we have the well defined functor nondegenerate A-modules → unital AG/P -modules Z → Z ⊗A Z. Remark 7.67. The functor of restricting G-equivariant sheaves on G/P to the open cell C is faithful and detects isomorphisms. Proof. Any sheaf homomorphism which is the zero map, resp. an isomorphism, on sections on any compact open subset of C has, by G-equivariance, the same property on any standard compact open subset and hence, by Proposition 7.61, on any compact open subset of G/P. Proposition 7.68. The above functor Z → Z ⊗A Z is an equivalence of categories; a quasi-inverse functor is given by sending the AG/P -module Y to 0 V ⊂C 1V Y where V runs over all compact open subsets in C. Proof. We abbreviate the asserted candidate for the quasi-inverse functor by 0 R(Y ) := V ⊂C 1V Y . It immediately follows from Remark 7.67 that the functor R, which in terms of sheaves is the functor of restriction, is faithful and detects isomorphisms. By a slight abuse of notation we identify in the following a function f ∈ C ∞ (G/P, A) with the element 1 f ∈ AG/P , where 1 ∈ G denotes the unit

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element. Let V ⊂ C be a compact open subset. Then 1V AG/P 1V is a subring of AG/P (with the unit element 1V ), which we compute as follows: 1V AG/P 1V = 1V gC ∞ (V , A) = g1g −1 V C ∞ (V , A) g∈G

=

g∈G

gC ∞ (g −1 V ∩ V, A).

g∈G

We note: – If U ⊂ V ⊂ C are two compact open subsets then 1V AG/P 1V ⊃ 1U AG/P 1U . – Let f ∈ C c∞ (g −1 C ∩ C, A) be supported on the compact open subset U ⊂ g −1 C ∩C. Then V := U ∪ gU is compact open in C as well, and U ⊂ 0 g −1 V ∩V . This shows that Cc∞ (g −1 C∩C, A) = V ⊂C C ∞ (g −1 V ∩V , A). We deduce that

1V AG/P 1V = AC ⊂G/P = A.

V ⊂C

A completely analogous computation shows that 1V Z = 1V A. Given a nondegenerate A-module Z the map 1V (Z ⊗A Z ) = (1V Z) ⊗A Z = (1V A) ⊗A Z → 1V Z 1V a ⊗ z = 1V ⊗ 1V az → 1V az therefore is visibly an isomorphism of 1V AG/P 1V -modules. In the limit with respect to V we obtain a natural isomorphism of A-modules ∼ =

R(Z ⊗A Z ) − → Z. On the other hand, for any unital AG/P -module Y there is the obvious natural homomorphism of AG/P -modules Z ⊗A R(Y ) → Y a ⊗ z → az. It is an isomorphism because applying the functor R, which detects isomorphisms, to it gives the identity map. Remark 7.69. Let Z be a nondegenerate A-module. Viewed as an AC -module it corresponds to a P-equivariant sheaf Z on C. On the other hand, the AG/P on G/P. We module Y := Z ⊗A Z corresponds to a G-equivariant sheaf Y |C = extends the sheaf have Y Z , i.e., the sheaf Y Z.

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We have now seen that the step of going from A to AG/P is completely formal. On the other hand, for any topologically étale A[P+ ]-module M, the P-equivariance of Res together with Proposition 7.52 imply that Res extends to the A-algebra homomorphism Res :

P AC → Endcont A (M ) b fb → b ◦ Res( f b ).

b∈P

b∈P

When M is compact it is relatively easy, as we will show in the next section, to further extend this map from AC to A. This makes crucial use of the full topological module M P and not only its submodule McP of sections with compact support. When M is not compact this extension problem is much more subtle and requires more facts about the ring A. We introduce the compact open subset C0 := N0 w0 P/P of C, and we consider the unital subrings A0 := 1C0 AG/P 1C0 = gC ∞ (g −1 C0 ∩ C0 , A) g∈G

and AC 0 := 1C0 AC 1C0 =

bC ∞ (b−1 C0 ∩ C0 , A)

b∈P

of A and AC , respectively. Obviously AC 0 ⊆ A0 with the same unit element 1C0 . Since g −1 C0 ∩ C0 is nonempty if and only if g ∈ N0 P N0 we in fact have A0 = gC ∞ (g −1 C0 ∩ C0 , A). g∈N0 P N0

The map A[G] −→ AG/P sending g to g1G/P is a unital ring homomorphism. Hence we may view AG/P as an A[G]-module for the adjoint action G × AG/P −→ AG/P (g, y) −→ (g1G/P )y(g1G/P )−1 . One checks that AC ⊆ A are A[P]-submodules, that AC 0 ⊆ A0 are A[P+ ]submodules, and that the map Res : AC −→ E cont is a homomorphism of A[P]-modules. Proposition 7.70. The homomorphism of A[P]-modules ∼ =

A[P] ⊗ A[P+ ] A0 −→ A b ⊗ y −→ (b1G/P )y(b1G/P )−1

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299 ∼ =

is bijective; it restricts to an isomorphism A[P] ⊗ A[P+ ] AC 0 −→ AC . Proof. Since P = s −N P+ the assertion amounts to the claim that

A= (s −n 1G/P )A0 (s n 1G/P ) n≥0

and correspondingly for AC . But we have (s −n 1G/P ) gC ∞ (g −1 C0 ∩ C0 , A) (s n 1G/P ) = s −n gs n C ∞ ((s −n g −1 s n )s −n C0 ∩ s −n C0 , A) for any n ≥ 0 and any g ∈ G. P Suppose that we may extend the map Res : AC 0 −→ Endcont A (M ) to an A[P+ ]-equivariant (unital) A-algebra homomorphism

R0 : A0 −→ End A (M P ). By the above Proposition 7.70 it further extends uniquely to an A[P]equivariant map R : A −→ End A (M P ). Lemma 7.71. The map R is multiplicative. Proof. Using Proposition 7.70 we have that two arbitrary elements y, z ∈ A are of the form y = (s −m 1G/P )y0 (s m 1G/P ), z = (s −n 1G/P )z 0 (s n 1G/P ) with m, n ∈ N and y0 , z 0 ∈ A0 . We can choose m = n. It follows that yz = (s −m 1G/P )y0 z 0 (s m 1G/P ) = (s −m 1G/P )x0 (s m 1G/P ) with x0 := y0 z 0 ∈ A0 , and that R(yz) = R((s −m 1G/P )x0 (s m 1G/P )) = s −m ◦ R0 (x0 ) ◦ s m = s −m ◦ R0 (y0 ) ◦ R0 (z 0 ) ◦ s m = (s −m ◦ R0 (y0 ) ◦ s m ) ◦ (s −m ◦ R0 (z 0 ) ◦ s m ) = R(y) ◦ R(z).

Note that the images Res(AC 0 ) and R0 (A0 ) necessarily lie in the image of End A (M) = End A (Res(1C0 )(M P )) by the natural embedding into End A (M P ). This reduces us to search for an A[P+ ]-equivariant (unital) A-algebra homomorphism R0 : A0 −→ End A (M)

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which extends Res |AC 0 . In fact, since for g ∈ N0 P N0 and f ∈ C ∞ (g −1 C0 ∩ C0 , A) we have g f = (g1g −1 C0 ∩C0 )(1 f ) with 1 f ∈ AC 0 it suffices to find the elements Hg = R0 (g1g −1 C0 ∩C0 ) ∈ End A (M)

for g ∈ N0 P N0 .

Note that P+ = N0 L + is contained in N0 P N0 = N0 L N N0 . Proposition 7.72. We suppose given, for any g ∈ N0 P N0 , an element Hg ∈ End A (M). Then the map R0 :

g∈N0 P N0

A0 −→ End A (M) g f g −→ Hg ◦ res( f g ) g∈N0 P N0

is an A[P+ ]-equivariant (unital) A-algebra homomorphism which extends Res |AC 0 if and only if, for all g, h ∈ N0 P N0 , b ∈ P ∩ N0 P N0 , and all compact open subsets V ⊂ C0 , the relations H1. res(1V ) ◦ Hg = Hg ◦ res(1g −1 V ∩C0 ), H2. Hg ◦ Hh = Hgh ◦ res(1(gh)−1 C0 ∩h −1 C0 ∩C0 ), H3. Hb = b ◦ res(1b−1 C0 ∩C0 ). hold true. When H1 is true, H2 can equivalently be replaced by Hg ◦ Hh = Hgh ◦ res(1h −1 C0 ∩C0 ). Proof. Necessity of the relations is easily checked. Vice versa, the first two relations imply that R0 is multiplicative. The third relation says that R0 extends Res |AC 0 . The last sentence of the assertion derives from the fact that we have Hgh ◦ res(1(gh)−1 C0 ∩h −1 C0 ∩C0 ) = Hgh ◦ res(1(gh)−1 C0 ∩C0 ) ◦ res(1h −1 C0 ∩C0 ) = Hgh ◦ res(1h −1 C0 ∩C0 ) since Hgh ◦ res(1(gh)−1 C0 ∩C0 ) = Hgh by the first relation. The P+ -equivariance is equivalent to the identity R0 ((c1G/P )g f g (c1G/P )−1 ) = ϕc ◦ R0 (g f g ) ◦ ψc where c ∈ P+ and f g is any function in C ∞ (g −1 C0 ∩ C0 ). By the definition of R0 and the P+ -equivariance of res the left-hand side is equal to Hcgc−1 ◦ ϕc ◦ res( f g ) ◦ ψc whereas the right-hand side is ϕc ◦ Hg ◦ res( f g ) ◦ ψc .

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Since ψc is surjective and res( f g ) = res(1g −1 C0 ∩C0 ) ◦ res( f g ) we see that the P+ -equivariance of R0 is equivalent to the identity Hcgc−1 ◦ ϕc ◦ res(1g−1 C0 ∩C0 ) = ϕc ◦ Hg ◦ res(1g −1 C0 ∩C0 ). But as a special case of the first relation we have Hg ◦ res(1g−1 C0 ∩C0 ) = Hg . Hence the latter identity coincides with the relation Hcgc−1 ◦ ϕc ◦ res(1g −1 C0 ∩C0 ) = ϕc ◦ Hg . This relation holds true because ϕc = Hc and by the second relation Hcgc−1 ◦ Hc = Hcg and Hc ◦ Hg = Hcg ◦ res(1g −1 C0 ∩C0 ).

5.3. Integrating α when M is compact Let M be a compact topologically étale A[P+ ]-module. Then M P is compact, hence the continuous action of P on M P (Proposition 7.52) induces a continuous map P → E cont . We will construct an extension R es of Res to AC ⊂G/P by integration. For any g ∈ G, we consider the continuous map α(g,.)

αg g −1 C ∩ C −−−→ P → E cont . We introduce the A-linear maps ρ A = AC ⊂G/P → Cc (C, E cont ) g f g → αg f g g∈G

g∈G

and R es A = AC ⊂G/P → E cont / a → ρ(a)d Res . C

For b ∈ P the map αb is the constant map on C with value b (Lemma 5.3 iii). It follows that R es | AC = Res is an extension as we want it.

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Theorem 7.73. R es is a homomorphism of A-algebras. Proof. Let g, h ∈ G and let Vg and Vh be compact open subsets of g −1 C ∩ C and h −1 C ∩ C, respectively. We have to show that R es((g1Vg )(h1Vh )) = R es(g1Vg ) ◦ R es(h1Vh ) holds true. This is, by definition of R es, equivalent to the identity / / / αgh 1h −1 Vg ∩Vh dRes = αg 1Vg dRes ◦ αh 1Vh dRes. C

C

C

Let Ug , Uh be the open compact subsets of N corresponding to Vg , Vh and let f be the map αg 1Vg , seen as a map on N with support on Ug . Let L ⊂ M P be an open A-submodule and let VL be chosen as in Lemma 7.57, with λ = Res. If we let Nk,v = α(h, xv ).(v Nk ), then the P-equivariance of Res combined with Remark 7.58 yield, for v ∈ Uh and k ≥ 1 / / αg 1Vg dRes ◦ α(h, xv ) ◦ Res(1v Nk ) = f dRes ◦ Res(1 Nk,v ) ◦ α(h, xv ) C

N

/ = N

( f 1 Nk,v )dRes ◦ α(h, xv ).

Writing α(h, xv ) = n v tv with n v ∈ N and tv ∈ L, for k large enough we have Ug tv Nk tv−1 ⊂ Ug and tv Nk tv−1 ⊂ VL for all v ∈ Uh (by compactness of (tv )v∈Uh ). Since Nk,v = (α(h, xv ).v)tv Nk tv−1 , we deduce that Nk,v ∩ Ug = ∅ ⇔ α(h, x v ).v ∈ Ug ⇔ hx v ∈ Vg and hence, by Lemma 7.57 for all sufficiently large k we have, uniformly in v ∈ Uh , / f 1 Nk,v dRes ≡ 1xv ∈h −1 Vg ∩Vh f (α(h, xv ).v) ◦ Res(1 Nk,v ) = N

cont = 1xv ∈h −1 Vg ∩Vh α(g, hx v ) ◦ Res(1 Nk,v ) (mod E L ).

Combining the last two relations with Lemma 7.63, and using again the P-equivariance of Res, we obtain for k large enough and for all v ∈ Uh / αg 1Vg dRes ◦ α(h, xv ) ◦ Res(1v Nk ) C

cont ≡ 1xv ∈h −1 Vg ∩Vh α(gh, xv ) ◦ Res(1v Nk ) (mod E L ).

The result follows by summing over v and letting k → ∞ (Corollary 7.56).

5.4. G-equivariant sheaf on G/ P Let M be a compact topologically étale A[P+ ]-module. We briefly survey our construction of a G-equivariant sheaf on G/P functorially associated with M.

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From Proposition 7.32 we obtain an A-algebra homomorphism Res Cc∞ (C, A)#P → E cont which gives rise to a P-equivariant sheaf on C as described in detail in Theorem 7.38. By Theorem 7.73, it extends to an A-algebra homomorphism R es AC ⊂G/P → E cont . This homomorphism defines on the global sections with compact support McP of the sheaf on C the structure of a nondegenerate AC ⊂G/P -module. The latter leads, by Proposition 7.68, to the unital Cc∞ (G/P, A)#G-module Z ⊗A McP which corresponds to a G-equivariant sheaf on G/P extending the earlier sheaf on C (Remark 7.39). We will denote the sections of this latter sheaf on an open subset U ⊂ G/P by M U . The restriction maps in this sheaf, for open subsets V ⊂ U ⊂ G/P, will simply be written as ResU V M U → M V. We observe that for a standard compact open subset U ⊂ G/P with g ∈ G such that gU ⊂ C the action of the element g on the sheaf induces an iso∼ =

morphism of A-modules M U − → M gU = Mg U . Being the image of a continuous projector on M P (Proposition 7.52), Mg U is naturally a compact topological A-module. We use the above isomorphism to transport this topology to M U. The result is independent of the choice of g since, if gU = hU for some other h ∈ G, then hU ⊂ (gh −1 )−1 C ∩ C and, by construction, the endomorphism gh −1 of M hU is given by the continuous map R es(gh −1 1h U ). A general compact open subset U ⊂ G/P is the disjoint union U = ˙ ...∪ ˙ Um of standard compact open subsets Ui (Proposition 7.61). We U1 ∪ equip M U = M U1 ⊕ . . . ⊕ M Um with the direct product topology. One easily verifies that this is independent of the choice of the covering. Finally, for an arbitrary open subset U ⊂ G/P we have M U = lim M V, ← − where V runs over all compact open subsets V ⊂ U, and we equip M U with the corresponding projective limit topology. It is straightforward to check that all restriction maps are continuous and that any g ∈ G acts by continuous homomorphisms. We see that (M U)U is a G-equivariant sheaf of compact topological A-modules. Lemma 7.74. For any compact open subset U ⊂ G/P the action G †U × (M U) → M U of the open subgroup G †U (Lemma 7.65) on the sections on U is continuous. Proof. Using Proposition 7.61, it suffices to consider the case that U ⊂ C. Note that G †U acts by continuous automorphisms on M U = MU . By (7.24) the map

304

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is continuous. Hence ([3] TG X.28 Th. 3) the corresponding map G †U → C(U, E cont ) is continuous, where we always equip the module C(U, E cont ) of E cont -valued continuous maps on U with the compact-open topology. On the other hand it is easy to see that, for any measure λ on C with values in E cont , the map / . dλ C(U, E cont ) → E cont U

is continuous. It follows that the map G †U → E cont g → R es(g1U ) is continuous. The direct decomposition M P = MU ⊕ MC −U gives a natcont through which the above map ural inclusion map Endcont A (MU ) → E factorizes. The resulting map G †U → Endcont A (MU ) is continuous and coincides with the G †U -action on MU . As MU is compact this continuity implies the continuity of the action G †U × MU → MU . The same construction can be done, starting from the compact topologically étale A[PU ]-module MU , for any compact open subgroup U ⊂ N . Proposition 7.75. Let U ⊂ N be a compact open subgroup. The Gequivariant sheaves on G/P associated to (N0 , M) and to (U, MU ) are equal. Proof. As the P-equivariant sheaves on the open cell associated to (N0 , M) and to (U, MU ) are equal by Proposition 7.43, and as the function αg depends only on the open cell, our formal construction gives the same G-equivariant sheaf.

6. Integrating α when M is non compact Recall that we have chosen a certain element s ∈ Z(L) such that L = L − s Z and (Nk = s k N0 s −k )k∈Z is a decreasing sequence with union N and trivial

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intersection. We now suppose in addition that (N k := s −k w0 N0 w0−1 s k )k∈Z is a decreasing sequence with union N = w0 N w0−1 and trivial intersection. We have chosen A and M in Section 4.1. We suppose now in addition that M is a topologically étale A[P+ ]-module which is Hausdorff and complete. Definition 7.76. A special family of compact sets in M is a family C of compact subsets of M satisfying: C(1) C(2) C(3) C(4)

Any compact subset of a compact set in C also lies in C. 0 If C1 , C2 , . . . , C n ∈ C then in=1 Ci is in C, as well. For all C ∈ C we have N0 C ∈ C. 0 M(C) := C∈C C is an étale A[P+ ]-submodule of M.

Note that M is the union of its compact subsets, and that the family of all compact subsets of M satisfies these four properties. Let C be a special family of compact sets in M. A map from M(C) to M is called C-continuous if its restriction to any C ∈ C is continuous. We equip the A-module HomCont A (M(C), M) of C-continuous A-linear homomorphisms from M(C) to M with the C-open topology. The A-submodules E(C, M) := { f ∈ HomCont (M(C), M) : f (C) ⊆ M}, A for any C ∈ C and any open A-submodule M ⊆ M, form a fundamental system of open neighbourhoods of zero in HomCont (M(C), M). Indeed, A this system is closed for finite intersection by C(2). Since N0 is compact the E(C, M) for C such that N0 C ⊆ C and M an A[N0 ]-submodule still form a fundamental system of open neighbourhoods of zero (Lemma 7.50 and C(3)). We have: – HomCont (M(C), M) is a topological A-module. A – HomCont (M(C), M) is Hausdorff, since C covers M(C) by C(4) and M is A Hausdorff. – HomCont (M(C), M) is complete ([3] TG X.9 Cor.2). A

6.1. (s, res, C)-integrals We have the P+ -equivariant measure res : C ∞ (N0 , A) −→ Endcont A (M) on N0 . If M is not compact then it is no longer possible to integrate any map in the A-module C(N0 , Endcont A (M)) of all continuous maps on N0 with values in Endcont (M) against this measure. This forces us to introduce a notion of A integrability with respect to a special family of compact sets in M.

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Definition 7.77. A map F : N0 → HomCont (M(C), M) is called integrable A with respect to (s, res, C) if the limit / Fd res := lim F(u) ◦ res(1u Nk ), k→∞

N0

u∈J (N0 /Nk )

where J (N0 /Nk ) ⊆ N0 , for any k ∈ N, is a set of representatives for the cosets in N0 /Nk , exists in HomCont A (M(C), M) and does not depend on the choice of the sets J (N0 /Nk ). We suppress C from the notation when C is the family of all compact subsets of M. Note that we regard res(1u Nk ) as an element of Endcont A (M(C)). This makes sense as the algebraically étale submodule M(C) of the topologically étale module M is topologically étale. One easily sees that the set C int (N0 , HomCont (M(C), M)) of integrable A maps is an A-module. The A-linear map / .d res : C int (N0 , HomCont (M(C), M)) −→ HomCont A A (M(C), M) N0

will be called the (s, res, C)-integral. We give now a general integrability criterion. Proposition 7.78. A map F : N0 −→ HomCont (M(C), M) is (s, res, C)A integrable if, for any C ∈ C and any open A-submodule M ⊆ M, there exists an integer kC,M ≥ 0 such that (F(u) − F(uv)) ◦ res(1u Nk+1 ) ∈ E(C, M) for any k ≥ kC,M , u ∈ N0 , and v ∈ Nk . Proof. Let J (N0 /Nk ) and J (N0 /Nk ), for k ≥ 0, be two choices of sets of representatives. We put sk (F) := F(u) ◦ res(1u Nk ) and sk (F) u∈J (N0 /Nk )

:=

F(u ) ◦ res(1u Nk ).

u ∈J (N0 /Nk )

Since HomCont (M(C), M) is Hausdorff and complete it suffices to show that, A given any neighbourhood of zero E(C, M), there exists an integer k0 ≥ 0 such that sk (F) − sk+1 (F), sk (F) − sk (F) ∈ E(C, M)

for any k ≥ k0 .

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For u ∈ J (N0 /Nk+1 ) let u¯ ∈ J (N0 /Nk ) and u ∈ J (N0 /Nk+1 ) be the unique elements such that u Nk = u¯ Nk and u Nk+1 = u Nk+1 , respectively. Then sk (F) = F(u) ¯ ◦ res(1u Nk+1 ) u∈J (N0 /Nk+1 )

and hence sk (F) − sk+1 (F) =

(F(u(u −1 u)) ¯ − F(u)) ◦ res(1u Nk+1 ). (7.26)

u∈J (N0 /Nk+1 )

Since u −1 u¯ ∈ Nk it follows from our assumption that the right-hand side lies in E(C, M) for k ≥ kC,M . Similarly sk+1 (F) − sk+1 (F) = (F(u) − F(u(u −1 u ))) ◦ res(1u Nk+1 ); u∈J (N0 /Nk+1 )

again, as u −1 u ∈ Nk+1 ⊆ Nk , the right-hand sum is contained in E(C, M) for k ≥ kC,M .

6.2. Integrability criterion for α Let Ug ⊆ N0 be the compact open subset such that Ug w0 P/P = g −1 C0 ∩ C0 . This intersection is nonempty if and only if g ∈ N0 P N0 , which we therefore assume in the following. We consider the map αg,0 : N0 −→ Endcont A (M) $ Res(1 N0 ) ◦ αg (xu ) ◦ Res(1 N0 ) if u ∈ Ug , u −→ 0 otherwise cont under the natural (where we identify Endcont A (M) with its image in E embedding (7.20) using that Res(1 N0 ) = σ0 ◦ ev0 ). Restricting αg,0 (u) ∈ Endcont A (M) to M(C) for any u ∈ N0 we may view αg,0 as a map from N0 to Endcont A (M(C)) since M(C) is an étale A[P+ ]-submodule of M. However, as we do not assume M(C) to be complete, it will be more convenient for the purpose of integration to regard αg,0 as a map into HomCont A (M(C), M). We want to establish a criterion for the (s, res, C)-integrability of the map αg,0 . By the argument in the proof of Lemma 7.64 (applied to V = g −1 C0 ∩ C0 ) we may choose an integer k g(0) ≥ 0 such that, for any k ≥ k g(0) , we have Ug Nk ⊆ Ug and

α(g, xu ).u Nk ⊆ gUg

for any u ∈ Ug .

(7.27)

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Lemma 7.79. For u ∈ Ug and k ≥ k g we have αg,0 (u) ◦ res(1u Nk ) = α(g, xu ) ◦ Res(1u Nk ). Proof. Using the P-equivariance of Res we have α(g, xu ) ◦ Res(1u Nk ) = Res(1α(g,xu ).u Nk ) ◦ α(g, xu ) ◦ Res(1u Nk ) = Res(1 N0 ) ◦ Res(1α(g,xu ).u Nk ) ◦ α(g, xu ) ◦ Res(1u Nk ) = Res(1 N0 ) ◦ α(g, xu ) ◦ Res(1 N0 ) ◦ Res(1u Nk ) = αg,0 (u) ◦ res(1u Nk ) where the second identity follows from (7.27). (0)

For u ∈ U g and k ≥ k g we put Hg,J (N0 /Nk ) :=

α(g, xu ) ◦ Res(1u Nk ).

(7.28)

u∈Ug ∩J (N0 /Nk )

By Lemma 7.79, each summand on the right-hand side belongs to End A (M(C)). If αg,0 is (s, res, C)-integrable, the limit Hg :=

lim

(0)

k≥k g ,k→∞

Hg,J (N0 /Nk )

(7.29)

exists in HomCont (M(C), M) and is equal to the (s, res, C)-integral of αg,0 A / αg,0 d res = Hg . (7.30) N0

We investigate the integrability criterion of Proposition 7.78 for the function αg,0 . We have to consider the elements g (u, k, v) := (αg,0 (u) − αg,0 (uv)) ◦ res(1u Nk+1 ),

(7.31)

(0)

for u ∈ Ug , k ≥ k g , and v ∈ Nk . By Lemma 7.79, we have g (u, k, v) = (αg,0 (u) ◦ res(1u Nk ) − αg,0 (uv) ◦ res(1uv Nk )) ◦ res(1u Nk+1 ) = (α(g, xu ) ◦ Res(1u Nk ) − α(g, xuv ) ◦ Res(1uv Nk )) ◦ Res(1u Nk+1 ) = (α(g, xu ) − α(g, xuv )) ◦ Res(1u Nk+1 ) = (α(g, xu ) − α(g, xuv )) ◦ u ◦ Res(1 Nk+1 ) ◦ u −1 . Recall that N g ⊂ N is the subset such that N g w0 P/P = g −1 C ∩ C. Lemma 7.80. For u ∈ Ng and v ∈ N such that uv ∈ N g we have: i. v ∈ Nn(g,u) ; ¯ ii. α(g, xuv ) = α(g, xu )uα(n(g, ¯ u), xv )u −1 .

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Proof. i. Because of gu = α(g, xu )u n(g, ¯ u) we have α(g, xu )u n(g, ¯ u)v = guv ∈ α(g, xuv )uv N and hence n(g, ¯ u)vw0 P = u −1 α(g, xu )−1 α(g, xuv )uvw0 P ∈ Pw0 P. ii. By i. the equation n(g, ¯ u)vw0 N = α(n(g, ¯ u), xv )vw0 N holds. Hence guvw0 N = α(g, xu )u n(g, ¯ u)vw0 N = α(g, x u )uα(n(g, ¯ u), xv )vw0 N and therefore α(g, xuv )uv = α(g, xu )uα(n(g, ¯ u), xv )v. Let f : Ug → P be the map u → α(g, xu )u. The previous computation shows that for all u ∈ Ug and v ∈ Nk we have g (u, k, v) = ( f (u) − f (uv)v −1 ) ◦ Res(1 Nk+1 ) ◦ u −1 .

(7.32)

Let f (u) = n(g, u)t (g, u), with n(g, u) ∈ N0 and t (g, u) ∈ L. Also, write gu = f (u)n(g, u) with n(g, u) ∈ N . Since t (g, Ug ) ⊂ L and n(g, Ug ) ⊂ N (1) (0) are compact subsets, there is k g ≥ k g such that (1)

g := t (g, Ug )s k g ⊂ L + ,

n(g, Ug ) ⊂ N −k (1) . g

(7.33)

Proposition 7.81. For any compact open subgroup P1 of P0 there is (2) (1) (2) k g (P1 ) ≥ k g such that for all k ≥ k g (P1 ), u ∈ Ug and v ∈ Nk (1)

f (u) − f (uv)v −1 ∈ N0 s k−k g (1 − P1 ) g s −k . Proof. We abbreviate n(u) = n(g, u) and similarly for t (u) and n(u). Since f (u)n(u)v = guv = f (uv)n(uv), we have f (u) − f (uv)v −1 = f (u)(1 − n(u)vn(uv)−1 v −1 ) = n(u)(1 − t (u)n(u)vn(uv)−1 v −1 t (u)−1 )t (u). (1)

Since n(u) ∈ N0 , t (u) ∈ s −k g g and (t (u))u∈Ug is compact, it suffices to prove that for any compact open subgroup P2 of P0 we have n(u)vn(uv)−1 v −1 ∈ s k P2 s −k for sufficiently large k. But if v = s k n 0 s −k , we can write −k n(u)vn(uv)−1 v −1 = s k (s −k n(u)s k )n 0 (s −k n(uv)−1 s k )n −1 0 s

∈ s k N k−k (1) ( g

n 0 ∈N0

−k n 0 N k−k (1) n −1 0 )s . g

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The result follows from the compactness of N0 and the fact that the N k ’s shrink to {1} as k → ∞. (2)

Corollary 7.82. For any compact open subgroup P1 of P0 and k ≥ k g (P1 ) (1)

g (Ug , k, Nk ) ⊂ N0 s k−kg (1 − P1 ) g sψ k+1 N0 . Proof. Proposition 7.81 and relation (7.32) show that (1)

g (Ug , k, Nk ) ⊂ N0 s k−k g (1 − P1 ) g s ◦ s −(k+1) ◦ Res(1 Nk+1 ) ◦ N0 . The P-equivariance of Res yields s −(k+1) ◦ Res(1 Nk+1 ) = Res(1 N0 ) ◦ s −k−1 , cont P and this is the image of ψ k+1 ∈ Endcont A (M) in End A (M ). The result follows. This leads to an integrability criterion for αg,0 , which depends only on (s, M, C). Proposition 7.83. We suppose that (s, M, C) satisfies: C(5) For any C ∈ C the compact subset ψ(C) ⊆ M also lies in C. T(1) For any C ∈ C such that C = N0 C, any open A[N0 ]-submodule M of M, and any compact subset C+ ⊆ L + there exists a compact open subgroup P1 = P1 (C, M, C+ ) ⊆ P0 and an integer k(C, M, C+ ) ≥ 0 such that s k (1 − P1 )C+ ψ k (C) ⊆ M

for any k ≥ k(C, M, C+ ).

(7.34)

Then the map αg,0 : N0 → HomCont (M(C), M) is (s, res, C)-integrable for all A g ∈ N0 P N0 . Proof. By the general integrability criterion of Proposition 7.78, the map αg,0 is integrable if for any (C, M) as above, there exists kC,M,g ≥ 0 such that g (u, k, v) ∈ E(C, M)

for any k ≥ kC,M,g , u ∈ Ug , and v ∈ Nk . (7.35) (2) By Corollary 7.82, this is true if kC,M,g ≥ k g (P1 ) and (1)

s k−kg (1 − P1 ) g sψ k+1 (C) ⊂ M,

(7.36)

because N0 M = M and N0 C = C. We note that the set C+ = g s is contained in L + by (7.33) and is compact, (1)

that the set C = ψ kg +1 (C) ⊂ M is compact and N0 C = C because the map ψ is continuous and N0 ψ(C) = ψ(s N0 s −1 C) = ψ(C), and that (7.36) is equivalent to (1)

(1)

s k−k g (1 − P1 )C+ ψ k−kg ⊂ E(C , M).

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By our hypothesis, there exists an open subgroup P1 ⊂ P0 such that this (1) inclusion is satisfied when k ≥ k g + k(C , M, C + ). For kC,M,g := max(k g(1) + k(C , M, C + ), k g(2) (P1 )) (1)

(7.35) is satisfied. By construction, P1 depends on ψ k g only on C, M, g.

+1 (C), M,

(7.37) g s, hence

Later, under the assumptions of Proposition 7.83, we will use the argument in the previous proof in the following slightly more general form: for C, M, C+ as in the proposition and an integer k ≥ 0 we have

s k−k (1 − P1 (ψ k (C), M, C+ ))C+ ψ k ⊆ E(C, M)

(7.38)

k

for any k ≥ k + k(ψ (C), M, C + ).

6.3. Extension of Res Proposition 7.84. Suppose that (s, M, C) satisfies the assumptions of Proposition 7.83 and that the (s, res, C)-integral Hg of αg,0 is contained in End A (M(C)) for all g ∈ N0 P N0 . In addition we assume that: C(6) For any C ∈ C the compact subset ϕ(C) ⊆ M also lies in C. T(2) Given a set J (N0 /Nk ) ⊂ N0 of representatives for cosets in N0 /Nk , for k ≥ 1, for any x ∈ M(C) and g ∈ N0 P N0 there exists a compact A-submodule C x,g ∈ C and a positive integer k x,g such that Hg,J (N0 /Nk ) (x) ⊆ C x,g for any k ≥ k x,g . Then the Hg satisfy the relations H1, H2, H3 of Proposition 7.72. Remark 7.85. The properties C(3), C(5), C(6) imply that for any u ∈ N0 , k ≥ 1, and C ∈ C also res(1u Nk )(C) lies in C. Indeed, res(1u Nk ) = u ◦ ϕ k ◦ ψ k ◦ u −1 . We prove now H1 and H3, which do not use the last assumption. The proof of H2 is postponed to the next subsection. Proof. For the proof of H1 let V ⊂ C0 be a compact open subset and let U1 , U2 be the compact open subsets of N0 corresponding to V and g −1 V ∩C0 . To prove that res(1V ) ◦ Hg = Hg ◦ res(1g −1 V ∩C0 ), it suffices to verify that if k is large enough, then for all u ∈ U g we have Res(1U1 ) ◦ α(g, xu ) ◦ Res(1u Nk ) = α(g, xu ) ◦ Res(1u Nk ) ◦ Res(1U2 ). (7.39) If Nk,u = α(g, xu ).(u Nk ), then by P-equivariance of Res, (7.39) is equivalent to

312

Peter Schneider, Marie-France Vigneras, and Gergely Zabradi Res(1U1 ∩Nk,u ) ◦ α(g, xu ) = α(g, xu ) ◦ Res(1u Nk ∩U2 ).

(7.40)

Write α(g, xu ) = n u tu with n u ∈ N and tu ∈ L. If k is large enough, then for all u ∈ U g we have U2 Nk ⊂ U2 and U1 tu Nk tu−1 ⊂ U1 . Since Nk,u = (α(g, xu ).u)tu Nk tu−1 , we obtain U1 ∩ Nk,u = ∅ ⇔ α(g, xu ).u ∈ U1 ⇔ gx u ∈ V ⇔ xu ∈ g −1 V ∩ C0 ⇔ u ∈ U2 ⇔ u Nk ⊂ U2 . Hence (7.40) is equivalent to 0 = 0 or to Res(1 Nk,u ) ◦ α(g, xu ) = α(g, xu ) ◦ Res(1u Nk ), which is true as Res is P-equivariant. H3. For b ∈ P ∩ N0 P N0 we have αb,0 = constant map on N0 with value res(1C0 ) ◦ b ◦ res(1C0 ) and hence Hb = res(1C0 ) ◦ b ◦ res(1C0 ) = b ◦ res(1b−1 C0 ∩C0 ).

6.4. Proof of the product formula We invoke now the full set of assumptions of Proposition 7.84 and we prove the product formula Hg ◦ Hh = Hgh ◦ res(1h −1 C0 ∩C0 ) for g, h ∈ N0 P N0 . This suffices by Proposition 7.72. (0) (1) (0) Let k0 := max(k g , kh , k gh ) + 1 and let k ≥ k0 . (0)

(1)

(0)

As k ≥ kh (because kh ≥ kh (7.33)), the set Uh is a disjoint union of cosets u Nk . We choose a set J (N0 /Nk ) ⊂ N0 of representatives of the cosets in N0 /Nk and for each u ∈ J (N0 /Nk ) ∩ Uh a set Ju (N0 /Nk−k0 ) ⊂ N0 of representatives of the cosets in N0 /Nk−k0 with n(g, u) ∈ Ju (N0 /Nk−k0 ) (see (7.23)). We write Hg ◦ Hh − Hgh ◦ res(1h −1 C0 ∩C0 ) as the sum over u ∈ J (N0 /Nk ) ∩Uh of (Hg ◦ Hh − Hgh ◦ Res(1Uh )) ◦ Res(1u Nk ) = ak,u + bk,u + ck,u ,

(7.41)

where ak,u :=(Hg ◦ Hh − Hg,Ju (N0 /Nk−k0 ) ◦ Hh,J (N0 /Nk ) ) ◦ Res(1u Nk ) bk,u :=(Hg, Ju (N0 /Nk−k0 ) ◦ Hh,J (N0 /Nk ) −Hgh,J (N0 /Nk ) ) ◦ Res(1Uh )◦ Res(1u Nk ) ck,u :=(Hgh,J (N0 /Nk ) − Hgh ) ◦ Res(1Uh ) ◦ Res(1u Nk ).

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The product formula follows from the claim that bk,u = 0 and that for an arbitrary compact subset C ∈ C such that N0 C = C, and an arbitrary open A[N0 ]-module M ⊂ M, ak,u and ck,u lie in E(C, M) when k is very large, independently of u. The claim results from the following three propositions. Because (s, M, C) satisfies Proposition 7.83, we associate to (C, M, g) the integer kC,M,g defined in (7.37) which is independent of the choice of the J (N0 /Nk ). For the sake of simplicity, we write (k) (k+1) H(k) − H(k) g := Hg, J (N0 /Nk ) , sg := Hg g .

(7.42)

(0)

By (7.26), we have, for k ≥ k g , sg(k) =

g (u, k, vu )

u∈Ug ∩J (N0 /Nk+1 )

for some vu ∈ Nk . It follows from Corollary 7.82 that, for any given compact open subgroup P1 ⊂ P0 , we have (1)

sg(k) ∈ < N0 s k−k g (1 − P1 ) g sψ k+1 N0 > A

for k ≥ k g(2) (P1 ),

(7.43)

where we use the notation < X > A for the A-submodule in End A (M) gener(k) ated by X . We deduce from the proof of Proposition 7.83, that sg ∈ E(C, M) for any k ≥ kC,M,g . Proposition 7.86. (Hg − Hg,J (N0 /Nk ) ) ◦ Res(1u Nk ) ∈ E(C, M) for any k ≥ kC,M,g . (0)

Proof. When k ≥ 0, k2 ≥ max(k − 1, k g ), u ∈ Ug , v ∈ Nk we have that g (u , k2 , v) ◦ Res(1u Nk ) is equal either to g (u , k2 , v) or to 0. If follows that sg(k2 ) ◦ Res(1u Nk ) ⊆ E(C, M)

for any k2 ≥ max(k − 1, kC,M,g ) and k ≥ 0.

Now we fix k ≥ kC,M,g . Note that Res(1u Nk )(C) is contained in C by the (k ) stability of C by ψ, ϕ, and u ±1 . Therefore the sequence (Hg 2 ◦ Res(1u Nk ))k2 converges to Hg ◦ Res(1u Nk ) in HomCont (M(C), M). In particular, we have A 2) (Hg − H(k g ) ◦ Res(1u Nk ) ⊆ E(C, M)

for any k2 ≥ max(k − 1, kC,M,g ) and k ≥ 0. The statement follows by taking k2 = k. This establishes that ck,u lies in E(C, M) when k ≥ kC,M,gh . Note that the proposition is true also for any other system J (N0 /Nk ) ⊂ N0 of representatives for the cosets in N0 /Nk for the same integer kC,M,g . We (k) write H(k) g and sg for the elements defined in (7.42) for J (N0 /Nk ).

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Proposition 7.87. There exists an integer kC,M,g,h,k0 ∈ N, independent of the choices of J (N0 /Nk ) and J (N0 /Nk ), such that: (k+1−k )

(k−k )

0 i. Hg ◦ Hh(k+1) − Hg 0 ◦ Hh(k) ∈ E(C, M), for all k ≥ (k−k ) (k) kC,M,g,h,k0 , and the sequence (Hg 0 ◦ Hh ) converges to Hg ◦ Hh in Cont Hom A (M(C), M). (k−k ) (k) ii. (Hg ◦ Hh − Hg 0 ◦ Hh ) ◦ Res(1u Nk ) ∈ E(C, M), for all k ≥ kC,M,g,h,k0 .

Proof. i. To prove the first assertion, we write (k+1)

(k)

0) ◦ H H(k+1−k g h

0) ◦ H − H(k−k g h

(k)

(k)

(k−k 0 ) 0) ◦ s = H(k+1−k ◦ Hh . g h + sg (1)

(k)

(k)

Note that, when k ≥ k g , the endomorphisms Hg in the A-module < N0 s

(1) k−k g

(7.44)

and Hg are contained

g ψ k N0 > A , because

α(g, xu ) ◦ Res(1u Nk ) (1)

= n(g, u)t (g, u)u −1 us k ψ k u −1 ⊂ N0 s k−k g g ψ k N0

for u ∈ Ug .

We consider any compact open subgroup P1 ⊂ P0 and we assume k ≥ (2) (2) max(k g (P1 )+k0 , kh (P1 )). With (7.43) we obtain that (7.44) is contained in (1)

(1)

< N0 s k+1−k0 −kg g ψ k+1−k0 N0 s k−kh (1 − P1 ) h sψ k+1 N0 > A (1)

(1)

+ < N0 s k−k0 −kg (1 − P1 ) g sψ k−k0 +1 N0 s k−kh h ψ k N0 > A . Recalling that ψ a (N0 ϕ a+b (m)) = ψ a (N0 )ϕ b (m) = N0 ϕ b (m) for a, b ∈ N and m ∈ M, we see that this is contained in (1)

(1)

< N0 s k+1−k0 −kg g N0 s k0 −kh

−1

(1)

(1 − P1 ) h sψ k+1 N0 > A (1)

+ < N0 s k−k0 −kg (1 − P1 ) g N0 s k0 −kh h ψ k N0 > A . As k + 1 − k0 − k g(1) ≥ k g(2) (P1 ) + 1 − k g(1) ≥ 1 and as g ⊂ L + , we have (1)

(1)

N0 s k+1−k0 −kg g N0 ⊂ N0 s k+1−k0 −k g g , and this is contained in (1)

(1)

< N0 s k+1−k0 −k g g s k0 −kh (1)

−1

(1 − P1 ) h sψ k+1 N0 > A (1)

+ < N0 s k−k0 −k g (1 − P1 ) g s k0 −kh h ψ k N0 > A .

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We assume, as we may, that the compact open subgroup P1 of P0 satisfies (1) t P1 t −1 ⊆ P1 for all t in the compact set g s k0 −kh −1 of L + . Then we finally obtain that (7.44) is contained in (1)

(1)

< N0 s k+1−k0 −k g (1 − P1 ) g s k0 −kh h ψ k+1 N0 > A (1)

(1)

+ < N0 s k−k0 −k g (1 − P1 ) g s k0 −kh h ψ k N0 > A . This subset of End A (M) is contained in E(C, M) when (1)

(1)

s k+1−k0 −k g (1 − P1 ) g s k0 −kh h ψ k+1 (C) (1)

(1)

and s k−k0 −k g (1 − P1 ) g s k0 −kh h ψ k (C) are contained in E(C, M) because N0 C = C and M is an A[N0 ]-module. By (7.38), this is true when P1 is contained in (1) (1) P1 (ψ k0 +kg (C), M, g s k0 −kh h ) and k ≥ kC,M,g,h,k0 where (2)

(1)

kC,M,g,h,k0 := max(k g(2) (P1 ) + k0 , kh (P1 ), k(ψ k0 +k g (C), (1)

M, g s k0 −kh h )).

(7.45)

The first assertion of i. is proved. We deduce the second assertion from the following claim and the last assumption of Proposition 7.84: Let ( An )n∈N and (Bn )n∈N be two convergent sequences in HomCont A (M(C), M) with limits A and B, respectively; assume that (Bn )n∈N and B are in End A (M(C)) and that, for any x ∈ C there exists an A-submodule C ∈ C such that Bn (x) ∈ C for any large n. Then, if the sequence (An ◦ Bn )n∈N is convergent, its limit is A ◦ B. Let D be the limit of the sequence (An ◦ Bn )n . It suffices to show that, for any open A-submodule M ⊆ M and any element x ∈ M(C) we have (D − A ◦ B)(x) ∈ M. We write D − A ◦ B = (D − An ◦ Bn ) − (A − An ) ◦ Bn − A ◦ (B − Bn ). Obviously (D − An ◦ Bn )(x) ∈ M for large n. Secondly, the elements Bn (x) for any large n are contained in some compact A-submodule C ∈ C, hence also (B − Bn )(x). Moreover A − An ∈ E(C, M) for large n. Hence (A − An ) ◦ Bn (x) ∈ M for large n. Finally, A being C-continuous there is an open A-submodule M ⊆ M such that A(M ∩ C) ⊆ M. Furthermore (B − Bn )(x) ∈ M ∩C for large n. Hence A ◦(B − Bn )(x) ∈ M for large n. ii. This follows from the second assertion in i. together with Remark 7.85. We have now proved that ak,u ∈ E(C, M) when k ≥ kC,M,g,h,k0 .

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Proposition 7.88. For u ∈ J (N0 /Nk ) ∩ Uh , we have Hg,Ju (N0 /Nk−k0 ) ◦Hh,J (N0 /Nk ) ◦Res(1u Nk ) = Hgh, J (N0 /Nk ) ◦Res(1u Nk ). (7.46) Proof. The left side of (7.46) is α(g, xv ) ◦ Res(1v Nk−k0 ) ◦ α(h, xu ) ◦ Res(1u Nk ). v∈Ug ∩Ju (N0 /Nk−k0 )

The right side of (7.46) is α(gh, x u ) ◦ Res(1u Nk ) if u ∈ J (N0 /Nk ) ∩ Uh ∩ Ugh and is 0 if u does not belong to Ugh . We recall that α(h, xu )u = n(h, u)t (h, u)

(1)

with n(h, u) ∈ N0 and t (h, u) ∈ L + s −kh .

It follows that α(h, xu )u Nk w0 P ⊆ n(h, u)Nk−k (1) w0 P ⊂ n(h, u)Nk−k0 w0 P. h

We obtain Res(1v Nk−k0 ) ◦ α(h, xu ) ◦ Res(1u Nk ) $ α(h, xu ) ◦ Res(1u Nk ) if v Nk−k0 = n(h, u)Nk−k0 , = 0 otherwise. We check now that u ∈ U gh ∩ Uh if and only if n(h, u) ∈ Ug . Indeed xu = uw0 P/P belongs to h −1 C0 ∩ C0 = Uh w0 P/P, xu ∈ (gh)−1 C0 ∩ h −1 C0 ∩ C0

if and only if

hx u ∈ g −1 C0 ∩ C0

and hxu = α(h, x u )xu = n(h, u)w0 P/P. It follows that u ∈ Ugh ∩ Uh if and only if n(h, u) ∈ Ug . As Ju (N0 /Nk−k0 ) contains n(h, u), we have v = n(h, u) when v Nk−k0 = n(h, u)Nk−k0 . We deduce that the left side of (7.46) is 0 when u does not belong to Ugh and otherwise is equal to α(g, hx u ) ◦ α(h, xu ) ◦ Res(1u Nk ) = α(gh, xu ) ◦ Res(1u Nk ), where the last equality follows from the product formula for α (Lemma 7.63). We have proved that bk,u = 0, therefore ending the proof of the product formula.

6.5. Reduction modulo pn We investigate now the situation that will appear for generalized (ϕ, )modules M, where the reduction modulo a power of p allows us to reduce to the simpler case where M is killed by a power of p. We will use later

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317

this section to get a special family Cs in M such that the (s, res, Cs )-integrals Hg exist for all g ∈ N0 P N0 and satisfy the relations H1, H2, H3 of Proposition 7.72. We assume now that (A, M) satisfies: a. A is a commutative ring with the p-adic topology (the ideals pn A for n ≥ 1 form a basis of neighbourhoods of 0) and is Hausdorff. b. M is a linearly topological A-module with a topology weaker than the p-adic topology (a neighbourhood of 0 contains some p n M) and M is a Hausdorff and topological A[P+ ]-module as in Section 6 (we do not suppose that M is complete). c. The submodules pn M, for n ≥ 1, are closed in M. d. M is p-adically complete: the linear map M → limn≥1 (M/ p n M) is ← − bijective. For all n ≥ 1, we equip M/ p n M with the quotient topology so that the quotient map pn : M → M/ p n M is continuous. The natural homomorphism ∼ =

M −→ lim(M/ p n M) ← − n≥1

is a homeomorphism, and the natural homomorphism ∼ =

n Endcont Endcont A (M) −→ lim A (M/ p M) ← − n≥1

is bijective. We have: – For a subset C of M, let C be the closure of C. Then C = lim p (C) ← −n≥1 n and if C is closed, C = limn≥1 pn (C). If C is p-compact (i.e. pn (C) are ← − compact for all n ≥ 1), then C is compact, and conversely ([2] I.29 Cor. and I.64 Prop.8). – An endomorphism f of M which is p-continuous (i.e. the endomorphism f n induced by f on M/ p n M is continuous for all n ≥ 1) is continuous, and conversely. – An action of a topological group H on M which is p-continuous (i.e. the induced action of H on M/ p n M is continuous for all n ≥ 1) is continuous, and conversely. – If the M/ p n M are complete for all n ≥ 1, then M is complete. – The image Cn in M/ p n M, for all n ≥ 1, of a special family C of compact subsets in M such that, for all positive integers n, p n M ∩ M(C) = p n M(C) is a special family. In this case, one has M(Cn ) = M(C)/ p n M(C).

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– M is a topologically étale A[P+ ]-module if and only if M/ p n M is a topologically étale A[P+ ]-module, for all n ≥ 1. If we replace “topologically” by “algebraically”, this is the same proof as for classical (ϕ, )-modules (see Subsection 7.3). The canonical inverse ψs of the action ϕs of s is continuous if and only if it is p-continuous. We introduce now our setting which will be discussed in this section. We suppose that: – M is a topologically étale A[P+ ]-module, and M/ p n M is complete for all n ≥ 1. – We are given, for n ≥ 1, a special family Cn of compact subsets in Mn = M/ p n M such that Cn contains the image of Cn+1 in Mn for all n ≥ 1. Let C be the set of compact subsets C ⊂ M such that pn (C) ∈ Cn for all n ≥ 1. Lemma 7.89. C is a special family in M and M(C) = limn≥1 M(Cn ). ← − Proof. C(1) It is obvious that a compact subset C of C ∈ C is in C because pn is continuous and pn (C ) is compact. C(2) pn commutes with finite union hence C is stable by finite union. C(3) pn commutes with the action of N0 hence C ∈ C implies N0 C ∈ C. C(4) By definition x ∈ M(C) if and only if pn (x) ∈ M(Cn ) for all n > 1. The compatibility of the Cn implies that the M(Cn ) form a projective system. We deduce M(C) = limn≥1 M(Cn ). As the latter ones are topolog← − ically étale, the topological A[P+ ]-module M(C) is topologically étale by Remark 7.24. We have the natural map lim Hom A (M(Cn ), M/ p n M) → Hom A (lim M(Cn ), lim M/ p n M) ← − ← − ← − n

n

n

= Hom A (M(C), M). Lemma 7.90. The above map induces a continuous map lim HomCAn cont (M(Cn ), M/ p n M) → HomCcont (M(C), M), A ← −

(7.47)

n

for the projective limit of the Cn -open topologies on the left-hand side. Proof. Let f = lim f n be a map in the image, and let C ∈ C. Then f |C is the ← − projective limit of the f n | pn (C) hence is continuous. This means that the map

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319

in the assertion is well defined. For the continuity, let C ∈ C and M ⊂ M be an open A-submodule. The preimage of E(C, M) is equal to lim HomCAn cont (M(Cn ), M/ p n M) ∩ E( pn (C), M + p n M/ p n M) . ← − n

n

Since M contains some

p n o M, this intersection is equal to the open submodule

{( f n ) ∈ lim HomCAn cont (M(Cn ), M/ pn M) : f n ∈ E( pn (C), M + p n M/ p n M) ← − n

for n ≤ n 0 }.

Proposition 7.91. In the above setting assume that all the assumptions of Proposition 7.84 are satisfied for (s, M/ p n M, Cn ) and for all n ≥ 1. Then, for all g ∈ N0 P N0 , the functions αg,0 : N0 → HomCont (M(C), M) A are (s, r es, C)-integrable, their (s, r es, C)-integrals Hg belong to End A (M(C)) and satisfy the relations H1, H2, H3 of Proposition 7.72. Proof. In the following we indicate with an extra index n that the corresponding notation is meant for the module M/ p n M with the special family Cn . Then αg,0 (u) is the image of (αg,0,n (u))n by the map (7.47), for u ∈ N0 . It follows that Hg,J (N0 /Nk ) is the image of (Hg,J (N0 /Nk ),n )n for g ∈ N0 P N0 . By assumption the integral Hg,n = limk→∞ Hg,J (N0 /Nk ),n exists, lies in HomCAn ont (M(Cn ), M/ p n M), and satisfies the relations H1, H2, H3 of Proposition 7.72. The continuity of the map (7.47) implies that the image of (Hg,n )n is equal to the limit limk→∞ Hg,J (N0 /Nk ) , therefore is the integral Hg of αg,0 . The additional properties for Hg are inherited from the corresponding properties of the Hg,n . Under the assumptions of Proposition 7.91, we associate to (s, M, C), an A-algebra homomorphism R es AC ⊂G/P → End A (M(C) P ) via Propositions 7.72, 7.70, which extends the A-algebra homomorphism Res Cc∞ (C, A)#P → End A (M(C) P ) constructed in Proposition 7.32. The homomorphism Res gives rise to a P-equivariant sheaf on C as described in detail in Theorem 7.38. The homo morphism R es defines on the global sections with compact support M(C)cP

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of the sheaf on C the structure of a nondegenerate AC ⊂G/P -module. The latter leads, by Proposition 7.68, to the unital Cc∞ (G/P, A)#G-module Z ⊗A M(C)cP which corresponds to a G-equivariant sheaf on G/P extending the earlier sheaf on C (Remark 7.69).

7. Classical (ϕ, )-modules on OE 7.1. The Fontaine ring OE Let K /Q p be a finite extension of ring of integers o, of uniformizer p K and residue field k. By definition the Fontaine ring OE over o is the p-adic completion of the localization of the Iwasawa o-algebra (Z p ) := o[[Z p ]] with respect to the multiplicative set of elements which are not divisible by p. We choose a generator γ of Z p of image [γ ] in OE and we denote X = [γ ] − 1 ∈ OE . The Iwasawa o-algebra (Z p ) is a local noetherian ring of maximal ideal M(Z p ) generated by p K , X . It is a compact ring for the M(Z p )-adic topology. The ring OE can be viewed as the ring of infinite Laurent series n∈Z an X n over o in the variable X with limn→−∞ an = 0, and (Z p ) as the subring o[[X ]] of Taylor series. The Fontaine ring OE is a local noetherian ring of maximal ideal p K OE and residue field isomorphic to k((X )); it is a pseudocompact ring for the p-adic (= strong) topology and a complete ring (with continuous multiplication) for the weak topology. A fundamental system of open neighbourhoods of 0 for the weak topology of OE is given by (On,k = p n OE + M(Z p )k )n,k∈N or by (Bn,k = p n OE + X k (Z p )n,k∈N . Other fundamental systems of neighbourhoods of 0 for the weak topology are (On := On,n )n≥1

or

(Bn := Bn,n )n≥1 .

7.2. The group G L(2, Q p ) We consider the group G = G L(2, Q p ) and , ∗ Z∗p Zp 0 1 Zp N0 := , := , L 0 := 0 1 0 0 1 Z p − {0} 0 L ∗ := , 0 1

0 , Z∗p

G-equivariant sheaves on G/P

Nk :=

1 0

pk Z p 1 + pk Z p , L k := 1 0

0 1 + pk Z p

321 for k ≥ 1,

Pk = L k Nk for k ∈ N, the upper triangular subgroup P, the diagonal subgroup L, the upper unipotent subgroup N , the center Z , the mirabolic monoid P∗ = N0 L ∗ , and the monoids L + = L ∗ Z , P+ = N0 L + . The subset of non invertible elements in the monoid L ∗ is a 0 N−{0} s p = {sa := for a ∈ pZ p − {0}}. 0 1 N−{0}

An element s ∈ s p Z is called strictly dominant. In the following we identify the group Z p with N0 . The action of P+ on N0 induces an étale ring action of P+ (trivial on Z ) on (N0 ) which respects the ideal generated by p. This action extends first to the localization and then to the completion to give an étale ring action of P+ on OE determined by its restriction to P∗ . For the weak topology (and not for the p-adic topology), the action P+ × OE → OE of the monoid P+ on OE is continuous (see Lemma 8.24.i in [12]). For t ∈ L + the canonical left inverse ψt of the action ϕt of t is continuous (this is proved in a more general setting later in Proposition 7.119).

7.3. Classical étale (ϕ, )-modules N−{0} s p Z.

Let s ∈ A finitely generated étale ϕs -module D over OE is a finitely generated OE -module with an étale semilinear endomorphism ϕs . These modules form an abelian category Met OE (ϕs ). We fix such a module D. In the following, the topology of D is its weak topology. For any surjective OE -linear map f : ⊕d OE → D, the image in D of a fundamental system of neighbourhoods of 0 in ⊕d OE for the weak topology is a fundamental system of neighbourhoods of 0 in D. Finitely generated (N0 )-submodules of D generating the OE -module D will be called lattices. The map f sends ⊕d (Z p ) onto a lattice D 0 of D. We note On,k := p n D + M(Z p )k D 0 and Bn,k := p n D + X k D 0 . Writing On := On,n and Bn := Bn,n , (On )n and (Bn )n are two fundamental systems of neighbourhoods of 0 in D. The topological OE -module D is Hausdorff and complete. A treillis D0 in D is a compact (N0 )-submodule D0 such that the image of D0 in the finite dimensional k((X ))-vector space D/ p K D is a k[[X ]]-lattice ([5] Déf. I.1.1). A lattice is a treillis and a treillis contains a lattice. For n ≥ 1, the reduction modulo p n of D is the finitely generated OE module D/ p n D with the induced action of ϕs . The action remains étale, because the multiplication by p n being a morphism in Met OE (ϕs ) its cokernel

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belongs to the category. The reduction modulo p n of ψs is the canonical left inverse of the reduction modulo p n of ϕs . The reduction modulo p n of a treillis of D is a treillis of D/ p n D. Conversely, if the reduction modulo p n of a finitely generated ϕs -module D over OE is étale for all n ≥ 1, then D is an étale ϕs -module over OE because D = lim D/ p n D. ← −n The weak topology of D is the projective limit of the weak topologies of D/ p n D. When D is killed by a power of p and D0 is a treillis of D, we have: 1. D0 is open and closed in D. 2. (M(Z p )n D0 )n∈N and (X n D0 )n∈N form two fundamental systems of open neighbourhoods of zero in D. 3. Any treillis of D is contained in X −n D0 for some n ∈ N. 0 4. D = k∈N X −k D0 . 5. D0 is a lattice. The first four properties are easy; a reference is [5] Prop. I.1.2. To show that D0 is a lattice, we pick some lattice D 0 then D0 is contained in the lattice X −n D 0 for some n ∈ N by property 3. Since the ring (N0 ) is noetherian the assertion follows. When D is killed by a power of p, the weak topology of D is locally compact (by properties 2 and 5). Proposition 7.92. Let D be a finitely generated étale ϕs -module over OE . Then ϕs and its canonical inverse ψs are continuous. Proof. (a) The above OE -linear surjective map f : ⊕d OE → D sends (ai )i to i ai di for some elements di ∈ D. As ϕs is étale, the map (ai )i → i ai ϕs (di ) also gives an OE -linear surjective map ⊕d OE → D. Both surjections are topological quotient maps by the definition of the topology on D, and the morphism ϕs of OE is continuous. We deduce that the morphism ϕs of D is continuous. (b) The image of ⊕d (N0 ) by f is a lattice D0 of D. For any k ∈ N the

(N0 )-submodule D0,k of D generated by (ϕs (X k ei ))1≤i≤d also is a treillis of D because ϕs is étale. Here {ei | 1 ≤ i ≤ d} is a generating family of D0 . We have ψs (D0,k ) = X k D0 (cf. Lemma 7.19). (c) When D is killed by a power of p, we deduce that ψs is continuous by the properties 1 and 2 of the treillis. When D is not killed by a power of p, we deduce that the reduction modulo p n of ψs is continuous for all n; this implies that ψs is continuous because (A = o, D) satisfy the properties a,

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b, c, d of Section 6.5, and D/ p n D is a (finitely generated) étale ϕs -module over OE . We put D + := {x ∈ D : the sequence (ϕsk (x))k∈N is bounded in D} and D ++ := {x ∈ D | lim ϕsk (x) = 0}. k→∞

(7.48)

Proposition 7.93. (i) If D is killed by a power of p, then D + and D ++ are lattices in D. (ii) There exists a unique maximal treillis D ' such that ψs (D ' ) = D ' . (iii) The set of ψs -stable treillis in D has a unique minimal element D & ; it satisfies ψs (D & ) = D & . (iv) X −k D ' is a treillis stable by ψs for all k ∈ N. Proof. The references given in the following are stated for étale (ϕs p , )modules but the proofs never use that there exists an action of and they are valid for étale ϕs p -modules. (i) For s = s p this is [5] Prop. II.2.2(iii) and Lemma II.2.3. The properties of s p which are needed for the argument are still satisfied for general s in the following form: – ϕs (X ) ∈ ϕsmp (X ) (Z p )× where s = s0 s m p z with s0 ∈ , m ≥ 1, and z ∈ Z. k k – (ϕs (X )X −1 ) p ∈ p k+1 (Z p ) + X ( p−1) p (Z p ) for any k ∈ N. (ii) and (iii) For any finitely generated OE -torsion module M we denote its Pontrjagin dual of continuous o-linear maps from M to K /o by M ∨ := ∨ Homcont o (M, K /o). Obviously, M again is an OE -module by (λ f )(x) := ∨ f (λx) for λ ∈ OE , f ∈ M , and x ∈ M. It is shown in [5] Lemma I.2.4 that: – M ∨ is a finitely generated OE -torsion module, – the topology of pointwise convergence on M ∨ coincides with its weak topology as an OE -module, and – M ∨∨ = M. Now let D be as in the assertion but killed by a power of p. One checks that D ∨ also belongs to Met OE (ϕs ) with respect to the semilinear map ϕs ( f ) := f ◦ ψs for f ∈ D ∨ ; moreover, the canonical left inverse is ψs ( f ) = f ◦ ϕs . Next, [5] Lemma I.2.8 shows that:

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– If D0 ⊂ D is a lattice then D0⊥ := {d ∈ D ∨ : f (D0 ) = 0} is a lattice in D ∨ , and D0∨∨ = D0 . We now define D & := (D ∨ )+ and D ' := (D ∨ )++ . The purely formal arguments in the proofs of [5] Prop. II.6.1, Lemma II.6.2, and Prop. II.6.3 show that D & and D ' have the asserted properties. For a general D in Met OE (ϕs ) the (formal) arguments in the proof of [5] Prop. II.6.5 show that ((D/ p n D)& )n∈N and ((D/ p n D)' )n∈N are well defined projective systems of compact (Z p )-modules (with surjective transition maps). Hence D & := lim(D/ p n D)& ← −

and

D ' := lim(D/ p n D)' ← −

have the asserted properties. (iv) X −k D ' is clearly a treillis. As X divides ϕs (X ) = (1 + X )a − 1 in

(Z p ) = o[[X ]] (when s ∈ sa Z for some a ∈ pZ p \ {0}), there exists f (X ) ∈ o[[X ]] such that ϕs (X k ) = X k f (X )k . So we have ψs (X −k D ' ) = ψs (ϕs (X −k ) f (X )k D ' ) = X −k ψs ( f (X )k D ' ) ⊂ X −k ψs (D ' ) ⊂ X −k D ' since D ' is ψs -stable by definition. Proposition 7.94. Let D be a finitely generated étale ϕs -module over OE killed by a power of p. For any compact subset C ⊆ D, there exists an r ∈ N such that

ψsk (N0 C) ⊆ X −r D ++ . k≥0

Proof. Since N0 C is compact and D ++ and D ' are treillis, there exists l ∈ N such that N0 C ⊂ X −l D ' ⊂ X −2l D ++ . By (iv) of Proposition 7.93 we obtain for all k ≥ 0 ψsk (N0 C) ⊂ ψsk (X −l D ' ) ⊂ X −l D ' ⊂ X −2l D ++ and we can take r = 2l. Corollary 7.95. Let D be a finitely generated étale ϕs -module over OE . For any compact subset C ⊆ D and any n ∈ N, there exists k0 ∈ N such that

ψsk (N0 C) ⊆ D ' + p n D. k≥k0

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Proof. We may assume that D is killed by a power of p. In view of (the proof of) Proposition 7.94 it suffices to show that for all l > 0 there exists a k0 such k that ψs 0 (X −l D ' ) = D ' . By Proposition 7.93(ii) and (iv) we have D ' = ψsk+1 (D ' ) ⊆ ψsk+1 (X −l D ' ) ⊆ ψsk (X −l D ' ) ⊆ X −l D ' 6 for any k ≥ 0. Hence k ψsk (X −l D ' ) is a treillis in D on which ψs is surjective. Therefore it coincides with D ' by the maximality of D ' (Proposition 7.93(ii)). On the other hand, the Z p [[X ]]-module (X −l D ' )/D ' is killed by both X l and p n and hence is finite. So there exists a k0 such that ψsk (X −l D ' ) = D ' for all k ≥ k0 . For any submonoid L ⊂ L + containing a strictly dominant element, an étale L -module over OE is a finitely generated OE -module with an étale semilinear action of L . A topologically étale L -module over OE will be an étale L -module D over OE such that the action L × D → D of L on D is continuous. This terminology is provisional since we will show later on (Corollary 7.125) in a more general context that any étale L -module over OE in fact is topologically étale and, in particular, is a complete topologically étale o[N0 L ]-module in our previous sense. a b Let D be a topologically étale L + -module over OE and let g = ∈ c d GL2 (Q p ). Denote 0 1 1 b w0 = , u(b) = . 1 0 0 1 Using the formula

ar + b gu(r )w0 = cr + d

a , c

one checks that the set U g defined by g −1 C0 ∩ C0 = Ug w0 P/P is Ug = u(X g ) where

ar + b ∈ Z p }. cr + d

X g = {r ∈ Z p |cr + d = 0,

For each r ∈ X g we can write

gu(r )w0 = u(g[r ])t (g, r )w0 u where ar + b g[r ] = ∈ Zp, cr + d

, t (g, r ) =

c cr + d

det g cr +d

0

,

0 . cr + d

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We deduce that for u(r ) ∈ Ug we have α(g, xu(r) ) = u(g[r ])t (g, r). Let now s = sa z ∈ L + be strictly dominant, with z ∈ Z and a ∈ pZ p − {0}. There exists a positive integer k g,s such that for all k ≥ k g,s we have t (g, r )s k ∈ L + . Note that Nk = s k N0 s −k = u(a k Z p ). We deduce that for k ≥ k g,s the operators Hg, J (N0 /Nk ) introduced in (7.28) are equal to the operators Hg,s,J (Z p /a k Z p ) = (1 + X )g[r] ϕt (g,r)s k ψsk ((1 + X )−r ). r∈X g ∩J (Z p /a k Z p )

Proposition 7.96. Let D be a topologically étale L + -module over OE . For the cont compact open topology in Endcont o (D), the maps αg,0 : N0 −→ Endo (D), for g ∈ N0 P N0 , are integrable with respect to s and res, for all s ∈ L + strictly dominant, i.e. s = sa z with a ∈ pZ p − {0} and z ∈ Z , their integrals / Hg = αg,0 d res = lim Hg,s,J (Z p /a k Z p ) N0

k→∞

for any choices of J (Z p /a k Z p ) ⊂ Z p , do not depend on the choice of s and satisfy the relations H1, H2, H3 of Proposition 7.72. Proof. By Proposition 7.91, we reduce to the case that D is killed by a power of p and to showing the assumptions of Proposition 7.84 for the family of all compact subsets of D. The axioms Ci , for 1 ≤ i ≤ 6, are obviously satisfied by continuity of ϕs , ψs , and of the action of n ∈ N0 on D. i. We check first the convergence criterion of Proposition 7.83, using the theory of treillis, i.e. of lattices, in D. Given a lattice M ⊆ D, a compact subset C ⊆ D such that N0 C ⊆ C, and a compact subset C+ ⊆ L + , we want to find a compact open subgroup P1 ⊂ P0 and an integer k0 ∈ N such that s k (1 − P1 )C+ ψsk (C) ⊆ M

(7.49)

for all k ≥ k0 . We choose r0 ∈ N with ϕsk (D ++ ) ⊂ M for all k ≥ r0 , as we may by the definition of D ++ . We choose r, k0 ∈ N such that k0 ≥ r0 and

ψsk (C) ⊆ X −r D ++ , k≥k0

as we may by Proposition 7.94. Applying C+ we obtain

C+ ψsk (C) ⊆ C+ (X −r D ++ ). k≥k0

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The continuity of the action of P+ on D implies that C+ (X −r D ++ ) is com pact. Hence we can choose r ∈ N such that C+ (X −r D ++ ) ⊆ X −r D ++ and we obtain

C+ ψsk (C) ⊆ X −r D ++ . k≥k0

As X −r D ++ is compact and D ++ an open neighbourhood of 0, the continuity of the action of P+ on D there exists a compact open subgroup P1 ⊆ P+ such that

(1 − P1 )X −r D ++ ⊆ D ++ . Hence we have s k (1 − P1 )C+ ψsk (C) ⊂ ϕsk (D ++ ) ⊂ M for all k ≥ k0 . ii. To obtain all the assumptions of Proposition 7.84 for the family of all compact subsets of D, it remains to prove that, given x ∈ D and g ∈ N0 P N0 , s = sa z with a ∈ pZ p − {0} and z ∈ Z, and (J (Z p /a k Z p ))k , there exists a compact C x,g,s ⊂ D and a positive integer k x,g,s such that Hg,s,J (Z p /a k Z p ) (x) ∈ C x,g,s for any k ≥ k x,g,s . This is clear because D is locally compact (by hypothesis D is killed by a power of p) and the sequence (Hg,s,J (Z p /a k Z p ) (x))k converges. iii. The independence of the choice of s ∈ L + strictly dominant results from the fact that, for z ∈ Z , e ∈ Z∗p , and a positive integer r , we have kr k k r k kr (zs pr e )k N0 (zs pr e )−k = s kr p N0 s p and ϕzs pr e ψzs pr e = ϕs p ψs p as ψzse is the right and left inverse of ϕzse . Remark 7.97. Let D be a topologically étale L + -module over OE , on which .Z acts through a character ω. The pointwise convergence of the integrals N0 αg,0 dres is a basic theorem of Colmez, allowing him the construction of the representation of GL2 (Q p ) that he denotes D ω P1 . Our construction coincides with Colmez’s construction because our Hg ∈ Endcont o (D) are the same as the Hg of Colmez given in [6] lemma II.1.2 (ii). Indeed, α(g, xu(r) ) = u(g[r])t (g, r ) = , ω(cr + d)u(g[r ]) where g [r] =

det g . (cr+d)2

det g (cr +d)2

0

0 g [r ] = ω(cr + d) 0 1

g[r ] , 1

This coincides with Colmez’s formula.

The major goal of the paper is to generalize Proposition 7.96. See Proposition 7.141.

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8. A generalization of (ϕ, )-modules We return to a general group G. We denote G (2) := G L(2, Q p ) and the objects relative to G (2) will be affected with an upper index (2) . (a) We suppose that N0 has the structure of a p-adic Lie group and that we have a continuous surjective homomorphism : N0 → N0(2) . (2)

We choose a continuous homomorphism ι : N0 → N0 which is a section of (2) (which is possible because N0 Z p ). (2) We have N0 = N ι(N0 ) where N is the kernel of . We denote by L ,+ := {t ∈ L | t N t −1 ⊂ N , t N0 t −1 ⊂ N0 } the stabilizer of N in the L-stabilizer of N0 , and by L ,ι := {t ∈ L | t N t −1 ⊂ (2) (2) (2) N , tι(N0 )t −1 ⊂ ι(N0 )} the stabilizer of N in the L-stabilizer of ι(N0 ). We have L ,ι ⊂ L ,+ . (b) We suppose given a submonoid L ∗ of L ,ι containing s and a continuous (2) homomorphism : L ∗ → L + such that (, ι) satisfies (tut −1 ) = (t)(u)(t)−1 , tι(y)t −1 = ι((t)y(t)−1 ), (2)

for u ∈ N0 , y ∈ N0 , t ∈ L ∗ . (2)

The sequence (s n N0 s −n ) = (s)n N0 (s)−n in N (2) is decreasing with trivial intersection. The maps in (a) and (b) combine to a unique continuous homomorphism (2)

P∗ := N0 L ∗ → P+ .

8.1. The microlocalized ring (N0 ) The ring (N0 ), denoted by N (N0 ) in [12], is a generalization of the ring (2) OE , which corresponds to id (N0 ). We refer the reader to [12] for the proofs of some claims in this section. The maximal ideal M(N ) of the completed group o-algebra (N ) = o[[N ]] is generated by p K and by the kernel of the augmentation map o[N ] → o. The ring (N0 ) is the M(N )-adic completion of the localization of

(N0 ) with respect to the Ore subset S (N0 ) of elements which are not in M(N ) (N0 ). The ring (N0 ) can be viewed as the ring (N )[[X ]] of skew Taylor series over (N ) in the variable X = [γ ] − 1 where γ ∈ N0 and

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(γ ) is a topological generator of (N0 ). Then (N0 ) is viewed as the ring of infinite skew Laurent series n∈Z an X n over (N ) in the variable X with limn→−∞ an = 0 for the pseudocompact topology of (N ). The ring (N0 ) is strict-local noetherian of maximal ideal M (N0 ) generated by M(N ). It is a pseudocompact ring for the M(N )-adic topology (called the strong topology). It is a complete Hausdorff ring for the weak topology ([12] Lemma 8.2) with fundamental system of open neighbourhoods of 0 given by On,k := M (N0 )n + M(N0 )k for n ∈ N, k ∈ N. In the computations it is sometimes better to use the fundamental systems of open neighbourhoods of 0 defined by Bn,k := M (N0 )n + X k (N0 ) for n ∈ N, k ∈ N, and Cn,k := M (N0 )n + (N0 )X k for n ∈ N, k ∈ N, which are equivalent due to the two formulas X k (N0 ) ⊆ (N0 )X k +M(N0 )k

and (N0 )X k ⊆ X k (N0 )+M(N0 )k ,

We write On := On,n , Bn := Bn,n , and Cn = Cn,n . Then (On )n , (Bn )n , and (Cn )n are also fundamental system of open neighbourhoods of 0 in (N0 ). The action (b = ut, n 0 ) → b.n 0 = utn 0 t −1 of the monoid P,+ = N0 L ,+ on N0 induces a ring action (t, x) → ϕt (x) of L ,+ on the oalgebra (N0 ) respecting the ideal (N0 )M(N ), and the Ore set S (N0 ) hence defines a ring action of L ,+ on the o-algebra (N0 ). This action respects the maximal ideals M(N0 ) and M (N0 ) of the rings (N0 ) and

(N0 ) and hence the open neighbourhoods of zero On,k . Lemma 7.98. For t ∈ L ,+ , a fundamental system of open neighbourhoods of 0 in (N0 ) is given by (ϕt (On,k ) (N0 ))n,k∈N . Proof. We have just seen ϕt (On,k ) (N0 ) ⊂ On,k . Conversely, given n, k ∈ N, we have to find n , k ∈ N such that On ,k ⊂ ϕt (On,k ) (N0 ). This can be deduced from the following fact. Let H ⊂ H be an open subgroup. Then given k ∈ N, there is k ∈ N such that

M(H )k (H ) ⊃ M(H )k .

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Indeed by taking a smaller H we can suppose that H ⊂ H is open normal. Then M(H )k (H ) is a two-sided ideal in (H ) and the factor ring (H )/M(H ) (H ) is an artinian local ring with maximal ideal M(H )/M(H ) (H ). It remains to observe that in any artinian local ring the maximal ideal is nilpotent. Proposition 7.99. The action of L ,+ on (N0 ) is étale: for any t ∈ L ,+ , the map (λ, x) → λϕt (x) : (N0 ) ⊗ (N0 ),ϕt (N0 ) → (N0 ) is bijective. Proof. We follow ([12] Prop. 9.6, Proof, Step 1). (1) The conjugation by t gives a natural isomorphism

(N0 ) → t N t −1 (t N0 t −1 ). (2) Obviously t N t −1 (t N0 t −1 ) = (t N0 t −1 ) ⊗ (t N0 t −1 ) t N t −1 (t N0 t −1 ), and the map

(t N0 t −1 ) ⊗ (t N0 t −1 ) t N t −1 (t N0 t −1 ) → (N0 )⊗ (t N0 t −1 )

t N t −1 (t N0 t −1 )

is injective because t N t −1 (t N0 t −1 ) is flat on (t N0 t −1 ). (3) The natural map

(N0 ) ⊗ (t N0 t −1 ) t N t −1 (t N0 t −1 ) → (N0 ) is bijective. (4) The ring action ϕt : (N0 ) → (N0 ) of t on (N0 ) is the composite of the maps of (1), (2), (3), hence is injective. (5) The proposition is equivalent to (3) and ϕt injective.

Remark 7.100. The proposition is equivalent to: for any t ∈ L ,+ , the map (u, x) → uϕt (x) : o[N0 ] ⊗o[N0 ],ϕt (N0 ) → (N0 ) is bijective. et et 8.2. The categories M (L ∗ ) and MO (L ∗ ) (N0 ) E ,

By the universal properties of localization and adic completion the continuous homomorphisms and ι between N0 and N0(2) extend to continuous o-linear homomorphisms of pseudocompact rings,

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: (N0 ) → OE , ι : OE → (N0 ), ◦ ι = id .

(7.50)

If we view the rings as rings of Laurent series, (X ) = X (2) , ι(X (2) ) = X , and is the augmentation map (N ) → o and ι is the natural injection o →

(N ), on the coefficients. We have for n, k ∈ N, (2) (M (N0 )) = p K OE , (Bn,k ) = Bn,k ,

(7.51)

(2)

ι( p K OE ) = M (N0 ) ∩ ι(OE ), ι(Bn,k ) = Bn,k ∩ ι(OE ). (2)

We denote by J (N0 ) the kernel of : (N0 ) → (N0 ) and by J (N0 ) the kernel of : (N0 ) → OE . They are the closed two-sided ideals generated (as left or right ideals) by the kernel of the augmentation map o[N ] → o. We have

(N0 ) = ι(OE ) ⊕ J (N0 ), M (N0 ) = p K ι(OE ) ⊕ J (N0 ), (2)

X k (N0 ) = ι((X (2) )k (N0 ) ⊕ X k J (N0 ),

(7.52)

(2) Bn,k = ι(Bn,k ) ⊕ (J (N0 ) ∩ Bn,k ).

The maps and ι are L ∗ -equivariant: for t ∈ L ∗ , ◦ ϕt = ϕ(t) ◦ ,

ι ◦ ϕ(t) = ϕt ◦ ι,

(7.53)

thanks to the hypothesis (b) made at the beginning of this chapter. The map ι is equivariant for the canonical action of the inverse monoid L −1 ∗ , but not the map as the following lemma shows. Lemma 7.101. For t ∈ L ∗ , we have ι◦ψ(t) = ψt ◦ι. We have ◦ψt = ψ(t) ◦ if and only if N = t N t −1 . (2)

(2)

Proof. Clearly N0 = N ι(N0 ) and t N0 t −1 = t N t −1 tι(N0 )t −1 for t ∈ L. We choose, as we may, for t ∈ L ,ι , a system J (N0 /t N0 t −1 ) of representatives of N0 /t N0 t −1 containing 1 such that (2)

(2)

J (N0 /t N0 t −1 ) = {uι(v) |u ∈ J (N /t N t −1 ), v ∈ J (N0 /(t)N0 (t −1 ))}. (7.54) We have ι ◦ ψ(t) = ψt ◦ ι because, for λ ∈ OE , we have on one hand (7.12) λ= vϕ(t) (λv,(t) ), λv,(t) = ψ(t) (v −1 λ), (2)

(2)

v∈J (N0 /(t)N0 (t)−1 )

ι(λ) = (2)

ι(v)ϕt (ι(λv,(t) )), (2)

v∈J (N0 /(t)N0 (t)−1 )

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and on the other hand (7.12) ι(λ) =

uι(v)ϕt (ι(λ)uι(v),t ),

(2) (2) u∈J (N /t N t −1 ),v∈J (N0 /(t)N0 )(t)−1 )

where ι(λ)uι(v),t = ψt (ι(v)−1 u −1 ι(λ)). By the uniqueness of the decomposition, ι(λ)ι(v),t = ι(λv,(t) ), ι(λ)uι(v),t = 0 if u = 1. Taking u = 1, v = 1, we get ψt (ι(λ)) = ι(ψ(t) (λ)). A similar argument shows that ◦ ψt = ψ(t) ◦ if and only if N = t N t −1 . For λ ∈ (N0 ), λ= uϕt (λu,t ), λu,t = ψt (u −1 λ), u∈J (N0 /t N0 t −1 )

(λ) =

u∈ J (N0 /t N0

(u)ϕ(t) ((λu,t )) = t −1 )

vϕ(t) ((λ)v,(t) ).

(2) (2) v∈J (N0 /(t)N0 )

By the uniqueness of the decomposition, (λ)v,(t) =

(λuι(v),t ).

u∈J (N /t N t −1 )

We deduce that ◦ ψt = ψ(t) ◦ if and only if N = t N t −1 . Remark 7.102. ◦ ψs = ψ(s) ◦ , except in the trivial case where : N0 → (2) N0 is an isomorphism, because s N s −1 = N as the intersection of the decreasing sequence s k N s −k for k ∈ N is trivial. For future use, we note: Lemma 7.103. The left or right o[N0 ]-submodule generated by ι(OE ) in

(N0 ) is dense. Proof. As o[N0 ] is dense in (N0 ) it suffices to show that the left or right

(N0 )-submodule generated by ι(OE ) in (N0 ) is dense. This will be shown even with respect to the M (N0 )-adic topology. View λ ∈ (N0 ) as an infinite Laurent series λ = n∈Z λn X n with λn ∈

(N ) and limn→−∞ λn = 0 in the M(N )-adic topology of (N ). Further, note that the left, resp. right, (N0 )-submodule of (N0 ) generated by ι(OE ) contains (N0 )X −m , resp. X −m (N0 ), for any positive integer m. Finally, for each n ∈ N there exists μn in (N0 )X −m , resp. X −m (N0 ), for some large m, such that λ − μn ∈ M (N0 )n .

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n Let M be a finitely generated (N0 )-module and let f : ⊕i=1

(N0 ) → M be a (N0 )-linear surjective map. We put on M the quotient topology of the weak topology on ⊕in=1 (N0 ); this is independent of the choice of f . Then M is a Hausdorff and complete topological (N0 )-module and every submodule is closed ([12] Lemma 8.22). In the same way we can equip M with the pseudocompact topology. Again M is Hausdorff and complete and every submodule is closed in the pseudocompact topology, because (N0 ) is noetherian. The weak topology on M is weaker than the pseudocompact topology which is weaker than the p-adic topology. In particular the intersection of the submodules p n M for n ∈ N is 0. By [9] IV.3.Prop. 10, M is p-adically complete, i.e., the natural map M → limn M/ p n M is bijective. ← − Unless otherwise indicated, M is always understood to carry the weak topology.

Lemma 7.104. The properties a,b,c,d of Section 6.5 are satisfied by (o, M) and M is complete. Definition 7.105. A finitely generated module M over (N0 ) with an étale semilinear action of a submonoid L of L ,+ is called an étale L -module over

(N0 ). We denote by Met

(N0 ) (L ) the category of étale L -modules on (N0 ). Lemma 7.106. The category Met

(N0 ) (L ) is abelian.

Proof. As in the proof of Proposition 7.23 and using that the ring (N0 ) is noetherian. (2)

The continuous homomorphism : L ∗ → L + defines an étale semilinear (2) action of L ∗ on the ring id (N0 ) isomorphic to OE . Definition 7.107. A finitely generated module D over OE with an étale semilinear action of L ∗ is called an étale L ∗ -module over OE . An element t ∈ L ∗ in the kernel L =1 of acts trivially on OE hence ∗ bijectively on an étale L ∗ -module over OE . Remark 7.108. The action of L =1 on D extends to an action of the subgroup ∗ =1 is commutative or if we assume that for each of L generated by L =1 if L ∗ ∗ t ∈ L =1 there exists an integer k > 0 such that s k t −1 ∈ L ∗ . The assumption ∗ is trivially satisfied whenever L ∗ = H ∩ L + for some subgroup H ⊂ L. Indeed, the subgroup generated by L =1 is the set of words of the form ∗ ±1 x1 . . . xn±1 with xi ∈ L =1 for i = 1, . . . , n. So if we have an action of all ∗ the elements and all the inverses, then we can take the products of these, as

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well. We need to show that this action is well defined, i.e., whenever we have a relation x1±1 . . . xn±1 = y1±1 . . . yr±1

(7.55)

in the group then the action we just defined is the same using the x’s or the y’s. If L =1 is commutative, this is easily checked. In the second case, we can ∗ n r ki −1 ∈ L and choose a big enough k = ∗ i=1 ki + j =1 k j such that s x i −1 k k j s y j ∈ L ∗ . Then multiplying the relation (7.55) by s we obtain a relation in L ∗ so the two sides will define the same action on D. This shows that the actions defined using the two sides of (7.55) are equal on ϕsk (D) ⊂ D. (2) However, they are also equal on group elements u ∈ N0 hence on the whole ) k D = u∈J (N (2) /ϕ k (N (2) )) uϕs (D). 0

s

0

We denote by MOE, (L ∗ ) the category of étale L ∗ -modules on OE . et

Lemma 7.109. The category Met O

E,

(L ∗ ) is abelian.

Proof. As in the proof of Proposition 7.23 and using that the ring OE is noetherian. et We will prove later that the categories Met OE, (L ∗ ) and M (N0 ) (L ∗ ) are equivalent.

8.3. Base change functors We recall a general argument of semilinear algebra (see [12]). Let A be a ring with a ring endomorphism ϕ A , let B be another ring with a ring endomorphism ϕ B , and let f : A → B be a ring homomorphism such that f ◦ ϕ A = ϕ B ◦ f . When M is an A-module with a semilinear endomorphism ϕ M , its image by base change is the B-module B ⊗ A, f M with the semilinear endomorphism ϕ B ⊗ ϕ M . The endomorphism ϕ M of M is called étale if the natural map a ⊗ m → aϕ M (m) : A ⊗ A,ϕ A M → M is bijective. Lemma 7.110. When ϕ M is étale, then ϕ B ⊗ ϕ M is étale. Proof. We have B ⊗ B,ϕ B (B ⊗ A, f M) = B ⊗ A,ϕ B ◦ f M = B ⊗ f ◦ϕ A M ∼ B ⊗ A, f M. = B ⊗ A, f (A ⊗ A,ϕ A M) =

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Applying these general considerations to the L ∗ -equivariant maps :

(N0 ) → OE and ι : OE → (N0 ) satisfying ◦ ι = id (see (7.50), (7.53)), we have the base change functors M → D(M) := OE ⊗ (N0 ), M from the category of (N0 )-modules to the category of OE -modules, and D → M(D) := (N0 ) ⊗OE ,ι D in the opposite direction. Obviously these base change functors respect the property of being finitely generated. By the general lemma we obtain: Proposition 7.111. The above functors restrict to functors et D : Met

(N0 ) (L ∗ ) → MOE, (L ∗ ) and

et M : Met OE, (L ∗ ) → M (N0 ) (L ∗ ).

When M ∈ Met

(N0 ) (L ∗ ), the diagonal action of L ∗ on D(M) is: ϕt (μ ⊗ m) = ϕ(t) (μ) ⊗ ϕt (m) for t ∈ L ∗ , μ ∈ OE , m ∈ M.

(7.56)

When D ∈ Met OE, (L ∗ ), the diagonal action of L ∗ on M(D) is: ϕt (λ ⊗ d) = ϕt (λ) ⊗ ϕt (d) for t ∈ L ∗ , λ ∈ (N0 ), d ∈ D.

(7.57)

The natural map M : M → D(M), M (m) = 1 ⊗ m is surjective, L ∗ -equivariant, with a P∗ -stable kernel M := J (N0 )M. The injective L ∗ -equivariant map ι D : D → M(D), ι D (d) = 1 ⊗ d is ψt -equivariant for t ∈ L ∗ (same proof as Lemma 7.101). For future use we note the following property. Lemma 7.112. Let d ∈ D and t ∈ L ∗ . We have $ (2) ι D (ψt (v −1 d)) if u = ι(v) with v ∈ N0 , ψt (u −1 ι D (d)) = 0 if u ∈ N0 \ ι(N0(2) )t N0 t −1 . Proof. We choose a set J ⊂ N0(2) of representatives for the cosets in (2) (2) N0 /(t)N0 (t)−1 . The semilinear endomorphism ϕt of D is étale hence d= vϕt (dv,t ) where dv,t = ψt (v −1 d). v∈J

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Applying ι D we obtain ι D (d) = ι(v)ι D (ϕt (dv,t )) = ι(v)ϕt (ι D (dv,t )) v

=

v

ι(v)ϕt (ψt (ι D (v

−1

d))).

v

The map ι induces an injective map from J into N0 /t N0 t −1 with image included in a set J (N0 /t N0 t −1 ) ⊂ N0 of representatives for the cosets in N0 /t N0 t −1 . As the action ϕt of t in M(D) is étale, we have (7.12) m= uϕt (m u,t )) where m u,t = ψt (u −1 m) u∈J (N0 /t N0 t −1 )

for any m ∈ M(D). We deduce that ψt (ι(v −1 )ι D (d)) = ι D (dv,t ) when v ∈ J and ψt (u −1 ι D (d)) = 0 when u ∈ J (N0 /t N0 t −1 ) \ ι( J ). As any element of N0(2) can belong to a set of representatives of N0(2) /(t)N0(2) (t)−1 , we (2) deduce that ψt (ι(v −1 )ι D (d)) = ι D (dv,t ) for any v ∈ N0 . For the same −1 reason ψt (ι(u )ι D (d)) = 0 for any u ∈ N0 which does not belong to (2) ι(N0 )t N0 t −1 .

8.4. Equivalence of categories Let D ∈ Met OE , (L ∗ ). By definition D(M(D)) = OE ⊗ (N0 ), ( (N0 ) ⊗OE ,ι D), and we have a natural map μ ⊗ (λ ⊗ d) → μ(λ)d : OE ⊗ (N0 ), ( (N0 ) ⊗OE ,ι D) → D. Proposition 7.113. The natural map D(M(D)) → D is an isomorphism in Met OE , (L ∗ ). Proof. The natural map is bijective because ◦ι = id : OE → (N0 ) → OE , and L ∗ -equivariant because the action of t ∈ L ∗ satisfies ϕt (μ ⊗ (λ ⊗ d)) = ϕ(t) (μ) ⊗ ϕt (λ ⊗ d) = ϕ(t) (μ) ⊗ (ϕt (λ) ⊗ ϕt (d)), ϕt (μ(λ)d) = ϕ(t) (μ((λ))ϕt (d) = ϕ(t) (μ)(ϕt (λ))ϕt (d), by (7.56), (7.57). The kernel N of : N0 → Z p being a closed subgroup of N0 is also a padic Lie group, hence contains an open pro- p-subgroup H with the following property ([11] Remark 26.9 and Thm. 27.1): n For any integer n ≥ 1, the map h → h p is an homeomorphism of H onto an open subgroup Hn ⊆ H , and (Hn )n≥1 is a fundamental system of open neighbourhoods of 1 in H .

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The groups s k N s −k for k ≥ 1 are open and form a fundamental system of neighbourhoods of 1 in N . For any integer n ≥ 1 there exists a positive integer k such that any element in s k N s −k is contained in Hn , hence is a p n -th power of some element in N . We denote by kn the smallest positive integer such that any element in s kn N s −kn is a p n -th power of some element in N . Lemma 7.114. For any positive integers n and k ≥ kn , we have ϕ k (J (N0 )) ⊂ M (N0 )n+1 . Proof. For u ∈ N , and j ∈ N, the value at u of the p j -th cyclotomic polynomial p j (u) lies in M (N0 ) and n

up − 1 =

n

p j (u)

j=0

lies in M (N0 )n+1 . An element v ∈ s k N s −k is a p n -th power of some element in N hence v −1 lies in M (N0 )n+1 . The ideal J (N0 ) of (N0 ) is generated by u − 1 for u ∈ N and ϕ k (J (N0 )) is contained in the ideal generated by v − 1 for v ∈ s k N s −k . As M (N0 ) is an ideal of (N0 ) we deduce that ϕ k (J (N0 )) ⊂ M (N0 )n+1 . Lemma 7.115. i. The functor D is faithful. ii. The functor M is fully faithful. Proof. Obviously ii. follows from i. by Proposition 7.113. To prove i. let f : M1 → M2 be a morphism in Met

(N0 ) (L ∗ ) such that D( f ) = 0, i.e., such that f (M1 ) ⊆ J (N0 )M2 . Since M1 is étale we deduce that f (M1 ) ⊆ 6 k 6 (N )n M2 . Since the k ϕ (J (N0 ))M2 and hence, by Lemma 7.114, in n M 6 0 pseudocompact topology on M2 is Hausdorff we have n M (N0 )n M2 = 0. It follows that f = 0. Let M ∈ Met

(N0 ) (L ∗ ). By definition, MD(M) = (N0 ) ⊗OE ,ι (OE ⊗ (N0 ), M) = (N0 ) ⊗ (N0 ),ι◦ M. In the particular case where L ∗ = s N is the monoid generated by s, we et et denote the category Met

(N0 ) (L ∗ ) (resp. MOE, (L ∗ )), by M (N0 ) (ϕ) (resp. et et Met OE, (ϕ)). The category M (N0 ) (L ∗ ) (resp. MOE, (L ∗ )) is a subcategory of et Met

(N0 ) (ϕ) (resp. MOE, (ϕ)). Proposition 7.116. For any M ∈ Met

(N0 ) (ϕ) there is a unique morphism % M : M → MD(M)

in Met

(N0 ) (ϕ)

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D(% M ) such that the composed map D (% M ) : D(M) −−−−→ DMD(M) ∼ = D(M) is the identity. The morphism % M , in fact, is an isomorphism.

Proof. The uniqueness follows immediately from Lemma 7.115.i. The construction of such an isomorphism % M will be done in three steps. Step 1: We assume that M is free over (N0 ), and we start with an arbitrary finite (N0 )-basis (i )i∈I of M. By (8.2), we have M = (⊕i ∈I ι(OE )i ) ⊕ (⊕i ∈I J (N0 )i ). The (N0 )-linear map from M to MD(M) sending i to 1 ⊗ (1 ⊗ i ) is bijective. If ⊕i ∈I ι(OE )i is ϕ-stable, the map is also ϕ-equivariant and is an isomorphism in the category Met

(N0 ) (ϕ). We will construct a (N0 )-basis (ηi )i∈I of M such that ⊕i ∈I ι(OE )ηi is ϕ-stable. We have ϕ(i ) = (ai, j + bi, j ) j where ai, j ∈ ι(OE ), bi, j ∈ J (N0 ). j∈I

If the bi, j are not all 0, we will show that there exist elements xi, j ∈ J (N0 ) such that (ηi )i∈I defined by ηi := i + xi, j j ,

j∈I

satisfies ϕ(ηi ) = j∈I ai, j η j for i ∈ I . By the Nakayama lemma ([1] II §3.2 Prop. 5), the set (ηi )i ∈I is a (N0 )-basis of M, and we obtain an isomorphism in Met

(N0 ) (ϕ), % M M → MD(M), %(ηi ) = 1 ⊗ (1 ⊗ ηi ) for i ∈ I, such that D (% M ) is the identity morphism of D(M). The conditions on the matrix X := (xi, j )i, j ∈I are: ϕ(Id + X )(A + B) = A(Id + X ) for the matrices A := (ai, j )i, j∈I , B := (bi, j )i, j ∈I . The coefficients of A belong to the commutative ring ι(OE ). The matrix A is invertible because the

(N0 )-endomorphism f of M defined by f (i ) = ϕ(i ) for i ∈ I is an automorphism of M as ϕ is étale. We have to solve the equation A−1 B + A−1 ϕ(X )(A + B) = X. For any k ≥ 0 define Uk = A−1 ϕ(A−1 ) . . . ϕ k−1 (A−1 ) ϕ k (A−1 B) ϕ k−1 (A+B) . . . ϕ( A+B)(A+B).

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We have A−1 ϕ(Uk )(A + B) = Uk+1 . Hence X := k≥0 Uk is a solution of our equation provided this series converges with respect to the pseudocompact topology of (N0 ). The coefficients of A−1 B belong to the two-sided ideal J (N0 ) of (N0 ). Therefore the coefficients of Uk belong to the two-sided ideal generated by ϕ k (J (N0 )). Hence the series converges (Lemma 7.114). The coefficients of every term in the series belong to J (N0 ) and J (N0 ) is closed in (N0 ), hence xi, j ∈ J (N0 ) for i, j ∈ I . Step 2: We show that any module M in Met

(N0 ) (ϕ) is the quotient of another module M1 in Met (ϕ) which is free over

(N0 ).

(N0 ) Let (m i )i ∈I be a minimal finite system of generators of the (N0 )-module M. As ϕ is étale, (ϕ(m i ))i∈I is also a minimal system of generators. We denote by (ei )i∈I the canonical (N0 )-basis of ⊕i∈I (N0 ), and we consider the two surjective (N0 )-linear maps f, g : ⊕i ∈I (N0 ) → M, f (ei ) = m i , g(ei ) = ϕ(m i ). In particular, we find elements m i ∈ M, for i ∈ I , such that g(m i ) = ϕ(m i ). By the Nakayama lemma ([1] II §3.2 Prop. 5) the (m i )i∈I form another

(N0 )-basis of ⊕i∈I (N0 ). The ϕ-linear map ⊕i ∈I (N0 ) → ⊕i∈I (N0 ), ϕ( λi ei ) := ϕ(λi )m i i∈I

i∈I

therefore is étale. With this map, M1 := ⊕i∈I (N0 ) is a module in Met

(N0 ) (ϕ) which is free over (N0 ), and the surjective map f is a morphism in Met

(N0 ) (ϕ). Step 3: As (N0 ) is noetherian, we deduce from Step 2 that for any module M in Met

(N0 ) (ϕ) we have an exact sequence f

f

M2 − → M1 − →M →0 in Met

(N0 ) (ϕ) such that M1 and M2 are free over (N0 ). We now consider the diagram MD(M O 2)

MD( f )

% M2 ∼ =

M2

f

/ MD(M1 ) MD( f )/ MD(M) O O % M1 ∼ % = M f / M1 /M

/0. /0

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Since the functors M and D are right exact both rows of the diagram are exact. By Step 1 the left two vertical maps exist and are isomorphisms. Since D(MD( f ) ◦ % M2 − % M1 ◦ f ) = D( f ) ◦ D (% M2 ) − D (% M1 ) ◦ D( f ) = 0 it follows from Lemma 7.115.i that the left square of the diagram commutes. Hence we obtain an induced isomorphism % M as indicated, which moreover by construction satisfies D (% M ) = idD(M) . Theorem 7.117. The functors et et et M : Met OE , (L ∗ ) → M (N0 ) (L ∗ ), D : M (N0 ) (L ∗ ) → MOE , (L ∗ ),

are quasi-inverse equivalences of categories. Proof. By Proposition 7.113 and Lemma 7.115.ii it remains to show that the functor M is essentially surjective. Let M ∈ Met

(N0 ) (L ∗ ). We have to find a D ∈ Met (L ) together with an isomorphism M ∼ = M(D) in OE, ∗ et M (N0 ) (L ∗ ). It suffices to show that the morphism % M in Proposition 7.116 is L ∗ -equivariant. We want to prove that (% M ◦ ϕt − ϕt ◦ % M )(m) = 0 for any m ∈ M and t ∈ L ∗ . Since D (% M ) = idD(M) we certainly have (% ◦ ϕt − ϕt ◦ %)(m) ∈ J (N0 )MD(M) for any m ∈ M and t ∈ L ∗ . We choose for any positive integer r a set J (N0 /Nr ) ⊆ N0 of representatives for the cosets in N0 /Nr . Writing (7.12) m= uϕ r (m u,s r ), m u,s r = ψ r (u −1 m) u∈J (N0 /Nr )

and using that st = ts we see that (% M ◦ ϕt − ϕt ◦ % M )(m) =

ϕt (u)ϕr ((% M ◦ ϕt − ϕt ◦ % M )(m u,sr ))

u∈J (N0 /Nr )

lies, for any r , in the (N0 )-submodule of MD(M) generated by ϕ r (J (N0 ))MD(M). As in the proof of Lemma 7.115.ii we obtain 6 r r >0 ϕ (J (N0 ))MD(M) = 0. Since the functors M and D are right exact they commute with the reduction modulo p n , for any integer n ≥ 1.

8.5. Continuity In this section we assume that L ∗ contains a subgroup L 1 which is open in L ∗ and is a topologically finitely generated pro- p-group.

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We will show that the L ∗ -action on any étale L ∗ -module over (N0 ) is automatically continuous. Our proof is highly indirect so that we temporarily will have to make some definitions. But first a few partial results can be established directly. Let M be a finitely generated (N0 )-module. Definition 7.118. A lattice in M is a (N0 )-submodule of M generated by a finite system of generators of the (N0 )-module M. The lattices of M are of the form M 0 = ri=1 (N0 )m i for a set (m i )1≤i≤r of generators of the (N0 )-module M. We have the three fundamental systems of neighbourhoods of 0 in M: (

r i =1 r

(

i =1 r

(

On,k m i = M (N0 )n M + M(N0 )k M 0 )n,k∈N ,

(7.58)

Bn,k m i = M (N0 )n M + X k M 0 )n,k∈N ,

(7.59)

Cn,k m i = M (N0 )n M + Mk0 )n,k∈N ,

(7.60)

i =1

where Mk0 is the lattice ri=1 (N0 )X k m i , and is different from the set X k M0 when N0 is not commutative. If M is an étale L ∗ -module over (N0 ), for any fixed t ∈ L ,+ we have a fourth fundamental system of neighbourhoods of 0 in M: r ( ϕt (On,k ) (N0 )ϕt (m i ))n,k∈N , i=1

given by Lemma 7.98, because (ϕt (m i )1≤i ≤r is also a system of generators of the (N0 )-module M. Proposition 7.119. Let L be a submonoid of L ,+ . Let M be an étale L module over (N0 ). Then the maps ϕt and ψt , for any t ∈ L , are continuous on M. Proof. The ring endomorphisms ϕt of (N0 ) are continuous since they preserve M(N0 ) and M(N ). The continuity of the ϕt on M follows as in part (a) of the proof of Proposition 7.92. The continuity of the ψt follows from ψt (

r i=1

ϕt (On,k ) (N0 )ϕt (m i )) =

r i=1

On,k ψt ( (N0 ))m i =

r i=1

On,k m i .

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The same proof shows that, for any D ∈ Met OE , (L ∗ ), the maps ϕt and ψt , for any t ∈ L ∗ , are continuous on D. Proposition 7.120. The L ∗ -action L ∗ × D → D on an étale L ∗ -module D over OE is continuous. Proof. Let D be in Met OE , (L ∗ ). Since we already know from Proposition 7.119 that each individual ϕt , for t ∈ L ∗ , is a continuous map on D and since L 1 is open in L ∗ it suffices to show that the action L 1 × D → D of L 1 on D is continuous. As D is p-adically complete with its weak topology being the projective limit of the weak topologies on the D/ p n D we may further assume that D is killed by a power of p. In this situation the weak topology on D is locally compact. By Ellis’ theorem ([8] Thm. 1) we therefore are reduced to showing that the map L 1 × D → D is separately continuous. Because of Proposition 7.119 it, in fact, remains to prove that, for any d ∈ D, the map L 1 −→ D, g −→ gd is continuous at 1 ∈ L 1 . This amounts to finding, for any d ∈ D and any lattice D0 ⊂ D, an open subgroup H ⊂ L 1 such that (H − 1)d ⊂ D0 . We observe that (X m D++ )m∈Z is a fundamental system of L 1 -stable open 0 neighbourhoods of zero in D such that m X m D++ = D. We now choose an m ≥ 0 large enough such that d ∈ X −m D++ and X m D++ ⊂ D0 . The L 1 action on D induces an L 1 -action on X −m D++ / X m D++ which is o-linear hence given by a group homomorphism L 1 → Auto (X −m D++ / X m D++ ). Since D++ is a finitely generated o[[X ]]-module which is killed by a power of p we see that X −m D++ / X m D++ is finite. It follows that the kernel H of the above homomorphism is of finite index in L 1 . Our assumption that L 1 is a topologically finitely generated pro- p-group finally implies, by a theorem of Serre ([7] Thm. 1.17), that H is open in L 1 . We obtain (H − 1)d ⊂ (H − 1)X −m D++ ⊂ X m D++ ⊂ D0 .

In the special case of classical (ϕ, )-modules on OE the proposition is stated as Exercise 2.4.6 in [10] (with the indication of a totally different proof). Proposition 7.121. Let L be a submonoid of L ,+ containing an open subgroup L 2 which is a topologically finitely generated pro- p-group. Then the L -action L × (N0 ) → (N0 ) on (N0 ) is continuous. Proof. Since we know already from Propositions 7.99 and 7.119 that each individual ϕt , for t ∈ L , is a continuous map on (N0 ) and since L 2 is open

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in L it suffices to show that the action L 2 × (N0 ) → (N0 ) of L 2 on

(N0 ) is continuous. The ring (N0 ) is M (N0 )-adically complete with its weak topology being the projective limit of the weak topologies on the

(N0 )/M (N0 )n (N0 ). It suffices to prove that the induced action of L 2 on = (N0 )/M (N0 )n is continuous. The weak topology on is locally compact since (Bk = (X k (N0 ) + M (N0 )n )/M (N0 )n )k∈Z forms a fundamental system of compact neighbourhoods of 0. By Ellis’ theorem ([8] Thm. 1) we therefore are reduced to showing that the map L 2 × → is separately continuous. Because of Proposition 7.119 it, in fact, remains to prove that, for any x ∈ , the map L 2 −→ , g −→ gx is continuous at 1 ∈ L 2 . This amounts to finding, for any x ∈ and any large k ≥ 1, an open subgroup H ⊂ L 2 such that (H − 1)x ⊂ Bk . We observe that the Bk , for k ∈ Z, are L 2 -stable of union . We now choose an . The L -action on induces an L m ≥ k large enough such that x ∈ B−m 2 2 action on B−m /Bm which is o-linear hence given by a group homomorphism /B ). Since B is isomorphic to o[[X ]] ⊗ (N )/M(N )n L 2 → Auto (B−m o m 0 /B is as an o[[X]]-module, and (N )/M(N )n is finite, we see that B−m m finite. It follows that the kernel H of the above homomorphism is of finite index in L 2 . Our assumption that L 2 is a topologically finitely generated prop-group finally implies, by a theorem of Serre ([7] Thm. 1.17), that H is open in L 2 . We obtain (H − 1)x ⊂ (H − 1)B−m ⊂ Bm ⊂ Bk .

Lemma 7.122. i. For any M ∈ Met

(N0 ) (L ∗ ) the weak topology on D(M) is the quotient topology, via the surjection M : M → D(M), of the weak topology on M. ii. For any D ∈ Met OE , (L ∗ ) the weak topology on M(D) induces, via the injection ι D : D → M(D), the weak topology on D. Proof. i. If we write M as a quotient of a finitely generated free (N0 )module then we obtain an exact commutative diagram of surjective maps of the form // M . ⊕n (N0 ) i=1

⊕i

⊕ni=1 OE

M

/ / D(M)

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The horizontal maps are continuous and open by the definition of the weak topology. The left vertical map is continuous and open by direct inspection of the open zero neighbourhoods Bn,k (see (7.51)). Hence the right vertical map M is continuous and open. ii. An analogous argument as for i. shows that ι D is continuous. Moreover ι D has the continuous left inverse M(D) . Any continuous map with a continuous left inverse is a topological inclusion. An étale L ∗ -module M over (N0 ), resp. over OE , will be called topologically étale if the L ∗ -action L ∗ × M → M is continuous. Let Met,c

(N0 ) (L ∗ ) et,c and MOE , (L ∗ ) denote the corresponding full subcategories of Met

(N0 ) (L ∗ ) and Met (L ), respectively. Note that, by construction, all morphisms in ∗ OE , et Met (L ) and in M (L ) are automatically continuous. Also note that ∗ ∗

(N0 ) OE , by Proposition 7.119 any object in these categories is a complete topologically étale o[N0 L ∗ ]-module in our earlier sense. Proposition 7.123. The functors M and D restrict to quasi-inverse equivalences of categories et,c et,c et,c M : Met,c OE , (L ∗ ) → M (N0 ) (L ∗ ), D : M (N0 ) (L ∗ ) → MOE , (L ∗ ).

Proof. It is immediate from Lemma 7.122.i that if L ∗ acts continuously on M ∈ Met

(N0 ) (L ∗ ) then it also acts continuously on D(M). On the other hand, let D ∈ Met OE , (L ∗ ) such that the action of L ∗ on D is continuous. We choose a lattice D0 in D with a finite system (di ) of generators. (2) Given t ∈ L ∗ we introduce Dt := i (N0 )t.di which is a lattice in D since the action of t on D is étale. Also D0 + Dt is a lattice in D. The (N0 )-module M(D) is generated by ι D (D0 ) as well as by ι D (D0 + Dt ) and both (Cn ι D (D0 ))n∈N

and (Cn ι D (D0 + Dt ))n∈N

are fundamental systems of neighbourhoods of 0 in M(D) for the weak topology. To show that the action of L ∗ on M(D) is continuous, it suffices to find for any t ∈ L ∗ , λ0 ∈ (N0 ), d0 ∈ D0 , n ∈ N a neighbourhood L t ⊂ L ∗ of t and n ∈ N such that L t .(λ0 ι D (d0 ) + Cn ι D (D0 )) ⊂ t.λ0 ι D (d0 ) + Cn ι D (D0 + Dt ). The three maps λ → λι D (d0 ) : (N0 ) → M(D) d → λ0 ι D (d) : D → M(D) (λ, d) → λι D (d) : (N0 × D) → M(D)

(7.61)

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are continuous because ι D is continuous. The action of L ∗ on D and on (N0 ) is continuous (Proposition 7.121). Altogether this implies that we can find a small L t such that L t .λ0 ι D (d0 ) ⊂ t.λ0 ι D (d0 ) + Cn ι D (D0 + Dt ). Since ι D is L ∗ -equivariant we have, for any n ∈ N, L t .Cn ι D (D0 ) = (L t .Cn ) ι D (L t .D0 ). The continuity of the action of L ∗ on (N0 ) shows that L t .Cn ⊂ Cn when L t is small enough and n is large enough. (2) (2) For d ∈ D0 we have L t . (N0 )d ⊂ (N0 )(L t .d). The action of L ∗ on D is continuous hence, for any n , we can choose a small L t such that (2) L t .d ⊂ t.d + Cn D0 . We can choose the same L t for each di and we obtain L t .D0 ⊂

(2)

(2)

(N0 )t.di + Cn D0 .

i

Applying ι D , we obtain ι D (L t .D0 ) ⊂ ι D (Dt ) + Cn ι D (D0 ) and then (L t .Cn ) ι D (L t .D0 ) ⊂ Cn ι D (Dt ) + Cn C n ι D (D0 ). We check that C n C n ⊂ Cn,n+n ⊂ Cn when n ≥ n. Hence when n is large enough, L t .(Cn ι D (D0 )) ⊂ Cn ι D (Dt + D0 ). This ends the proof of (7.61). et,c et Proposition 7.124. We have Met,c OE , (L ∗ ) = MOE , (L ∗ ) and M (N0 ) (L ∗ ) = Met

(N0 ) (L ∗ ).

Proof. The first identity was shown in Proposition 7.120 and is equivalent to the second identity by Theorem 7.117 and Proposition 7.123. Corollary 7.125. Any étale L ∗ -module over (N0 ), resp. over OE , is a complete topologically étale o[N0 L ∗ ]-module in our sense. Proof. Use Propositions 7.119 and 7.124.

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9. Convergence in L + -modules on (N0 ) In this section, we use the notation of Section 8 where we assume that N is a padic Lie group. We assume that and ι are continuous group homomorphisms : P → P (2) , ι : N (2) → N , ◦ ι = id, (2) such that (L + ) ⊂ L (2) + , (N ) = N , (ι ◦ )(N0 ) ⊂ N0 , and

tι(y)t −1 = ι((t)y(t)−1 ) for y ∈ N (2) , t ∈ L.

(7.62)

The assumptions of Section 8 are naturally satisfied with L ∗ = L + . Indeed, the compact open subgroup N0 of N is a compact p-adic Lie group, the group (N0 ) is a compact non-trivial subgroup N0(2) of N (2) Q p hence N0(2) is isomorphic to Z p and is open in N (2) , the kernel of | N0 is normalized by L ,+ . (2) (2) Note that L + normalizes ι(N0 ) since (L + ) normalises N0 and (7.62). et et Let M ∈ M (N0 ) (L + ) and D ∈ MOE , (L + ) be related by the equivalence of categories (Theorem 7.117), M = (N0 ) ⊗OE ,ι D = (N0 )ι D (D). We will exhibit in this section a special family Cs of compact subsets in M such that M(Cs ) is a dense o-submodule of M, and such that the P-equivariant sheaf on C associated to the étale o[P+ ]-module M(Cs ) by Theorem 7.38 extends to a G-equivariant sheaf on G/P. We will follow the method explained in Subsection 6.5 which reduces the most technical part to the easier case where M is killed by a power of p.

9.1. Bounded sets Definition 7.126. A subset A of M is called bounded if for any open neighbourhood B of 0 in M there exists an open neighbourhood B of 0 in (N0 ) such that B A ⊂ B. Compare with [12] Def. 8.5. The properties satisfied by bounded subsets of M can be proved directly or deduced from the properties of bounded subsets of

(N0 ) ([15] §12). Using the fundamental system (7.59) of neighbourhoods of 0, the set A is bounded if and only if for any large n there exists n > n such that

(M (N0 )n + X n (N0 )) A ⊂ M (N0 )n M + X n M 0 ,

G-equivariant sheaves on G/P

347

equivalently X n −n A ⊂ M (N0 )n M + M 0 . We obtain (compare with [12] Lemma 8.8): Lemma 7.127. A subset A of M is bounded if and only if for any large positive n there exists a positive integer n such that

A ⊂ M (N0 )n M + X −n M 0 . The following properties of bounded subsets will be used in the construction of a special family Cs in the next subsection. – Let f : ⊕ri=1 (N0 ) → M be a surjective homomorphism of (N0 )modules. The image by f of a bounded subset of ⊕ri=1 (N0 ) is a bounded subset of M. For 1 ≤ i ≤ r , the i-th projections Ai ⊂ (N0 ) of a subset A of ⊕ri=1 (N0 ) are all bounded if and only if A is bounded. – A compact subset is bounded. – The (N0 )-module generated by a bounded subset is bounded. – The closure of a bounded subset is bounded. – Given a compact subset C in (N0 ) and a bounded subset A of M, the subset C A of M is bounded. – The image of a bounded subset by f ∈ Endcont o (M) is bounded. The image by M of a bounded subset in M is bounded in D. – A subset A of D is bounded if and only if the image An of A in D/ p n D is bounded for all large n. – When D is killed by a power of p, a subset A of D is bounded if and only if A is contained in a lattice, i.e. if A is contained in a compact subset (by the properties of lattices given in Section 7.3). Lemma 7.128. The image by ι D of a bounded subset in D is bounded in M. Proof. Let A ⊂ D be a bounded subset and let D 0 be a fixed lattice in D. For all n ∈ N there exists n ∈ N such that A ⊂ p n D + (X (2) )−n D 0 by Lemma 7.127. Applying ι D we obtain

ι D (A) ⊂ p n ι D (D) + X −n ι D (D 0 ) ⊂ M (N0 )n M + X −n M 0 where M 0 = (N0 )ι D (D 0 ) is a lattice in M. By the same lemma, this means that ι D (A) is bounded in M.

9.2. The module Msbd Definition 7.129. Msbd is the set of m ∈ M such that the set of M (ψ k (u −1 m)) for k ∈ N, u ∈ N0 is bounded in D.

348

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The definition of Msbd depends on s because ψ is the canonical left inverse of the action ϕ of s on M. We recognize m u,s k = ψ k (u −1 m) appearing in the expansion (7.12). Proposition 7.130. Msbd is an étale o[P+ ]-submodule of M. Proof. (a) We check first that Msbd is P+ -stable. As Msbd is N0 -stable and P+ = N0 L + , it suffices to show that tm = ϕt (m) ∈ Msbd when t ∈ L + and m ∈ Msbd . Using the expansion (7.12) of m and st = ts, for k ∈ N and n 0 ∈ N0 , we write ψ k (n −1 0 tm) as the sum over u ∈ J (N0 /Nk ) of k k −1 −1 k ψ k (n −1 0 tuϕ (m u,s k )) = ψ (n 0 tut ϕ (ϕt (m u,s k ))) −1 = ψ k (n −1 0 tut )ϕt (m u,s k ),

and M (ψ k (n −1 0 ϕt (m))) as the sum over u ∈ J (N0 /Nk ) of −1 M (ψ k (n −1 0 tut )ϕt (m u,s k )) = vk,n 0 M (ϕt (m u,s k )) = vk,n 0 ϕt ( M (m u,s k )), (2)

−1 bd where vk,n 0 := (ψ k (n −1 0 tut )) belongs to N0 or is 0. As m ∈ Ms , the set of M (m u,s k ) for k ∈ N and u ∈ N0 is bounded in D. Its image (2) by the continuous map ϕt is bounded and generates a bounded o[N0 ]bd submodule of D. Hence ϕt (m) ∈ Ms . (b) The o[P+ ]-module Msbd is ψ-stable (hence Msbd is étale by Corollary 7.45) because we have, for m ∈ Msbd , u ∈ N0 , k ∈ N,

ψ k (u −1 ψ(m)) = ψ k+1 (ϕ(u −1 )m).

(7.63)

The goal of this section is to show that the P-equivariant sheaf on C associated to the étale o[P+ ]-module Msbd extends to a G-equivariant sheaf on G/P. We will follow the method explained in Subsection 6.5. Let pn : M → M/ p n M be the reduction modulo p n for a positive integer n. Recall that M is p-adically complete. Lemma 7.131. The o-submodule Msbd ⊂ M is closed for the p-adic topology, in particular Msbd = lim(Msbd / pn Msbd ). ← − n

Moreover is the set of m ∈ M such that pn (m) belongs to (M/ p n M)bd s for all n ∈ N, and we have Msbd

Msbd = lim(M/ p n M)bd s . ← − n

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349

Proof. (a) Let m be an element in the closure of Msbd in M for the p-adic topology. For any r ∈ N, we choose m r ∈ Msbd with m − m r ∈ pr M. For each r , we choose r ≥ 1 such that M (ψ k (u −1 m r )) ∈ pr D + X −r D 0 for all k ∈ N, u ∈ N0 , applying Lemma 7.127. We have M (ψ k (u −1 m)) ∈ M (ψ k (u −1 m r ) + pr M)

= M (ψ k (u −1 m r )) + pr D ⊂ pr D + X −r D 0 . By the same lemma, m ∈ Msbd . This proves that Msbd is closed in M hence p-adically complete. (b) The reduction modulo p n commutes with M , ψ, and the action of N0 . The following properties are equivalent: – m ∈ Msbd , – { M (ψ k (u −1 m)) for k ∈ N, u ∈ N0 } ⊂ D is bounded, – { M/ pn M (ψ k (u −1 pn (m))) for k ∈ N, u ∈ N0 } ⊂ D/ p n D is bounded for all positive integers n, a. pn (m) ∈ (M/ p n M)bd s for all positive integers n. We deduce that m → ( pn (m))n : Msbd → limn (M/ p n M)bd s is an ← − isomorphism. Proposition 7.132. D = Dsbd and Msbd contains ι D (D). Proof. (i) We show that D = Dsbd . By Lemma 7.131, we can suppose that D is killed by a power of p. Let d ∈ D. By Corollary 7.95, for n ∈ N, there (2) exists k0 ∈ N such that ψ k (v −1 d) ∈ D ' for k ≥ k0 , v ∈ N0 . As D ' ⊂ D (2) is bounded, and as the set of ψ k (v −1 d) for all 0 ≤ k < k0 , v ∈ N0 , is (2) also bounded because the set of v −1 d for v ∈ N0 is bounded and ψ k is bd continuous, we deduce that d ∈ Ds . (ii) We show that Msbd contains ι D (D) by showing { M (ψ k (u −1 ι D (d))) for k ∈ N, u ∈ N0 } (2)

= {ψ k (v −1 d) for k ∈ N, v ∈ N0 } when d ∈ D (the right-hand side is bounded in D by (i)). We write an (2) element of N0 as ι(v)u for u in N and v ∈ N0 . By Lemma 7.112, ψ k (u −1 ι(v)−1 ι D (d)) = ψ k (u −1 ι D (v −1 d)) = s −k u −1 s k ψ k (ι D (v −1 d)) when u ∈ s k N s −k and is 0 when u is not in s k N s −k . When u ∈ s k N s −k we have M (s −k u −1 s k ψ k (ι D (v −1 d))) = ψ k (v −1 d) as ι D is ψ-equivariant.

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Proposition 7.133. Msbd is dense in M. Proof. Msbd ⊂ M is an o[N0 ]-submodule, which by Proposition 7.132 contains ι D (D). The o[N0 ]-submodule of M generated by ι D (D) is dense by Lemma 7.103. We summarize: we proved that Msbd ⊂ M is a dense o[N0 ]-submodule, stable by L + , and the action of L + on Msbd is étale. Remark 7.134. It follows from Lemma 7.131 and the subsequent Proposition 7.135 that Msbd is a (N0 )-submodule of M.

9.3. The special family Cs when M is killed by a power of p We suppose that M is killed by a power of p. Proposition 7.135. 1. For any lattice D0 in D, the o-submodule Msbd (D0 ) := {m ∈ M | M (ψ k (u −1 m)) ∈ D0 for all u ∈ N0 and k ∈ N} of M is compact, and is a ψ-stable (N0 )-submodule. 2. The family Cs of compact subsets of M contained in Msbd (D0 ) for some lattice D0 of D, is special (Definition 7.76), satisfies C(5) (Proposition 7.83) and C(6) (Proposition 7.84), and M(Cs ) = Msbd is a

(N0 )-submodule of M. Proof. 1. (a) As and ψ are continuous (Proposition 7.119) and D0 ⊂ D is closed, it follows that Msbd (D0 ) is an intersection of closed subsets in M, hence Msbd (D0 ) is closed in M. As Msbd (D0 ) is an o[N0 ]submodule of M and o[N0 ] is dense in (N0 ) we deduce that Msbd (D0 ) is a (N0 )-submodule. It is ψ-stable by (7.63). The weak topology on M is the projective limit of the weak topologies on M/M (N0 )n M, and we have ([2] I.29 Corollary) Msbd (D0 ) = lim(Msbd (D0 ) + M (N0 )n M)/M (N0 )n M. ← − n≥1

Therefore it suffices to show that (Msbd (D0 ) + M (N0 )n M)/M (N0 )n M is compact for each large n. We will show the stronger property that it is a finitely generated (N0 )-module. (b) We prove first that Msbd (D0 ) is the intersection of the (N0 )-modules generated by the image by ϕ k of the inverse image −1 M (D0 ) of D0 in M, for k ∈ N,

G-equivariant sheaves on G/P Msbd (D0 ) =

(N0 )ϕ k (−1 M (D0 )).

351

(7.64)

k∈N

The inclusion from left to right follows from the expansion (7.12), as m ∈ Msbd (D0 ) is equivalent to m u,s k = ψ k (u −1 m) ∈ −1 M (D0 ) for all u ∈ N0 and k ∈ N. The inclusion from right to left follows from M ψ k u −1 ( (N0 )ϕ k (−1 M (D0 ))) = D0 . (c) We pick a lattice M0 of M such that −1 M (D0 ) = M0 + J (N0 )M, as J (N0 )M is the kernel of M . By Lemma 7.114 we can choose for each n ∈ N a large integer r such that ϕ r (J (N0 )M) ⊆ M (N0 )n M. Therefore we have Msbd (D0 ) ⊆ (N0 )ϕ r (M0 + J (N0 )M) ⊆ (N0 )ϕ r (M0 ) + M (N0 )n M. We deduce (Msbd (D0 ) + M (N0 )n M)/M (N0 )n M ⊆ ( (N0 )ϕr (M0 ) + M (N0 )n M)/M (N0 )n M. The right term is a finitely generated (N0 )-module hence the left term is finitely generated as a (N0 )-module since (N0 ) is noetherian. 2. The family is stable by finite union because a finite sum of lattices is a lattice. If C ∈ Cs then N0 C ∈ Cs because Msbd (D0 ) is a (N0 )-module. We have M(Cs ) = ∪ D0 Msbd (D0 ) = Msbd , when D0 runs over the lattices of D, the last follows from the fact that a bounded subset of D is contained in a lattice (this is the only part in the proof where the assumption that M is killed by a power of p is used). Apply Proposition 7.130. Property C(5) is immediate because Msbd (D0 ) is ψ-stable. Property C(6) follows from ϕ(Msbd (D0 )) ⊂ Msbd (Ds ) where Ds is the lattice of D generated by ϕ(D0 ) (this uses part (a) of the proof of Proposition 7.130). Consider the lattice M ++ = (N0 )i D (D ++ ) of M. Since (N0 ) and D ++ are ϕ-stable and since ϕ and ι D commute, M ++ is ϕ-stable and M (M ++ ) = D ++ . Hence for a subset S ⊂ M we have S ⊂ X r M ++ + J (N0 )M ⇔ M (S) ⊂ (X (2) )r D ++ .

(7.65)

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Proposition 7.136. Let r ∈ N, C+ ⊂ L + a compact subset and let D0 be a lattice in D. There is a compact open subgroup P1 ⊂ P0 and k0 ≥ 0 such that for all k ≥ k0 s k (1 − P1 )C+ Msbd (D0 ) ⊂ X r M ++ + M (N0 )r M. Proof. Denote for simplicity S = Msbd (D0 ). By definition M (S) ⊂ D0 . Since (2) P+ acts continuously on D, (C+ )D0 is compact and (X (2) )r D ++ is open, (2) (2) (2) there is a compact open subgroup P1 ⊂ P+ such that (1 − P1 )(C+ )D0 ⊂ (X (2) )r D ++ . We may choose a compact open subgroup P1 of P0 such that (2) (P1 ) ⊂ P1 , hence M ((1 − P1 )C+ S) ⊂ (X (2) )r D ++ . Relation (7.65) yields (1 − P1 )C+ S ⊂ J (N0 )M + X r M ++ . Choosing k0 such that ϕ k (J (N0 )) ⊂ M (N0 )r for k ≥ k0 (as we may by Lemma 7.114), the result follows from the ϕ-stability of X r M ++ (which follows from the fact that ϕ(X r ) ∈ X r (N0 ) and ϕ(M ++ ) ⊂ M ++ ). Corollary 7.137. Property T(1) in Proposition 7.83 is satisfied. Proof. Let C, C+ , M as in Proposition 7.83 and choose r such that M (N0 )r M + X r M ++ ⊂ M. Choose a lattice D0 such that C ⊂ Msbd (D0 ). As Msbd (D0 ) is ψ-stable (Proposition 7.135), we can choose the subgroup P1 and k(C, M, C+ ) = k0 given by Proposition 7.136. bd Recall that we defined (7.42) operators sg(k) = H(k+1) − H(k) g g on Ms = M(Cs ). From now on we fix a lattice D0 in D and g ∈ N0 P N0 . We denote S = Msbd (D0 ).

Corollary 7.138. There is k g ≥ 0 such that for all x ∈ N and k ≥ x + k g ++ ψ x ◦ N0 ◦ sg(k) (S) ⊂ −1 ). M (D

Proof. Let r = 1, C+ = g s and choose P1 and k0 as in Proposition 7.136, so ++ ) for k ≥ k . Since S is (N )[ψ]-stable and that s k (1 − P1 )C+ S ⊂ −1 0 0 M (D ++ ) is o[N ]-stable, relation (7.43) and the inequality k (2) (P ) ≥ k (1) −1 (D 0 1 g g M ++ ) for k ≥ x + k + k (2) (P ). Applying ψ x ◦ N yield sg(k) (S) ⊂ N0 ϕ x −1 (D 0 1 0 g M (2) yields the desired result, with k g = k0 + k g (P1 ). Lemma 7.139. There is a lattice D1 in D such that for all u ∈ Ug , k ≥ k g ≥ (1) k g and x ≥ k − k g ψ x ◦ N0 ◦ α(g, xu ) ◦ Res(1u Nk )(S) ⊂ −1 M (D1 ).

(7.66)

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353 (1)

= s k g t (g, U ). This is a compact subset of L , since k ≥ k . Proof. Let C+ g + g g Since S is (N0 )[ψ]-stable, we have Res(1u Nk )(S) ⊂ u ◦ ϕ k (S), hence α(g, xu ) ◦ Res(1u Nk )(S) ⊂ N0 ◦ t (g, u) ◦ ϕ k (S) ⊂ N0 ϕ k−k g (C+ S). S), Hence the left-hand-side of (7.66) is contained in ψ x−k+kg ( (N0 )C+ −1 −1 (D ), because S ⊂ (N )ϕ x−k+k g ( (D )) which is a subset of (N0 )C+ 0 0 0 M M and C+ (N0 ) ⊂ (N0 )C+ . Thus (2)

M (ψ x ◦ N0 ◦ α(g, xu ) ◦ Res(1u Nk )(S)) ⊂ (N0 )(C+ )(D0 )

and the last subset of D is compact, hence contained in some lattice D1 . bd ++ ). Corollary 7.140. For all k ≥ k g we have H(k) g (S) ⊂ Ms (D1 + D bd ++ Moreover, Hg (S) ⊂ Ms (D1 + D ).

Proof. The second assertion follows from the first by letting k → ∞, since Msbd (D1 + D ++ ) is closed in M. For the first assertion, we need to prove that (k) ++ ) for all x ≥ 0 and k ≥ k . Fix x ≥ 0. If ψ x (N0 Hg (S)) ⊂ −1 g M (D1 + D k ≤ k g + x, simply add all relations (7.66) for u ∈ J (Ug /Nk ). If k > k g + x, (x+k ) ( j) (k) the equation Hg = Hg g + k−1 j =x+k g sg and Corollary 7.138 show that (x+k g )

x ψ x (N0 H(k) g (S)) ⊂ ψ (N0 Hg

(x+k g )

But we have already seen that ψ x (N0 Hg

++ (S)) + −1 ). M (D

++ ). (S)) ⊂ −1 M (D1 + D

Proposition 7.141. All the assumptions of Proposition 7.84 are satisfied. Proof. Property T(1) was checked in Corollary 7.137. Property T(2) and the fact that Hg preserves Msbd (for g ∈ N0 P N0 ) follow from Corollary 7.140 and the fact that any m ∈ Msbd is in S = Msbd (D0 ) for some lattice D0 in D.

9.4. Functoriality and dependence on s Let Z (L)†† ⊂ Z (L) be the subset of elements s such that L = L − s N and (s k N0 s −k )k∈Z and (s −k w0 N0 w0−1 s k )k∈Z are decreasing sequences of trivial intersection and union N and w0 N w0−1 , respectively (see Section 6). Let M be a topologically etale L + -module over (N0 ) and let D := D(M). We have D/ p n D = D(M/ p n M) for n ≥ 1. By Lemma 7.104, M satisfies the properties a,b,c,d of Subsection 6.5 and is complete (the same is true for M/ p n M). The image D0,n in D/ p n D of any lattice D0,n+1 in D/ pn+1 D is a lattice and the maps and ψ commute with the reduction modulo p n , hence n bd (M/ p n+1 M)bd s (D0,n+1 ) maps into (M/ p M)s (D0,n ). Therefore the special n+1 family Cs,n+1 in M/ p M maps to the special family Cs,n in M/ p n M. As

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in Lemma 7.89 we define the special family Cs in M to consist of all compact subsets C ⊂ M such that pn (C) ∈ Cs,n for all n ≥ 1. By Proposition 7.135 and Lemma 7.131 we have M(Cs ) = Msbd . Theorem 7.142. Let s ∈ Z (L)†† and M ∈ Met

(N0 ) (L + ). (i) The (s, res, Cs )-integrals Hg,s of the functions αg,0 | Msbd for g ∈ N0 P N0 exist, lie in Endo (Msbd ), and satisfy the relations H1, H2, H3 of Proposition 7.72. (ii) The map M → (Msbd , (Hg,s )g∈N0 P N0 ) is functorial. Proof. (i) By Proposition 7.141 the assumptions of Proposition 7.91 are satisfied. (ii) Let f : M → M be a morphism in Met

(N0 ) (L + ). For m ∈ M we denote E s (m) = { M (ψsk u −1 m) for u ∈ N0 , k ∈ N}. We have D( f )(E s (m)) = E s ( f (m)) when m ∈ M,

(7.67)

because the maps M : M → D and M : M → D sending x to 1⊗x for x ∈ M or x ∈ M satisfy M ◦ f = D( f )◦ M , and f is P − -equivariant by Lemma 7.22. Any morphism between finitely generated modules on OE is continuous for the weak topology (cf. [12] Lemma 8.22). The image of a bounded subset by a continuous map is bounded. We deduce from (7.67) that E s (m) bounded implies E s ( f (m)) bounded, equivalently m ∈ Msbd implies f (m) ∈ Ms bd . For m ∈ Msbd we have f (Hg,s (m)) = Hg,s ( f (m)) where Hg,s (.) = lim n(g, u)ϕt (g,u)s k ψsk u −1 (.), k→∞

u∈J (N0 /s k N0 s −k )

because f is P+ and P− -equivariant by Lemma 7.22. We investigate now the dependence on s ∈ Z (L)†† of the dense subset Msbd ⊆ M and of the (s, res, Cs )-integrals Hg,s . Lemma 7.143. Z (L)†† is stable by product. Proof. Let s, s ∈ Z(L)†† . Clearly L − s n = L − s −n s n s n ⊂ L − (ss )n because L − is a monoid and s −1 ∈ Z(L)− = Z (L) ∩ L − . Therefore L = L − (ss )N . The sequence ((ss )k N0 (ss )−k )k∈Z is decreasing because s k+1 s k+1 N0 s −k−1 s −k−1 ⊂ s k s k+1 N0 s −k−1 s −k ⊂ s k s k N0 s −k s −k .

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The intersection is trivial and the union is N because s k s k N0 s −k s −k ⊂ s k N0 s −k when k ∈ N and s k s k N0 s −k s −k ⊃ s k N0 s −k when −k ∈ N. One makes the same argument with w0 N0 w0−1 . (2)

Lemma 7.144. (i) The action of t0 ∈ −1 (L 0 ) ∩ L + on D is invertible. (ii) There exists a treillis D0 in D which is stable by −1 (L (2) 0 ) ∩ L +. Proof. (i) is true because the action of t0 on D is étale and N0(2) = (2) (t0 )N0 (t0 )−1 . (ii) Let s ∈ Z (L)†† and let ψs be the canonical inverse of the étale action ϕs of s on D. We show that the minimal ψs -stable treillis D & of D (2) (Proposition 7.93(iii)) is stable by −1 (L 0 ) ∩ L + . (2) For t0 ∈ −1 (L 0 ) ∩ L + we claim that ϕt0 (D & ) is also a ψs -stable treillis in D. We have ψs ψt0 = ψt0 ψs as t0 ∈ Z (L). Multiplying by ϕt0 on both sides, one gets ϕt0 ψs ψt0 ϕt0 = ϕt0 ψt0 ψs ϕt0 . Since ψt0 is the two-sided inverse of ϕt0 by (i) we get that ϕt0 and ψs commute. Hence ϕt0 (D & ) is (2) a compact o-module which is ψs -stable. It is a (N0 )-module because (2) (2) any λ ∈ (N0 ) is of the form λ = ϕ(t0 ) (μ) for some μ ∈ (N0 ) and & λϕt0 (d) = ϕt0 (μd) for all d ∈ D. As D contains a lattice and ϕt0 is étale, we deduce that ϕt0 (D & ) contains a lattice and therefore is a treillis. By the minimality of D & we must have D & ⊂ ϕt0 (D & ). Similarly one checks that ψt0 (D & ) is a treillis. It is ψs -stable because ψs and ψt0 commute. Hence D & ⊂ ψt0 (D & ). Applying ϕt0 which is the two-sided inverse of ψt0 we obtain ϕt0 (D & ) ⊂ D & hence D & = ϕt0 (D & ). We denote by Z(L)† ⊂ Z(L) the monoid of z ∈ Z (L)+ = Z (L) ∩ L + such that z −1 w0 N0 w0−1 z ⊂ w0 N0 w0−1 . We have Z (L)†† Z (L)† ⊂ Z (L)†† . (2) (2) Note that L (2) 0 contains the center of G L(2, Q p ) and that Z(L )† = L + . For m ∈ M, t ∈ L + , u ∈ U, and a system of representatives J (N0 /t N0 t −1 ) ⊂ N0 for the cosets in N0 /t N0 t −1 we have (7.12) m= uμt,u , μt,u := ϕt ψt (u −1 m). (7.68) u∈J (N0 /t N0 t −1 )

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For g ∈ N0 P N0 and s ∈ Z (L)†† , we have the smallest positive integer k g,s as (0) in (7.27). For k ≥ k g,s , we have Hg,s,J (N0 /Nk ) ∈ Endcont o (M) where (compare with (7.28)) Hg,s,J (N0 /Nk ) (m) = n(g, u)t (g, u)μs k ,u . (7.69) u∈J (U g /Nk )

When m ∈ Msbd , the integral Hg,s (m) is the limit of Hg,s,J (N0 /Nk ) (m) by Theorem 7.142 and (7.29). Proposition 7.145. Let s ∈ Z (L)†† , t0 ∈ −1 (L (2) 0 ) ∩ Z (L)† and r a positive integer. (i) We have Mstbd0 ⊆ Msbd = Msbd r . (ii) For g ∈ N0 P N0 we have Hg,s = Hg,st0 on Mstbd0 and Hg,s = Hg,sr on Msbd . Proof. (a) Note that st0 and s r in proposition belong also to Z(L)†† . (2) For a treillis D0 in D which is stable by −1 (L 0 ) ∩ L + (Lemma 7.144), (2) (X (2) )−r D0 is a treillis in D; it is also stable by t0 ∈ −1 (L 0 ) ∩ L + because (2)

(2)

ϕ(t0 ) ((X (2) )−r (N0 )) = ϕ(t0 ) ((X (2) )−r )ϕ(t0 ) ( (N0 )) (2)

= (X (2) )−r (N0 ). When M is killed by a power of p, this implies with Proposition 7.135 that Msbd is the union of Msbd (D0 ) when D0 runs over the lattices of D which (2) are stable by −1 (L 0 ) ∩ L + . (b) We suppose from now on, as we can by Lemma 7.131, that M is killed by bd a power of p to prove Mstbd0 ⊂ Msbd = Msbd r . Let m ∈ Mst (D0 ) where D0 0 (2)

is a −1 (L 0 )∩ L + -stable lattice of D. For u ∈ N0 and k ∈ N, using (7.12) for t = t0k we obtain that M (ψsk (u −1 m)) = M ( v ◦ ϕtk0 ◦ ψtk0 ◦ v −1 ◦ ψsk (u −1 m)) v∈J (N0 /t0k N0 t0−k )

=

(v)ϕtk0 ( M (ψstk 0 (ϕsk (v −1 )u −1 m)))

v∈J (N0 /t0k N0 t0−k )

lies in D0 , since D0 is both N0(2) - and ϕt0 -invariant and M (ψstk 0 (u m)) ∈ D0 for u ∈ N0 . Therefore Mstbd0 (D0 ) ⊂ Msbd (D0 ) and by (a) we deduce Mstbd0 ⊂ Msbd .

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For any m ∈ M we observe that { M (ψskr (u −1 m)) for k ∈ N, u ∈ N0 } ⊂ { M (ψsk (u −1 m)) for k ∈ N, u ∈ N0 }, as ψskr = ψsr k . We deduce that Msbd (D0 ) ⊂ Msbd r (D0 ) for any lattice D0 of D hence Msbd ⊂ Msbd . Conversely, for k ∈ N we write k1 = r k − k2 with r 1 k ∈ N and 0 ≤ k2 < r and we observe that M (ψsk1 (u −1 m)) = M ( v ◦ ϕsk2 ◦ ψsr k (ϕsk1 (v −1 )u −1 m)) v∈J (N0 /s k2 N0 s −k2 )

=

(v)ϕsk2 ( M (ψskr (ϕsk1 (v −1 )u −1 m))).

v∈J (N0 /s k2 N0 s −k2 )

r−1 i (2) The (N0 )-submodule Dr generated by i=1 ϕs (D0 ) is a lattice because the action ϕs of s on D is étale. We deduce that Msbd r (D0 ) ⊂ Msbd (Dr ) since M (ψskr (u m)) ∈ D0 for u ∈ N0 , m ∈ Msbd r (D0 ). bd bd bd Therefore Msbd r (D0 ) ⊂ Ms (Dr ) hence Ms r ⊂ Ms . It is obvious that Hg,s = Hg,s r on Msbd . (0) (2) (c) Let g ∈ N0 P N0 , k ≥ k g,s , t0 ∈ −1 (L 0 ) ∩ Z (L)† and r ≥ 1. We have (0)

(0) k g,st0 ≤ k g,s ,

(0)

(0) k g,s r ≤ k g,s

because (st0 )k N0 (st0 )−k ⊂ Nk and (s r )k N0 (s r )−k = Nkr ⊂ Nk . Let d in D and v ∈ N0 . By (7.12) we have k d= uϕst ◦ ψstk 0 (u −1 d) 0 (2)

(2)

u∈J (N0 /(st0 )k N0 (st0 )−k )

= (2)

uϕsk ◦ ψsk (u −1 d), (2)

u∈J (N0 /(s)k N0 (s)−k )

with the second equality holding true summand per summand, because ψt0 is the left and right inverse of ϕt0 on D (Lemma 7.144 (i)) and (2) (2) (t0 )N0 (t0 )−1 = N0 . Since ι D commutes with ϕt and ψt for t ∈ L + , this implies k vι D (d) = vι(u)ϕst ◦ ψstk 0 (ι(u)−1 ι D (d)) 0 (2)

(2)

u∈J (N0 /(st0 )k N0 (st0 )−k )

=

vι(u)ϕsk ◦ ψsk (ι(u)−1 ι D (d)),

(2) (2) u∈J (N0 /(s)k N0 (s)−k )

again with the second equality holding true summand per summand. We choose, as we can, systems of representatives J (N0 /(st0 )k N0 (st0 )−k ) and

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Peter Schneider, Marie-France Vigneras, and Gergely Zabradi (2)

(2)

(0)

J (N0 /s k N0 s −k ) containing ι( J (N0 /(s)k N0 (s)−k )). For k ≥ k g,s ≥ (0) k g,st0 , we obtain Hg,st0 ,v J (N0 /(st0 )k N0 (st0 )−k ) (vι D (d)) = Hg,s,v J (N0 /s k N0 s −k ) (vι D (d)). Passing to the limit when k goes to infinity, and using linearity we deduce that Hg,st0 = Hg,s on the o[N0 ]-submodule < N0 ι D (D) >o generated by ι D (D) in Mstbd0 . (d) Let m ∈ Msbd (D1 ) with D1 ⊂ D a ψs -stable lattice (Proposition 7.93 (iv)). For a positive integer k, and a set of representatives J (N0 /s k N0 s −k ), we write m in the form (7.12) m= uϕsk (ι D (d(s, u)) + m(s, u)) u∈J (N0 /s k N0 s −k )

with m(s, u) in J (N0 )M and d(s, u) = M (ψsk (u −1 m)) in D1 . Then m(s) := uϕsk (ι D (d(s, u))) lies in < N0 ι D (D) >o u∈J (N0 /s k N0 s −k )

because ι D is L + -equivariant. Moreover m − m(s) is contained in the o[N0 ]-submodule N0 ϕsk (J (N0 )M) generated by ϕsk (J (N0 )M). We show that m(s) ∈ Msbd (D1 ). For v ∈ N0 and r ≤ k we have

ψsr (v −1 (m − m(s))) = ψsr (v −1

(7.70)

uϕsk (m(s, u)))

u∈J (N0 /s k N0 s −k )

=

ψsr (v −1 u)ϕsk−r (m(s, u))

u∈J (N0 /s k N0 s −k )

which lies in J (N0 )M since m(s, u) is in J (N0 )M and J (N0 )M is N0 and ϕs -stable. This shows that M (ψsr (v −1 m(s))) = M (ψsr (v −1 m)) lies in D1 . On the other hand, for r > k we have M (ψsr (v −1 m(s))) = M (ψsr (v −1 uϕsk (ι D (d(s, u))))) =

u∈J (N0 /s k N0 s −k )

M (ψsr−k (ψsk (v −1 u)ι D (d(s, u))))

u∈J (N0 /s k N0 s −k )

which lies in D1 . Indeed, since D1 is ψs -stable the formula in part (ii) of the proof of Proposition 7.132 implies that ι D (D1 ) ⊆ Msbd (D1 ); hence the ι D (d(s, u)) lie in the ψs - and N0 -invariant subspace Msbd (D1 ). We conclude that m(s) ∈ Msbd (D1 ).

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Therefore, for any ψst0 -stable lattice D1 ⊂ D, any k ≥ 1, and any set of representatives J (N0 /(st0 )k N0 (st0 )−k ), we have defined an o-linear homomorphism m → m(st0 )

Mstbd0 (D1 ) → Mstbd0 (D1 ) ∩ < N0 ι D (D) >o

such that k m − m(st0 ) ∈ Mstbd0 (D1 ) ∩ ϕst (J (N0 )M). 0

(7.71)

By (c) we have Hg,st0 (m(st0 )) = Hg,s (m(st0 )) for m ∈ Mstbd0 (D1 ). (e) To end the proof that Hg,st0 = Hg,s on Mstbd0 (D1 ) we use the Cs -uniform convergence of (Hg,s,J (N0 /s k N s −k ) )k . We fix systems of representatives J (N0 /(st0 )k N0 (st0 )−k ) and J (N0 /s k N s −k ), for any k ≥ 1. We also (2) choose a lattice D0 ⊂ D which is stable by −1 (L 0 ) ∩ L + and such that D1 ⊂ D0 . We recall that Mstbd0 (D1 ) is compact (Proposition 7.135 i)) and that Mstbd0 (D1 ) ⊂ Mstbd0 (D0 ) ⊂ Msbd (D0 ) by (b). For any open (N0 )submodule in the weak topology M0 ⊂ M, there exists a common constant (0) (0) k0 ≥ k g,s ≥ k g,st (by (c)) such that for k ≥ k0 , 0 Hg,st0 , J (N0 /(st0 )k N (st0 )−k ) ∈ Hg,st0 + E(Mstbd0 (D1 ), M0 ) Hg,s,J (N0 /s k N s −k ) ∈ Hg,s +

E(Mstbd0 (D1 ),

M0 ).

(7.72) (7.73)

On the left-hand side of (7.72), (7.73), we have continuous endomorphisms of M. By Lemma 7.114, there exists an integer k1 ≥ k0 such that they send k N0 ϕst10 (J (N0 )M) into M0 . Therefore, for m ∈ Mstbd0 (D1 ), they send the element m − m(st0 ) associated to k1 and J (N0 /((st0 )k1 N0 (st0 )−k1 ) as in (d) (7.71) into M0 hence Hg,st0 (m − m(st0 )) and Hg,s (m − m(st0 )) lie in M0 . By (d) we obtain that Hg,st0 (m) − Hg,s2 (m) lies in M0 for m ∈ Mstbd0 (D1 ). The statement follows since we chose M0 to be an arbitrary open neighbourhood of zero in the weak topology of M. Definition 7.146. We define the transitive relation s1 ≤ s2 on Z(L)†† generated by (2)

s1 = s2 t0 for t0 ∈ −1 (L 0 ) ∩ Z (L)† or s1r1 = s2r2 for positive integers r1 , r2 . Proposition 7.145 admits the following corollary. Corollary 7.147. Let s1 , s2 ∈ Z (L)†† . (i) When s1 ≤ s2 we have Msbd ⊆ Msbd and Hg,s1 = Hg,s2 on Msbd . 1 2 1

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(ii) When the relation ≤ on Z (L)†† is right filtered, we have Hg,s1 = Hg,s2 on Msbd ∩ Msbd . 1 2 Proof. (i) If s1 ≤ s2 then there exists, by definition, a sequence s1 = s1 ≤ s2 ≤ . . . ≤ sm = s2 in Z (L)†† such that each pair si , si+1 satisfies one of the two conditions in Definition 7.146. Hence we may assume, by induction, that the pair s1 , s2 satisfies one of these conditions, and we apply Proposition 7.145. (ii) When there exists s3 ∈ Z (L)†† such that s1 ≤ s3 and s2 ≤ s3 , by (i) Msbd 1 bd and H bd ∩ M bd . and Msbd are contained in M = H = H on M g,s g,s g,s s s s 1 2 3 2 3 1 2 Proposition 7.148. We assume that the relation ≤ on Z(L)†† is right filtered. Then, the intersection and the union

M∩bd := Msbd ⊂ M∪bd := Msbd s∈Z (L)††

s∈Z (L)††

are dense étale L + -submodules of M over (N0 ). For g ∈ N0 P N0 the endomorphisms Hg ∈ Endo (M∪bd ) equal to Hg,s on bd Ms for each s ∈ Z(L)†† , are well defined, stabilize M∩bd and satisfy the relations H1, H2, H3 of Proposition 7.72. Proof. M∩bd is an L + -submodule of M over (N0 ) by Proposition 7.130 and Remark 7.134. It is dense in M by Proposition 7.132 and Lemma 7.103. The action of L + on M∩bd is étale because M∩bd is L − -stable. When ≤ is right filtered, M∪bd is a (N0 )-module by Corollary 7.147(i). For the same reasons as for M∩bd , it is an étale L + -submodule of M over (N0 ). By Corollary 7.147 the Hg are well defined and stabilize M∩bd . They satisfy the relations H1, H2, H3 of Proposition 7.72 because the Hg,s satisfy them (Theorem 7.142). We summarize our results and give our main theorem. Theorem 7.149. For any s ∈ Z(L)†† , we have a faithful functor Ys : Met

(N0 ) (L + )

→

G-equivariant sheaves on G/P,

which associates to M ∈ Met

(N0 ) (L + ) the G-equivariant sheaf Ys on G/P such that Ys (C0 ) = Msbd . When the relation ≤ on Z (L)†† is right filtered, we have faithful functors Y∩ , Y∪ : Met

(N0 ) (L + )

→

G-equivariant sheaves on G/P,

which associate to M ∈ Met

(N0 ) (L + ) the G-equivariant sheaves Y∩ and Y∪ on G/P with sections on C0 equal to Y∩ (C0 ) = M∩bd and Y∪ (C0 ) = M∪bd .

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Proof. The existence of the functors results from Proposition 7.148, Theorem 7.142, Proposition 7.72, and Remark 7.69. We show the faithfulness of the functors. For a non zero morphism f : M → bd M in Met

(N0 ) (L + ), we have f (M∩ ) = 0 because f is continuous ([12] Lemma 8.22) and M∩bd containing (N0 )ι D (D) is dense (Proposition 7.148). We deduce Y∩ ( f ) = 0 since it is nonzero on sections on C0 . A fortiori Ys ( f ) = 0, and Y∪ ( f ) = 0.

10. Connected reductive split group We explain how our results apply to connected reductive groups. (a) Let F be a locally compact non archimedean field of ring of integers o F and uniformizer p F . Let G be a connected reductive F-group, let S be a maximal F-split subtorus of G and let P be a parabolic F-subgroup of G with Levi component L containing S and unipotent radical N . Let X ∗ (S) be the group of characters of S, let L , resp. , be the subset of roots of S in L, resp. G, and let +,N be the subset of roots of S in N (we suppress the index N if P is a minimal parabolic F-subgroup of G). Let s be any element of S(F) such that α(s) = 1 for α ∈ L and the p-valuation of α(s) ∈ F ∗ is positive for all roots α ∈ +,N . For any compact open subgroup N0 of N (F), the data (P(F), L(F), N (F), N0 , s) satisfy all the conditions introduced in the section on étale P+ -modules, Section 3 and Subsection 3.2, the assumptions introduced in Section 6, and in Section 9. (b) We suppose that P is a minimal parabolic F-subgroup. Let W ⊂ NG (L) be a system of representatives of the Weyl group N G (L)/L and let w0 be the longest element of the Weyl group. The data (G(F), P(F), W ) satisfy the assumptions of Section 5 on G-equivariant sheaves on G/P. (c) We suppose until the end of this chapter that F = Q p , G is Q p -split and P is a Borel Q p -subgroup. The Levi subgroup L = T of P is a split Q p -torus. The monoid of dominant elements and the submonoid without unit of strictly dominant elements are T (Q p )+ = {t ∈ T (Q p ), α(t) ∈ Z p for all α ∈ }, T (Q p )++ = {t ∈ T (Q p ), α(t) ∈ pZ p − {0} for all α ∈ }.

362

Peter Schneider, Marie-France Vigneras, and Gergely Zabradi With our former notation Z(L) = T (Q p ), Z (L)†† = T (Q p )++ . For each root α ∈ , let u α : Q p → Nα (Q p ), tu α (x)t −1 = u α (α(t)x) for x ∈ Q p , t ∈ T (Q p ), (7.74) be a continuous isomorphism from Q p onto the root subgroup Nα (Q p ) of N (Q p ) normalized by T (Q p ). We can write an element u ∈ N (Q p ) in the form u= u α (x α ) α∈+

for any ordering of + . The coordinates x α = xα (u) ∈ Q p of u are determined by the ordering of the roots, but for a simple root α, the coordinate xα : N (Q p ) → Q p

(7.75)

is independent of the choice of the ordering, and satisfies u α ◦ xα = 1. We suppose, as we can, that the u α have been chosen such that the product N0 = u α (Z p ) α∈+

is a group for some ordering of + . Then N0 is the product of the u α (Z p ) = Nα (Z p ) for any ordering of + . We choose a simple root α. We consider the continuous homomorphisms α : P(Q p ) → P (2) (Q p ), ια : N (Q p )(2) → N (Q p ), α ◦ ια = 1, defined by α(t) α (ut) := 0

xα (u) 1 (2) (2) , ια (u (x)) := u α (x) for u (x) := 1 0

x , 1

for t ∈ T (Q p ), u ∈ N (Q p ), x ∈ Q p . They satisfy the functional equation tια (y)t −1 = ια (α (t)yα (t)−1 ) for y ∈ N (Q p )(2) and t ∈ T (Q p ). The data (N0 , α , ια ) satisfies the assumptions introduced in Sections 8 and 9. We consider the binary relation s1 ≤ s2 on T (Q p )++ generated by s1 = s2 s0 with s0 ∈ T (Q p )+ , α(s0 ) ∈ Z∗p , or s1n = s2m with n, m ≥ 1. Lemma 7.150. The relation s1 ≤ s2 on T (Q p )++ is right filtered.

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Proof. Let = {α = α1 , . . . , αn }. The image of T (Q p )++ by A = (val p (αi (.))αi ∈ is contained in (N − {0})n and s1 ≤ s2 depends only on the cosets s1 T (Q p )0 and s1 T (Q p )0 , where T (Q p )0 = {t ∈ T (Q p ), α(t) ∈ Z∗p for all α ∈ }. (a) First we assume that, for any positive integer k, there exists s[k] ∈ T (Q p ) such that A(s[k] ) = (k, 1, . . . , 1). Then we have s[k] ≤ s[k+1] , and s ≤ s[k(s)] for s ∈ T (Q p )++ with k(s) = val p (α(s)). For any s1 , s2 in T (Q p )++ we deduce that s1 ≤ s[k(s1 )+k(s2 )] and s2 ≤ s[k(s1 )+k(s2 )] . Hence the relation ≤ on T (Q p )++ is right filtered. (b) When G is semi-simple and adjoint the dominant coweights ωα1 , . . . , ωαn for = {α = α1 , . . . , αn } form a basis of Y = Hom(Gm , T ), and A(T (Q p )++ ) = (N − {0})n . Hence s[k] exists for any k ≥ 1. (c) When G is semi-simple we consider the isogeny π : G → G ad from G onto the adjoint group G ad ([13] 16.3.5). The image Tad of T is a maximal split Q p -torus in G ad . The isogeny gives a homomorphism T (Q p ) → Tad (Q p ), inducing an injective map between the cosets T (Q p )++ /T (Q p )0 → Tad (Q p )++ /Tad (Q p )0 respecting ≤, and such that for any tad ∈ Tad (Q p ) there exists an inten ger n ≥ 1 such that tad ∈ π(T (Q p )). Given s1 , s2 ∈ T (Q p )++ there exists sad ∈ Tad (Q p )++ such that π(s1 ), π(s2 ) ≤ sad by (b) and (a). Let n n n ≥ 1 such that sad = π(s3 ) for s3 ∈ T (Q p ). We have sad ≤ sad hence π(s1 ), π(s2 ) ≤ π(s3 ). This is equivalent to s1 , s2 ≤ s3 . (d) When G is reductive let π : G → G = G/Z 0 be the natural Q p homomorphism from G to the quotient of G by its maximal split central torus Z 0 . The group G is semi-simple, π(T ) = T is a maximal split Q p -torus in G , π |T gives an exact sequence 1 → Z 0 (Q p ) → T (Q p ) → T (Q p ) → 1, inducing a bijective map between the cosets T (Q p )++ /T (Q p )0 → T (Q p )++ /T (Q p )0 respecting ≤. By (c), ≤ is right filtered on T (Q p )++ . We deduce that ≤ is right filtered on T (Q p )++ . By Theorem 7.117 and Theorem 7.149, we can associate functorially to an étale T+ -module D over OE ,α different sheaves: • For any s ∈ T++ , a G(Q p )-equivariant sheaf Ys on G(Q p )/P(Q p ) with sections on C0 equal to M(D)bd s .

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• The G(Q p )-equivariant sheaves Y∩ and Y∪ on G(Q p )/P(Q p ) with secbd tions on C0 equal to ∩s∈T++ M(D)bd s and ∪s∈T++ M(D)s . In general M(D) is different from ∪s∈T++ M(D)bd s , by the following proposition. Proposition 7.151. Let M be an étale T+ -module M over α (N0 ). When the root system of G is irreducible of positive rank r k(G), we have: (i) If r k(G) = 1, the G(Q p )-equivariant sheaf on G(Q p )/P(Q p ) with sections Msbd over C0 does not depend on the choice of s ∈ T++ , and M = Msbd . (ii) If r k(G) > 1, a G(Q p )-equivariant sheaf of o-modules Y on G(Q p )/P(Q p ) such that Y(C0 ) ⊂ M and (u α (1) − 1) is bijective on Y(C0 ), is zero. Proof. We prove (i). If r k(G) = 1, then OE = α (N0 ) and M = D is an étale T+ -module over OE . With the same proof as in Proposition 7.96, we have Msbd = M for any s ∈ T++ and the integrals Hg for g ∈ N0 P N0 do not depend on the choice of s. (ii) is equivalent to the property: an étale o[P+ ]-submodule M of M which is also a R = o[N0 ][(u α (1) − 1)−1 ]-submodule of M, and is endowed with endomorphisms Hg ∈ Endo (M), for all g ∈ N0 P(F)N0 , satisfying the relations H1, H2, H3 (Proposition 7.72), is 0. (a) Preliminaries. As r k(G) ≥ 2 and the root system is irreducible, there exists a simple root β such that α + β is a root. The elements n α := u α (1) and n β := u β (1) do not commute. By the commutation formulas, n α n β = n β n α h for some h = 1 in the group H = γ Nγ (Z p ) for all positive roots of the form γ = i α + jβ ∈ + with i, j > 0. Note that H is normalized by Nα (Z p ). Let s ∈ T++ . We have the expansion (7.12) (n α h − 1)−k = uϕs (ψs (u −1 (n α h − 1)−k )) (7.76) u∈J (Nα (Z p )H/s Nα (Z p )H s −1 )

in R. We choose, as we can, a lift wβ of sβ in the normalizer of T (Q p ) such that – wβ n β ∈ n β P(Q p ) – wβ normalizes the group N+ −β (Z p ) = γ Nγ (Z p ) for all positive roots γ = β. The subset Nβ (Z p ) ⊂ Nβ (Z p ) of u β (b) such that wβ u β (b) ∈ u β (Z p )P(Q p ), contains n β but does not contain 1. The subset Uwβ ⊂ N0 of u such that wβ u ∈ N0 P(Q p ) is equal to

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Uwβ = Nβ (Z p )N+ −β (Z p ) = N+ −β (Z p )Nβ (Z p ). Hence Uwβ = uUwβ , i.e. wβ−1 C0 ∩C0 = uwβ−1 C0 ∩C0 , for any u ∈ N+ −β (Z p ). (b) Let M be an R = o[N0 ][(n α − 1)−1 ]-module of M, which is also an étale o[P+ ]-submodule, and is endowed with endomorphisms Hg ∈ Endo (M), for all g ∈ N0 P(F)N0 , satisfying the relations H1, H2, H3 (Proposition 7.72), and let m ∈ M be an arbitrary element. We want to prove that m = 0. The idea of the proof is that, for s ∈ T++ , we have m = 0 if Hwβ (n β ϕs (m)) = 0 and that Hwβ (n β ϕs (m)) = 0 because it is infinitely divisible by n γ − 1, where γ = sβ (α). An element in M with this property is 0 because n γ − 1 lies in the maximal ideal of α (N0 ). Let a ∈ Z p . The product formula in Proposition 7.84ii implies Hwβ ◦ Hn aα ◦ res(1w−1 C β

0 ∩C0

) = Hwβ n aα ◦ res(1w−1 C β

0 ∩ C0

)=

Hn aγ wβ ◦ res(1w−1 C0 ∩C0 ) = Hn aγ ◦ Hwβ ◦ res(1w−1 C0 ∩C0 ) β

β

−1 −1 −1 −a since n −a α wβ C0 ∩ C0 = wβ C0 ∩ C0 = wβ n γ C0 ∩ C0 . For all k ∈ N, the elements

m k :=(n α − 1)−k n β ϕs (m) = n β (n α h − 1)−k ϕs (m)

(7.77)

lie in the image of the idempotent res(1w−1 C0 ∩C0 ) ∈ Endo (M), because β

mk =

n β uϕs (ψs (u −1 (n α h − 1)−k m))

u∈J (Nα (Z p )H/s Nα (Z p

(7.78)

)H s −1 )

by (7.76), (7.77). Therefore the product relations between Hwβ , Hn aα and Hn aγ imply Hwβ (n β ϕs (m)) = Hwβ ((n α − 1) m k ) = k

k a=0

=

k

k a=0

k Hwβ ◦ Hn aα (m k ) a

(−1)

k Hwβ ◦ Hn aα ◦ res(1w−1 C ∩C )(m k ) 0 0 β a

(−1)k−a

k Hn aγ ◦ Hwβ ◦ res(1w−1 C0 ∩C0 )(m k ) β a

k−a

a=0

=

(−1)

k−a

= (n γ − 1)k Hwβ (m k ). Hence Hwβ (n β ϕs (m)) = 0 since it is infinitely divisible by n γ − 1 which lies in the maximal ideal of α (N0 ). We also have

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n β ϕs (m) = H1 ◦ res(1w−1 C0 ∩C0 )(n β ϕs (m)) = Hwβ ◦ Hwβ (n β ϕs (m)) = 0. β

As n β ◦ ϕs ∈ Endo

(M )

is injective, we deduce m = 0.

Corollary 7.152. There exists a G(Q p )-equivariant sheaf on G(Q p )/P(Q p ) with sections M on C0 if and only if rk(G) = 1.

References [1] [2] [3] [4] [5] [6] [7]

[8] [9] [10] [11] [12]

[13] [14] [15] [16]

Bourbaki N., Algèbre commutative. Ch. 1 à 4. Masson 1985. Bourbaki N., Topologie générale. Ch. 1 à 4. Hermann 1971. Bourbaki N., Topologie générale. Ch. 5 à 10. Hermann 1974. Bosch S., Güntzer U., Remmert R., Non-Archimedean analysis. Springer 1984. Colmez P., (ϕ, )-modules et représentations du mirabolique de G L 2 (Q p ). Astérisque 330, 2010, 61–153. Colmez P., Représentations de G L 2 (Q p ) et (ϕ, )-modules. Astérisque 330, 2010, 281–509. Dixon J. D., du Sautoy M. P. F., Mann A., Segal D., Analytic pro- p groups. Second edition. Cambridge Studies in Advanced Mathematics, 61. Cambridge University Press, 1999. Ellis R., Locally compact transformation groups. Duke Math. J. 24, 1957 119–125. Gabriel P., Des catégories abéliennes. Bull. Soc. Math. France 90, 1962, 323–448. Kedlaya K., New methods for (ϕ, )-modules, preprint (2011), http://math.mit. edu/kedlaya/papers/new-phigamma.pdf Schneider P., p-Adic Lie groups. Springer Grundlehren, Vol. 344, Springer, 2011. Schneider P., Vigneras M.-F., A functor from smooth o-torsion representations to (ϕ, )-modules. Volume in honour of F. Shahidi. Clay Mathematics Proceedings, Volume 13, 525–601, 2011. Springer T. A., Linear Algebraic Groups. Second edition. Birkhäuser, 2009. Vigneras M.-F., Représentations -modulaires d’un groupe réductif p-adique avec = p. PM 137, Birkhäuser, 1996. Warner S.: Topological rings. Elsevier, 1993. Zábrádi G., Exactness of the reduction of étale modules. J. Algebra 331, 2011, 400–415.

8 Intertwining of ramified and unramified zeros of Iwasawa modules Chandrashekhar Khare and Jean-Pierre Wintenberger

1. Introduction Let F be a totally real field, p > 2 a prime, F∞ ⊂ F(μ p∞ ) the cyclotomic Z p -extension of F with Galois group = Z p = γ . This short note is a sequel to [5]. The sole aim is to point out the ubiquity of a phenomenon dicussed in a particular case in our previous paper. Namely, the (arithmetic) eigenvalues of γ acting on Galois groups of maximal p-abelian unramified extensions of F(μ p∞ ) intertwine with the eigenvalues acting on inertia subgroups of ramified p-abelian extensions of F(μ p∞ ). We make this vague philosophy precise in the text below after alluding for mise-en-scène to the p-adic L-functions that lurk suggestively in the wings, but do not play an explicit role in the algebraic computations of this note. Let ψ be an even Dirichlet character of F. Consider the p-adic L function ζ F, p (s, ψ) = L p (s, ψ), s ∈ Z p , which is characterised by the interpolation property L p (1 − n, ψ) = L(1 − n, ψω−n )v| p (1 − ψω−n (v)N (v)n−1 ), for n ≥ 1 a positive integer. When ψ = 1, we denote the corresponding L-function by ζ F, p (s). It is known that L p (s, ψ) is holomorphic outside s = 1, is holomorphic everywhere when ψ = 1, and otherwise has at most a simple pole at s = 1. This pole is predicted to exist by the conjecture of Leopoldt, which asserts the non-vanishing of the the p-adic regulator of units of F. The residue of ζ F, p at s = 1 has been computed by Pierre Colmez: Ress=1 ζ F, p (s) =

2d R F, p h F , √ 2 DF

CK was partially supported by NSF grants. JPW is member of the Institut Universitaire de France. Automorphic Forms and Galois Representations, ed. Fred Diamond, Payman L. Kassaei and c Cambridge University Press 2014. Minhyong Kim. Published by Cambridge University Press.

367

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where R F, p is the p-adic regulator for F. The non-vanishing of R F, p is the Leopoldt conjecture for F and p. We recall a folklore conjecture that is a very particular case of the general conjectures of Jannsen ([4]) about the non-vanishing of higher regulators. Conjecture 8.1. (Non-vanishing of higher p-adic regulators) For an integer m = 0, L F, p (m, ψ) = 0 if either m = 1 or ψ = 1. Furthermore, ζ F, p (s) has a pole at s = 1. As a supplement in the case m = 0, the case of “trivial zeros”, the multiplicity of the zero at s = 0 of L p (s, ψ) is conjectured to be given by the number of v| p such that (1 − ψω−1 (v)) = 0. We call the zeros of ζ F, p (s) unramified zeros following a similar usage in [8]. The main conjecture of Iwasawa theory realises the zeros of the L-functions as roots of the characteristic polynomial of γ acting on “unramified arithmetic spaces”. Nevertheless it gives no direct information about the zeros, belying the Hilbert–Polya philosophy in this case! Our basic observation, coming from [5] that we reinforce here, is that the unramified zeros (of the p-adic L-function) always intertwine with (the eigenvalues of γ acting on) ramification at p. Further the Leopoldt zeros, i.e. eigenvalues of that correspond on a finite index subgroup to its action on p-power roots of unity, intertwine with ramification at Q, for finite sets of primes away from p, for a generic choice of Q. This shows that the non-vanishing of p-adic regulators is equivalent to splitting of ramification in naturally occuring exact sequences of Iwasawa modules. The reader is referred to Theorems 8.5, 8.9 and 8.10 for precise statements. The proofs of these theorems rely on numerical coincidences between – dimensions of certain Galois cohomology groups whose computation result from the work of Soulé and Poitou–Tate duality, – dimensions of Iwasawa modules that follow from theorems of Iwasawa describing the structure of inertia at p (resp. at a set Q of auxiliary primes) in the Galois group of the maximal odd abelian p-extension of F(μ p∞ ) that is unramified outside p (resp. Q).

2. Galois cohomology 2.1. In our previous paper [5] we paid attention to integral questions, while here we work over Q p exclusively. We consider F a totally real field, an odd prime p.

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We consider a sufficiently large finite extension K of Q p with ring of integers O (that will contain values of the character under consideration). Let S p be the set of places of F above p and ∞ and let S be a finite subset of places of F containing S p . Let G S be a Galois group of the maximal algebraic extension of F unramified outside S. We consider a potentially crystalline, or arithmetic, character χ of G S , of (parallel) weight m and thus of the form ηχ pm where χ p is the p-adic cyclotomic character, and η a finite order character. We impose that χ (c) is independent of the choice of complex conjugation c ∈ G F , and call it odd or even according as this value is either −1 or 1. We denote by ω the Teichmüller character. We consider the cohomology subgroup H(1p) f (G S , K (ηχ pm )) of H 1 (G S , K (ηχ pm )) defined by imposing for v ∈ S, v ∈ / S p the condition to be unramified (although χ might be ramified at these v). We denote by H 1f (G S , K (ηχ pm )) their Bloch–Kato subgroups, where for v primes over p, we impose the Bloch–Kato finiteness condition ([2]). We denote the dimensions over K of H(1p) f (G S , K (ηχ pm )) and H 1f (G S , K (ηχ pm )) by h 1 (χ ) and h 1f (χ ). We use h 1split (χ ) to denote dimensions of cohomology groups where we ask that the classes are split locally at all places above p and unramified at other primes. We have the tables.

2.2. Odd χ χ = ηχ pm , χ odd h 1 (χ ) h 1f (χ )

m>1 d d

d+

v| p

m=1 −1 v , η ) − δχ ,χ p d − δχ ,χ p

h 0 (G

m≤0 d 0

2.3. Even χ χ = ηχ pm , χ even h 1 (χ ) h 1f (χ )

m>1 0 0

m=1 0 (G , η−1 ) h v v| p 0

m≤0 δχ ,id + h 1split (χ p χ −1 ) 0

Remark. The situation for even χ is thus not satisfactory as we do not have an explicit formula in all cases for h 1 (χ ). The conjectures in [4], concerning non-vanishing of higher p-adic regulators, predict the vanishing of h 1split (χ p χ −1 ) for χ an even arithmetic character of G F (see also [1] 5.2). This follows from the main conjecture of Iwasawa theory if the weight of χ is > 0.

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2.4. Ingredients of the computation We justify the values in the tables. We need the following ingredients: • (Bloch–Kato) h 1f (χ ) = dim K K (χ )G F + ords=1−m L(η−1 , s). • (global duality) χ even: h 1 (χ ) = h 1split (χ p χ −1 ) + δχ ,id +

h 0 (G v , χ p χ −1 ).

v| p

• (global duality) χ odd: h 1 (χ ) = d + h 1split (χ p χ −1 ) − δχ ,χ p +

h 0 (G v , χ p χ −1 ).

v| p

The (global duality) equalities follow from Theorem 8.7.9. of [6] and: • (local Euler Poincaré characteristic) h 1 (G v , V ) = h 0 (G v , V ) + h 2 (G v , V ) + [Fv : Q p ]dim K (V ) where v| p. • (local duality) h 0 (G v , V ∗ (1)) = h 2 (G v , V ). The Bloch–Kato formula, which directly implies the bottom row of both tables, follows from a theorem of Soulé (Theorem 1 of [7]) and duality as we justify. – The theorem of Soulé directly implies Bloch–Kato formula for h 1f (χ ) for m > 1. – The case of m = 1 for the bottom rows follows from Kummer theory. – We have the equality for m ≤ 0 and χ even: h 1f (χ ) = h 1f (χ p χ −1 ) − d + δχ ,id , and for m ≤ 0 and χ odd: h 1f (χ ) = h 1f (χ p χ −1 ). These equalities follow from Theorem 8.7.9 of [6] and the fact that for m ≤ 0, 1 (G , χ ). These two H 1f (G v , χ ) coincides with the unramified cohomology Hur v equalities allow us to deduce the case m ≤ 0 of the Bloch–Kato formula from the Theorem of Soulé (see also [1], §4.3.1). This checks the second rows of both the tables.

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Let us check the first row. For the first column in the case m > 1, we use the fact that h 1 (G v , χ ) = h 1f (G v , χ ). The case m ≤ 0 even χ follows from (global duality) as h 0 (G v , χ p χ −1 ) = 0 for all v above p. For the case m = 1, we use global duality and that h 1split (η−1 ) = 0 which follows easily from the fact that a Z p -extension of a number field has to be ramified at a place above p. For χ odd and m ≤ 0, we use global duality and the fact that h 1split (χ p χ −1 ) ≤ h 1f (χ p χ −1 ) = 0.

2.5. Galois groups We consider ε totally odd and ψ totally even finite order characters of G F , such that εψ = ω. We set ψ(n) = ψ(ω−1 χ p )n := ψκ n , and likewise for ε(n). We consider characters ψζ of that send a chosen generator γ of to a p-power root of unity ζ and consider ψ(n)ψζ , ε(n)ψζ . We assume that ψ is of type S, i.e. we assume that the field Fψ cut out by ψ is linearly disjoint from the cyclotomic Z p -extension F∞ of F. We denote by the Galois group of F∞ /F with choice of generator γ , consider

the completed group algebra Z p [[]] that is isomorphic to Z p [[T ]] via the homomorphism which sends γ → 1 + T . We let K = ⊗ K . We consider the Galois group Gψ = Gε = Gal(Fψ (μ p )/F) × Gal(F∞ /F) of Fψ (μ p∞ ) over F. A continuous character χ of Gψ with values in O∗ is the product of a finite character χψ of Gal(Fψ (μ p )/F) and a character χ of . The character χ induces a K -algebra map K → K . We denote by Pχ the corresponding prime ideal of K kernel of this map. It is generated by γ − χ (γ ). The character χψ induces a morphism f χ : Z p [Gal(Fψ (μ p )/F)] → O ⊂ K . The morphism Z p [[Gψ ]] → K induced by χ is the composite of the map Z p [[Gψ ]] → K induced by f χ and the morphism K → K . We consider the maximal, abelian pro- p extension L∞ of Fψ (μ p∞ ) unram and set X ified everywhere: we denote its Galois group by X ∞ ∞ = X∞ ⊗ K . We set X ∞,ε to be the maximal quotient of X ∞ on which Gal(Fψ (μ p )/F) acts by ε. We also consider the analogous extensions L∞,ε (Q) and corresponding Galois group X ∞,ε,Q when Q is any set of places of F and we allow ramification above Q. There are two natural cases to consider: • Q is all the places above p, and then we replace Q by p in the notation; • Q is a finite set of places disjoint from the places above p. Let Fψ,∞ be the cyclotomic Z p -extension of the totally real field Fψ . We the Galois group of the maximal abelian prop- p extension of denote by Y∞

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. As above, we denote Fψ,∞ that is unramified outside p, and Y∞ = Q p ⊗ Y∞ by Y∞,ψ the related quotient on which Gal(Fψ (μ p )/F) acs by ψ, and recall the perfect -equivariant Iwasawa pairing

Y∞,ψ × X ∞,ε → K (1). Lemma 8.2. Let ψ be an even character of G F of type S and n ∈ Z. (1) We have h 1 (ψζ ψ(n)) = dim K (Y∞,ψ /Pψζ ψ(n) ) if ψζ ψ(n) is not trivial, and h 1 (ψζ ε(n)) = dim K (X ∞,ε, p /Pψζ ε(n) ). (2) We have that dim K ((X ∞,ε ) Pψζ ε(−n) /Pψζ ε(−n) ) = dim K ((Y∞,ψ ) P

ψζ−1 ψ(n+1)

/Pψ −1 ψ(n+1) ). ζ

Proof. For (i) we use the inflation-restriction sequence relative to G S → Gψ . Recall that the unramified condition for η ∈ H(1p) f (G S , K (εχ pm )) at v not above p and such that ψ is ramified at v (Section 2.1). We check that this condition is equivalent to that the restriction of η to the kernel of the map G S → Gψ is unramified at v. For (ii), we invoke the pairing of Iwasawa. We use the cyclicity of to identify dimensions of twisted invariants and covariants for . We rederive a standard result about trivial zeros of p-adic L-functions, usually proved using genus theory, which is a corollary to the lemma. Proposition 8.3. Suppose ε is an odd character of Gal(Fψ (μ p )/F) as before. Then dim K ((X ∞,ε ) Pψζ ε /Pψζ ε ) = h 1 (χ p ψζ−1 ε−1 ) = h 0 (G v , ψζ−1 ε−1 ). v| p

Proof. We deduce the first equality using (2) of Lemma 8.2 for n = 0 and the second equality using the above table for m = 1 even χ. Note that v| p h 0 (G v , ψζ−1 ε−1 ) is the number of places v| p such that the

Euler factor (1−ψζ−1 ε−1 (v)) is 0. It is conjectured by Greenberg that the trivial zeros occur semisimply i.e. dim K (X ∞,ε /Pψζ ε ) = dim K (X ∞,ε ) Pψζ ε .

3. Main conjecture and higher regulators We consider the Deligne–Ribet p-adic L-function ζ F, p (s, ψ) for an even character ψ of F of type S. It is defined on Z p when ψ is non-trivial, and on

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373

Z p \{1} when ψ is trivial. It is characterised by the interpolation formula that for integers n ≥ 1, ζ F, p (1 − n, ψ) = L p (1 − n, ψω−n ), where the superscript denotes that we have dropped the Euler factors at p. There is a power series Wψ (T ) in K = Z p [[]] ⊗ K (the latter by the isomorphism that sends a chosen generator γ of = Gal(F∞ /F) to 1 + T and u := χ p (γ )), with the property that: Wψ (u s − 1) = ζ F, p (s, ψ) u 1−s − 1 when ψ is trivial, and Wψ (u s − 1) = ζ F, p (s, ψ) otherwise. Furthermore we have: Wψψζ (T ) = Wψ (ζ −1 (1 + T ) − 1); see the introduction of [8], where the notation is G ψ (T ) for Wψ (u (1 + T )−1 − 1). Then the main conjecture asserts for characters ψ of type S that (Wψ (T )) = char K (X ∞,ψ −1 ω ), i.e. the characteristic polynomial of the action of γ on the finite dimensional K -vector space X ∞,ε generates the same ideal as Wψ (T ). (We ignore μ-invariants via this formulation.) When m is an integer = 0, Conjecture 8.1 is equivalent via the main conjecture to the statement that the generalised ζ u m -eigenspace of X ∞,ψ −1 ω ⊗ Q p for the action of γ is trivial . When m = 0, via the main conjecture, we have that the generalised ζ -eigenspace is of dimension given by the number of v| p of F such that (1 − ψζ−1 ε−1 (v)) = 0.

4. Intertwining of ramified and unramified zeros 4.1. Ramification at p We consider the exact sequences: 0 → I Q → X ∞,ε,Q → X ∞,ε → 0, of finitely generated K -modules with Q = p (see 2.5).

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4.1.1. Ramification at p and intertwining with non-trivial unramified zeros Next theorem follows from Th. 25 of [3] : Theorem 8.4. We have an ismorphism of K -modules ) s { dK ⊕ j =1 IndG ℘ K (1)} j Ip = , K (1) where G ℘ are the decomposition groups of the places ℘ above p in . We deduce from this and the computations in Galois cohomology earlier: Theorem 8.5. Let χ = εψζ κ m be an odd arithmetic character of Gε of weight m, and let Pχ be the corresponding prime ideal of O for O such that χ is valued in O ∗ . Then the exact sequence 0 → (I p ) Pχ → (X ∞,ε, p ) Pχ → (X ∞,ε ) Pχ → 0, of finitely generated K -modules splits if and only if (X ∞,ε ) Pχ vanishes. This is equivalent to that ζ F, p (m, ψζ−1 ε−1 ω) = 0 when χ = χ p , and when χ = χ p , to that ζ F, p (s) has a pole at s = 1. Proof. One direction is trivial. For the other direction, one notes by the theorem of Iwasawa that dim K ((I p ) Pχ /Pχ ) is d if m = 1 and d + 0 −1 v| p h (G v , ε ) − δχ ,χ p if m = 1. We have the numerical coincidence: dim K ((I p ) Pχ /Pχ ) = h 1 (χ ). Further one notes from Lemma 8.2 that h 1 (χ ) = dim K (X ∞,ε, p ) Pχ /Pχ . Thus if the exact sequence in the theorem splits, we deduce that (X ∞,ε ) Pχ /Pχ = 0, which is equivalent to the vanishing of (X ∞,ε ) Pχ . By the main conjecture proved by Wiles, the vanishing of (X ∞,ε ) Pχ /Pχ is equivalent to the vanishing of Wψ at ζ u m − 1. It is equivalent to the vanishing of ζ p (m, ψψζ−1 ) = ζ F, p (m, ψζ−1 ε−1 ω). Remarks. 1. We also deduce that an equivalent formulation of the non-vanishing of the higher regulator conjecture is the conjecture that the exact sequence 0 → (I p ) Pχ → (X ∞,ε, p ) Pχ → (X ∞,ε ) Pχ → 0, of finitely generated K -modules splits for all odd arithmetic characters χ of G of non-zero weight. 2. As trivial zeros (in weight 0) do occur we get examples of Iwasawa modules in which ramification is allowed at p such that γ acts non-semisimply.

Ramified and unramified zeros

375

We have a conditional result for any character χ = εκ s of Gε not necessarily arithmetic, and that follows by similar arguments. Proposition 8.6. Let χ = εκ s be any odd character of Gε . Assume further that 1 (G −1 Hsplit F, p , χ p χ ) is trivial. Then the exact sequence 0 → (I p ) Pχ → (X ∞,ε, p ) Pχ → (X ∞,ε ) Pχ → 0, of compact finitely generated K -modules splits if and only if (X ∞,ε ) Pχ = 0. 1 (G −1 Remark. Note that the vanishing of Hsplit F, p , χ p χ ) = 0 is predicted by Greenberg’s conjecture that the p-part of the class group of the cyclotomic Z p -extension F∞ of F is finite. The above proposition suggests that there may be a formulation of the main conjecture using Ext K (X ∞,ε , I p ).

4.2. Ramification away from p 4.2.1. ψ and ψζ trivial Consider the maximal abelian p-extension L ∞ of F , and denote its Z p -rank by 1 + δ. The Leopoldt conjecture asserts that δ = 0. Definition 8.7. (generic sets Q) We say that a finite set of primes Q of cardinality r away from p is generic if the rank r Q of the subgroup generated by the Frobq ’s for q ∈ Q in Gal(L ∞ /F) is min(r, 1 + δ). The terminology is meant to reflect the fact that when δ > 0, the Frobq1 , Frobq2 will be linearly independent in Z1+δ = Gal(L ∞ /F) for most p choices of q1 , q2 . If r = 2 and we choose a prime q1 freely, then for a density one set of primes q2 , the set Q = {q1 , q2 } is generic. We now show the intertwining of the Leopoldt zero u = u γ with the ramification at Q provided Q is a generic set of primes with |Q| > 1. Proposition 8.8. Let Q be a finite set of primes of F away from p. Then the subgroup I Q of X ∞,ω,Q generated by the conjugacy class of the inertia groups Iq for q ∈ Q is isomorphic as a Gal(F∞ /F) = Gal(F(μ p )/F) ×-module to (q∈Q IndG q K (1)) K (1) where G q is the decomposition subgroup at q in = Gal(F∞ /F), and where we declare that Gal(F(μ p )/F ) acts by ω. Proof. This follows from class field theory. Note that when the primes q are inert in F∞ /F then I Q = K (1)r−1 .

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Theorem 8.9. Let Q be generic set of primes Q of cardinality r ≥ 2. Then the exact sequence 0 → (I Q ) Pχ p → (X ∞,ω,Q ) Pχ p → (X ∞,ω ) Pχ p → 0 splits if and only if (X ∞,ω ) Pχ p = 0, i.e., if and only if the Leopoldt conjecture is true. Thus if there is a Leopoldt zero, then it intertwines with the ramified zeros at Q for a generic set of primes (away from p) with |Q| ≥ 2. Proof. If the Leopoldt conjecture is true, (X ∞,ω ) Pχ p = 0 and the exact sequence splits. Let us prove the converse. For a finite dimensional vector space V over K endowed with a continuous action of G F that is unramified outside a finite set of places, and a set of Selmer conditions L = {Lv } for Lv ⊂ H 1 (Fv , V ) where Lv is outside a finite set of places the unramified subgroup, we have the formula : h 1L (F, V ) − h 1L⊥ (F, V ∗ (1)) =h 0 (F, V ) − h 0 (F, V ∗ (1)) + (dim K Lv − h 0 (Fv , V )). v

We apply this formula for V = K (1), and with the Selmer conditions L to be unramified everywhere. In particular the Selmer condition is trivial at places v above p as V Iv = 0 for v above p. We get: h 1L (F, V ) − h 1L⊥ (F, V ∗ (1)) = −1. Furthermore, we have h 1L⊥ (F, K ) = 1 + δ. Consider the Selmer conditions L Q that arise when we allow ramification at Q, i.e., (L Q )v = Lv for v ∈ / Q and (L Q )v = H 1 (G v , K (1)) for v ∈ Q. We get: h 1L Q (F, V ) − h 1L⊥ (F, V ∗ (1)) = −1 + r. Q

Furthermore, we have h 1L⊥ (F, K ) = 1 + δ − r Q . We see that: Q

h 1L Q (F, K (1)) = h 1L (F, K (1)) + r − r Q . If the exact sequence splits, it remains exact after reduction modulo Pχ p , hence we have: h 1L Q (F, K (1)) = h 1L (F, K (1)) + r − 1.

Ramified and unramified zeros

377

Thus we get r Q = 1. As Q is generic and | Q |≥ 2, i.e., r Q = min(r, 1 + δ) with r ≥ 2, we get that 1 + δ = 1, thus δ = 0 and the Leopoldt conjecture is true. 4.2.2. Weight 1, ψ or ψζ non-trivial Theorem 8.10. Consider the exact sequence 0 → (Iq ) Pψζ εκ → (X ∞,ε,q ) Pψζ εκ → (X ∞,ε ) Pψζ εκ → 0. Assume ε = ω or that ζ = 1. Then the sequence splits for all choices of primes q of F away from p, if and only if (X ∞,ε ) Pψζ εκ = 0. Proof. We only sketch the proof as its very similar to the proof of Theorem 8.9. By (2) of Lemma 8.2, if (X ∞,ε ) Pψζ εκ = 0, the maximal abelian p-extension L of Fε−1 ψ −1 ω = Fψψ −1 on which Gal(Fψψ −1 /F) acts by ψψζ−1 and which ζ

ζ

ζ

is unramified outside p, has Galois group that is of positive rank as a Z p module. Let q be a prime of F away from p, that splits in Fψψ −1 , such ζ

that the Frobenius at a prime above q of Fψψ −1 is a non-torsion element in ζ

Gal(L/Fψψ −1 ). With V = K (ψζ εκ) and the Selmer conditions as above, it ζ

follows that h 1L⊥ − h 1L⊥ = 1. q

Using the above formula, we get h 1Lq − h 1L⊥ − h 1L + h 1L⊥ = 0. If the exact q

sequence were to split, we would have h 1Lq = h 1L , which is a contradiction.

References [1] Joël Bellaiche. An introduction to the conjecture of Bloch and Kato. Lectures at the Clay Mathematical Institute summer School, Honolulu, Hawaii , 2009. [2] Spencer Bloch, Kazuya Kato. L-functions and Tamagawa numbers of motives. The Grotendieck Festschrift, vol. 1, 333-400, Prog. Math. 86, Birkhäuser Boston, Boston, MA, 1990. [3] Kenkichi Iwasawa. On Z -extensions of algebraic number fields. Ann. of Math. (2) 98 (1973), 246–326. [4] Uwe Jannsen. On the -adic cohomology of varieties over number fields and its Galois cohomology, in Galois groups over Q (Berkeley, CA, 1987), 315–360, Math. Sci. Res. Inst. Publ., 16, Springer, New York, 1989. [5] Chandrashekhar Khare, Jean-Pierre Wintenberger. Ramification in Iwasawa modules and splitting conjectures. To appear in International Mathematics Research Notices.

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[6] Jürgen Neukirch. Cohomology of number fields. 2nd edition. Grundlehren des mathematischen Wissenschaften, 323, Springer, 2008. [7] Christophe Soulé. On higher p-adic regulators, in Algebraic K -theory. Evanston 1980 Lecture Notes in Math., 854, Springer, 1981. [8] Andrew Wiles. The Iwasawa conjecture for totally real fields. Ann. of Math. (2) 131 (1990), no. 3, 493–540.

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London Mathematical Society Lecture Note Series: 415

Automorphic Forms and Galois Representations Volume 2

Edited by

FRED DIAMOND King’s College London PAY MA N L . KASSAEI McGill University, Montréal M I N H YO N G K I M University of Oxford

University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107693630 c Cambridge University Press 2014 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 Printed in the United Kingdom by CPI Group Ltd, Croydon CR0 4YY A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Automorphic forms and galois representations / edited by Fred Diamond, King’s College London, Payman L. Kassaei, McGill University, Montréal, Minhyong Kim, University of Oxford. volumes cm. – (London Mathematical Society lecture note series ; 414, 415) Papers presented at the London Mathematical Society, and EPSRC (Great Britain Engineering and Physical Sciences Research Council), Symposium on Galois Representations and Automorphic Forms, held at the University of Durham from July 18–28, 2011. ISBN 978-1-107-69192-6 (v. 1) – ISBN 978-1-107-69363-0 (v. 2) 1. Automorphic forms–Congresses. 2. Automorphic functions–Congresses. 3. Forms (Mathematics)–Congresses. 4. Galois theory–Congresses. I. Diamond, Fred, editor of compilation. II. Kassaei, Payman L., 1973– editor of compilation. III. Kim, Minhyong, editor of compilation. IV. Symposium on Galois Representations and Automorphic Forms (2011 : Durham, England) QA353.A9A925 2014 515 .9–dc23 2014001841 ISBN 978-1-107-69363-0 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

List of contributors Preface

page vi vii

1 On the local structure of ordinary Hecke algebras at classical weight one points Mladen Dimitrov 2 Vector bundles on curves and p-adic Hodge theory Laurent Fargues and Jean-Marc Fontaine

1 17

3 Around associators Hidekazu Furusho

105

4 The stable Bernstein center and test functions for Shimura varieties Thomas J. Haines

118

5 Conditional results on the birational section conjecture over small number fields Yuichiro Hoshi

187

6 Blocks for mod p representations of GL2 (Q p ) Vytautas Pašk¯unas

231

7 From étale P+ -representations to G-equivariant sheaves on G/P Peter Schneider, Marie-France Vigneras, and Gergely Zabradi

248

8 Intertwining of ramified and unramified zeros of Iwasawa modules Chandrashekhar Khare and Jean-Pierre Wintenberger

367

v

Contributors

Mladen Dimitrov, UFR Mathématiques, Université Lille 1, Lille, 59655 Villeneuve d’Ascq Cedex, France. Laurent Fargues, CNRS, Institut de Mathématiques de Jussieu, Paris, 75252, France. Jean-Marc Fontaine, Laboratoire de Mathématiques d’Orsay, Université Paris Sud, Paris 91405 Orsay, France. Hidekazu Furusho, Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan. Thomas J. Haines, Department of Mathematics, Univeristy of Maryland, College Park, MD 20742-4015, USA. Yuichiro Hoshi, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan. Chandrashekhar Khare, Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA. Vytautas Pašk¯unas, Fakultät für Mathematik, Universität Duisburg–Essen, Essen 45127, Germany. Peter Schneider, Mathematischen Instituts, Westfälische Wilhelms-Universität Münster, 48149 Germany. Marie-France Vignéras, Institut de Mathématiques de Jussieu, Université Paris Diderot (Paris 7), 775205 Paris Cedex 13, France. Jean-Pierre Wintenberger, Département de Mathématiques, Université de Strasbourg, 67084 Strasbourg Cedex, France. Gergely Zabradi, Institute of Mathematics, Eötvös Loránd University, H-1518 Budapest, Pf. 120, Hungary. vi

Preface

The London Mathematical Society Symposium – EPSRC Symposium on Galois Representations and Automorphic Forms was held at the University of Durham from 18th July until 28th July 2011. These topics have been playing an important role in present-day number theory, especially via the Langlands program and the connections it entails. The meeting brought together researchers from around the world on these and related topics, with lectures on a variety of recent major developments in the area. Roughly half of these talks were individual lectures, while the rest constituted series on the following themes: • • • • •

p-adic local Langlands Curves and vector bundles in p-adic Hodge theory The fundamental lemma and trace formula Anabelian geometry Potential automorphy

These Proceedings present much of the progress described in those lectures. The organizers are very grateful to all the speakers and to others who contributed articles. We also wish to thank the London Mathematical Society and EPSRC for the financial support that made the meeting possible. We warmly appreciate the assistance and hospitality provided by the University of Durham’s Department of Mathematics and Grey College. These institutions have helped to make the Symposia such a well established and highly valued event in the number theory community. Fred Diamond Payman Kassaei Minhyong Kim

vii

1 On the local structure of ordinary Hecke algebras at classical weight one points Mladen Dimitrov

Abstract The aim of this chapter is to explain how one can obtain information regarding the membership of a classical weight one eigenform in a Hida family from the geometry of the Eigencurve at the corresponding point. We show, in passing, that all classical members of a Hida family, including those of weight one, share the same local type at all primes dividing the level.

1. Introduction Classical weight one eigenforms occupy a special place in the correspondence between Automorphic Forms and Galois Representations since they yield two dimensional Artin representations with odd determinant. The construction of those representations by Deligne and Serre [5] uses congruences with modular forms of higher weight. The systematic study of congruences between modular forms has culminated in the construction of the p-adic Eigencurve by Coleman and Mazur [4]. A p-stabilized classical weight one eigenform corresponds then to a point on the ordinary component of the Eigencurve, which is closely related to Hida theory. An important result of Hida [11] states that an ordinary cuspform of weight at least two is a specialization of a unique, up to Galois conjugacy, primitive Hida family. Geometrically this translates into the smoothness of the Eigencurve at that point (in fact, Hida proves more, namely that the map The author is partially supported by Agence Nationale de la Recherche grants ANR-10-BLAN-0114 and ANR-11-LABX-0007-01. Automorphic Forms and Galois Representations, ed. Fred Diamond, Payman L. Kassaei and c Cambridge University Press 2014. Minhyong Kim. Published by Cambridge University Press.

1

2

Mladen Dimitrov

to the weight space is etale at that point). Whereas Hida’s result continues to hold at all non-critical classical points of weight two or more [13], there are examples where this fails in weight one [6]. The purely quantitive question of how many Hida families specialize to a given classical p-stabilized weight one eigenform, can be reformulated geometrically as to describe the local structure of the Eigencurve at the corresponding point. An advantage of the new formulation is that it provides group theoretic and homological tools for the study of the original question thanks to Mazur’s theory of deformations of Galois representations. Moreover, this method gives more qualitative answers, since the local structure of the Eigencurve at a given point contains more information than the collection of all Hida families passing through that point. The local structure at weight one forms with RM was first investigated by Cho and Vatsal [3] in the context of studying universal deformation rings, who showed that in many cases the Eigencurve is smooth, but not etale over the weight space, at those points. The main result of a joint work with Joël Bellaïche [1] states that the p-adic Eigencurve is smooth at all classical weight one points which are regular at p and gives a precise criterion for etalness over the weight space at those points. The author has learned recently that the work [10] of Greenberg and Vatsal contains a slightly weaker version of this result. It would be interesting to describe the local structure at irregular points, to which we hope to come back in a future work. The chapter is organized as follows. Section 2 describes some p-adic aspects in the theory of weight one eigenforms. Sections 3 and 4 introduce, respectively, the ordinary Hecke algebras and primitive Hida families, which are central objects in Hida theory [12]. In Section 5 various Galois representations are studied with emphasis on stable lattices, leading to the construction of a representation (1.10) which is a bridge between a primitive Hida family and its classical members. This is used in Section 6 to establish the rigidity of the local type in a Hida family, including in weight one (see Proposition 1.8). Section 7 quotes the main results of [1] and describes their consequences in classical Hida theory (see Corollary 1.15). The latter would have been rather straightforward, should the Eigencurve have been primitive, in the sense that the irreducible component of its ordinary locus would have corresponded (after inverting p) to primitive Hida families. Lacking a reference for the construction of such an Eigencurve, we establish a local isomorphism, at the points of interest, between the reduced Hecke algebra, used in the definition of the Eigencurve, and the new quotient of the full Hecke algebra, used in the definition of primitive Hida families (see Corollary 1.14).

Ordinary Hecke algebras at classical weight one points

3

Acknowledgements. The author would like to thank Joël Bellaïche for many helpful discussions, as well as the referee for his careful reading of the manuscript and for pointing out some useful references.

2. Artin modular forms and the Eigencurve ¯ ⊂ C be the field of algebraic numbers, and denote by Gal(Q/ ¯ Q) the We let Q absolute Galois group of Q. For a prime we fix a decomposition subgroup G ¯ Q) and denote by I its inertia subgroup and by Frob the arithmetic of Gal(Q/ Frobenius in G /I . ¯ → Q ¯ p. We fix a prime number p and an embedding Q Let f (z) = n≥1 an q n be a newform of weight one, level M and central character . Thus a1 = 1 and for every prime M (resp. | M) f is an eigenvector with eigenvalue a for the Hecke operator T (resp. U ). By a theorem of Deligne and Serre [5] there exists a unique continuous irreducible representation: ¯ Q) → GL2 (C), ρ f : Gal(Q/

(1.1)

such that its Artin L-function L(ρ f , s) equals L( f, s) =

an n

ns

=

M

(1 − a −s + ()−2s )−1

(1 − a −s )−1 .

|M

It follows that if a = 0 for | M, then a is the eigenvalue of ρ f (Frob ) acting on the unique line fixed by I . Since ρ f has finite image, a is an eigenvalue of a finite order matrix, hence it is a root of unity. Similarly, for M the characteristic polynomial X 2 − a X + () of ρ f (Frob ) has two (possibly equal) roots α and β which are both roots of unity. In order to deform f p-adically, one should first choose a p-stabilization of f with finite slope, that is an eigenform of level 1 (M) ∩ 0 ( p) sharing the same eigenvalues as f away from p and having a non-zero U p -eigenvalue. By the above discussion if such a stabilization exists, then it should necessarily be ordinary. We distinguish two cases: If p does not divide M, then f has two p-stabilizations f α (z) = f (z) − β p f ( pz) and f β (z) = f (z) − α p f ( pz) with U p -eigenvalue α p and β p , respectively. If p divides M and a p = 0, then f is already p-stabilized. We let then α p = a p and f α = f . Denote by N the prime to p-part of M.

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Definition 1.1. We say that f α is regular at p if either p divides M and a p = 0, or p does not divide M and α p = β p . The Eigencurve C of tame level 1 (N ) is a rigid analytic curve over Q p parametrizing systems of eigenvalues for the Hecke operators T ( N p) and U p appearing in the space of finite slope overconvergent modular forms of tame level dividing N . We refer to the original article of Coleman and Mazur [4] for the case N = 1 and p > 2, and to Buzzard [2] for the general case. Recall that C is reduced and endowed with a flat and locally finite weight map κ : C → W, where W is the rigid space over Q p representing homomorphisms × Z× p ×(Z /N Z) → Gm . The p-stabilized newform f α defines a point on the ordinary component of C, whose image by κ is a character of finite order. Theorem 1.2. [1] Let f be a classical weight one cuspidal eigenform form which is regular at p. Then the Eigencurve C is smooth at the point defined by f α , so there is a unique irreducible component of C containing that point. In particular, if f has CM by a quadratic field in which p splits, then all classical points of that component also have CM by the same field. Moreover, C is etale over the weight space W at the point defined by f α , unless f has RM by a quadratic field in which p splits. In Section 7 we will revisit this theorem from the perspective of Hida families.

3. Ordinary Hecke algebras The results in this and the following two sections are due to Hida [11, 12] when p is odd and have been completed for p = 2 by Wiles [18] and Ghate–Kumar [8]. Let = Z p [[Gal(Q∞ / Q)]] Z p [[1 + p ν Z p ]] be the Iwasawa algebra of the cyclotomic Z p extension Q∞ of Q, where ν = 2 if p = 2 and ν = 1 otherwise. It is a complete local Z p -algebra which is an integral domain of Krull dimension 2. Let χcyc be the universal -adic cyclotomic character obtained by ¯ Q) Gal(Q∞ / Q) with the canonical continuous group composing Gal(Q/ homomorphism from Gal(Q∞ / Q) to the units of its completed group ring . We say that a height one prime ideal p of a finite -algebra T is of weight k (an integer ≥ 1) if P = p ∩ is the kernel of a Z p -algebra homomor¯ p whose restriction to a finite index subgroup of 1 + p ν Z p phism → Q is given by x → x k−1 . Such an ideal p induces a Galois orbit of Z p -algebra ¯ p called specializations in weight k. homomorphisms T → T / p → Q By definition a -adic ordinary cuspform of level N (a positive integer not divisible by p) is a formal q-expansion with coefficients in the integral closure

Ordinary Hecke algebras at classical weight one points

5

of in some finite extension of its fraction field, whose specialization in any weight k ≥ 2 yield the q-expansion of a p-stabilized, ordinary, normalized cuspform of tame level N and weight k. However, specializations in weight one are not always classical. The ordinary Hecke algebra T N of tame level N is defined as the -algebra generated by the Hecke operators U (resp. T , ) for primes dividing N p (resp. not dividing N p) acting on the space of -adic ordinary cuspforms of tame level N . Hida proved that T N is free of finite rank over and its height one primes of weight k ≥ 2 are in bijection with the (Galois orbits of) classical ordinary eigenforms of weight k and tame level dividing N . A -adic ordinary cuspform of level N is said to be N -new if all specializations in weights ≥ 2 are p-stabilized, ordinary cuspforms of tame level N which are N -new. Define Tnew N as the quotient of T N acting faithfully on the space of -adic ordinary cuspforms of level N , which are N -new. A result of Hida (see [12, Corollaries 3.3 and 3.7]) states that Tnew N is a finite, reduced, torsion free algebra, whose height one primes of weight k ≥ 2 are in bijection with the Galois orbits of classical ordinary eigenforms of weight k and tame level N which are N -new.

4. Primitive Hida families A primitive Hida family F = n≥1 An q n of tame level N is by definition a adic ordinary cuspform, new of level N and which is a normalized eigenform for all the Hecke operators, i.e., a common eigenvector of the operators U , T and as above. The relations between coefficients and eigenvalues for the Hecke operators are the usual ones for newforms. One can see from [12, p. 265] that primitive Hida families can be used to write down a basis of the space of

-adic ordinary cuspforms in the same fashion as classically newforms can be used to write down a basis of the space of cuspforms. The central character ψ F : (Z /N pν )× → C× of the family is defined by ψ F () = eigenvalue of . Galois orbits of primitive Hida families of level N are in bijection with the minimal primes of T = Tnew N . More precisely, a primitive Hida family determines and is uniquely determined by a -algebra homomorphism T → Frac( ), sending each Hecke operator to its eigenvalue on F, whose kernel is a minimal prime a ⊂ T. Since T is a finite and reduced -algebra, its localization Ta is a finite field extension of Frac( ). Hence, we obtain the following homomorphisms of -algebras: ∼ T T / a → T / a → Ta → K F ⊂ Frac( ),

(1.2)

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where T / a denotes the integral closure of the domain T / a in its field of fractions Ta . In particular, the image K F of Ta in Frac( ) is a finite extension of Frac( ) generated by the coefficients of F. By definition all specializations of F in weight k ≥ 2 yield p-stabilized, ordinary newforms of tame level N and weight k. In weight one, there are only finitely many classical specializations, unless F has CM by a quadratic field in which p splits (see [9] and [6]). Nevertheless, a theorem of Wiles [18] asserts that any p-stabilized newform of weight one occurs as a specialization of a primitive Hida family. n Given a primitive Hida family F = n≥1 An q of level N , Hida constructed in [11, Theorem 2.1] an absolutely irreducible continuous representation: ¯ Q) → GL2 (K F ), ρ F : Gal(Q/

(1.3)

unramified outside N p, such that for all not dividing N p the trace of the image of Frob equals A . Moreover det ρ F = ψ F χcyc . Finally by Wiles [18, Theorem 2.2.2] the space of I p -coinvariants is a line on which Frob p acts by A p .

5. Galois representations 5.1. Minimal primes

The total quotient field of T is given by T ⊗ Frac( ) a Ta where the product is taken over all minimal primes of T. The representation (1.3) can be rewritten as ¯ Q) → GL2 (Ta ) ρa : Gal(Q/

(1.4)

and by putting those together we obtain a continuous representation ¯ Q) → GL2 (T ⊗ Frac( )) ρT : Gal(Q/

(1.5)

unramified outside N p, such that for all not dividing N p the trace of the image of Frob equals T . Moreover the space of I p -coinvariants is free of rank one and Frob p acts on it as U p .

5.2. Maximal primes Since T is a finite -algebra, it is semi-local, and is isomorphic to the direct product m Tm where the product is taken over all maximal primes. By composing (1.5) with the canonical projection, one obtains:

Ordinary Hecke algebras at classical weight one points ¯ Q) → GL2 (Tm ⊗ Frac( )). ρm : Gal(Q/

7

(1.6)

The composition: Tr(ρm )

¯ Q) −→ Tm → T / m Gal(Q/

(1.7)

is a pseudo-character taking values in a field and sending the complex conjugation to 0. By a result of Wiles [18, §2.2] it is the trace of a unique semi-simple representation: ¯ Q) → GL2 (T / m). ρ¯m : Gal(Q/

(1.8)

Note that whereas each minimal prime a ⊂ T is contained in a unique maximal prime, there may be several minimal primes contained in a given maximal prime m, those corresponding to primitive Hida families sharing the same residual Galois representation ρ¯m .

5.3. Galois stable lattices A lattice over a noetherian domain R (or R-lattice) is a finitely generated R-submodule of a finite dimensional Frac(R)-vector space which spans the latter. This definition extends to a noetherian reduced ring R and its total quotient field a Ra , where a runs over the (finitely many) minimal primes of R. The continuity of ρa implies the existence of a Galois stable T / a-lattice 2 in Ta , and similar statements hold for ρ F , ρT and ρm . It is worth mentioning that ρT cannot necessarily be defined over the normalization of T in a Ta . In other words ρa does not necessarily stabilize a free T / a-lattice. There is an exception: if K F = Frac( ) and p > 2 the regularity of implies that ρ F always admits a Galois stable free -lattice (see [11, §2]). If m is a maximal prime such that the residual Galois representation ρ¯m is absolutely irreducible, then by a result of Nyssen [15] and Rouquier [16] ρm stabilizes a free Tm -lattice. It follows that for every minimal prime a ⊂ m, the representation ρa stabilizes a free lattice over Tm / a = T / a.

5.4. Height one primes Let f be a p-stabilized, ordinary, newform of tame level N and weight k. It determines uniquely a height one prime p ⊂ T and an embedding of Tp / p ¯ p , although not every height one prime of T of weight one is obtained into Q in this way. Our main interest is in the structure of the P -algebra Tp , where P = p ∩ . The ring Tp is local, noetherian, reduced of Krull dimension 1, but is not necessarily integrally closed. It might even not be a domain, since f

8

Mladen Dimitrov

could be a specialization of several, non Galois conjugate, Hida families (see [6, §7.4]), hence there may be several minimal primes a of T contained in p. ¯ Q) → GL2 (Q ¯ p ) be the continuous irreducible representaLet ρ f : Gal(Q/ tion attached to f by Deligne when k ≥ 2 and by (1.1) when k = 1 via the ¯ → Q ¯ p . Since ρ f is odd, it can be defined over the fixed embeddings C ⊃ Q ¯ p generated by its coefficients, and hence ring of integers of the subfield of Q defines an isomorphic representation: ¯ Q) → GL2 (Tp / p), ρ¯p : Gal(Q/

(1.9)

admitting a model over the integral closure of T / p in its field of fractions Tp / p. The normalization of Tp in its total quotient field a⊂p Ta is given by a⊂p Tp / a, where Tp / a (T / a)p is the integral closure of Tp / a

(T / a)p in Ta . Denote by T p / a the completion of the discrete valuation ring T p / a. Note

that they share the same residue field which is a finite extension of Tp / p and that there is a natural bijection between the set of T p / a-lattices in a given Ta -vector space V and the set of T / a-lattices in V ⊗ p Ta Frac(Tp / a). Since ρ¯p is absolutely irreducible and Tp / a is local and complete, by a result of Nyssen [15] and Rouquier [16] the representation ρa ⊗Ta Frac(T p / a) sta bilizes a free Tp / a-lattice. The latter lattice yields (by intersection) a free T p / a-lattice stable by ρa . In other terms there exists a unique, up to conjugacy, continuous representation: ¯ Q) → GL2 (T ρpa : Gal(Q/ p / a),

(1.10)

such that ρpa ⊗T Ta ρa and ρpa mod p ρ¯p . p /a This representation is a bridge between a form and a family and will be used in Section 6 to transfer properties in both directions. The exact control theorem for ordinary Hecke algebras, proved by Hida for p > 2 and by Ghate–Kumar [8] for p = 2, has the following consequence: Theorem 1.3. [11, Corollary 1.4] Assume that k ≥ 2. Then the local algebra Tp is etale over the discrete valuation ring P . In particular, f is a specialization of a unique, up to Galois conjugacy, Hida family corresponding to a minimal prime a. Assume for the rest of this section that Tp is a domain. Then the field of fractions of Tp is isomorphic to Ta , where a is the unique minimal prime of T contained in p. Since normalization and localization commute, we have

Ordinary Hecke algebras at classical weight one points

9

p . Therefore, the collection of representations (1.10) / a)p (T / a)p T (T are replaced by a unique, up to conjugacy, continuous representation: ¯ Q) → GL2 (T p ), ρp : Gal(Q/

(1.11)

such that ρp ⊗Tp Ta ρa and ρp mod p ρ¯p . If we further assume that Tp is etale over P , then Tp is itself a discrete p . valuation ring, hence Tp T

6. Rigidity of the automorphic type in a Hida family By definition, all specializations in weight at least two of a primitive Hida family F of level N share the same tame level. Also, by [7, Proposition 2.2.4], the tame conductor of ρ F equals N . The aim of this section is to show that the tame level of all classical weight one specializations of F is also N , and to show that all classical specializations of F (including those of weight one) share the same automorphic type at all primes dividing N .

6.1. Minimally ramified Hida families Recall that a newform f is said to be minimally ramified if it has minimal level amongst the underlying newforms of all its twists by Dirichlet characters. Lemma 1.4. Let F be a primitive Hida family and let χ be a Dirichlet character of conductor prime to p. There exists a unique primitive Hida family Fχ underlying F ⊗ χ , in the sense that the p-stabilized, ordinary newform underlying a given specialization of F ⊗χ can be obtained by specializing Fχ . Proof. By [12, p. 250] one can write any -adic ordinary cuspform as a linear combination of translates of primitive Hida families of lower or equal level. Since F ⊗χ is an eigenform for all but finitely many Hecke operators, it is necessarily a linear combination of translates of the same primitive Hida family, denoted Fχ . It follows that any specialization of Fχ in weight at least two is the p-stabilized, ordinary newform underlying the corresponding specialization of F ⊗ χ. Definition 1.5. We say that a primitive Hida family F of level N is minimally ramified if for every Dirichlet character χ of conductor prime to p, the level of Fχ is a multiple of N . As for newforms, it is clear that any primitive Hida family admits a unique twist which is minimally ramified.

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Lemma 1.4 implies that being minimally ramified is pure with respect to specializations in weight at least two, that is to say, all specializations of a minimally ramified primitive Hida family are minimally ramified, and a primitive Hida family admitting a minimally ramified specialization is minimally ramified. This observation together with the classification of the admissible representations of GL2 (Q ), easily implies: Lemma 1.6. Let F = n≥1 An q n be a minimally ramified, primitive Hida family of level N and let be a prime dividing N . Denote by unr(C) the unramified character of G sending Frob to C. (i) If ψ F is unramified at and 2 does not divide N , then every specialization in weight at least two corresponds to an automorphic form which is special at . In particular A = 0 and the restriction of ρ F to G is an unramified twist of an extension of 1 by unr(). (ii) If the conductor of ψ F and N share the same -part, then every specialization in weight at least two corresponds to an automorphic form which is a ramified principal series at . In particular A = 0 and the restriction of ρ F to G equals unr(A ) ⊕ unr(B )ψ F , for some B ∈ K F . (iii) In all other cases, every specialization in weight at least two corresponds to an automorphic form which is supercuspidal at . In particular A = 0 and the restriction of ρ F to G is irreducible.

6.2. General case Definition 1.7. Let F be a primitive Hida family of level N and let be a prime dividing N . We say that F is special (resp. ramified principal series or supercuspidal) at , if a minimally ramified twist of F falls in case (i) (resp. (ii) or (iii)) of Lemma 1.6. It follows from Lemma 1.6, that being special, principal series or supercuspidal is pure with respect to specializations, that is to say, all specializations in weight at least two are of the same type. We will now describe the local automorphy type in greater detail and deduce information about classical weight one specializations. Proposition 1.8. Let F be a primitive Hida family of level N and let be a prime dividing N . If F is special at , so are all its specializations in weight at least two and F does not admit any classical weight one specialization. Otherwise, ρ F (I ) is a finite group invariant under any classical specialization, including in weight one. More precisely

Ordinary Hecke algebras at classical weight one points

11

(i) If F is a ramified principal series at , then the restriction of ρ F to G is isomorphic to ϕ ⊕ ϕ , where ϕ and ϕ are characters whose restrictions to inertia have finite order. (ii) If F is supercuspidal at , then either the restriction of ρ F to G is induced from a character of an index two subgroup of G whose restriction to inertia has finite order, or = 2 and all classical specializations of F are extraordinary supercuspidal representations at 2. In particular, all classical weight one specializations of F have tame level N . Proof. Although parts of the proposition seem to be well-known to experts, for the benefit of the reader, we will give a complete proof. If F is special at , then the claim about specializations in weight at least two follows directly from Lemma 1.6(i). Moreover in this case ρ F |G is by definition reducible and the quotient of the two characters occurring in its semisimplification equals unr(). Since is not a root of unity, F does not admit any classical weight one specializations. Suppose now that F is not special at . Since = p and ρ F is continuous, Grothendieck’s -adic monodromy theorem implies that ρ F (I ) is finite. Let p be a height one prime of T corresponding to a classical cusp form f of weight k, containing the minimal prime a defined by F. Denote by L the free rank two a a T p / a-lattice on which ρp acts (see (1.10)). Recall that ρp mod p ρ¯p and consider the natural projection: ρ F (I ) ρpa (I ) ρ¯p (I ) ρ f (I ),

(1.12)

which we claim is an isomorphism. In fact, an eigenvalue ζ of an element of the kernel has to be a root of unity since the latter is a finite group, in particular ¯ p . Since by assumption (ζ −1)2 ∈ p, the product of all its G p -conjugates ζ ∈Q belongs to p ∩ Q p , which is {0} because Tp is a Q p -algebra. Hence ζ = 1 which implies that the kernel is trivial. The claim (i) follows directly from 1.6(ii), so we can assume for the rest of the proof that F is supercuspidal at . Denote by W the wild inertia subgroup of I . Suppose first that ρa |W is reducible, isomorphic to ⊕ with = . Since W is normal in G , it follows easily that extends to an index two subgroup of G , as claimed. If ρa |W is reducible and isotropic, then by taking an eigenvector for a topological generator of I / W one sees that ρa | I is reducible too, which allows us to conclude as in the previous case. Suppose finally that ρa |W is irreducible. Then by a classical result on hyper-solvable groups its image is a dihedral group, hence = 2. Assume further ρa |G is not dihedral, since this case can be handled as above. Then, any specialization in weight at least two of F yields an eigenform f which

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Mladen Dimitrov

is an extraordinary supercuspidal representation at = 2. The isomorphism (1.12) implies that all other classical specializations of F are also extraordinary supercuspidal representations at = 2 (we refer to [17] and [14, §5.1] for a detailed analysis of this case).

7. Local structure of the ordinary Hecke algebras at classical weight one points 7.1. A deformation problem Let f α be a weight one p-stabilized newform of tame level N as in Section 2. Assume that f is regular at p. By ordinarity the restriction of ρ f to G p is a sum of two characters ψ1 and ψ2 , and by regularity exactly one of those characters, say ψ1 , is the unramified character sending Frob p to α p . By (1.9) the Galois representation ρ f is defined over a finite extension E = Tp / p of Q p , where p denotes the height one prime of T determined by f . Consider the functor D sending a local Artinian ring A with maximal ideal m A and residue field A/ m A = E to the set of strict equivalence classes of ¯ Q) → GL2 (A) such that ρ representations ρ : Gal(Q/ mod m A ρ f and fitting in an exact sequence 2 → ρ 1 → 0 0→ψ → ψ 1 is an unramified character of A[G p ]-modules, free over A, and such that ψ whose reduction modulo m A equals ψ1 . We define D as the subfunctor of D consisting of deformation with constant determinant. Finally, define Dmin ) as the subfunctor of D (resp. D ) of deformations ρ (resp. Dmin such that for all dividing N such that a = 0, the I -invariants in ρ are a free A-module of rank one. The functors D and Dmin are pro-representable by local noetherian complete E-algebras R and Rmin , while D and Dmin are representable by local Artinian E-algebras R and Rmin . Denote by tD the tangent space of D, etc. Using the interpretation of tD and tD in terms of Galois cohomology groups, the main technical result of [1] states: Theorem 1.9. If f is regular at p, then dim tD = 1. If we further assume that f does not have RM by a quadratic field in which p splits, then tD = 0.

7.2. Modular deformations p the completion of Tp The composition: Denote by T T) ¯ Q) Tr(ρ p Gal(Q/ −→ T → Tp → T

(1.13)

Ordinary Hecke algebras at classical weight one points

13

is a two dimensional pseudo-character taking values in a complete local ring and whose reduction modulo the maximal ideal is the trace of the absolutely irreducible representation ρ¯p . By a result of Nyssen [15] and Rouquier [16] the pseudo-character (1.13) is the trace of a two dimensional irreducible representation: ¯ Q) → GL2 (T p ). ρ

p : Gal(Q/ (1.14) This representation contains more information than the collection (ρpa )a⊂p and plays a central role in the analysis of the p-adic deformations of f . Note that

P is formal power series over its residue field

P /P P /P which is a finite extension of Q p . Consider the local Artinian Q p -algebra p ⊗

T =T P /P.

P

(1.15)

Reducing (1.14) modulo P yields a continuous representation: ¯ Q) → GL2 (T ), ρT : Gal(Q/

(1.16)

such that det(ρT ) = det(ρ f ). In fact, for all ∈ (Z /N p)× , the image of in T is given by the henselian lift of its image in T / p, hence is fixed. As in [1], one can describe the local behavior of ρ

p at bad primes. Proposition 1.10. (i) For all dividing N such that a = 0, the space of I -invariants in ρ

p is free of rank one and Frob acts on it as U . (ii) Assume that f is regular at p. Then ρ

p is ordinary, in the sense that the space of I p -coinvariants is free of rank one and Frob p acts on it as U p . By Proposition 1.10 the Galois representation (1.14) (resp. (1.16)) defines a ). One deduces the following surjective homomorpoint of Dmin (resp. of Dmin phisms of local reduced

P -algebras (resp. local Artinian E-algebras): p , and R Rmin T

(1.17)

R Rmin T .

7.3. Smoothness and etaleness Lemma 1.11. The P -algebra Tp is etale if, and only if, T is a field. Proof. Since T is flat over , so is T ⊗ P over P . The algebra T ⊗ P is unramified over P if, and only if, T ⊗ P /P is unramified over P /P, that is to say is a product of fields. Since T ⊗ P = Tp , we have

p ∩ =P

Tp ⊗ P P /P = T ⊗ P /P T ⊗

P /P =

p ∩ =P

p ∩ =P

p ⊗

T

P P /P.

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p is unramified One deduces that Tp is unramified over P if, and only if, T over

if, and only if, T = T ⊗

/P is a field. P p

P P Proposition 1.12. Suppose that f is regular at p. Then Tp is a discrete valuation ring and the homomorphisms in (1.17) are isomorphisms. Moreover, if f does not have RM by a quadratic field in which p splits, then Tp is etale over P , and otherwise, under the additional assumption that dim E R ≤ 2, the ramification index of Tp over P equals 2. Proof. Since f is regular at p, Theorem 1.9 implies that dim tD = 1. Since p > 0, one deduces that the natural surjective homomorphism R T p dim T

is an isomorphism of discrete valuation rings, hence Tp is a discrete valuation p modulo P we obtain the ring too. By reducing the isomorphism R T isomorphism R T . If tD = 0, by Nakayama’s lemma the structural homomorphisms E → R and

P → R are isomorphisms. By Lemma 1.11, it follows that Tp is etale over P , as claimed. Assume now that dim tD = 1, in which case Theorem 1.9 implies that f has RM by a quadratic field in which p splits. Since dim E R ≤ 2 by assumption and dim E T ≥ 2 by [6, Proposition 2.2.4], we deduce that the ramification index is 2. Remark 1.13. Cho and Vatsal [3] have proved that dim E R ≤ 2 under some additional assumptions, and their method is expected to continue to work under the only assumption of regularity at p.

7.4. Reduced Hecke algebras Define the reduced ordinary Hecke algebra T = TN of tame level N as the subalgebra of T N generated over by the Hecke operators U p , T and for primes not dividing N p. By the theory of newforms, the natural composition: TN → T N → Tnew (1.18) N , N |N

is injective, in particular TN is reduced. A classical result from Hida theory says that the localization of (1.18) at any height one prime of weight at least two yields an isomorphism. Let p be a height one prime T corresponding to a p-stabilized classical weight one eigenform f α and denote by p the corresponding height one prime of T . Corollary 1.14. Suppose that f is regular at p. Then the localization of (1.18) yields an isomorphism Tp Tp .

Ordinary Hecke algebras at classical weight one points

15

Proof. Proposition 1.8 implies Tnew N ,p = {0} for all N < N , hence localizing (1.18) at p yields an injective homomomorphism Tp → Tp . Since Tp and Tp are finite over the Zariski ring P , it is enough to check the surjectivity after completion. This follows from Proposition 1.12, since the surjective p factors through T homomorphism R T p .

We will conclude this chapter, by giving a partial answer to the original question that motivated this research. Corollary 1.15. Let f be a classical weight one cuspidal eigenform form which is regular at p. Then there exists a unique Hida family specializing to f α . In particular, if f has CM by a quadratic field in which p splits, then the family has also CM by the same field.

References [1] J. B ELLAÏCHE AND M. D IMITROV . On the Eigencurve at classical weight one points, arXiv:1301.0712, submitted. [2] K. B UZZARD, Eigenvarieties, in L-functions and Galois Representations, vol. 320 of London Math. Soc. Lecture Note Ser., Cambridge University Press, Cambridge, 2007, pp. 59–120. [3] S. C HO AND V. VATSAL, Deformations of induced Galois representations, J. Reine Angew. Math., 556 (2003), 79–98. [4] R. C OLEMAN AND B. M AZUR, The eigencurve, in Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996), vol. 254 of London Math. Soc. Lecture Note Ser., Cambridge University Press, Cambridge, 1998, pp. 1–113. [5] P. D ELIGNE AND J.-P. S ERRE, Formes modulaires de poids 1, Ann. Sci. École Norm. Sup. 7 (4), (1974), 507–530. [6] M. D IMITROV AND E. G HATE, On classical weight one forms in Hida families, J. Théor. Nombres Bordeaux, 24, 3 (2012), 669–690 [7] M. E MERTON , R. P OLLACK AND T. W ESTON, Variation of Iwasawa invariants in Hida families, Invent. Math., 163 (2006), 523–580. [8] E. G HATE AND N. K UMAR, Control theorems for ordinary 2-adic families of modular forms, to appear in the Proceedings of the International Colloquium on Automorphic Representations and L-functions, TIFR, 2012. [9] E. G HATE AND V. VATSAL, On the local behaviour of ordinary -adic representations, Ann. Inst. Fourier, Grenoble, 54, 7 (2004), 2143–2162. [10] R. G REENBERG AND V. VATSAL, Iwasawa theory for Artin representations, in preparation. [11] H. H IDA, Galois representations into GL2 (Z p [[X ]]) attached to ordinary cusp forms, Invent. Math., 85 (1986), 545–613. [12] , Iwasawa modules attached to congruences of cusp forms, Ann. Sci. Ecole Norm. Sup. 19 (4), (1986), 231–273. [13] M. K ISIN, Overconvergent modular forms and the Fontaine-Mazur conjecture, Invent. Math., 153 (2003), 373–454.

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[14] P. K UTZKO, The Langlands conjecture for Gl2 of a local field, Ann. of Math. 112 (2), (1980), pp. 381–412. [15] L. N YSSEN, Pseudo-représentations, Math. Ann., 306 (1996), 257–283. [16] R. ROUQUIER, Caractérisation des caractères et pseudo-caractères, J. Algebra, 180 (1996), 571–586. [17] A. W EIL, Exercices dyadiques, Invent. Math., 27 (1974), 1–22. [18] A. W ILES, On ordinary λ-adic representations associated to modular forms, Invent. Math., 94 (1988), 529–573.

2 Vector bundles on curves and p-adic Hodge theory Laurent Fargues and Jean-Marc Fontaine

Contents Introduction 1 Holomorphic functions of the variable π 2 The space |Y | 3 Divisors on Y 4 Divisors on Y/ϕ Z 5 The curve 6 Vector bundles 7 Vector bundles and ϕ-modules References

page 17 18 30 44 48 54 62 83 103

Introduction This text is an introduction to our work [12] on curves and vector bundles in p-adic Hodge theory. This is a more elaborate version of the reports [13] and [14]. We give a detailed construction of the “fundamental curve of p-adic Hodge theory” together with sketches of proofs of the main properties of the objects showing up in the theory. Moreover, we explain thoroughly the classification theorem for vector bundles on the curve, giving a complete proof for rank two vector bundles. The applications to p-adic Hodge theory, the theorem “weakly admissible implies admissible” and the p-adic monodromy theorem, are not given here but can be found in [13] and [14]. We would like to thank the organizers of the EPSRC Durham Symposium “Automorphic forms and Galois representations” for giving us the opportunity to talk about this subject. Automorphic Forms and Galois Representations, ed. Fred Diamond, Payman L. Kassaei and c Cambridge University Press 2014. Minhyong Kim. Published by Cambridge University Press.

17

18

Laurent Fargues and Jean-Marc Fontaine

1. Holomorphic functions of the variable π 1.1. Background on holomorphic functions in a p-adic punctured disk after Lazard ([25]) 1.1.1. The Frechet algebra B Let F be a complete non-archimedean field for a non trivial valuation v : F −→ R ∪ {+∞}, with characteristic p residue field. We note |·| = p −v(·) the associated absolute value. Consider the open punctured disk D∗ = {0 < |z| < 1} ⊂ A1F as a rigid analytic space over F, where z is the coordinate on the affine line. If I ⊂]0, 1[ is a compact interval set D I = {|z| ∈ I } ⊂ D∗ , ∗ an annulus that is an affinoïd domain in D if I = [ρ1 , ρ2 ] with ρ1 , ρ2 ∈ |F × |. There is an admissible affinoïd covering

D∗ = DI I ⊂]0,1[

where I goes through the preceding type of compact intervals. Set now B = O(D∗ ) = an z n | an ∈ F, ∀ρ ∈]0, 1[ n∈Z

lim |an |ρ n = 0

|n|→+∞

the ring of holomorphic functions on D∗ . In the preceding description of B, one checks the infinite set of convergence conditions associated to each ρ ∈]0, 1[ can be rephrased in the following two conditions ⎧ ⎨lim inf v(an n ) ≥ 0 n→+∞ (2.1) ⎩ lim v(a−n ) = +∞. n For ρ ∈]0, 1[ and f =

n→+∞

n an z

n

∈ B set

| f |ρ = sup{|an |ρ n }. n∈Z

If ρ = p −r with r > 0 one has | f |ρ = p −vr ( f ) with vr ( f ) = inf {v(an ) + nr }. n∈Z

Vector bundles on curves and p-adic Hodge theory

19

Then | · |ρ is the Gauss supremum norm on the annulus {|z| = ρ}. It is in fact a multiplicative norm, that is to say vr is a valuation. Equipped with the set of norms (| · |ρ )ρ∈]0,1[ , B is a Frechet algebra. The induced topology is the one of uniform convergence on compact subsets of the Berkovich space associated to D∗ . If I ⊂]0, 1[ is a compact interval then B I = O(D I ) equipped with the set of norms (| · |ρ )ρ∈I is a Banach algebra. In fact if I = [ρ1 , ρ2 ] then by the maximum modulus principle, for f ∈ B I sup | f ρ | = sup{| f |ρ1 , | f |ρ2 }. ρ∈I

One then has B = lim B I ←− I ⊂]0,1[

as a Frechet algebra written as a projective limit of Banach algebras. For f = n an z n ∈ B one has | f |1 = lim | f |ρ = sup |an | ∈ [0, +∞]. ρ→1

n∈Z

We will later consider the following closed sub-O F -algebra of B B+ = { f ∈ B | f 1 ≤ 1} = an z n ∈ B | an ∈ O F n∈Z

=

an z n | an ∈ O F ,

n∈Z

v(a−n ) = +∞ . n→+∞ n lim

Set now Bb = { f ∈ B | ∃N ∈ N, z N f ∈ O(D) and is bounded on D} = an z n | an ∈ F, ∃C ∀n |an | ≤ C . n−∞

This is a dense sub-algebra of B. In particular one can find back B from Bb via completion with respect to the norms (| · |ρ )ρ∈]0,1[ . In the same way b Bb,+ = B ∩ B+ = an z n | an ∈ O F

= is dense in

B+

n−∞ O F z[ 1z ]

and thus one can find back B+ from Bb,+ via completion.

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Laurent Fargues and Jean-Marc Fontaine

1.1.2. Zeros and growth of holomorphic functions Recall the following properties of holomorphic functions overs C. Let f be holomorphic on the open disk of radius 1 (one could consider the punctured disk but we restrict to this case to simplify the exposition). For ρ ∈ [0, 1[ set M(ρ) = sup | f (z)|. |z|=ρ

The following properties are verified: • the function ρ → − log M(ρ) is a concave function of log ρ (Hadamard), • if f (0) = 0, f has no zeros on the circle of radius ρ and (a1 , . . . , an ) are its zeros counted with multiplicity in the disk of radius ρ, then as a consequence of Jensen’s formula − log | f (0)| ≥

n

(− log |ai |) − nρ − log M(ρ).

i=1

In the non-archimedean setting we have an exact formula linking the growth of an holomorphic function and its zeros. For this, recall the formalism of the Legendre transform. Let ϕ : R −→] − ∞, +∞] be a convex decreasing function. Define the Legendre transform of ϕ as the concave function (see Figure 1) L (ϕ) : ]0, +∞[ −→ [−∞, +∞[ λ −→ inf {ϕ(x) + λx}. x∈R

L(ϕ)(λ) ϕ(x) x

λ

Figure 1 The Legendre transform of ϕ evaluated at the slope λ where by definition the slope is the opposite of the derivative (we want the slopes to be the valuations of the roots for Newton polygons).

Vector bundles on curves and p-adic Hodge theory

21

If ϕ1 , ϕ2 are convex decreasing functions as before define ϕ1 ∗ ϕ2 as the convex decreasing function defined by (ϕ1 ∗ ϕ2 )(x) = inf {ϕ1 (a) + ϕ2 (b)}. a+b=x

We have the formula L (ϕ1 ∗ ϕ2 ) = L (ϕ1 ) + L (ϕ2 ). One can think of the Legendre transform as being a “tropicalization” of the Laplace transform: tropicalization

(R, +, ×) −−−−−−−−→ (R, inf, +) Laplace transform /o /o /o / Legendre transform usual convolution ∗ /o /o o/ / tropical ∗ just defined. The function ϕ is a polygon, that is to say piecewise linear, if and only if L (ϕ) is a polygon. Moreover in this case: • the slopes of L (ϕ) are the x-coordinates of the breakpoints of ϕ, • the x-coordinates of the breakpoints of L (ϕ) are the slopes of ϕ. Thus L and its inverse give a duality slopes ←→ x-coordinates of breakpoints. From these considerations one deduces that if ϕ1 and ϕ2 are convex decreasing polygons such that ∀i = 1, 2, ∀λ ∈]0, +∞[, L (ϕi )(λ) = −∞ then the slopes of ϕ1 ∗ ϕ2 are obtained by concatenation from the slopes of ϕ1 and the ones of ϕ2 . For f = n∈Z an z n ∈ B set now Newt(f) = decreasing convex hull of {(n, v(an ))}n∈Z . This is a polygon with integral x-coordinate breakpoints. Moreover the function r → vr ( f ) defined on ]0, +∞[ is the Legendre transform of Newt(f). Then, the statement of the “ p-adic Jensen formula” is the following: the slopes of Newt(f) are the valuations of the zeros of f (with multiplicity). Example 2.1. Take f ∈ O(D), f (0) = 0. Let ρ = p −r ∈]0, 1[ and (a1 , . . . , an ) be the zeros of f in the ball {|z| ≤ ρ} counted with multiplicity. Then, as a consequence of the fact that r → vr ( f ) is the Legendre transform

22

Laurent Fargues and Jean-Marc Fontaine

v(f(0)) v(a1) v(a2) vr(f)

r

1

v(an)

2

n

Figure 2 An illustration of the “ p-adic Jensen formula”. The numbers over each line are their slopes.

of a polygon whose slopes are the valuations of the roots of f , one has the formula n v( f (0)) = vr ( f ) − nr + v(ai ) i =1

(see Figure 2). Finally, remark that B+ is characterized in terms of Newton polygons: B+ = { f ∈ B | Newt(f) ⊂ upper half plane y ≥ 0}. 1.1.3. Weierstrass products For a compact interval I = [ρ1 , ρ2 ] ⊂]0, 1[ with ρ1 , ρ2 ∈ |F × | the ring B I is a P.I.D. with Spm(B I ) = |D I |. In particular Pic(D I ) = 0. Now let’s look at Pic(D∗ ). In fact in the following we will only be interested in the submonoid of effective line bundles Pic+ (D∗ ) = {[L ] ∈ Pic(D∗ ) | H 0 (D∗ , L ) = 0}. Set

Div+ (D∗ ) = D = m x [x] | m x ∈ N, ∀I ⊂]0, 1[ x∈|D∗ |

compact supp(D) ∩ D I is finite

the monoid of effective divisors on D∗ . There is an exact sequence div

0 → B \ {0}/B× −−→ Div+ (D∗ ) −→ Pic+ (D∗ ) −→ 0. We are thus led to the question: for D ∈ Div+ (D∗ ), does there exist f ∈ B\{0} such that div( f ) = D ?

Vector bundles on curves and p-adic Hodge theory

23

This is of course the case if supp(D) is finite. Suppose thus it is infinite. We will suppose moreover F is algebraically closed (the discrete valuation case is easier but this is not the case we are interested in) and thus |D∗ | = m F \ {0} where m F is the maximal ideal of O F . Suppose first there exists ρ0 ∈]0, 1[ such that supp(D) ⊂ {0 < |z| ≤ ρ0 }. Then we can write D= [ai ], ai ∈ m F \ {0}, lim |ai | = 0. i→+∞

i ≥0

The infinite product

+∞

1−

i =0

ai z

,

converges in the Frechet algebra B and its divisor is D. We are thus reduced to the case supp(D) ⊂ {ρ0 < |z| < 1} for some ρ0 ∈]0, 1[. But if we write D= [ai ], lim |ai | = 1 i≥0

i→+∞

then neither of the infinite products “ i≥0 1 − azi ” or “ i ≥0 1 − azi ” converges. Recall that over C this type of problem is solved by introducing renormalization factors. Typically, if we are looking for a holomorphic func tion f on C such that div( f ) = n∈N [−n] then the product “z n∈N 1 + nz ” z does not converge but z n∈N 1 + nz e− n = eγ z 1(z) does. In our nonarchimedean setting this problem has been solved by Lazard. Theorem 2.2 (Lazard [25]). If F is spherically complete then there exists a sequence (h i )i≥0 of elements of B× such that the product i≥0 [(z − ai ).h i ] converges. Thus, if F is spherically complete Pic+ (D∗ ) = 0 (and in fact Pic(D∗ ) = 0). In the preceding problem, it is easy to verify that for any F there always exist renormalization factors h i ∈ B\{0} such that i≥0 [(z − ai ).h i ] converges and thus an f ∈ B \ {0} such that div( f ) ≥ D. The difficulty is thus to introduce renormalization factors that do not add any new zero. Let’s conclude this section with a trick that sometimes allows us not to introduce any renormalization factors. Over C this is the following. Suppose we are looking for a holomorphic on C whose divisor is n∈Z [n]. The infi function nite product “z n∈Z\{0} 1 − nz ” does not converge. Nevertheless, regrouping the terms, the infinite product z n≥1 1 − nz 1 + nz = sinππ z converges.

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Laurent Fargues and Jean-Marc Fontaine

In the non-archimedean setting there is a case where this trick works. This is the following. Suppose E|Q p is a finite extension and E is an algebraic closure of E. Let LT be a Lubin–Tate group law over O E . Its logarithm logLT is a rigid analytic function on the open disk D with zeros the torsion points of the Lubin–Tate group law, LT [π ∞ ] = {x ∈ m E | ∃n ≥ 1, [π n ]LT (x) = 0}. The infinite product z

1−

ζ ∈LT [π ∞ ]\{0}

z ζ

does not converge since in this formula |ζ | → 1. Nevertheless, the infinite product z z 1− ζ n n−1 n≥1

ζ ∈LT [π ]\LT [π

]

converges in the Frechet algebra of holomorphic functions on D and equals logLT . This is just a reformulation of the classical formula logLT = lim π −n [π n ]LT . n→+∞

1.2. Analytic functions in mixed characteristic 1.2.1. The rings B and B+ Let E be a local field with uniformizing element π and finite residue field Fq . Thus, either E is a finite extension of Q p or E = Fq ((π )). Let F|Fq be a valued complete extension for a non trivial valuation v : F → R ∪ {+∞}. Suppose moreover F is perfect (in particular v is not discrete). Let E |E be the unique complete unramified extension of E inducing the extension F|Fq on the residue fields, OE /π OE = F. There is a Teichmüller lift [·] : F → OE and E = [xn ]π n | xn ∈ F (unique writing). n−∞

If charE = p then [·] is additive, E |F, and E = F((π )). If E|Q p then E = WO E (F)[ π1 ] = W (F) ⊗W (Fq ) E the ramified Witt vectors of F. There is a Frobenius ϕ acting on E , q ϕ [xn ]π n = [xn ]π n . n

n

Vector bundles on curves and p-adic Hodge theory

25

f

If E|Q p then on W (F) ⊗W (Fq ) E one has ϕ = ϕQ p ⊗ Id where in this formula

q = p f and ϕQ p is the usual Frobenius of the Witt vectors. In this case the addition law of WO E (F) is given by [xn ]π n + [yn ]π n = [Pn (x0 , . . . , xn , y0 , . . . , yn )]π n (2.2) n≥0

n≥0

n≥0

q i−n q j −n where Pn ∈ Fq X i , Y j are generalized polynomials. The 0≤i, j ≤n multiplication law is given in the same way by such kind of generalized polynomials. Definition 2.3. (1) Define Bb = [xn ]π n ∈ E | ∃C, ∀n |xn | ≤ C n−∞

b,+

B

=

[xn ]π n ∈ E | xn ∈ O F

n−∞

= WO E (O F ) π1 if E|Q p (2) For x =

= O F π [ π1 ] if E = Fq ((π )). n [x n ]π

n

∈ Bb and r ≥ 0 set vr (x) = inf {v(xn ) + nr }. n∈Z

If ρ = q −r ∈]0, 1] set |x|ρ = q −vr (x) . (3) For x = n [xn ]π n ∈ Bb set Newt(x) = decreasing convex hull of {(n, v(xn ))}n∈Z . In the preceding definition one can check that the function vr does not depend on the choice of a uniformizing element π . In the equal characteristic case, that is to say E = Fq ((π )), setting z = π one finds back the rings defined in Section 1.1. For x ∈ Bb the function r → vr (x) defined on ]0, +∞[ is the Legendre transform of Newt(x). One has v0 (x) = lim vr (x). The Newton polygon of r→0

x is +∞ exactly on ] − ∞, vπ (x)[ and moreover lim Newt(x) = v0 (x). One +∞

has to be careful that since the valuation of F is not discrete, this limit is not always reached, that is to say Newt(x) may have an infinite number of strictly positive slopes going to zero. One key point is the following proposition whose proof is not very difficult but needs some work. Proposition 2.4. For r ≥ 0, vr is a valuation on Bb .

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Laurent Fargues and Jean-Marc Fontaine

Thus, for all ρ ∈]0, 1], | · |ρ is a multiplicative norm. One deduces from this that for all x, y ∈ Bb , Newt(xy) = Newt(x) ∗ Newt(y) (see 1.1.2). In particular the slopes of Newt(xy) are obtained by concatenation from the slopes of Newt(x) and the ones of Newt(y). For example, as a consequence, if a1 , . . . , an ∈ m F \ {0}, then Newt (π − [a1 ]) . . . (π − [an ]) is +∞ on ] − ∞, 0[, 0 on [n, +∞[ and has non-zero slopes v(a1 ), . . . , v(an ). Definition 2.5. Define • B = completion of Bb with respect to (| · |ρ )ρ∈]0,1[ , • B+ = completion of Bb,+ with respect to (| · |ρ )ρ∈]0,1[ , • for I ⊂]0, 1[ a compact interval B I = completion of Bb with respect to (| · |ρ )ρ∈I . The rings B and B+ are E-Frechet algebras and B+ is the closure of Bb,+ in B. Moreover, if I = [ρ1 , ρ2 ] ⊂]0, 1[, for all f ∈ B sup | f |ρ = sup{| f |ρ1 , | f |ρ2 } ρ∈I

because the function r → vr ( f ) is concave. Thus, B I is an E-Banach algebra. As a consequence, the formula B = lim B I ←− I ⊂]0,1[

expresses the Frechet algebra B as a projective limit of Banach algebras. Remark 2.6. Of course, the preceding rings are not new and appeared for example under different names in the work of Berger ([2]) and Kedlaya ([21]). The new point of view here is to see them as rings of holomorphic functions of the variable π . In particular, the fact that vr is a valuation (Proposition 2.4) had never been noticed before. The Frobenius ϕ extends by continuity to automorphisms of B and B+ , and ∼ for [ρ1 , ρ2 ] ⊂]0, 1[ to an isomorphism ϕ : B[ρ1 ,ρ2 ] − → B[ρ q ,ρ q ] . 1

2

Remark 2.7. In the case E = Fq ((π )), setting z = π as in Section 1.1, the q n n Frobenius ϕ just defined is given by ϕ n x n z = n x n z. This is thus an arithmetic Frobenius, the geometric one being n xn z n → n xn z qn .

Vector bundles on curves and p-adic Hodge theory

27

The ring B+ satisfies a particular property. In fact, if x ∈ Bb,+ and r ≥ r > 0 then vr (x) ≥

r vr (x). r

(2.3)

Thus, if 0 < ρ ≤ ρ < 1 we have + + B+ [ρ,ρ ] = Bρ ⊂ Bρ b,+ where for a compact interval I ⊂]0, 1[ we note B+ I for the completion of B + + with respect to the (| · |ρ )ρ∈I , and Bρ := B{ρ} . One deduces that for any ρ0 ∈]0, 1[, B+ ρ0 is stable under ϕ and B+ = ϕ n B+ ρ0 n≥0

the biggest sub-algebra of B+ ρ0 on which ϕ is bijective. Suppose E = Q p and choose ρ ∈ |F × |∩]0, 1[. Let a ∈ F such that |a| = ρ. Define B+ cris,ρ = p-adic completion of the P.D. hull of the ideal W (O F )[a] of W (O F ) ⊗Z p Q p . This depends only on ρ since W (O F )[a] = {x ∈ W (O F ) | |x|0 ≥ ρ}. We thus have n 1 [a ] B+ = W (O ) . F cris,ρ n! n≥1 p One has + + B+ ρ p ⊂ Bcris,ρ ⊂ Bρ p−1 .

From this one deduces B+ =

ϕ n B+ cris,ρ ,

n≥0 + the ring usually denoted “Brig ” in p-adic Hodge theory. The ring B+ cris,ρ appears naturally in comparison theorems where it has a natural interpretation in terms of crystalline cohomology. But the structure of the ring B+ is simpler as we will see. Moreover, if k ⊂ O F is a perfect sub-field, K 0 = W (k)Q and (D, ϕ) a k-isocrystal, then ϕ=Id ϕ=Id D ⊗ K 0 B+ = D ⊗ K 0 B+ cris,ρ + because ϕ is bijective on D. Replacing B+ cris,ρ by B is thus harmless most of the time.

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Laurent Fargues and Jean-Marc Fontaine

Remark 2.8. Suppose E|Q p and let (xn )n∈Z be a sequence of O F such that lim v(xn−n ) = +∞. Then, the series n→+∞

[xn ]π n

n∈Z

converges in

B+ .

But:

• we don’t know if each element of B+ is of this form, • for such an element of B+ we don’t know if such a writing is unique, • we don’t know if the sum or product of two element of this form is again of this form. The same remark applies to B. Nevertheless, there is a sub E-vector space of B+ where the preceding remark does not apply. One can define for any O E -algebra R the group of (ramified) Witt bivectors BWO E (R). Elements of BWO E (R) have a Teichmüller expansion that is infinite on the left and on the right. One has BWO E (O F ) :=

BWO E (O F /a)

lim ←−

a⊂O F non zero ideal

=

n∈Z

=

Vπn [xn ] | xn ∈ O F , liminf v(xn ) > 0 n→−∞

[yn ]π n | yn ∈ O F , liminf q n v(x n ) > 0 ⊂ B+ . n→−∞

n∈Z

The point is that in BWO E (O F ) [xn ]π n + [yn ]π n = lim Pk (xn−k , . . . , xn , yn−k , . . . , yn ) π n n∈Z

n∈Z

n∈Z

k→+∞

where the generalized polynomials q i −k q j −k Pk ∈ Fq X i , Y j 1≤i, j ≤k give the addition law of the Witt vectors as in formula (2.2) and the limits in the preceding formulas exist thanks to [15], prop.1.1 chap. II, and the convergence condition appearing in the definition of the bivectors. In fact, periods of π -divisible O E -modules (that is to say p-divisible groups when E = Q p ) lie in BWO E (O F ), and BWO E (O F ) contains all periods whose Dieudonné–Manin slopes lie in [0, 1]. In equal characteristic, when E = Fq ((π )), there is no restriction on Dieudonné–Manin slopes of formal O E -modules (what we call here a formal O E -module is a Drinfeld module in dimension 1). This gives a meta-explanation to the fact that in equal

Vector bundles on curves and p-adic Hodge theory

29

characteristic all elements of B have a unique power series expansion and the fact that this may not be the case in unequal characteristic. 1.2.2. Newton polygons Since the elements of B may not be written uniquely as a power series n n∈Z [x n ]π , we need a trick to define the Newton polygon of such elements. The following proposition is an easy consequence of the following Dini type theorem: if a sequence of concave functions on ]0, +∞[ converges point-wise then the convergence is uniform on all compact subsets of ]0, +∞[ (but not on all ]0, +∞[ in general). Proposition 2.9. If (xn )n≥0 is a sequence of Bb that converges to x ∈ B \ {0} then for all I ⊂]0, 1[ compact, ∃N, n ≥ N and q −r ∈ I =⇒ vr (xn ) = vr (x). One deduces immediately: Corollary 2.10. For x ∈ B the function r → vr (x) is a concave polygon with integral slopes. This leads us to the following definition. Definition 2.11. For x ∈ B, define Newt(x) as the inverse Legendre transform of the function r → vr (x). Thus, Newt(x) is a polygon with integral x-coordinate breakpoints. Moreover, if (λi )i ∈Z are its slopes, where λi is the slope on the segment [i, i + 1] (we set λi = +∞ if Newt(x) is +∞ on this segment), then lim λi = 0 and lim λi = +∞.

i →+∞

i→−∞

In particular lim Newt(x) = +∞. Those properties of Newt(x) are the only −∞

restrictions on such polygons. Remark 2.12. If xn −→ x in B with xn ∈ Bb and x = 0 then one checks n→+∞

using Proposition 2.9 that in fact for any compact subset K of R, there exists an integer N such that for n ≥ N , Newt(xn )|K = Newt(x)|K . The advantage of Definition 2.11 is that it makes it clear that Newt(x) does not depend on the choice of a sequence of Bb going to x. Example 2.13. (1) If (xn )n∈Z is a sequence of F satisfying the two conditions of formula n is the decreasing convex hull (2.1) then the polygon Newt [x ]π n∈Z n of {(n, v(xn ))}n∈N .

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Laurent Fargues and Jean-Marc Fontaine

(2) If (an )n≥0 is a sequence of F × going to zero then the infinite product [an ] converges in B and its Newton polygon is zero on [0, +∞[ n≥0 1− π and has slopes the (v(an ))n≥0 on ] − ∞, 0]. Of course the Newton polygon of x does not give more information than the polygon r → vr (x). But it is much easier to visualize and its interest lies in the fact that we can appeal to our geometric intuition from the usual case of holomorphic functions recalled in Section 1.1 to guess and prove results. Here is a typical application: the proof of the following proposition is not very difficult once you have convinced yourself it has to be true by analogy with the usual case of holomorphic functions. Proposition 2.14. We have the following characterizations: (1) B+ = {x ∈ B | Newt(x) ≥ 0}. (2) Bb = {x ∈ B | Newt(x) is bounded below and ∃A, Newt(x)|]−∞,A] = +∞}. (3) The algebra {x ∈ B | ∃ A, Newt(x)|]−∞,A] = +∞} is a subalgebra of ! v(x n ) n WO E (F)[ π1 ] equal to n−∞ [x n ]π | liminf n ≥ 0 . n→+∞

This has powerful applications that would be difficult to obtain without Newton polygons. For example one obtains the following. Corollary 2.15. × ! (1) B× = Bb = x ∈ Bb | Newt(x) has 0 as its only non infinite slope . d (2) One has Bϕ=π = 0 for d < 0, Bϕ=Id = E and for d ≥ 0, ϕ=π d d Bϕ=π = B+ . Typically, the second point is obtained in the following way. If x ∈ B satisfies ϕ(x) = π d x then Newt(ϕ(x)) = Newt(π d x) that is to say Newt(x) satisfies the functional equation qNewt(x)(t) = Newt(x)(t − d). By solving this functional equation and applying Proposition 2.14 one finds the results.

2. The space |Y | 2.1. Primitive elements We would like to see the Frechet algebra B defined in the preceding section as an algebra of holomorphic functions on a “rigid analytic space” Y . This is of course the case if E = Fq ((π )) since we can take Y = D∗ a punctured disk as in Section 1.1. This is not the case anymore when E|Q p , at least as a Tate rigid space. But nevertheless we can still define a topological space |Y | that embeds

Vector bundles on curves and p-adic Hodge theory

31

in the Berkovich space M(B) of rank 1 continuous valuations on B. It should be thought of as the set of classical points of this “space” Y that would remain to construct. To simplify the exposition, in the following we always assume E|Q p , that is to say we concentrate on the most difficult case. When E = Fq ((π )), all stated results are easy to obtain by elementary manipulation and are more or less already contained in the backgrounds of Section 1.1. Definition 2.16. (1) An element x = n≥0 [xn ]π n ∈ WO E (O F ) is primitive if x0 = 0 and there exists an integer n such that xn ∈ O× F . For such a primitive element x we define deg(x) as the smallest such integer n. (2) A primitive element of strictly positive degree is irreducible if it can not be written as a product of two primitive elements of strictly lower degree. If k F is the residue field of O F there is a projection WO E (O F ) WO E (k F ). Then, x is primitive if and only if x ∈ / π WO E (O F ) and its projection x˜ ∈ WO E (k F ) is non-zero. For such an x, deg(x) = vπ (x). ˜ We deduce from this that the product of a degree d by a degree d primitive element is a degree d +d primitive element. Degree 0 primitive elements are the units of WO E (O F ). Any primitive degree 1 element is irreducible. In terms of Newton polygons, x ∈ WO E (O F ) is primitive if and only if Newt(x)(0) = +∞ and Newt(x)(t) = 0 for t 0. Definition 2.17. Define |Y | to be the set of primitive irreducible elements modulo multiplication by an element of WO E (O F )× . There is a degree function deg : |Y | → N≥1 given by the degree of any representative of a class in |Y |. If x is primitive note x¯ ∈ O F for its reduction modulo π. We have |x| ¯ = | y¯ | if y ∈ WO E (O F )× .x. There is thus a function · : |Y | −→ ]0, 1[ WO E (O F )× .x −→ |x| ¯ 1/ deg(x) . A primitive element x ∈ WO E (O F ) of strictly positive degree is irreducible if and only if the ideal generated by x is prime. In fact, if x = yz with y, z ∈ WO E (O F ) and x primitive then projecting to W O E (k F ) and O F the

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preceding equality one obtains that y and z are primitive. There is thus an embedding |Y | ⊂ Spec(WO E (O F )). The Frobenius ϕ induces a bijection ∼

ϕ : |Y | − → |Y | that leaves invariant the degree and satisfies ϕ(y) = yq . Remark 2.18. When E = Fq ((π )), replacing WO E (O F ) by O F z in the preceding definitions (we set z = π ) there is an identification |Y | = |D∗ |. In fact, according to Weierstrass, any irreducible primitive f ∈ O F z has a unique irreducible unitary polynomial P ∈ O F [z] in its O F z× -orbit satisfying: P(0) = 0 and the roots of P have absolute value < 1. Then for y ∈ |D∗ |, deg(y) = [k(y) : F] and y is the distance from y to the origin of the disk D.

2.2. Background on the ring R For an O E -algebra A set q R(A) = x (n) n≥0 | x (n) ∈ A, x (n+1) = x (n) . If A is π -adic, I is a closed ideal of A such that A is I + (π )-adic, then the reduction map induces a bijection ∼

R(A) −→ R(A/I ) with inverse given by

x (n)

n≥0

−→

lim

k→+∞

x (n+k)

q k n≥0

,

where x (n+k) ∈ A is any lift of x (n+k) ∈ A/I , and the preceding limit is for the I + (π )-adic topology. In particular, applying this for I = (π ), we deduce that the set valued functor R factorizes canonically as a functor R : π -adic O E -algebras −→ perfect Fq -algebras. If WO E stands for the (ramified) Witt vectors there is then a couple of adjoint functors π -adic O E -algebras o

R WO E

/

perfect Fq -algebras

Vector bundles on curves and p-adic Hodge theory

33

where WO E is left adjoint to R and the adjunction morphisms are given by: ∼ A −→ R WO E (A) −n a −→ a q n≥0

and θ : WO E (R(A)) −→ A [xn ]π n −→ x n(0) π n . n≥0

n≥0

If L|Q p is a complete valued extension for a valuation w : L → R ∪ {+∞} extending a multiple of the p-adic valuation, then R(L) equipped with the valuation x −→ w(x (0) ) is a characteristic p perfect complete valued field with ring of integers R(O L ) (one has to be careful that the valuation on R(L) may be trivial). It is not very difficult to prove that if L is algebraically closed then R(L) is too. A reciprocal to this statement will be stated in the following sections.

2.3. The case when is F algebraically closed Theorem 2.19. Suppose F is algebraically closed. Let p ∈ Spec(WO E (O F )) generated by a degree one primitive element. Set A = WO E (O F )/p and θ : WO E (O F ) A the projection. The following properties are satisfied: (1) There is an isomorphism ∼

O F −→ R(A) −n x −→ θ x q

n≥0

.

(2) The map O F −→ A x −→ θ ([x]) is surjective. (3) There is a unique valuation w on A such that for all x ∈ O F , w(θ ([x])) = v(x).

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Moreover, A[ π1 ] equipped with the valuation w is a complete algebraically closed extension of E with ring of integers A. There is an identification of valued fields F = R A[ π1 ] . (4) If π ∈ R(A) is such that π (0) = π then p = ([π] − π ). Remark 2.20. One can reinterpret points (2) and (4) of the preceding theorem in the following way. Let x be primitive of degree 1. Then: • we have a Weierstrass division in WO E (O F ): given y ∈ WO E (O F ), there exists z ∈ WO E (O F ) and a ∈ O F such that y = zx + [a], • we have a Weierstrass factorization of x: there exists u ∈ WO E (O F )× and b ∈ O F such that x = u.(π − [b]). One has to be careful that, contrary to the classical situation, the remainder term a is not unique in such a Weierstrass division. Similarly, b is not uniquely determined by x in the Weierstrass factorization. Indications on the proof of Theorem 2.19. Statement (1) is an easy consequence of general facts about the ring R recalled in Section 2.2: if p = (x) ∼

R(A) −→ R(A/π A) = R(O F /x) ¯ = OF since O F is perfect. According to point (1), point (2) is reduced to proving that any element of A has a q-th root. If E 0 = W (Fq )Q , the norm map N E/E 0 induces a norm map WO E (O F ) → W (O F ) sending a primitive element of degree 1 to a primitive element of degree 1. Using this one can reduce the problem to the case E = Q p . Suppose p = 2 to simplify. Then any element of 1 + p 2 W (O F ) has a p-th root. Using this fact plus some elementary manipulations one is reduced to solving some explicit equations in the truncated Witt vectors of length 2, W2 (O F ). One checks this is possible, using the fact that F is algebraically closed (here this hypothesis is essential, the hypothesis F perfect is not sufficient). In fact, the preceding proof gives that for any integer n, any element of A has an n-th root, the case when n is prime to p being easier than the case n = p we just explained (since then any element of 1 + pWO E (O F ) has an n-th root).

Vector bundles on curves and p-adic Hodge theory

35

Point (4) is an easy consequence of the following classical characterization (0) n of ker θ: an element y = ∈ π A× is a n≥0 [yn ]π ∈ ker θ such that y0 generator of ker θ . In fact, if y is such an element then ker θ = (y) + π ker θ and one concludes ker θ = (y) by applying the π -adic Nakayama lemma (ker θ is π -adically closed). In point (3), the difficulty is to prove that the complete valued field L = A[ π1 ] is algebraically closed; other points following easily from point (2). Using the fields of norms theory one verifies that L contains an algebraic closure of Q p . More precisely, one can suppose thanks to point (4) that p = (π − [π ]). There is then an embedding Fq ((π )) ⊂ F that induces F := Fq ((π )) ⊂ F. This induces a morphism L = WO E (O F ) π1 /(π −[π]) → L. But thanks to the fields of norms theory, L is the completion of an algebraic closure of E. In particular, L contains all roots of unity. Let us notice that since O L /π O L = O F /πO F , the residue field of L is the same as the one of F and is thus algebraically closed. Now, we use the following proposition that is well known in the discrete valuation case thanks to the theory of ramification groups (those ramification groups do not exist in the non discrete valuation case, but one can define some ad hoc one to obtain the proposition). Proposition 2.21. Let K be a complete valued field for a rank 1 valuation and K |K a finite degree Galois extension inducing a trivial extension on the residue fields. Then the group Gal(K |K ) is solvable. Since for any integer n any element of L has an n-th root, one concludes L is algebraically closed using Kümmer theory. Note now OC = WO E (O F )/p with fraction field C. One has to be careful that the valuation w on C extends only a multiple of the π-adic valuation of E: q −w(π ) = p. The quotient morphism Bb,+ −→ C extends in fact by continuity to surjective morphisms / Bb / B /B Bb,+E EE nn I ~ n ~ n EE ~~ nnn EE EE" ~ ~~n~nnnn " ~~ wnw n C where I ⊂]0, 1[ is such that p ∈ I . This is a consequence of the inequality q −w( f ) ≤ | f |ρ for f ∈ Bb and ρ = p. If p = (x) then all kernels of those surjections are the principal ideals generated by x in those rings.

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Convention: we will now see |Y |deg=1 as a subset of Spm(B). For m ∈ |Y |deg=1 , we note Cm = B/m, θm : B Cm and vm the valuation such that vm (θm ([x])) = v(x).

Theorem 2.22. If F is algebraically closed then the primitive irreducible elements are of degree one. Remark 2.23. One can reinterpret the preceding theorem as a factorization statement: if x ∈ WO E (O F ) is primitive of degree d then x = u.(π − [a1 ]) . . . (π − [ad ]), u ∈ WO E (O F )× , a1 , . . . , ad ∈ m F \ {0}. Indications on the proof of theorem 2.22. Let f ∈ WO E (O F ) be primitive. The theorem is equivalent to saying that f “has a zero in |Y |deg=1 ” that is to say there exists m ∈ |Y |deg=1 such that θm ( f ) = 0. The method to construct such a zero is a Newton type method by successive approximations. To make it work we need to know it converges in a sense that has to be specified. We begin by proving the following. Proposition 2.24. For m1 , m2 ∈ |Y |deg=1 set d(m1 , m2 ) = q −vm1 (a) if θm1 (m2 ) = OCm1 a. Then: deg=1 . (1) This defines an ultrametric distance on |Y | (2) For any ρ ∈]0, 1[, |Y |deg=1,·≥ρ , d is a complete metric space.

Remark 2.25. In equal characteristic, if E = Fq ((π )), then |Y | = |D∗ | = m F \{0} and this distance is the usual one induced by the absolute value |·| of F. The approximation algorithm then works like this. We define a sequence (mn )n≥1 of |Y |deg=1 such that: • (mn )n≥1 is constant, • it is a Cauchy sequence, • lim vmn ( f ) = +∞. n→+∞

Write f = k≥0 [xk ]π k . The Newton polygon of f as defined in Section 1.2.2 k is the same as the Newton polygon of g(T ) = k≥0 x k T ∈ O F T . Let z ∈ m F be a root of g(T ) with valuation the smallest one among the valuations

Vector bundles on curves and p-adic Hodge theory

37

of the roots of g(T ) (that is to say the smallest non-zero slope of Newt(f)). Start with m1 = (π − [z]) ∈ |Y |deg=1 . If mn is defined, mn = (ξ ) with ξ primitive of degree 1, we can write f = [ak ]ξ k k≥0

in WO E (O F ) (this is a consequence of point (2) of Theorem 2.19 and the fact that WO E (O F ) is ξ -adic). We check the power series h(T ) = k≥0 ak T k ∈ O F T is primitive of degree d. Let z be a root of h(T ) of maximal valuation. Then ξ − [z] is primitive of degree 1 and we set mn+1 = (ξ − [z]). We then prove the sequence (mn )n≥1 satisfies the required properties.

2.4. Parametrization of |Y | when F is algebraically closed Suppose F is algebraically closed. We see |Y | as a subset of Spm(B). As we saw, for any m ∈ |Y | there exists a ∈ m F \ {0} such that m = (π − [a]). The problem is that such an a is not unique. Moreover, given a, b ∈ m F \ {0}, there is no simple rule to decide whether (π − [a]) = (π − [b]) or not. Here is a solution to this problem. Let LT be a Lubin–Tate group law over O E . We note Q = [π]LT ∈ O E T and G the associated formal group on Spf(O E ). We have G(O F ) = m F , + . LT

The topology induced by the norms (| · |ρ )ρ∈]0,1[ on WO E (O F ) is the “weak topology” on the coefficients of the Teichmüller expansion, that is to say the product topology via the bijection ∼

ON F −→ WO E (O F ) (xn )n≥0 −→ [xn ]π n . n≥0

If a ∈ m F \ {0}, this coincides with the ([a], π )-adic topology. Moreover WO E (O F ) is complete, that is to say closed in B. If WO E (O F )00 = [xn ]π n | x0 ∈ m F = {x ∈ WO E (O F ) | x mod π ∈ m F }, n≥0

then

G WO E (O F ) = WO E (O F )00 , + . LT

One verifies the following proposition.

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Laurent Fargues and Jean-Marc Fontaine

Proposition 2.26. For x ∈ m F , the limit [x] Q := lim [π n ]LT

n→+∞

xq

−n

exists in WO E (O F ), reduces to x modulo π , and defines a “lift” [·] Q : G(O F ) → G(WO E (O F )). The usual Teichmüller lift [·] is well adapted to the multiplicative group law: [x y] = [x].[y]. The advantage of the twisted Teichmüller lift [·] Q is that it is more adapted to the Lubin–Tate one: [x] Q + [y] Q = x + y Q . LT

LT

When E = Q p and LT = Gm one has [x] Q = [1 + x] − 1. Definition 2.27. For ∈ m F \ {0} define u =

[] Q ∈ WO E (O F ). [ 1/q ] Q

This is a primitive degree one element since it is equal to the power series For example, if E = Q p and LT = Gm , setting

Q(T ) evaluated at [ 1/q ] Q . T = 1 + one has

1 p−1 u = 1 + p + · · · + p .

Proposition 2.28. There is a bijection ∼ G(O F ) \ {0} /O× E −→ |Y |

O× E . −→ (u ).

The inverse of this bijection is given by the following rule. For m ∈ |Y |, define ! X (G)(OCm ) = (xn )n≥0 | xn ∈ G(OCm ), π.xn+1 = xn . More generally, X (G) will stand for the projective limit “lim G” where the ←− n≥0

transition mappings are multiplication by π (one can give a precise geometric meaning to this but this is not our task here, see [12] for more details). The reduction modulo π map induces a bijection ∼

X (G)(OCm ) −→ X (G)(OCm /πOCm ) with inverse given by (xn )n →

lim π k xˆn+k

k→+∞

n

Vector bundles on curves and p-adic Hodge theory

39

where xˆn+k is any lift of xn+k . But [π]LT modulo π is the Frobenius Frobq . We thus have X (G)(OCm /πOCm ) = G R(OCm /π OCm ) = G(O F ) since ∼

∼

O F −−→ R(OCm ) −−→ R(OCm /π OCm ). θm ◦[·]

The Tate module Tπ (G) = {(xn )n≥0 ∈ X (G)(OCm ) | x0 = 1} ⊂ X (G)(OCm ) embeds thus in G(O F ). This is a rank 1 sub-O E -module. The inverse of the bijection of Proposition 2.28 sends m to this O E -line in G(O F ).

2.5. The general case: Galois descent Let now F be general (but still perfect), that is to say not necessarily algebraically closed. Let F be an algebraic closure of F with Galois group G F = Gal(F |F). Since the field F will vary we now " put " a subscript in the " is equipped with an preceding notation to indicate this variation. The set "Y

F action of G F . Theorem 2.29. For p ∈ |Y F | ⊂ Spec(WO E (O F )) set L = WO E (O F )/p π1 and θ : Bb,+ F L. Then: (1) There is a unique valuation w on L such that for x ∈ O F , w(θ ([x])) = v(x). (2) (L , w) is a complete valued extension of E. (3) (L , w) is perfectoid in the sense that the Frobenius of O L /π O L is surjective. (4) Via the embedding O F −→ R(L) −n a −→ θ a q

n≥0

one has R(L)|F and this extension is of finite degree [R(L) : F] = deg p.

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Laurent Fargues and Jean-Marc Fontaine

Remark 2.30. What we call here perfectoid is what we called “strictly p-perfect” in [13] and [14]. The authors decided to change their terminology because meanwhile the work [29] appeared. Indications on the proof. The "proof " is based on Theorems 2.19 and 2.22 via " " to |Y F |. For this we need the following. a Galois descent argument from Y

F Theorem 2.31. One has m F .H 1 G F , O

= 0. F Using Tate’s method ([31]) this theorem is a consequence of the following “almost etalness” statement whose proof is much easier than in characteristic 0. Proposition 2.32. If L|F is a finite degree extension then m F ⊂ tr L/F (O L ). Sketch of proof. The trace tr L/F commutes with the Frobenius ϕ = Frobq . Choosing x ∈ O L such that tr L/F (x) = 0 one deduces that lim |tr L/F (ϕ −n (x))| = 1.

n→+∞

From Theorem 2.31 one deduces that H 1 G F , m

= 0 which implies F H 1 G F , 1 + WO E m

=0 (2.4) F where

× × 1 + WO E m

= ker W O −→ W kF .

O O E E F F

Finally, let us notice that thanks to Ax’s theorem (that can be easily deduced from 2.32) one has G F WO E O

= WO E (O F ). F Proposition 2.33. Let x ∈ WO E O

be a primitive element such that ∀σ ∈ F G F , (σ (x)) = (x) as an ideal of WO E O

. Then, there exists y ∈ WO E (O F ) F such that (x) = (y). Proof. If x˜ ∈ WO E (k F ) is the projection of x via WO E (O

) → WO E (k F ), up F deg x to multiplying x by a unit, one can suppose x˜ = π . Looking at the cocycle σ → σ (x) , the proposition is then a consequence of the vanishing (2.4). x Let now x ∈ WO E (O F ) be primitive irreducible of degree d. There exist y1 , . . . , yr ∈ WO E O

primitive of degree 1 satisfying (yi ) = (y j ) for i = j, F a1 , . . . , ar ∈ N≥1 and u ∈ WO E (O

)× such that F x = u.

r i=1

yiai .

Vector bundles on curves and p-adic Hodge theory

The finite subset (yi )

! 1≤i ≤r

41

⊂ |Y

| F

is stable under G F . Using Proposition 2.33 and the irreducibility of x one verifies that this action is transitive and a1 = · · · = ar = 1. In particular one has r = d. Note mi = (yi ) and C mi the associated algebraically closed residue field. Let K |F be the degree d extension of F in F such that G K = StabG F (m1 ). One has Bb,+

/(x) = F

and thus

d

Cmi

i =1

G F GK Bb,+ = Cm .

/(x) 1 F

Now, one verifies using that x is primitive that if a ∈ m F \ {0} then 1 WO E (O F )/(x) π1 = WO E (O F )/(x) [a] . Theorem 2.31 implies that for such an a, [a].H 1 G F , WO E O

= 0. F One deduces from this that

G F b,+ L = Bb,+ /(x) = B /(x)

F F

that is thus a complete valued field. Moreover GK GK R(L) = R Cm = R(Cm1 )G K =

F = K. 1 Other statements of Theorem 2.29 are easily deduced in the same way. In the preceding theorem the quotient morphism θ : Bb,+ F L extends by continuity to a surjection B F L with kernel the principal ideal B F p. From now on, we will see |Y F | as a subset of Spm(B). If m ∈ |Y F | we note L m = B F /m, θm : B L m . The preceding arguments give the following. " "G F −fin " " " " whose G F -orbit is Theorem 2.34. Let "Y

be the elements of "Y

F F finite. There is a surjection " "G F −fin " β : "Y

−→ |Y F | F

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Laurent Fargues and Jean-Marc Fontaine

whose fibers are the G F -orbits: " "G F −fin ∼ "Y " /G F −→ |Y F |. F Moreover: • for m ∈ |Y F |, one has #β −1 (m) = [R(L m ) : F] = deg m, " "G F −fin " • for n ∈ "Y

, if m = β(n), one has an extension Cn |L m that identifies F Cn with the completion of the algebraic closure L m of L m and ∼ Gal F |R(L m ) −→ Gal L m |L m . Remark 2.35. (1) One has to be careful that θm (WO E (O F )) ⊂ O L m is only an order. It is equal to O L m if and only if deg m = 1. Using Theorem 2.31 one can verify that this order contains the maximal ideal of O L m . (2) Contrary to the case when F is algebraically closed, in general an m ∈ |Y F | of degree 1 is not generated by an element of the form π −[a], a ∈ m F \{0}.

2.6. Application to perfectoid fields Reciprocally, given a complete valued field L|E for a rank 1 valuation, it is perfectoid if and only if the morphism θ : WO E (R(O L )) −→ O L is surjective. In this case one can check that the kernel of θ is generated by a primitive degree 1 element. The preceding considerations thus give the following. Theorem 2.36. (1) There is an equivalence of categories between perfectoid fields L|E and the category of couples (F, m) where F|Fq is perfectoid and m ∈ |Y F | is of degree 1. (2) In the preceding equivalence, if L corresponds to (F, m), the functor R induces an equivalence between finite étale L-algebras and finite étale F-algebras. The inverse equivalence sends the finite extension F |F to B F /B F m.

Vector bundles on curves and p-adic Hodge theory

43

Example 2.37. (1) The perfectoid field L = Q p (ζ p∞ ) corresponds to F = F p ((T ))perf and p−1 m = 1 + [T 1/ p + 1] + · · · + [T p + 1] .

with (2) Choose p ∈ R(Q p ) such that p (0) = p. The perfectoid field L = M (n) perf M = ∪n≥0 Q p ( p ) corresponds to F = F p ((T )) and m = ([T ] − p). Remark 2.38. In the preceding correspondence, F is maximally complete if and only if L is. In particular, one finds back the formula given 1 in [26] for p-adic maximally complete fields: they are of the form W (O F ) p /([x] − p) where F is maximally complete of characteristic p and x ∈ F × satisfies v(x) > 0. Remark 2.39. Suppose F = F p ((T )). One can ask what are the algebraically closed residue fields Cm up to isomorphism when m goes through |Y F |. Let us note C p the completion of an algebraic closure of Q p . Thanks to the fields of norms theory it appears as a residue field Cm for some m ∈ |Y F |. The question is: is it true that for all m ∈ |Y F |, Cm C p ? The authors do not know the answer to this question. They know that for each integer n ≥ 1, OCm / p n OCm OC p / p n OC p but in a non canonical way. As a consequence of the preceding theorem we deduce almost étalness for characteristic 0 perfectoïd fields. Corollary 2.40. For L|E a perfectoïd field and L |L a finite extension we have m L ⊂ tr L |L (O L ). Proof. Set F = R(L ) and F = R(L). If L corresponds to m ∈ |Y F |, a = {x ∈ O F | |x| ≤ m} and a = {x ∈ O F | |x| ≤ m} we have identifications O L /πO L = O F /a O L /π O L = O F /a . According to point (2) of the preceding theorem, with respect to those identifications the map tr L |L modulo π is induced by the map tr F |F . The result is thus a consequence of Proposition 2.32. Remark 2.41. (1) Let K be a complete valued extension of Q p with discrete valuation and perfect residue field and M|K be an algebraic infinite degree arithmetically profinite extension. Then, by the fields of norms theory ([32]),

is perfectoid and point (2) of the preceding theorem is already L = M contained in [32].

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Laurent Fargues and Jean-Marc Fontaine

(2) In [29] Scholze has obtained a different proof of point (2) of Theorem 2.36 and of Corollary 2.40. Using Sen’s method ([30]), Corollary 2.40 implies the following. Theorem 2.42. Let L|E be a perfectoid field with algebraic closure L. Then the functor V → V ⊗ L

L induces an equivalence of categories between finite dimensional L-vector spaces and finite dimensional

L-vector spaces equipped with a continuous semi-linear action of Gal(L|L). An inverse is given by the functor W → W Gal(L|L) . Using this theorem one deduces by dévissage the following that we will use later. Theorem 2.43. Let m ∈ |Y F | and note B+ = B+

F ,d R,m

m ∈|Y | F β(m )=m

F ,d R,m

the B

m-adic completion of B

(see Definition 2.44). Then the functor F F M −→ M ⊗B+

F,d R,m

B+

F ,d R,m

induces an equivalence of categories between finite type B+ F,d R,m -modules and + finite type B

-modules equipped with a continuous semi-linear action of F ,d R,m

Gal(F |F). An inverse is given by the functor W → W Gal( F |F ) .

3. Divisors on Y 3.1. Zeros of elements of B We see |Y | as a subset of Spm(B). For m ∈ |Y |, we set L m = B/m and θm : B L m . We note vm the valuation on L m such that vm (θm ([a])) = v(a). One has m = q −vm (π )/ deg m where . was defined after Definition 2.17. Definition 2.44. For m ∈ |Y | define Bd+R,m as the m-adic completion of B.

Vector bundles on curves and p-adic Hodge theory

45

The ring B+ d R,m is a discrete valuation ring with residue field L m and the + natural map B → B+ d R,m is injective. We note again θm : Bd R,m → L m . We note ordm : B+ d R,m −→ N ∪ {+∞} its normalized valuation. Example 2.45. If E = Fq ((π )) then |Y | = |D∗ | and if m ∈ |Y | corresponds ∗ to x ∈ |D∗ | then B+ d R,m = OD ,x . Theorem 2.46. For f ∈ B, the non-zero finite slopes of Newt(f) are the − logq m with multiplicity ordm ( f ) deg(m) where m goes through the elements of |Y | such that θm ( f ) = 0. Indications on the proof. It suffices to prove that for any finite non-zero slope λ of Newt(f) there exists m ∈ |Y | such that q −λ = m and θm ( f ) = 0. As in Proposition 2.24 there is a metric d on |Y | such that for all ρ ∈]0, 1[, {m ≥ ρ} is complete. For m1 , m2 ∈ |Y |, if θm1 (m2 ) = O L m1 x then d(m1 , m2 ) = q −vm1 (x)/ deg m2 . We begin with the case when f = [xn ]π n ∈ WO E (O F ). n≥0

If d ≥ 0 set fd =

d

[xn ]π n .

n=0

For d 0, λ appears as a slope of Newt(fd ) with the same multiplicity as in Newt(f). For each d, f d = [ad ].gd for some ad ∈ O F and gd ∈ WO E (O F ) primitive. Thanks to the preceding results, we already know the result for each gd . Thus, setting X d = {m ∈ |Y | | m = q −λ and θm ( f d ) = 0}, we know that for d 0, X d = ∅. Moreover #X d is bounded when d varies. Now, if m ∈ X d , looking at vm (θm ( f d+1 )), one verifies that there exists m ∈ X d+1 such that d(m, m ) ≤ q

−

(d+1)λ−v(x0 ) #X d+1

.

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From this one deduces there exists a Cauchy sequence (md )d0 where md ∈ X d . If m = lim md then θm ( f ) = 0. This proves the theorem when d→+∞

f ∈ WO E (O F ) and thus when f ∈ Bb . The general case is now obtained in the same way by approximating f ∈ B by a converging sequence of elements of Bb . Example 2.47. As a corollary of the preceding theorem and Proposition 2.14, for f ∈ B \ {0} one has f ∈ B × if and only if for all m ∈ |Y |, θm ( f ) = 0.

3.2. A factorization of elements of B when F is algebraically closed Set B[0,1[ = { f ∈ B | ∃A, Newt(f)|]−∞,A] = +∞} "" v(x n ) = [xn ]π n ∈ WO E (F) π1 " liminf ≥0 n→+∞ n n−∞ (see Proposition 2.14 for this equality). Suppose F is algebraically closed. Given f ∈ B, applying Theorem 2.46, if λ1 , . . . , λr > 0 are some slopes of Newt(f) one can write f = g.

r

1−

i=1

[ai ] π

where v(ai ) = λi . Using this one proves the following. Theorem 2.48. Suppose F is algebraically closed. For f ∈ B there exists a sequence (ai )i≥0 of elements of m F going to zero and g ∈ B[0,1[ such that f = g.

+∞ i=0

1−

[ai ] . π

If moreover f ∈ B+ there exists such a factorization with g ∈ WO E (O F ).

3.3. Divisors and closed ideals of B Definition 2.49. Define " Div+ (Y ) = am [m] " ∀I ⊂]0, 1[ m∈|Y |

compact {m | am = 0 and m ∈ I } is finite .

Vector bundles on curves and p-adic Hodge theory For f ∈ B \ {0} set div( f ) =

47

ordm ( f )[m] ∈ Div+ (Y ).

m∈|Y |

Definition 2.50. For D ∈ Div+ (Y ) set a−D = { f ∈ B | div( f ) ≥ D}, an ideal of B. For each m ∈ |Y |, the function ordm : B → N ∪ {+∞} is upper semicontinuous. From this one deduces that the ideal a−D is closed in B. Theorem 2.51. The map D → a−D induces an isomorphism of monoids between Div+ (Y ) and the monoid of closed non-zero ideals of B. Moreover, D ≤ D if and only if a−D ⊂ a−D . If a is a closed ideal of B then ∼

a −→ lim B I a. ←− I ⊂]0,1[

The result is thus a consequence of the following. Theorem 2.52. For a compact non-empty interval I ⊂]0, 1[: • if I = {ρ} with ρ ∈ / |F × | then B I is a field • if not then the ring B I is a principal ideal domain with maximal ideals {B I m | m ∈ |Y |, m ∈ I }. Sketch of proof. The proof of this theorem goes as follows. First, given f ∈ B I , one can define a bounded Newton polygon NewtI (f). If f ∈ Bb then NewtI (f) is obtained from Newt(f) by removing the slopes λ such that q −λ ∈ / I (if there is no such slope we define NewtI (f) as the empty polygon). Now, if f ∈ B the method used in Definition 2.11 to define the Newton polygon does not apply immediately. For example, if I = {ρ} and f ∈ Bb then | f |ρ does not determine NewtI (f) (more generally, if I = [q −λ1 , q −λ2 ] one has the same problem with the definition of the pieces of NewtI (f) where the slopes are λ1 and λ2 ). But one verifies that if f n −→ f in B I with n→+∞

f n ∈ Bb then for n 0, NewtI (fn ) is constant and does not depend on the sequence of Bb going to f . Then, if f ∈ B I we prove a theorem that is analogous to Theorem 2.46: the slopes of NewtI (f) are the − logq m with multiplicity ordm ( f ) deg m where

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Laurent Fargues and Jean-Marc Fontaine

m ∈ I and θm ( f ) = 0. This gives a factorization of any f ∈ B I as a product r f = g. ξi i =1

where the ξi are irreducible primitive, ξi ∈ I , and g ∈ B I satisfies NewtI (g) = ∅. Finally, we prove that if f ∈ B I satisfies NewtI (f) = ∅ then f ∈ B× I . For f ∈ Bb this is verified by elementary manipulations. Then if f n −→ f in n→+∞

B I with f n ∈ Bb , since NewtI (f) = ∅ for n 0 one has NewtI (fn ) = ∅. But then for n 0 and ρ ∈ I , −1 −1 | f n+1 − f n−1 |ρ = | f n+1 |−1 ρ .| f n |ρ .| f n+1 − f n |ρ −→ 0. n→+∞

Thus the sequence

( f n−1 )n0

of B I converges towards an inverse of f .

Example 2.53. For f, g ∈ B \ {0}, f is a multiple of g if and only if div( f ) ≥ div(g). In particular there is an injection of monoids div : B \ {0}/B× −→ Div+ (Y ). Corollary 2.54. The set |Y | is the set of closed maximal ideals of B. Remark 2.55. Even when F is spherically complete we do not know whether div : B× → Div+ (Y ) is surjective or not (see 2.2).

4. Divisors on Y/ϕ Z 4.1. Motivation Suppose we want to classify ϕ-modules over B, that is to say free B-modules equipped with a ϕ-semi-linear automorphism. This should be the same as vector bundles on Y /ϕ Z where Y is this “rigid” space we did not really define but that should satisfy • (Y, OY ) = B • |Y | is the set of “classical points” of Y . Whatever this space Y is, since ϕ(m) = mq , ϕ acts in a proper discontinuous way without fixed point on it. Thus, Y /ϕ Z should have a sense as a “rigid” space. Let’s look in more details at what this space Y /ϕ Z should be.

Vector bundles on curves and p-adic Hodge theory

49

It is easy to classify rank 1 ϕ-modules over B. They are parametrized by Z: to n ∈ Z one associates the ϕ-module with basis e such that ϕ(e) = π n e. We thus should have ∼

Z −→ Pic(Y/ϕ Z ) n −→ L ⊗n where L is a line bundle such that for all d ∈ Z, H 0 (Y/ϕ Z , L ⊗d ) = Bϕ=π . d

If E = Fq ((π )) and F is algebraically closed Hartl and Pink classified in [20] the ϕ-modules over B, that is to say ϕ-equivariant vector bundles on D∗ . The first step in the proof of their classification ([20] theo.4.3) is that if (M, ϕ) is such a ϕ-module then for d 0 M ϕ=π = 0. d

The same type of result appears in the context of ϕ-modules over the Robba ring in the work of Kedlaya (see for example [22] prop. 2.1.5). From this one deduces that the line bundle L should be ample. We are thus led to study the scheme # d Proj Bϕ=π d≥0

for which one hopes it is “uniformized by Y ” and allows us to study ϕ-modules over B. In fact if (M, ϕ) is such a ϕ-module, we hope the quasi-coherent sheaf # d M ϕ=π d≥0

is the vector bundle associated to the ϕ-equivariant vector bundle on Y attached to (M, ϕ).

4.2. Multiplicative structure of the graded algebra P Definition 2.56. Define P=

#

Bϕ=π

d

d≥0

as a graded E-algebra. We note Pd = Bϕ=π the degree d homogeneous elements. d

In fact we could replace B by B+ in the preceding definition since Bϕ=π = (B+ )ϕ=π d

(Corollary 2.15). One has P0 = E.

d

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Laurent Fargues and Jean-Marc Fontaine

Definition 2.57. Define Div+ (Y/ϕ Z ) = {D ∈ Div+ (Y ) | ϕ ∗ D = D}. There is an injection |Y |/ϕ Z −→ Div+ (Y /ϕ Z ) m −→ [ϕ n (m)] n∈Z

that makes Div+ (Y/ϕ Z ) a free abelian monoid on |Y |/ϕ Z . If x ∈ Bϕ=π \ {0} then div(x) ∈ Div+ (Y /ϕ Z ). d

Theorem 2.58. If F is algebraically closed the morphism of monoids div : Pd \ {0} /E × −→ Div+ (Y/ϕ Z ) d≥0

is an isomorphism. Let us note the following important corollary. Corollary 2.59. If F is algebraically closed the graded algebra P is graded factorial with irreducible elements of degree 1. In the preceding theorem, the injectivity is an easy application of Theorem 2.51. In fact, if x ∈ Pd and y ∈ Pd are non-zero elements such that div(x) = div(y) then x = uy with u ∈ B× . But B× = (Bb )× (see the comment after Proposition 2.14). Thus, $ 0 if d = d b ϕ=π d−d u ∈ (B ) = E if d = d . The surjectivity uses Weierstrass products. For this, let x ∈ WO E (O F ) be a primitive degree d element and D = div(x) its divisor. We are looking for f ∈ Pd \ {0} satisfying div( f ) = ϕ n (D). n∈Z

Up to multiplying x by a unit we can suppose x ∈ π d + WO E (m F ). Then the infinite product + (x) =

ϕ n (x) n≥0

πd

Vector bundles on curves and p-adic Hodge theory

51

converges. For example, if x = π − [a], + (π − [a]) =

n [a q ] 1− . π

n≥0

One has div(+ (x)) =

ϕ n (D).

n≥0

We then would like to define “− (x) =

ϕ n (x)”

n0

n≥0

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for any ρ0 . Moreover, for any x ∈ B[ρ,1[ there is defined a Newton polygon Newt(x) and ! B+ ρ = x ∈ B[ρ,1[ | Newt(x) ≥ 0 ! B+ = x ∈ B | Newt(x) ≥ 0 . In fact, by concavity of the Gauss valuation r → vr (x), for any x ∈ B[ρ,1[ the limit |x|1 := lim |x|ρ ρ→1

exists in [0, +∞] and equals q −v0 (x) for x ∈ Bb and ! B+ ρ = x ∈ B[0,ρ[ | |x|1 ≤ 1 . Let us note |x|1 := q −v0 (x) . One has v0 (x) = lim Newt(x) +∞

B+ ρ , v0

and on is a valuation extending the valuation previously defined on Bb . One has to be careful that, contrary to B, the Frechet algebra B+ is not Frechet–Stein since the rings B+ ρ are not noetherian and it is not clear whether + + + ϕ : Bρ → Bρ (that is to say the inclusion B+ ρ q ⊂ Bρ ) is flat or not. Note ϕ-ModB+ and ϕ-ModB+ρ for the associated categories of finite rank free modules equipped with a semi-linear isomorphism. The category ϕ-ModB+ρ does not depend on ρ. There is a scalar extension functor ϕ-ModB+ −→ ϕ-ModB and, using Proposition 2.123, a functor ϕ-ModB+ρ −→ ϕ-ModB (M, ϕ) −→ ϕ n M ⊗B+ρ B[ρ,1[ . n≥0

Proposition 2.129. The functors ϕ-ModB+ → ϕ-ModB and ϕ-ModB+ρ → ϕ-ModB are fully faithful. Proof. Let’s treat the case of ϕ-ModB+ , the case of ϕ-ModB+ρ being identical. Using internal Hom’s this is reduced to proving that for (M, ϕ) ∈ ϕ-ModB+ one has ϕ=I d ∼ M ϕ=I d −→ M ⊗B+ B . Let us fix a basis of M and for x = (x1 , . . . , xn ) ∈ Bn M ⊗ B and r > 0 set Wr (x) = inf vr (xi ). 1≤i ≤n

Vector bundles on curves and p-adic Hodge theory

99

An element a ∈ B+ satisfies vr (a) ≥ 0 for r 0. Let us fix r0 0 such that all the coefficients (ai j )i, j of the matrix of ϕ in the fixed basis of M satisfy vr0 (ai, j ) ≥ 0. Then if x ∈ M ⊗ B satisfies ϕ(x) = x one has qW r0 (x) = Wr0 (ϕ(x)) ≥ Wr0 (x) q

and thus for k ≥ 1 W r0n (x) ≥ q

1 Wr (x). qn 0

Taking the limit when n → +∞ one obtains W0 (x) ≥ 0 that is to say x ∈ (B+ )n M. The preceding proposition together with Theorem 2.126 then gives the following. Theorem 2.130. Suppose F is algebraically closed. For A ∈ {B+ , B+ ρ } there is a bijection ∼

{λ1 ≥ · · · ≥ λn | n ∈ N, λi ∈ Q} −→ ϕ-Mod A / ∼ n # (λ1 , . . . , λn ) −→ A(−λi ) . i =1

One deduces there are equivalences of categories ∼

∼

∼

ϕ-ModB+ −→ ϕ-ModB+ρ −→ ϕ-ModB −→ ϕ-ModR F where an inverse of the first equivalence is given by M → ∩n≥0 ϕ n (M).

7.8. Another proof of the classification of ϕ-modules over B+ and B+ ρ We explain how to give a direct proof of Theorem 2.130 without using Kedlaya’s Theorem 2.121. This proof is much simpler and in fact applies even if the field F is not algebraically closed: Theorem 2.131. Theorem 2.130 remains true for any F with algebraically closed residue field. This relies on the introduction of a new ring called B. Set ! P = x ∈ Bb,+ | v0 (x) > 0 ! = [xn ]π n |x n ∈ O F , ∃C > 0, ∀n, v(xn ) ≥ C n−∞

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and B = Bb,+ /P. If k stands for the residue field of O F , there is a reduction morphism Bb,+ −→ WO E (k)Q . Then, B is a local ring with residue field WO E (k)Q . Let’s begin by classifying ϕ-modules over B. Theorem 2.132. Any ϕ-module over B is isomorphic to a direct sum of B(λ), λ ∈ Q. Sketch of proof. One first proves that for λ, μ ∈ Q, Ext1 (B(λ), B(μ)) = 0. For two ϕ-modules M and M one has Ext1 (M, M ) = H 1 (M ∨ ⊗ M ) where for a ϕ-module M , H 1 (M ) = coker(I d − ϕ M ). Up to replacing ϕ by a power of itself, that is to say replacing E by an unramified extension, we are thus reduced to proving that for any d ∈ Z, I d − π d ϕ : B −→ B is surjective. For d > 0, this is a consequence of the fact that I d − π d ϕ : WO E (O F ) −→ WO E (O F ) is surjective since π d ϕ is topologically nilpotent on WO E (O F ) for the π -adic topology. For d < 0 this is deduced in the same way using I d − π −d ϕ −1 . For d = 0, this is a consequence of the fact that I d − ϕ is bijective on WO E (m F ) since ϕ is topologically nilpotent on WO E (m F ) for the ([a], π )-adic topology for any a ∈ m F \ {0} (the topology induced by the Gauss norms (|.|ρ )ρ∈]0,1[ ) and since the residue field k of F is algebraically closed. Let (M, ϕ) ∈ ϕ-ModB and Mk = M ⊗ WO E (k)Q be the associated isocrystal. Let λ be the smallest slope of (Mk , ϕ). It suffices now to prove that M has a sub ϕ-module isomorphic to B(λ) whose underlying B-module is a direct factor. Up to raising ϕ to a power and twisting one is reduced to the case λ = 0. Then Mk has a sub-lattice stable under ϕ. Let us remark that any element of Bb,+ whose image in WO E (k)Q lies in WO E (k) is congruent modulo p to an element of WO E (O F ). Lifting the basis of to a basis of M (recall B is a local ring), one then checks there is a free ϕ-module N over WO E (O F ) together with ∼ a morphism N → M inducing an isomorphism N ⊗WO E (O F ) B − → M and such ∼

that N ⊗ WO E (k) − → . For such an N , there is an isomorphism ∼

N ϕ=I d −→ ϕ=I d

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101

such that N ϕ=I d ⊗ E WO E (O F ) is a direct factor in N . In fact, after fixing a basis of N , N WO E (O F )n is complete with respect to the family of norms (.ρ )ρ∈]0,1[ where (x1 , . . . , xn )ρ = sup1≤i≤n |xi |ρ . Moreover the Frobenius of N is topologically nilpotent on WO E (m F )n for this set of norms. The result is deduced (for the direct factor assertion, one has to use that WO E (m F ) is contained in the Jacobson radical of WO E (O F ) together with Nakayama lemma). To make the link between B+ , B+ ρ and B we need the following. Lemma 2.133. For any a ∈ m F \ {0}, B+ = [a]B+ + Bb,+ and B+ ρ = b,+ . [a]B+ + B ρ In fact, for any x = n−∞ [x n ]π n let us note x + = ]π n and n≥0 [x n − n + + x = n 0} + + and the same for B+ ρ . From this we deduce surjections B → B and Bρ → B. Now, Theorem 2.130 is a consequence of Theorem 2.132 and the following.

Proposition 2.134. The reduction functor ϕ-ModB+ → ϕ-ModB is fully faithful. The same holds for B+ ρ. Sketch of proof. We treat the case of B+ , the case of B+ ρ being identical. This is reduced to proving that for (M, ϕ) ∈ ϕ-ModB+ , if M is the associated mod∼ ϕ=I d ule over B, then M ϕ=I d − →M . If = pM this is reduced to proving that ∼

I d − ϕ : −→ . Fix a basis of M (B+ )n and note A = (ai j )i, j ∈ GLn (B+ ) the matrix of ϕ in this base. For each r ≥ 0 and m = (x1 , . . . , xn ) ∈ M set mr = inf1≤i≤n vr (xi ). Let r0 > 0 be fixed. Then = pn is complete with respect to the set of additive norms (.)r>0 . Note Ar = infi, j vr (ai, j ). Fix an r > 0. One first checks that for any m ∈ M and k ≥ 1 ϕ k (m)r ≥ q k m

r qk

+

k−1 i=0

q i A ri . q

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Now, according to inequality (2.3) of Section 1.2.1 (the inequality is stated for Bb,+ but extends by continuity to B+ ), for any r ≤ r one has Ar ≥

r Ar r

and thus k−1

q i A

i =0

r qk

≥ αk + β

for some constants α, β ∈ R. But now, if m ∈ , lim mr = m0 > 0 and r →0

thus

lim ϕ k (m)r = +∞.

k→+∞

We deduce that ϕ is topologically nilpotent on and thus I d − ϕ is bijective on it. Remark 2.135. The preceding proof does not use the fact that F is algebraically closed and thus Theorem 2.130 remains true when F is any perfectoid field with algebraically closed residue field. On this point, there is a big difference between ϕ-modules over B+ and the ones over B. In fact, one can prove that ϕ-modules over B satisfy Galois descent like vector bundles (Theorem 2.103) for any perfectoid field F and thus for a general F Theorem 2.126 is false. Remark 2.136. As a consequence of the classification theorem and the first part of the proof of Theorem 2.132, for any M, M ∈ ϕ-ModB+ , Ext1 (M, M ) = 0. This is not the case for ϕ-modules over B. The equivalence 2.127 is an equivalence of exact categories and for example Ext1ϕ -ModB (B, B(1)) = 0. In fact, although the scalar extension functor ∼

ϕ-ModB+ − → ϕ-ModB is exact, its inverse is not. Let us conclude with a geometric interpretation of the preceding result. Set = Spec(Z p ) and for an F p -scheme S note F -IsocS/ for the category of F-isocrystals. If S → S is a thickening then F-Isoc S/ F-Isoc S / . Let now a ∈ m F \ {0}, ρ = |a| and Sρ = Spec(O F /O F a). The category F-Isoc Sρ / does not depend on the choice of ρ ∈]0, 1[. The crystalline site Cris(Sρ /) has an initial object Acris,ρ such that Acris,ρ 1p = Bcris,ρ (see Section 1.2.1). We thus have an equivalence F-Isoc Sρ / ϕ-ModB+

cris,ρ

.

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103

+ + But since B+ ρ p ⊂ Bcris,ρ ⊂ Bρ p−1 we have an equivalence

ϕ-ModB+

cris,ρ

ϕ-ModB+ρ .

Moreover, one can think of ϕ-ModB+ as being the category of “convergent F-isocrystals on Sρ ”. We thus have proved the following. Theorem 2.137. Suppose F is a perfectoid field with algebraically closed residue field. Then any F-isocrystal, resp. convergent F-isocrystal, on Sρ is isotrivial that is to say comes from the residue field of k after the choice of a s / / k. splitting O F

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[16] J.-M. Fontaine. Le corps des périodes p-adiques. Astérisque, (223):59–111, 1994. With an appendix by Pierre Colmez, Périodes p-adiques (Bures-sur-Yvette, 1988). [17] B. Gross and M. Hopkins. Equivariant vector bundles on the Lubin-Tate moduli space. In Topology and representation theory (Evanston, IL, 1992), volume 158 of Contemp. Math., pages 23–88. Amer. Math. Soc., 1994. [18] A. Grothendieck. Sur la classification des fibrés holomorphes sur la sphère de Riemann. Amer. J. Math., 79:121–138, 1957. [19] A. Grothendieck. Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I. Inst. Hautes Études Sci. Publ. Math., (11):167, 1961. [20] U. Hartl and R. Pink. Vector bundles with a Frobenius structure on the punctured unit disc. Compos. Math., 140(3):689–716, 2004. [21] K. Kedlaya. Slope filtrations revisited. Doc. Math., 10:447–525, 2005. [22] K. Kedlaya. Slope filtrations for relative Frobenius. Astérisque, (319):259–301, 2008. Représentations p-adiques de groupes p-adiques. I. Représentations galoisiennes et (φ, )-modules. [23] K. Kedlaya, J. Pottharst, and L. Xiao. Cohomology of arithmetic families of (phi,gamma)-modules. arXiv:1203.5718v1. [24] G. Laffaille. Groupes p-divisibles et corps gauches. Compositio Math., 56(2):221–232, 1985. [25] M. Lazard. Les zéros des fonctions analytiques d’une variable sur un corps valué complet. Inst. Hautes Études Sci. Publ. Math., (14):47–75, 1962. [26] B. Poonen. Maximally complete fields. Enseign. Math. (2), 39(1-2):87–106, 1993. [27] M. Rapoport and T. Zink. Period spaces for p-divisible groups. Number 141 in Annals of Mathematics Studies. Princeton University Press, 1996. [28] P. Schneider and J. Teitelbaum. Algebras of p-adic distributions and admissible representations. Invent. Math., 153(1):145–196, 2003. [29] P. Scholze. Perfectoid spaces. Preprint. [30] S. Sen. Continuous cohomology and p-adic Galois representations. Invent. Math., 62(1):89–116, 1980/81. [31] J. T. Tate. p-divisible groups. In Proc. Conf. Local Fields (Driebergen, 1966), pages 158–183. Springer, 1967. [32] J.-P. Wintenberger. Le corps des normes de certaines extensions infinies de corps locaux; applications. Ann. Sci. École Norm. Sup. (4), 16(1):59–89, 1983.

3 Around associators Hidekazu Furusho

Abstract This is a concise exposition of recent developments around the study of associators. It is based on the author’s talk at the Mathematische Arbeitstagung in Bonn, June 2011 (cf. [F11b]) and at the Automorphic Forms and Galois Representations Symposium in Durham, July 2011. The first section is a review of Drinfeld’s definition [Dr] of associators and the results [F10, F11a] concerning the definition. The second section explains the four pro-unipotent algebraic groups related to associators; the motivic Galois group, the Grothendieck– Teichmüller group, the double shuffle group and the Kashiwara–Vergne group. Relationships, actually inclusions, between them are also discussed.

1. Associators We recall the definition of associators [Dr] and explain our main results in [F10, F11a] concerning the defining equations of associators. The notion of associators was introduced by Drinfeld in [Dr]. They describe monodromies of the KZ (Knizhnik–Zamolodchikov) equations. They are essential for the construction of quasi-triangular quasi-Hopf quantized universal enveloping algebras ([Dr]), for the quantization of Lie-bialgebras (Etingof–Kazhdan quantization [EtK]), for the proof of formality of chain operad of little discs by Tamarkin [Ta] (see also Ševera and Willwacher [SW]) and also for the combinatorial reconstruction of the universal Vassiliev knot invariant (the Kontsevich invariant [Kon, Ba95]) by Bar-Natan [Ba97], Cartier [C], Kassel and Turaev [KssT], Le and Murakami [LM96a] and Piunikhin [P]. Automorphic Forms and Galois Representations, ed. Fred Diamond, Payman L. Kassaei and c Cambridge University Press 2014. Minhyong Kim. Published by Cambridge University Press.

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Notation 3.1. Let k be a field of characteristic 0 and k¯ be its algebraic closure. Denote by U F2 = k X 0 , X 1

the non-commutative formal power series ring defined as the universal enveloping algebra of the completed free Lie algebra F2 with two variables X 0 and X 1 . An element ϕ = ϕ(X 0 , X 1 ) of U F2 is called group-like1 if it satisfies (ϕ) = ϕ ⊗ ϕ and ϕ(0, 0) = 1

(3.1)

ˆ F2 is given by (X 0 ) = X 0 ⊗ 1 + 1 ⊗ X 0 and where : U F2 → U F2 ⊗U (X 1 ) = X 1 ⊗ 1 + 1 ⊗ X 1 . For any k-algebra homomorphism ι : U F2 → S, the image ι(ϕ) ∈ S is denoted by ϕ(ι(X 0 ), ι(X 1 )). Denote by U a3 (resp. U a4 ) the universal enveloping algebra of the completed pure braid Lie algebra a3 (resp. a4 ) over k with 3 (resp. 4) strings, which is generated by ti j (1 i, j 3 (resp. 4)) with defining relations tii = 0, ti j = t ji , [ti j , tik + t jk ] = 0 (i, j ,k: all distinct) and [ti j , tkl ] = 0 (i, j ,k,l: all distinct). Note that X 0 → t12 and X 1 → t23 give an isomorphism U F2 U a3 . Definition 3.2 ([Dr]). A pair (μ, ϕ) with a non-zero element μ in k and a group-like series ϕ = ϕ(X 0 , X 1 ) ∈ U F2 is called an associator if it satisfies one pentagon equation ϕ(t12 , t23 + t24 )ϕ(t13 + t23 , t34 ) = ϕ(t23 , t34 )ϕ(t12 + t13 , t24 + t34 )ϕ(t12 , t23 ) (3.2) in U a4 and two hexagon equations μ(t13 + t23 ) μt13 μt23 } = ϕ(t13 , t12 ) exp{ }ϕ(t13 , t23 )−1 exp{ }ϕ(t12 , t23 ), 2 2 2 (3.3) μ(t12 + t13 ) μt μt 13 12 exp{ } = ϕ(t23 , t13 )−1 exp{ }ϕ(t12 , t13 )exp{ }ϕ(t12 , t23 )−1 2 2 2 (3.4) in U a3 . exp{

Remark 3.3. (i) Drinfeld [Dr] proved that such a pair always exists for any field k of characteristic 0. (ii) The equations (3.2)–(3.4) reflect the three axioms of braided monoidal categories [JS]. We note that for any k-linear infinitesimal tensor category C, each associator gives a structure of a braided monoidal category on C[[h]] (cf. [C, Dr, KssT]). Here C[[h]] denotes the category whose set of objects is equal to that of C and whose set of morphisms MorC [[h]] (X, Y ) is MorC (X, Y ) ⊗ k[[h]] (h: a formal parameter). 1 It is equivalent to ϕ ∈ exp F . 2

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Actually, the two hexagon equations are a consequence of the one pentagon equation: Theorem 3.4 ([F10]). Let ϕ = ϕ(X 0 , X 1 ) be a group-like element of U F2 . Suppose that ϕ satisfies the pentagon equation (3.2). Then there always exists μ ∈ k¯ (unique up to signature) such that the pair (μ, ϕ) satisfies two hexagon equations (3.3) and (3.4). Recently several different proofs of the above theorem were obtained (see [AlT, BaD, Wi]). One of the nicest examples of associators is the Drinfeld associator: Example 3.5. The Drinfeld associator Φ K Z = Φ K Z (X 0 , X 1 ) ∈ C X 0 , X 1

is defined to be the quotient Φ K Z = G 1 (z)−1 G 0 (z) where G 0 and G 1 are the solutions of the formal KZ-equation, which is the following differential equation for multi-valued functions G(z) : C\{0, 1} → C X 0 , X 1

X0 d X1 G(z) = + G(z), dz z z−1 such that G 0 (z) ≈ z X 0 when z → 0 and G 1 (z) ≈ (1 − z) X 1 when z → 1 (cf. √ [Dr]). It is shown in [Dr] (see also [Wo]) that the pair (2π −1, Φ K Z ) √ forms an associator for k = C. Namely Φ K Z satisfies (3.1)∼(3.4) with μ = 2π −1. Remark 3.6. (i) The Drinfeld associator is expressed as follows: k −1 Φ K Z (X 0 , X 1 ) = 1 + (−1)m ζ (k1 , · · · , km )X 0km −1 X 1 · · · X 01 X 1 m,k1 ,...,km ∈N km >1

+ (regularized terms). Here ζ (k1 , · · · , km ) is the multiple zeta value (MZV in short), the real number defined by the following power series ζ (k1 , · · · , km ) :=

0 1, ζ (a)ζ (b) =

a−1 b−1+i i=0

+

i

ζ (a − i, b + i)

b−1 a−1+ j ζ (b − j, a + j ), j j=0

ζ (a)ζ (b) = ζ (a, b) + ζ (a + b) + ζ (b, a). The former follows from (3.1) and the latter follows from (3.6).

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The regularized double shuffle relations are also a consequence of the pentagon equation: Theorem 3.10 ([F11a]). Let ϕ = ϕ(X 0 , X 1 ) be a group-like element of U F2 . Suppose that ϕ satisfies the pentagon equation (3.2). Then it also satisfies the regularized double shuffle relations (3.6). This result attains the final goal of the project posed by Deligne– Terasoma [Te]. Their idea is to use some convolutions of perverse sheaves, whereas our proof is to use Chen’s bar construction calculus. It would be our next project to complete their idea and to get another proof of Theorem 3.10. Remark 3.11. Our Theorem 3.10 was extended cyclotomically in [F12]. The following Zagier’s relation which is essential for Brown’s proof of Theorem 3.17 might be also one of the most fascinating ones. Theorem 3.12 ([Z]). For a, b 0 ζ (2{a} , 3, 2{b} ) = 2 with

Ara,b

=

2r 2a+2

a+b+1

r (−1)r (Ara,b − Ba,b )ζ (2r + 1)ζ (2{a+b+1−r} )

r=1

r = (1 − 2−2r ) 2r . and Ba,b 2b+1

2. Four groups We explain recent developments on the four pro-unipotent algebraic groups related to associators; the motivic Galois group, the Grothendieck–Teichmüller group, the double shuffle group and the Kashiwara–Vergne group, all of which are regarded as subgroups of Aut exp F2 . In the end of this section we discuss natural inclusions between them.

2.1. Motivic Galois group We review the formulations of the motivic Galois groups (consult also [An] as a nice exposition). Notation 3.13. We work in the triangulated category D M(Q)Q of mixed motives over Q (a part of an idea of mixed motives is explained in [De] §1) constructed by Hanamura, Levine and Voevodsky. Tate motives Q(n) (n ∈ Z) are (Tate) objects of the category. Let D M T (Q)Q be the triangulated subcategory of D M(Q)Q generated by Tate motives Q(n) (n ∈ Z). By the work

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of Levine a neutral tannakian Q-category M T (Q) = M T (Q)Q of mixed Tate motives over Q is extracted by taking the heart with respect to a t-structure of D M T (Q)Q . Deligne and Goncharov [DeG] introduced the full subcategory M T (Z) = M T (Z)Q of unramified mixed Tate motives inside of M T (Q)Q , All objects there are mixed Tate motives M (i.e. an object of M T (Q)) such that for each subquotient E of M which is an extension of Q(n) by Q(n + 1) for n ∈ Z, the extension class of E in Ext1M T (Q) (Q(n), Q(n + 1)) = Ext1M T (Q) (Q(0), Q(1)) = Q× ⊗ Q is equal to in Z× ⊗ Q = {0}. In the category M T (Z) of unramified mixed Tate motives, the following holds: $ 1 (m = 3, 5, 7, . . . ), 1 dimQ Ext M T (Z) (Q(0), Q(m)) = (3.7) 0 (m : others), dimQ Ext2M T (Z) (Q(0), Q(m)) = 0.

(3.8)

The category M T (Z) forms a neutral tannakian Q-category (consult [DeM]) with the fiber functor ωcan : M T (Z) → VectQ (VectQ : the category of Q-vector spaces) sending each motive M to W ⊕n Hom(Q(n), Gr−2n M). Definition 3.14. The motivic Galois group here is defined to be the Galois group of M T (Z), which is the pro-Q-algebraic group defined by GalM (Z) := Aut⊗ (M T (Z) : ωcan ). By the fundamental theorem of tannakian category theory, ωcan induces an equivalence of categories M T (Z) Rep GalM (Z)

(3.9)

where the right-hand side of the isomorphism denotes the category of finite dimensional Q-vector spaces with GalM (Z)-action. Remark 3.15. The action of GalM (Z) on ωcan (Q(1)) = Q defines a surjection GalM (Z) → Gm and its kernel GalM (Z)1 is the unipotent radical of GalM (Z). There is a canonical splitting τ : Gm → GalM (Z) which gives a negative grading on its associated Lie algebra LieGalM (Z)1 . From (3.7) and (3.8) it follows that the Lie algebra is the graded free Lie algebra generated by one element in each degree −3, −5, −7, . . . . (consult [De] §8 for the full story).

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→ − The motivic fundamental group π1M (P1 \{0, 1, ∞} : 01) constructed in [DeG] §4 is a (pro-) object of M T (Z). The Drinfeld associator (cf. Example 3.5) is essential in describing the Hodge realization of the motive (cf. [An, DeG, F07]). By our tannakian equivalence (3.9), it gives a (pro-) object of the right-hand side of (3.9), which induces a (graded) action : GalM (Z)1 → Aut exp F2 .

(3.10)

Remark 3.16. For each σ ∈ GalM (Z)1 (k), its action on exp F2 is described by e X 0 → e X 0 and e X 1 → ϕσ−1 e X 1 ϕσ for some ϕσ ∈ exp F2 . The following has been conjectured (Deligne–Ihara conjecture) for a long time and finally proved by Brown by using Zagier’s relation (Theorem 3.12). Theorem 3.17 ([Br]). The map " is injective. It is a pro-unipotent analogue of the so-called Bely˘ı’s theorem [Bel] in the pro-finite group setting. The theorem says that all unramified mixed Tate motives are associated with MZVs.

2.2. Grothendieck–Teichmüller group The Grothendieck–Teichmüller group was introduced by Drinfeld [Dr] in his study of deformations of quasi-triangular quasi-Hopf quantized universal enveloping algebras. It was defined to be the set of “degenerated” associators. The construction of the group was also stimulated by the previous idea of Grothendieck, un jeu de Lego–Teichmüller, which was posed in his article Esquisse d’un programme [G]. Definition 3.18 ([Dr]). The Grothendieck–Teichmüller group G RT1 is defined to be the pro-algebraic variety whose set of k-valued points consists of degenerated associators, which are group-like series ϕ ∈ U F2 satisfying the defining equations (3.2)–(3.4) of associators with μ = 0. Remark 3.19. (i) By Theorem 3.4, G RT1 is reformulated to be the set of group-like series satisfying (3.2) without quadratic terms. (ii) It forms a group [Dr] by the multiplication below ϕ2 ◦ ϕ1 := ϕ1 (ϕ2 X 0 ϕ2−1 , X 1 ) · ϕ2 = ϕ2 · ϕ1 (X 0 , ϕ2−1 X 1 ϕ2 ).

(3.11)

By the map X 0 → X 0 and X 1 → ϕ −1 X 1 ϕ, the group G RT1 is regarded as a subgroup of Aut exp F2 . (iii) Ihara came to the Lie algebra of G RT1 independently of Drinfeld’s work in his arithmetic study of Galois action on fundamental groups (cf. [Iy90]).

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(iv) The cyclotomic analogues of associators and that of the Grothendieck– Teichmüller group were introduced by Enriquez [En]. Some elimination results on their defining equations in special case were obtained in [EnF]. Geometric interpretation (cf. [Dr, Iy90, Iy94]) of equations (3.2)–(3.4) implies the following (for a proof, see also [An, F07]) Theorem 3.20. Im ⊂ G RT1 . Related to the questions posed in [De, Dr, Iy90], it is expected that they are isomorphic. Remark 3.21. (i) The Drinfeld associator Φ K Z is an associator (cf. Example 3.5) but is not a degenerated associator, i.e. Φ K Z ∈ G RT1 (C). p (ii) The p-adic Drinfeld associator Φ K Z introduced in [F04] is not an p associator but a degenerated associator, i.e. Φ K Z ∈ G RT1 (Q p ) (cf. [F07]).

2.3. Double shuffle group The double shuffle group was introduced by Racinet as the set of solutions of the regularized double shuffle relations with “degeneration” condition (no quadratic terms condition). Definition 3.22 ([R]). The double shuffle group D M R0 is the pro-algebraic variety whose set of k-valued points consists of the group-like series ϕ ∈ U F2 satisfying the regularized double shuffle relations (3.6) without linear terms and quadratic terms. Remark 3.23. (i) We note that D M R stands for double mélange regularisé ([R]). (ii) It was shown in [R] that it forms a group by the operation (3.11). (iii) In the same way as in Remark 3.19 (ii), the group D M R0 is regarded as a subgroup of Aut exp F2 . It is also shown that Im" is contained in D M R0 (cf. [F07])). Actually it is expected that they are isomorphic. Theorem 3.10 follows the inclusion between G RT1 and D M R0 : Theorem 3.24 ([F11a]). G RT1 ⊂ D M R0 . It is also expected that they are isomorphic.

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Remark 3.25. (i) The Drinfeld associator Φ K Z satisfies the regularized double shuffle relations (cf. Remark 3.8) but it is not an element of the double shuffle group, i.e. Φ K Z ∈ D M R0 (C), because its quadratic term is non-zero, actually is equal to ζ (2)X 1 X 0 − ζ (2)X 0 X 1 . p (ii) The p-adic Drinfeld associator Φ K Z satisfies the regularized double shuffle relations (cf. [BeF, FJ]) and it is an element of the double shuffle group, p i.e. Φ K Z ∈ D M R0 (Q p ), which also follows from Remark 3.21.(ii) and Theorem 3.24.

2.4. Kashiwara–Vergne group In [KswV] Kashiwara and Vergne proposed a conjecture related to the Campbell–Baker–Hausdorff series which generalizes Duflo’s theorem (Duflo isomorphism) to some extent. The conjecture was settled generally by Alekseev and Meinrenken [AlM]. The Kashiwara–Vergne group was introduced as a “degeneration” of the set of solutions of the conjecture by Alekseev and Torossian in [AlT], where they gave another proof of the conjecture by using Drinfeld’s [Dr] theory of associators. The following is one of the formulations of the conjecture stated in [AlET]. Generalized Kashiwara–Vergne problem: Find a group automorphism P : exp F2 → exp F2 such that P belongs to T Aut exp F2 (that is, P ∈ Aut exp F2 such that P(e X 0 ) = p1 e X 0 p1−1 and P(e X 1 ) = p2 e X 1 p2−1 for some p1 , p2 ∈ exp F2 ) and P satisfies P(e X 0 e X 1 ) = e(X 0 +X 1 ) and the coboundary Jacobian condition δ ◦ J (P) = 0. Here J stands for the Jacobian cocycle J : T Aut exp F2 → tr2 and δ denotes the differential map δ : trn → trn+1 for n = 1, 2, . . . (for their precise definitions see [AlT]). We note that P is uniquely determined by the pair ( p1 , p2 ). The following is essential for the proof of the conjecture. Theorem 3.26 ([AlT, AlET]). Let (μ, ϕ) be an associator. Then the pair ( p1 , p2 ) = ϕ(X 0 /μ, X ∞ /μ), e X ∞ /2 ϕ(X 1 /μ, X ∞ /μ) with X ∞ = −X 0 − X 1 gives a solution to the above problem.

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The Kashiwara–Vergne group is defined to be the set of solutions of the problem with “degeneration condition” (“the condition μ = 0”): Definition 3.27 ([AlT, AlET]). The Kashiwara–Vergne group K RV is defined to be the set of P ∈ Aut exp F2 which satisfies P ∈ T Aut exp F2 , P(e(X 0 +X 1 ) ) = e(X 0 +X 1 ) and the coboundary Jacobian condition δ ◦ J (P) = 0. The above K RV forms a subgroup of Aut exp F2 . We denote by K RV0 the subgroup of K RV consisting of P without linear terms in both p1 and p2 . Theorem 3.26 yields the inclusion below. Theorem 3.28 ([AlT, AlET]). G RT1 ⊂ K RV0 . Actually it is expected that they are isomorphic (cf. [AlT]). A recent result of Schneps in [S] also leads to Theorem 3.29 ([S]). D M R0 ⊂ K RV0 .

2.5. Comparison By Theorem 3.17, 3.20, 3.24, 3.28 and 3.29, we obtain Theorem 3.30. GalM (Z)1 ⊆ G RT1 ⊆ D M R0 ⊆ K RV0 . We finish our exposition by posing the following question: Question 3.31. Are they all equal? Namely, GalM (Z)1 = G RT1 = D M R0 = K RV0 ? These four groups were constructed independently and there are no philosophical reasonings why we expect that they are all equal. Though it might be not so good mathematically to believe such equalities without any strong conceptual support, the author believes that it might be good at least spiritually to imagine a hidden theory to relate them.

References [An] André, Y.; Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses, 17, Société Mathématique de France, Paris, 2004. [AlET] Alekseev, A., Enriquez, B. and Torossian, C.; Drinfeld associators, braid groups and explicit solutions of the Kashiwara-Vergne equations, Publ. Math. Inst. Hautes Études Sci. 112 (2010), 143–189.

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[AlM] [AlT] [Ba95] [Ba97]

[BaD] [Bel] [BeF]

[Br] [C]

[De]

[DeG] [DeM]

[Dr] [En] [EnF] [EtK] [Eu]

[F03] [F04] [F07]

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and Meinrenken, E.; On the Kashiwara-Vergne conjecture, Invent. Math. 164 (2006), no. 3, 615–634. and Torossian, C.; The Kashiwara-Vergne conjecture and Drinfeld’s associators, Ann. of Math. (2) 175 (2012), no. 2, 415–463. Bar-Natan, D.; On the Vassiliev knot invariants, Topology 34 (1995), no. 2, 423–472. ; Non-associative tangles, Geometric topology (Athens, GA, 1993), 139–183, AMAPIP Stud. Adv. Math., 2.1, Amer. Math. Soc., Providence, RI, 1997. and Dancso, Z.; Pentagon and hexagon equations following Furusho, Proc. Amer. Math. Soc. 140 (2012), no. 4, 1243–1250. Bely˘ı, G. V., Galois extensions of a maximal cyclotomic field, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 2, 267–276, 479. Besser, A. and Furusho, H.; The double shuffle relations for p-adic multiple zeta values, Primes and Knots, AMS Contemporary Math, Vol 416, 2006, 9–29. Brown, F.; Mixed Tate Motives over Spec(Z ), Annals of Math., 175, (2012) no. 2, 949–976. Cartier, P.; Construction combinatoire des invariants de Vassiliev-Kontsevich des nœuds, C. R. Acad. Sci. Paris Ser. I Math. 316 (1993), no. 11, 1205–1210. Deligne, P.; Le groupe fondamental de la droite projective moins trois points, Galois groups over Q (Berkeley, CA, 1987), 79–297, Math. S. Res. Inst. Publ., 16, Springer, New York-Berlin, 1989. and Goncharov, A.; Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. Ecole Norm. Sup. (4) 38 (2005), no. 1, 1–56. and Milne, J.; Tannakian categories, in Hodge cycles, motives, and Shimura varieties (P.Deligne, J.Milne, A.Ogus, K.-Y.Shih editors), Lecture Notes in Mathematics 900, Springer-Verlag, 1982. Drinfel’d, V. G.; On quasitriangular quasi-Hopf algebras and a group closely connected with Gal(Q/Q), Leningrad Math. J. 2 (1991), no. 4, 829–860. Enriquez, B.; Quasi-reflection algebras and cyclotomic associators, Selecta Math. (N.S.) 13 (2007), no. 3, 391–463. and Furusho, H.; Mixed Pentagon, octagon and Broadhurst duality equation, J. Pure Appl. Algebra, 216, (2012), no. 4, 982–995. no. 4, Etingof, P. and Kazhdan, D.; Quantization of Lie bialgebras. II, Selecta Math. 4 (1998), no. 2, 213–231. Euler, L., Meditationes circa singulare serierum genus, Novi Commentarii academiae scientiarum Petropolitanae 20, 1776, pp. 140–186 and Opera Omnia: Series 1, Volume 15, pp. 217–267 (also available from www.math.dartmouth.edu/euler/). Furusho, H.; The multiple zeta value algebra and the stable derivation algebra, Publ. Res. Inst. Math. Sci. 39, (2003), no. 4, 695–720. ; p-adic multiple zeta values I – p-adic multiple polylogarithms and the p-adic K Z equation, Invent. Math. 155, (2004), no. 2, 253–286. ; p-adic multiple zeta values II – tannakian interpretations, Amer. J. Math. 129, (2007), no 4, 1105–1144.

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[F10] [F11a] [F11b] [F12]

[FJ]

[G]

[IkKZ] [Iy90]

[Iy94]

[JS] [KswV] [KssT] [Kon] [LM96a] [LM96b] [P] [R] [S] [SW] [Ta] [Te]

; Pentagon and hexagon equations, Annals of Math. 171 (2010), no. 1, 545–556. ; Double shuffle relation for associators, Annals of Math. 174 (2011), no. 1, 341–360. ; Four groups related to associators, report of the Mathematische Arbeitstagung in Bonn, June 2011, arXiv:1108.3389. ; Geometric interpretation of double shuffle relation for multiple L-values, Galois-Teichmüller theory and Arithmetic Geometry, Advanced Studies in Pure Math 63 (2012), 163–187. and Jafari, A.; Regularization and generalized double shuffle relations for p-adic multiple zeta values, Compositio Math. 143, (2007), 1089–1107. Grothendieck, A.; Esquisse d’un programme, 1983, available on pp. 243–283. London Math. Soc. LNS 242, Geometric Galois actions, 1, 5–48, Cambridge University Press. Ihara, K., Kaneko, M. and Zagier, D.; Derivation and double shuffle relations for multiple zeta values, Compos. Math. 142 (2006), no. 2, 307–338. Ihara, Y.; Braids, Galois groups, and some arithmetic functions, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 99–120, Math. Soc. Japan, Tokyo, 1991. London Math. Soc. LNS ; On the embedding of Gal(Q/Q) into GT, 200, The Grothendieck theory of dessins d’enfants (Luminy, 1993), 289–321, Cambridge University Press, Cambridge, 1994. Joyal, A. and Street, R.; Braided tensor categories, Adv. Math. 102 (1993), no. 1, 20–78. Kashiwara, M. and Vergne, M.; The Campbell-Hausdorff formula and invariant hyperfunctions, Invent. Math. 47 (1978), no. 3, 249–272. Kassel, C. and Turaev, V.; Chord diagram invariants of tangles and graphs, Duke Math. J. 92 (1998), no. 3, 497–552. Kontsevich, M.; Vassiliev’s knot invariants, I. M. Gelfand Seminar, 137–150, Adv. Soviet Math., 16, Part 2, Amer. Math. Soc., Providence, RI, 1993. Le, T.Q.T. and Murakami, J.; The universal Vassiliev-Kontsevich invariant for framed oriented links, Compositio Math. 102 (1996), no. 1, 41–64. ; Kontsevich’s integral for the Kauffman polynomial, Nagoya Math. J. 142 (1996), 39–65. Piunikhin, S.; Combinatorial expression for universal Vassiliev link invariant, Comm. Math. Phys. 168 (1995), no. 1, 1–22. Racinet, G.; Doubles melangés des polylogarithmes multiples aux racines de l’unité, Publ. Math. Inst. Hautes Etudes Sci. 95 (2002), 185–231. Schneps, L.; Double shuffle and Kashiwara-Vergne Lie algebra, J. Algebra 367 (2012), 54–74. Ševera, P and Willwacher, T.; Equivalence of formalities of the little discs operad, Duke Math. J. 160 (2011), no. 1, 175–206. Tamarkin, D. E.; Formality of chain operad of little discs, Lett. Math. Phys. 66 (2003), no. 1–2, 65–72. Terasoma, T.; Geometry of multiple zeta values, International Congress of Mathematicians. Vol. II, 627–635, Eur. Math. Soc., Zürich, 2006.

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[Wi] Willwacher, T.; M. Kontsevich’s graph complex and the GrothendieckTeichmueller Lie algebra, arXiv:1009.1654, preprint (2010). [Wo] Wojtkowiak, Z.; Monodromy of iterated integrals and non-abelian unipotent periods, Geometric Galois actions, 2, 219–289, London Math. Soc. LNS 243, Cambridge University Press, Cambridge, 1997. [Z] Zagier, D.; Evaluation of the multiple zeta values ζ (2, ..., 2, 3, 2, ..., 2), Annals of Math., 175, (2012), no. 2, 977–1000.

4 The stable Bernstein center and test functions for Shimura varieties Thomas J. Haines

Abstract We elaborate the theory of the stable Bernstein center of a p-adic group G, and apply it to state a general conjecture on test functions for Shimura varieties due to the author and R. Kottwitz. This conjecture provides a framework by which one might pursue the Langlands–Kottwitz method in a very general situation: not necessarily PEL Shimura varieties with arbitrary level structure at p. We give a concrete reinterpretation of the test function conjecture in the context of parahoric level structure. We also use the stable Bernstein center to formulate some of the transfer conjectures (the “fundamental lemmas”) that would be needed if one attempts to use the test function conjecture to express the local Hasse–Weil zeta function of a Shimura variety in terms of automorphic L-functions.

Contents 1 2 3 4 5 6 7 8

Introduction page 119 Notation 122 Review of the Bernstein center 123 The local Langlands correspondence 129 The stable Bernstein center 131 The Langlands–Kottwitz approach for arbitrary level structure 142 Test functions in the parahoric case 153 Overview of evidence for the test function conjecture 159

Research partially supported by NSF DMS-0901723. Automorphic Forms and Galois Representations, ed. Fred Diamond, Payman L. Kassaei and c Cambridge University Press 2014. Minhyong Kim. Published by Cambridge University Press.

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9 Evidence for conjectures on transfer of the Bernstein center 10 Explicit computation of the test functions 11 Appendix: Bernstein isomorphisms via types References

119

161 163 166 183

1. Introduction The main purpose of this chapter is to give precise statements of some conjectures on test functions for Shimura varieties with bad reduction. In the Langlands–Kottwitz approach to studying the cohomology of a Shimura variety, one of the main steps is to identify a suitable test function that is “plugged into” the counting points formula that resembles the geometric side of the Arthur–Selberg trace formula. To be more precise, let (G, h −1 , K p K p ) denote Shimura data where p is a fixed rational prime such p that the level-structure group factorizes as K p K p ⊂ G(A f )G(Q p ). This data gives rise to a quasi-projective variety Sh K p := Sh(G, h −1 , K p K p ) over a number field E ⊂ C. Let p ∈ Gal(Q p /Q p ) denote a geometric Frobenius element. Then one seeks to prove a formula for the semi-simple Lefschetz number Lefss (rp , Sh K p ) trss (rp , H•c (Sh K p ⊗E Q p , Q )) = c(γ0 ; γ , δ) Oγ (1 K p ) TOδθ (φr ), (γ0 ;γ ,δ)

(4.1) (see §6.1 for more details). The test function φr appearing here is the most interesting part of the formula. Experience has shown that we may often find a test function belonging to the center Z(G(Q pr ), K pr ) of the Hecke algebra H(G(Q pr ), K pr ), in a way that is explicitly determined by the E-rational conjugacy class {μ} of 1-parameter subgroups of G associated to the Shimura data. In most PEL cases with good reduction, where K p ⊂ G(Q p ) is a hyperspecial maximal compact subgroup, this was done by Kottwitz (cf. e.g. [Ko92a]). When K p is a parahoric subgroup of G(Q p ) and when GQ p is unramified, the Kottwitz conjecture preK

p dicts that we can take φr to be a power of p times the Bernstein function z −μ, j arising from the Bernstein isomorphism for the center Z(G(Q pr ), K pr ) of the parahoric Hecke algebra H(G(Q pr ), K pr ) (see Conjecture 4.43 and §11). In fact Kottwitz formulated (again, for unramified groups coming from Shimura data) a closely related conjecture concerning nearby cycles on Rapoport–Zink local models of Shimura varieties, which subsequently played an important role in the study of local models (Conjecture 4.44). It also inspired important developments in the geometric Langlands program,

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e.g. [Ga]. Both versions of Kottwitz’ conjectures were later proved in several parahoric cases attached to linear or symplectic groups (see [HN02a, H05]). In a recent breakthrough, Pappas and Zhu [PZ] defined group-theoretic versions of Rapoport–Zink local models for quite general groups, and proved in the unramified situations the analogue of Kottwitz’ nearby cycles conjecture for them. These matters are discussed in more detail in §7 and §8. Until around 2009 it was still not clear how one could describe the test functions φr in all deeper level situations. In the spring of 2009 the author and Kottwitz formulated a conjecture predicting test functions φr for general level structure K p . This is the test function conjecture, Conjecture 4.30. It postulates that we may express φr in terms of a distribution Z E j0 in the Bernstein center Z(G(Q pr )) associated to a certain represenV−μ, j

E

j0 L tation V−μ, j (defined in (4.16)) of the Langlands L-group (G Q pr ). Let d = dim(Sh K p ). Then Conjecture 4.30 asserts that we may take

φr = pr d/2 Z

E j0 V−μ, j

∗ 1 K pr ∈ Z(G(Q pr ), K pr )

the convolution of the distribution Z

E j0

V−μ, j

with the characteristic function 1 K pr

of the subgroup K pr . As shown in §7, this specializes to the Kottwitz conjecture for parahoric subgroups in unramified groups. Conjecture 4.30 was subsequently proved for Drinfeld case Shimura varieties with 1 ( p)-level structure by the author and Rapoport [HRa1], and for modular curves and for Drinfeld case Shimura varieties with arbitrary level structure by Scholze [Sch1, Sch2]. The distributions in Conjecture 4.30 are best seen as examples of a construction V ; Z V which attaches to any algebraic representation V of the Langlands dual group L G (for G any connected reductive group over any p-adic field F), an element Z V in the stable Bernstein center of G/F. This chapter elaborates the theory of the stable Bernstein center, following the lead of Vogan [Vo]. The set of all infinitesimal characters,

i.e. the set of all G-conjugacy classes of admissible homomorphisms λ : W F → L G (where W F is the Weil group of the local field F), is given the structure of an affine algebraic variety over C, and the stable Bernstein center Zst (G/F) is defined to be the ring of regular functions on this variety.1 In order to describe the precise conjectural relation between the Bernstein and stable Bernstein centers of a p-adic group, it was necessary to formulate an 1 The difference between our treatment and Vogan’s is in the definition of the variety structure

on the set of infinitesimal characters.

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enhancement LLC+ of the usual conjectural local Langlands correspondence LLC for that group. Having this relation in hand, the construction V ; Z V provides a supply of elements in the usual Bernstein center of G/F, which we call the geometric Bernstein center. It is for such distributions that one can formulate natural candidates for (Frobenius-twisted) endoscopic transfer, which we illustrate for standard endoscopy in Conjecture 4.35 and for stable base change in Conjecture 4.36. These form part of the cadre of “fundamental lemmas” that one would need to pursue the “pseudostabilization” of (4.1) and thereby express the cohomology of Sh K p in terms of automorphic representations along the lines envisioned by Kottwitz [Ko90] but for places with arbitrary bad reduction. In the compact and non-endoscopic situations, we prove in Theorem 4.40 that the various Conjectures we have made yield an expression of the semisimple local Hasse–Weil zeta function in terms of semi-simple automorphic L-functions. Earlier unconditional results in this direction, for nice PEL situations, were established in [H05], [HRa1], [Sch1, Sch2]. We stress that the framework here should not be limited to PEL Shimura varieties, but should work more generally. In recent work of Scholze and Shin [SS], the connection of the stable Bernstein center with Shimura varieties helped them to give nearly complete descriptions of the cohomology of many compact unitary Shimura varieties with bad reduction at p; they consider the “EL cases” where GQ p is a product of Weil restrictions of general linear groups. It would be interesting to extend the connection to further examples. Returning to the original Kottwitz conjecture for parahoric level structure, Conjecture 4.30 in some sense subsumes it, since it makes sense for arbitrary level structure and without the hypothesis that GQ pr be unramified. However, Conjecture 4.30 has the drawback that it assumes LLC+ for GQ pr . Further, it is still of interest to formulate the Kottwitz conjecture in the parahoric cases for arbitrary groups in a concrete way that can be checked (for example) by explicit comparison of test functions with nearby cycles. In §7 we formulate the Kottwitz conjecture for general groups, making use of the transfer homomorphisms of the Appendix §11 to determine test functions on arbitrary groups from test functions on their quasi-split inner forms. The definition of transfer homomorphisms requires a theory of Bernstein isomorphisms more general than what was heretofore available. Therefore, in the Appendix we establish these isomorphisms in complete generality in a nearly self-contained way, and also provide some related structure theory results that should be of independent interest.

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Here is an outline of the contents of this chapter. In §3 we review the Bernstein center of a p-adic group, including the algebraic structure on the Bernstein variety of all supercuspidal supports. In §4 we recall the conjectural local Langlands correspondence (LLC), and discuss additional desiderata we need in our elaboration of the stable Bernstein center in §5. In particular in §5.2 we describe the enhancement (LLC+) which plays a significant role throughout the chapter, and explain why it holds for general linear groups in Remark 4.8 and Corollary 4.11. The distributions Z V are defined in §5.7, and are used to formulate the test function conjecture, Conjecture 4.30, in §6.1. In the rest of §6, we describe the nearby cycles variant Conjecture 4.31 along with some of the endoscopic transfer conjectures needed for the “pseudostabilization”, and assuming these conjectures we prove in Theorem 4.40 the expected form of the semi-simple local Hasse–Weil zeta functions, in the compact and nonendoscopic cases. In §7 we give a concrete reformulation of the test function conjecture in parahoric cases, recovering the Kottwitz conjecture and generalizing it to all groups using the material from the Appendix. The purpose of §8 and §9 is to list some of the available evidence for Conjectures 4.30 and 4.36. In §10 certain test functions are described very explicitly. Finally, the Appendix gives the treatment of Bernstein isomorphisms and the transfer homomorphisms, alluded to above. Acknowledgments. I am very grateful to Guy Henniart for supplying the proof of Proposition 4.10 and for allowing me to include his proof in this chapter. I warmly thank Timo Richarz for sending me his unpublished article [Ri] and for letting me quote a few of his results in Lemma 4.56. I am indebted to Brooks Roberts for proving Conjecture 4.7 for GSp(4) (see Remark 4.8). I thank my colleagues Jeffrey Adams and Niranjan Ramachandran for useful conversations. I also thank Robert Kottwitz for his influence on the ideas in this chapter and for his comments on a preliminary version. I thank Michael Rapoport for many stimulating conversations about test functions over the years. I am grateful to the referee for helpful suggestions and remarks.

2. Notation If G is a connected reductive group over a p-adic field F, then R(G) will denote the category of smooth representations of G(F) on C-vector spaces. We will write π ∈ R(G)irred or π ∈ (G/F) if π is an irreducible object in R(G). If G as above contains an F-rational parabolic subgroup P with F-Levi factor M and unipotent radical N , define the modulus function δ P : M(F) →

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δ P (m) = |det(Ad(m) ; Lie(N (F)))| F 1/2

where | · | F is the normalized absolute value on F. By δ P (m) we mean the positive square-root of the positive real number δ P (m). For σ ∈ R(M), we frequently consider the normalized induced representation G(F)

1/2

i PG (σ ) = Ind P(F) (δ P σ ). We let 1 S denote the characteristic function of a subset S of some ambient space. If S ⊂ G, let g S = gSg −1 . If f is a function on S, define the function g f on g S by g f (·) = f (g −1 · g). Throughout the chapter we use the Weil form of the local or global Langlands L-group L G.

3. Review of the Bernstein center We shall give a brief synopsis of [BD] that is suitable for our purposes. Other useful references are [Be92], [Ren], and [Roc]. The Bernstein center Z(G) of a p-adic group G is defined as the ring of endomorphisms of the identity functor on the category of smooth representations R(G). It can also be realized as an algebra of certain distributions, as the projective limit of the centers of the finite-level Hecke algebras, and as the ring of regular functions on a certain algebraic variety. We describe these in turn.

3.1. Distributions We start by defining the convolution algebra of distributions. We write G for the rational points of a connected reductive group over a p-adic field. Thus G is a totally disconnected locally compact Hausdorff topological group. Further G is unimodular; fix a Haar measure d x. Let Cc∞ (G) denote the set of C-valued compactly supported and locally constant functions on G. Let H(G, d x) = (Cc∞ (G), ∗d x ), the convolution product ∗d x being defined using the Haar measure d x. A distribution is a C-linear map D : Cc∞ (G) → C. For each f ∈ C ∞ (G) we define f˘ ∈ C ∞ (G) by f˘(x) = f (x −1 ) for x ∈ G. We set ˘ f ) := D( f˘). D( We can convolve a distribution D with a function f ∈ Cc∞ (G) and get a new function D ∗ f ∈ C ∞ (G), by setting ˘ · f ), (D ∗ f )(g) = D(g

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where (g · f )(x) := f (xg). The function D ∗ f does not automatically have compact support. We say D is essentially compact provided that D ∗ f ∈ Cc∞ (G) for every f ∈ Cc∞ (G). We define g f by g f (x) := f (g −1 xg) for x, g ∈ G. We say that D is G of G-invariant G-invariant if D( g f ) = D( f ) for all g, f . The set D(G)ec ∞ essentially compact distributions on Cc (G) turns out to have the structure of a commutative C-algebra. We describe next the convolution product and its properties. Given distributions D1 , D2 with D2 essentially compact, we define another distribution D1 ∗ D2 by (D1 ∗ D2 )( f ) = D˘ 1 (D2 ∗ f˘). Lemma 4.1. The convolution products D ∗ f and D1 ∗ D2 have the following properties: (a) For φ ∈ Cc∞ (G) let Dφ d x (sometimes abbreviated φ d x) denote the . essentially compact distribution given by f → G f (x)φ(x) d x. Then D φ d x ∗ f = φ ∗d x f . (b) If f ∈ Cc∞ (G), then D ∗ ( f d x) = (D ∗ f ) d x. In particular, Dφ1 d x ∗ Dφ2 d x = Dφ1 ∗d x φ2 d x . (c) If D2 is essentially compact, then (D1 ∗ D2 ) ∗ f = D1 ∗ (D2 ∗ f ). If D1 and D2 are each essentially compact, so is D1 ∗ D2 . (d) If D2 and D3 are essentially compact, then (D1 ∗D2 )∗D3 = D1 ∗(D2 ∗D3 ). (e) An essentially compact distribution D is G-invariant if and only if D ∗ (1U g d x) = (1U g d x) ∗ D for all compact open subgroups U ⊂ G and g ∈ G. Here 1U g is the characteristic function of the set U g. (f) If D is essentially compact and f 1 , f 2 ∈ C c∞ (G), then D ∗ ( f 1 ∗d x f 2 ) = (D ∗ f 1 ) ∗d x f 2 . G , ∗) is a commutative and associative Corollary 4.2. The pair (D(G)ec C-algebra.

3.2. The projective limit Let J ⊂ G range over the set of all compact open subgroups of G. Let H(G) denote the convolution algebra of compactly-supported measures on G, and let H J (G) ⊂ H(G) denote the ring of J -bi-invariant compactly-supported measures, with center Z J (G). The ring H J (G) has as unit e J = 1 J d x J , where 1 J is the characteristic function of J and d x J is the Haar measure with vold x J (J ) = 1. Note that if J ⊂ J , then d x J = [J : J ] d x J .

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Let Z(G, J ) denote the center of the algebra H(G, J ) consisting of compactly-supported J -bi-invariant functions on G with product ∗d x J . There is an isomorphism Z(G, J ) → Z J (G) by z J → z J d x J . For J ⊂ J there is an algebra map Z(G, J ) → Z(G, J ), given by z J → z J ∗d x J 1 J . Equivalently, we have Z J (G) → Z J (G) given by z J d x J → z J d x J ∗ (1 J d x J ). We can view R(G) as the category of non-degenerate smooth H(G)modules, and any element of lim Z(G, J ) acts on objects in R(G) in a way that ← − commutes with the action of H(G). Hence there is a canonical homomorphism lim Z(G, J ) → Z(G). ← − There is also a canonical homomorphism G lim Z (G, J ) → D(G)ec ← − since Z = (z J ) J ∈ lim Z(G, J ) gives a distribution on f ∈ Cc∞ (G) as ← − follows: choose J ⊂ G sufficiently small that f is right-J -invariant, and set / Z( f ) = z J (x) f (x) d x J . (4.2) G

This is independent of the choice of J . Note that for f ∈ H(G, J ) we have Z ∗ f = z J ∗d x J f , and in particular Z ∗ 1 J = z J , for all J . To see Z = (z J ) J as a distribution is really G-invariant, note that for f ∈ H(G, J ), the identities Z ∗ f = z J ∗d x J f = f ∗d x J z J imply that Z ∗ ( f d x) = ( f d x) ∗ Z . This in turn shows that Z is G-invariant by Lemma 4.1(e). Now §1.4–1.7 of [BD] show that the above maps yield isomorphisms G Z(G) ← lim Z(G, J ) → D(G)ec . (4.3) ← − Corollary 4.3. Let Z ∈ Z(G), and suppose a finite-length representation π ∈ R(G) has the property that Z acts on π by a scalar Z (π ).

(a) For every compact open subgroup J ⊂ G, Z ∗ 1 J acts on the left on π J by the scalar Z (π ). (b) For every f ∈ H(G), tr(Z ∗ f | π ) = Z (π ) tr( f | π ).

3.3. Regular functions on the variety of supercuspidal supports 3.3.1. Variety structure on set of supercuspidal supports We describe the variety of supercuspidal supports in some detail. Also we will describe it in a slightly unconventional way, in that we use the Kottwitz homomorphism to parametrize the (weakly) unramified characters on G(F). This will be useful later on, when we compare the Bernstein center with the stable Bernstein center.

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Let us recall the basic facts on the Kottwitz homomorphism [Ko97]. Let

un of the maximal unramified extension F un in some L be the completion F algebraic closure of F, and let L¯ ⊃ F¯ denote an algebraic closure of L. Let ∼ ¯ ¯ un ) denote the inertia group. Let ∈ Aut(L/F) be I = Gal( L/L) = Gal( F/F the inverse of the Frobenius automorphism σ . In [Ko97] is defined a functorial surjective homomorphism for any connected reductive F-group H

)) I , κ H : H (L) X ∗ (Z ( H

(4.4)

= H

(C) denotes the Langlands dual group of H . By [Ko97, §7], it where H remains surjective on taking -fixed points:

)) κ H : H (F) X ∗ (Z( H I . We define H (L)1 := ker(κ H ) H (F)1 := ker(κ H ) ∩ H (F). We also define H (F)1 ⊇ H (F)1 to be the kernel of the map H (F) →

)) /tor s derived from κ H . X ∗ (Z ( H I If H is anisotropic modulo center, then H (F)1 is the unique maximal compact subgroup of H (F) and H (F)1 is the unique parahoric subgroup of H (F) (see e.g. [HRo]). Sometimes the two subgroups coincide: for example if H is any unramified F-torus, then H (F)1 = H (F)1 . We define X (H ) := Homgrp (H (F)/H (F)1 , C× ), the group of unramified characters on H (F). This definition of X (H ) agrees with the usual one as in [BD]. We define X w (H ) := Homgrp (H (F)/H (F)1 , C× ) and call it the group of weakly unramified characters on H (F). We follow the notation of [BK] in discussing supercuspidal supports and inertial equivalence classes. As indicated earlier in §3.1, for convenience we will sometimes write G when we mean the group G(F) of F-points of an F-group G. A cuspidal pair (M, σ ) consists of an F-Levi subgroup M ⊆ G and a supercuspidal representation σ on M. The G-conjugacy class of the cuspidal pair (M, σ ) will be denoted (M, σ )G . We define the inertial equivalence classes: we write (M, σ ) ∼ (L , τ ) if there exists g ∈ G such that g Mg −1 = L and g σ = τ ⊗ χ for some χ ∈ X (L). Let [M, σ ] denote the equivalence class of G (M, σ )G .

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If π ∈ R(G)irred , then the supercuspidal support of π is the unique element (M, σ )G such that π is a subquotient of the induced representation i PG (σ ), where P is any F-parabolic subgroup having M as a Levi subgroup. Let XG denote the set of all supercuspidal supports (M, σ )G . Denote by the symbol s = [M, σ ]G a typical inertial class. For an inertial class s = [M, σ ]G , define the set Xs = {(L , τ )G | (L , τ ) ∼ (M, σ )}. We have * XG = Xs . s

We shall see below that XG has a natural structure of an algebraic variety, and the Bernstein components Xs form the connected components of that variety. First we need to recall the variety structure on X (M). As is well-known, X (M) has the structure of a complex torus. To describe this, we first consider the weakly unramified character group X w (M). This is a diagonalizable group over C. In fact, by Kottwitz we have an isomorphism

I ) = X ∗ ((Z ( M)

I ) ). M(F)/M(F)1 ∼ = X ∗ (Z ( M) This means that

I ) )], C), X w (M)(C) = Homgrp (M(F)/M(F)1 , C× ) = Homalg (C[X ∗ ((Z( M)

in other words,

I ) . X w (M) = (Z ( M)

(4.5)

Another way to see (4.5) is to use Langlands’ duality for quasicharacters, which is an isomorphism

Homcont (M(F), C× ) ∼ = H 1 (W F , Z ( M)). (Here W F is the Weil group of F; see §4.) Under this isomorphism, X w (M)

I) → is identified with the image of the inflation map H 1 (W F /I, Z ( M)

that is, with H 1 ( , Z ( M)

I ) = (Z ( M)

I ) . This last idenH 1 (W F , Z ( M)), cocyc

I ) to tification is given by the map sending a cocycle ϕ ∈ Z 1 ( , Z( M) cocyc I w

) . The two ways of identifying X (M) with (Z ( M)

I ) , ϕ () ∈ (Z ( M) that via the Kottwitz isomorphism and that via Langlands duality, agree.2 For a more general result which implies this agreement, see [Kal], Prop. 4.5.2. Now we can apply the same argument to identify the torus X (M). We first get X (M)(C) = Homgrp (M(F)/M(F)1 , C× ). 2 We normalize the Kottwitz homomorphism as in [Ko97], so that κ × Gm : L → Z is the

valuation map sending a uniformizer $ to 1. Then the claimed agreement holds provided we normalize the Langlands duality for tori as in (4.7).

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Since M(F)/M(F)1 is the quotient of M(F)/M(F)1 by its torsion, it follows that

I ) )◦ =: (Z ( M)

I )◦ , X (M) = ((Z ( M) the neutral component of X w (M). Now we turn to the variety structure on Xs . We fix a cuspidal pair (M, σ ) representing s M = [M, σ ] M and s = [M, σ ]G . Let the corresponding Bernstein components be denoted Xs and Xs M . As sets, we have Xs = {(M, σ χ )G } and Xs M = {(M, σ χ ) M }, where χ ∈ X (M). The torus X (M) acts on Xs M by χ → (M, σ χ ) M . The isotropy group is stabσ := {χ | σ ∼ = σ χ }. Let Z (M)◦ denote the neutral component of the center of M. Then stabσ belongs to the kernel of the map X (M) → X (Z(M)◦ ), χ → χ | Z (M)◦ (F) , hence stabσ is a finite subgroup of X (M). Thus Xs M is a torsor under the torus X (M)/stabσ , and thus has the structure of an affine variety over C. There is a surjective map Xs M → Xs (M, σ χ ) M → (M, σ χ )G , χ ∈ X (M)/stabσ . ∼ σ χ for some χ ∈ X (M)}. Then Let N G ([M, σ ] M ) := {n ∈ N G (M) | n σ = G the fibers of Xs M → Xs are precisely the orbits the finite group W[M,σ ] M := N G ([M, σ ]G )/M on Xs M . Via G Xs = W[M,σ ] M \Xs M

the set Xs acquires the structure of an irreducible affine variety over C. Up to isomorphism, this structure does not depend on the choice of the cuspidal pair (M, σ ). 3.3.2. The center as regular functions on X G An element z ∈ Z(G) determines a regular function X: for a point (M, σ )G ∈ Xs , z acts on i G P (σ ) by a scalar z(σ ) and the function (M, σ )G → z(σ ) is a regular function on XG . This is the content of [BD, Prop. 2.11]. In fact we have by [BD, Thm. 2.13] an isomorphism Z(G) → C[XG ].

(4.6)

Together with (4.3) this gives all the equivalent ways of realizing the Bernstein center of G.

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4. The local Langlands correspondence We need to recall the general form of the conjectural local Langlands correspondence (LLC) for a connected reductive group G over a p-adic field F. Let ¯ F¯ denote an algebraic closure of F. Let W F ⊂ Gal( F/F) =: F be the Weil group of F. It fits into an exact sequence of topological groups 1

/ IF

/ WF

val

/Z

/ 1,

where I F is the inertia subgroup of F and where, if ∈ W F is a geometric Frobenius element (the inverse of an arithmetic Frobenius element), then val() = −1. Here I F has its profinite topology and Z has the discrete topology. Sometimes we write I for I F in what follows. Recall the Weil–Deligne group is W F := W F C, where wzw−1 = |w|z val(w) for w ∈ W F and z ∈ C, with |w| := q F for q F = #(O F /($ F )), the cardinality of the residue field of F. A Langlands parameter is an admissible homomorphism ϕ : W F → L G,

W F . This means: where L G := G • ϕ is compatible with the projections W F → W F and ν : L G → W F ; • ϕ is continuous and respects Jordan decompositions of elements in W F and L G (cf. [Bo79, §8], for the definition of Jordan decomposition in the group W F C and what it means to respect Jordan decompositions here); • if ϕ(W F ) is contained in a Levi subgroup of a parabolic subgroup of L G, then that parabolic subgroup is relevant in the sense of [Bo79, §3.3]. (This condition is automatic if G/F is quasi-split.)

Let (G/F) denote the set of G-conjugacy classes of admissible homomor L phisms ϕ : W F → G and let (G/F ) = R(G(F))irred the set of irreducible smooth (or admissible) representations of G(F) up to isomorphism. Conjecture 4.4 (LLC). There is a finite-to-one surjective map (G/F) → (G/F), which satisfies the desiderata of [Bo79, §10]. The fiber ϕ over ϕ ∈ (G/F) is called the L-packet for ϕ. We mention a few desiderata of the LLC that will come up in what follows. First, LLC for Gm is nothing other than Langlands duality for Gm , which we

, normalize as follows: for T any split torus torus over F, with dual torus T

) Homconts (T (F), C× ) = Homconts (W F , T ξ ↔ ϕξ

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) and w ∈ W F , satisfies, for every ν ∈ X ∗ (T ) = X ∗ (T ν(ϕξ (w)) = ξ(ν(Art−1 F (w))).

(4.7)

ab × Here Art−1 F : W F → F is the reciprocity map of local class field theory which sends any geometric Frobenius element ∈ W F to a uniformizer in F. Next, we think of Langlands parameters in two ways, either as continuous L-homomorphisms

ϕ : W F →

L

G

modulo G-conjugation, or as continuous 1-cocycles

ϕ cocyc : W F → G

via the projection W → W F ). modulo 1-coboundaries (where W F acts on G F The dictionary between these is

WF ϕ(w) = (ϕ cocyc (w), w) ¯ ∈G for w ∈ W F and w¯ the image of w under W F → W F . The desideratum we will use explicitly is the following (a special case of [Bo79, 10.3(2)]: given any Levi pair (M, σ ) (where σ ∈ (M/F)) with repcocyc

and any unramified 1-cocycle z χcocyc : resenting 1-cocycle ϕσ : W F → M,

I representing χ ∈ X (M) via the Langlands correspondence for W F → Z ( M) quasi-characters on M(F), we have ϕσcocyc = ϕσcocyc · z χcocyc χ

We may view z cocyc modulo 1-coboundaries with values in M. as a 1-cocycle χ

the righton W F which is trivial on C; since it takes values in the center of M, hand side is a 1-cocycle whose cohomology class is independent of the choices cocyc cocyc of 1-cocycles ϕσ and z χ in their respective cohomology classes. Hence the condition just stated makes sense. Concretely, if χ ∈ X (M) lifts to an

I , then up to M-conjugacy

element z ∈ Z( M) we have

WF . ϕσ χ () = (z, 1)ϕσ () ∈ M

(4.8)

Remark 4.5. There is a well-known dictionary between equivalence classes of admissible homomorphisms ϕ : W F C → L G and equivalence classes of admissible homomorphisms W F × SL2 (C) → L G. For a complete explanation, see [GR, Prop. 2.2]. Because of this equivalence, it is common in the literature for the Weil–Deligne group W F to sometimes be defined as W F C, and sometimes as W F × SL2 (C).

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5. The stable Bernstein center 5.1. Infinitesimal characters

Following Vogan [Vo], we term a G-conjugacy class of an admissible3 homomorphism λ : WF →

L

G

an infinitesimal character. Denote the G-conjugacy class of λ by (λ)G . In this section we give a geometric structure to the set of all infinitesimal characters for a group G. It should be noted that the variety structure we define here differs from that put forth by Vogan in [Vo, §7]. If ϕ : W F → L G is an admissible homomorphism, then its restriction ϕ|W F represents an infinitesimal character. Here it is essential to consider restriction along the proper embedding W F → W F : if W F is thought of as W F C, then this inclusion is w → (w, 0); if W F is thought of as W F × SL2 (C), then the inclusion is w → (w, diag(|w|1/2 , |w|−1/2 )). If ϕπ ∈ (G/F) is attached by LLC to π ∈ (G/F), then following Vogan [Vo] we shall call the

G-conjugacy class (ϕπ |W F )G

the infinitesimal character of π . If G is quasi-split over F, then conjecturally every infinitesimal character λ is represented by a restriction ϕπ |W F : W F → L G for some π ∈ (G/F). Assume LLC holds for G/F. Let λ be an infinitesimal character for G. Define the infinitesimal class to be the following finite union of L-packets * λ := ϕ . ϕ ;λ

Here ϕ ranges over G-conjugacy classes of admissible homomorphisms W F → L G such that (ϕ|W F )G = (λ)

, and ϕ is the corresponding L-packet G of smooth irreducible representations of G(F).

5.2. LLC+ In order to relate the Bernstein variety X with the variety Y of infinitesimal characters, we will assume the Local Langlands Correspondence (LLC) for G and all of its F-Levi subgroups. We assume all the desiderata listed by Borel in [Bo79]. There are two additional desiderata of LLC we need. 3 “Admissible” is defined as for the parameters ϕ : W → L G (e.g. λ(W ) consists of F F semisimple elements of L G) except that we omit the “relevance condition”. This is because

the restriction ϕ|W F of a Langlands parameter could conceivably factor through a non-relevant Levi subgroup of L G (even though ϕ does not) and we want to include such restrictions in what we call infinitesimal characters.

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Definition 4.6. We will declare that G satisfies LLC+ if the LLC holds for G and its F-Levi subgroups, and these correspondences are compatible with normalized parabolic induction in the sense of the Conjecture 4.7 below, and invariant under certain isomorphisms in the sense of Conjecture 4.12 below. Let M ⊂ G denote an F-Levi subgroup. Then the inclusion M → G

induces an embedding L M → L G which is well-defined up to G-conjugacy (cf. [Bo79, §3]). Conjecture 4.7. (Compatibility of LLC with parabolic induction) Let σ ∈ (M/F) and π ∈ (G/F) and assume π is an irreducible subquotient of i PG (σ ), where P = M N is any F-parabolic subgroup of G with F-Levi factor M. Then the infinitesimal characters ϕπ | W F : W F →

L

G

and ϕσ | W F : W F →

L

M →

L

G

are G-conjugate. Remark 4.8. (1) The conjecture implies that the restriction ϕπ |W F depends only on the supercuspidal support of π. This latter statement is a formal consequence of Vogan’s Conjecture 7.18 in [Vo], but the Conjecture 4.7 is slightly more precise. In Proposition 4.23 we will give a construction of the map f in Vogan’s Conjecture 7.18, by sending a supercuspidal support (M, σ )G (a “classical infinitesimal character” in [Vo]) to the infinitesimal character (ϕσ |W F )G . With this formulation, the condition on f imposed in Vogan’s Conjecture 7.18 is exactly the compatibility in the conjecture above. (2) The conjecture holds for GLn , and is implicit in the way the local Langlands correspondence for GLn is extended from supercuspidals to all representations (see Remark 13.1.1 of [HRa1]). It was a point of departure in Scholze’s new characterization of LLC for GLn [Sch3], and that paper also provides another proof of the conjecture in that case. (3) I was informed by Brooks Roberts (private communication), that the conjecture holds for GSp(4). (4) Given a parameter ϕ : W F → L G, there exists a certain P = M N and a certain tempered parameter ϕ M : W F → L M and a certain real-valued unramified character χ M on M(F) whose parameter is in the interior of the Weyl chamber determined by P, such that the L-packet φ consists of Langlands quotients J (π M ⊗ χ M ), for π M ranging over the packet

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ϕ M . The parameter ϕ is the twist of ϕ M by the parameter associated to the character χ M . This reduces the conjecture to the case of tempered representations. One can further reduce to the case of discrete series representations. The following is a very natural kind of functoriality which should be satisfied for all groups. Conjecture 4.9. (Invariance of LLC under isomorphisms) Suppose φ : (G, π ) → (G , π ) is an isomorphism of connected reductive F-groups together with irreducible smooth representations on them. Then the induced isomorphism L φ : L G → L G (well-defined up to an inner automorphism of

G), takes the G -conjugacy class of ϕπ : W F → L G to the G-conjugacy L class of ϕπ : W F → G. Proposition 4.10. Conjecture 4.9 holds when G = GLn . Proof. (Guy Henniart). It is enough to consider the case where G = GLn and φ is an F-automorphism of GLn . The functorial properties in the Langlands correspondence for GLn are: (i) Compatibility with class field theory, that is, with the case where n = 1. (ii) The determinant of the Weil–Deligne group representation corresponds to the central character: this is Langlands functoriality for the homomorphism det : GLn (C) → GL1 (C). (iii) Compatibility with twists by characters, i.e., Langlands functoriality for the obvious homomorphism of dual groups GL1 (C) × GLn (C) → GLn (C). (iv) Compatibility with taking contragredients: this is Langlands functoriality with respect to the automorphism g → t g −1 (transpose inverse), since it is known that for GLn (F) this sends an irreducible representation to a representation isomorphic to its contragredient. These properties are enough to imply the desired functoriality for F-automorphisms of GLn . When n = 1, the functoriality is obvious for any F-endomorphism of GL1 . When n is at least 2, an F-automorphism of GLn induces an automorphism of SLn hence an automorphism of the Dynkin diagram which must be the identity or, (when n ≥ 3) the opposition automorphism. Hence up to conjugation by GLn (F), the F-automorphism is the identity on SLn , or possibly (when n ≥ 3) transpose inverse. Consequently the F-automorphism can be reduced (by composing with an inner automorphism or possibly with transpose inverse) to one which is the identity on SLn , hence is of the form g → g · c(det(g)) where c ∈ X ∗ (Z (GLn )). But this

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implies that it is the identity unless n = 2, in which case it could also be g → g · det(g)−1 . In that exceptional case, the map induced on the dual group GL2 (C) is also g → g · det(g)−1 , and the desired result holds by invoking (ii) and (iii) above. Corollary 4.11. Let M = GLn 1 × · · · × GLnr ⊂ GLn be a standard Levi subgroup. Let g ∈ GLn (F). Then Conjecture 4.9 holds for the isomorphism cg : M → g M given by conjugation by g. Proof. It is enough to consider the case where g belongs to the normalizer of M in GLn . Let T ⊂ M be the standard diagonal torus in GLn . Then g ∈ N G (T )M. Thus composing g with a permutation matrix which normalizes M we may assume that cg preserves each diagonal factor GLn i . The desired functoriality follows by applying Proposition 4.10 to each GLn i . For the purposes of comparing the Bernstein center and the stable Bernstein center as in Proposition 4.23, we need only this weaker variant of Conjecture 4.9. Conjecture 4.12. (Weak invariance of LLC) Let M ⊆ G be any F-Levi subgroup and let g ∈ G(F). Then Conjecture 4.9 holds for the isomorphism cg : M → g M.

5.3. Variety structure on the set of infinitesimal characters

⊃

It is helpful to rigidify things on the dual side by choosing the data G B⊃T of a Borel subgroup and maximal torus which are stable under the action of

and which form part of the data of a F -invariant splitting for G

F on G (cf. [Ko84a, §1]). The variety structure we will define will be independent

of this choice, up to isomorphism, since different choices such that

B ⊃ T L F

([Ko84a, Cor. 1.7]). Let B :=

are conjugate under G B W F and LT = T

WF . Following [Bo79, §3.3], we say a parabolic subgroup P ⊆ L G is standard

is a W F -stable standard if P ⊇ L B. Then its neutral component P ◦ := P ∩ G

parabolic subgroup of G (containing B), and P = P ◦ W F . Every parabolic

subgroup in L G is G-conjugate to a unique standard parabolic subgroup. Assume P is standard and let M◦ ⊂ P ◦ be the unique Levi factor with

; it is W F -stable. Then M := NP (M◦ ) is a Levi subgroup of P in M◦ ⊇ T the sense of [Bo79, §3.3], and M = M◦ W F . The Levi subgroups M ⊂ L G which arise this way are called standard. Every Levi subgroup in L G is

G-conjugate to at least one standard Levi subgroup; two different standard Levi

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135

Denote by {M} the set of standard Levi subgroups may be conjugate under G.

subgroups in L G which are G-conjugate to a fixed standard Levi subgroup M. L Now suppose λ : W F → G is an admissible homomorphism. Then there exists a minimal Levi subgroup of L G containing λ(W F ). Any two such are

commuting conjugate by an element of Cλ◦ , where Cλ is the subgroup of G with λ(W F ), by (the proof of) [Bo79, Prop. 3.6].

Suppose λ1 , λ2 : W F → L G are G-conjugate. Then there exists a + +

G-conjugate λ1 (resp. λ2 ) of λ1 (resp. λ2 ) and a standard Levi subgroup M1 + + −1 (resp. M2 ) containing λ+ = 1 (W F ) (resp. λ2 (W F )) minimally. Write gλ1 g + −1

Then the Levi subgroups gM1 g and M2 contain λ2 for some g ∈ G. λ+ 2 (W F ) minimally, hence by [Bo79, Prop. 3.6] are conjugate by an element s ∈ C ◦+ . Then sg(M1 )(sg)−1 = M2 , and thus {M1 } = {M2 }. λ2

Hence any G-conjugacy class (λ)G gives rise to a unique class of standard Levi subgroups {Mλ }, with the property that the image of some element λ+ ∈ (λ)G

is contained minimally by Mλ for some Mλ in this class. A similar argument shows the following lemma. + + Lemma 4.13. Let λ+

= 1 and λ2 be admissible homomorphisms with (λ1 )G + + + (λ2 )G , and suppose λ (W ) and λ (W ) are contained minimally by a

F F 1 2 standard Levi subgroup M. Then there exists n ∈ NG (M) such that n λ + = λ+ . 1 2

The following lemma is left to the reader. Lemma 4.14. If M ⊆

LG

is a standard Levi subgroup, then

◦ ◦ NG

(M) = {n ∈ N G

(M ) | nM is W F -stable}.

Consequently, conjugation by n ∈ NG (M) preserves the set (Z (M◦ ) I )◦ . More generally, if M1 and M2 are standard Levi subgroups of L G and if we define the transporter subset by

| gM1 g −1 = M2 }, TransG (M1 , M2 ) := {g ∈ G then ◦ ◦ ◦ ◦ TransG

(M1 , M2 ) = {g ∈ TransG

(M1 , M2 ) | gM1 = M2 g is W F -stable}.

Consequently, conjugation by g ∈ TransG (M1 , M2 ) sends (Z (M◦1 ) I )◦ into (Z (M◦2 ) I )◦ . We can now define the notion of inertial equivalence (λ1 )G

∼ (λ2 )G

of infinitesimal characters.

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Definition 4.15. We say (λ1 )G and (λ2 )G

are inertially equivalent if • {Mλ1 } = {Mλ2 }; + • there exists M ∈ {Mλ1 }, and λ+

and λ2 ∈ (λ2 )G

whose images 1 ∈ (λ1 )G are minimally contained by M, and an element z ∈ (Z (M◦ ) I )◦ , such that + (zλ+ 1 )M◦ = (λ2 )M◦ .

We write [λ]G for the inertial equivalence class of (λ)G

. Note that M automatically contains (zλ+ 1 )(W F ) minimally if it contains minimally.

λ+ 1 (W F )

Lemma 4.16. The relation ∼ is an equivalence relation on the set of infinitesimal characters. Proof. Use Lemmas 4.13 and 4.14.

⊃ B

⊃T

Remark 4.17. To define (λ1 )G ∼ (λ2 )G we used the choice of G

(which was assumed to form part of a F -invariant splitting for G) in order to define the notion of standard Levi subgroup of L G. However, the equivalence relation ∼ is independent of this choice, since as remarked above, any two

are conjugate under G

F , by [Ko84a, Cor. 1.7]. F -invariant splittings for G Remark 4.18. The property we need of standard Levi subgroups M ⊆ L G

is W F -stable, and is that they are decomposable, that is, M◦ := M ∩ G ◦ M = M W F . Any standard Levi subgroup is decomposable. In our discussion, we could have avoided choosing a notion of standard Levi, by associating to each (λ)G

a unique class of decomposable Levi subgroups {M},

all of which are G-conjugate, such that λ factors minimally through some M ∈ {M}. Now fix a standard Levi subgroup M ⊆ L G. We write tM◦ for an inertial equivalence class of admissible homomorphisms W F → M. We write YtM◦ for the set of M◦ -conjugacy classes contained in this inertial class. We want to give this set the structure of an affine algebraic variety over C. Define the torus Y (M◦ ) := (Z(M◦ ) I )◦ . Then Y (M◦ ) acts transitively on YtM◦ . Fix a representative λ : WF → M for this inertial class, so that tM◦ = [λ]M◦ .

(4.9)

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137

Lemma 4.19. The Y (M◦ )-stabilizer stabλ := {z ∈ Y (M◦ ) | (zλ)M◦ = (λ)M◦ } is finite. Proof. There exists an integer r ≥ 1 such that r acts trivially on M◦ . The group stabλ is contained in the preimage of the finite group Z (M◦ ) F ∩ (M◦ )der under the norm homomorphism Nr : (Z (M◦ ) I ) → Z (M◦ ) F , z → z(z) · · · r−1 (z) and the kernel of this homomorphism is finite. Then YtM◦ is a torsor under the quotient torus YtM◦ ∼ = Y (M◦ )/stabλ . In this way the left-hand side acquires the structure of an affine algebraic variety. Up to isomorphism, this structure is independent of the choice of λ representing tM◦ . Now let t denote an inertial class of infinitesimal characters for G, and let Yt denote the set of infinitesimal characters in t. Recall t gives rise to a unique class of standard Levi subgroups {M}, having the property that some representative λ for t factors minimally through some M ∈ {M}. Fix such a representative λ : W F → M → L G for t, so that t = [λ]G and tM◦ = [λ]M◦ . By our previous work, there is a surjective map YtM◦ → Yt (zλ)M◦ → (zλ)G . where z ∈ Y (M◦ )/stabλ . Let n ◦ NG

(M, [λ]M◦ ) = {n ∈ N G

(M) | ( λ)M◦ = (zλ)M◦ , for some z ∈ Y (M )}.

From the above discussion we see the following. Lemma 4.20. The fibers of YtM◦ → Yt are precisely the orbits of the finite

G ◦ group W[λ] := N G

(M, [λ]M◦ )/M on YtM◦ . M◦

Hence Yt = WtGM◦ \YtM◦ acquires the structure of an affine variety over + C. Thus Y = t Yt is an affine variety over C and each Yt is a connected component. Let Zst (G) denote the ring of regular functions on the affine variety Y. We call this ring the stable Bernstein center of G/F.

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5.4. Base change homomorphism of the stable Bernstein center Let E/F be a finite extension in F/F with ramification index e and residue field extension k E /k F of degree f . Then W E ⊂ W F and I E ⊆ I F . Further, we f can take E := F as a geometric Frobenius element in W E . Let YG/F resp. G/E Y denote the variety of infinitesimal characters associated to G resp. G E . Proposition 4.21. The map (λ)G → (λ|W E )G determines a morphism of algebraic varieties YG/F → YG/E . Definition 4.22. We call the corresponding map b E/F : Zst (G E ) → Zst (G) the base change homomorphism for the stable Bernstein center.

W F factors minimally through the standard Proof. Suppose λ : W F → G

WE Levi subgroup M ⊂ G W F and that its restriction λ|W E : W E → G

factors minimally through the standard Levi subgroup M E ⊂ G W E . We may assume M◦E ⊆ M◦ and thus Z(M◦ ) ⊂ Z(M◦E ). There is a homomorphism of tori Y (M◦ ) = (Z(M◦ ) I F )◦ F −→ (Z(M◦E ) I E )◦

f

F

= Y (M◦E )

(4.10) f −1

z −→ z f := N f (z) := z · F (z) · · · F

(z).

Recall that z ∈ (Z (M◦ ) I F ) F is identified with the image of the element z( F ) ∈ Z(M◦ ) I F , where z is viewed as a cohomology class z ∈ H 1 ( F , Z (M◦ ) I F ). Using the same fact for E in place of F, it follows that (zλ)|W E = z f λ|W E , where z f is defined as above. Thus the map (zλ)G

→ ((zλ)|W E )G lifts to the map (zλ)M◦ → (z f λ|W E )M◦ → (z f λ|W E )G

, and being induced by (4.10), the latter is an algebraic morphism.

5.5. Relation between the Bernstein center and the stable Bernstein center The varieties X and Y are defined unconditionally. In order to relate them, we need to assume LLC+ holds. Proposition 4.23. Assume LLC+ holds for the group G. Then the map (M, σ )G → (ϕσ |W F )G defines a quasi-finite morphism of affine algebraic varieties f : X → Y. It is surjective if G/F is quasi-split.

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The reader should compare this with Conjecture 7.18 in [Vo]. Our variety structure on the set Y is different from that put forth by Vogan, and our f is given by a simple and explicit rule. In view of LLC+ our f automatically satisfies the condition which Vogan imposed on the map in his Conjecture 7.18: if π has supercuspidal support (M, σ )G , then the infinitesimal character of π is f ((M, σ )G ). Proof. It is easy to see that the map (M, σ )G → (ϕσ |W F )G is well-defined. We need to show that an isomorphism cg : (M, σ ) → ( g M, g σ ) given by conjugation by g ∈ G(F) gives rise to parameters ϕσ : W F → L M → L G and ϕ g σ : W F → L ( g M) → L G which differ by an inner automorphism of

In view of Conjecture 4.12 applied to M, the isomorphism L ( g M) → G. LM

takes ϕ g σ to an M-conjugate of ϕσ . On the other hand the embeddings L M → L G and L ( g M) → L G are defined using based root systems in such a way

that it is obvious that they are G-conjugate. To examine the local structure of this map, we first fix a λ and a standard Mλ through which λ factors minimally. Let t = [λ]G . Then over Yt the map f takes the form * Xs M → Yt . (4.11) s M ;t

Here s M ranges over the inertial classes [M, σ ]G such that (ϕσ |W F )G is inertially equivalent to (λ)G . We now fix a representative (M, σ ) for s M . Given

such a ϕσ , its restriction ϕσ |W F factors through a G-conjugate of L M. But

ϕσ |W F factors mini(ϕσ |W F )G

∼ (λ)G

implies that (up to conjugation by G) L mally through Mλ . Thus we may assume that Mλ ⊆ M. The corresponding

→ Z (M◦ ) induces a morphism of algebraic tori inclusion Z ( M) λ

= (Z ( M)

I )◦ → (Z (M◦λ ) I )◦ = Y (M◦λ ). Y ( M)

by the Kottwitz isomorphism (or the Langlands Further, recall X (M) ∼ = Y ( M) cocyc duality for quasi-characters), by the rule χ → z χ (). Taking (4.8) into account, we see that (4.11) on Xs M , given by (M, σ χ )G → (ϕσ χ |W F )G

for χ ∈ X (M)/stabσ , lifts to the map X (M)/stabσ → Y (M◦λ )/stabλ ,

(4.12)

→ Y (M◦ ), up to transwhich is the obvious map induced by X (M) → Y ( M) λ ◦ lation by an element in Y (Mλ ) measuring the difference between (ϕσ |W F )M◦λ and (λ)M◦λ . The map (4.12) is clearly a morphism of algebraic varieties. Hence the map f is a morphism of algebraic varieties.

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The fibers of f are finite by a property of LLC. Finally, if G/F is quasi-split, the morphism f is surjective by another property of LLC. Corollary 4.24. Assume G/F satisfies LLC+, so that the map f in Proposition 4.23 exists. Then f induces a C-algebra homomorphism Zst (G) → Z(G). It is injective if G/F is quasi-split. Remark 4.25. For the group GLn the constructions above are unconditional because the local Langlands correspondence and its enhancement LLC+ are known (cf. Remark 4.8(2) and Corollary 4.11). One can see that XGLn → YGLn is an isomorphism and hence Zst (GLn ) = Z(GLn ). Remark 4.26. As remarked by Scholze and Shin [SS, §6], one may conjecturally characterize the image of Zst (G) → Z(G) in a way that avoids direct mention of L-parameters. According to them it should consist of the distriG such that, for any function f ∈ C ∞ (G(F) whose stable butions D ∈ D(G)ec c orbital integrals vanish at semi-simple elements, the function D∗ f also has this property. See [SS, §6] for further discussion of this. From conjectured relations between stable characters and stable orbital integrals, one can conjecturally rephrase the condition on D in terms of stable characters, as SOϕ (D ∗ f ) = 0, ∀ϕ, if SOϕ ( f ) = 0, ∀ϕ.

(4.13)

An element of Zst (G) acts by the same scalar on all π ∈ ϕ , and so the above condition holds if D ∈ f (Zst (G)). The converse direction is much less clear, and implies non-trivial statements about the relation between supercuspidal supports, L-packets, and infinitesimal classes. Indeed, suppose we are given D ∈ Z(G) that satisfies (4.13). This should mean that it acts by the same scalar on all π ∈ ϕ . On the other hand, saying D comes from Zst (G) would mean that D acts by the same scalar on all π ∈ λ and those scalars vary algebraically as λ ranges over Y. So if for some λ the infinitesimal class * λ = ϕ ϕ ;λ

contains an L-packet ϕ0 λ such that the set of supercuspidal supports coming from ϕ0 does not meet the set of those coming from any ϕ with ϕ not conjugate to ϕ0 , then one could construct a regular function D ∈ Z(G) which is constant on the L-packets ϕ but not constant on λ , and thus not in f (Zst (G)). In that case (4.13) would not be sufficient to force D ∈ f (Zst (G)), and the conjecture of Scholze–Shin would be false. In that case, one could define the subring Zst∗ (G) ⊆ Z(G) of regular functions on the Bernstein variety which take the same value on all supercuspidal supports of representations in the same L-packet. This would then perhaps better deserve the title “stable

Stable Bernstein center and test functions

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Bernstein center” and it would be strictly larger than f (Zst (G)) at least in some cases. To illustrate this in a more specific setting, suppose G/F is quasi-split and λ does not factor through any proper Levi subgroup of L G. Then by Proposition 4.27 below, we expect λ to consist entirely of supercuspidal representations. If λ contains at least two L-packets ϕ , then there would exist a D ∈ Z(G) which is constant on the ϕ ’s yet not constant on λ , and the Scholze–Shin conjecture should be false. Put another way, if the Scholze–Shin conjecture is true, we expect that whenever λ does not factor through a proper Levi in L G, the infinitesimal class λ consists of at most one L-packet.4

5.6. Aside: when does an infinitesimal class consist only of supercuspidal representations? Proposition 4.27. Assume G/F is quasi-split and LLC+ holds for G. Then λ consists entirely of supercuspidal representations if and only if λ does not factor through any proper Levi subgroup L M L G. Proof. If λ contains a nonsupercuspidal representation π with supercuspidal support (M, σ )G for M G, then by LLC+, we may assume ϕπ |W F , and hence λ, factors through the proper Levi subgroup L M L G. Conversely, if λ factors minimally through a standard Levi subgroup Mλ L G, then we must show that contains a nonsupercuspidal representation λ of G. Since G/F is quasi-split, we may identify Mλ = L Mλ for an F-Levi subgroup Mλ G. Now for t = [λ]G , the map (4.11) is surjective. For any F-Levi subgroup

I )◦ < M Mλ , a component of the form X[M,σ ]G has dimension dim (Z ( M) I ◦ dim (Z ( Mλ ) ) = dim Yt . Thus the union of the components of the form X[M,σ ]G with M Mλ cannot surject onto Yt . Thus there must be a component of the form X[Mλ ,σλ ] appearing in the left-hand side of (4.11). We may cocyc assume ϕσλ factors through L Mλ along with λ. Writing (λ) M ϕσλ ) M

λ = (z χ

λ for some χ ∈ X (Mλ ), it follows that λ contains the nonsupercuspidal representations with supercuspidal support (Mλ , σλ χ )G .

5.7. Construction of the distributions ZV Let (r, V ) be a finite-dimensional algebraic representation of L G on a complex vector space. Given a geometric Frobenius element ∈ W F and an admissible homomorphism λ : W F → L G, we may define the semi-simple trace 4 In fact this statement holds: if λ does not factor through a proper Levi subgroup of L G, then there is at most one way to extend it to an admissible homomorphism ϕ: W F → L G.

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Thomas J. Haines trss (λ(), V )) := tr(r λ(), V rλ(I F ) ).

Note this is independent of the choice of . This notion was introduced by Rapoport [Ra90] in order to define semisimple local L-functions L(s, π p , r ), and is parallel to the notion for -adic Galois representations used in [Ra90] to define semi-simple local zeta functions ζpss (X, s); see also [HN02a], [H05]. The following result is an easy consequence of the material in §5.3. Proposition 4.28. The map λ → trss (λ(), V ) defines a regular function on the variety Y hence defines an element Z V ∈ Zst (G) by ss Z V ((λ)G

) = tr (λ(), V ).

We use the same symbol Z V to denote the corresponding element in Z(G) given via Zst (G) → Z(G). The latter has the property Z V (π ) = trss (ϕπ (), V )

(4.14)

for every π ∈ (G/F), where Z V (π ) stands for Z V ((M, σ )G ) if (M, σ )G is the supercuspidal support of π . Remark 4.29. One does not really need the full geometric structure on the set Y in order to construct Z V ∈ Z(G): one may show directly, assuming that LLC and Conjecture 4.7 hold, that π → trss (ϕπ (), V ) descends to give a regular function on X and hence (4.14) defines an element Z V ∈ Z(G). Using the map f simply makes the construction more transparent (but has the drawback that we also need to assume Conjecture 4.12).

6. The Langlands–Kottwitz approach for arbitrary level structure 6.1. The test functions Let (G, X ) be a Shimura datum, where X is the G(R)-conjugacy class of an R-group homomorphism h : RC/R Gm → GR . This gives rise to the reflex field E ⊂ C and a G(C)-conjugacy class {μ} ⊂ X ∗ (GC ) which is defined over E. Choose a quasi-split group G∗ over Q and an inner twisting ψ : G∗ → G of Q-groups. In particular we get an inner twisting G∗E → GE as well as an isomorphism of L-groups L (GE ) → L (G∗E ). Let Q ⊂ C denote the algebraic numbers, so that we have an inclusion E ⊂ Q and we can regard {μ} as a G(Q)-conjugacy class in X ∗ (GQ ) which is

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defined over E (cf. [Ko84b, Lemma 1.1.3]). Using ψ regard {μ} as a G∗ (Q)conjugacy class in X ∗ (G∗ ), defined over E. By Kottwitz’ lemma ([Ko84b, Q 1.1.3]), {μ} is represented by an E-rational cocharacter μ : Gm → G∗E . Following Kottwitz’ argument in [Ko84b, 2.1.2], it is easy to show that there exists a unique representation (r−μ , V−μ ) of L (GE ) such that as a represen ∗ it is an irreducible representation with extreme weight −μ and the tation of G Weil group WE acts trivially on the highest-weight space corresponding to any

∗ . E -fixed splitting for G E Using ψ we can regard (r−μ , V−μ ) as a representation of L (GE ). The isomorphism class of this representation depends only on the equivalence class of the inner twisting ψ, thus only on G and {μ}. Now we fix a rational prime p and set G := GQ p . Choose a prime ideal p ⊂ E lying above p, and set E := Ep . Choose an algebraic closure Q p of Q p and fix henceforth an isomorphism of fields C ∼ = Q p such that the embedding E → C ∼ = Q p corresponds to the prime ideal p. This gives rise to an embedding Q → Q p extending E → Q p , and thus to an embedding W E → WE . We get from this an embedding L (G E ) → L (GE ). Via this embedding we can regard (r−μ , V−μ ) as a representation (r−μ , V−μ ) of L (G E ). Associated to (r−μ , V−μ ) ∈ Rep( L (G E )) we have an element Z V−μ in the Bernstein center Z(G E ). Of course here and in what follows, we are assuming LLC+ holds for G E . Now we review briefly the Langlands–Kottwitz approach to studying the local Hasse–Weil zeta functions of Shimura varieties. Let Sh K p = Sh(G, h −1 , K p K p ) denote the canonical model5 over E for the Shimura variety attached to the data (G, h −1 , K p K p ) for some sufficiently small compact p open subgroup K p ⊂ G(A f ) and some compact open subgroup K p ⊂ G(Q p ). We limit ourselves to constant coefficients Q in the generic fiber of Sh K p (here = p is a fixed rational prime). Let p denote any geometric Frobenius element in Gal(Q p /Q p ). Then in the Langlands–Kottwitz approach to the semi-simple local zeta function ζpss (s, Sh K p ), one needs to prove an identity of the form trss (rp , H•c (Sh K p ⊗E Q p , Q )) =

c(γ0 ; γ , δ) Oγ (1 K p ) TOδθ (φr ).

(γ0 ;γ ,δ)

(4.15) Here the semi-simple Lefschetz number Lefss (rp , Sh K p ) on the left-hand side is the alternating semi-simple trace of Frobenius on the compactly-supported 5 We use this term in the same sense as Kottwitz [Ko92a], comp. Milne [Mil, §1, esp. 1.10].

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-adic cohomology groups6 of Sh K p (see [Ra90] and [HN02a] for the notion of semi-simple trace). The expression on the right has precisely the same form as the counting points formula proved by Kottwitz in certain good reduction cases (PEL type A or C, K p hyperspecial; cf. [Ko92a, (19.6)]). The integer r ≥ 1 ranges over integers of the form j · [k E0 : F p ], j ≥ 1, where E 0 /Q p is the maximal unramified subextension of E/Q p and k E0 is its residue field. Thus j rp = p where p is a geometric Frobenius element in Gal(Q p /E). Finally, φr is an element in the Hecke algebra H(G(Q pr ), K pr ) with values in Q , where Q pr is the unique degree r unramified extension in Q p /Q p , and where K pr ⊂ G(Q pr ) is a suitable compact open subgroup which is assumed to be a natural analogue of K p ⊂ G(Q p ). To be more precise about K pr , in practice there is a smooth connected Z p -model G for G, such that K p = G(Z p ). In that case, we always take K pr = G(Z pr ), where Z pr is the ring of integers in Q pr . In forming TOδσ (φr ), the Haar measure on G(Q pr ) is normalized to give K pr measure 1. Let E j /E be the unique unramified extension of degree j in Q p /E. Let E j0 /Q p be the maximal unramified subextension of E j /Q p . So E/E 0 and E j /E j0 are totally ramified of the same degree, and E j 0 = Q pr . √ 1/2 We make the choice of p ∈ Q , and use it to define δ P as a function √ with values in Q( p) ⊂ Q . We can now specify the test function φr ∈ Z(G(E j0 ), K j 0 ), which will take values in Q . In the construction of the elements Z V ∈ Zst (G), everything works the same

) W F on a Q -vector space. way for (r, V ) a representation of L G := G(Q We henceforth take this point of view. Let (r−μ, j , V−μ, j ) ∈ Rep( L (G E j )) denote the restriction of (r−μ , V−μ ) ∈ Rep( L (G E )) via L (G E j ) → L (G E ). E

E

j0 j0 L We can then induce to get a representation (r−μ, j , V−μ, j ) of (G E j0 ). By Mackey theory, we get the same representation if we first induce to L (G E0 ) and then restrict to L (G E j0 ), that is, we have

E

E

GW E j0

j0 j0 (r−μ,

j , V−μ, j ) := IndGW

Ej

GW E0

E ResGW r−μ = ResGW

GW Ej

GW E

E j0

0 IndGW r−μ .

E

(4.16) This gives rise to Z

E j0

V−μ, j

∈

Zst (G

E j0 ). By abuse of notation, we use the same

symbol to denote the image of this in the Bernstein center: Z

E j0

V−μ, j

∈ Z(G E j 0 ).

6 The Langlands–Kottwitz method really applies to the middle intersection cohomology groups

of the Baily–Borel compactification and not just to the cohomology groups with compact supports; see [Ko90] and [Mor] for some general conjectures and results in this context, at primes of good reduction. The identity (4.15) corresponds to the contribution of the interior, at primes of arbitrary reduction, and is a first step toward understanding the intersection cohomology groups.

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Of course here we are viewing Z(G E j 0 ) as Q -valued regular functions on the Bernstein variety, or equivalently as Q -valued invariant essentially compact distributions: the topology on C playing no role, it is harmless to identify it with Q . The following is the conjecture formulated jointly with R. Kottwitz. Conjecture 4.30. (Test function conjecture) Let d = dim(Sh K p ). The test r d/2 function φr in (4.15) may be taken to be p Z E j 0 ∗ 1 K pr . In particular, V−μ, j

φr may be taken in the center Z(G(Q pr ), K pr ) of H(G(Q pr ), K pr ) and these test functions vary compatibly with change in the level K p in an obvious sense. The same test functions should be used when one incorporates arbitrary Hecke operators away from p into (4.15). Following Rapoport’s strategy (cf. [Ra90], [Ra05], [H05]), one seeks to find a natural integral model M K p over O E for Sh K p , and then rephrase the above conjecture using the method of nearby cycles R" := R" M K p (Q ). Conjecture 4.31. There exists a natural integral model M K p /O E for Sh K p , such that trss (rp , R"x ) = c(γ0 ; γ , δ) O(1 K p ) TOδθ (φr ), (4.17) x∈M K p (k E j0 )

where φr = pr d/2 Z

(γ0 ;γ ,δ)

E j0 V−μ, j

∗ 1 K pr

as in Conjecture 4.30.

Remark 4.32. Implicit in this conjecture is that the method of nearby cycles can be used for compactly-supported cohomology. In fact we could conjecture there exists a suitably nice compactification of M K p /O E so that the natural map Hic (M K p ⊗O E F p , R"(Q )) → Hic (Sh K p ⊗ E Q p , Q ) is a Galois-equivariant isomorphism. For G = GSp2g and where M K p is the natural integral model for Sh K p for K p an Iwahori subgroup, this was proved by Benoit Stroh. Of course, one is really interested in intersection cohomology groups of the Baily–Borel compactification (see footnote 5), and in fact Stroh [Str] computed the nearby cycles and verified the analogue of the Kottwitz conjecture on nearby cycles (see Conjecture 4.44 below) for these compactifications. Remark 4.33. Some unconditional versions of Conjectures 4.30 and 4.31 have been proved. See §8.

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6.2. Endoscopic transfer of the stable Bernstein center Part of the Langlands–Kottwitz approach is to perform a “pseudostabilization” of (4.15), and in particular prove the “fundamental lemmas” that are required for this. The stabilization expresses (4.15) in the form H i(G, H) STe∗ (h), the sum over global Q-elliptic endoscopic groups H for G of the (G, H)regular Q-elliptic part of the geometric side of the stable trace formula for (H, h) (cf. notation of [Ko90]), for a certain transfer function h ∈ Cc∞ (H(A)). (By contrast in “pseudostabilization” which is used in certain situations, one instead writes (4.15) in terms of the trace formula for G and not its quasisplit inner form, and this is sometimes enough, as in e.g. Theorem 4.40 below.) For stabilization one needs to produce elements h p ∈ Cc∞ (H(Q p )) which are Frobenius-twisted endoscopic transfers of φr . The existence of such transfers h p is due mainly to the work of Ngô [Ngo] and Waldspurger [Wal97, Wal04, Wal08]. But we hope to have a priori spectral information about the transferred functions h p . A guiding principle is that the nearby cycles on an appropriate “local model” for Sh K p should naturally produce a central element as a test function φr , which should coincide with that given by the test function conjecture (cf. Conjecture 4.31); then its spectral behavior is known by construction. In that case one can formulate a conjectural endoscopic transfer h p of φr with known spectral behavior. General Frobenius-twisted endoscopic transfer homomorphisms Zst (G Q pr ) → Zst (HQ p ) will be described elsewhere. Here for simplicity we content ourselves to describe two special cases: standard (untwisted) endoscopic transfer of the geometric Bernstein center, and the base change transfer for the stable Bernstein center. 6.2.1. Endoscopic transfer of the geometric Bernstein center Let us fix an endoscopic triple (H, s, η0 ) for G over a p-adic field F (cf. [Ko84a, §7]), and suppose we have fixed an extension η : L H → L G of

→ G

(we suppose we are in a situation, e.g. G der = G sc , where such η0 : H extensions always exist). We could hope the natural map Y H/F −→ YG/F (λ) H −→ (η ◦ λ)G

would be algebraic and hence would induce an endoscopic transfer homomorphism Zst (G) → Zst (H ). By invoking further expectations about endoscopic lifting, one would then formulate a map on the level of Bernstein centers, Z(G) → Z(H ), which we could write as Z → Z |η . But these assertions

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147

are not obvious. Fortunately, in practice we need this construction rather on the geometric Bernstein center. Definition 4.34. Assume LLC+ holds for G/F. We define the geometric Bernstein center Zgeom (G) to be the subalgebra of Zst (G) generated by the elements Z V as V ranges over Rep( L G). The terminology geometric Bernstein center is motivated by § 6.4 below. Let V |η ∈ Rep( L H ) denote the restriction of V ∈ Rep( L G) along η. Further assume LLC+ also holds for H/F. Then Z V → Z V |η determines a map Zgeom (G) → Zgeom (H ). Write Z VG (resp. Z VH|η ) for the image of Z V (resp. Z V |η ) in Z(G) (resp. Z(H )). Conjecture 4.35. Assume LLC+ holds for both G and H . Then in the above situation the distribution Z VH|η ∈ Z(H ) is the endoscopic transfer of Z VG ∈

Z(G) in the following sense: whenever a function φ H ∈ Cc∞ (H (F)) is a transfer of a function φ ∈ Cc∞ (G(F)), then Z VH|η ∗ φ H is a transfer of Z VG ∗ φ. This conjecture and its Frobenius-twisted analogue were announced by the author in April 2011 at Princeton [H11]. A very similar statement subsequently appeared as Conjecture 7.2 in [SS]. Considering the untwisted case for simplicity, the difference is that in [SS], the authors take in place of Z V an element in the stable Bernstein center essentially of the form (λ)G → tr(λ( F ), V−μ ), where here the usual trace, not the semi-simple trace, is used. That conjecture is proved in [SS] in all EL or quasi-EL cases, by invoking special features of general linear groups such as the existence of base change representations. Formally, Conjecture 4.35 contains as a special case the “fundamental lemma implies spherical transfer” result of Hales [Hal] (see also Waldspurger [Wal97]). Indeed if K , K H are hyperspecial maximal compact subgroups in G(F), H (F), then 1 K H is a transfer of 1 K by the fundamental lemma, and hence Z VH|η ∗ 1 K H is a transfer of Z VG ∗ 1 K . But by the Satake isomorphism, every K -spherical function on G(F) is a linear combination of functions of the form Z VG ∗ 1 K for some representation V (comp. §6.4). Even in more general situations, Conjecture 4.35 is most useful when applied to a pair φ, φ H of unit elements in appropriate Hecke algebras. At least when G splits over F un , Kazhdan–Varshavsky proved in [KV] that for some explicit scalar c, the Iwahori unit c1 I H is a transfer of the Iwahori unit 1 I . As another example, if K nG ⊂ G(F) is the n-th principal congruence subgroup in G(F), then for some explicit scalar c the function c1 K nH is a transfer

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of 1 K nG (proved by Ferrari [Fer] under some mild restrictions on the residue characteristic of F), and thus c(Z VH|η ∗ 1 K nH ) should be an explicit transfer of Z VG ∗ 1 K nG . A Frobenius-twisted analogue of Ferrari’s theorem together with the Frobenius-twisted analogue of Conjecture 4.35 would give an explicit Frobenius-twisted transfer of the test function φr from Conjecture 4.30, if K p is a principal congruence subgroup.

6.2.2. Base change of the stable Bernstein center We return to the situation of Proposition 4.21, but we specialize it to cyclic Galois extensions of F and furthermore we assume G/F is quasi-split. Let E/F be any finite cyclic Galois subextension of F/F with Galois group θ , and with corresponding inclusion of Weil groups W E → W F . If φ ∈ H(G(E)) and f ∈ H(G(F)) are functions in the corresponding Hecke algebras of locally constant compactly-supported functions, then we say φ, f are associated (or f is a base-change transfer of φ), if the following result holds for the stable (twisted) orbital integrals: for every semisimple element γ ∈ G(F), we have SOγ ( f ) = (γ , δ) SOδθ (φ) (4.18) δ

where the sum is over stable θ -conjugacy classes δ ∈ G(E) with semisimple norm N δ, and where (γ , δ) = 1 if N δ = γ and (γ , δ) = 0 otherwise. See e.g. [Ko86], [Ko88], [Cl90], or [H09] for further discussion. Conjecture 4.36. In the above situation, consider Z ∈ Zst (G E ), and consider its image, also denoted by Z , in Z(G E ). Consider b E/F (Z) ∈ Zst (G) (cf. Definition 4.22) and also denote by b E /F (Z ) its image in Z(G). Then b E/F (Z) is the base-change transfer of Z ∈ Z(G E ), in the following sense: whenever a function f ∈ Cc∞ (G(F)) is a base-change transfer of φ ∈ Cc∞ (G(E)), then b E/F (Z ) ∗ f is a base-change transfer of Z ∗ φ. Proposition 4.37. Conjecture 4.36 holds for GLn . Proof. The most efficient proof follows Scholze’s proof of Theorem C in [Sch2] which makes essential use of the existence of cyclic base change lifts for GLn . Let π ∈ (GLn /F) be a tempered irreducible representation with base change lift ∈ (GLn /E), a tempered representation which is characterized by the character identity % ((g, θ )) = %π (N g) for all elements g ∈ GLn (E) with regular semisimple norm N g ([AC, Thm. 6.2, p. 51]). Here (g, θ ) ∈ GLn (E) Gal(E/F) and θ acts on by the normalized intertwiner Iθ : → of [AC, p. 11].

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Suppose f is a base-change transfer of φ. Using the Weyl character formula and its twisted analogue (cf. [AC, p. 36]), we see that tr((φ, θ ) | ) = tr( f | π ). Multiplying by the constant Z () = b E/F (Z )(π ), we get tr((Z ∗ φ, θ ) | ) = tr(b E/F (Z) ∗ f | π ). (Use Corollary 4.3 and its twisted analogue.) There exists a base-change transfer h ∈ Cc∞ (GLn (F)) of Z ∗ φ ([AC, Prop. 3.1]). Using the same argument as above for the pair Z ∗φ and h, we conclude that tr(b E/F (Z)∗ f −h | π ) = 0 for every tempered irreducible π ∈ (GLn /F). By Kazhdan’s density theorem (Theorem 1 in [Kaz]) the regular semi-simple orbital integrals of b E/F (Z ) ∗ f and h agree. Thus the (twisted) orbitals integrals of b E /F (Z ) ∗ f and φ match at all regular semi-simple elements, and hence at all semi-simple elements by Clozel’s Shalika germ argument [Cl90, Prop. 7.2]. Remark 4.38. Unconditional versions of Conjecture 4.36 are available for parahoric and pro-p Iwahori–Hecke algebras, when G/F is unramified.7 See §9.

6.3. Application: local Hasse–Weil zeta functions By Kottwitz’ base change fundamental lemma for units [Ko86], we know 1 K p is a base-change transfer of 1 K pr whenever K p = G(Z p ) and K pr = G(Z pr ) for a smooth connected Z p -model G for G. Then Conjectures 4.30 and 4.36 together say that ( j)

fp

:= pr d/2 b E j 0 /Q p (Z

E j0

V−μ, j

) ∗ 1K p

(4.19)

is a base-change transfer of a test function φr that satisfies (4.15). Setting E

GW E

E

0 (r−μ0 , V−μ0 ) := IndGW r−μ

(4.20)

E

we have for any admissible parameter ϕ : WQ p → ϕ (G/Q p ) the identity ( j)

K

L (G

Qp )

E

and any π p ∈

tr( f p | π p ) = pr d/2 dim(π p p ) trss (ϕ(rp ) , V−μ0 ),

(4.21)

7 The pro-p Sylow subgroup of an Iwahori subgroup I ⊂ G(F) coincides with its pro-unipotent radical I + , and it has become conventional to term the Hecke algebra Cc∞ (I + \G(F)/I + ) the

pro- p Iwahori–Hecke algebra.

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where r = j[E 0 : Q p ]. In the compact and non-endoscopic cases, the above discussion allows us to express ζpss (s, Sh K p ) in terms of semi-simple automorphic L-functions. To explain this we need a detour on the point of view taken in [L1, L2] (comp. [Ko84b, §2.2]). Recall (r−μ , V−μ ) ∈ Rep( L (G E )). Consider the Langlands representation

GW

r := IndG WQ r−μ , E

and for each prime p of E dividing p, consider

GW Qp

rp := IndGW

Ep

GW Q

GW

p E ResGW r−μ = IndGW r−μ .

Ep

Mackey theory gives

W G

Q ResGW r=

Qp

Ep

#

rp .

p| p

If p is understood, let Ep0 /Q p denote the maximal unramified subextension of Ep /Q p , and set E = Ep and E 0 := Ep0 . Then we have

WQ G

WQ G

E

p p rp = IndGW r−μ = IndGW r−μ0 .

E

E0

(4.22)

Lemma 4.39. Suppose π p ∈ ϕ (G/Q p ). Then $ j E trss (ϕ(p ), r−μ0 ) if r = j [E 0 : Q p ] −1 ss r [E 0 : Q p ] tr (ϕ( p ), rp ) = 0, if [E 0 : Q p ] |r . (4.23)

WQ p -modules Proof. There is an isomorphism of G E0

WQ p ] ⊗C[G W ] r−μ rp ∼ , = C[G E 0

and

ϕ(IQ ) rp p

has a C-basis of the form {ϕ(ip ) ⊗ wk } where 0 ≤ i ≤ [E 0 :

Q p ]−1 and {wk } comprises a C-basis for (r−μ0 )ϕ(IQ p ) . The lemma follows. E

The following result shows the potential utility of Conjectures 4.30 and 4.36. It applies not just to PEL Shimura varieties, but to any Shimura variety where these conjectures are known. Similar results will hold when incorporating Hecke operators away from p. Theorem 4.40. Suppose Gder is anisotropic over Q, so that the associated Shimura variety Sh K p = Sh(G, h −1 , K p K p ) is proper over E. Suppose G has “no endoscopy”, in the sense that the group K(Gγ0 /Q) is trivial for every semisimple element γ0 ∈ G(Q), as in e.g. [Ko92b]. Let p be a prime ideal of E

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dividing p. Assume (LLC+) (cf. §5.2), and Conjectures 4.30 and 4.36 hold for all groups GQ pr . Then in the notation above, we have K d ζpss (s, Sh K p ) = L ss (s − , π p , rp )a(π f ) dim(π f ) , (4.24) 2 π f

πp

where π f = ⊗ π p runs over irreducible admissible representations of G(A f ) and the integer a(π f ) is given by a(π f ) = m(π f ⊗ π∞ ) tr( f ∞ |π∞ ), π∞ ∈∞

where m(π f ⊗π∞ ) is the multiplicity of π f ⊗ π∞ in L2 (G(Q)AG (R)◦ \G(A)). Here AG is the Q-split component of the center of G (which we assume is also its R-split component). Further ∞ is the set of irreducible admissible representations of G(R) which have trivial infinitesimal and central characters, and f ∞ is defined as in [Ko92b] to be (−1)dim(Sh K ) times a pseudo-coefficent 0 ∈ . of an essentially discrete series member π∞ ∞ Proof. The method follows closely the argument of Kottwitz in [Ko92b] (comp. [HRa1, §13.4]), so we just give an outline. We will use freely the notation of Kottwitz and [HRa1]. Set f = [Ep0 : Q p ]. By definition we have log ζpss (s, Sh K p ) =

∞

j

Lefss (p , Sh K p )

j =1

p− j f s . j

(4.25)

By using (4.15) together with Conjectures 4.30 and 4.36, the arguments of Kottwitz [Ko92b] show that for each j ≥ 1 j ( j) Lefss (p , Sh K p ) = τ (G) SOγ0 ( f p f p f ∞ ), (4.26) γ0

( j)

where f p is defined as in (4.19) and f p is the characteristic function of K p ⊂ p G(A f ). Here γ0 ranges over all stable conjugacy classes in G(Q). Since Gder is anisotropic over Q, the trace formula for any f ∈ Cc∞ (AG (R)◦ \G(A)) takes the simple form τ (Gγ )Oγ ( f ) = m(π ) tr( f |π ), (4.27) γ

π

where γ ranges over conjugacy classes in G(Q) and π ranges over irreducible representations in L2 (G(Q)AG (R)◦ \G(A)). By [Ko92b, Lemma 4.1], the vanishing of all K(Gγ0 /Q) means that

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τ (Gγ ) Oγ ( f ) = τ (G)

γ

SOγ0 ( f ).

γ0

It follows that j

Lefss (p , Sh K p ) ( j) = m(π ) tr( f p f p f ∞ |π ) π

=

π f π∞ ∈∞

=

πf

( j)

p

m(π f ⊗ π∞ ) · tr( f p |π f ) · tr( f p |π p ) · tr( f ∞ |π∞ ) j

E

a(π f ) dim(π Kf ) p j f d/2 trss (ϕπ p (p ), V−μ0 ),

the last equality by (4.21). By definition we have log L ss (s, π p , rp ) =

∞

trss (ϕπ p (rp ), rp )

r =1

p −r s . r

Now (4.24) follows by invoking (4.23). Remark 4.41. Unconditional versions of Theorem 4.40 are available for some parahoric or pro-p-Iwahori level cases, or for certain compact “Drinfeld case” Shimura varieties with arbitrary level; these cases are alluded to in §8.

6.4. Relation with geometric Langlands For simplicity, assume G is split over a p-adic or local function field F. Assume G satisfies LLC+. From the construction of Z V in Proposition 4.28, we have a map

→ Z(G, J ) K 0 RepC (G)

(4.28)

V → Z V ∗ 1 J for any compact open subgroup J ⊂ G(F), which gives rise to a commutative diagram Z(G, J) 7 −∗ J 1 I

Z(G, I) 5 k k k Bernkkkk −∗ I 1 K kkk∼ kkk Sat / H(G, K )

K 0 Rep(G) ∼

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whenever J ⊆ I ⊂ K where I resp. K is an Iwahori resp. special maximal compact subgroup, and where the bottom two arrows are the Bernstein resp. Satake isomorphisms. We warn the reader that the oblique arrow

→ Z(G, J ) is injective but not surjective in general, and also it is K 0 Rep(G) additive but not an algebra homomorphism in general. Gaitsgory [Ga] constructed the two arrows Sat and Bern geometrically when F is a function field, using nearby cycles for a degeneration of the affine Grassmannian GrG to the affine flag variety Fl for G. One can hope that, as in the

→ Z(G, J ) catIwahori case [Ga], one can construct the arrow K 0 Rep(G) egorically using nearby cycles for a similar degeneration of GrG to a “partial affine flag variety”, namely an fpqc-quotient LJ/L + J where J is a smooth connected group scheme over F p [[t]] with generic fiber JF p ((t)) = G F p ((t)) and J(F p [[t]]) = J . Here LJ (resp. L + J) is the ind-scheme (resp. scheme) over F p representing the sheaf of groups for the fpqc-topology whose sections for an F p -algebra R are given by LJ(Spec R) = J(R[[t]][ 1t ]) (resp. L + J(Spec R) = J(R[[t]])). At least for J = I + , the pro-p Iwahori subgroup, this will be realized in forthcoming joint work of the author and Benoit Stroh. In a related vein, the geometric Satake equivalence of Mirkovic–Vilonen [MV] is a categorical version of the Satake isomorphism Sat, and this is usually stated when G is a split group over F = F p ((t)). One can ask for a version of this when G is nonsplit, possibly not even quasi-split, over such a field F. The correct Satake isomorphism to “categorify” appears to be the one described in [HRo]. In many cases where G is quasi-split and split over a tamely ramified extension of F, this has been carried out in recent work of X. Zhu [Zhu].

7. Test functions in the parahoric case We fix r = j[E 0 : Q p ] for some j ∈ N. We assume K p is a parahoric subgroup of G(Q p ), and we let K pr denote the corresponding parahoric subgroup of G(Q pr ). Assuming LLC+ holds for G Q pr , we can speak of the test function φr = pr d/2 Z E j 0 ∗ 1 K pr ∈ Z(G(Q pr ), K pr ). (4.29) V−μ, j

We wish to give a more concrete description of this function, making use of Bernstein’s isomorphism for Z(G(Q pr ), K pr ) which is detailed in the Appendix, §11. In the next two subsections, we are concerned with the case where G Q pr is quasi-split. We write F := Q pr . Choose a maximal F-split torus A in G, and let T denote its centralizer in G. Fix an F-rational Borel subgroup B containing T . Let K F ⊂ G(F) denote the parahoric subgroup corresponding to K p .

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By Kottwitz [Ko84b, Lemma (1.1.3)], the G(Q p )-conjugacy class {μ} is represented by an F-rational cocharacter μ ∈ X ∗ (T ) F = X ∗ (A). It is clear that E, the field of definition of {μ}, is contained in any subfield of Q p which splits G. Given π ∈ (G/F) with π K F = 0, to understand (4.29) we need to compute the scalar tr(ϕπ ( F ), (V−μ0 )ϕπ (I F ) ). E

(4.30)

There is an unramified character χ of T (F) such that π is a subquotient of i BG (χ ), and we may assume ϕπ |W F = ϕχ |W F . Since χ is unramified, ϕχ (I F ) =

W F . Regarding χ as an element of T

, (4.7) implies that we may 1 IF ⊂ T

write ϕχ ( F ) = χ F ∈ T W F . Then we need to compute E

tr(χ F , (V−μ0 )1I F ).

(4.31)

7.1. Unramified groups and the Kottwitz conjecture Let us consider the case where G Q pr is unramified. Since we are assuming G splits over an unramified extension of Q p , it follows that E/Q p is unramified, E i.e. E = E 0 and V−μ0 = V−μ . Moreover F = E j0 contains E with degree j.

1I F Further, since G splits over F un , we have V−μ = V−μ . So we are reduced to computing tr(χ F , V−μ ). Exactly as in Kottwitz’ calculation of the Satake transform in [Ko84b, p. 295], we see that (4.30) is tr(χ F , V−μ ) = (−λ)(χ ). (4.32) λ∈W (F)·μ

Here W (F) = W (G, A) is the relative Weyl group for G/F, and we view

. This proves the following result. λ ∈ X ∗ (A) = X ∗ (T ) F as a character on T Lemma 4.42. In the above situation, Z

E j0

V−μ, j

∗ 1 K pr = z −μ, j ,

(4.33)

where the Bernstein function z −μ, j (cf. Definition 4.72) is the unique element of Z(G(F), K F ) which acts (on the left) on the normalized induced representation i BG (χ ) K F by the scalar λ∈W (F)·μ (−λ)(χ ), for any unramified character χ : T (F) → C× . Of course the advantage of z −μ, j is that unlike the left-hand side of (4.33), it is defined unconditionally. A relatively self-contained, elementary, and efficient approach to Bernstein functions is given in §11. Thus Conjecture 4.30 in this situation is equivalent to the Kottwitz Conjecture.

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Conjecture 4.43. (Kottwitz conjecture) In the situation where GQ pr is unramified and K p is a parahoric subgroup, the function φr in (4.15) may be taken to be pr d/2 z −μ, j . Conjecture 4.43 was formulated by Kottwitz in 1998, about 11 years earlier than Conjecture 4.30. There is a closely related conjecture of Kottwitz concerning nearby cycles on Rapoport–Zink local models Mloc K p for Sh K p . We refer to [RZ, Ra05] for definitions of local models, and to [H05, HN02a] for further details about the following conjecture in various special cases. Conjecture 4.44. (Kottwitz Conjecture for Nearby Cycles) Write G for the Bruhat–Tits parahoric group scheme over Z pr with generic fiber G Q pr and with G(Z pr ) = K pr . Let Gt denote the analogous parahoric group scheme over F pr [[t]] with the “same” special fiber as G. Then there is an L + Gt,F pr -equivariant embedding of Mloc K p ,F r into the affine flag variety p

LGt,F pr /L + Gt,F pr , via which we can identify the semisimple trace of FrobeMloc Kp

nius function x → trss (Fr pr , R"x

r ) on x ∈ Mloc K p (F p ) with the function

p dr/2 z −μ, j ∈ Z(Gt (F pr ((t))) , Gt (F pr [[t]])).

7.2. The quasi-split case

I F is a possibly disconnected reductive group, with maximal torus The group G I ◦

F ) (see the proof of Theorem 8.2 of [St]). Now we may restrict the rep(T E

IF W F ⊂ G

W F . Let χ be a weakly resentation V−μ0 to the subgroup G

I F ) F . The only unramified character of T (F); by (4.5) we can view χ ∈ (T E 0 1I F I F

T -weight spaces of (V−μ ) which contribute to (4.31) are indexed by the

I F ) F . (It is important to note that it is F -fixed weights, i.e. by those in X ∗ (T

I F , and not for the maximal the weight spaces for the diagonalizable group T I ◦ F

torus (T ) , which come in here.) This is consistent with Theorem 4.71 of the Appendix, and may be expressed as follows. Proposition 4.45. In the general quasi-split situation, Z

E j0

V−μ, j

∗ 1 K pr is the

unique function in Z(G(Q pr ), K pr ) which acts on the left on each weakly unramified principal series representation i BG (χ ) K F by the scalar (4.31), and thus is a certain linear combination of Bernstein functions z −λ, j where −λ ∈ E0

I F ) F ranges over the W (G, A)-orbits of F -fixed T

I F -weights in V−μ X ∗ (T . It is an interesting exercise to write out the linear combinations of Bernstein functions explicitly in each given case. Once this is done, the result can be used to find explicit descriptions of test functions for inner forms of quasi-split groups. This is the subject of the next subsection.

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7.3. Passing from quasi-split to general cases via transfer homomorphisms 7.3.1. Test function conjecture via transfer homomorphisms We use freely the notation and set-up explained in the Appendix §11.12. Let G ∗ be a quasi-split F-group with an inner twisting ψ : G → G ∗ . Let J ∗ ⊂ G ∗ (F) resp. J ⊂ G(F) be parahoric subgroups and consider the normalized transfer homomorphism t˜ : Z(G ∗ (F), J ∗ ) → Z(G(F), J ) from Definition 4.78. The following conjecture indicates that test functions for the quasi-split group G ∗ should determine test functions for G. This is compatible with the global considerations which led to Theorem 4.40. Conjecture 4.46. Let K pr resp. K ∗pr be parahoric subgroups of G(Q pr ) resp. G ∗ (Q pr ), with corresponding normalized transfer homomorphism t˜ : Z(G ∗ (Q pr ), K ∗pr ) → Z(G(Q pr ), K pr ). If φr∗ ∈ Z(G ∗ (Q pr ), K ∗pr ) is the ∗ function pr d/2 (Z G E j 0 ∗ 1 K ∗r ) described in Proposition 4.45 for the data V−μ, j (G ∗Q r , {−μ}, K ∗pr ), then p

p

φr := t˜(φr∗ ) ∈ Z(G(Q pr ), K pr ) is a test function satisfying (4.15) for the original data (G Q pr , {−μ}, K pr ). Assuming Conjecture 4.30 holds, another way to formulate this is that the ∗ normalized transfer homomorphism t˜ takes the function Z G E j 0 ∗ 1 K ∗r ∈ Z(G ∗ (Q

pr ),

K ∗pr )

to the function Z

V−μ , j

G E j0

V−μ, j

p

∗ 1 K pr ∈ Z(G(Q pr ), K pr ). But

the point of Conjecture 4.46 is to provide an explicit test function for the non-quasi-split data (G Q pr , {−μ}, K pr ) which can be compared with direct geometric calculations of the nearby cycles attached to this data, and thus to provide a method to prove an unconditional analogue of Conjecture 4.30 for such data. This is illustrated in §7.3.3 below. The next two paragraphs show that Conjecture 4.46 is indeed reasonable. 7.3.2. A calculation for GL2 Take = GL2,F and G = D × , where D is the central simple division algebra over F of dimension 4. Here we will explicitly calculate and compare the test functions associated to (GL2,F , {−μ}, I F ) and (D × , {−μ}, O× D ), where μ = (1, 0), and where I F ⊂ × × GL2 (F) and O D ⊂ D are the standard Iwahori subgroups. This calculation will show that the normalized transfer homomorphism takes one test function to the other. This is required in order for both Conjectures 4.30 and 4.46 to hold true. G∗

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157 ×

∗ Write z −μ = Z V−μ2 ∗ 1 I F ∈ Z(GL2 (F), I F ) and z −μ = Z VD−μ ∗ 1O× ∈ D H(D × , O× ) ∼ = C[Z]. The last isomorphism is induced by the Kottwitz homoGL

D

morphism, which in this case is the normalized valuation valF ◦ Nrd D : D × Z, where val F is the normalized valuation for F and Nrd D : D → F is the reduced norm. Write μ¯ = (0, 1) and let B ∗ denote the Borel subgroup of lower triangular ∗ acts on the left on i GL2 (χ ) I F by the scalar matrices in GL2 . Then z −μ B∗ tr(χ F , V−μ ) = (−μ − μ)(χ ¯ ), for any unramified character χ ∈ Hom(T ∗ (F)/T ∗ (F)1 , C× ). We may view χ as a diagonal 2 × 2 complex matrix χ = diag(χ1 , χ2 ). To calculate z −μ we need a few preliminary remarks. First we parametrize × unramified characters η ∈ Hom(D × /O× D , C ) by writing η = η0 ◦ Nrd D , where η0 ∈ C× is viewed as the unramified character on F × which sends $ F → η0 . The map Nrd D : D × → F × is Langlands dual to the diagonal embedding Gm (C) → GL2 (C), and it follows that the cocycles z η and z η0 attached by Langlands duality to the quasicharacters η and η0 satisfy z η0 0 η0 0 zη = , and thus, z η ( F ) = . 0 z η0 0 η0 On the other hand, if 1 denotes the trivial 1-dimensional representation of D × , × × then its Langlands parameter ϕ1D satisfies ϕ1D ( F ) = diag(q −1/2 , q 1/2 ) (see, e.g., [PrRa, Thm. 4.4]). So using (4.8), we obtain ×

tr(ϕηD ( F ), V−μ ) η q −1/2 0 0 = tr , V −μ 0 η0 q 1/2 =

−1/2 (δ B ∗ ($ −μ ) · −μ|

Z

+

−1/2 δ B ∗ ($ −μ¯ ) · −μ| ¯

Z)

η

0

0

0 . η0

∗ . Using the definition of t˜ we deduce the following Here

Z is the center of G result. Proposition 4.47. The normalized transfer homomorphism t˜ : Z(GL2 (F), ∗ I F ) → H(D × , O× D ) sends z −μ to z −μ . 7.3.3. Compatibility with nearby cycles in some anisotropic cases Suppose we are in a situation where E = Q p . As before, write F = Q pr . Suppose G F = (D ⊗ F)× × Gm , where D is a central division algebra over E of degree n 2 , for n > 2. This situation arises in the setting of “fake unitary”

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simple Shimura varieties (see, e.g. [H01, §5]). Let G ∗ = GLn × Gm , a split inner form of G over Q p . ∗ = Suppose that V−μ = ∧m (Cn ) for 0 < m < n, i.e. the representation of G × GLn (C) × C where the first factor acts via the irreducible representation with highest weight (1m , 0n−m ) and the second factor acts via scalars. Consider the local models M∗loc = Mloc (G ∗ , {−μ}, K ∗p ) and Mloc = Mloc (G, {−μ}, K p ), where K ∗p ⊂ G ∗ (F) and K p ⊂ G(F) are Iwahori subgroups. We can choose the inner twist G → G ∗ and the subgroups K ∗p and K p so that M∗loc (F p ) = Mloc (F p ) and where the action of geometric Frobenius p on the right-hand side is given by p = Ad(c p ) · ∗p where ∗p is the usual Frobenius action (on the lefthand side) and where τ → Ad(cτ ) represents the class in H 1 (Q p , PGLn ) corresponding to the inner twist G → G ∗ . Assume (r, n) = 1 and set q = pr . Then Mloc (G, {−μ}, K p )(Fq ) consists of a single point. To understand the corresponding test function we may ignore the Gm -factor and pretend that G = D × and G ∗ = GLn . Then the Kottwitz homomorphism κG : G(F) → Z induces an isomorphism H(G(Q pr ), K pr ) ∼ = C[Z]. The test function for the Shimura variety giving rise to the local Shimura data (G, {−μ}, K p ) should be calculated by understanding the function trace of Frobenius on nearby cycles on Mloc , similarly to Conjecture 4.44 in the unramified case. The test function should be of the form Cq · 1m ∈ C[Z] = H(G(Q p ), K pr ) for some scalar Cq . Proposition 4.48. In the above situation, Conjecture 4.46 predicts that the coefficient Cq is given by Cq = #Gr(m, n)(Fq ), the number of Fq -rational points on the Grassmannian variety Gr(m, n) parametrizing m-planes in n-space. This is compatible with calculations of Rapoport of the trace of Frobenius on nearby cycles of the local models for such situations, see [Ra90]. Thus the normalized transfer homomorphism gives a group-theoretic framework with which we could make further predictions about nearby cycles on the local models attached to non-quasiplit groups G, assuming we know explicitly the corresponding test function for a quasi-split inner form of G. Proof. By the final sentence of Proposition 4.79, we simply need to inte∗ grate the function pr d/2 z −μ ∈ Z(GLn (Q pr ), K ∗pr ) over the fiber of the

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159

Kottwitz homomorphism val ◦ det over 1m ∈ C[Z]. This is a combinatorial ∗ explicitly. However, it problem which could be solved since we know pr d/2 z −μ is easier to use geometry. Translating “integration over the fiber of the Kottwitz homomorphism” in terms of local models gives us the equality ∗loc Cq = Tr(rp , R" M (Q )x ). x∈M∗loc (Fq )

(Here is a rational prime with = p.) But the special fiber of M∗loc embeds into the affine flag variety FlGLn for GLn /F p , and under the projection p : FlGLn → GrGLn to the affine Grassmannian, M∗loc maps onto Gr(m, n) and ∗loc Rp∗ (R" M (Q )) = Q , the constant -adic sheaf on Gr(m, n) in degree 0. Thus we obtain Cq = Tr(rp , (Q )x ) = #Gr(m, n)(Fq ) x∈Gr(m,n)(Fq )

as desired. (The reader should note the similarity with Prop. 3.17 in [Ra90], which is justified in a slightly different way.)

8. Overview of evidence for the test function conjecture 8.1. Good reduction cases In case GQ p is unramified and K p is a hyperspecial maximal compact subgroup of G(Q p ), we expect Sh(G, h −1 , K p K p ) to have good reduction over OEp . In PEL cases this was proved by Kottwitz [Ko92a]. In the same paper for PEL cases of type A or C, it is proved that the function φr = 1 K pr μ( p−1 )K pr satisfies (4.15), which can be viewed as verifying Conjecture 4.30 for these cases.

8.2. Parahoric cases Assume K p is a parahoric subgroup. We will discuss only PEL Shimura varieties. Here the approach is via the Rapoport–Zink local model Mloc K p for a suitable integral model M K p for Sh K p and the main ideas are due to Rapoport. We refer to the survey articles [Ra90], [Ra05], and [H05] for more about how local models fit in with the Langlands–Kottwitz approach. For much more about the geometry of local models, we refer the reader to the survey article of Pappas– Rapoport–Smithing [PRS] and the references therein.

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Using local models, the first step to proving Conjecture 4.31 is to prove Conjecture 4.44. The first evidence was purely computational: in [H01], z −μ, j was computed explicitly in the Drinfeld case and the result was compared with Rapoport’s computation of the nearby cycles in that setting, proving Conjecture 7.1.3 directly. This result motivated Kottwitz’ more general conjecture and also inspired Beilinson and Gaitsgory to construct the center of an affine Hecke algebra via a nearby cycles construction, a feat carried out in [Ga]. Then in [HN02a] Gaitsgory’s method was adapted to prove Conjecture 4.44 for the split groups GLn and GSp2n . This in turn was used to demonstrate Conjecture 4.43 for certain special Shimura varieties in [H05], and then to prove the analogue of Theorem 4.40 for those special Shimura varieties with parahoric level structure at p. The harmonic analysis ingredient needed for the latter was provided by [H09]. In his 2011 PhD thesis, Sean Rostami proved Conjecture 4.44 when G is an unramified unitary group. In a recent breakthrough, Pappas and Zhu defined group-theoretic local models Mloc K p whenever G splits over a tamely ramified extension, and for unramified groups G proved Conjecture 4.44, see [PZ], esp. Theorem 10.16.

8.3. Deeper level cases We again limit our discussion to PEL situations, where progress to date has occurred. It is again natural to study directly the nearby cycles relative to a suitable integral model for the Shimura variety and hope that it gives rise to a test distribution in the Bernstein center. For Shimura varieties in the “Drinfeld case” with K p a pro-p Iwahori subgroup of G(Q p ) = GLn (Q p ) × Q× p (“1 ( p)-level structure at p”), one may use Oort–Tate theory to define suitable integral models and prove Conjectures 4.31 and 4.30 for them. This was done by the author and Rapoport [HRa1] (and [H12] provided the harmonic analysis ingredient needed to go further and prove Theorem 4.40 in this case). Around the same time as [HRa1], Scholze studied in [Sch1] nearby cycles on suitable integral models for the modular curves with arbitrarily deep full level structure at p. In this way he proved Conjectures 4.31 and 4.30 in these cases, taking the compactifications also into account, and thereby proved the analogue of Theorem 4.40 for the compactified modular curves at nearly all primes of bad reduction. The nearby cycles on his integral models naturally gave rise to some remarkable distributions in the Bernstein center, for which he gave explicit formulas (see § 10).

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161

Then in [Sch2] Scholze generalized the approach of [Sch1] to compact Shimura varieties in the Drinfeld case, again finding an explicit description of nearby cycles. In this case, he was still able to produce a test function to plug into (4.15), or rather, simultaneously incorporating the base-change transfer results he needed in precisely this case, he found a test function that goes directly into the pseudostabilization of (4.15). This allowed him to prove Theorem 4.40. In contrast to the modular curve situation, in higher rank the nearby cycles on Scholze’s integral models do not directly produce distributions in the Bernstein center, and an explicit description of his test functions seems hopeless. But nevertheless Scholze was able to prove by indirect means Conjecture 4.30 in this case. The description of the nearby cycles in [Sch2] provided one ingredient for Scholze’s subsequent paper [Sch3] which gave a new and streamlined proof of the local Langlands conjecture for general linear groups. In later work Scholze [Sch4] formalized his method of producing test functions in many cases, using deformation spaces of p-divisible groups, and this is used to give a nearly complete description of the cohomology groups of many compact unitary Shimura varieties in his joint work [SS] with S. W. Shin; their main assumption at p is that GQ p is a product of Weil restrictions of general linear groups. The advantage of what we could call the Langlands–Kottwitz– Scholze approach in this situation is that it yields in [SS] a new construction of the Galois representations constructed earlier by Shin [Sh], in a shorter way that avoids Igusa varieties. In these later developments, Conjecture 4.30 does not play a central part, but the stable Bernstein center does nevertheless still play a clarifying role in the pseudostabilization process (e.g. in [SS]). It seems that only certain integral models, such as those we see in many parahoric or pro-p Iwahori level cases, have the favorable property that their nearby cycles naturally give rise to distributions in the Bernstein center. It remains an interesting problem to find such integral models in more cases, and to better understand the role of the Bernstein center in the study of Shimura varieties.

9. Evidence for conjectures on transfer of the Bernstein center Here we present some evidence for the general principle that the (stable/geometric) Bernstein center is particularly well-behaved with respect to (twisted) endoscopic transfer. The primary evidence thus far consists of some unconditional analogues of Conjecture 4.36.

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Let G/F be an unramified group, and let Fr /F be the degree r unramified extension of F in some algebraic closure of F. In [H09, H12], the author defined base change homomorphisms br : Z(G(Fr ), Jr ) → Z(G(F), J ), where J ⊂ G(F) is either a parahoric subgroup or a pro-p Iwahori subgroup, and where Jr is the corresponding subgroup of G(Fr ). Then we have “basechange fundamental lemmas” of the following form.8 Theorem 4.49. For any φr ∈ Z(G(Fr ), Jr ), the function br (φr ) is a basechange transfer of φr in the sense of (4.18). By Kottwitz [Ko86], the function 1 J is a base-change transfer of 1 Jr . Hence for any Vr ∈ Rep( L (G Fr )), Conjecture 4.36 predicts that b Fr /F (Z Vr ) ∗ 1 J is a base-change transfer of Z Vr ∗ 1 Jr . This is a consequence of Theorem 4.49, because of the following compatibility between the base-change operations in [H09, H12] and in the context of stable Bernstein centers (cf. Prop. 4.21). Lemma 4.50. In the above situations, br (Z Vr ∗ 1 Jr ) = b Fr /F (Z Vr ) ∗ 1 J . Proof. First assume J is a parahoric subgroup. Let χ be any unramified character of T (F). It is enough to show that the two functions act on the left by the same scalar on every unramified principal series representation i BG (χ ) J . Let Nr : T (Fr ) → T (F) be the norm homomorphism. By the definition of br in [H09], br (Z Vr ∗ 1 Jr ) acts by the scalar by which Z Vr ∗ 1 Jr acts on i BGrr (χ ◦ Nr ) Jr . This is the scalar by which Z Vr acts on i BGrr (χ ◦ Nr ), which in view of LLC+ is Tr trss (ϕχ◦N (rF ), Vr ) = trss (ϕχT (rF ), Vr ). r

(4.34)

But the right-hand side is the scalar by which b Fr /F (Z Vr ) ∗ 1 J acts on i BG (χ ) J . Tr The equality ϕχ◦N (rF ) = ϕχT (rF ) we used in (4.34) follows from the r commutativity of the diagram of Langlands dualities for tori Homconts (T (F), C× ) Nr

Homconts (T (Fr ), C× )

/ H 1 (W F , L T ) conts Res

/ H 1 (W F , L Tr ) conts r

which was proved in [KV, Lemma 8.1.3]. 8 Relating to pro-p Iwahori level, a much stronger result is proved in [H12] concerning the

base-change transfer of Bernstein centers of Bernstein blocks for depth-zero principal series representations.

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163

Now suppose J = I + is a pro-p Iwahori subgroup. Then the same argument works given the following fact: for any depth-zero character χ : T (F)1 → C× and any extension of it to a character χ˜ on T (F), and any zr ∈ Z(G(Fr ), Ir+ ), + the function br (z r ) acts on i BG (χ) ˜ I by the same scalar by which zr acts on + i BGrr (χ˜ ◦ Nr ) Ir . This follows from the definition of br given in Definition 10.0.3 of [H12], using [H12, Lemma 4.2.1]. Let us also mention again Scholze’s Theorem C in [Sch2], which essentially proves Conjecture 4.36 for GLn (see Proposition 4.37).

10. Explicit computation of the test functions 10.1. Parahoric cases Conjecture 4.30 implies that test functions are compatible with change of level. Therefore for the purposes of computing them for parahoric level, the key case is where K p is an Iwahori subgroup. Thus, for the rest of this subsection we consider only Iwahori level structure. Since test functions attached to quasisplit groups should determine, in an computable way, those for inner forms (by Conjecture 4.46 and Proposition 4.79), it is enough to understand quasi-split groups. Via Proposition 4.45 this boils down to giving explicit descriptions of the Bernstein functions z −λ, j , assuming we have already expressed the test function explicitly in term of these – this is automatic for unramified groups using the Kottwitz Conjecture (Conjecture 4.43). Let us therefore consider the problem of explicitly computing Bernstein functions z μ attached to any group G/F and an Iwahori subgroup I ⊂ G(F) (F being any local nonarchimedean field). For simplicity consider the case where G/F is unramified, and regard μ as a dominant coweight in X ∗ (A). The μ which arise in Conjecture 4.43 are minuscule; however, we consider μ denote the extended affine which are not necessarily minuscule here. Let W Weyl group of G over F (cf. §11). Attached to μ is an the μ-admissible set , defined by Adm(μ) ⊂ W | x ≤ tλ , for some λ ∈ W (G, A) · μ}, Adm(μ) = {x ∈ W determined by the Iwahori subwhere ≤ denotes the Bruhat order on W corresponding group I and where tλ denotes the translation element in W to λ ∈ X ∗ (A). The μ-admissible set has been studied for its relation to the stratification by Iwahori-orbits in the local model Mloc K p ; for much information see [KR], [HN02b], [Ra05]. The strongest combinatorial results relating local models and Adm(μ) are due to Brian Smithling, see e.g. [Sm1, Sm2, Sm3].

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For our purposes, the set Adm(μ) enters because it is the set indexing the double cosets in the support of z μ . 0 Proposition 4.51. The support of z μ is the union x∈Adm(μ) I x I . Proof. This was proved using the theory of alcove walks as elaborated by Görtz [G], in the Appendix to [HRa1]. It applies to affine Hecke algebras with arbitrary parameters, hence the corresponding result holds for arbitrary groups, not just unramified groups. The following explicit formula was proved in [H01] and in [HP]. Let Tx = . In the formulas here and below, q = pr is the cardinality of 1 I x I for x ∈ W the residue field of F. Proposition 4.52. Assume μ is minuscule. Assume the parameters for the Iwahori Hecke algebra are all equal. Then q (tμ )/2 z μ = (−1)(tμ ) (−1)(x) R x,tλ(x) (q) Tx , x∈Adm(μ)

and where Rx,y (q) where x decomposes as x = tλ(x) w ∈ X ∗ (A) W = W denotes the R-polynomial of Kazhdan–Lusztig [KL], and the length function, . A similar formula holds in the context of affine for the quasi-Coxeter group W Hecke algebras with arbitrary parameters. In the Drinfeld case G = GLn and μ = (1, 0n−1 ), the coefficient of Tx is (1 − q)(tμ )−(x) . There are also explicit formulas for Bernstein functions z μ when μ is not minuscule, but they tend to be much more complicated. For related computations see [HP] and [GH].

10.2. A Pro-p Iwahori level case In the Drinfeld case where GQ p = GLn × Gm and μ = (1, 0n−1 ) × 1, and where K p is a pro-p Iwahori subgroup, an explicit formula for the test function φr for Sh(G, h −1 , K p K p ) was found by the author and Rapoport. We shall rephrase this slightly by ignoring the Gm factor and giving the formula for G = GLn . Proposition 4.53 ([HRa1], Prop. 12.2.1). Let q = pr . Let Ir+ denote the standard pro-p Iwahori subgroup of G r := GLn (Q pr ). Let T denote the standard diagonal torus in GLn . In terms of natural embeddings T (Fq ) → G r , and → G r giving elements tw−1 ∈ G r representing Ir+ \G r /Ir+ , we have w∈W

Stable Bernstein center and test functions

φr (Ir+ tw−1 Ir+ ) =

⎧ ⎪ ⎪ ⎨0,

165

if w ∈ / Adm(μ)

if w ∈ Adm(μ) but Nr (t) ∈ / TS(w) (F p ) ⎪ ⎪ ⎩(−1)n ( p − 1)n−|S(w)| (1 − q)|S(w)|−n−1 , otherwise. 0,

(4.35) Here S(w) is the set of critical indices for w, equivalently S(w) is the set of standard basis vectors e j ∈ Zn such that w ≤ te j in the Bruhat order on W determined by the standard Iwahori subgroup of GLn .

10.3. Deeper level structures Here the known explicit descriptions pertain only to G = GL2 and first were proved by Scholze [Sch1]. It remains an interesting question whether one can find explicit descriptions of test functions in higher rank groups with arbitrary level structure: even the Drinfeld case G = GLn , μ = (1, 0n−1 ) looks difficult, cf. [Sch2]. To state Scholze’s result, we need some notation. As usual let F be a nonarchimedean local field with ring of integers O, uniformizer $ , and residue field cardinality q. Let B denote the O-subalgebra M2 (O) of M2 (F). For any j ∈ Z set B j = $ j B. Let K = B × , the standard maximal compact subgroup of G = GL2 (F). For n ≥ 1, let K n = 1 + Bn ; so K n is a principal congruence subgroup and is a normal subgroup of K . Scholze defines a (compatible) family of functions φn ∈ H(G, K n ) for n ≥ 1. His definition uses two functions, : G → Z ∪ {∞} and k : G → Z. Let (g) = val ◦ det(1 − g). Let k(g) be the unique integer k such that g ∈ Bk and g ∈ / Bk+1 . By definition φn is 0 unless val ◦ det(g) = 1, tr(g) ∈ O, and g ∈ B1−n . Assume these conditions, in which case one can check that 1 − n ≤ k(g) ≤ 0 and (g) ≥ 0. Now define ⎧ ⎪ if tr(g) ∈ $ O, ⎪ ⎨−1 − q, φn (g) := 1 − q 2(g) , if tr(g) ∈ O× and (g) < n + k(g), ⎪ ⎪ ⎩1 + q 2(n+k(g))−1 , if tr(g) ∈ O× and (g) ≥ n + k(g). Proposition 4.54. (Scholze [Sch1]) For each n ≥ 1, the function z n := q−1 [K :K n ] · φn belongs to the center Z(GL2 (F), K n ), and the family (z n )n is compatible with change of level and thus defines a distribution in the sense of (4.2). This distribution is q 1/2 Z V where V is the standard representation C2 of the Langlands dual group GL2 (C). In an unpublished work, Kottwitz gave another proof of this proposition and also described the same distribution in terms of a family (n )n of functions

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n ∈ Z(GL2 (F), In ) where In ranges over the “barycentric” Moy–Prasad filtration in the standard Iwahori subgroup I ⊂ GL2 (F). By a completely different technique, in [Var] Sandeep Varma extended both the results of Scholze and Kottwitz stated above, by describing the distributions attached to V = Symr (C2 ) where r is any odd natural number less than p, the residual characteristic of F.

11. Appendix: Bernstein isomorphisms via types 11.1. Statement of Purpose Alan Roche proved the following beautiful result in [Roc, Theorem 1.10.3.1]. Theorem 4.55 (Roche). Let e be an idempotent in the Hecke algebra H = H(G(F)). View H as a smooth G(F)-module via the left regular repre sentation, and write e = s∈S es according to the Bernstein decomposition ) H = Se = {s ∈ S | es = 0}, and consider the cates∈S Hs . Let Se = gory RSe (G(F)) = s∈Se Rs (G(F)) and its categorical center Z s∈Se Zs . Let Z(eHe) denote the center of the algebra eHe. Then the map z → z(e) defines an algebra isomorphism ZSe → Z(eHe). Roche’s proof is decidedly non-elementary: besides the material developed in [Roc], it relies on some deep results of Bernstein cited there, most importantly Bernstein’s Second Adjointness Theorem and the construction of an explicit progenerator for each Bernstein block Rs (G(F)). In this chapter we use only the very special case of Roche’s theorem where e = e J for a parahoric subgroup J ⊂ G(F). We will explain a more elementary approach to this special case. It will rely only on the part of Bernstein’s theory embodied in Proposition 4.68 below. Formally, the inputs needed are, first, the existence of Bernstein’s categorical decomposition R(G) = s Rs (G), which is proved for instance in [Roc, Theorem 1.7.3.1], in an elementary way, and, second, the internal structure of the Bernstein block Rs (G) associated to a cuspidal pair s = [(M(F), χ˜ ]G where M is a minimal F-Levi subgroup of G and χ˜ is a character on M(F) which is trivial on its unique parahoric subgroup. For such components, progenerators can be constructed in an elementary way, without using Bernstein’s Second Adjointness Theorem. In fact in what follows we describe this internal structure using a few straightforward elements of the theory of Bushnell–Kutzko types, all of which are contained in [BK]. For e = e J Roche’s theorem gives the identification of the center of the parahoric Hecke algebra, in other words a Bernstein isomorphism for the most

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general case, where G/F is arbitrary and J ⊂ G(F) is an arbitrary parahoric subgroup. However we will provide a proof only for the crucial case of J = I , an Iwahori subgroup of G(F). The general parahoric case should follow formally from the Iwahori case, following the method of Theorem 3.1.1 of [H09], provided one is willing to rely on some basic properties of intertwiners for principal series representations (a purely algebraic theory of such intertwiners was detailed for split resp. unramified groups in [HKP] resp. [H07], and the extension to arbitrary groups should be similar to [H07]). The Iwahori case is approached in a different way by S. Rostami [Ro]. Rostami’s proof yields more information, describing the Iwahori–Matsumoto and Bernstein presentations for the Iwahori–Hecke algebra and deducing the description of its center from its Bernstein presentation.

11.2. Some notation un and let σ ∈ The notation will largely come from [HRo]. Recall L = F Aut(L/F) the Frobenius automorphism, which has fixed field F. We use the

σ . Moreover, if S denotes a maxisymbol G as an abbreviation for X ∗ (Z(G)) IF mal L-split torus in G which is defined over F, with centralizer T = CentG (S), un = N G (S)(L)/T (L)1 , the extended then G will denote the subgroup of W affine Weyl group for G/L, which preserves the alcove a in the apartment S corresponding to S, in the building B(G, L) of G over L. ◦& As always, we let I be the Iwahori subgroup I = Ga (O F ), which we recall we have chosen to be in good position relative to A: the corresponding alcove aσ ⊂ B(G, F) is required to belong in the apartment A corresponding to A. Let v F ∈ aσ be a special vertex with corresponding special maximal ◦& parahoric subgroup K = Gv F (O F ). Thus K ⊃ I . Recall M is a minimal F-Levi subgroup of G. Further, if I is an Iwahori subgroup of G(F), then I M := M(F) ∩ I = M(F)1 is the corresponding Iwahori subgroup of M(F) (cf. [HRo, Lemma 4.1.1]). Use the symbol 1 to denote the trivial 1-dimensional representation of any group.

11.3. Preliminary structure theory results Several of the results discussed here were proved independently by S. Rostami and will appear with somewhat different proofs in [Ro]. 11.3.1. Iwahori–Weyl group over F The following lemma concerns variations on well-known results, and were first proved by Timo Richarz [Ri].

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Let G 1 denote the subgroup of G(F) generated by all parahoric subgroups of G(F). By [HRa1, Lemma 17] and [Ri], we have G 1 = G(F)1 . Let N1 = N G (A)(F) ∩ G 1 , and let S denote the set of reflections through the walls of a. Then by [BT2, Prop. 5.2.12], the quadruple (G 1 , I, N1 , S) is a (double) Tits system with affine Weyl group Waff = N1 /I ∩ N1 , and the inclusion G 1 → G(F) is B N -adapted9 of connected type. Lemma 4.56 (T. Richarz [Ri]). (a) Let M1 = M(F)1 . Define the Iwahori– := NG (A)(F)/M1 . Then there is an isomorphism W = Weyl group as W Waff G , which is canonical given the choice of base alcove a. This gives the structure of a quasi-Coxeter group. W (b) If S ⊂ G is a maximal L-split torus which is F-rational and contains un := N G (S)(L)/T (L)1 , then A, and if we set T := CentG (S) and W σ the natural map NG (S)(L) → N G (A)(F) induces an isomorphism un )σ → . (W W Thus, in light of (b) we may reformulate the Bruhat–Tits decomposition of [HRa1, Prop. 8 and Rem. 9], as follows. Lemma 4.57. The map N G (A)(F) → G(F) induces a bijection ∼ W = I \G(F)/I.

(4.36)

Further, the Bruhat order ≤ and length function on Waff extend in the usual , and we have for w ∈ W and s ∈ Waff representing a simple affine way to W reflection, the usual BN-pair relations $ I swI, if w < sw IsIwI = (4.37) I wI ∪ I sw I, if sw < w. 11.3.2. Iwahori factorization Let P = M N be an F-rational parabolic subgroup with Levi factor M, unipotent radical N and opposite unipotent radical N . LetI H = I ∩ H for H = N , N , or M. Lemma 4.58. In the above situation, we have the Iwahori factorization I = IN · IM · IN .

(4.38)

Proof. We use the notation of [BT2]. By [BT2, 5.2.4] with := a, we have +&

−&

◦&

G◦a (O& ) = Ua Ua Na , 9 In [BT2] the symbol B is used in place of I .

Stable Bernstein center and test functions ◦&

&

169

&

where Na := N & ∩ Z◦ (O& )Ua . Since Z◦ (O & )Ua ⊂ G◦a (O& ), we have ◦&

Na = N & ∩ G◦a (O& ) = Z◦ (O& ). The key inclusion here, N & ∩ G◦a (O& ) ⊆ Z◦ (O& ), translates in our notation to N G (A)(F) ∩ I ⊆ M(F)1 , which can be deduced from Lemma 4.56(a). Translating again back to our notation we get I = I N · I N · I M which is the desired equality since I M normalizes I N . 11.3.3. On M(F)1 /M(F)1 Lemma 4.59. The following basic structure theory results hold: (a) In the notation of [HRo], we have M(F)1 /M(F)1 = K˜ /K , which injects into G(F)1 /G(F)1 . Thus M(F)1 = M(F)1 ∩ G(F)1 . (b) The Weyl group W (G, A) acts trivially on M(F)1 /M(F)1 . (c) Let a ⊂ B(G, L) denote the alcove invariant under the group Aut(L/F) ⊃ σ which corresponds to the Iwahori I ⊂ G(F). We assume I ⊂ K . Then the naive Iwahori I˜ := G(F)1 ∩ Fix(aσ ) has the following properties • M(F)1 /M(F)1 = I˜/I = K˜ /K . • I˜ = I · M(F)1 . Proof. Part (a): in the notation of [HRo], we know that M,tors = K˜ /K ([HRo, Prop. 11.1.4]). Applying this to G = M we get M,tors = M(F)1 /M(F)1 . So M(F)1 /M(F)1 = K˜ /K . By (8.0.1) and Lemma 8.0.1 in [HRo], the latter injects into G(F)1 /G(F)1 . The final statement follows. Part (b): By [HRo, Lemma 5.0.1], W (G, A) has representatives in K ∩ NG (A)(F). Thus it is enough to show that if n ∈ K ∩ N G (A)(F) and m ∈ M(F)1 , then nmn −1 m −1 ∈ M(F)1 . This follows from (a), since we clearly have nmn −1 m −1 ∈ M(F)1 ∩ G(F)1 . Part (c): First note that M(F)1 ⊂ I˜ and M(F)1 ⊂ I . Thus there is a commutative diagram / K˜ /K 8 q q q q q qq qqq M(F)1 /M(F)1 I˜/I O

The oblique arrow is bijective by (a). We claim the horizontal arrow is injective, that is, I˜ ∩ K = I . But I˜ ∩ K = G(F)1 ∩ Fix(aσ ) ∩ G(F)1 ∩ Fix(v F ), where v F is the special vertex in B(G ad , F) corresponding to K (cf. [HRo, Lem. 8.0.1]). Thus I˜ ∩ K = G(F)1 ∩ Fix(aσ ) = I by Remark 8.0.2 of [HRo].

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It now follows that all arrows in the diagram are bijective. This implies both statements in (c). Remark 4.60. Let P = M N be as above. We deduce from (c) and (4.38) the Iwahori factorization for I˜ I˜ = I N · M(F)1 · I N ,

(4.39)

using the fact that M(F)1 normalizes I N and I N . 11.3.4. Iwasawa decomposition Next we need to establish a suitable form of the Iwasawa decomposition. Let P = M N be as above. Lemma 4.61. The inclusion N G ( A)(F) → G(F) induces bijections := NG (A)(F)/M(F)1 → W M(F)1 N (F)\G(F)/I W (G, A) = NG (A)(F)/M(F) → P(F)\G(F)/I.

(4.40) (4.41)

= σ W (G, A) (cf. Lemma Proof. In view of the decomposition W M 3.0.1(III) of [HRo] plus Lemma 4.56(b)) and the Kottwitz isomorphism σM → M(F)/M(F)1 (cf. Lemma 3.0.1 [HRo]), it suffices to prove (4.40). For x ∈ B(G, F), let Px ⊂ G(F) denote the subgroup fixing x. By [Land, Prop. 12.9], we have G(F) = N (F) · N G ( A)(F) · Px . For sufficiently generic points x ∈ aσ , we have Px = I˜, which is M(F)1 I by Lemma 4.59(c). Since M(F)1 ⊂ NG (A)(F), we have G(F) = N (F) · N G (A)(F) · I and the map (4.40) is surjective. To prove injectivity, assume n 1 = um 0 · n 2 · j for u ∈ N (F), m 0 ∈ M(F)1 , n 1 , n 2 ∈ N G ( A)(F), and j ∈ I . There exists z ∈ Z (M)(F) such that zuz −1 ∈ I N (cf. e.g. [BK, Lem. 6.14]). Then zn 2 = (zuz −1 )m 0 · zn 2 · j ∈ I zn 2 I, and so by (4.36), zn 2 ≡ zn 2 modulo M(F)1 . and M(F)1 N (F)x I ∩ I y I = ∅, then x ≤ y in the Lemma 4.62. If x, y ∈ W determined by I . Bruhat order on W Proof. This follows from the BN-pair relations (4.37) as in the proof of the Claim in Lemma 1.6.1 of [HKP].

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11.3.5. The universal unramified principal series module M Define M = Cc (M(F)1 N (F)\G(F)/I ). The subscript “c” means we are considering functions supported on finitely many double cosets. Some basic facts about M were given in [HKP] for the special case where G is split, and here we need to state those facts in the current general situation. Abbreviate by setting H = H(G(F), I ) and R = C[ M ]. Then f ∈ H acts on the left on M by right convolutions by f˘, which is defined by f˘(g) = f (g −1 ). The same goes for the normalized induced representation 1/2 i PG (χ) ˜ I = IndGP (δ P χ) ˜ I , where χ˜ is a character on M(F)/M(F)1 . Moreover, R acts on the left on M by normalized left convolutions: for r ∈ R and φ ∈ M, m ∈ M(F), / 1/2 (r · φ)(m) = r (y)δ P (y)φ(y −1 m) dy M(F)

where voldy (M(F)1 ) = 1. The actions of R and H commute, so M is an (R, H )-bimodule. Lemma 4.63. The following statements hold. (a) Any character χ˜ −1 : M(F)/M(F)1 → C× extends to an algebra homomorphism χ˜ −1 : R → C, and there is an isomorphism of left H -modules C ⊗ R,χ˜ −1 M = i PG (χ˜ ) I . (b) For w ∈ W (G, A) =: W , set vw = 1 M(F)1 N (F)w I ∈ M. Then M is free of rank 1 over H with canonical generator v1 . (c) M is free as an R-module, with basis {vw }w∈W . Proof. The proofs for (a–b) are nearly identical to their analogues for split groups in [HKP]. Part (a) is formal. Part (b) relies on the Bruhat–Tits decomposition (4.36), the Iwasawa decomposition (4.40), and Lemma 4.62. Part (c) was not explicitly mentioned in [HKP]. But it can be proved using (4.40) along with the relations analogous to [HKP, (1.6.1–1.6.2)], for which the Iwahori factorization (4.38) is the main ingredient.

11.4. Why (M(F)1 , 1) is an S M -type We let χ range over the characters of M(F)1 /M(F)1 . Let χ˜ denote any extension to a character of the finitely generated abelian group M(F)/M(F)1 . Fix one such extension χ˜ 0 . Note that the inertial class [M(F), χ˜ 0 ] M consists of all

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pairs (M(F), χ), ˜ since M(F)-conjugation does not introduce any new characters on M(F). Therefore we may abuse notation and denote this inertial class by [M(F), χ] ˜ M =: sχM . Let S M := {[M(F), χ˜ ] M | χ ranges as above}. This is a finite set of inertial classes, in bijective correspondence with M(F)1 /M(F)1 . Proposition 4.64. The pair (M(F)1 , 1) is a Bushnell–Kutzko type for S M . Note: This proposition simply makes precise the last paragraph of [BK, 9.2]. Proof. Let σ be an irreducible smooth representation of M(F). We must show that σ = χ˜ for some χ˜ iff σ | M(F )1 ⊃ 1. (⇒): Obvious. (⇐): We see that σ . v = 0 on which M(F)1 acts trivially. Since σ is irreducible, it coincides with the smallest M(F)-subrepresentation containing v, and then since M(F)1 / M(F), we see that M(F)1 acts trivially on all of σ ; further, σ is necessarily finite-dimensional over C. Since M(F)/M(F)1 is abelian, σ contains an M(F)-stable line, since a commuting set of matrices can be simultaneously triangularized. This line is all of σ since σ is irreducible. Thus σ is 1-dimensional, and so σ = χ˜ for some χ˜ .

11.5. Why ( I, 1) is an S G -type We define SG = {[t]G | [t] M ∈ S M }. The map [M, χ] ˜ M → [M, χ˜ ]G is injective: if [M, χ˜1 ]G = [M, χ˜2 ]G , then there exists n ∈ NG (A)(F) such that n (χ˜ ) = χ˜ η for some character η on M(F)/M(F)1 . Restricting to M(F)1 1 2 and using n (χ1 ) = χ1 (Lemma 4.59(b)), we see χ1 = χ2 . So S M ∼ = SG via [t] M → [t]G . We saw above that (M(F)1 , 1) is an S M -type. The fact that (I, 1) is an SG -type follows from [BK], Theorem 8.3, once we check the following proposition. Proposition 4.65. The pair (I, 1) is a G-cover for (M(F)1 , 1) in the sense of [BK, Definition 8.1]. Proof. We need to check the three conditions (i–iii) of Definition 8.1. First (i), the fact that (I, 1) is decomposed with respect to (M, P) in the sense of [BK, (6.1)], follows from the Iwahori factorization I = I N · I M · I N discussed in Remark 4.60. The equality I ∩ M(F) = M(F)1 gives condition (ii). Finally we must prove (iii): for every F-parabolic P with Levi factor M, there exists an invertible element of H(G(F), I ) supported on I z P I , where z P belongs to Z (M)(F) and is strongly (P, I )-positive. The existence of elements z P ∈ Z (M)(F) which are strongly (P, I )-positive is proved in [BK, Lemma 6.14]. Any corresponding characteristic function 1 I z P I is invertible

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in H(G(F), I ), as follows from the Iwahori–Matsumoto presentation of H(G(F), I ). (This presentation itself is easy to prove using (4.37).)

11.6. Structure of the Bernstein varieties Let R(G) denote the category of smooth representations of G(F), and let Rχ (G) denote the full subcategory corresponding to the inertial class [M, χ˜ ]G . That is, a representation (π, V ) ∈ R(G) is an object of Rχ (G) if and only if for each irreducible subquotient π of π , there exists an extension χ˜ of χ such 1/2 that π is a subquotient of IndGP (δ P χ˜ ). We review the structure of the Bernstein varieties XχG and XχM . In this discussion, for each χ we fix an extension χ˜ of χ once and for all – the structures we define will be independent of the choice of χ, ˜ i.e. uniquely determined by (M, χ ) up to a unique isomorphism. As a set XχG (resp. XχM ) consists of the elements (M, χ˜ η)G (resp. (M, χ˜ η) M ) belonging to the inertial equivalence class [M, χ] ˜ G (resp. [M, χ] ˜ M ) as η × ranges over the set X (M) of unramified C -valued characters on M(F) (unramified means it factors through M(F)/M(F)1 ). The map X (M) → XχM , η → (M, χ˜ η) M , is a bijection. Since X (M) is a complex torus, this gives XχM the structure of a complex torus. More canonically, XχM is just the variety of all extensions χ˜ of χ , and it is a torsor under the torus X (M). Now fix χ˜ again. There is a surjective map XχM → XχG (M, χ˜ η) M → (M, χ˜ η)G . Since W := W (G, A) acts trivially on M(F)1 /M(F)1 (Lemma 4.59), one can prove that the fibers of this map are precisely the W -orbits on XχM . Thus as sets W \XχM = XχG , and this gives XχG the structure of an affine variety over C. Having chosen the isomorphism X (M) ∼ = XχM as above, we can transport the W -action on XχM over to an action on X (M). This action depends on the choice of χ˜ and is not the usual action unless χ˜ is W -invariant. We obtain XχG = W \χ˜ X (M), where the latter denotes the quotient with respect to this unusual action on X (M). Let C[XχG ] denote the ring of regular functions on the variety XχG . The algebraic morphism XχM → XχG induces an isomorphism of algebras C[XχG ] → C[XχM ]W .

(4.42)

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11.7. Consequences of the theory of types Let us define convolution in H(G(F), I ) using the Haar measure d x on G(F) which gives I volume 1. Let Z(G(F), I ) denote the center of H(G(F), I ). We define for each χ ∈ (M(F)1 /M(F)1 )∨ a function eχ ∈ H(G(F), I ) by requiring it to be supported on I˜, and by setting eχ (y) = vold x ( I˜)−1 χ( y¯ )−1 if y ∈ I˜. Here we regard χ as a character on I˜/I (cf. Lemma 4.59) and let y¯ ∈ I˜/I denote the image of y. If y = n + · m 1 · n − ∈ I N · M(F)1 · I N , then eχ (y) = vold x ( I˜)−1 χ (m 1 )−1 . Lemma 4.66. The functions {eχ }χ give a complete set of central orthogonal idempotents for H(G(F), I ): (a) eχ ∈ Z(G(F ), I ); (b) eχ eχ = δχ ,χ eχ , there δχ ,χ is the Kronecker delta function; (c) 1 I = χ eχ . Proof. The proof is a straightforward exercise for the reader. For parts (a–b), use the fact that M(F)1 normalizes I , that G(F) = I · NG (A)(F) · I , and that W (G, A) acts trivially on M(F)1 /M(F)1 (cf. Lemma 4.59). Proposition 4.67. The idempotents eχ are the elements in the Bernstein center which project the category R(G) onto the various Bernstein components Rχ (G). That is, there is a canonical isomorphism of algebras H(G(F), I ) = eχ H(G(F), I )eχ χ

and, for any smooth representation (π, V ) ∈ R(G), the G(F)-module spanned by the χ -isotypical vectors V χ = π(eχ )V is the component of V lying in the subcategory Rχ (G). Finally, eχ H(G(F), I )eχ = H(G(F), I˜, χ ), the right-hand side being the algebra of I -bi-invariant C-valued functions f ∈ Cc (G(F)) such that f (i˜1 g i˜2 ) = χ (i˜1 )−1 f (g)χ (i˜2 )−1 for all g ∈ G(F) and i˜1 , i˜2 ∈ I˜. The following records the standard consequences of the fact that (I, 1) is an SG -type (see [BK, Theorem 4.3]). Let R I (G) denote the full subcategory of R(G) whose objects are generated as G-modules by their I -invariant vectors. Proposition 4.68. As subcategories of R(G), we have the equality R I (G) = χ Rχ (G). In particular, an irreducible representation (π, V ) ∈ R(G)

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belongs to R I (G) if and only if π ∈ Rχ (G) for some χ . Furthermore, there is an equivalence of categories R I (G) → H(G(F), I )-Mod (π, V ) → V I . Finally, Z(G(F), I ) is isomorphic with the center of the category χ Rχ (G), which according to Bernstein’s theory is the ring χ C[XχG ]. Concretely, the map Z(G(F), I ) → χ C[XχG ], z → zˆ , is characterized as follows: for every χ and every (M, χ˜ )G ∈ XχG , z ∈ Z(G(F), I ) acts on 1/2

IndGP (δ P χ˜ ) I by the scalar zˆ (χ). ˜ Let us single out what happens in the special case of G = M. We can identify H(M(F), M(F)1 ) = C[ M ]. Let eχM denote the idempotent in H(M(F), M(F)1 ) analogous to eχ , for the case G = M. By Propositions 4.67 and 4.68 for G = M, we have H(M(F), M(F)1 ) = eχM H(M(F), M(F)1 )eχM = C[XχM ], (4.43) χ

χ

the last equality holding since H(M(F), M(F)1 ) is already commutative. Thus, the ring eχM H(M F), M(F)1 )eχM can be regarded as the ring of regular functions on the variety XχM of all extensions χ˜ of χ .

11.8. The embedding of C[ M ]W into Z(G(F), I) We make use of the following special case of a general construction of Bushnell–Kutzko [BK]: for any F-parabolic P with Levi factor M, there is an injective algebra homomorphism t P : H(M(F), M(F)1 ) → H(G(F), I ) which is uniquely characterized by the property that for each (π, V ) ∈ R I (G), v ∈ V I , and h ∈ H(M(F), M(F)1 ), we have the identity qπ (t P (h) · v) = h · qπ (v).

(4.44)

Here qπ : V I → VNM(F)1 is an isomorphism, which is induced by the canonical projection V → V N to the (unnormalized) Jacquet module. See [BK, Thm. 7.9].

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It turns out that it is better to work with a different normalization. We define another injective algebra homomorphism θ P : H(M(F), M(F)1 ) → H(G(F), I ) −1/2

h → t P (δ P

h).

Then using (4.44) θ P satisfies −1/2

qπ (θ P (h) · v) = (δ P

h) · qπ (v).

(4.45)

We view χ˜ as a varying element of the Zariski-dense subset S of the variety of all characters on the finitely generated abelian group M(F)/M(F)1 , conG 1/2 sisting of those regular characters χ˜ such that V (χ˜ ) := i G ˜ P (χ˜ ) := Ind P (δ P χ) is irreducible as an object of R(G). We apply the above discussion to the representations V := V (χ) ˜ with χ˜ ∈ S. By a result of Casselman [Cas], we know that as M(F)-modules # 1/2 VN = δ P ( w χ˜ ) w∈W

and that M(F)1 acts trivially on this module. Now suppose h ∈ C[ M ]W . −1/2 M(F)1 Then δ P h acts on VN = V N by the scalar h(χ) ˜ (viewing h as a regular M function on Xχ ). It follows from (4.45) that θ P (h) acts by the scalar h(χ) ˜ on i PG (χ) ˜ I , for χ˜ ∈ S. Now let f ∈ H be arbitrary, and set := f ∗ θ P (h) − θ P (h) ∗ f ∈ H . We see that acts by zero on i PG (χ˜ ) I for every χ˜ ∈ S.

(4.46)

We claim that = 0. Recall that ∈ H gives an R-linear endomorphism of M, hence by Lemma 4.63(c) may be represented by an |W | × |W | matrix E with entries in R. Now Spec(R) = Spec(C[ M ]) is a diagonalizable group scheme over C with character group M . Hence R is a reduced finite-type C-algebra and its maximal ideals are precisely the kernels of the C-algebra homomorphisms χ˜ −1 : R → C coming into Lemma 4.63(a). By that Lemma and (4.46), we see that E ≡ 0 (mod m) for a Zariski-dense set of maximal ideals m in Spec(R). Since R is reduced and finite-type over C, this implies that the entries of E are identically zero. This proves the claim because M is free of rank 1 over H (Lemma 4.63(b)). Since f was arbitrary, we get θ P (h) ∈ Z(G(F), I ), as desired. We have proved the following result. Lemma 4.69. The map θ P : C[ M ] → H(G(F), I ) restricts to give an embedding C[ M ]W → Z(G(F), I ).

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11.9. The center of the Iwahori–Hecke algebra Theorem 4.70. The map θ P gives an algebra isomorphism C[ M ]W → Z(G(F), I ). Further, this isomorphism is independent of the choice of parabolic P containing M as a Levi factor. Proof. The description of θ P above, and the preceding discussion, show that we have a commutative diagram ∼ / M W C[ M ]W χ C[Xχ ] O θP

Z(G(F), I )

∼

/

χ

C[XχG ] .

The left vertical arrow is bijective because the right vertical arrow is, by (4.42).

11.10. Bernstein isomorphisms and functions Putting together Roche’s theorem 4.55 with Theorem 4.70, we deduce a more general result that holds for any parahoric subgroup J ⊇ I . Theorem 4.71. The composition C[ M ]W

θP

/ Z(G(F), I )

−∗ I 1 J

/ Z(G(F), J )

(4.47)

is an isomorphism. We call this map the Bernstein isomorphism. Definition 4.72. Given μ ∈ M , we define the Bernstein function z μ ∈ Z(G(F), J ) to be the image of the symmetric monomial function λ∈W ·μ λ ∈ C[ M ]W under the Bernstein isomorphism (4.47).

11.11. Compatibility with constant terms Recall M = CentG (A) is a minimal F-Levi subgroup of G and P = M N is a minimal F-parabolic subgroup with Levi factor M and unipotent radical N . Let Q = L R be another F-parabolic subgroup with F-Levi factor L and unipotent radical R. Assume Q ⊇ P; then L ⊇ M and R ⊆ N . Further L contains a minimal F-parabolic subgroup L ∩ P = M · (L ∩ N ), and N = L ∩ N · R. If J ⊂ G(F) is a parahoric subgroup corresponding to a facet in the apartment of the Bruhat–Tits building of G(F) corresponding to A, then JL := J ∩ L is a parahoric subgroup of L(F) (by [HRo, Lem. 4.1.1]).

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Given f ∈ H(G(F), J ), define f (Q) ∈ H(L(F), JL ) by / / 1/2 −1/2 f (Q) (l) = δ Q (l) f (lr ) dr = δ Q (l) f (rl) dr, R

R

where voldr (J ∩ R) = 1. An argument similar to Lemma 4.7.2 of [H09] shows that f → f (Q) sends Z(G(F), J ) into Z(L(F), JL ), and determines a map cG L making the following diagram commute: (G, A) C[ M ]W _

C[ M ]W (L ,A)

∼

/ Z(G(F), J )

(4.48)

cG L

∼

/ Z(L(F), JL ).

The diagram shows that c G L is indeed an (injective) algebra homomorphism and, as the notation suggests, is independent of the choice of parabolic subgroup Q having L as a Levi factor. We call cG L the constant term homomorphism. By its very construction, the map θ M : C[ M ] → H(M(F), M(F)1 ) has its inverse induced by the Kottwitz isomorphism κ M (F) : M(F)/M(F)1 → M . By taking L = M in (4.48), this remark allows us to write down the inverse of θ P in general. W (G, A) Corollary 4.73. The composition κ M (F) ◦ c G M takes values in C[ M ] and is the inverse of the Bernstein isomorphism θ P .

11.12. Transfer homomorphisms 11.12.1. Construction Transfer homomorphisms were defined for special maximal parahoric Hecke algebras in [HRo]. By virtue of the Bernstein isomorphisms (4.47), we can now define these homomorphisms on the level of centers of arbitrary parahoric Hecke algebras. Let us recall the general set-up from [HRo, §11.2]. Let G ∗ be a quasi-split group over F. Let F s denote a separable closure of F, and set = Gal(F s /F). Recall that an inner form of G ∗ is a pair (G, ") consisting of a connected reductive F-group G and a -stable G ∗ad (F s )-orbit " of F s -isomorphisms ψ : G → G ∗ . The set of isomorphism classes of pairs (G, ") corresponds bijectively to H 1 (F, G ∗ad ). Fix once and for all parahoric subgroups J ⊂ G(F) and J ∗ ⊂ G ∗ (F). Choose any maximal F-split tori A ⊂ G and A∗ ⊂ G ∗ such that the facet fixed by J (resp. J ∗ ) is contained in the apartment of the building B(G, F)

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179

(resp. B(G ∗ , F)) corresponding to the torus A (resp. A∗ ). Let M = CentG (A) and T ∗ = CentG ∗ (A∗ ), a maximal F-torus in G ∗ . Now choose an F-parabolic subgroup P ⊂ G having M as Levi factor, and an F-rational Borel subgroup B ∗ ⊂ G ∗ having T ∗ as Levi factor. Then there exists a unique parabolic subgroup P ∗ ⊂ G ∗ such that P ∗ ⊇ B ∗ and P ∗ is G ∗ (F s )-conjugate to ψ(P) for every ψ ∈ ". Let M ∗ be the unique Levi factor of P ∗ containing T ∗ . Then define " M = {ψ ∈ " | ψ(P) = P ∗ , ψ(M) = M ∗ }. ∗ (F s )-orbit of F s -isomorphisms M → The set " M is a nonempty -stable Mad ∗ M , and so (M, " M ) is an inner form of M ∗ . Choose any ψ0 ∈ " M . Then since ψ0 |A is F-rational, ψ0 (A) is an F-split torus in Z(M ∗ ) and hence ψ0 (A) ⊆ A∗ .

→ ∗ and hence a Now ψ0 induces a -equivariant map Z ( M) Z( M ∗ ) → T homomorphism ∗

∗ ) ∗F ]W (G t A∗ ,A : C[X ∗ (T I

∗ , A∗ )

F

F ]W (G,A) , −→ C[X ∗ (Z( M)) IF

where (·)∗ designates the Galois action on G ∗ (for Weyl-group equivariance ∗ this homomorphism does not see [HRo, §12.2]). Since " M is a torsor for Mad depend on the choice of ψ0 ∈ " M . Further, it depends only on the choice of A∗ and A, and not on the choice of the parabolic subgroups P ⊃ M and B ∗ ⊃ T ∗ we made in constructing it. Definition 4.74. Let J ⊂ G(F) and J ∗ ⊂ G ∗ (F ) be any parahoric subgroups and choose compatible maximal F-split tori A resp. A∗ as above. Then we define the transfer homomorphism t : Z(G ∗ (F), J ∗ ) → Z(G(F), J ) to be the unique homomorphism making the following diagram commute t

Z(G ∗ (F ), J ∗ ) 0

/ Z(G(F), J ) 0

∗

∗ )∗F ]W (G ∗ , A∗ ) C[X ∗ (T I

t A∗ , A

F

/ C[X ∗ (Z ( M))

F ]W (G, A) , IF

where the vertical arrows are the Bernstein isomorphisms. By [BT2, 4.6.28], any two choices for A (resp. A∗ ) are J -(resp. J ∗ -) conjugate. Using Corollary 4.73 it follows that t is independent of the choice of A and A∗ and is a completely canonical homomorphism. Remark 4.75. The map ∗

∗ )∗F → X ∗ (Z ( M))

F t A∗ ,A : X ∗ (T IF I F

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Thomas J. Haines

is surjective. Via the Kottwitz homomorphism we may view this as the composition / M ∗ (F)/M ∗ (F)1

T ∗ (F)/T ∗ (F)1

ψ0−1 ∼

/ M(F)/M(F)1

(4.49)

where the first arrow is induced by the inclusion T ∗ → M ∗ . It is enough to observe that M ∗ (F) = T ∗ (F) · M ∗ (F)1 , which in turn follows from the Iwasawa decomposition (cf. (4.41)) for M ∗ (F), which states that M ∗ (F) = ∗ (F) · K ∗ for an F-rational Borel subgroup B ∗ = T ∗ · U ∗ and T ∗ (F) · U M ∗ M M∗ M∗ a special maximal parahoric subgroup K M ∗ in M ∗ , and from the vanishing of ∗ (F) · K ∗ . the Kottwitz homomorphism on U M ∗ M 11.12.2. Normalized transfer homomorphism The transfer homomorphism is slightly too naive, and it is necessary to normalize it in order to get a homomorphism which has the required properties. We need to define normalized homomorphisms t A∗ , A on Weyl-group invariants, for which the following lemma is needed. Lemma 4.76. Recall that T ∗ = CentG ∗ (A∗ ) is a maximal torus in G ∗ defined over F; let S ∗ be the F un -split component of T ∗ , a maximal F un -split torus in G ∗ defined over F and containing A∗ . We have T ∗ = CentG ∗ (A∗ ) = CentG ∗ (S ∗ ). Choose a maximal F un -split torus S ⊂ G which is defined over F and which contains A, and set T = CentG (S). Choose ψ0 ∈ " M such that ψ0 is defined over F un and satisfies ψ0 (S) = S ∗ and hence ψ0 (T ) = T ∗ . Then the diagram ψ

&

0 W (G, A) _ _ _ _ _ ∼_ _/ W (G ∗ , A∗ )/ W (M ∗ , A∗ )

0

W (G, S)/W (M, S) F

0

ψ0 ∼

∗ / W (G ∗ , S ∗ )/ W (M ∗ , S ∗ ) F

&

defines a bijective map ψ0 . It depends on the choice of the data P, B ∗ used to define " M and M ∗ , but it is independent of the choices of S and ψ0 ∈ " M with the stated properties. Proof. The left vertical arrow is [HRo, Lem. 6.1.2]. The right vertical arrow is described in [HRo, Prop. 12.1.1]. The proof of the latter justifies the lower horizontal arrow. Indeed, given w ∈ W (G, A) we may choose a representative n ∈ NG (S)(L) F (cf. [HRo]). We have ψ0−1 ◦ ∗F ◦ ψ0 ◦ −1 F = Int(m ) for s some m ∈ N M (S)(F ). Since n is F -fixed, we get ∗F (ψ0 (n)) = ψ0 (n) · ψ0 (n)−1 ψ0 (m )ψ0 (n)ψ0 (m )−1 .

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181

As n normalizes M and hence ψ0 (n) normalizes M ∗ , this shows that ψ0 (n)W (M ∗ , S ∗ ) is ∗F -fixed. There exists m ∗n ∈ N M ∗ (S ∗ )(L) such that ψ0 (n)m ∗n ∈ N G ∗ (A∗ )(F). Then & ψ0 (w) is the image of ψ0 (n)m ∗n in W (G ∗ , A∗ )/W (M ∗ , A∗ ). The independence statement is proved using this description. Via the Kottwitz homomorphism we can view t A∗ , A as induced by the composition (4.49). We now alter this slightly. Lemma 4.77. Given the choices of P ⊃ M and B ∗ ⊃ T ∗ needed to define " M and given any F un -rational ψ0 ∈ " M , we define an algebra homomorphism C[T ∗ (F)/T ∗ (F)1 ] −→ C[M(F)/M(F)1 ] (4.50) −1/2 1/2 at ∗ t ∗ −→ at ∗ δ B ∗ (t ∗ )δ P (m) · m, t∗

m

t ∗ →m

where t ∗ ranges over T ∗ (F)/ T ∗ (F)1 and m ranges over M(F)/M(F)1 and t ∗ → m means that ψ0−1 (t ∗ ) ∈ m M(F)1 , (cf. (4.49)). Then (4.50) takes W (G ∗ , A∗ )-invariants to W (G, A)-invariants, and the resulting homomorphism t˜A∗ , A : C[T ∗ (F)/T ∗ (F)1 ]W (G

∗ ,A∗ )

→ C[M(F)/M(F)1 ]W (G, A)

is independent of the choices of P, B ∗ , and F un -rational ψ0 ∈ " M . Proof. To check the Weyl-group invariance, we may fix P and B ∗ , and choose S and ψ0 as in Lemma 4.76. Suppose t ∗ at ∗ t ∗ is W (G ∗ , A∗ )-invariant. We need to show that the function on M(F)/M(F)1 −1/2 1/2 m → at ∗ δ B ∗ (t ∗ )δ P (m) (4.51) t ∗ →m

is W (G, A)-invariant, and independent of the choice of P and B ∗ . For w ∈ W (G, A) choose n and m ∗n as in the proof of Lemma 4.76, and define n by ψ0 (n ) = ψ0 (n)m ∗n . Thus ψ0 (n ) ∈ NG ∗ (A∗ )(F) and hence it normalizes T ∗ (F). We claim that (4.51) takes the same values on m M(F)1 and on n m M(F)1 . First we observe that n m M(F)1 = n m M(F)1 . Setting m n := ψ0−1 (m ∗n ), it is enough to prove m n m M(F)1 = m M(F)1 . But as ψ0 is L-rational we have m n ∈ M(L) and so conjugation by m n induces the identity map on M(L)/M(L)1 and hence on its subset M(F)/M(F)1 as well. Now we write the value of (4.51) on n m M(F)1 as −1/2 1/2 at ∗∗ δ B ∗ (t ∗∗ )δ P ( n m).

t ∗∗ → n m M(F)1

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Thomas J. Haines −1

Setting t ∗ = ψ0 (n ) t ∗∗ and using at ∗ = at ∗∗ (which follows from W (G ∗ , A∗ )invariance), we write the above as −1/2 1/2 at ∗ δ B ∗ ( ψ0 (n ) t ∗ ) δ P ( n m). t ∗ →m

The index set is stable under the W (M ∗ , A∗ )-action on T ∗ (F)/T ∗ (F)1 . If we look at the sum over the W (M ∗ , A∗ )-orbit of a single element t0∗ , with stabilizer group Stab(t0∗ ), we get −1/2 1 · at0∗ · δ B ∗ ( ψ0 (n )y t0∗ ), ∗ M∗ |Stab(t0 )| y

(4.52)

where y ranges over W (M ∗ , A∗ ). Now n ∈ NG (S)(L) F ⊆ NG (A)(F) = N G (M)(F). Hence ψ0 (n)m ∗n = ψ0 (n ) normalizes M ∗ as well as T ∗ , and thus ∗ to another F-rational Borel subgroup of M ∗ conjugation by ψ0 (n ) takes B M ∗ ∗ containing T . Using this it is clear that (4.52) is unchanged if the superscript ψ0 (n ) is omitted, and this proves our claim. For the same reason (4.51) is independent of the choice of P and B ∗ , and similarly t˜A∗ , A is independent of the choice of P and B ∗ , and of the choice of F un -rational ψ0 ∈ " M . Now we give a normalized version of Definition 4.74. Definition 4.78. We define the normalized transfer homomorphism t˜ : Z(G ∗ (F), J ∗ ) → Z(G(F), J ) to be the unique homomorphism making the following diagram commute Z(G ∗ (F), J ∗ ) 0

∗ ) C[X ∗ (T

/ Z(G(F), J ) 0

∗F I F∗

t˜

]W (G

∗ ,A∗ )

t˜A∗ , A

/ C[X ∗ (Z ( M))

F ]W (G, A) , IF

where the vertical arrows are the Bernstein isomorphisms. As was the case for t, the homomorphism t˜ is independent of the choice of A and A∗ , and it is a completely canonical homomorphism. The following shows it is compatible with constant term homomorphisms. 11.12.3. Normalized transfer homomorphisms and constant terms We use the notation of §11.11. Write L = CentG (A L ) for some torus A L ⊆ A. Let L ∗ = CentG ∗ (A∗L ∗ ) for a subtorus A∗L ∗ ⊆ A∗ . Without loss of generality, we may assume that our inner twist G → G ∗ restricts to give an inner twist L → L ∗ .

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Proposition 4.79. In the above situation, the following diagram commutes: Z(G ∗ (F), J ∗ )

t˜

/ Z(G(F), J )

∗

cG L

cG L∗

Z(L ∗ (F), JL∗∗ )

(4.53)

t˜

/ Z(L(F), JL ).

Taking L = M, the diagram shows in order to compute t˜ it is enough to compute it in the case where G ad is anisotropic. In that case if z ∈ Z(G ∗ (F), J ∗ ), the function t˜(z) is given by integrating z over the fibers of the Kottwitz homomorphism κG ∗ (F). Proof. The commutativity boils down to the fact that the quantities (4.52) do not depend on the ambient group G. Remark 4.80. The final sentence in Proposition 4.79 is the key to explicit computation of t˜(z) given z, and is illustrated in §7.3.3. This final sentence was incorrectly asserted to hold for the unnormalized transfer homomorphisms (for special maximal parahoric Hecke algebras) in Prop. 12.3.1 of [HRo].

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5 Conditional results on the birational section conjecture over small number fields Yuichiro Hoshi

Abstract In the present chapter, we give necessary and sufficient conditions for a birational Galois section of a projective smooth curve over either the field of rational numbers or an imaginary quadratic field to be geometric. As a consequence, we prove that, over such a small number field, to prove the birational section conjecture for projective smooth curves, it suffices to verify that, roughly speaking, for any birational Galois section of the projective line, the local points associated to the birational Galois section avoid three distinct rational points, and, moreover, a certain Galois representation determined by the birational Galois section is unramified at all but finitely many primes.

Contents Introduction 0 Notations and conventions 1 Birational Galois sections and their geometricity 2 Local geometricity of birational Galois sections 3 Galois sections of tori that locally arise from points 4 Conditional results on the birational section conjecture Appendix A Ramification of Galois sections References

page 188 193 195 199 205 210 221 229

2000 Mathematics Subject Classification. 14H30. Automorphic Forms and Galois Representations, ed. Fred Diamond, Payman L. Kassaei and c Cambridge University Press 2014. Minhyong Kim. Published by Cambridge University Press.

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Introduction Let k be a field of characteristic 0, k an algebraic closure of k, and X a projecdef

tive smooth geometrically connected curve over k. Write G k = Gal(k/k) for the absolute Galois group of k determined by the fixed algebraic closure k of k. Now we have a natural surjection π1 (X ) G k from the étale fundamental group π1 (X ) of X to G k induced by the structure morphism of X . Then Grothendieck’s section conjecture may be stated as follows: If k is finitely generated over the field of rational numbers, and X is of genus ≥ 2, then any section of this surjection π1 (X ) G k arises from a k-rational point of X , i.e., the image of any section of this surjection coincides with, or, equivalently, is contained in, a decomposition subgroup of π1 (X ) associated to a k-rational point of X . In the present chapter, we discuss the birational version of this conjecture, i.e., the birational section conjecture. Denote by k(X ) the function field of X . Fix an algebraic closure k(X ) of k(X ) def

containing k and write G k(X ) = Gal(k(X )/k(X )). Then the natural inclusions k → k(X ), k → k(X ) determine a surjection G k(X ) G k , which factors through the above surjection π1 (X ) G k . We shall refer to a section of this surjection G k(X ) G k as a [pro-Primes] birational Galois section of X/k [cf. Definition 5.2]. In the present chapter, we discuss the geometricity of birational Galois sections. Let x be a closed point of X and D x ⊆ G k(X ) a decomposition subgroup of G k(X ) associated to x. Then, as is well-known, the image of the composite Dx → G k(X ) G k coincides with the open subgroup G k(x) ⊆ G k of G k corresponding to the residue field k(x) of X at x, and, moreover, the resulting surjection Dx G k(x) admits a [not necessarily unique] section. In particular, if, moreover, k(x) = x, i.e., x ∈ X (k), then the closed subgroup Dx ⊆ G k(X ) of G k(X ) contains the image of a [not necessarily unique] birational Galois section of X/k. We shall say that a birational Galois section of X/k is geometric if its image is contained in a decomposition subgroup of G k(X ) associated to a [necessarily k-rational] closed point of X [cf. Definition 5.3]. The birational section conjecture over local fields has been solved affirmatively. In [8], Koenigsmann proved that if k is either a p-adic local field for some prime number p [i.e., a finite extension of the p-adic completion of the field of rational numbers] or the field of real numbers, then any birational Galois section of X/k is geometric [cf. [8] Proposition 2.4, (2)]. Moreover,

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in [12], Pop obtained, by a refined discussion of Koenigsmann’s discussion, a result concerning birational Galois sections over p-adic local fields [cf. [12], Theorem A], which leads naturally to a proof of the geometrically pro- p version of Koenigsmann’s result over p-adic local fields [cf. Proposition 5.10]. In [18], Wickelgren proved a strong version of the birational section conjecture over the field of real numbers [cf. [18], Corollary 1.2]. In the remainder of the Introduction, we discuss the geometricity of birational Galois sections over number fields; suppose that k is a number field [i.e., a finite extension of the field of rational numbers]. First, let us recall that, in [1], Esnault and Wittenberg proved that if the Shafarevich–Tate group of the Jacobian variety of X over k is finite, then the existence of a birational Galois section of X/k implies the existence of a divisor of degree 1 on X ; more precisely, the existence of a section of the natural surjection G k(X ) /[G k·k(X ) , G k·k(X ) ] G k , where we write def

G k·k(X ) = Gal(k(X )/k · k(X )) and [G k·k(X ) , G k·k(X ) ] for the closure of the commutator subgroup of G k·k(X) , is equivalent to the existence of a divisor of degree 1 on X [cf. [1], Theorem 2.1]. Next, let us recall that, in [4], Harari and Stix proved, as a consequence of results obtained by Stoll in [16], that if there exist an abelian variety A over k and a nonconstant morphism X → A over k such that both the Mordell–Weil group and the Shafarevich–Tate group of A/k are finite, then any birational Galois section of X/k is geometric [cf. [4], Theorem 17]. This result of Harari and Stix gives us some examples of X/k for which any birational Galois section is geometric [cf. [4], Remark 18, (1)]. To state our main results, let us discuss local points associated to a biraf tional Galois section. Write Pk for the set of nonarchimedean primes of k. For f each p ∈ Pk , fix an algebraic closure k p of the p-adic completion kp of k con def f taining k and write G p = Gal(k p /kp ) ⊆ G k . Finally, write Ak ⊆ p∈P f kp k for the finite part of the adele ring of k, i.e., the subring of p∈P f kp conk sisting of elements (ap )p∈P f ∈ p∈P f kp such that ap is contained in the k

k

f

ring of integers of kp for all but finitely many p ∈ Pk . Then it follows from f a result obtained in [8], as well as [12], that, for each p ∈ Pk , a birational Galois section s of X/k uniquely determines a kp -valued point xp of X such that, for any open subscheme U ⊆ X of X , the image of the homomorphism ∼ G p → π1 (U ⊗k kp ) naturally determined by the isomorphism π1 (U ⊗k kp ) → s π1 (U ) ×G k G p and the composite G p → G k → G k(X ) π1 (U ) is contained in a decomposition subgroup of π1 (U ⊗k kp ) associated to xp [cf. Proposition 5.24]; we shall refer to the kp -valued point xp as the kp -valued point of X associated to s [cf. Definition 5.20]. In particular, [since X is projective f over k] the birational Galois section s uniquely determines an Ak -valued point

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Yuichiro Hoshi def

f

xA = (x p )p∈P f ∈ X (Ak ) ⊆ f Ak -valued

k

f X (kp ) of X ; we shall p∈Pk f Ak -valued point of X associated to

refer to the

point xA as the s [cf. Definition 5.20]. Note that if the birational Galois section s is geometric, then there exists a [necessarily unique] k-rational point x ∈ X (k) of X such that, for each f p ∈ Pk , the kp -valued point of X determined by x is the kp -valued point of X associated to s [cf. Remark 5.21]. The following result is the main result, which gives necessary and sufficient conditions for a birational Galois section of a projective smooth geometrically connected curve over a small number field, i.e., either the field of rational numbers or an imaginary quadratic field, to be geometric [cf. Theorem 5.40 in the case where C consists of all finite groups]. Theorem A. Let k be either the field of rational numbers or an imaginary quadratic field, X a projective smooth geometrically connected curve over k, and s a [pro-Primes] birational Galois section of X/k [cf. Definition 5.2]. Then the following conditions are equivalent: (1) s is geometric [cf. Definition 5.3]. (2) The following two conditions are satisfied: (2-i) There exists a finite morphism φ : X → P1k over k such that, for f each p ∈ Pk , the composite φ

xp

Spec kp −→ X −→ P1k determines a kp -valued point of P1k \ {0, 1, ∞} ⊆ P1k . (2-ii) For each open subscheme U ⊆ X of X which is a hyperbolic curve over k [cf. §0], there exists a prime number lU such that the pro-lU Galois section of U/k [cf. Definition 5.2] naturally determined by s is either cuspidal [cf. Definition 5.37, (i)] or unramified almost everywhere [cf. Definition 5.37, (ii)]. (3) There exists a finite morphism φ : X → P1k over k such that the composite f

xA

φ

Spec Ak −→ X −→ P1k f

determines an Ak -valued point of P1k \ {0, 1, ∞} ⊆ P1k . f f (4) There exist a finite subset T ⊆ Pk of Pk and a closed subscheme Z ⊆ X f of X which is finite over k such that, for each p ∈ Pk \ T , [the image of] the kp -valued point xp of X is contained in Z ⊆ X . The outline of the proof of Theorem A is as follows: The implications (1) ⇒ (2) ⇒ (3) follow immediately from the various definitions involved, together with some results that are proved in Appendix and derived from the discussion

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given in [7]. Next, to verify the implications (3) ⇒ (4) ⇒ (1), let us observe that since k is either the field of rational numbers or an imaginary quadratic field, the following assertion (†) holds [cf. Lemma 5.32]: f

(†): for every [pro-Primes] birational Galois section of P1k /k, if the associated Ak def f valued point of P1k determines an Ak -valued point of Gm,k = P1k \ {0, ∞}, then the induced [pro-Primes] Galois section of Gm,k /k arises from a k-rational point of Gm,k .

Then the implication (3) ⇒ (4) follows immediately from (†). Thus, it remains to verify the implication (4) ⇒ (1). Since X is projective, we have a closed immersion X → PkN for some positive integer N . Now by condition (4), we may assume without loss of generality that, for every nonarchimedean prime p of k, the kp -valued point of PkN determined by this closed immersion X → PkN and the kp -valued point of X associated to the given birational Galois section s lies on the open subscheme Gm,k ×k · · · ×k Gm,k ⊆ AkN ⊆ PkN of PkN ; in particular, we have a kp -valued point of Gm,k ×k · · · ×k Gm,k . Moreover, again by condition (4), together with the above assertion (†), one verifies easily that, for each p, any one of the N factors of the coordinate of the kp valued point of Gm,k ×k · · · ×k Gm,k is k-rational. In particular, it follows that X admits a k-rational point. Thus, by applying this observation to the various open subgroups of G k(X ) that contain the image of s, we conclude that s is geometric. This completes the explanation of the outline of the proof of Theorem A. Note that Theorem A is a result without any assumption on the finiteness of a Shafarevich–Tate group. Next, let us observe that the equivalence (1) ⇔ (3) of Theorem A may be regarded as a tripod analogue of the result due to Harari and Stix discussed above, i.e., [4], Theorem 17. The condition that k is either the field of rational numbers or an imaginary quadratic field [i.e., the assumption that the group of units of the ring of integers of k is finite] in the statement of Theorem A may be regarded as an analogue of the finiteness condition on the Mordell–Weil group in the statement of [4], Theorem 17; on the other hand, since any abelian variety is proper, in the case of [4], Theorem 17, the condition corresponding that the birational Galois sec to our condition f tion determines [not only a f k p -valued point but also] an Ak -valued p∈P k

point of the tripod P1k \ {0, 1, ∞} in Theorem A is automatically satisfied. Let us also observe that our main result leads naturally to some examples – which, however, were already essentially obtained by Stoll – of projective smooth curves for which any prosolvable birational Galois section is geometric [cf. Remarks 5.35; 5.41, (iv)].

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As a corollary of Theorem A, we prove the following result [cf. Corollary 5.42]. Theorem B. Let k be either the field of rational numbers or an imaginary quadratic field. Then the following assertions are equivalent: (1) Any [pro-Primes] birational Galois section [cf. Definition 5.2] of any projective smooth geometrically connected curve over k is geometric [cf. Definition 5.3]. (2) Any [pro-Primes] birational Galois section of P1k /k is geometric. (3) Any [pro-Primes] birational Galois section s of P1k /k satisfies the following two conditions: (3-i) There exist three distinct elements a, b, c ∈ P1k (k) of P1k (k) such that, for any nonarchimedean prime p of k, the kp -valued point of P1k associated to s is ∈ {a, b, c} ⊆ (P1k (k) ⊆) P1k (kp ). (3-ii) There exists a prime number l such that the pro-l Galois section of P1k \ {0, 1, ∞} [cf. Definition 5.2] naturally determined by s is either cuspidal [cf. Definition 5.37, (i)] or unramified almost everywhere [cf. Definition 5.37, (ii)]. (4) Any [pro-Primes] birational Galois section s of P1k /k satisfies the following two conditions: (4-i) There exist three distinct elements a, b, c ∈ P1k (k) of P1k (k) such that, for any nonarchimedean prime p of k, the kp -valued point of P1k associated to s is ∈ {a, b, c} ⊆ (P1k (k) ⊆) P1k (kp ). (4-ii) Write s P for the pro-Primes Galois section of P1k \ {0, 1, ∞} [cf. Definition 5.2] naturally determined by s. Then it holds either that s P is cuspidal [cf. Definition 5.37, (i)], or that there exists a prime number l such that the l-adic Galois representation sP

G k −→ π1 (P1k \ {0, 1, ∞}) −→ GL2 (Zl ) – where the second arrow π1 (P1k \ {0, 1, ∞}) → GL2 (Zl ) is the l-adic representation of π1 (P1k \ {0, 1, ∞}) determined by the Legendre family of elliptic curves over P1k \ {0, 1, ∞}, i.e., the elliptic curve over P1k \ {0, 1, ∞} = Spec k[u ±1 , (1 − u)−1 ] determined by the equation “y 2 = x(x − 1)(x − u)” – is unramified at all but finitely many primes of k. As a consequence [cf. the equivalences (1) ⇔ (3) and (1) ⇔ (4) of Theorem B], for a number field k which is either the field of rational numbers or an imaginary quadratic field, to prove the birational section conjecture over k [i.e., assertion (1) of Theorem B], it suffices to verify that, roughly

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speaking, for any birational Galois section of the projective line over k, the local points associated to the birational Galois section avoid three distinct rational points [cf. conditions (3-i), (4-i)], and, moreover, a certain Galois representation determined by the birational Galois section is unramified at all but finitely many primes [cf. conditions (3-ii), (4-ii)]. However, it is not clear to the author at the time of writing whether or not these are always satisfied. Finally, the author should mention that the referee pointed out that, after the present paper, Stix presented, in [15], results that are similar to and stronger than some results of the present paper. In fact, for instance, a similar result to the equivalence (1) ⇔ (2) of Theorem A (respectively, the equivalence (1) ⇔ (4) of Theorem B; Lemma 5.32; Theorem 5.33; Corollary 5.34) of the present paper may be found as [15], Theorem A (respectively, [15], Theorem C; [15], Proposition 4; [15], Corollary 9; [15], Corollary 11).

Acknowledgements The author would like to thank Akio Tamagawa, Takahiro Tsushima, and Seidai Yasuda for their helpful comments and, especially, discussions concerning §3. The author also would like to thank the referee for some comments and suggestions. This research was supported by Grant-in-Aid for Young Scientists (B), No. 22740012, Japan Society for the Promotion of Science.

0. Notations and conventions Numbers: The notation Primes will be used to denote the set of all prime numbers. The notation Z will be used to denote the ring of rational integers. If ⊆ Primes, then we shall refer to a nonzero integer whose prime divisors are contained in as a -integer, and we shall write

Z for the pro- completion def of Z, i.e.,

Z = lim Z/nZ, where the projective limit is over all positive ← − integers n. We shall refer to a finite (respectively, finitely generated) extension of the field of rational numbers as a number field (respectively, finitely generated field of characteristic 0). If p ∈ Primes, then the notation Z p will be used to denote the p-adic completion of Z, and we shall refer to a finite extension of the p-adic completion of the field of rational numbers as a p-adic local field. Profinite groups: Let G be a profinite group and H ⊆ G a closed subloc (H ) the centralizer, group of G. Then we shall denote by Z G (H ), NG (H ), Z G normalizer, local centralizer of H in G, respectively, i.e.,

194

Yuichiro Hoshi Z G (H ) = { g ∈ G | ghg −1 = h for any h ∈ H } , def

NG (H ) = { g ∈ G | g · H · g −1 = H } , def

def

loc ZG (H ) = lim Z G (U ) − → U

– where the injective limit is over all open subgroups U ⊆ H of H . We shall def

def

loc (G) as the center, local center refer to Z (G) = Z G (G), Z loc (G) = Z G of G, respectively. We shall say that G is center-free, slim if Z (G) = {1}, Z loc (G) = {1}, respectively. Let ⊆ Primes be a nonempty subset of Primes [where we refer to the discussion entitled “Numbers” concerning the set Primes]. Then we shall say that a finite group G is a -group if the cardinality of G is a -integer [where we refer to the discussion entitled “Numbers” concerning the term “-integer”]. Let C be a full formation [i.e., a family of finite groups that is closed under taking quotients, subgroups, and extensions]. We shall say that a finite group is a C-group if [a finite group which is isomorphic to] the finite group is contained in C. We shall say that a profinite group is a pro-C group if every finite quotient of the profinite group is a C-group. We shall write (C) ⊆ Primes for the set of prime numbers p ∈ Primes such that Z/ pZ is a C-group. Here, we note that one verifies easily that (C) = Primes if and only if C contains all finite solvable groups. If C consists of all -groups for some nonempty subset ⊆ Primes, then we shall refer to a pro-C group as a pro- group. Let G be a profinite group. Then we shall write Aut(G) for the group of [continuous] automorphisms of G, Inn(G) ⊆ Aut(G) for the group of inner automorphisms of G, and def

Out(G) = Aut(G)/Inn(G) . If, moreover, G is topologically finitely generated, then one verifies easily that the topology of G admits a basis of characteristic open subgroups, which thus induces a profinite topology on the group Aut(G), hence also a profinite topology on the group Out(G). Curves: Let S be a scheme and X a scheme over S. Then we shall say that X is a smooth curve over S if there exist a scheme X cpt which is smooth, proper, geometrically connected, and of relative dimension 1 over S and a closed subscheme D ⊆ X cpt of X cpt which is finite and étale over S such that the complement X cpt \ D of D in X cpt is isomorphic to X over S. Note

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that, as is well-known, if X is a smooth curve over [the spectrum of] a field k, then the pair “(X cpt , D)” is uniquely determined up to canonical isomorphism over k; we shall refer to X cpt as the smooth compactification of X over k and to a geometric point of X cpt whose image lies on D as a cusp of X . Let S be a scheme. Then we shall say that a smooth curve X over S is a hyperbolic curve (respectively, tripod) over S if there exist a pair (X cpt , D) satisfying the condition in the above definition of the term “smooth curve” and a pair (g, r ) of nonnegative integers such that 2g − 2 + r > 0 (respectively, (g, r ) = (0, 3)), any geometric fiber of X cpt → S is [a necessarily smooth, proper, and connected curve] of genus g, and the degree of D ⊆ X cpt over S is r. Let S be a scheme, U ⊆ S an open subscheme of S, and X a hyperbolic curve over U . Then we shall say that X admits good reduction over S if there exists a hyperbolic curve X S over S such that X S × S U is isomorphic to X over U .

1. Birational Galois sections and their geometricity In the present §1, we discuss the notion of a birational Galois section. In the present §1, let C be a full formation, k a field of characteristic 0, and k an algebraic closure of k. For a finite extension k (⊆ k) of k, write def

G k = Gal(k/k ). Definition 5.1. Let X be a quasi-compact scheme which is geometrically integral over k. (i) We shall write k(X ) for the function field of X . (ii) We shall write CX/k for the pro-C geometric fundamental group of X , i.e., the maximal pro-C quotient of π1 (X ⊗k k), and CX/k for the geometrically pro-C fundamental group of X , i.e., the quotient of π1 (X) by the kernel of the natural surjection π1 (X ⊗k k) CX/k . If X is the spectrum of a k-algebra R, then we shall write

196

Yuichiro Hoshi CR/k = CX/k ; CR/k = CX/k . def

def

Thus, we have a commutative diagram of profinite groups 1 −−−−→ Ck(X )/k −−−−→ Ck(X )/k −−−−→ ⏐ ⏐ ⏐ ⏐ 3 3 1 −−−−→ CX/k

−−−−→ CX/k

G k −−−−→ 1 4 4 4

−−−−→ G k −−−−→ 1

[cf. (i)] – where the horizontal sequences are exact [cf. [3], Exposé IX, Théorème 6.1]. If C consists of all -groups [cf. §0] for some nonempty subset ⊆ Primes [cf. §0], then we shall write C C X/k = X/k ; X/k = X/k . def

def

Definition 5.2. Let X be a quasi-compact scheme which is geometrically integral over k. Then we shall refer to a section of the upper (respectively, lower) exact sequence of the commutative diagram of Definition 5.1, (ii), as a pro-C birational Galois section (respectively, pro-C Galois section) of X/k. The Ck(X )/k -conjugacy (respectively, CX/k -conjugacy) class of a pro-C birational Galois section (respectively, pro-C Galois section) of X/k as the conjugacy class of the pro-C birational Galois section (respectively, pro-C Galois section). If C consists of all -groups for some nonempty subset ⊆ Primes, then we shall refer to a pro-C birational Galois section (respectively, pro-C Galois section) of X/k as a pro- birational Galois section (respectively, pro- Galois section) of X/k. Definition 5.3. Let X be a smooth curve over k [cf. §0] and s a pro-C birational Galois section (respectively, pro-C Galois section) of X/k [cf. Definition 5.2]. Then we shall say that s is geometric if the image of s is contained in a decomposition subgroup of Ck(X )/k (respectively, CX/k ) associated to a [necessarily k-rational] closed point of the [uniquely determined] smooth compactification of X over k. Remark 5.4. Let X be a smooth curve over k. Then it follows immediately from the various definitions involved that the geometricity of a pro-C birational Galois section (respectively, pro-C Galois section) of X/k depends only on its conjugacy class [cf. Definition 5.2]. Remark 5.5. Let X , Y be smooth curves over k and Y → X a dominant morphism over k, which thus determines a finite extension k(X ) → k(Y ) over k. If a pro-C birational Galois section (respectively, pro-C Galois section) s

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of Y/k is geometric, then it follows immediately from the various definitions involved that the pro-C birational Galois section (respectively, pro-C Galois section) of X/k determined by s and the morphism Y → X [i.e., the pro-C birational Galois section (respectively, pro-C Galois section) of X/k obtained as the composite of s and the natural open homomorphism Ck(Y )/k → Ck(X )/k (respectively, CY/k → CX/k ) induced by Y → X ] is geometric.

Remark 5.6. Let X be a projective smooth curve over k, U ⊆ X an open subscheme of X , and s a pro-C birational Galois section of X/k. Then it follows immediately from the various definitions involved that if s is geometric, then the pro-C Galois section of U/k naturally determined by s [i.e., the pro-C Galois section of U/k obtained as the composite of s and the natural surjection C ] is geometric. Ck(X )/k U/k Lemma 5.7. Let X be a hyperbolic curve over k [cf. §0] and x, y closed points of the [uniquely determined] smooth compactification of X . Suppose that k is generalized sub- p-adic [i.e., k is isomorphic to a subfield of a finitely generated extension of the p-adic completion of the maximal unramified extension of the p-adic completion of the field of rational numbers – cf. [11], Definition 4.11] for some p ∈ (C) [cf. §0]. Then the following conditions are equivalent: (1) x = y. (2) There exist respective decomposition subgroups Dx , D y ⊆ CX/k of CX/k associated to x, y such that the image of the composite Dx ∩ D y → CX/k G k is open. Proof. The implication (1) ⇒ (2) is immediate. Next, we verify the implication (2) ⇒ (1). Suppose that condition (2) is satisfied. Then it is immediate that, to verify the implication (2) ⇒ (1), by replacing CX/k by an open subgroup of CX/k , we may assume without loss of generality that X is of genus ≥ 2, and, moreover, the displayed composite of condition (2) is surjective, hence also that x and y are k-rational. Thus, to verify the implication (2) ⇒ (1), by replacing X by its smooth compactification, we may assume without loss of generality that x, y ∈ X (k). Then, by considering the quo{ p} tient CX/k X/k of CX/k , the implication (2) ⇒ (1) follows immediately from [11], Theorem 4.12 [cf. also [11], Remark following Theorem 4.12], together with a similar argument to the argument used in the proof of [10], Theorem C. This completes the proof of the implication (2) ⇒ (1), hence also of Lemma 5.7.

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Lemma 5.8. Let X be a hyperbolic curve over k, s a pro-C Galois section of X/k [cf. Definition 5.2], and k (⊆ k) a finite extension of k. Suppose that k is either (a) a finitely generated field of characteristic 0 [cf. §0] or (b) a p-adic local field for some p ∈ (C) [cf. §0]. Then the following conditions are equivalent: (1) s is geometric [cf. Definition 5.3]. (2) The pro-C Galois section s|G k of X ⊗k k /k determined by s is geometric. (3) For any open subgroup H ⊆ CX/k of CX/k containing the image of s, the [uniquely determined] smooth compactification of the finite étale covering of X corresponding to H ⊆ CX/k admits a k -valued point. Proof. The equivalence (1) ⇔ (2) follows immediately from a similar argument to the argument used in the proof of [7], Lemma 54. Here, we note that if k satisfies the condition (b), then, in order to apply a similar argument to the argument used in the proof of [7], Lemma 54, we have to replace “[Moc99, Theorem C]” (respectively, the finiteness of the set “(X n )cpt (k)” obtained by Mordell–Faltings’s theorem) in the proof of [7], Lemma 54, by Lemma 5.7 (respectively, the compactness of the set “(X n )cpt (k)” obtained by the consideration of a suitable model of “(X n )cpt ” over the ring of integers of k). The equivalence (2) ⇔ (3) follows immediately from a similar argument to the argument applied in the proof of [17], Proposition 2.8, (iv). This completes the proof of Lemma 5.8. Lemma 5.9. Let X be a smooth curve over k, s a pro-C birational Galois section of X/k [cf. Definition 5.2], and k (⊆ k) a finite extension of k. Suppose that k is either (a) a finitely generated field of characteristic 0 [cf. §0] or (b) a p-adic local field [cf. §0] for some p ∈ (C) [cf. §0]. Then the following conditions are equivalent: (1) s is geometric [cf. Definition 5.3]. (2) The pro-C birational Galois section s|G k of X ⊗k k /k determined by s is geometric. (3) For any open subgroup H ⊆ Ck(X )/k of Ck(X)/k containing the image of s, the [uniquely determined] smooth compactification of the normalization of X in the finite extension of k(X ) corresponding to H ⊆ Ck(X )/k admits a k -valued point.

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Proof. This follows immediately from a similar argument to the argument applied in the proof of Lemma 5.8. The following result was essentially proved in [12] by a refined discussion of the discussion given in [8]. Proposition 5.10. Let p be a prime number and X a smooth curve over k. Suppose that p ∈ (C) [cf. §0], and that k is a p-adic local field. Then any pro-C birational Galois section of X/k [cf. Definition 5.2] is geometric [cf. Definition 5.3]. Proof. It follows from the equivalence (1) ⇔ (3) of Lemma 5.9 that, to verify Proposition 5.10, it suffices to verify that, for any open subgroup H ⊆ Ck(X )/k

of Ck(X )/k containing the image of s, the [uniquely determined] smooth compactification of the normalization of X in the finite extension of k(X ) corresponding to H ⊆ Ck(X )/k admits a k(ζ p )-valued point, where we use the notation ζ p ∈ k to denote a primitive p-th root of unity. On the other hand, this follows immediately from [12], Theorem A, (2). This completes the proof of Proposition 5.10.

2. Local geometricity of birational Galois sections In the present §2, we discuss the notion of the local geometricity of birational Galois sections of smooth curves over number fields. In the present §2, let C be a full formation, k a number field [cf. §0], k an algebraic closure of k, and X a smooth curve over k [cf. §0]. Write ok ⊆ k for the ring of integers of k, f

Pk

for the set of all nonarchimedean primes of k, and X cpt for the [uniquely determined] smooth compactification of X over k. Moreover, f for each p ∈ Pk , write kp for the p-adic completion of k and op ⊆ kp

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for the ring of integers of kp . For each p ∈ Pk , let us fix an algebraic closure k p of kp containing k and write def

def

G p = Gal(k p /kp ) ⊆ G k = Gal(k/k) . Definition 5.11. Let s be a pro-C Galois section of X/k [cf. Definition 5.2]. f For a nonarchimedean prime p ∈ Pk of k, we shall say that s is geometric at p if the pro-C Galois section of X ⊗k kp /kp naturally determined by s [i.e., the pro-C Galois section of X ⊗k kp /kp determined by the natural isomorphism ∼

CX ⊗k kp /kp −→ CX/k ×G k G p and the composite G p → G k → CX/k ] s

f

f

is geometric [cf. Definition 5.3]. For a subset S ⊆ Pk of Pk , we shall say that s is geometric at S if, for each p ∈ S, s is geometric at p. Finally, we shall say f that s is locally geometric if s is geometric at Pk . Remark 5.12. In the notation of Definition 5.11, it is immediate that if s is geometric [cf. Definition 5.3], then s is locally geometric. f

f

Definition 5.13. Let S ⊆ Pk be a subset of Pk . Then we shall write def f Ak | S = kp ; p∈S f

def

Ak | S =

" " f (ap )p∈S ∈ Ak | S " ap ∈ op for all but finitely many p ∈ S ; f def f f def f Ak = Ak | P f ; Ak = Ak | P f . k

k

f

f

Remark 5.14. Since X cpt is proper over k, for any subset S ⊆ Pk of Pk , the f f natural injection X cpt (Ak | S ) → X cpt ( Ak | S ) is bijective. Definition 5.15. Let s be a pro-C Galois section of X/k [cf. Definition 5.2]. If f s is geometric at a nonarchimedean prime p ∈ Pk of k [cf. Definition 5.11], cpt i.e., there exists a kp -valued point xp ∈ X (kp ) = (X cpt ⊗k kp )(kp ) of X cpt such that the image of the pro-C Galois section of X ⊗k kp /kp naturally determined by s is contained in a decomposition subgroup of CX⊗k kp /kp associated to xp , then we shall refer to such a kp -valued point “xp ” of X cpt as a kp -valued f f point of X cpt associated to s. If s is geometric at a subset S ⊆ Pk of Pk , f then we shall refer to an Ak | S -valued point, or, equivalently [cf. Remark 5.14],

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f

an Ak | S -valued point, of X cpt determined by kp -valued points of X cpt associf ated to s – where p ranges over elements of S – as an Ak | S -valued point, or, f equivalently, an Ak | S -valued point, of X cpt associated to s. Remark 5.16. In the notation of Definition 5.15, suppose that s is geometric [cf. Definition 5.3], hence also locally geometric [cf. Definition 5.11; Remark 5.12]. Then it is immediate that there exists a k-rational point x ∈ f X cpt (k) of X cpt such that, for each p ∈ Pk , the kp -valued point of X cpt cpt determined by x is a kp -valued point of X associated to s. In particular, f f the Ak -valued point of X cpt determined by x is an Ak -valued point of X cpt associated to s. Note that if C contains all finite solvable groups, and X is a hyperbolic curve over k [cf. §0], then it follows from Theorem 5.33 below that the converse holds, i.e., if s is locally geometric, and there exists a k-rational point f x ∈ X cpt (k) of X cpt such that, for each p ∈ Pk , the kp -valued point of cpt cpt X determined by x is a kp -valued point of X associated to s, then s is geometric. Lemma 5.17. Let s be a pro-C birational Galois section of X/k [cf. Deff inition 5.2] and p ∈ Pk . For an open subscheme U ⊆ X cpt of X cpt , write s[U ] for the pro-C Galois section of U/k [cf. Definition 5.2] naturally determined by s [i.e., the pro-C Galois section of U/k obtained as the composite of s and C ]; the natural surjection Ck(X )/k U/k s[U, p] for the pro-C Galois section of U ⊗k kp /kp naturally determined by s [i.e., the pro-C Galois section of U ⊗k kp /kp determined by the natural isomorphism ∼

C C U ⊗k kp /kp −→ U/k ×G k G p

and the composite C G p → G k → Ck(X )/k U/k ]. s

Then the following conditions are equivalent: (1) There exists a kp -valued point xp ∈ X cpt (kp ) = (X cpt ⊗k kp )(kp ) of X cpt such that, for any open subscheme U ⊆ X cpt of X cpt , the image of the proC Galois section s[U, p] of U ⊗k kp /kp is contained in a decomposition C subgroup of U ⊗k kp /kp associated to x p .

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(2) For any open subscheme U ⊆ X cpt of X cpt , the pro-C Galois section s[U ] of U/k is geometric at p [cf. Definition 5.11], i.e., the pro-C Galois section s[U, p] of U ⊗k kp /kp is geometric [or, equivalently, the image of s[U, p] C is contained in a decomposition subgroup of U ⊗k k p /kp associated to a kp -rational point of X cpt ⊗k kp ]. (3) For any open subgroup H ⊆ Ck(X )/k of Ck(X)/k containing the image of s, the [uniquely determined] smooth compactification of the normalization of X in the finite extension of k(X ) corresponding to H ⊆ Ck(X )/k admits a kp -valued point. (4) The image of the homomorphism G p → Ck(X )/k ×G k G p induced by s is contained in the image of a decomposition subgroup of Ckp (X ⊗k kp )/kp associated to a [necessarily kp -rational] closed point of X cpt ⊗k kp by the natural surjection Ckp (X ⊗k kp )/kp Ck(X )/k ×G k G p .

Proof. The implications (4) ⇒ (1) ⇒ (2) ⇒ (3) are immediate. Finally, the implication (3) ⇒ (4) follows immediately from a similar argument to the argument applied in the proof of [17], Proposition 2.8, (iv). This completes the proof of Lemma 5.17. Definition 5.18. Let s be a pro-C birational Galois section of X/k [cf. Deff inition 5.2]. For a nonarchimedean prime p ∈ Pk of k, we shall say that s is geometric at p if the pair (s, p) satisfies equivalent conditions (1), (2), (3), f f and (4) of Lemma 5.17. For a subset S ⊆ Pk of Pk , we shall say that s is geometric at S if, for each p ∈ S, s is geometric at p. Finally, we shall say that f s is locally geometric if s is geometric at Pk . Remark 5.19. In the notation of Definition 5.18, it is immediate that if s is geometric [cf. Definition 5.3], then s is locally geometric. Definition 5.20. Let s be a pro-C birational Galois section of X/k [cf. Deff inition 5.2]. If s is geometric at a nonarchimedean prime p ∈ Pk of k [cf. Definition 5.18], i.e., the pair (s, p) satisfies condition (1) of Lemma 5.17, then we shall refer to a kp -valued point “xp ” of X cpt appearing in condition (1) of Lemma 5.17 as a kp -valued point of X cpt associated to s. If s is geometric f f f at a subset S ⊆ Pk of Pk , then we shall refer to an Ak | S -valued point, or, f equivalently [cf. Remark 5.14], an Ak | S -valued point, of X cpt determined by kp -valued points of X cpt associated to s – where p ranges over elements of f f S – as an Ak | S -valued point, or, equivalently, an Ak | S -valued point, of X cpt associated to s. Remark 5.21. In the notation of Definition 5.20, suppose that s is geometric [cf. Definition 5.3], hence also locally geometric [cf. Definition 5.18;

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Remark 5.19]. Then it is immediate that there exists a k-rational point x ∈ f X cpt (k) of X cpt such that, for each p ∈ Pk , the kp -valued point of X cpt determined by x is a kp -valued point of X cpt associated to s. In particular, f f the Ak -valued point of X cpt determined by x is an Ak -valued point of X cpt associated to s. Note that if C contains all finite solvable groups, then it follows from Theorem 5.33 below that the converse holds, i.e., if s is locally geometric, and there f exists a k-rational point x ∈ X cpt (k) of X cpt such that, for each p ∈ Pk , the cpt cpt kp -valued point of X determined by x is a kp -valued point of X associated to s, then s is geometric. Lemma 5.22. Let s be a pro-C birational Galois section (respectively, prof f C Galois section) of X/k [cf. Definition 5.2] and S ⊆ Pk a subset of Pk . Suppose that s is geometric at S [cf. Definition 5.18 (respectively, Definition 5.11)], and that, for each p ∈ S, the residue characteristic of p is ∈ (C) [cf. §0]. Suppose, moreover, that if s is a pro-C Galois section of X/k, then f X is a hyperbolic curve over k [cf. §0]. Then an Ak | S -valued point of X cpt associated to s [cf. Definition 5.20 (respectively, Definition 5.15)] is uniquely determined by s. Proof. Observe that, to verify Lemma 5.22, by replacing S by a subset of S of cardinality 1, we may assume without loss of generality that S = {p} for f some p ∈ Pk . Then the uniqueness in question follows immediately from Lemma 5.7. This completes the proof of Lemma 5.22. Lemma 5.23. Let s be a pro-C birational Galois section of X/k [cf. Definif f f f tion 5.2], S ⊆ Pk a subset of Pk , and xA ∈ X cpt (Ak | S ) an Ak | S -valued point cpt of X . Suppose that s is geometric at S [cf. Definition 5.18]. Write s[X ] for the pro-C Galois section of X/k [cf. Definition 5.2] naturally determined by s. Then the following hold: (i) s[X ] is geometric at S [cf. Definition 5.11]. f f (ii) If xA ∈ X cpt (Ak | S ) is an Ak | S -valued point of X cpt associated to s [cf. f f Definition 5.20], then xA ∈ X cpt (Ak | S ) is an Ak | S -valued point of X cpt associated to s[X ] [cf. Definition 5.15; assertion (i)]. (iii) Suppose, moreover, that, for each p ∈ S, the residue characteristic of p is ∈ (C) [cf. §0], and that X is a hyperbolic curve over k. Then it holds f f that xA ∈ X cpt (Ak | S ) is an Ak | S -valued point of X cpt associated to s if f f and only if xA ∈ X cpt (Ak | S ) is an Ak | S -valued point of X cpt associated to s[X ] [cf. assertion (i)].

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Proof. Assertions (i), (ii) follow immediately from the various definitions involved. Assertion (iii) follows immediately from Lemma 5.22, together with assertion (ii). This completes the proof of Lemma 5.23. The following result was essentially proved in [12] by a refined discussion of the discussion given in [8]. Proposition 5.24. Let s be a pro-C birational Galois section of X/k [cf. Defif f nition 5.2] and S ⊆ Pk a subset of Pk such that, for each p ∈ S, the residue characteristic of p is ∈ (C) [cf. §0]. Then s is geometric at S [cf. Definif tion 5.18]. In particular, s determines a unique Ak | S -valued point of X cpt [cf. Definition 5.20]. f

Proof. If s is geometric at S, then the uniqueness of an Ak | S -valued point of X cpt associated to s follows from Lemma 5.22. Thus, to verify Proposition 5.24, it suffices to verify that s is geometric at S. Moreover, it follows immediately from the various definitions involved that, to verify that s is geometric at S, by replacing S by a subset of S of cardinality 1, we may assume f without loss of generality that S = {p} for some p ∈ Pk , whose residue characteristic we denote by p. Thus, it follows from Lemma 5.25 below [cf. condition (5) of Lemma 5.25 below] that, to verify Proposition 5.24, it suffices to verify that, for any open subgroup H ⊆ Ck(X)/k of Ck(X )/k containing the image of s, the [uniquely determined] smooth compactification Y of the normalization of X in the finite extension of k(X ) corresponding to H ⊆ Ck(X )/k admits a kp (ζ p )-valued point, where we use the notation ζ p ∈ k to denote a primitive p-th root of unity. On the other hand, by considering the restriction of the pro-C birational Galois section of Y/k naturally determined by s to the closed subgroup Gal(k/k(ζ p )h ) of G k , where we write k(ζ p )h ⊆ k for the algebraic closure of k(ζ p ) in kp (ζ p ), we conclude from [12], Theorem B, (2), that Y (k(ζ p )h ) = ∅, hence also that Y (kp (ζ p )) = ∅. This completes the proof of Proposition 5.24. Lemma 5.25. In the notation of Lemma 5.17, suppose, moreover, that the residue characteristic of p is ∈ (C) [cf. §0]. Let kp (⊆ k p ) be a finite extension of kp . Then equivalent conditions (1), (2), (3), and (4) of Lemma 5.17 are equivalent to the following conditions: (5) For any open subgroup H ⊆ Ck(X )/k of Ck(X)/k containing the image of s, the [uniquely determined] smooth compactification of the normalization of X in the finite extension of k(X ) corresponding to H ⊆ Ck(X )/k admits a kp -valued point.

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(6) The image of the composite of the natural inclusion Gal(k p /kp ) → G p and the homomorphism G p → Ck(X )/k ×G k G p induced by s is contained in the image of a decomposition subgroup of Ckp (X ⊗k kp )/kp associated to a closed point of X cpt ⊗k kp [necessarily defined over a subfield of kp ] by the natural surjection Ckp (X⊗k kp )/kp Ck(X )/k ×G k G p .

Proof. The implication (3) ⇒ (5) is immediate. Moreover, by applying the implication (3) ⇒ (4) of Lemma 5.17 to the restriction of s to a suitable open subgroup of G k , we conclude that the implication (5) ⇒ (6) holds. Finally, the implication (6) ⇒ (4) follows immediately from a similar argument to the argument applied in the proof of [7], Lemma 54, by replacing “[Moc99, Theorem C]” (respectively, the finiteness of the set “(X n )cpt (k)” obtained by Mordell–Faltings’s theorem) in the proof of [7], Lemma 54, by Lemma 5.7 (respectively, the compactness of the set “(X n )cpt (k)” obtained by the consideration of a suitable model of “(X n )cpt ” over the ring of integers of k). This completes the proof of Lemma 5.25. Proposition 5.26. Suppose that X is a hyperbolic curve over k. Let s be a f pro-C Galois section of X/k [cf. Definition 5.2] and S ⊆ Pk a subset of f Pk such that, for each p ∈ S, the residue characteristic of p is ∈ (C) [cf. §0]. Suppose that s arises from a pro-C birational Galois section of X/k [cf. Definition 5.2]. Then s is geometric at S [cf. Definition 5.11]. In particular, s f determines a unique Ak | S -valued point of X cpt [cf. Definition 5.15]. Proof. The fact that s is geometric at S follows immediately from Proposition 5.24, together with Lemma 5.23, (i). The fact that s determines a f unique Ak | S -valued point of X cpt follows immediately from Lemma 5.22. This completes the proof of Proposition 5.26.

3. Galois sections of tori that locally arise from points In the present §3, we discuss Galois sections of tori that locally arise from points. We maintain the notation of the preceding §2. Let ⊆ Primes be a nonempty subset of Primes [cf. §0]. Write def # def Div(ok ) = Z Pic(ok ) = Pic(Spec ok ) ; f

p∈Pk

dk for the minimal positive integer such that dk · Pic(ok ) = {0}; Gm,Z = P1Z \ {0, ∞} = Spec Z[u ±1 ] ; def

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for each p ∈ Pk , vp : k × −→ Z for the p-adic valuation which induces a surjection kp× Z; divk : k × a

−→ Div(ok ) → f vp (a) · p . p∈P k

[Thus, we have an exact sequence of modules divk

× 0 −→ o× k −→ k −→ Div(ok ) −→ Pic(ok ) −→ 0 .]

Write, moreover, for a ring R, def

Gm,R = Gm,Z ⊗Z R . Let us identify Gm,R (R) with R × by the invertible function u ∈ R[u ±1 ]× : Gm,R (R) R × . Definition 5.27. Let M be a module. Then we shall write def

M[] = lim M/n M ← − – where n ranges over positive -integers [cf. §0]. Lemma 5.28. (i) For every [not necessarily algebraic] extension k of k, if we write GS (Gm,k /k ) for the set of conjugacy classes of pro- Galois sections of Gm,k /k [cf. Definition 5.2], then the map GS (Gm,k /k ) −→ H 1 (k , Gm,k /k ) determined by the natural isomorphism G

m,k /k

∼

−→ Gm,k /k

induced by k → k and the map GS (Gm,k /k ) −→ H 1 (k , G

m,k /k

)

given by mapping an element s ∈ GS (Gm,k /k ) to the element of H 1 (k , G /k ) obtained by considering the difference of s and the elem,k

ment of GS (Gm,k /k ) arising from the k-rational point 1 ∈ (k )×

Gm,k (k ) is bijective.

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∼

(ii) There exists a natural isomorphism Gm,k /k → Z (1) [where “(1)” denotes a Tate twist] such that, for every [not necessarily algebraic] extension k of k, the following diagram of sets commutes: ∼

Gm,k (k ) −−−−→ GS (Gm,k /k ) −−−−→ H 1 (k , G /k ) ⏐ ⏐ m,k ⏐ ⏐ 03 03 (k )×

−−−−→

∼

−−−−→ H 1 (k ,

Z (1)) .

(k )× []

Here, the left-hand upper horizontal arrow is the natural map given by mapping a k -rational point of Gm,k to the conjugacy class of a pro- Galois section of Gm,k /k associated to the k -rational point, the righthand upper horizontal arrow is the bijection of (i), the left-hand vertical arrow is the natural identification by the fixed invertible function u, the right-hand vertical arrow is the isomorphism induced by the isomorphism ∼

in question Gm,k /k → Z (1), the left-hand lower horizontal arrow is the natural homomorphism [cf. Definition 5.27], and the right-hand lower horizontal arrow is the natural isomorphism given by the Kummer theory. f f (iii) Let S ⊆ Pk be a subset of Pk . Then there exists a natural isomorphism between the commutative diagram of sets Gm,k (k) ⏐ ⏐ 3

Gm,k (k)

Gm,k (Ak | S ) −−−−→ Gm,k ( Ak | S ) −−−−→ f

GS (Gm,k /k) ⏐ ⏐ 3

−−−−→

f

p∈S

GS (Gm,kp /kp )

and the commutative diagram of modules k× ⏐ ⏐ 3

k×

−−−−→

f f (Ak | S )× −−−−→ ( Ak | S )× −−−−→

k × [] ⏐ ⏐ 3 × p∈S (k p []) .

Proof. Assertion (i) follows immediately from the various definitions involved. Next, we verify assertion (ii). For a positive integer n, write μn ⊆ k for the group of n-th roots of unity. Then, as is well-known, for a positive -integer n, there exist natural isomorphisms ∼

∼

1 ±1 × ±1 × n H 1 ( Gm,k /k , μn ) −→ H (Gm,k , μn ) ←− k[u ] /(k[u ] ) .

Thus, the invertible function u ∈ k[u ±1 ]× determines an element of ∼

H 1 ( Gm,k /k , Z (1)) −→ Hom(Gm,k /k , Z (1)) .

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On the other hand, one verifies easily that the resulting homomorphism

Gm,k /k → Z (1) is an isomorphism and satisfies the condition in the statement of assertion (ii). This completes the proof of assertion (ii). Assertion (iii) follows immediately from assertion (ii). This completes the proof of Lemma 5.28. Lemma 5.29. The following hold: (i) The exact sequence of modules divk

× 1 −→ o× k −→ k −→ Div(ok )

determines an exact sequence of modules divk []

× 1 −→ o× k [] −→ k [] −→ Div(ok )[] .

(ii) There is no nontrivial element of the cokernel of the natural homomorphism k × → k × [] which is annihilated by a -integer. def

Proof. First, we verify assertion (i). Write M = Im(divk ) ⊆ Div(ok ) for the image of divk . Then since M is a free Z-module, there exists a section of the natural surjection k × M; thus, we obtain a noncanonical isomorphism ∼ × × × o× k × M → k . In particular, the natural homomorphism ok [] → k [] is injective. Thus, to verify assertion (i), it suffices to verify that the kernel of × divk [] is contained in o× k [] ⊆ k [], or, equivalently [by the existence of ∼

× the noncanonical isomorphism o× k × M → k ], the natural homomorphism M[] → Div(ok )[] is injective. f For an element ai · pi ∈ Div(ok ), where ai ∈ Z and pi ∈ Pk , write [ ai · pi ] ∈ Pic(ok ) = Div(ok )/M for the element of the cokernel of divk determined by ai · pi ∈ Div(ok ). Now, for an element a ∈ Pic(ok ), let us f fix a nonarchimedean prime qa ∈ Pk of k such that a = [1 · qa ]. [Note that it follows immediately from Chebotarev’s density theorem that the subset of f f Pk consisting of p ∈ Pk such that a = [1 · p] is of density 1/'Pic(ok ); in def

f

particular, such a qa always exists.] Write T = { qa ∈ Pk | a ∈ Pic(ok ) }. f

def

Moreover, for each p ∈ Pk \ T , write xp = 1 · p − 1 · q[1·p] ∈ Div(ok ). Then one verifies easily that the Z-submodule N ⊆ Div(ok ) generated by f { xp | p ∈ Pk \ T } is contained in M and determines a section of the natural ) projection Div(ok ) Z. In particular, we obtain a commutative f p∈Pk \T diagram of free Z-modules

Conditional results on birational section conjecture 0 −−−−→ N −−−−→ 4 4 4

−−−−→

M ⏐ ⏐ 3

M/N ⏐ ⏐ 3

209

−−−−→ 0

0 −−−−→ N −−−−→ Div(ok ) −−−−→ Div(ok )/N −−−−→ 0 – where the horizontal sequences are exact, and the vertical arrows are injective. On the other hand, since Pic(ok ), hence also T , is finite, M/N and Div(ok )/N are finitely generated free Z-modules. In particular, one verifies easily that the natural homomorphism (M/N )[] → (Div(ok )/N )[] is injective. Thus, it follows immediately that the natural homomorphism in question M[] → Div(ok )[] is injective. This completes the proof of assertion (i). Next, we verify assertion (ii). Now let us observe that one verifies easily that there is no nontrivial element of the cokernel of the natural homomorphism Z →

Z which is annihilated by a -integer. Thus, assertion (ii) follows immediately from the existence of the [noncanonical] isomorphism o× k × ∼ M → k × obtained in the proof of assertion (i), together with the well-known fact that o× k is finitely generated. This completes the proof of assertion (ii). Remark 5.30. The observation given in the proof of Lemma 5.29 was related to the author by A. Tamagawa and S. Yasuda. Lemma 5.31. By applying Lemma 5.28, (iii), let us identify Gm,k (k) f (respectively, Gm,k (Ak ); GS (Gm,k /k); p∈P f GS (Gm,kp /kp )) with k × k f (respectively, (Ak )× ; k × []; p∈P f (kp× [])). Suppose that k is either the k field of rational numbers or an imaginary quadratic field. Let (ap )p∈P f ∈ (Ak )× Gm,k (Ak ) f

f

k

a ∈ k × [] GS (Gm,k /k) be such that their images in p∈P f (kp× []) p∈P f GS (Gm,kp /kp ) [cf. k k the diagrams of Lemma 5.28, (iii)] coincide. Then the following hold: (i) a dk ∈ k × [] GS (Gm,k /k) is contained in the image of the natural homomorphism k × Gm,k (k) → k × [] GS (Gm,k /k). (ii) If dk is a -integer, then a ∈ k × [] GS (Gm,k /k) is contained in the image of the natural homomorphism k × Gm,k (k) → k × []

GS (Gm,k /k). (iii) If dk is a -integer, and we fix an element a ∈ k × Gm,k (k) whose × image in k [] GS (Gm,k /k) coincides with a [cf. (ii)], then, for each f p ∈ Pk whose residue characteristic is ∈ , the difference ap · a −1 ∈ kp× is a root of unity whose order is a (Primes \ )-integer.

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Proof. First, we verify assertion (i). Since the image of a ∈ k × []

GS (Gm,k /k) in p∈P f (kp× [])

f GS (Gm,kp /k p ) is contained p∈P k

k

in the image of the natural homomorphism (Ak )× Gm,k (Ak ) → × f (k p [])

f GS (Gm,kp /kp ), one verifies easily that the image p∈P p∈P k

f

f

k

of a ∈ k × [] by the homomorphism divk [] : k × [] → Div(ok )[] is contained in the Z-submodule Div(ok ) ⊆ Div(ok )[]. Thus, it follows immediately from the definition of dk that there exists b ∈ k × such that the images d k b and a in Div(ok )[] coincide. On the other hand, since k is either the field of rational numbers or an imaginary quadratic field, it holds that o× k is finite, × which thus implies that o× → o [] is surjective. Thus, it follows immek k diately from Lemma 5.29, (i), that, by replacing b by a suitable element of k × , we conclude that a dk coincides with the image of b ∈ k × in k × []. This completes the proof of assertion (i). Assertion (ii) follows immediately from Lemma 5.29, (ii), together with assertion (i); our assumption that dk is a -integer. Finally, we verify assertion f (iii). One verifies easily that, for each p ∈ Pk whose residue characteristic is ∈ , the kernel of the natural homomorphism kp× → kp× [] consists of roots of unity in kp whose orders are (Primes \ )-integers. Thus, assertion (iii) follows immediately from assertion (ii). This completes the proof of assertion (iii). Lemma 5.32. In the notation of Lemma 5.31, if = Primes, then the commutative diagram of sets Gm,k (k) −−−−→ ⏐ ⏐ 3 f

Gm,k (Ak ) −−−−→

GS (Gm,k /k) ⏐ ⏐ 3 f

p∈Pk

GS (Gm,kp /kp )

is cartesian. Proof. This follows immediately from Lemma 5.31, (ii), (iii).

4. Conditional results on the birational section conjecture In the present §4, we prove conditional results on the birational section conjecture for projective smooth curves over number fields. We maintain the notation of the preceding §3. First, let us recall the following result that was essentially proved in [16]. It seems to the author that [at least, a similar result to] the following result

Conditional results on birational section conjecture

211

is likely to be well-known to experts. Since, however, the result could not be found in the literature, the author decided to give a proof. Theorem 5.33. Let C be a full formation that contains all finite solvable groups, k a number field [cf. §0], X a projective smooth curve (respectively, hyperbolic curve) over k [cf. §0], and s a pro-C birational Galois section (respectively, locally geometric pro-C Galois section) of X/k [cf. Defif nition 5.2 (respectively, Definitions 5.2; 5.11)]. Write Pk for the set of nonarchimedean primes of k and X cpt for the [uniquely determined] smooth f compactification of X over k. For each p ∈ Pk , write kp for the p-adic completion of k. Then the following conditions are equivalent: (1) s is geometric [cf. Definition 5.3]. f f (2) There exist a subset T ⊆ Pk of Pk of density 0 and a closed subscheme f Z ⊆ X cpt of X cpt which is finite over k such that, for each p ∈ Pk \ T , the [image of the uniquely determined – cf. Lemma 5.22] kp -valued point of X cpt associated to s [cf. Definition 5.20; Proposition 5.24 (respectively, Definition 5.15)] is contained in Z ⊆ X cpt . Proof. First, we verify Theorem 5.33 in the case where s is a locally geometric pro-C Galois section. The implication (1) ⇒ (2) is immediate [cf. also Remark 5.16]. Next, we verify the implication (2) ⇒ (1). Now observe that it follows from the equivalence (1) ⇔ (2) of Lemma 5.8 that, to verify the implication (2) ⇒ (1), by replacing k by a suitable finite extension of k, we may assume without loss of generality that k is totally imaginary. Next, observe that, for each open subgroup H ⊆ CX/k of CX/k containing the image of s, if we write Y for the connected finite étale covering of X corresponding to H ⊆ CX/k [thus, CY /k = H ⊆ CX/k ], then since the morphism Y → X is finite, one verifies easily that the pro-C Galois section of Y/k naturally determined by s [which is necessarily locally geometric by the various definitions involved] satisfies condition (2). Thus, to verify the implication (2) ⇒ (1), by replacing X by such a suitable Y , we may assume without loss of generality that X is of genus ≥ 2; moreover, it follows from the equivalence (1) ⇔ (3) of Lemma 5.8 that, to verify the implication (2) ⇒ (1), by applying the conclusion to various open subgroups of CX/k containing the image of s, it suffices to verify that X cpt (k) = ∅. In particular, since X is of genus ≥ 2, and [one verifies easily that] the pro-C Galois section of X cpt /k naturally determined by s is locally geometric and satisfies condition (2), to verify that X cpt (k) = ∅, by replacing X by X cpt , we may assume without loss of generality that X cpt = X . Now since s is locally geometric, and k is totally imaginary, it follows immediately from the definition of “X (Ak )f-ab • ” [cf. [16], Definition 5.4, (3)] that the

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[uniquely determined] kp -valued points of X associated to s – where p ranges over nonarchimedean primes of k – form a part of an element of X (Ak )f-ab • . Thus, it follows immediately from [16], Theorem 8.2, together with condition (2), that Z (k) = ∅, hence also X (k) = ∅. This completes the proof of the implication (2) ⇒ (1), hence also of Theorem 5.33 in the case where s is a locally geometric pro-C Galois section. Next, we verify Theorem 5.33 in the case where s is a pro-C birational Galois section. The implication (1) ⇒ (2) is immediate [cf. also Remark 5.21]. Next, we verify the implication (2) ⇒ (1). First, observe that it follows immediately from a similar argument to the argument applied in the proof of Theorem 5.33 in the case where s is a locally geometric pro-C Galois section that, to verify Theorem 5.33 in the case where s is a pro-C birational Galois section, by replacing Ck(X )/k by an open subgroup of Ck(X )/k containing the image of s, we may assume without loss of generality that X is of genus ≥ 2; moreover, it follows from the equivalence (1) ⇔ (3) of Lemma 5.9 that, to verify the implication (2) ⇒ (1), by applying the conclusion to various open subgroups of Ck(X )/k containg the image of s, it suffices to verify that X (k) = ∅. On the other hand, since X is of genus ≥ 2, in light of Proposition 5.26, by applying Theorem 5.33 in the case where s is a locally geometric pro-C Galois section to the pro-C Galois section of X/k naturally determined by s, we conclude that X (k) = ∅. This completes the proof of the implication (2) ⇒ (1), hence also of Theorem 5.33 in the case where s is a pro-C birational Galois section. Theorem 5.33 naturally leads to the following corollary that was essentially proved by Stoll [cf., e.g., [16], Theorem 8.6]. Corollary 5.34. Let C be a full formation that contains all finite solvable groups, k a number field [cf. §0], and X a projective smooth curve (respectively, hyperbolic curve) over k [cf. §0]. Suppose that there exist an abelian variety A over k and a nonconstant morphism X → A over k such that both the Mordell–Weil group and the Shafarevich–Tate group of A/k are finite. Then any pro-C birational Galois section (respectively, any locally geometric pro-C Galois section) of X/k [cf. Definition 5.2 (respectively, Definitions 5.2; 5.11)] is geometric [cf. Definition 5.3]. Proof. Write f (A/k) def Sel = lim Ker H 1 (k, A(k)[n]) → H 1 (kp , A(k)) ← − n

f

p∈Pk

Conditional results on birational section conjecture

213

– where the projective limit is over all positive integers n, and A(k)[n] is the subgroup of A(k) consisting of elements of A(k) that are annihilated by n. Then the well-known natural G k -equivariant isomorphism ∼ Primes Primes f (A/k) → A(k)[n] → A/k /n A/k induces a natural injection Sel Primes

H 1 (k, A/k ); moreover, it follows immediately from the various definitions involved that the pro-Primes Kummer homomorphism A(k) → Primes H 1 (k, A/k ) associated to A [cf., e.g., [6], Remark 1.1.4, (iii)] factors f Primes through Sel (A/k) ⊆ H 1 (k, ), which thus implies that we have a A/k

f (A/k). [Here, this injectivity is a formal consenatural injection A(k) → Sel quence of the well-known fact that there is no nontrivial divisible element of A(k).] On the other hand, since the Shafarevich–Tate group of A/k is finite, in light of the fact that the absolute Galois group of the completion of k at an archimedean prime is either Z/2Z or {1}, one verifies easily that, for each positive integer n, the cokernel of the natural homomorphism A(k)/n A(k) −→ Ker H 1 (k, A(k)[n]) → H 1 (kp , A(k)) f

p∈Pk

is annihilated by a positive integer which does not depend on n. Thus, since the Mordell–Weil group of A/k is finite, it follows immediately that the resulting f (A/k) is an isomorphism. injection A(k) → Sel Let s be a pro-C birational Galois section (respectively, locally geometric pro-C Galois section) of X/k. Write s A for the pro-Primes Galois section of A/k obtained as the composite Primes

G k −→ Ck(X )/k −→ CX/k −→ CA/k = A/k s

Primes

(respectively, G k −→ CX/k −→ CA/k = A/k s

)

– where the third (respectively, second) arrow is the homomorphism over G k induced by the nonconstant morphism X → A over k. Then s A natuPrimes rally determines an element of H 1 (k, A/k ) [cf., e.g., [6], Remark 1.1.4, (ii)]; moreover, it follows immediately from Proposition 5.24 (respectively, our assumption that s is locally geometric), together with the various defi∼ f (A/k) ⊆ nitions involved, that this element is contained in A(k) → Sel Primes 1 H (k, A/k ). In particular, since X → A is nonconstant, and the Mordell–Weil group A(k) is finite, it follows immediately from the injectivity of the pro-Primes Kummer homomorphism associated to A ⊗k kp [that is a formal consequence of the well-known fact that there is no nontrivial divisible element of A(kp )], together with [6], Remark 1.1.4, (iii), that s satisfies condition (2) of Theorem 5.33. Thus, it follows from the implication

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Yuichiro Hoshi

(2) ⇒ (1) of Theorem 5.33 that s is geometric. This completes the proof of Corollary 5.34. Remark 5.35. As in the cases of [4], Theorem 17; [16], Theorem 8.6, one may apply Corollary 5.34 to obtain some examples of projective smooth curves over number fields for which any prosolvable birational Galois section [i.e., any pro-C birational Galois section in the case where C consists of all finite solvable groups] is geometric [cf., e.g., the discussions in [4], Remark 18, (1); [16], Example 8.7; [16], Corollary 8.8]. Remark 5.36. The observation given in the proof of Corollary 5.34 was related to the author by A. Tamagawa and S. Yasuda. Definition 5.37. Suppose that X is a hyperbolic curve over [the number field] k. Let s be a pro-C Galois section of X/k [cf. Definition 5.2]. (i) We shall say that s is cuspidal if the image of s is contained in a decomposition subgroup of CX/k associated to a cusp of X/k. (ii) We shall say that s is unramified almost everywhere if the composite G k −→ CX/k −→ Aut(CX/k ) s

– where the second arrow is the action of CX/k on CX/k obtained by f

conjugation – is unramified for all but finitely many p ∈ Pk . Remark 5.38. In the notation of Definition 5.37, it is immediate that if s is cuspidal [cf. Definition 5.37, (i)], then s is geometric [cf. Definition 5.3]. Proposition 5.39. Suppose that is finite. Then any geometric [cf. Definition 5.3] pro- Galois section [cf. Definition 5.2] of a hyperbolic curve over a number field is either cuspidal [cf. Definition 5.37, (i)] or unramified almost everywhere [cf. Definition 5.37, (ii)]. Proof. This follows immediately from Proposition 5.A.50. Next, we prove the main result of the chapter. Theorem 5.40. Let C be a full formation, k either the field of rational numbers or an imaginary quadratic field, X a projective smooth curve over k [cf. §0], and s a pro-C birational Galois section of X/k [cf. Definition 5.2]. f Write ok for the ring of integers of k and Pk for the set of nonarchimedean f primes of k. For each p ∈ Pk , write kp for the p-adic completion of k and f op for the ring of integers of kp . Write, moreover, Ak for the finite part of the adele ring of k, i.e.,

Conditional results on birational section conjecture f def

Ak =

(ap )p∈P f ∈ k

215

" " kp " ap ∈ op for all but finitely many p .

f

p∈Pk

Suppose that the following three conditions are satisfied: (a) The pro-C birational Galois section s is locally geometric [cf. Definition 5.18]. (b) (C) [cf. §0] is cofinite, i.e., Primes \ (C) [cf. §0] is finite. def

(c) Pic(ok ) = Pic(Spec ok ) is annihilated by a (C)-integer [cf. §0]. [Note that it follows from Proposition 5.24 that if (C) = Primes, or, equivalently [cf. §0], C contains all finite solvable groups, then the above three conditions are satisfied.] Then the following conditions are equivalent: (1) The pro-C birational Galois section s is geometric [cf. Definition 5.3]. (2) The following two conditions are satisfied: (2-i) There exist a finite morphism φ : X → P1k over k and, for each f p ∈ Pk , a kp -valued point xp of X associated to s [cf. Definition 5.20; condition (a)] [note that if the residue characteristic of p is ∈ (C), then the kp -valued point x p of X associated to s is uniquely determined – cf. Lemma 5.22] such that the composite φ

xp

Spec kp −→ X −→ P1k determines a kp -valued point of P1k \ {0, 1, ∞} ⊆ P1k . (2-ii) For each open subscheme U ⊆ X of X which is a hyperbolic curve over k [cf. §0], there exists a prime number lU ∈ (C) contained in (C) such that the pro-lU Galois section of U/k [cf. Definition 5.2] naturally determined by s is either cuspidal [cf. Definition 5.37, (i)] or unramified almost everywhere [cf. Definition 5.37, (ii)]. f (3) There exist a finite morphism φ : X → P1k over k and an Ak -valued point xA of X associated to s [cf. Definition 5.20; condition (a)] [note that if f (C) = Primes, then the Ak -valued point x A of X associated to s is uniquely determined – cf. Lemma 5.22] such that the composite f

xA

φ

Spec Ak −→ X −→ P1k f

determines an Ak -valued point of P1k \ {0, 1, ∞} ⊆ P1k . f f (4) There exist a finite subset T ⊆ Pk of Pk and a closed subscheme Z ⊆ X f of X which is finite over k such that, for each p ∈ Pk \ T whose residue characteristic is ∈ (C), the [image of the uniquely determined – cf. Lemma 5.22] kp -valued point x p of X associated to s [cf. Definition 5.20; condition (a)] is contained in Z ⊆ X .

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Proof. The implication (1) ⇒ (2) follows immediately from Proposition 5.39, together with Remark 5.21. Next, we verify the implication (2) ⇒ (3). Supf pose that condition (2) is satisfied. Then, by condition (2-i), for each p ∈ Pk , the composite φ

xp

Spec kp −→ X −→ P1k determines a kp -valued point of P1k \ {0, 1, ∞}. Thus, to verify the implication (2) ⇒ (3), it suffices to verify that the above kp -valued point of P1k \ {0, 1, ∞} obtained as the composite φ ◦ xp determines an op -valued point f of P1op \ {0, 1, ∞} for all but finitely many p ∈ Pk . Write U ⊆ X for the open subscheme of X obtained as the inverse image of P1k \ {0, 1, ∞} ⊆ P1k by φ. Then, by condition (2-ii), there exists a prime number lU ∈ (C) contained in (C) such that the pro-lU Galois section s U of U/k obtained as the composite {l }

U G k −→ Ck(X )/k −→ U/k

s

is either cuspidal or unramified almost everywhere. Write s P for the pro-lU Galois section of P1k \ {0, 1, ∞} obtained as the composite {l }

U G k −→ Ck(X )/k −→ U/k −→

s

{lU } (P1k \{0,1,∞})/k

– where the third arrow is the homomorphism over G k induced by φ. Then since the morphism U → P1k \ {0, 1, ∞} induced by φ is finite, one verifies {lU } {l } easily that the homomorphism U/k → U1 maps injectively any {l }

(Pk \{0,1,∞})/k

U cuspidal decomposition subgroup of U/k associated to a cusp of U/k to a

cuspidal decomposition subgroup of

{lU } (P1k \{0,1,∞})/k

associated to a cusp of

P1k \ {0, 1, ∞}. Thus, it follows immediately that if s U is cuspidal, then s P is f cuspidal. On the other hand, by applying Lemma 5.7 [to “φ ◦ xp ” for p ∈ Pk whose residue characteristic is = lU ], it follows immediately from condition (2-i) that s P is not cuspidal. Thus, we conclude that s U is not cuspidal, hence also [by condition (2-ii)] unramified almost everywhere. In particular, it follows Proposition 5.A.55, (ii), that s P is unramified almost everywhere. Therefore, it follows immediately from Proposition 5.A.50, together with condition (2-i), that the kp -valued point of P1k obtained as the composite φ ◦ xp determines an op -valued point of P1op \ {0, 1, ∞} for all but finitely many f

p ∈ Pk . This completes the proof of the implication (2) ⇒ (3). Next, we verify the implication (3) ⇒ (4). Suppose that condition (3) is def

satisfied. Write s G for the pro-(C) Galois section of Gm,k = P1k \ {0, ∞} over k obtained as the composite

Conditional results on birational section conjecture

217

(C )

G k → Ck(X )/k → Ck(P1 )/k CGm,k /k = Gm,k /k s

k

– where the second arrow is the homomorphism over G k induced by φ – and t G for the pro-(C) Galois section of Gm,k over k obtained as the composite ∼

(C )

G k → Ck(X )/k → Ck(P1 )/k → Ck(P1 )/k CGm,k /k = Gm,k /k s

k

k

– where the second arrow is the homomorphism over G k induced by φ, and the third arrow is the automorphism over G k induced by the automorphism of P1k over k given by “u → 1 − u”. Then it follows immediately f from condition (3) that there exists an element (ap )p∈P f ∈ ( Ak )× such that k

(ap )p∈P f , (1 − ap )p∈P f ∈ (Ak )× Gm,k (Ak ), and, moreover, the respecf

k

f

k

tive images of the pro-(C) Galois sections s G , t G ∈ GS(C ) (Gm,k /k)

k × [(C)] [cf. Lemma 5.28, (i), (ii)] in the set p∈P f GS(C ) (Gm,kp /kp )

k × f (k p [(C)]) [cf. the diagrams of Lemma 5.28, (iii)] coincide with the p∈P k

respective images of the elements (ap )p∈P f , (1 − ap )p∈P f ∈ (Ak )×

k k f Gm,k (Ak ) in the set p∈P f (kp× [(C)]) p∈P f GS(C ) (Gm,kp /kp ). Thus, k k it follows from Lemma 5.31, (ii), (iii); together with condition (c), that f

(∗): there exist as , at ∈ k × such that, for p ∈ Pk , if we write u p = ap · as−1 , def

f

vp = (1 − ap ) · at−1 ∈ kp× , and the residue characteristic of p is ∈ (C), then u p , vp are roots of unity of kp whose orders are (Primes \ (C))-integers. def

f

Now let us observe that, for p ∈ Pk , the pair (u p , vp ) satisfies the equation 1 = as · u p + at · v p . Thus, it follows immediately from [2], Theorem 1.1, together with condition (b), that the set {(u p , vp )}p∈P f , hence also the set {u p }p∈P f , is finite. In partick k ular, since ap = as ·u p [cf. (∗)], it follows immediately that the pro-C birational Galois section of P1k /k obtained as the composite G k −→ Ck(X )/k −→ Ck(P1 )/k s

k

– where the second arrow is the homomorphism over G k induced by φ – satisfies condition (4). Therefore, since φ is finite, one verifies easily that the pro-C birational Galois section s satisfies condition (4). This completes the proof of the implication (3) ⇒ (4). Finally, we verify the implication (4) ⇒ (1). Suppose that condition (4) is f f satisfied. Let us fix an element p0 ∈ Pk \ T of Pk \ T such that the residue

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Yuichiro Hoshi

characteristic of p0 is ∈ (C) [note that, by condition (b), such a p0 always exists] and write r (p0 ) for the cardinality of the set of roots of unity of kp0 . Now observe that, for any open subgroup H ⊆ Ck(X )/k of Ck(X )/k containing the image of s, if we write Y for the normalization of X in the finite extension of k(X ) corresponding to H ⊆ Ck(X)/k [thus, Ck(Y )/k = H ⊆ Ck(X )/k ], then since the morphism Y → X is finite, the pro-C birational Galois section of Y/k f determined by s satisfies condition (4) relative to the finite subset “T ” ⊆ Pk appearing in condition (4). Thus, it follows from the equivalence (1) ⇔ (3) of Lemma 5.9 that, to verify condition (1), by applying the conclusion to various such H ’s, it suffices to verify that X admits a k(ζr(p0 ) )-valued point – where we use the notation ζr(p0 ) ∈ k to denote a primitive r(p0 )-th root of unity. f For each p ∈ Pk , let us fix a kp -valued point xp of X associated to s [cf. condition (a)]. Now since X is projective, there exists a closed immersion X → PkN over k for some positive integer N . Then it follows immediately from condition (4) that there exists a hyperplane H ⊆ PkN defined over k such f that, for any p ∈ Pk , [the image of] the fixed kp -valued point xp of X is contained in X \ (X ∩ H ) ⊆ PkN \ H AkN . Moreover, again by condition (4) – by considering a suitable automorphism of A1k over k – we may assume without f loss of generality that, for each i ∈ {1, · · · , N } and p ∈ Pk , the kp -valued point of A1k obtained as the composite xp

pri

Spec kp → X \ (X ∩ H ) → PkN \ H AkN → A1k def

factors through Gm,k = A1k \{0} ⊆ A1k . Therefore, we conclude that there exist an open subscheme U ⊆ X of X and a closed immersion U → Gm,k ×k · · ·×k def f Gm,k over k such that the A -valued point xA = (xp ) f of X determined by k

p∈Pk

the fixed kp -valued points xp lies on U . On the other hand, again by condition f (4), one verifies easily that, for each i ∈ {1, · · · , N }, the Ak -valued point of Gm,k obtained as the composite f xA

pri

Spec Ak → U → Gm,k ×k · · · ×k Gm,k → Gm,k f

determines an Ak -valued point of Gm,k . Thus, it follows immediately from Lemma 5.31, (ii), (iii); condition (c), that, for each i ∈ {1, · · · , N }, the kp0 valued point of Gm,k obtained as the composite x p0

pri

Spec kp0 → U → Gm,k ×k · · · ×k Gm,k → Gm,k determines a k(ζr(p0 ) )-valued point of Gm,k . In particular, since U → Gm,k ×k · · · ×k Gm,k is a closed immersion, one verifies easily that the kp0 -valued

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point xp0 of U , hence also X , determines a k(ζr(p0 ) )-valued point. This completes the proof of the implication (4) ⇒ (1), hence also of Theorem 5.40. Remark 5.41. (i) Theorem 5.40 is a result without any assumption on the finiteness of a Shafarevich–Tate group. (ii) The equivalence (1) ⇔ (3) of Theorem 5.40 may be regarded as a tripod analogue of [4], Theorem 17. The condition that k is either the field of rational numbers or an imaginary quadratic field [i.e., the assumption that o× k is finite] in the statement of Theorem 5.40 may be regarded as an analogue of the finiteness condition on the Mordell–Weil group in the statement of [4], Theorem 17; the condition that Pic(ok ) is annihilated by a (C)-integer in the statement of Theorem 5.40 may be regarded as an analogue of the finiteness condition on the Shafarevich–Tate group in the statement of [4], Theorem 17. On the other hand, since any abelian variety is proper, in the case of [4], Theorem 17, the condition corresponding to our condition that the birational Galois section determines [not only an f f Ak -valued point but also] an Ak -valued point of the tripod P1k \ {0, 1, ∞} in Theorem 5.40 is automatically satisfied. (iii) If C contains all finite solvable groups, then Theorem 5.33 implies the equivalence (1) ⇔ (4) of Theorem 5.40. (iv) One verifies easily that the proof of the equivalence (1) ⇔ (4) of Theorem 5.40 gives us an alternative proof of Corollary 5.34 in the case where s is a pro-C birational Galois section, and k is either the field of rational numbers or an imaginary quadratic field. Indeed, in the notation of Corollary 5.34, it follows from the argument given in the proof of Corollary 5.34 that every pro-C birational Galois section s of X/k satisfies condition (4) of Theorem 5.40. Thus, it follows from the equivalence (1) ⇔ (4) of Theorem 5.40 that s is geometric. Corollary 5.42. Let k be either the field of rational numbers or an imagidef

nary quadratic field and k an algebraic closure of k. Write G k = Gal(k/k) f f and Pk for the set of nonarchimedean primes of k. For each p ∈ Pk , write kp for the p-adic completion of k. Then the following assertions are equivalent: (1) Any pro-Primes birational Galois section [cf. Definition 5.2] of any projective smooth curve over k [cf. §0] is geometric [cf. Definition 5.3]. (2) Any pro-Primes birational Galois section of P1k /k is geometric. (3) Any pro-Primes birational Galois section s of P1k /k satisfies the following two conditions:

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(3-i) There exist three distinct elements a, b, c ∈ P1k (k) of P1k (k) such that, for any nonarchimedean prime p of k, the [uniquely determined – cf. Lemma 5.22] kp -valued point of P1k associated to s [cf. Definition 5.20; Proposition 5.24] is ∈ {a, b, c} ⊆ (P1k (k) ⊆) P1k (kp ). (3-ii) There exists a prime number l such that the pro-l Galois section of P1k \ {0, 1, ∞} [cf. Definition 5.2] naturally determined by s is either cuspidal [cf. Definition 5.37, (i)] or unramified almost everywhere [cf. Definition 5.37, (ii)]. (4) Any pro-Primes birational Galois section s of P1k /k satisfies the following two conditions: (4-i) There exist three distinct elements a, b, c ∈ P1k (k) of P1k (k) such that, for any nonarchimedean prime p of k, the [uniquely determined – cf. Lemma 5.22] kp -valued point of P1k associated to s [cf. Definition 5.20; Proposition 5.24] is ∈ {a, b, c} ⊆ (P1k (k) ⊆) P1k (kp ). (4-ii) Write s P for the pro-Primes Galois section of P1k \ {0, 1, ∞} naturally determined by s. Then it holds either that s P is cuspidal, or that there exists a prime number l such that the l-adic Galois representation sP

Primes (P1k \{0,1,∞})/k

G k −→

−→ GL2 (Zl )

– where we refer to Definition 5.1, (ii), concerning the profiPrimes Primes nite group 1 ; the second arrow 1 → (Pk \{0,1,∞})/k

(Pk \{0,1,∞})/k

GL2 (Zl ) is the l-adic representation determined by the Legendre family of elliptic curves over P1k \ {0, 1, ∞}, i.e., the elliptic curve over P1k \ {0, 1, ∞} = Spec k[u ±1 , (1 − u)−1 ] determined by the equation “y 2 = x(x − 1)(x −u)” – is unramified at all but finitely f many p ∈ Pk . Proof. The implications (1) ⇒ (2) ⇒ (4) are immediate [cf. also Remark 5.21]. On the other hand, the implication (2) ⇒ (1) follows immediately from the fact that any projective smooth curve over k may be obtained as the normalization of P1k in the finite extension of k(P1k ) corresponding to an Primes open subgroup of 1 . Moreover, let us observe that it follows immedik(Pk )/k

ately from Proposition 5.39, together with Remark 5.21, that the implications (2) ⇒ (3) holds. Finally, we verify the implication (3) ⇒ (2) (respectively, (4) ⇒ (2)). Suppose that assertion (3) (respectively, assertion (4)) holds. Let s be a pro-Primes birational Galois section of P1k /k. For each nonarchimedean prime p of k, write xp for the [uniquely determined] kp -valued point of P1k associated to s. Then it

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follows from condition (3-i) (respectively, condition (4-i)) that, by considering a suitable automorphism of P1k over k, we may assume without loss of generalf ity that, for any p ∈ Pk , xp ∈ P1k (kp ) is ∈ {0, 1, ∞} ⊆ P1k (kp ). Thus, for any f prime number l, by applying Lemma 5.7 [to “x p ” for p ∈ Pk whose residue characteristic is = l], it follows immediately that the pro-l Galois section s P,{l} of P1k \ {0, 1, ∞} obtained as the composite s

G k −→

Primes k(P1k )/k

−→

{l} (P1k \{0,1,∞})/k

,

hence also the pro-Primes Galois section s P of P1k \ {0, 1, ∞} obtained as the composite s

G k −→

Primes k(P1k )/k

−→

Primes (P1k \{0,1,∞})/k

,

is not cuspidal. Thus, by condition (3-ii) (respectively, condition (4-ii)), we conclude that there exists a prime number l0 such that s P,{l0 } is unramified almost everywhere (respectively, the l0 -adic Galois representation obtained as the displayed composite of condition (4-ii) is unramified at all but finitely many f f p ∈ Pk ). Thus, since [we have assumed that] for any p ∈ Pk , xp ∈ P1k (kp ) 1 is ∈ {0, 1, ∞} ⊆ Pk (kp ), it follows immediately from Proposition 5.A.50 (respectively, [14], Theorem 1) that the birational pro-Primes Galois section s of P1k /k satisfies condition (3) of Theorem 5.40, hence also [by the equivalence (1) ⇔ (3) of Theorem 5.40] that s is geometric. This completes the proof of the implication (3) ⇒ (2) (respectively, (4) ⇒ (2)), hence also of Corollary 5.42.

Appendix A. Ramification of Galois sections In the present §A, we discuss the ramification of Galois sections of hyperbolic curves over p-adic local fields. In the present §A, let ⊆ Primes be a nonempty subset of Primes [cf. §0], k a p-adic local field for some prime number p [cf. §0], k an algebraic closure of k, and X a hyperbolic curve over k [cf. §0]. For a finite extension k (⊆ k) of k, write G k = Gal(k/k ) , def

Ik ⊆ G k for the inertia subgroup of G k , and ok ⊆ k

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for the ring of integers of k . Write, moreover, X cpt for the [uniquely determined] smooth compactification of X over k; X/k for the pro- geometric fundamental group of X , i.e., the maximal pro- quotient of π1 (X ⊗k k); X/k for the geometrically pro- fundamental group of X , i.e., the quotient of π1 (X ) by the kernel of the natural surjection π1 (X ⊗k k) X/k . Thus, we have an exact sequence of profinite groups [cf. [3], Exposé IX, Théorème 6.1] 1 −→ X/k −→ X/k −→ G k −→ 1 .

Let s be a pro- Galois section of X/k [cf. [6], Definition 1.1, (i)], i.e., a section of the above exact sequence of profinite groups. Definition 5.A.43. We shall say that s is unramified (respectively, potentially unramified) if the image of the composite s

Ik −→ G k −→ X/k −→ Aut( X/k ) – where the third arrow is the action of X/k on X/k obtained by conjugation – is trivial (respectively, finite).

Proposition 5.A.44. The following hold: (i) If p ∈ , then any pro- Galois section of X/k [cf. [6], Definition 1.1, (i)] is not potentially unramified, hence also not unramified [cf. Definition 5.A.43]. (ii) If X does not admit good reduction over ok [cf. §0], then any pro- Galois section of X/k is not unramified. Proof. Let s be a pro- Galois section of X/k. First, we verify assertion (i). Now one verifies easily that there exists a characteristic open subgroup H ⊆ X/k of X/k such that the connected finite étale covering of X corresponding to the open subgroup H · Im(s) of X/k topologically generated by H and Im(s) is of genus ≥ 1. On the other hand, since H ⊆ X/k is characteristic, and [as is well-known] X/k is slim [cf. §0], it follows from [7], Lemma 5, that we have a natural injection Aut( X/k ) → Aut(H ). Thus, to verify assertion (i), by replacing by the open subgroup H ·Im(s), we may assume without X/k

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loss of generality that X cpt is of genus ≥ 1. Next, let us observe that, as is well-known, since p ∈ , and X cpt is of genus ≥ 1, there exist G k -equivariant isomorphisms 2 cpt H 2 ( ⊗k k, Z p ) Z p (1) X cpt /k , Z p ) H (X

– where “(1)” denotes a Tate twist. In particular, [the restriction to Ik of] the p-adic cyclotomic representation χ p : Ik → Aut(Z p (1)) factors through the displayed composite of Definition 5.A.43. On the other hand, one may verify easily that the image of χ p is infinite. Thus, s is not potentially unramified. This completes the proof of assertion (i). Next, we verify assertion (ii). Suppose that X does not admit good reduction over ok . Now it follows from assertion (i) that, to verify assertion (ii), we may assume without loss of generality that p ∈ . Then it follows immediately from [17], Theorem 0.8, that the image of the composite Ik −→ Aut( X/k ) −→ Out( X/k )

– where the first arrow is the displayed composite of Definition 5.A.43 – is nontrivial, hence that s is not unramified. This completes the proof of assertion (ii). Definition 5.A.45. If X admits good reduction X over ok [cf. §0], then we shall write (π (X ) ) -ét 1

X/k

X/k

for the quotient of π1 (X ) by the normal closed subgroup topologically normally generated by the kernels of the natural surjections π1 (X ) X/k , π1 (X ) π1 (X ). Thus, the natural surjection X/k G k determines a -ét surjection X/k G k /Ik . We shall write -ét X/k -ét for the kernel of the surjection X/k G k /Ik . Thus, we have a commutative diagram of profinite groups 1 −−−−→ → −−−→ X/k −−−− X/k − ⏐ ⏐ ⏐ ⏐ 3 3

Gk ⏐ ⏐ 3

−−−−→ 1

-ét -ét −−−→ G /I −−−−→ 1 1 −−−−→ → k k X/k −−−− X/k − – where the horizontal sequences are exact.

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Remark 5.A.46. In the notation of Definition 5.A.45, as is well-known, if -ét p ∈ , then the left-hand vertical arrow X/k → X/k of the commutative diagram of Definition 5.A.45 is an isomorphism. In particular, the right-hand upper horizontal arrow X/k → G k induces an isomorphism ∼ -ét Ker( ) → I , and the right-hand square of the commutative X/k

k

X/k

diagram of Definition 5.A.45 is cartesian. Proposition 5.A.47. The following conditions are equivalent: (1) s is unramified [cf. Definition 5.A.43]. (2) p ∈ , X admits good reduction over ok [cf. §0], and the image of the composite s I → G → -ét k

k

X/k

X/k

[cf. Definition 5.A.45] is trivial. (3) p ∈ , X admits good reduction over ok , and the composite s

Ik → G k → X/k determines an isomorphism ∼ -ét Ik −→ Ker( X/k X/k ) . (4) p ∈ , and, for any open subgroup H ⊆ X/k of X/k containing the image of s, the connected finite étale covering of X corresponding to H ⊆ X/k admits good reduction over ok .

Proof. First, we verify the equivalence (1) ⇔ (2). It follows immediately from Proposition 5.A.44 that both (1) and (2) imply that p ∈ , and that X admits good reduction over ok . Thus, suppose that these conditions are satisfied. Write J for the image of the displayed composite of condition (2). Then it follows immediately from the existence of the commutative diagram -ét -ét of Definition 5.A.45 that J ⊆ X/k ⊆ X/k . Thus, it follows immediately from Remark 5.A.46 that the displayed composite Ik → Aut( X/k ) of Definition 5.A.43 factors as -ét -ét ∼ Ik J → X/k → Aut( X/k ) ← Aut( X/k ) -ét -ét – where the third arrow is the action of X/k on X/k obtained by conju∼ gation. Now since, as is well-known, → -ét is center-free, the third X/k

X/k

arrow of this composite is injective. Therefore, it follows immediately that the condition that s is unramified is equivalent to the condition that J = {1}. This completes the proof of the equivalence (1) ⇔ (2).

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The equivalence (2) ⇔ (3) follows immediately from Remark 5.A.46. Next, we verify the implication (3) ⇒ (4). Suppose that condition (3) is satisfied. Then it is immediate that if an open subgroup of X/k contains the image of -ét ; thus, it follows immediately s, then it arises from an open subgroup of X/k from the various definitions involved that the corresponding connected finite étale covering of X admits good reduction over ok . This completes the proof of the implication (3) ⇒ (4). Finally, we verify the implication (4) ⇒ (3). Suppose that condition (4) is satisfied. Let H ⊆ X/k be an open subgroup of X/k containing the image of s. Write Y → X for the connected finite étale covering of X corresponding to H ⊆ X/k ; thus, Y /k = H ⊆ X/k . Then it follows from condition (4) that Y admits good reduction over ok . Thus, it follows from [9], Lemma 8.3, that the morphism Y → X extends to a morphism between their [uniquely determined] smooth models. In particular, it follows immediately -ét -ét from the definitions of X/k and Y /k that the inclusion Y/k ⊆ X/k -ét -ét determines an inclusion Ker( Y/k Y /k ) ⊆ Ker( X/k X/k ). Thus, -ét it follows immediately from Remark 5.A.46 that Ker(Y /k Y /k ) = -ét -ét Ker( X/k X/k ), hence that Ker( X/k X/k ) ⊆ Y/k = H . Therefore, by considering the intersection of such H ’s, we obtain that Ker( X/k -ét X/k ) ⊆ Im(s). Thus, again by Remark 5.A.46, we conclude that condition (3) holds. This completes the proof of the implication (4) ⇒ (3), hence also of Proposition 5.A.47. Lemma 5.A.48. Suppose that p ∈ , and that X admits good reduction -ét -ét over ok [cf. §0]. Let ⊆ X/k be an open subgroup of X/k . Write so s -ét for the composite G k → X/k X/k , k (⊆ k) for the [necessarily unramified] finite extension of k corresponding to the image of the compos-ét ite → X/k G k /Ik , Y → X for the connected finite étale covering of -ét cpt for the [uniquely X corresponding to the open subgroup ⊆ X/k , and Y determined] smooth compactification of Y over k . [Here, it follows immediately from the various definitions involved that Y is a hyperbolic curve over -ét -ét k ; Y , hence also Y cpt , admits good reduction over ok ; Y /k = ⊆ X/k .] Suppose, moreover, that Y is of genus ≥ 2. Then the image of the composite s (I ) ∩ -ét → -ét -ét ( -ét )ab o

k

Y/k

Y /k

Y cpt /k

Y cpt /k

– where the second arrow is the surjection induced by the open immersion Y → Y cpt – is trivial. Proof. This follows immediately from a similar argument to the argument used in the proof of assertion (ii) in the proof of [6], Lemma 3.3.

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Proposition 5.A.49. Suppose that p ∈ , and that X admits good reduction over ok [cf. §0]. Then the following conditions are equivalent: (1) s is ramified, i.e., not unramified [cf. Definition 5.A.43]. (2) The image of the composite s -ét Ik → G k → X/k X/k

-ét is a nontrivial closed subgroup of a cuspidal inertia subgroup of X/k associated to a cusp of X/k. (3) The image of the composite s

Ik → G k → X/k X/k – where X/k is the quotient of X/k defined in [7], Definition 1, (iv), i.e., the quotient of X/k by the kernel Z ( X/k ) of the homomorphism X/k

X/k → Aut( X/k ) obtained by conjugation – is a nontrivial closed subgroup of a cuspidal inertia subgroup of X/k associated to a cusp of X/k. (4) There exists an element l ∈ of such that the pro-l Galois section of X/k [cf. [6], Definition 1.1, (i)] naturally determined by s is ramified.

Proof. First, we verify the equivalence (1) ⇔ (2). It follows from the equivalence (1) ⇔ (2) of Lemma 5.A.47 that s is ramified if and only if the image of the composite of condition (2) is nontrivial. On the other hand, it follows immediately from the existence of the commutative diagram of Definition 5.A.45, together with Remark 5.A.46, that the composite of condition (2) factors through the maximal pro- quotient of Ik , which is, as is wellknown, procyclic. Thus, it follows immediately from Lemma 5.A.48, together with [5], Lemma 1.6, that the equivalence (1) ⇔ (2) holds. This completes the proof of the equivalence (1) ⇔ (2). Next, let us observe that the implication (3) ⇒ (1) follows immediately from the various definitions involved [cf. also the definition of the quotient X/k ]. Next, we verify the implication (2) ⇒ (3). Since, as is well-known, X/k is slim [cf. §0], it follows from [7], Proposition 6, (ii), together with Remark 5.A.46, that we have a -ét sequence of natural surjections X/k X/k X/k , which induces an ∼ injection → -ét → . Thus, one verifies easily that the implicaX/k

X/k

X/k

tion (2) ⇒ (3) holds. This completes the proof of the implication (2) ⇒ (3). Finally, we verify the equivalence (1) ⇔ (4). For each nonempty subset ⊆ of , write J for the image of the composite

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s

227

-ét Ik → G k → X/k X/k X/k .

Then it follows immediately from the verified equivalence (1) ⇔ (2), together with the well-known structure of the maximal pro- quotient of the fundamental group of a smooth curve over an algebraically closed field of characteristic ∈ , that, for each nonempty subset ⊆ of , the image J is procyclic, and, moreover, J is the maximal pro- quotient of J [relative to the natural surjection J J ]. In particular, we conclude that J = {1} if and only if J{l} = {1} for any l ∈ , i.e., the equivalence (1) ⇔ (4) holds. This completes the proof of the equivalence (1) ⇔ (4), hence also of Proposition 5.A.49. Proposition 5.A.50. Suppose that p ∈ , and that s is geometric [cf. [6], Definition 1.1, (iii)]. Let x ∈ X cpt (k) be a k-rational point of X cpt such that a decomposition subgroup of X/k associated to x contains the image of s. Consider the following three conditions: (1) s is unramified [cf. Definition 5.A.43]. (2) x ∈ X (k), and the hyperbolic curve X \ {x}, hence also the hyperbolic curve X , over k admits good reduction over ok [cf. §0]. (3) x ∈ X (k). Then we have implications (2) =⇒ (1) =⇒ either (2) or (3) . In particular, if x ∈ X (k), then we have an equivalence (1) ⇐⇒ (2) . Proof. To verify Proposition 5.A.50, it is immediate that it suffices to verify that if x ∈ X (k), then condition (1) is equivalent to condition (2). Thus, suppose that x ∈ X (k). Now let us observe that it follows immediately from Proposition 5.A.44, (ii), that both (1) and (2) imply that X admits good reduction over ok . Thus, we may assume without loss of generality that X admits good reduction over ok . Moreover, observe that it follows immediately from the equivalence (1) ⇔ (4) of Proposition 5.A.49 that, to verify the equivalence (1) ⇔ (2), by considering the pro-l Galois section of X/k naturally determined by s – where l ranges over elements of – we may assume without loss of generality that is of cardinality 1. On the other hand, since is of cardinality 1, it follows immediately from [7], Proposition 19, (ii), that s the kernel of the composite G k → X/k → Aut( X/k ) coincides with the kernel of the pro- outer Galois representation associated to the hyperbolic curve X \ {x} over k. Thus, the equivalence (1) ⇔ (2) follows immediately from [17], Theorem 0.8. This completes the proof of Proposition 5.A.50.

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Yuichiro Hoshi

Proposition 5.A.51. Suppose that p ∈ , that X admits good reduction over ok [cf. §0], and that X is proper over k. Then any pro- Galois section of X/k [cf. [6], Definition 1.1, (i)] is unramified [cf. Definition 5.A.43]. Proof. This follows immediately from the equivalence (1) ⇔ (2) of Proposition 5.A.49. Remark 5.A.52. Proposition 5.A.51 may be regarded as a Galois section version of the valuative criterion for properness of morphisms of schemes. Remark 5.A.53. In [13], Saïdi proved the existence of a nongeometric pro Galois section in the situation of Proposition 5.A.51 [cf. [13], Proposition 4.2.1]. Proposition 5.A.54. Suppose that p ∈ , and that X admits good reduction over ok [cf. §0]. Then s is unramified if and only if s is potentially unramified [cf. Definition 5.A.43]. Proof. This follows immediately from the equivalence (1) ⇔ (2) of Proposition 5.A.49, together with the well-known fact that any cuspidal inertia -ét

subgroup of X/k associated to a cusp of X/k is isomorphic to Z as an abstract profinite group. Proposition 5.A.55. Let Y be a hyperbolic curve over k and X → Y a dominant morphism over k. Write sY for the pro- Galois section of Y /k [cf. [6], s Definition 1.1, (i)] determined by s, i.e., the composite G k → X/k → Y/k . Then the following hold: (i) Write X/k , Y /k for the respective quotients of X/k , Y /k defined in [7], Definition 1, (iv) [cf. also the statement of condition (3) of Proposi tion 5.A.49]. Then the natural homomorphism X/k → Y/k induces a homomorphism X/k → Y/k . (ii) If s is unramified (respectively, potentially unramified) [cf. Definition 5.A.43], then sY is unramified (respectively, potentially unramified). (iii) Suppose that X → Y is finite, and that X and Y admit good reduction over ok [cf. §0]. Then s is unramified if and only if sY is unramified.

Proof. First, we verify assertion (i). Now since, as is well-known, the profi nite group Y/k is slim [cf. §0], for any open subgroup H ⊆ Y /k of Y/k , it follows immediately from [7], Lemma 5, that N (H ) ∩ Z (Y /k ) = Y/k Y/k Z (H ). Thus, it follows immediately from the fact that the natural homoY/k

morphism X/k → Y /k is open that the natural homomorphism X/k →

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Y/k induces a homomorphism X/k → Y /k . This completes the proof of assertion (i). Assertion (ii) follows immediately from the various definitions involved, together with assertion (i) [cf. also the definitions of the quotients X/k , Y /k ]. Finally, we verify assertion (iii). It follows from Proposition 5.A.44, (i), that both the condition that s is unramified and the condition that sY is unramified imply that p ∈ . Thus, suppose that p ∈ . On the other hand, since X → Y is finite, one verifies easily that the restriction of the natural homomorphism X/k → Y /k to any cuspidal inertia sub group of X/k associated to a cusp of X/k is injective. Thus, assertion (iii) follows immediately from assertions (i), (ii); the equivalence (1) ⇔ (3) of Proposition 5.A.49, together with the fact that the sequences of natural surjec -ét -ét tions X/k X/k X/k , Y/k Y/k Y /k induce injections ∼ ∼ → -ét → , → -ét → , respectively [cf. the X/k

X/k

X/k

Y/k

Y /k

Y /k

proof of the implication (2) ⇒ (3) of Proposition 5.A.49]. This completes the proof of assertion (iii).

References [1] H. Esnault and O. Wittenberg, On abelian birational sections, J. Amer. Math. Soc. 23 (2010), no. 3, 713–724. [2] J.-H. Evertse, H. P. Schlickewei, and W. M. Schmidt, Linear equations in variables which lie in a multiplicative group, Ann. of Math. (2) 155 (2002), no. 3, 807–836. [3] A. Grothendieck et al., Revêtements Étales et Groupe Fondamental (SGA 1), Séminaire de géométrie algébrique du Bois Marie 1960–61, Documents Mathématiques, 3. Société Mathématique de France, Paris, 2003. [4] D. Harari and J. Stix, Descent obstruction and fundamental exact sequence, The Arithmetic of Fundamental Groups - PIA 2010, 147–166, Contributions in Mathematical and Computational Sciences, vol. 2, Springer-Verlag, Berlin Heidelberg, 2012. [5] Y. Hoshi and S. Mochizuki, On the combinatorial anabelian geometry of nodally nondegenerate outer representations, Hiroshima Math. J. 41 (2011), no. 3, 275– 342. [6] Y. Hoshi, Existence of nongeometric pro- p Galois sections of hyperbolic curves, Publ. Res. Inst. Math. Sci. 46 (2010), no. 4, 829–848. [7] Y. Hoshi, On monodromically full points of configuration spaces of hyperbolic curves, The Arithmetic of Fundamental Groups - PIA 2010, 167–207, Contributions in Mathematical and Computational Sciences, vol. 2, Springer-Verlag, Berlin Heidelberg, 2012. [8] J. Koenigsmann, On the ‘section conjecture’ in anabelian geometry, J. Reine Angew. Math. 588 (2005), 221–235.

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[9] S. Mochizuki, The profinite Grothendieck conjecture for closed hyperbolic curves over number fields, J. Math. Sci. Univ. Tokyo 3 (1996), no. 3, 571–627. [10] S. Mochizuki, The local pro- p anabelian geometry of curves, Invent. Math. 138 (1999), no. 2, 319–423. [11] S. Mochizuki, Topics surrounding the anabelian geometry of hyperbolic curves, Galois groups and fundamental groups, 119–165, Math. Sci. Res. Inst. Publ. 41, Cambridge University Press, Cambridge, 2003. [12] F. Pop, On the birational p-adic section conjecture, Compos. Math. 146 (2010), no. 3, 621–637. [13] M. Saïdi, Good sections of arithmetic fundamental groups, arXiv:1010.1313, 2010. [14] J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517. [15] J. Stix, On the birational section conjecture with local conditions, arXiv:1203.3236, 2012. [16] M. Stoll, Finite descent obstructions and rational points on curves, Algebra Number Theory 1 (2007), no. 4, 349–391. [17] A. Tamagawa, The Grothendieck conjecture for affine curves, Compositio Math. 109 (1997), no. 2, 135–194. [18] K. Wickelgren, 2-nilpotent real section conjecture, arXiv:1006.0265, 2010.

6 Blocks for mod p representations of GL2 (Q p ) Vytautas Pašk¯unas

Abstract Let π1 and π2 be absolutely irreducible smooth representations of G = GL2 (Q p ) with a central character, defined over a finite extension of F p . We show that if there exists a non-split extension between π1 and π2 then they both appear as subquotients of the reduction modulo p of a unit ball in a crystalline Banach space representation of G. The results of Berger–Breuil describe such reductions and allow us to organize the irreducible representation into blocks. The result is new for p = 2; the proof, which works for all p, is new.

1. Introduction Let L be a finite extension of Q p , with the ring of integers O, a uniformizer $ , and residue field k, and let G = GL2 (Q p ) and let B be the subgroup of upper-triangular matrices in G. Theorem 6.1. Let π1 , π2 be smooth, absolutely irreducible k-representations of G with a central character. Suppose that Ext1G (π2 , π1 ) = 0 then after replacing L by a finite extension, we may find integers (l, k) ∈ Z × N and × unramified characters χ1 , χ2 : Q× p → L with χ2 = χ1 | |, such that π1 and ss ss π2 are subquotients of , where is the semi-simplification of the reduction modulo $ of an open bounded G-invariant lattice in , where is the universal unitary completion of −1 l k−1 2 (IndG L . B χ1 ⊗ χ2 | | )sm ⊗ det ⊗ Sym

Automorphic Forms and Galois Representations, ed. Fred Diamond, Payman L. Kassaei and c Cambridge University Press 2014. Minhyong Kim. Published by Cambridge University Press.

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The results of Berger–Breuil [3], Berger [2], Breuil–Emerton [6] and [22] ss describe explicitly the possibilities for , see Proposition 6.13. These results and the Theorem imply that Ext1G (π2 , π1 ) vanishes in many cases. Let us make this more precise. Let Modsm G (O) be the category of smooth G-representation on O-torsion modules. It contains Modsm G (k), the category of smooth G-representations on k-vector spaces, as a full subcategory. Every irreducible object π of Modsm G (O) is killed by $ , and hence is an object of Modsm (k). Barthel–Livné [1] and G Breuil [4] have classified the absolutely irreducible smooth representations π admitting a central character. They fall into four disjoint classes: (i) (ii) (iii) (iv)

characters δ ◦ det; special series Sp ⊗δ ◦ det; principal series (IndG B δ1 ⊗ δ2 )sm , δ1 = δ2 ; supersingular representations;

where Sp is the Steinberg representation, that is the locally constant func× tions from P1 (Q p ) to k modulo the constant functions; δ, δ1 , δ2 : Q× p → k are smooth characters and we consider δ1 ⊗ δ2 as a character of B, which sends a0 db to δ1 (a)δ2 (d). Using their results and some easy arguments, see [25, §5.3], one may show that for an irreducible smooth representation π the following are equivalent: (1) π is admissible, which means that π H is finite dimensional for all open subgroups H of G; (2) EndG (π ) is finite dimensional over k; (3) there exists a finite extension k of k, such that π ⊗k k is isomorphic to a finite direct sum of distinct absolutely irreducible k -representations with a central character. Let Modl.adm (O) be the full subcategory of Modsm G G (O), consisting of representations, which are equal to the union of their admissible subrepresentations. l.adm The categories Modsm (O) are abelian, see [15, Prop.2.2.18]. G (O) and ModG l.adm We define ModG (k) in exactly the same way with O replaced by k. Let l.adm (O), then Irradm Irradm G be the set of irreducible representations in ModG G is sm the set of irreducible representations in ModG (O) satisfying the equivalent conditions described above. We define an equivalence relation ∼ on Irradm G : π ∼ τ , if there exists a sequence of irreducible admissible representations π = π1 , π2 , . . . , πn = τ , such that for each i one of the following holds: (1) πi ∼ = πi +1 ; (2) Ext1G (πi , πi+1 ) = 0; (3) Ext1G (πi +1 , πi ) = 0. We note that it does not matter for the definition of ∼, whether we compute Ext1G sm l.adm in Modsm (O) or Modl.adm (k), since we only care G (O), ModG (k), ModG G 1 about vanishing or non-vanishing of ExtG (πi , πi +1 ) for distinct irreducible representations. A block is an equivalence class of ∼.

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Corollary 6.2. The blocks containing an absolutely irreducible representation are given by the following: (i) B = {π } with π supersingular; −1 G −1 −1 (ii) B = {(IndG B δ1 ⊗ δ2 ω )sm , (Ind B δ2 ⊗ δ1 ω )sm } with δ2 δ1 ±1 ω , 1; −1 (iii) p > 2 and B = {(IndG B δ ⊗ δω )sm }; (iv) p = 2 and B = {1, Sp} ⊗ δ ◦ det; (v) p ≥ 5 and B = {1, Sp, (IndGB ω ⊗ ω−1 )sm } ⊗ δ ◦ det; (vi) p = 3 and B = {1, Sp, ω ◦ det, Sp ⊗ω ◦ det} ⊗ δ ◦ det;

=

× × × where δ, δ1 , δ2 : Q× p → k are smooth characters and where ω : Q p → k is the character ω(x) = x|x| (mod $ ).

One may view the cases (iii) to (vi) as degenerations of case (ii). A finitely generated smooth admissible representation of G is of finite length, [15, Thm.2.3.8]. This makes Modl.adm (O) into a locally finite category. It follows G from [17] that every locally finite category decomposes into blocks. In our situation we obtain: ∼ Modl.adm (O) Modl.adm (O)[B], (6.1) = G G B∈Irradm G /∼

where Modl.adm (O)[B] is the full subcategory of Modl.adm (O) consisting of G G representations, with all irreducible subquotients in B. One can deduce a similar result for the category of admissible unitary L-Banach space representations of G, see [25, Prop.5.32]. The result has been previously known for p > 2. Breuil and the author [7, §8], Colmez [8, §VII], Emerton [16, §4] and the author [23] have computed Ext1G (π2 , π1 ) by different characteristic p methods, which do not work in the exceptional cases, when p = 2. In this paper, we go via characteristic 0 and make use of a deep Theorem of Berger–Breuil. The proof is less involved, but it does not give any information about the extensions between irreducible representations lying in the same block. The motivation for these calculations comes from the p-adic Langlands correspondence for GL2 (Q p ). Colmez in [8] to a 2-dimensional absolutely irreducible L-representation of the absolute Galois group of Q p has associated an admissible unitary absolutely irreducible non-ordinary L-Banach space representation of G. He showed that his construction induces an injection on the isomorphism classes and asked whether it is a bijection, see [8, §0.13]. This has been answered affirmatively in [25] for p ≥ 5, where the knowledge of

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blocks has been used in an essential way. The results of this chapter should be useful in dealing with the remaining cases. Let us give a rough sketch of the argument. Let 0 → π1 → π → π2 → 0 be a non-split extension. The method of [7] allows us to embed π into , such that | K is admissible and an injective object in Modsm K (k), where K = GL2 (Z p ). Using the results of [24] we may lift to an admissible unitary L-Banach space representation E of G, in the sense that we may find a G-invariant unit ball E 0 in E, such that E 0 /$ E 0 ∼ = . Moreover, E| K is isomorphic to a direct summand of C(K , L)⊕r , where C(K , L) is the space of continuous function with the supremum norm. This implies, using an argument of Emerton, that the K -algebraic vectors are dense in E. As a consequence we find a closed G-invariant subspace of E, such that the reduction of ∩ E 0 modulo $ contains π as a subrepresentation, and con˜ c-IndG K Z 1i (T −ai )ni

⊗ detli ⊗ Symki −1 L 2 as a dense subrepresentation, where Z is the centre of G, 1˜ i : K Z → L × is a character, trivial on K , ai ∈ L, and ˜ c-IndG K Z 1i T is a certain Hecke operator in EndG (c-IndG 1˜ i ), such that is an

m tains ⊕i=1

KZ

(T −ai )

unramified principal series representation. Once we have this we are in a good shape to prove Theorem 6.1. Acknowledgements. I thank the anonymous referee for the comments, which led to an improvement of the exposition, and Jochen Heinloth for a stimulating discussion.

2. Notation Let L be a finite extension of Q p with the ring of integers O, uniformizer $ and residue field k. We normalize the valuation val on L so that val( p) = 1, and the norm | |, so that |x| = p − val(x) , for all x ∈ L. Let G = GL2 (Q p ); Z the centre of G; B the of upper triangular matrices; subgroup K = GL2 (Z p ); I = {g ∈ K : g ≡ ∗0 ∗∗ (mod p)}; I1 = {g ∈ K : g ≡ 10 ∗1 (mod p)}; let 0 × K be the G-normalizer of I ; let H = { [λ] 0 [μ] : λ, μ ∈ F p }, where [λ] is the Teichmüller lift of λ; let G be the subgroup of G generated by matrices 0p 0p , 0 1 + p 0 and H . Let G = {g ∈ G : val(det(g)) ≡ 0 (mod 2)}. Since we are working with representations of locally pro- p groups in characteristic p, these representations will not be semi-simple in general; socle is the maximal semisimple subobject. So for example, socG τ means the maximal semi-simple G-subrepresentation of τ . Let Banadm G (L) be the category of admissible unitary L-Banach space representations of G, studied in [26]. This category is abelian. Let be an admissible unitary L-Banach space representation of

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G, and let % be an open bounded G-invariant lattice in , then %/$ % is a smooth admissible k-representation of G. If %/$ % is of finite length as a ss G-representation, then we let be the semi-simplification of %/$ %. Since ss any two such %’s are commensurable, is independent of the choice of %. Universal unitary completions are discussed in [11, §1].

3. Main Let π1 , π2 be distinct smooth absolutely irreducible k-representations of G with a central character. It follows from [1] and [4] that π1 and π2 are admissible. We suppose that there exists a non-split extension in Modsm G (O): 0 → π1 → π → π2 → 0.

(6.2)

Since π1 and π2 are distinct and irreducible, by examining the long exact sequence induced by multiplication with $ , we deduce that π is killed by $ . A similar argument shows that the existence of a non-split extension implies that the central character of π1 is equal to the central character of π2 . Moreover, π also has a central character, which is then equal to the central character of π1 , see [23, Prop.8.1]. We denote this central character by ζ : Z → k × . After replacing L by a quadratic extension and twisting by a character we may assume that ζ ( 0p 0p ) = 1. Lemma 6.3. If π1I1 = π I1 then Theorem 6.1 holds for π1 and π2 . Proof. Since ζ is continuous, it is trivial on the pro- p group Z ∩ I1 . We thus may extend ζ to Z I1 , by letting ζ (zu) = ζ (z) for all z ∈ Z , u ∈ I1 . If τ is a smooth k-representation of G with a central character ζ then I1 is naturally an τ I1 ∼ = Hom I1 Z (ζ, τ ) ∼ = HomG (c-IndG K Z ζ, τ ). Thus τ G H := EndG (c-Ind I1 Z ζ ) module. Taking I1 -invariants of (6.2) we get an exact sequence of H-modules: 0 → π1I1 → π I1 → π2I1 .

(6.3)

Since π2 is irreducible, π2I1 is an irreducible H-module by [27]. Hence, if π1I1 = π I1 , then the last arrow is surjective. It is shown in [20], that if τ is a smooth k-representation of G, with a central character ζ , generated as a G-representation by its I1 -invariants, then the natural map τ I1 ⊗H c-IndG K Z ζ → τ is an isomorphism. This implies that the sequence 0 → π1I1 → π I1 → π2I1 → 0 is non-split, and hence defines a non-zero element of I Ext1H (π2I1 , π1I1 ). Since πi ∼ = πi 1 ⊗H c-IndG K Z ζ for i = 1, 2, the H-modules I1 I1 π1 and π2 are non-isomorphic. Non-vanishing of Ext1H (π2I1 , π1I1 ) implies

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that there exists a smooth character η : G → k × such that either (π1 ∼ = η and π2 ∼ = Sp ⊗η) or (π2 ∼ = η and π1 ∼ = Sp ⊗η), [25, Lem.5.24], where Sp is the Steinberg representation. In both cases the universal unitary completion of −1 (IndG ˜ where η˜ : G → O× is any smooth character lifting B | | ⊗ | | )sm ⊗ η, η, will satisfy the conditions of Theorem 6.1 by [13, 5.3.18]. Lemma 6.3 allows to assume that π I1 = π1I1 . We note that this implies that soc K π1 ∼ = soc K π , and, since I1 is contained in G + , the restriction of (6.2) to + G is a non-split extension of G + -representations. Now we perform a renaming trick, the purpose of which is to get around some technical issues, when p = 2. If either p > 2 or p = 2 and π1 is neither a special series nor a character then we let τ1 = π1 , τ = π and τ2 = π2 . If p = 2 and π1 is either a special series representation or a character, then we let 0 → τ1 → τ → τ2 → 0 be the exact sequence obtained by tensoring (6.2) with IndG 1. In particular, τ ∼ 1, which implies = π ⊗ IndG G+ G+ ∼ ∼ that τ |G + = π |G + ⊕ π |G + and τ1 |G + = π1 |G + ⊕ π1 |G + . Hence, τ I1 = τ1I1 and soc K τ ∼ = soc K τ1 ∼ = soc K π1 ⊕ soc K π1 . This implies that socG τ ∼ = socG τ1 . Lemma 6.4. socG τ ∼ = socG τ1 ∼ = π1 . Proof. We already know that socG τ ∼ = socG τ1 and we only need to consider the case p = 2 and π1 is either a special series or a character. The assumption on π1 implies that π1I1 is one dimensional. Let K be the normalizer of I1 in G, then I1 Z is a subgroup of K of index 2. We note that I = I1 as p = 2. Thus K acts on π1I1 by a character χ , such that the restriction of χ to I1 Z is equal to ζ . Since p = 2, we have an exact non-split sequence of G-representations 0 → 1 → IndG 1 → 1 → 0. We note that G + and hence Z I1 act trivially on all G+ the terms in this sequence. By tensoring with π1 we obtain an exact sequence 0 → π1 → τ1 → π1 → 0 of G-representations. Taking I1 -invariants, gives us I I an isomorphism of K-representations τ1 1 ∼ = π1 1 ⊗IndK Z I1 1. This representation is a non-split extension of χ by itself. Thus τ1 is a non-split extension of π1 by itself. Hence, socG τ1 ∼ = π1 . If p = 2 then τ1I1 is 2-dimensional and has a basis of the form {v, 0p 01 v}: if π1 is either a character or special series, this follows from the isomorphism I τ1I1 ∼ = π1 1 ⊗IndK Z I1 1, otherwise τ1 = π1 and the assertion follows from [7, Cor. 6.4 (i)] noting that the work of Bartel–Livné [1] and Breuil [4] on classification of irreducible representations of G implies that π I1 is isomorphic as a module of the pro- p Iwahori–Hecke algebra to M(r, λ, η) defined in [7, Def.6.2]. Since τ I1 = τ1I1 , [7, Prop.9.2] implies that the inclusion τ I1 → τ has a G-equivariant section.

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Proposition 6.5. There exists a G-equivariant injection τ → , where is a smooth k-representation of G, such that | K is an injective envelope of soc K τ p 0 K I1 ∼ in Modsm K (k), 0 p acts trivially on and |K = IndG . Proof. The existence of satisfying the first two conditions follows from [7, Cor.9.11]. The 0last condition is satisfied as a byproduct of the construction of 1 the action of p 0 in [7, Lem.9.6]. Corollary 6.6. Let be as above then soc K ∼ = soc K τ1 and socG ∼ = π1 . Proof. Since τ is a subrepresentation of , soc K τ is contained in soc K . Since | K is an injective envelope of soc K τ , every non-zero K -invariant subspace of intersects soc K τ non-trivially. This implies that soc K τ ∼ = soc K . ∼ This implies the first assertion, as soc K τ = soc K τ1 . Moreover, every G-invariant non-zero subspace of intersects τ non-trivially, since those are also K -invariant. This implies socG ∼ = socG τ ∼ = π1 , where the last isomorphism follows from Lemma 6.4. Lemma 6.7. Let κ be a finite dimensional k-representation of G on which p 0 0 p acts trivially. There exists an admissible unitary L-Banach space repre sentation (E, ) of K, such that E ⊂ |L|, 0p 0p acts trivially on E, and the reduction modulo $ of the unit ball in E is isomorphic to (IndK G κ)sm as a K-representation.

Proof. It is enough to prove the statement, when κ is indecomposable, which we now assume. Let p Z be the subgroup of G generated by 0p 0p . Since the order of H is prime to p, and H has index 2 in G/ pZ , κ is either a character or an induction of a character from H to G/ p Z . In both cases we may lift κ to a representation κ˜ 0 of G/ p Z on a free O-module of rank 1 or rank 2 respectively. Let κ˜ = κ˜ 0 ⊗O L and let be the gauge of κ˜ 0 . Then is G-invariant K/ pZ

and κ˜ 0 is the unit ball with respect to . Then (IndG / p Z κ) ˜ cont with the norm K/ p Z

f 1 := supg∈K/ p Z f (g) is a lift of (IndG / pZ κ)sm , where the subscript cont indicates continuous induction: the space of continuous functions with the right transformation property. Theorem 6.8. Let be any representation given by Proposition 6.5. Then there exists an admissibleunitary L-Banach space representation (E, ) of G, such that E ⊂ |L|, 0p 0p acts trivially on E, and the reduction modulo $ of the unit ball in E is isomorphic to as a G-representation. Proof. If p = 2 this is shown in [24, Thm.6.1]. We will observe that the renaming trick allows us to carry out essentially the same proof when p = 2. We make no assumption on p.

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We first lift | K to characteristic 0. Let σ be the K -socle of . Pontryagin duality induces an anti-equivalence of categories between Modsm K (k) and the pro.aug category of pseudocompact kK -modules, which we denote by Mod K (k). ∨ is a Since is an injective envelope of σ in Modsm (k), its Pontryagin dual K pro.aug σ ∨ be a projective envelope projective envelope of σ ∨ in Mod K (k). Let P σ ∨ /$ P σ ∨ is of σ ∨ in the category of pseudocompact OK -modules. Then P pro.aug ∨ a projective envelope of σ in Mod K (k). Since projective envelopes are σ ∨ /$ P σ ∨ . Since τ1 is admissiunique up to isomorphism, we obtain ∨ ∼ =P ∼ ble and σ = soc K τ1 by Corollary 6.6, σ is a finite dimensional k-vector space. In particular, σ ∨ is a finitely generated OK -module, and so there exists a surjection of OK -modules OK ⊕r σ ∨ . Since OK ⊕r is projective, and σ ∨ σ ∨ is essential, the surjection factors through OK ⊕r P σ ∨ , and so P Pσ ∨ is a finitely generated OK -module. Since Pσ ∨ is projective, we deduce that it is a direct summand of OK ⊕r , and hence it is O-torsion free. σ ∨ is an O-torsion free, finitely generated OK -module, and its Thus P pro.aug reduction modulo $ is isomorphic to ∨ in Mod K (k). Let E 0 = σ ∨ , L), and let 0 be the supremum norm. It follows from [26] Homcont ( P O that E 0 is an admissible unitary L-Banach space representation of K . More∨ over, the unit ball E 00 in E 0 is Homcont O ( Pσ , O) and cont cont ∼ ∼ Homcont O ( Pσ ∨ , O) ⊗O k = HomO ( Pσ ∨ , k) = Homk (Pσ ∨ , k) ∨ ∨ ∼ = ( ) ∼ = ,

see [24, §5] for details. We extend the action of K on E 0 to the action of K Z by letting 0p 0p act trivially. Since σ is finite dimensional, it follows from [21, Lem.6.2.4] that I1 is a I1 finite dimensional k-vector space. Since |K ∼ = (IndK G )sm by Proposition 6.5, Lemma 6.7 implies that there exists a unitary L-Banach space representation (E 1 , 1 ) of K, such that E 1 ⊆ |L|, 0p 0p acts trivially on E 1 and the reduction of the unit ball E 10 in E 1 modulo $ is isomorphic to |K . We claim that there exists an isometric, I Z -equivariant isomorphism ϕ : E 1 → E 0 such that the following diagram of I Z -representations: E 10 /$ E 10

ϕ

∼ =

/ E 0 /$ E 0 0 0

mod $

(6.4)

∼ =

id

/

commutes, where the left vertical arrow is the given K-equivariant isomorphism E 10 /$ E 10 ∼ = |K and the right vertical arrow is the given K Z equivariant isomorphism E 00 /$ E 00 ∼ = | K Z . Granting the claim, we may

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transport the action of K on E 0 by using ϕ to obtain a unitary action of K Z and K on E 0 , such that the two actions agree on K Z ∩ K, which is equal to I Z . The resulting action glues to the unitary action of G on E 0 , see [21, Cor.5.5.5], which is stated for smooth representations, but the proof of which works for any representation. The commutativity of the above diagram implies that E 00 ⊗O k ∼ = as a G-representation. 0 We will prove the claim now. Let M = Homcont O (E 1 , O) equipped with the pro.aug topology of pointwise convergence. Then M is an object of Mod I (O), pro.aug and M ⊗O k ∼ (k), see [24, Lem.5.4]. Since | K is injective = ∨ in Mod I sm in Modsm K (k), | I is injective in Mod I (k). Since I1 is a pro- p group, every non-zero I -invariant subspace of intersects I1 non-trivially. Thus | I is ∨ an injective envelope of I1 in Modsm I (k). Hence, is a projective envelope pro.aug I ∨ 1 of ( ) in Mod I (k). Since M is O-torsion free, and M ⊗O k is a propro.aug jective envelope of ( I1 )∨ in Mod I (k), [24, Prop.4.6] implies that M is pro.aug σ ∨ . a projective envelope of ( I1 )∨ in Mod I (O). The same holds for P Since projective envelopes are unique up to isomorphism, there exists an iso∼ = pro.aug σ ∨ → morphism ψ : P M in Mod I (O). It follows from [24, Cor.4.7] that σ ∨ ) → Autk I ( P σ ∨ /$ P σ ∨ ) is surjective. Using the natural map AutO I ( P pro.aug

this we may choose ψ so that the following diagram in Mod K σ ∨ /$ P σ ∨ P

ψ

∼ =

∨

/ M/$ M

mod $

(k):

(6.5)

∼ =

id

/ ∨

commutes. Dually we obtain an isometric I -equivariant isomorphism of unitary L-Banach space representations of I , ψ d : Homcont O (M, L) → σ ∨ , L). It follows from [26, Thm.1.2] that (E 1 , 1 ) is natHomcont ( P O urally and isometrically isomorphic to Homcont O (M, L) with the supremum norm. This gives our ϕ. The commutativity of (6.5) implies the commutativity of (6.4). Corollary 6.9. The Banach space representation (E, ) constructed in Theorem 6.8 is isometrically, K -equivariantly isomorphic to a direct summand of C(K , L)⊕r , where C(K , L) is the space of continuous functions from K to L, equipped with the supremum norm, and r is a positive integer. Proof. It follows from the construction of E, that (E, ) is isometrically, ∨ K -equivariantly isomorphic to Homcont O ( Pσ , L) with the supremum norm. σ ∨ is a direct sumMoreover, it follows from the proof of Theorem 6.8 that P mand of OK ⊕r . It is shown in [26, Lem.2.1, Cor.2.2] that the natural map

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K → OK , g → g induces an isometrical, K -equivariant isomorphism between C(K , L) and Homcont O (OK , L). If F is a finite extension of Q p then exactly the same proof works. We note that [7, Thm.9.8] is proved for GL2 (F). We record this as a corollary below. Let O F be the ring of integers of F, $ F a uniformizer, k F the let residue field, G F be the subgroup of GL2 (F) generated by the matrices $0F $0F , $0F 01 0 × and [λ] 0 [μ] , for λ, μ ∈ k F , where [λ] is the Teichmüller lift of λ. Let I1 be the standard pro- p Iwahori subgroup of G. Corollary 6.10. Let τ be an admissible smooth k-representation of GL2 (F), such that $0F $0F acts trivially on τ and if p = 2 assume that the inclusion τ I1 → τ has a G F -equivariant section. Then there exists a GL2 (F)-equivariant embedding τ → , such that |GL2 (O F ) is an injective envelope of GL2 (O F )-socle of τ in the category of smooth k-representations of GL2 (O F ) and $0F $0F acts trivially on . Moreover, we may lift to an admissible unitary L-Banach space representation of GL2 (F). Remark 6.11. We also note that one could work with a fixed central character throughout. Let Vl,k = detl ⊗ Symk−1 L 2 , for k ∈ N and l ∈ Z. Rather unfortunately k also denotes the residue field of L, we hope that this will not cause any confusion. Proposition 6.12. Let (E, ) be a unitary L-Banach space representation of K isomorphic in the category of unitary admissible L-Banach space representations of K to a direct summand of C(K , L)⊕r . The evaluation map # Hom K (Vl,k , E) ⊗ Vl,k → E (6.6) (l,k)∈Z×N

is injective and the image is a dense subspace of E. Moreover, the subspaces Hom K (Vl,k , E) are finite dimensional. Proof. The argument is the same as given in the proof of [14, Prop.5.4.1]. We have provided some details in the Appendix at the request of the referee. It is enough to prove the statement for C(K , L), since then it is true for C(K , L)⊕r and by applying the idempotent, which cuts out E, we may deduce the same statement for E. In the case E = C(K , L), the assertion follows from Proposition 6.A.17 applied to G = GL2 . We note that every rational irreducible representation of GL2 /L is isomorphic to Vl,k for a unique pair (l, k) ∈ Z×N. The last assertion follows from (6.A.13) below.

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−1 Proposition 6.13. Let ρ = (IndG B χ1 ⊗ χ2 | | )sm be a smooth principal × series representation of G, where χ1 , χ2 : Q p → L × smooth characters with χ1 || = χ2 . Let be the universal unitary completion of ρ ⊗Vl,k . Then is an admissible, finite length L-Banach space representation of G. Moreover, if ss is non-zero and we let be the semi-simplification of the reduction modulo ss $ of an open bounded G-invariant lattice in , then either is irreducible supersingular, or

ss

−1 ss G −1 ss ⊆ (IndG B δ1 ⊗ δ2 ω )sm ⊕ (Ind B δ2 ⊗ δ1 ω )sm ,

(6.7)

× for some smooth characters δ1 , δ2 : Q× p → k , where the superscript ss indicates the semi-simplification.

Proof. If = 0 then −(k + l) ≤ val(χ1 ( p)) ≤ −l, −(k + l) ≤ val(χ2 ( p)) ≤ −l and val(χ1 ( p)) + val(χ2 ( p)) = −(k + 2l), [24, Lem.7.9], [11, Lem.2.1]. If both inequalities are strict and χ1 = χ2 then it is shown in [3, §5.3] that is ss non-zero, admissible and absolutely irreducible. The assertion about then follows from [2]. If both inequalities are strict, χ1 = χ2 and is non-zero it is shown in [22, Prop.4.2] that there exist O-lattices M in ρ ⊗ Vl,k and M in ρ ⊗ Vl,k , −1 where ρ = (IndG B χ1 ⊗ χ2 | | )sm for some distinct smooth characters, × × χ1 , χ2 : Q p → L congruent to χ1 , χ2 modulo 1 + ($ ), such that both lattices are finitely generated O[G]-modules and their reductions modulo $ are isomorphic. Since M is O-torsion free, the completion of ρ ⊗ Vl,k with respect to the gauge of M is non-zero, and since M is a finitely generated O[G]module, the completion is the universal unitary completion, [11, Prop.1.17], thus is isomorphic to . Let 0 be the unit ball in with respect to the gauge of M. Then 0 /$ 0 ∼ = M/$ M ∼ = M /$ M . Now by the same argument the completion of ρ ⊗ Vl,k with respect to the gauge of M is the universal unitary completion of of ρ ⊗ Vl,k . Since χ1 = χ2 we may apply the results of Berger–Breuil [3] to conclude that the semi-simplification of M /$ M has the desired form. Suppose that either val(χ1 ( p)) = −l or val(χ2 ( p)) = −l. If χ1 = χ2 | | then this forces k = 1, so that Vl,k is a character and ρ ⊗ Vl,k ∼ = −1 ) × is a unitary character. It fol(IndG | | ⊗ | | ⊗ η, where η : G → L sm B lows from [13, Lem.5.3.18] that the universal unitary completion of ρ ⊗ Vl,k ss ss is admissible and of length 2. Moreover, ∼ = η ⊕ Sp ⊗η ∼ = (IndG B η ⊗ η)sm . If χ1 = χ2 | | then it follows from [6, Lem.2.2.1] that the universal unitary completion of ρ ⊗ Vl,k is isomorphic to a continuous induction of a unitary ss character. Hence is isomorphic to the semi-simplification of a principal series representation.

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Proof of Theorem 6.1. Let (E, ) be the unitary L-Banach space representation of G constructed in the proof of Theorem 6.8. Let E 0 be the unit ball in E, then by construction we have E 0 /$ E 0 ∼ = , where is a smooth k-representation of G, satisfying the conditions of Proposition 6.5. Let V = ⊕ Hom K (Vl,k , E)⊗Vl,k , where the sum is taken over all (l, k) ∈ Z×N. It follows from Corollary 6.9 and Proposition 6.12 that the natural map V → E is injective and the image is dense. Let {V i }i≥0 be any increasing, exhaustive filtration of V by finite dimensional K -invariant subspaces. Then V i ∩ E 0 i is a K -invariant O-lattice in V i , and we denote by V its reduction modulo $ . It follows from [24, Lem.5.5] that the reduction modulo $ induces a i i K -equivariant injection V → . The density of V in E implies that {V }i≥0 is an increasing, exhaustive filtration of by finite dimensional, K -invariant subspaces. Recall that contains τ as a subrepresentation, see Proposition 6.5. Now τ is finitely generated as a G-representation, since it is of finite length. Thus we may conclude, that there exists a finite dimensional K -invariant subspace W of V , such that τ is contained in the G-subrepresentation of generated by W . Let ϕ : Vl,k → E be a non-zero K -equivariant, L-linear homomorphism. Let R(ϕ) be the G-subrepresentation of E in the category of (abstract) G-representations on L-vector spaces, generated by the image of ϕ. Frobe˜ nius reciprocity gives us a surjection c-IndG R(ϕ), where KZ 1 ⊗ V l,k ˜1 : K Z → L × is an unramified character, such that p 0 acts trivially on 0 p ˜ Now EndG (c-IndG 1) ˜ is isomorphic to the ring of polynomials over L Vl,k ⊗1. KZ

in one variable T . It follows from the proof of [24, Cor.7.4] that the surjection c-IndG 1˜

factors through P(TK )Z ⊗ Vl,k R(ϕ), for some non-zero P(T ) ∈ L[T ]. Let R be the (abstract) G-subrepresentation of E generated by W , and let be the closure of R in E. Since W is isomorphic to a finite direct sum of Vl,k ’s, we deduce that if we replace L by a finite extension there exists a surjection: m ˜ # c-IndG K Z 1i ⊗ Vli ,ki R, n (T − ai ) i

(6.8)

i=1

c-IndG 1˜

KZ i for some ai ∈ L, n i ∈ N and (li , ki ) ∈ Z × N. Let ρi = T −a , then i using (6.8) we may construct a finite, increasing, exhaustive filtration {R j } j≥0 of R by G-invariant subspaces, such that for each j there exists a surjection ρi ⊗ Vli ,ki R j /R j −1 , for some 1 ≤ i ≤ m. Moreover, by choosing n i and m in (6.8) to be minimal, we may assume that HomG (ρi ⊗ Vli ,ki , R) is non-zero for all 1 ≤ i ≤ m. Let j be the closure of R j in E. We note that since E is admissible, j is an admissible unitary L-Banach space representation of

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j j G, moreover the category Banadm G (L) is abelian. Since R is dense in , its j j −1 image is dense in / . Hence, for each j there exists a G-equivariant map ϕ j : ρi ⊗ Vli ,ki → j / j −1 with a dense image. Let i be the universal unitary completion of ρi ⊗Vli ,ki . Since the target of ϕ j is unitary, we can extend it to a continuous G-equivariant map ϕ˜ j : i → j / j −1 . Moreover, since the target of ϕ j is admissible and the image is dense, ϕ˜ j is surjective. For each closed subspace U of E, we let U be the reduction of (U ∩ E 0 ) modulo $ . It follows from [24, Lem.5.5] that the reduction modulo $ induces an injection U → . Since contains W , will contain W . Since j is G-invariant, it will contain τ . Now { } j≥0 defines a finite, increasing, exhaustive filtration of by G-invariant subspaces. Since π2 is an irreducible subquotient of τ , there exists j , such that π2 is an irreducible subquotient of j j−1 / . Each representation ρi is an unramified principal series representation, considered in Proposition 6.13, see [5, Prop.3.2.1]. Hence, i is an admissible, ss finite length L-Banach space representation of G, moreover i is of finite length as described in Proposition 6.13. The surjection ϕ˜ j : i j / j−1 ss ss induces a surjection i ( j / j −1 ) . It follows from [24, Lem.5.5] that ss j j −1 the semi-simplification of / is isomorphic to ( j / j −1 ) . Thus π2 ss is a subquotient of i . Since HomG (ρi ⊗ Vli ,ki , ) is non-zero, there exists a non-zero continuous G-invariant homomorphism ϕ : i → . Let be the image of ϕ. Since i and are admissible, we have a surjection i and an injection → ss in the abelian category Banadm G (L). The surjection induces a surjection i ss . The injection induces an injection → → . Since socG ∼ = π1 ∼ by Corollary 6.6 and is non-zero, we deduce that π1 = socG . Hence, π1 ss is a subquotient of i .

Lemma 6.14. Let κ and λ be smooth k-representations of G and let l be a finite extension of k. Then ExtiG (κ, λ) ⊗k l ∼ = ExtiG (κ ⊗k l, λ ⊗k l), for all i ≥ 0, sm where the Ext groups are computed in Modsm G (k) and ModG (l), respectively. Proof. The assertion for i = 0 follows from [25, Lem.5.1]. Hence, it is enough to find an injective resolution of λ in Modsm G (k), which remains injective after G tensoring with l. Such resolution may be obtained by considering (Ind{1} V )sm , where {1} is the trivial subgroup of G and V is a k-vector space. We note that G G (Ind{1} V )sm ⊗k l ∼ V ⊗k l)sm , since l is finite over k. = (Ind{1} Proof of Corollary 6.2. Lemma 6.14 implies that replacing L by a finite extension does not change the blocks. It follows from Proposition 6.13 and Theorem 6.1 that an irreducible supersingular representation is in a block on

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its own. Let π{δ1 , δ2 } be the semi-simple representation defined by (6.7), × are smooth characters. We have to show that where δ1 , δ2 : Q× p → k all irreducible subquotients of π {δ1 , δ2 } lie in the same block. We adopt an argument used in [8]. It follows from [5, 5.3.3.1, 5.3.3.2, 5.3.4.1] that there exists an irreducible unitary L-Banach space representation of G, such that ss ∼ = π{δ1 , δ2 }, then [8, Prop.VII.4.5(i)] asserts that we may choose an open bounded G-invariant lattice % in such that %/$ % is indecomposable. It follows from (6.1) that all the irreducible subquotients of %/$ % lie in the same block. We will list explicitly the irreducible subquotients of π {δ1 , δ2 }. It is shown −1 in [1] that if δ2 δ1−1 = ω then (IndG B δ1 ⊗ δ2 ω )sm is absolutely irreducible, and there exists a non-split exact sequence 0 → δ1 ◦ det → (IndGB δ1 ⊗ δ2 ω−1 )sm → Sp ⊗δ1 ◦ det → 0

(6.9)

δ2 δ1−1

if = ω. Taking this into account there are the following possibilities for decomposing π {δ1 , δ2 } into irreducible direct summands depending on δ1 , δ2 and p: (i) If δ2 δ1−1 = ω±1 , 1 then −1 G −1 π {δ1 , δ2 } ∼ = (IndG B δ1 ⊗ δ2 ω )sm ⊕ (Ind B δ2 ⊗ δ1 ω )sm ;

(ii) if δ2 = δ1 (= δ) then (a) if p > 2 then π {δ, δ} ∼ = (IndGB δ ⊗ δω−1 )⊕2 sm ; (b) if p = 2 then π {δ, δ} ∼ = (Sp⊕2 ⊕1⊕2 ) ⊗ δ ◦ det. (iii) if δ2 δ1−1 = ω±1 then −1 (a) if p ≥ 5 then π{δ1 , δ2 } ∼ = (1 ⊕ Sp ⊕(IndG B ω ⊗ ω )sm ) ⊗ δ ◦ det; (b) if p = 3 then π {δ1 , δ2 } ∼ = (1 ⊕ Sp ⊕ω ◦ det ⊕ Sp ⊗ω ◦ det) ⊗ δ ◦ det; (c) if p = 2 then we are in the case (ii)(b), where δ is either δ1 or δ2 . Finally, we note that in the case (ii)(b) instead of using [5, 5.3.3.2], which is stated without proof, we could have observed that since (6.9) is non-split, Sp ⊗δ1 ◦ det and δ1 ◦ det lie in the same block.

Appendix A. Density of algebraic vectors Let X be an affine scheme of finite type over Z p and let A = (X, O X ). By choosing an isomorphism A ∼ = Z p [x1 , . . . , xn ]/( f 1 , . . . , f m ) we may identify the X (Z p ) with a closed subset of Znp . The induced topology on X (Z p ) is independent of a choice of the isomorphism, see [9, Prop.2.1]. Let C(X (Z p ), L) be the space of continuous functions from X (Z p ) to L. Since X (Z p ) is compact, C(X (Z p ), L) equipped with the supremum norm

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is an L-Banach space. Recall that X (Z p ) = HomZ p −alg (A, Z p ). We denote by C alg (X (Z p ), L) the functions f : X (Z p ) → L, which are obtained by evaluating elements of A ⊗Z p L at Z p -valued points of X . Lemma 6.A.15. C alg (X (Z p ), L) is a dense subspace of C(X (Z p ), L). Proof. We first look at the special case, when X = An , so that A = Z p [x1 , . . . , xn ] and X (Z p ) = Znp . Since addition and multiplication in Z p are continuous functions, we deduce that C alg (A(Z p ), L) is a subspace of C(A(Z p ), L). The density follows from the theory of Mahler expansions, see for example [19, III.1.2.4]. In the general case, we choose an isomorphism A ∼ = Z p [x1 , . . . , xn ]/( f 1 , . . . , f m ) and identify X (Z p ) with a closed subset of An (Z p ) = Znp . The restriction of functions to X (Z p ) induces a surjective map r : C(An (Z p ), L) → C(X (Z p ), L), see for example [10, Thm.3.1(1)]. Since C alg (An (Z p ), L) is dense in C(An (Z p ), L) and supx∈X (Z p ) |r ( f )(x)| ≤ supx∈An (Z p ) | f (x)| for all f ∈ C(An (Z p ), L) we deduce that r (C alg (An (Z p ), L)) is a dense subspace. Since it is equal to C alg (X (Z p ), L) we are done. Remark 6.A.16. If X is an affine scheme of finite type over O F , where O F is a ring of integers in a finite field extension F over Q p , then there are two ways to topologize X (O F ): as O F points of X and as Z p -points of the Weil restriction of X to Z p . However, they coincide, see [9, Ex.2.4]. Proposition 6.A.17. Let G be an affine group scheme of finite type over Z p such that G L is a split connected reductive group over L. Then the evaluation map # HomG(Z p ) (V , C(G(Z p ), L)) ⊗ V → C(G(Z p ), L), (6.A.10) [V ]

where the sum is taken over all the isomorphism classes of irreducible rational representations of G L , is injective and the image is equal to C alg (G(Z p ), L). In particular, the image of (6.A.10) is a dense subspace of C(G(Z p ), L). Proof. The category of rational representations of G L is semi-simple as L is of characteristic 0, see [18, II.5.6 (6)]. Hence, the regular representation O(G L ) decomposes into a direct sum of irreducible representations. Since we have assumed that G L is split, every irreducible rational representation V of G L is absolutely irreducible [18, II.2.9]. This implies that EndG L (V ) = L for every irreducible representation V . It follows from Frobenius reciprocity [18, I.3.7 (3)] and the semi-simplicity of O(G L ) that we have an isomorphism of G L -representations: # V∗ ⊗ V (6.A.11) O(G L ) ∼ = [V ]

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where the G L -action on V ∗ is trivial. The isomorphism (6.A.11) is G(L)equivariant, and hence G(Z p )-equivariant, which gives us an isomorphism of G(Z p )-representations: # C alg (G(Z p ), L) ∼ V ∗ ⊗ V. (6.A.12) = [V ]

The map ϕ → [v → ϕ(v)(1)] induces an isomorphism HomG(Z p ) (V, C(G(Z p ), L)) ∼ = V ∗,

(6.A.13)

with the inverse map given by → [v → [g → (gv)]]. Since every V is a finite dimensional L-vector space, we conclude from (6.A.12), (6.A.13) that the injection HomG(Z p ) (V , C alg (G(Z p ), L)) → HomG(Z p ) (V, C(G(Z p ), L)) (6.A.14) is an isomorphism. Moreover, as a byproduct we obtain that EndG(Z p ) (V ) = L and HomG(Z p ) (V , W ) = 0, if V and W are non-isomorphic irreducible representations of G L . We conclude that the evaluation map is injective, and the image is equal to C alg (G(Z p ), L). Lemma 6.A.15 implies the last assertion.

References [1] L. BARTHEL AND R. L IVNÉ, ‘Irreducible modular representations of GL2 of a local field’, Duke Math. J. 75, (1994) 261–292. [2] L. B ERGER , ‘Représentations modulaires de GL2 (Q p ) et représentations galoisiennes de dimension 2’, Astérisque 330 (2010), 263–279. [3] L. B ERGER AND C. B REUIL , ‘Sur quelques représentations potentiellement cristallines de GL2 (Q p )’, Astérisque 330 (2010) 155–211. [4] C. B REUIL, ‘Sur quelques représentations modulaires et p-adiques de GL2 (Q p ). I’, Compositio 138, (2003), 165–188. [5] C. B REUIL, ‘Sur quelques représentations modulaires et p-adiques de GL2 (Q p ). II’, J. Inst. Math. Jussieu 2, (2003), 1–36. [6] C. B REUIL AND M. E MERTON, ‘Représentations p-adiques ordinaires de GL2 (Q p ) et compatibilité local-global’, Astérisque 331 (2010), 255–315. ¯ [7] C. B REUIL AND V. PAŠK UNAS , ‘Towards a modulo p Langlands correspondence for GL2 ’, Memoirs of AMS, 216, 2012. [8] P. C OLMEZ, ‘Représentations de GL2 (Q p ) et (ϕ, )-modules’, Astérisque 330 (2010) 281–509. [9] B. C ONRAD, ‘Weil and Grothendieck approaches to adelic points’, Enseign. Math. (2) 58 (2012), no. 1–2, 61–97. [10] R.L. E LLIS , ‘Extending continuous functions on zero-dimensional spaces’, Math. Ann., 186, (1970), 114–122.

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[11] M. E MERTON, ‘ p-adic L-functions and unitary completions of representations of p-adic reductive groups’, Duke Math. J. 130 (2005), no. 2, 353–392. [12] M. E MERTON, ‘Locally analytic vectors in representations of locally p-adic analytic groups’, to appear in Memoirs of the AMS. [13] M. E MERTON, ‘Local-global compatibility conjecture in the p-adic Langlands programme for G L 2 /Q’, Pure and Applied Math. Quarterly 2 (2006), no. 2, 279–393. [14] M. E MERTON, ‘Local-global compatibility in the p-adic Langlands programme for G L 2 /Q’, Preprint 2011, available at www.math.uchicago.edu/~emerton/ preprints.html. [15] M. E MERTON, ‘Ordinary parts of admissible representations of p-adic reductive groups I. Definition and first properties’, Astérisque 331 (2010), 335–381. [16] M. E MERTON, ‘Ordinary parts of admissible representations of p-adic reductive groups II. Derived functors’, Astérisque 331 (2010), 383–438. [17] P. G ABRIEL, ‘Des catégories abéliennes’, Bull. Soc. Math. France 90 (1962) 323–448. [18] J.C. JANTZEN, Representations of algebraic groups, 2nd edn, Mathematical Surveys and Momographs, Vol. 107, AMS, 2003. [19] M. L AZARD, ‘Groupes analytiques p-adiques’, Publ. Math. IHES 26 (1965). [20] R. O LLIVIER, ‘Le foncteur des invariants sous l’action du pro-p-Iwahori de GL(2, F)’, J. für die Reine und Angewandte Mathematik 635 (2009) 149–185. ¯ [21] V. PAŠK UNAS , ‘Coefficient systems and supersingular representations of GL2 (F)’, Mémoires de la SMF, 99, (2004). ¯ [22] V. PAŠK UNAS ‘On some crystalline representations of GL2 (Q p )’, Algebra Number Theory 3 (2009), no. 4, 411–421. ¯ [23] V. PAŠK UNAS , ‘Extensions for supersingular representations of GL2 (Q p )’, Astérisque 331 (2010) 317–353. ¯ [24] V. PAŠK UNAS , ‘Admissible unitary completions of locally Q p -rational representations of GL2 (F)’, Represent. Theory 14 (2010), 324–354. ¯ [25] V. PAŠK UNAS , ‘The image of Colmez’s Montreal functor’, to appear in Publ. Math. IHES. DOI: 10.1007/s10240-013-0049-y. [26] P. S CHNEIDER AND J. T EITELBAUM, ‘Banach space representations and Iwasawa theory’, Israel J. Math. 127, (2002) 359–380. [27] M.-F. V IGNÉRAS, ‘Representations modulo p of the p-adic group G L(2, F)’, Compositio Math. 140 (2004) 333–358.

7 From étale P+-representations to G-equivariant sheaves on G/P Peter Schneider, Marie-France Vigneras, and Gergely Zabradi∗

Abstract Let K /Q p be a finite extension with ring of integers o, let G be a connected reductive split Q p -group of Borel subgroup P = T N and let α be a simple root of T in N . We associate to a finitely generated module D over the Fontaine ring over o endowed with a semilinear étale action of the monoid T+ (acting on the Fontaine ring via α), a G(Q p )-equivariant sheaf of o-modules on the compact space G(Q p )/P(Q p ). Our construction generalizes the representation D P1 of G L(2, Q p ) associated by Colmez to a (ϕ, )-module D endowed with a character of Q∗p .

Contents 1

2

Introduction 1.1 Notation 1.2 General overview 1.3 The rings α (N0 ) and OE ,α 1.4 Equivalence of categories 1.5 P-equivariant sheaves on C 1.6 Generalities on G-equivariant sheaves on G/P 1.7 (s, res, C)-integrals Hg 1.8 Main theorem 1.9 Structure of the chapter Induction IndG H for monoids H ⊂ G 2.1 Definition and remarks 2.2 From N to Z 2.3 (ϕ, ψ)-modules

page 250 250 251 252 252 253 254 255 257 258 261 261 262 264

∗ The third author was partially supported by OTKA Research grant no. K-101291.

Automorphic Forms and Galois Representations, ed. Fred Diamond, Payman L. Kassaei and c Cambridge University Press 2014. Minhyong Kim. Published by Cambridge University Press.

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3

Etale P+ -modules 3.1 Etale module M 3.2 Induced representation M P 3.3 Compactly induced representation McP 3.4 P-equivariant map Res : Cc∞ (N , A) → End A (M P ) 3.5 P-equivariant sheaf on N 3.6 Independence of N0 3.7 Etale A[P+ ]-module and P-equivariant sheaf on N 4 Topology 4.1 Topologically étale A[P+ ]-modules P 4.2 Integration on N with value in Endcont A (M ) 5 G-equivariant sheaf on G/P 5.1 Topological G-space G/P and the map α 5.2 Equivariant sheaves and modules over skew group rings 5.3 Integrating α when M is compact 5.4 G-equivariant sheaf on G/P 6 Integrating α when M is non compact 6.1 (s, res, C)-integrals 6.2 Integrability criterion for α 6.3 Extension of Res 6.4 Proof of the product formula 6.5 Reduction modulo pn 7 Classical (ϕ, )-modules on OE 7.1 The Fontaine ring OE 7.2 The group G L(2, Q p ) 7.3 Classical étale (ϕ, )-modules 8 A generalization of (ϕ, )-modules 8.1 The microlocalized ring (N0 ) et 8.2 The categories Met

(N0 ) (L ∗ ) and MOE , (L ∗ ) 8.3 Base change functors 8.4 Equivalence of categories 8.5 Continuity 9 Convergence in L + -modules on (N0 ) 9.1 Bounded sets 9.2 The module Msbd 9.3 The special family Cs when M is killed by a power of p 9.4 Functoriality and dependence on s 10 Connected reductive split group References

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266 267 271 274 275 279 281 282 283 283 287 290 292 295 301 302 304 305 307 311 312 316 320 320 320 321 328 328 330 334 336 340 346 346 347 350 353 361 366

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1. Introduction 1.1. Notation We fix a finite extension K /Q p of ring of integers o and an algebraic closure Q p of K . We denote by G p = Gal(Q p /Q p ) the absolute Galois group of Q p , by (Z p ) = o[[Z p ]] the Iwasawa o-algebra of maximal ideal M(Z p ), and by OE the Fontaine ring which is the p-adic completion of the localization of (Z p ) with respect to the elements not in p (Z p ). We put on OE the weak topology inducing the M(Z p )-adic topology on

(Z p ), a fundamental system of neighbourhoods of 0 being ( p n OE + M(Z p )n )n∈N . The action of Z p − {0} by multiplication on Z p extends to an action on OE . We fix an arbitrary split reductive connected Q p -group G and a Borel Q p -subgroup P = T N with maximal Q p -subtorus T and unipotent radical N . We denote by w0 the longest element of the Weyl group of T in G, by + the set of roots of T in N , and by u α : Ga → Nα , for α ∈ + , a Q p -homomorphism onto the root subgroup Nα of N such that tu α (x)t −1 = u α (α(t)x) for x ∈ Q p and t ∈ T (Q p ), and N0 = α∈+ u α (Z p ) is a subgroup of N (Q p ). We denote by T+ the monoid of dominant elements t in T (Q p ) such that val p (α(t)) ≥ 0 for all α ∈ + , by T0 ⊂ T+ the maximal subgroup, by T++ the subset of strictly dominant elements, i.e. val p (α(t)) > 0 for all α ∈ + , and we put P+ = N0 T+ , P0 = N0 T0 . The natural action of T+ on N0 extends to an action on the Iwasawa o-algebra

(N0 ) = o[[N0 ]]. The compact set G(Q p )/P(Q p ) contains the open dense subset C = N (Q p )w0 P(Q p )/P(Q p ) homeomorphic to N (Q p ) and the compact subset C0 = N0 w0 P(Q p )/P(Q p ) homeomorphic to N0 . We put P(Q p ) = w0 P(Q p )w0−1 . Each simple root α gives a Q p -homomorphism xα : N → Ga with section u α . We denote by α : N0 → Z p , resp. ια : Z p → N0 , the restriction of xα , resp. u α , to N0 , resp. Z p . For example, G = G L(n), P is the subgroup of upper triangular matrices, N consists of the strictly upper triangular matrices (1 on the diagonal), T is the diagonal subgroup, N0 = N (Z p ), the simple roots are α1 , . . . , αn−1 where αi (diag(t1 , . . . , tn )) = ti ti−1 +1 , x αi sends a matrix to its (i, i + 1)-coefficient, u αi (.) is the strictly upper triangular matrix, with (i, i + 1)-coefficient . and 0 everywhere else. We denote by C ∞ (X, o) the o-module of locally constant functions on a locally profinite space X with values in o, and by Cc∞ (X, o) the subspace of compactly supported functions.

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1.2. General overview Colmez established a correspondence V → (V ) from the absolutely irreducible K -representations V of dimension 2 of the Galois group G p to the unitary admissible absolutely irreducible K -representations of G L(2, Q p ) admitting a central character [6]. This correspondence relies on the construction of a representation D(V ) P1 of G L(2, Q p ) for any representation V (not necessarily of dimension 2) of G p and any unitary character δ : Q∗p → o∗ . When the dimension of V is 2 and when δ = (x|x|)−1 δV , where δV is the character of Q∗p corresponding to the representation det V by local class field theory, then D(V ) P1 is an extension of (V ) by its dual twisted by δ ◦ det. It is a general belief that the correspondence V → (V ) should extend to a correspondence from representations V of dimension d to representations of G L(d, Q p ). We generalize here Colmez’s construction of the representation D P1 of G L(2, Q p ), replacing G L(2) by the arbitrary split reductive connected Q p group G. More precisely, we denote by OE ,α the ring OE with the action of T+ via a simple root α ∈ (if the rank of G is 1, α is unique and we omit α). For any finitely generated OE ,α -module D with an étale semilinear action of T+ , we construct a representation of G(Q p ). It is realized as the space of global sections of a G(Q p )-equivariant sheaf on the compact quotient G(Q p )/P(Q p ). When the rank of G is 1, the compact space G(Q p )/P(Q p ) is isomorphic to P1 (Q p ) and when G = G L(2) we recover Colmez’s sheaf. We review briefly the main steps of our construction. 1. We show that the category of étale T+ -modules finitely generated over OE ,α is equivalent to the category of étale T+ -modules finitely generated over

α (N0 ), for a topological ring α (N0 ) generalizing the Fontaine ring OE , which is better adapted to the group G, and depends on the simple root α. 2. We show that the sections over C0 N0 of a P(Q p )-equivariant sheaf S of o-modules over C N is an étale o[P+ ]-module S(C0 ) and that the functor S → S(C0 ) is an equivalence of categories. 3. When S(C0 ) is an étale T+ -module finitely generated over α (N0 ), and the root system of G is irreducible, we show that the P(Q p )-equivariant sheaf S on C extends to a G(Q p )-equivariant sheaf over G(Q p )/P(Q p ) if and only if the rank of G is 1. 4. For any strictly dominant element s ∈ T++ , we associate functorially to an étale T+ -module M finitely generated over α (N0 ), a G(Q p )-equivariant sheaf Ys of o-modules over G(Q p )/P(Q p ) with sections over C0 a dense étale (N0 )[T+ ]-submodule Msbd of M. When the rank of G is 1, the sheaf

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Ys does not depend on the choice of s ∈ T++ , and Msbd = M; when G = G L(2) we recover the construction of Colmez. For a general G, the sheaf Ys depends on the choice of s ∈ T++ , the system (Ys )s∈T++ of sheaves is compatible, and we associate functorially to M the G(Q p )-equivariant sheaves Y∪ and Y∩ of o-modules over G(Q p )/P(Q p ) with sections over C0 equal to ∪s∈T++ Msbd and ∩s∈T++ Msbd , respectively.

1.3. The rings α (N0 ) and OE ,α Fixing a simple root α ∈ , the topological local ring α (N0 ), generalizing the Fontaine ring OE , is defined as in [11] with the surjective homomorphism α : N0 → Z p . We denote by M(Nα ) the maximal ideal of the Iwasawa o-algebra

(Nα ) = o[[Nα ]] of the kernel Nα of α . The ring α (N0 ) is the M(Nα )adic completion of the localization of (N0 ) with respect to the Ore subset of elements which are not in M(Nα ) (N0 ). This is a noetherian local ring with maximal ideal Mα (N0 ) generated by M(Nα ). We put on α (N0 ) the weak topology with fundamental system of neighbourhoods of 0 equal to (Mα (N0 )n + M(N0 )n )n∈N . The action of T+ on N0 extends to an action on

α (N0 ). We denote by OE ,α the ring OE with the action of T+ induced by (t, x) → α(t)x : T+ × Z p → Z p . The homomorphism α and its section ια induce T+ -equivariant ring homomorphisms α : α (N0 ) → OE ,α , ια : OE ,α → α (N0 ), such that α ◦ ια = id .

1.4. Equivalence of categories An étale T+ -module over α (N0 ) is a finitely generated α (N0 )-module M with a semi-linear action T+ × M → M of T+ which is étale, i.e. the action ϕt on M of each t ∈ T+ is injective and M = ⊕u∈J (N0 /t N0 t −1 ) uϕt (M), t −1 )

if J (N0 /t N0 ⊂ N0 is a system of representatives of the cosets N0 /t N0 t −1 ; in particular, the action of each element of the maximal subgroup T0 of T+ is invertible. We denote by ψt the left inverse of ϕt vanishing on uϕt (M) for u ∈ N0 not in t N0 t −1 . These modules form an abelian category Met

α (N0 ) (T+ ). We define analogously the abelian category Met (T ) of finitely generOE,α + ated OE ,α -modules with an étale semilinear action of T+ . The action ϕt of each element t ∈ T+ such that α(t) ∈ Z∗p is invertible. We show that the action T+ × D → D of T+ on D ∈ Met OE,α (T+ ) is continuous for the weak topology on D; the canonical action of the inverse T− of T is also continuous.

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Theorem 7.1. The base change functors OE ⊗α − and α (N0 )⊗ια − induce quasi-inverse isomorphisms D : Met

α (N0 )

et et (T+ ) → Met OE,α (T+ ), M : MOE,α (T+ ) → M

α (N0 )

(T+ ).

Using this theorem, we show that the action of T+ and of the inverse monoid T− (given by the operators ψ) on an étale T+ -module over α (N0 ) is continuous for the weak topology.

1.5. P-equivariant sheaves on C The o-algebra C ∞ (N0 , o) is naturally an étale o[P+ ]−module, and the monoid P+ acts on the o-algebra Endo M by (b, F) → ϕb ◦ F ◦ ψb . We show that there exists a unique o[P+ ]-linear map res : C ∞ (N0 , o) → Endo M sending the characteristic function 1 N0 of N0 to the identity id M ; moreover res is an algebra homomorphism which sends 1b.N0 to ϕb ◦ ψb for all b ∈ P+ acting on x ∈ N0 by (b, x) → b.x. For the sake of simplicity, we denote now by the same letter a group defined over Q p and the group of its Q p -rational points. Let M P be the o[P]-module induced by the canonical action of the inverse monoid P− of P+ on M; as a representation of N , it is isomorphic to the representation induced by the action of N0 on M. The value at 1, denoted by ev0 : M P → M, is P− -equivariant, and admits a P+ -equivariant splitting σ0 : M → M P sending m ∈ M to the function equal to n → nm on N0 and vanishing on N − N0 . The o[P]-submodule McP of M P generated by σ0 (M) is naturally isomorphic to A[P] ⊗ A[P+ ] M. When M = C ∞ (N0 , o) then McP = Cc∞ (N , o) and M P = C ∞ (N , o) with the natural o[P]-module structure. We have the natural o-algebra embedding F → σ0 ◦ F ◦ ev0 : Endo M → Endo M P , sending id M to the idempotent R0 = σ0 ◦ ev0 in Endo M P . Proposition 7.2. There exists a unique o[P]-linear map Res : Cc∞ (N , o) → Endo M P sending 1 N0 to R0 ; moreover Res is an algebra homomorphism. The topology of N is totally disconnected and by a general argument, the functor of compact global sections is an equivalence of categories from the

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P-equivariant sheaves on N C to the nondegenerate modules on the skew group ring Cc∞ (N , o)#P = ⊕b∈P bC c∞ (N, o) in which the multiplication is determined by the rule (b1 f 1 )(b2 f 2 ) = b b b1 b2 f 1 2 f 2 for bi ∈ P, f i ∈ Cc∞ (N, o) and f 1 2 (.) = f 1 (b2 .). Theorem 7.3. The functor of sections over N0 C0 from the P-equivariant sheaves on N C to the étale o[P+ ]-modules is an equivalence of categories. The space of global sections of a P-equivariant sheaf S on C is S(C) = S(C0 ) P .

1.6. Generalities on G-equivariant sheaves on G/ P The functor of global sections from the G-equivariant sheaves on G/P to the modules on the skew group ring AG/P = C ∞ (G/P, o)#G is an equivalence of categories. We have the intermediate ring A AC = C c∞ (C, o)#P ⊂ A = ⊕g∈G gCc∞ (g −1 C ∩ C, o) ⊂ AG/P , and the o-module Z = ⊕g∈G gC c∞ (C, o) which is a left ideal of AG/P and a right A-submodule. Proposition 7.4. The functor Z → Y (Z ) = Z ⊗A Z from the nondegenerate A-modules to the AG/P -modules is an equivalence of categories; moreover the G-sheaf on G/P corresponding to Y (Z ) extends the P-equivariant sheaf on C corresponding to Z |AC . Given an étale o[P+ ]-module M, we consider the problem of extending to A the o-algebra homomorphism Res : AC → Endo (McP ), b f b → b ◦ Res( f b ). b∈P

We introduce the subrings A0 = 1C0 A1C0 = ⊕g∈G gC ∞ (g −1 C0 ∩ C0 , o) ⊂ A, AC 0 = 1C0 AC 1C0 = ⊕b∈P bC ∞ (b−1 C0 ∩ C0 , o) ⊂ AC .

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The skew monoid ring AC0 = C ∞ (C0 , o)#P+ = ⊕b∈P+ bC ∞ (C0 , o) is contained in AC 0 . The intersection g −1 C0 ∩ C0 is not 0 if and only if g ∈ N0 P N0 . The subring Res(AC 0 ) of Endo (M P ) necessarily lies in the image of Endo (M). The group P acts on A by (b, y) → (b1G/P )y(b1G/P )−1 for b ∈ P, and the map b ⊗ y → (b1G/P )y(b1G/P )−1 gives o[P] isomorphisms o[P] ⊗o[P+ ] A0 → A and o[P] ⊗o[P+ ] AC 0 → AC . Proposition 7.5. Let M be an étale o[P+ ]-module. We suppose given, for any g ∈ N0 P N0 , an element Hg ∈ Endo (M). The map R0 : A0 → Endo (M), g f g → Hg ◦ res( f g ) g∈N0 P N0

g∈N0 P N0

is a P+ -equivariant o-algebra homomorphism which extends Res |AC0 if and only if, for all g, h ∈ N0 P N0 , b ∈ P ∩ N0 P N0 , and all compact open subsets V ⊂ C0 , the relations H1. res(1V ) ◦ Hg = Hg ◦ res(1g −1 V ∩C0 ), H2. Hg ◦ Hh = Hgh ◦ res(1h −1 C0 ∩C0 ), H3. Hb = b ◦ res(1b−1 C0 ∩C0 ) hold true. In this case, the unique o[P]-equivariant map R : A → End A (McP ) extending R0 is multiplicative. When these conditions are satisfied, we obtain a G-equivariant sheaf on G/P with sections on C0 equal to M.

1.7. (s, res, C)-integrals H g Let M be an étale T+ -module M over α (N0 ) with the weak topology. We denote by Endcont o (M) the o-module of continuous o-linear endomorphisms of M, and for g in N0 P N0 , by U g ⊆ N0 the compact open subset such that Ug w0 P/P = g −1 C0 ∩ C0 . For u ∈ Ug , we have a unique element α(g, u) ∈ N0 T such that guw0 N = α(g, u)uw0 N . We consider the map αg,0 : N0 → Endcont o (M) αg,0 (u) = Res(1C0 ) ◦ α(g, u) ◦ Res(1C0 ) for u ∈ Ug and αg,0 (u) = 0 otherwise.

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The module M is Hausdorff complete but not compact; also we introduce a notion of integrability with respect to a special family C of compact subsets C ⊂ M, i.e. satisfying: C(1) C(2) C(3) C(4)

Any compact subset of a compact set in C also lies in C. 0 If C1 , C2 , . . . , Cn ∈ C then in=1 Ci is in C, as well. For all C ∈ C we have N0 C ∈ C. 0 M(C) := C∈C C is an étale o[P+ ]-submodule of M.

A map from M(C) to M is called C-continuous if its restriction to any C ∈ C is continuous. The o-module HomCont (M(C), M) of C-continuous o-linear o homomorphisms from M(C) to M with the C-open topology, is a topological complete o-module. For s ∈ T++ , the open compact subgroups Nk = s k N0 s −k ⊂ N for k ∈ Z, form a decreasing sequence of union N and intersection {1}. A map F : N0 → HomCont A (M(C), M) is called (s, res, C)-integrable if the limit / Fd res := lim F(u) ◦ res(1u Nk ), N0

k→∞

u∈J (N0 /Nk )

where J (N0 /Nk ) ⊆ N0 , for any k ∈ N, is a set of representatives for the cosets in N0 /Nk , exists in HomCont A (M(C), M) and is independent of the choice of the sets J (N0 /Nk ). We denote by Hg,J (N0 /Nk ) the sum in the right-hand side when F = αg,0 (.)| M(C) . Proposition 7.6. For all g ∈ N0 P N0 , the map αg,0 (.)| M(C) : N0 → HomCont A (M(C), M) is (s, res, C)-integrable when C(5) For any C ∈ C the compact subset ψs (C) ⊆ M also lies in C. T(1) For any C ∈ C such that C = N0 C, any open A[N0 ]-submodule M of M, and any compact subset C+ ⊆ L + there exists a compact open subgroup P1 = P1 (C, M, C+ ) ⊆ P0 and an integer k(C, M, C+ ) ≥ 0 such that s k (1 − P1 )C+ ψsk ⊆ E(C, M)

for any k ≥ k(C, M, C+ ).

The integrals Hg of αg,0 (.)| M(C) satisfy the relations H1, H2, H3, when they belong to End A (M(C)), and when C(6) For any C ∈ C the compact subset ϕs (C) ⊆ M also lies in C. T(2) Given a set J (N0 /Nk ) ⊂ N0 of representatives for cosets in N0 /Nk , for k ≥ 1, for any x ∈ M(C) and g ∈ N0 P N0 there exists a compact A-submodule C x,g ∈ C and a positive integer k x,g such that Hg,J (N0 /Nk ) (x) ⊆ C x,g for any k ≥ k x,g .

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When C satisfies C(1), . . . , C(6) and the technical properties T(1), T(2) are true, we obtain a G-equivariant sheaf on G/P with sections on C0 equal to M(C).

1.8. Main theorem Let M be an étale T+ -module M over α (N0 ) with the weak topology and let s ∈ T++ . We have the natural T+ -equivariant quotient map M : M → D = OE ,α ⊗α M

,

m → 1 ⊗ m

from M to D = D(M) ∈ MOE,α (T+ ), of T+ -equivariant section ι D : D → M = α (N0 ) ⊗ια D,

d → 1 ⊗ d.

We note that o[N0 ]ι D (D) is dense in M. A lattice D0 in D is a (Z p )submodule generated by a finite set of generators of D over OE . When D is killed by a power of p, the o-module Msbd (D0 ) := {m ∈ M | M (ψsk (u −1 m)) ∈ D0 for all u ∈ N0 and k ∈ N} of M is compact and is a (N0 )-module. Let Cs be the family of compact subsets of M contained in Msbd (D0 ) for some lattice D0 of D, and let Msbd = ∪ D0 Msbd (D0 ) the union being taken over all lattices D0 in D. In general, M is p-adically complete, M/ p n M is an étale T+ -module over α (N0 ), and D/ p n D = D(M/ p n M). We denote by pn : M → M/ p n M the reduction modulo p n , and by Cs,n the family of compact subsets constructed above for M/ p n M. We define the family Cs of compact subsets C ⊂ M such that pn (C) ∈ Cs,n for all n ≥ 1, and the o-module Msbd of m ∈ M such that the set of M (ψsk (u −1 m)) for k ∈ N, u ∈ N0 is bounded in D for the weak topology. By reduction to the easier case where M is killed by a power of p, we show that Cs satisfies C(1), . . . , C(6) and that the technical properties T(1), T(2) are true. Proposition 7.7. Let M be an étale T+ -module M over α (N0 ) and let s ∈ T++ . (i) Msbd is a dense (N0 )[T+ ]-étale submodule of M containing ι D (D). (ii) For g ∈ N0 P N0 , the (s, res, Cs )-integrals Hg,s of αg,0 | Msbd exist, lie in Endo (Msbd ), and satisfy the relations H1, H2, H3. (iii) For s1 , s2 ∈ T++ , there exists s3 ∈ T++ such that Msbd contains Msbd ∪ 3 1 bd ∩ M bd . Msbd and H = H on M g,s1 g,s2 s1 s2 2

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The endomorphisms Hg,s ∈ Endo (Msbd ) induce endomorphisms of bd satisfying the relations ∩s∈T++ Msbd and of ∪s∈T++ Msbd = s∈T++ Ms H1, H2, H3. Moreover ∪s∈T++ Msbd and ∩s∈T++ Msbd are (N0 )[T+ ]-étale submodules of M containing ι D (D). Our main theorem is the following: Theorem 7.8. There are faithful functors Y∩ , (Ys )s∈T++ , Y∪ : Met OE,α (T+ ) −→ G-equivariant sheaves on G/P, sending D = D(M) to a sheaf with sections on C0 equal to the dense

(N0 )[T+ ]-submodules of M

Msbd , (Msbd )s∈T++ , and Msbd , s∈T++

s∈T++

respectively. When G = G L(2, Q p ), the sheaves Ys (D) are all equal to the G-equivariant sheaf on G/P P1 (Q p ) of global sections D P1 constructed by Colmez. When the root system of G is irreducible of rank > 1, we check that ∪s∈T++ Msbd is never equal to M.

1.9. Structure of the chapter In Section 2, we consider a general commutative (unital) ring A and A-modules M with two endomorphisms ψ, ϕ such that ψ ◦ ϕ = id. We show that the induction functor IndZ is exact and that the module A[Z] ⊗N,ϕ M N,ψ = lim ← −ψ is isomorphic to the subrepresentation of IndZ M generated by N,ψ (M) = lim ← −ψ k the elements of the form (ϕ (m))k∈N . In Section 3, we consider a general monoid P+ = N0 L + contained in a group P with the property that N0 is a group such that t N0 t −1 ⊂ N0 has a finite index for all t ∈ L + and we study the étale A[P+ ]-modules M. We show that the inverse monoid P− = L − N0 ⊂ P acts on M, the inverse of t ∈ L + acting by the left inverse ψt of the action ϕt of t with kernel uϕt (M) for u ∈ N0 not in t N0 t −1 . We add the hypothesis that L + contains a central element s such that the sequence (s k N0 s −k )k∈Z is decreasing of trivial intersection, of union a group N , and that P = N L is the semi-direct product of N and of L = ∪k∈N L − s k . An A[P+ ]-submodule of M is étale if and only if it is stable by ψs . The representation M P of P induced by M| P− , restricted to N is the representation induced from M| N0 , and restricted to s Z is the representation limψ M induced from M|s −N . The natural ← − s

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A[P+ ]-embedding M → M P generates a subrepresentation McP of M P isomorphic to A[P] ⊗ A[P+ ] M. When N is a locally profinite group and N0 an open compact subgroup, we show the existence and the uniqueness of a unitpreserving A[P+ ]-map res : C ∞ (N0 , A) → End A (M), we extend it uniquely to an A[P]-map Res : C ∞ (N , A) → End A (M P ), and we prove our first theorem: the equivalence between the P-equivariant sheaves of A-modules on N and the étale A[P+ ]-modules on N0 . In Section 4, we suppose that A is a linearly topological commutative ring, that P is a locally profinite group and that M is a complete linearly topological A-module with a continuous étale action of P+ such that the action of P− is also continuous, or equivalently ψs is continuous (we say that M is a topologically étale module). Then M P is complete for the compact-open topology and Res is a measure on N with values in the algebra E cont of continuous endomorphisms of M P . We show that E cont is a complete topocont logical ring for the topology defined by the ideals E L of endomorphisms P with image in an open A-submodule L ⊂ M , and that any continuous map N → E cont with compact support can be integrated with respect to Res. In Section 5, we introduce a locally profinite group G containing P as a closed subgroup with compact quotient set G/P, such that the double cosets P\G/P admit a finite system W of representatives normalizing L, of image in NG (L)/L equal to a group, and the image C = Pw0 P/P in G/P of a double coset (with w0 ∈ W ) is open dense and homeomorphic to N by the map n → nw0 P/P. We show that any compact open subset of G/P is a finite disjoint union of g −1 U w0 P/P for g ∈ G and U ⊂ N a compact open subgroup. We prove the basic result that the G-equivariant sheaves of A-modules on G/P identify with modules over the skew group ring C ∞ (G/P, A)#G, or with nondegenerate modules over a (non unital) subring A, and that an étale A[P+ ]-module M endowed with endomorphisms Hg ∈ End A (M), for g ∈ N0 P N0 , satisfying certain relations H1, H2, H3, gives rise to a nondegenerate A-module. For g ∈ G we denote Ng ⊂ N such that N g w0 P/P = g −1 C ∩ C. We study the map α from the set of (g, u) with g ∈ G and u ∈ N g to P defined by guw0 N = α(g, u)uw0 N . In particular, we show the cocycle relation α(gh, u) = α(g, h.u)α(h, u) when each term makes sense. When M is compact, then M P is compact and the action of P on M P induces a continuous map P → E cont . We show that the A-linear map A → E cont given by the integrals of α(g, .) f (.) with respect to Res, for f ∈ Cc∞ (Ng , A), is multiplicative. As explained above, we obtain a G-equivariant sheaf of A-modules on G/P with sections M on C0 .

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In Section 6, we do not suppose that M is compact and we introduce the notion of (s, res, C)-integrability for a special family C of compact subsets of M. We give an (s, res, C)-integrability criterion for the function αg,0 (u) = Res(1 N0 )α(gh, u) Res(1 N0 ) on the open subset Ug ⊂ N0 such that Ug w0 P/P = g −1 C0 ∩C0 , for g ∈ N0 w0 Pw0 N0 , a criterion which ensures that the integrals Hg of αg,0 satisfy the relations H1, H2, H3, as well as a method of reduction to the case where M is killed by a power of p. When these criteria are satisfied, as explained in Section 5, one gets a G-equivariant sheaf of A-modules on G/P with sections M on C0 . (2) Section 7 concerns classical (ϕ, )-modules over OE , seen as étale o[P+ ](2) module D, where the upper exponent indicates that P+ is the upper triangular monoid P+ of G L(2, Q p ). Using the properties of treillis we apply the method explained in Section 6 to this case and we obtain the sheaf constructed by Colmez. In Section 8 we consider the case where N0 is a compact p-adic Lie group (2) endowed with a continuous non-trivial homomorphism : N0 → N0 with a (2) section ι, that L ∗ ⊂ L is a monoid acting by conjugation on N0 and ι(N0 ), (2) that extends to a continuous homomorphism : P∗ = N0 L ∗ → P+ (2) sending L ∗ to L + and that ι is L ∗ equivariant. We consider the abelian categories of étale L ∗ -modules finitely generated over the microlocalized ring

(N0 ) resp. over OE (with the action of L ∗ induced by ). Between these categories we have the base change functors given by the natural L ∗ -equivariant algebra homomorphisms : (N0 ) → OE and ι : OE → (N0 ). We show our second theorem: the base change functors are quasi-inverse equivalences of categories. When L ∗ contains an open topologically finitely generated pro- p-subgroup, we show that an étale L ∗ -module over OE is automatically topologically étale for the weak topology; the result extends to étale L ∗ -modules over (N0 ), with the help of this last theorem. In Section 9, we suppose that : P → P (2) (Q p ) is a continuous homomorphism with (L) ⊂ L (2) (Q p ), and that ι : N (2) (Q p ) → N is a L-equivariant section of | N (as L acts on N (2) (Q p ) via ) sending (N0 ) in N0 . The assumptions of Section 8 are satisfied for L ∗ = L + . Given an étale L + -module M over

(N0 ), we exhibit a special family Cs of compact subsets in M which satisfies the criteria of Section 6 with M(Cs ) equal to a dense (N0 )[L + ]-submodule Msbd ⊂ M. We obtain our third theorem: there exists a faithful functor from the étale L + -modules over (N0 ) to the G-equivariant sheaves on G/P sending M to the sheaf with sections Msbd on C0 . In Section 10, we check that our theory applies to the group G(Q p ) of rational points of a split reductive group of Q p , to a Borel subgroup P(Q p )

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of maximal split torus T (Q p ) = L and to a natural homomorphism α : P(Q p ) → P (2) (Q p ) associated to a simple root α. We obtain our main theorem: there are compatible faithful functors from the étale T (Q p )+ -modules D over OE (where T (Q p )+ acts via α) to the G(Q p )-equivariant sheaves on G(Q p )/P(Q p ) sheaves with sections M(D)bd s on C0 , for all strictly dominant s ∈ T (Q p ). When the root system of G is irreducible of rank > 1, we show that ∪s Msbd = M = M(D). Acknowledgements: A part of the work on this article was done when the first and third authors visited the Institut Mathématique de Jussieu at the Universities of Paris 6 and Paris 7, and the second author visited the Mathematische Institut at the Universität Münster. We express our gratitude to these institutions for their hospitality. We thank heartily CIRM, IAS, the Fields Institute, as well as Durham, Cordoba and Caen Universities, for their invitations giving us the opportunity to present this work. Finally, we would like to thank the anonymous referee for a very careful reading of the manuscript and his suggestions for improving the presentation.

2. Induction IndG H for monoids H ⊂ G A monoid is supposed to have a unit.

2.1. Definition and remarks Let A be a commutative ring, let G be a monoid and let H be a submonoid of G. We denote by A[G] the monoid A-algebra of G and by M A (G) the category of left A[G]-modules, which has no reason to be equivalent to the category of right A[G]-modules. One can construct A[G]-modules starting from A[H ]-modules in two natural ways, by taking the two adjoints of the restriction functor ResG H : M A (G) → M A (H ) from G to H . For M ∈ M A (H ) and V ∈ M A (G) we have the isomorphism

Hom A[G] (A[G] ⊗ A[H ] M, V ) −−→ Hom A[H ] (M, V ) and the isomorphism

Hom A[G] (V, Hom A[H ] ( A[G], M)) −−→ Hom A[H ] (V, M).

(7.1)

For monoid algebras, restriction of homomorphisms induces the identification Hom A[H ] (A[G], M) = IndGH (M)

262

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where IndG H (M) is formed by the functions f : G → M such that f (hg) = h f (g) for any h ∈ H, g ∈ G; the group G acts by right translations, g f (x) = f (xg) for g, x ∈ G. The isomorphism (7.1) pairs φ of the left side and of the right side satisfying ([14] I.5.7) φ(v)(g) = (gv) for (v, g) ∈ V × G. It is well known that the left and right adjoint functors of ResG H are transitive (for monoids H ⊂ K ⊂ G), the left adjoint is right exact, the right adjoint is left exact. We observe important differences between monoids and groups: (a) The binary relation g ∼ g if g ∈ H g is not symmetric, there is no “quotient space” H \G, no notion of function with finite support modulo H in IndG H (M). (b) When h M = 0 for some h ∈ H such that hG = G, then IndG H (M) = 0. Indeed f (hg) = h f (g) implies f (hg) = 0 for any g ∈ G. (c) When G is a group generated, as a monoid, by H and the inverse monoid H −1 := {h ∈ G | h −1 ∈ H }, and when M in an A[H ]-module such that the action of any element h ∈ H on M is invertible, then f (g) = g f (1) for all g ∈ G and f ∈ IndG H (M). This can be seen by induction on the minimal number m ∈ N such that g = g1 . . . gm with gi ∈ H ∪ H −1 . Then g1 ∈ H implies f (g) = g1 f (g2 . . . gm ), and g1 ∈ H −1 implies f (g2 . . . gm ) = f (g1−1 g1 g2 . . . gm ) = g1−1 f (g). The representation IndGH (M) is isomorphic by f → f (1) to the natural representation of G on M.

2.2. From N to Z An A-module with an endomorphism ϕ is equivalent to an A[N]-module, ϕ being the action of 1 ∈ N, and an A-module with an automorphism ϕ is equivalent to an A[Z]-module. When ϕ is bijective, A[Z] ⊗ A[N] M and IndZ N (M) are isomorphic to M. In general, A[Z] ⊗ A[N] M is the limit of an inductive system and IndZ N (M) is the limit of a projective system. The first one is interesting when ϕ is injective, the second one when ϕ is surjective. For r ∈ N let Mr = M. The general element of Mr is written xr with x ∈ M. Let lim (M, ϕ) be the quotient of 2r∈N Mr by the − → equivalence relation generated by ϕ(x)r+1 ≡ xr , with the isomorphism

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induced by the maps xr → ϕ(x)r Mr → Mr of inverse induced by the maps xr → xr +1 Mr → Mr+1 . Let x → [x] : Z → A[Z] be the canonical map. The maps xr → [−r ] ⊗ x Mr → A[Z] ⊗ A[N] M for r ∈ N induce an isomorphism of A[Z]-modules →

lim M − →

A[Z] ⊗ A[N] M.

Let lim M := {x = (xm )m∈N ∈ ← −

M : ϕ(xm+1 ) = xm for any m ∈ N} (7.2)

m∈N

seen as an A[Z]-module via the automorphism x → (ϕ(x0 ), x0 , x1 , . . .) = (ϕ(x 0 ), ϕ(x1 ), ϕ(x2 ) . . .) of inverse x → (x1 , x2 , . . .). The map f → ( f (−m))m∈N is an isomorphism of A[Z]-modules IndZ N (M)

→

lim M. ← −

The submodules of M Mϕ

∞ =0

= ∪k∈N M ϕ

k =0

ϕ ∞ (M) = ∩n∈N ϕ n (M)

,

are stable by ϕ. The inductive limit sees only the quotient M/M ϕ projective limit sees only the submodule ϕ ∞ (M), lim M = lim (M/M ϕ − → − →

∞ =0

),

∞ =0

and the

lim M = lim (ϕ ∞ (M)). ← − ← −

Lemma 7.9. Let 0 → M1 → M2 → M3 → 0 be an exact sequence of A-modules with an endomorphism ϕ. (a) The sequence 0 → lim M1 → lim M2 → lim M3 → 0 − → − → − → is exact. (b) When ϕ is surjective on M1 , the sequence 0 → lim M1 → lim M2 → lim M3 → 0 ← − ← − ← − is exact. Proof. This is a standard fact on inductive and projective limits.

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2.3. (ϕ, ψ)-modules Let M be an A-module with two endomorphisms ψ, ϕ such that ψ ◦ ϕ = 1. Then ψ is surjective, ϕ is injective, the endomorphism ϕ ◦ ψ is a projector of M giving the direct decomposition M = ϕ(M) ⊕ M ψ=0 , and m ψ=0

m = (ϕ ◦ ψ)(m) + m ψ=0

(7.3)

M ψ=0

for m ∈ M ∈ the kernel of ψ. We consider the representation of Z induced by (M, ψ) as in Subsection 2.2, IndZ M. N,ψ (M) lim ← − ψ

On the induced representation ψ is an isomorphism and we introduce ϕ := ψ −1 . As ψ is surjective on M, the map ev0 IndZ N,ψ (M) → M, corresponding to the map lim M → M, (xm )m∈N → x0 ← − ψ

is surjective. A splitting is the map σ0 M → IndZ N,ψ (M) corresponding to M → lim M, ← −

x → (ϕ m (x))m∈N .

(7.4)

ψ

Obviously ev0 is ψ-equivariant, σ0 is ϕ-equivariant, ev0 ◦ σ0 = id M , and :=

R0

σ0 ◦ ev0 ∈ End A (IndZ N,ψ (M))

is an idempotent of image σ0 (M). Definition 7.10. The representation of Z compactly induced from (M, ψ) Z is the subrepresentation c-IndZ N,ψ (M) of IndN,ψ (M) generated by the image of σ0 (M). We note that, for any k ≥ 1, the endomorphisms ψ k , ϕ k satisfy the same properties as ψ, ϕ because ψ k ◦ ϕ k = 1. For any integer k ≥ 0, the value at k is a surjective map evk IndZ N,ψ (M) → M, corresponding to the map lim M → M, ← −

(xm )m∈N → xk

(7.5)

ψ

of splitting σk M → IndZ N,ψ (M) corresponding to the map M → lim M, ← −

x → (ψ k (x), . . . , ψ(x), x, ϕ(x), ϕ 2 (x), . . .) .

ψ

The following relations are immediate: evk = ev0 ◦ ϕ k = ψ ◦ evk+1 = evk+1 ◦ ψ, σk = ψ k ◦ σ0 = σk+1 ◦ ϕ = ϕ ◦ σk+1 .

(7.6)

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We deduce that σk (M) ⊂ σk+1 (M). Since σk (M) is ϕ-invariant we have

c-IndZ ψ k (σ0 (M)) = σk (M) = σk (M). (7.7) N,ψ (M) = k∈N

k∈N

k∈N

In lim (M) the subspace of (xm )m∈N such that xk+r = ϕ k (xr ) for all k ∈ N ← − ψ

Z and for some r ∈ N, is equal to c-IndZ N,ψ (M). The definition of c-IndN,ψ (M) is functorial. We get a functor c-IndZ N,ψ from the category of A-modules with two endomorphisms ψ, ϕ such that ψ ◦ ϕ = 1 (a morphism commutes with ψ and with ϕ) to the category of A[Z]-modules.

Proposition 7.11. The map A[Z] ⊗ A[N],ϕ M → Hom A[N],ψ (A[Z], M) = IndZ N,ψ (M) [k] ⊗ m → (ϕ k ◦ σ0 )(m) induces an isomorphism from the tensor product A[Z] ⊗ A[N],ϕ M to the compactly induced representation c-IndZ N,ψ (M) (note that ψ and ϕ appear). Proof. From (7.3) and the relations between the σk we have for m ∈ M, k ∈ N, k ≥ 1, σk (m) = σk−1 (ψ(m)) + σk (m ψ=0 ). By induction k∈N σk (M) = σ0 (M) + k≥1 σk (M ψ=0 ). Using (7.6) one checks that the sum is direct, hence by (7.7), ψ=0 c-IndZ )). N,ψ (M) = σ0 (M) ⊕ (⊕k≥1 σk (M

On the other hand, one deduces from (7.3) that A[Z] ⊗ A[N],ϕ M

=

([0] ⊗ M) ⊕ (⊕k≥1 ([−k] ⊗ M ψ=0 )).

With Lemma 7.9 we deduce: Corollary 7.12. The functor c-IndZ N,ψ is exact. We have two kinds of idempotents in End A (IndZ N,ψ (M)), for k ∈ N, defined by Rk = σ0 ◦ ϕ k ◦ ψ k ◦ ev0 ,

R−k := ψ k ◦ R0 ◦ ϕ k = σk ◦ evk .

(7.8)

The first ones are the images of the idempotents rk := ϕ k ◦ ψ k ∈ End A (M) via the ring homomorphism End A (M) → End A IndZ N,ψ (M),

f → σ0 ◦ f ◦ ev0 .

(7.9)

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The second ones give an isomorphism from IndZ N,ψ (M) to the limit of the projective system (σk (M), R−k : σk+1 (M) → σk (M)). Lemma 7.13. The map f IndZ N,ψ (M) to lim(σk (M)) ← −

:=

→ (R−k ( f ))k∈N is an isomorphism from

{( f k )k∈N | f k ∈ σk (M), f k = R−k ( f k+1 )

for k ∈ N }

R−k

of inverse ( f k )k∈N → f with evk ( f ) = evk ( f k ). Remark 7.14. As ϕ is injective, its restriction to ∩n∈N ϕ n (M) is an isomorphism and the following A[Z]-modules are isomorphic (Section 2.2): IndZ N,ϕ (M)

lim M ← − ϕ

∩n∈N ϕ n (M). ∞ =0

As ψ is surjective, its action on the quotient M/M ψ following A[Z]-modules are isomorphic (Section 2.2): A[Z] ⊗ A[N],ψ M

lim M − →

is bijective and the

M/M ψ

∞ =0

.

ψ

Remark 7.15. When the A-module M is noetherian, a ψ-stable submodule of M which generates M as a ϕ-module is equal to M. Proof. Let N be a submodule of M. As M is noetherian there exists k ∈ N such that the ϕ-stable submodule of M generated by N is the submodule Nk ⊂ M generated by N, ϕ(N ), . . . , ϕ k (N ). When N is ψ-stable we have ψ k (Nk ) = N and when N generates M as a ϕ-module we have M = Nk . In this case, M = ψ k (M) = ψ k (Nk ) = N .

3. Etale P+ -modules Let P = N L be a semi-direct product of an invariant subgroup N and of a group L and let N0 ⊂ N be a subgroup of N . For any subgroups V ⊂ U ⊂ N , the symbol J (U/V ) ⊂ U denotes a set of representatives for the cosets in U/V . The group P acts on N by (b = nt, x) → b.x = nt xt −1 for n, x ∈ N and t ∈ L. The P-stabilizer {b ∈ P | b.N0 ⊂ N0 } of N0 is a monoid P+ = N0 L +

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where L + ⊂ L is the L-stabilizer of N0 . Its maximal subgroup {b ∈ P | b.N0 = N0 } is the intersection P0 = N0 L 0 of P+ with the inverse monoid P− = L − N0 where L − is the inverse monoid of L + and L 0 is the maximal subgroup of L + . We suppose that the subgroup t.N0 = t N0 t −1 ⊂ N0 has a finite index, for all t ∈ L + . Let A be a commutative ring and let M be an A[P+ ]-module, equivalently an A[N0 ]-module with a semilinear action of L + . The action of b ∈ P+ on M is denoted by ϕb . If b ∈ P0 then ϕb is invertible and we also write ϕb (m) = bm, ϕb−1 (m) = b−1 m for m ∈ M. The action ϕt ∈ End A (M) of t ∈ L + is A[N0 ]-semilinear: ϕt (xm) = ϕt (x)ϕt (m)

for

x ∈ A[N0 ], m ∈ M.

(7.10)

3.1. Etale module M The group algebra A[N0 ] is naturally an A[P+ ]-module. For t ∈ L + the map ϕt is injective of image A[t N0 t −1 ], and A[N0 ] = ⊕u∈J (N0 /t N0 t −1 ) u A[t N0 t −1 ]. Definition 7.16. We say that M is étale if, for any t ∈ L + , the map ϕt is injective and M = ⊕u∈J (N0 /t N0 t −1 ) u ϕt (M).

(7.11)

An equivalent formulation is that, for any t ∈ L + , the linear map A[N0 ] ⊗ A[N0 ],ϕt M → M, x ⊗ m → xϕt (m) is bijective. For M étale and t ∈ L + , let ψt ∈ End A (M) be the unique canonical left inverse of ϕt of kernel M ψt =0 = uϕt (M). u∈(N0 −t N0 t −1 )

The trivial action of P+ on M is not étale, and obviously the restriction to P+ of a representation of P is not always étale. Lemma 7.17. Let M be an étale A[P+ ]-module. For t ∈ L + , the kernel M ψt =0 is an A[t N0 t −1 ]-module, the idempotents in End A M (u ◦ ϕt ◦ ψt ◦ u −1 )u∈J (N0 /t N0 t −1 ) are orthogonal of sum the identity. Any m ∈ M can be written uϕt (m u,t ) m= u∈J (N0 /t N0

t −1 )

for unique elements m u,t ∈ M, equal to m u,t = ψt (u −1 m).

(7.12)

268

Peter Schneider, Marie-France Vigneras, and Gergely Zabradi

Proof. The kernel M ψt =0 is an A[t N0 t −1 ]-module because N0 − t N0 t −1 is stable by left multiplication by t N0 t −1 . The endomorphism ϕt ◦ ψt is an idempotent because ψt ◦ ϕt = id M . Then apply (7.11) and notice that m ∈ M is equal to m= (u ◦ ϕt ◦ ψt ◦ u −1 )(m). u∈J (N0 /t N0 t −1 )

Remark 7.18. (1) An A[P+ ]-module M is étale when, for any t ∈ L + , the action ϕt of t admits a left inverse f t ∈ End A M such that the idempotents (u ◦ ϕt ◦ f t ◦ u −1 )u∈J (N0 /t N0 t −1 ) are orthogonal of sum the identity. The endomorphism f t is the canonical left inverse ψt . (2) The A[P+ ]-module A[N0 ] is étale. As A[N0 ] is a left and right free A[t N0 t −1 ]-module of rank [N0 : t N0 t −1 ] we have for x ∈ A[N0 ], x= uϕt (xu,t ) = ϕt (xu,t )u −1 u∈J (N0 /t N0 t −1 )

u∈J (N0 /t N0 t −1 )

= ψ (xu) and ψ is the left inverse of ϕ of where xu,t = ψt (u −1 x), xu,t t t t kernel u A[t N0 t −1 ] = A[t N0 t −1 ]u −1 . u∈N0 −t N0 t −1

u∈N0 −t N0 t −1

Let M be an étale A[P+ ]-module and t ∈ L + . We denote m → m ψt =0 : M → M ψt =0 the projector id M −ϕt ◦ψt along the decomposition M = ϕt (M)⊕ M ψt =0 . Lemma 7.19. Let x ∈ A[N0 ] and m ∈ M. We have ψt (ϕt (x)m) = xψt (m), (ϕt (x)m)ψt =0 = ϕt (x)(m ψt =0 ),

ψt (xϕt (m)) = ψt (x)m, (xϕt (m))ψt =0 = x ψt =0 ϕt (m).

Proof. We multiply m = (ϕt ◦ ψt )(m) + m ψt =0 on the left by ϕt (x). By the A[N0 ]-semilinearity of ϕt we have ϕt (x)m = ϕt (xψt (m)) + ϕt (x)(m ψt =0 ). As M ψt =0 is an A[t N0 t −1 ]-module, the uniqueness of the decomposition implies ψt (ϕt (x)m) = xψt (m) and (ϕt (x)m)ψt =0 = ϕt (x)(m ψt =0 ). We multiply x = (ϕt ◦ ψt )(x) + x ψt =0 on the right by ϕt (m). By the semilinearity of ϕt we have xϕt (m) = ϕt (ψt (x)m) + x ψt =0 ϕt (m). As A[N0 ]ψt =0 ϕt (M) = M ψt =0 the uniqueness of the decomposition implies ψt (xϕt (m)) = ψt (x)m, (xϕt (m))ψt =0 = x ψt =0 ϕt (m).

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Lemma 7.20. Let x ∈ A[N0 ] and m ∈ M. We have ψt (xm) = ψt (xu)ψt (u −1 m). u∈J (N0 /t N0 t −1 )

Proof. Using (7.12), replace m by u∈J (N0 /t N0 t −1 ) uϕt (m u,t ) in ψt (xm). We get ψt (xm) = ψt xuϕt (m u,t ) = ψt (xu)m u,t u∈J (N0 /t N0 t −1 )

=

u∈J (N0 /t N0 t −1 )

ψt (xu)ψt (u −1 m)

u∈J (N0 /t N0 t −1 )

using the first line of Lemma 7.19.

Proposition 7.21. Let M be an étale A[P+ ]-module. The map b−1 = (ut)−1 → ψb := ψt ◦ u −1 P− → End A (M) for t ∈ L + , u ∈ N0 , defines a canonical action of P− on M. Proof. We check that ψb1 b2 = ψb2 ◦ ψb1 for b1 = u 1 t1 , b2 = u 2 t2 ∈ P+ . We −1 have ψb1 b2 = ψt1 t2 ◦ (u 1 t1 u 2 t1−1 )−1 and ψb2 ◦ ψb1 = ψt2 ◦ u −1 2 ◦ ψt1 ◦ u 1 . −1 −1 −1 As u 2 ◦ ψt1 = ψt1 ◦ t1 u 2 t1 , it remains only to show ψt2 ψt1 = ψt1 t2 . For the sake of simplicity, we note ϕi = ϕti , ψi = ψti . For m ∈ M we have m = ϕ1 (ϕ2 ◦ ψ2 (ψ1 (m)) + ψ1 (m)ψ2 =0 ) + m ψ1 =0 . This is also m = (ϕt1 t2 ◦ ψ2 ◦ ψ1 )(m) + ϕ1 (ψ1 (m)ψ2 =0 ) + m ψ1 =0 because ϕ1 ◦ ϕ2 = ϕt1 t2 . By the uniqueness of the decomposition m = (ϕt1 t2 ◦ ψt1 t2 )(m) + m ψt1 t2 =0 we are reduced to show that M ψt1 t2 =0

=

ϕ1 (M ψ2 =0 ) + M ψ1 =0 .

It is enough to prove the inclusion M ψt1 t2 =0 ⊂ ϕ1 (M ψ2 =0 ) + M ψ1 =0 to get the equality because M = ϕt1 t2 (M) ⊕ V with V equal to any of them. Hence we want to show uϕt1 t2 (M) ⊂ ϕ1 uϕ2 (M) u∈N0 −t1 t2 N0 (t1 t2 )−1

+

u∈N0 −t2 N0 t2−1

uϕ1 (M).

(7.13)

u∈N0 −t1 N0 t1−1

As ϕ1 ◦ u ◦ ϕ2 = t1 ut1−1 ◦ ϕt1 t2 the right side of (7.13) is uϕt1 t2 (M) + uϕ1 (M). u∈t1 N0 t1−1 −t1 t2 N0 (t1 t2 )−1

u∈N0 −t1 N0 t1−1

As ϕt1 t2 = ϕ1 ◦ ϕ2 we have ϕt1 t2 (M) ⊂ ϕ1 (M). Hence (7.13) is true.

270

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Lemma 7.22. Let f : M → M be an A-morphism between two étale A[P+ ]-modules M and M . Then f is P+ -equivariant if and only if f is P− -equivariant (for the canonical action of P− ). Proof. Let t ∈ L + . We suppose that f is N0 -equivariant and we show that f ◦ ϕt = ϕt ◦ f is equivalent to f ◦ ψt = ψt ◦ f . Our arguments follow the proof of ([5] Prop. II.3.4). (a) We suppose f ◦ ϕt = ϕt ◦ f . Then f (ϕt (M)) = ϕt ( f (M)) is contained in ϕt (M ) and f (M ψt =0 ) = u∈N0 −t N0 t −1 uϕt ( f (M)) is contained in M ψt =0 . By Lemma 7.17, this implies f ◦ ϕt ◦ ψt = ϕt ◦ ψt ◦ f . As f ◦ ϕt = ϕt ◦ f and ϕt is injective this is equivalent to f ◦ ψt = ψt ◦ f . (b) We suppose f ◦ ψt = ψt ◦ f . Let m ∈ M. Then f (ϕt (m)) belongs to ϕt (M) because ϕt (M) is the subset of x ∈ M such that ψt (u −1 x) = 0 for all u ∈ N0 − t N0 t −1 and we have ψt (u −1 f (ϕt (m))) = f (ψt (u −1 (ϕt (m)))). Let x(m) ∈ M be the element such that f (ϕt (m)) = ϕt (x(m)). We have x(m) = ψt ϕt (x(m)) = ψt ( f (ϕt (m))) = f (ψt ϕt (m)) = f (m). Therefore f (ϕt (m)) = ϕt ( f (m)).

Proposition 7.23. The category M A (P+ )et of étale A[P+ ]-modules is abelian and has a natural fully faithful functor into the abelian category M A (P− ) of A[P− ]-modules. Proof. From Proposition 7.21 and Lemma 7.22, it suffices to show that the kernel and the image of a morphism f : M → M between two étale modules M, M , are étale. Since the ring homomorphism ϕt is flat, for t ∈ L + , the functor t := A[N0 ] ⊗ A[N0 ],ϕt − sends the exact sequence (E)

0 → Ker f → M → M → Coker f → 0

(7.14)

to an exact sequence 0 → t (Ker f ) → t (M) → t (M ) → t (Coker f ) → 0, (7.15) and the natural maps j− : t (−) → − define a map t (E) → (E). The maps j M and j M are isomorphisms because M and M are étale, hence the maps jKer f and jCoker f are isomorphisms, i.e. Ker f and Coker f are étale. (t (E))

Note that a subrepresentation of an étale representation of P+ is not necessarily étale nor stable by P− .

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Remark 7.24. An arbitrary direct product or a projective limit of étale A[P+ ]modules is étale. Proof. Since the A[t N0 t −1 ]-module A[N0 ] is free of finite rank, for t ∈ L + , the tensor product A[N0 ] ⊗ A[t N0 t −1 ] − commutes with arbitrary projective limits.

3.2. Induced representation M P Let P be a locally profinite group, semi-direct product P = N L of closed subgroups N , L, let N0 ⊂ N be an open profinite subgroup, and let s be an element of the centre Z(L) of L such that L = L − s Z (notation of Section 3) and (Nk := s k N0 s −k )k∈Z is a decreasing sequence of union N and trivial intersection. As the conjugation action L × N → N of L on N is continuous and N0 is compact open in N , the subgroups L 0 ⊂ L , P0 ⊂ P are open and the monoids P+ , P− are open in P. We have P = P− s Z

=

s Z P+

because, for n ∈ N and t ∈ L, there exists k ∈ N and n 0 ∈ N0 such that n = s −k n 0 s k and ts −k ∈ L − . Thus tn = ts −k n 0 s k ∈ P− s k and (tn)−1 ∈ s −k P+ . In particular P is generated by P+ and by its inverse P− . Let M be an étale left A[P+ ]-module. We denote by ϕ the action of s on M and by ψ the canonical left inverse of ϕ, by M P = Ind PP− (M) the A[P]-module induced from the canonical action of P− on M (Section 2.1). When f : P → M is an element of M P , the values of f on s N determine the values of f on N and reciprocally because, for any u ∈ N0 , k ∈ N, f (s −k us k ) = (ψ k ◦ u)( f (s k )), f (s k ) = (v ◦ ϕ k )( f (s −k v −1 s k )).

(7.16)

v∈J (N0 /Nk )

The first equality is obvious from the definition of Ind PP− , the second equality is obvious by the first equality as the idempotents (v ◦ ϕ k ◦ ψ k ◦ v −1 )v∈J (N0 /Nk ) are orthogonal of sum the identity, by Lemma 7.17. Proposition 7.25. (a) The restriction to s Z is an A[s Z ]-equivariant isomorphism MP

→

Z

Indss −N (M).

(b) The restriction to N is an N -equivariant bijection from M P to Ind N N0 (M).

272

Peter Schneider, Marie-France Vigneras, and Gergely Zabradi

Proof. (a) As P = P− s Z and s −N ⊂ P− ∩ s Z (it is an equality if N is not trivial), the restriction to s Z is a s Z -equivariant injective map M P → Z Z Indss −N (M). To show that the map is surjective, let φ ∈ Indss −N (M) and b ∈ P. Then, for b = b− s r with b− ∈ P− , r ∈ Z, f (b)

:=

b− φ(s r )

is well defined because the right side depends only on b, and not on the s r with b , b ∈ choice of (b− , r ). Indeed for two choices b = b− s r = b− − − P− , r ≥ r in Z, we have

r −r b− φ(s r ) = b− s φ(s r ) = b− φ(s r ).

The well defined function b → f (b) on P belongs obviously to M P and its restriction to s Z is equal to φ. (b) As P− ∩ N = N0 the restriction to N is an N -equivariant map M P → P Ind N N0 (M). The map is injective because the restriction to N of f ∈ M N determines the restriction of f to s by (7.16) which determines f by (a). We have the natural injective map f → φ f

Z

Indss −N (M) → M P → Ind N N0 (M)

:

φ f (s −k us k ) = (ψ k ◦ u)( f (s k ))

(7.17)

for

k ∈ N, u ∈ N0 ,

→

Indss −N (M)

and we have the map φ → f φ

:

Ind N N0 (M)

Z

defined by

f φ (s k ) =

(v ◦ ϕ k )(φ(s −k v −1 s k ))

for

k ∈ N.

v∈J (N0 /Nk )

Indeed the function f φ satisfies ψ( f φ (s k+1 )) = f φ (s k ): since ψ ◦ u ◦ ϕ k+1 = s −1 us ◦ ϕ k when u ∈ N1 and is 0 otherwise, we have ψ( f φ (s k+1 )) = ψ( (v ◦ ϕ k+1 )(φ(s −k−1 v −1 s k+1 )) v∈J (N0 /Nk+1 )

=

(s −1 vs ◦ ϕ k )(φ(s −k−1 v −1 s k+1 )).

v∈N1 ∩J (N0 /Nk+1 )

The last term is

(v ◦ ϕ k )(φ(s −k v −1 s k )) = f φ (s k )

v∈J (N0 /Nk )

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because s −1 (N1 ∩ J (N0 /Nk+1 ))s is a system of representatives of N0 /Nk and each term of the sum does not depend on the representative. Indeed for u ∈ N0 , (vs k us −k ◦ ϕ k )(φ(s −k (vs k us −k )−1 s k ) = (v ◦ ϕ k ◦ u)(φ(u −1 s −k v −1 s k )) = (v ◦ ϕ k )(φ(s −k v −1 s k )). For u ∈ N0 , k ∈ N, we have φ fφ (s −k us k ) = (ψ k ◦ u) f φ (s k ) = (ψ k ◦ uv ◦ ϕ k )(φ(s −k v −1 s k )) = φ(s −k us k ) v∈J (N0 /Nk )

k where the last equality comes from Ker ψ k = u∈N0 −Nk uϕ (M). Moreover, we have f φ f = f as a consequence of Lemma 7.17. Proposition 7.26. The induction functor Ind PP−

:

M A (P+ )et → M A (P− ) → M A (P)

is exact. Proof. The canonical action of any element of P− on an étale A[P+ ]-module is surjective. Apply Lemma 7.9. Proposition 7.27. Let f ∈ M P . Let n, n ∈ N and t ∈ L + and denote by k(n) the smallest integer k ∈ N such that n ∈ N−k . We have: (n f )(s m ) = (s m ns −m )( f (s m )) (t −1 f )(s m ) = ψt ( f (s m )) (s k f )(n ) =

for all m ≥ k(n),

(s f )(s m ) = f (s m+1 )

and k

vϕ ( f (s

−k −1 k

v

n s ))

for all m ∈ Z,

for all k ≥ 1,

v∈J (N0 /Nk )

(t −1 f )(n ) = ψt ( f (tn t −1 ))

and

(n f )(n ) = f (n n).

Proof. The formulas (s f )(s m ) = f (s m+1 ), (n f )(n ) = f (n n) are obvious. It is clear that (t −1 f )(s m ) = f (s m t −1 ) = f (t −1 s m ) = t −1 ( f (s m )) = ψt ( f (s m )), (t −1 f )(n ) = f (nt −1 ) = f (t −1 tn t −1 ) = t −1 ( f (tn t −1 )) = ψt ( f (tn t −1 )). n f (s m ) = f (s m n) = f (s m ns −m s m ) = (s m ns −m ) f (s m ).

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Using Lemma 7.17, we write (s k f )(n ) =

vϕ k (ψ k (v −1 ((s k f )(n )))),

v∈J (N0 /Nk )

ψ (v k

−1

((s f )(n ))) = ψ k (v −1 ( f (n s k ))) = ψ k ( f (v −1 n s k )) k

We obtain (s k f )(n ) =

= f (s −k v −1 n s k ).

v∈J (N0 /Nk ) vϕ( f (s

−k v −1 n s k )).

Definition 7.28. The s-model and the N -model of M P are the spaces Z Indss −N (M) lim M and Ind N N0 (M), respectively, with the action of P ← − ψ

described in Proposition 7.27.

3.3. Compactly induced representation McP The map ev0 M P → M,

f → f (1),

admits a splitting σ0 : M → M P . For m ∈ M, σ0 (m) vanishes on N − N0 and is equal to nm on n ∈ N0 and to ϕ k (m) on s k for k ∈ N. In particular, by Proposition 7.25.b, σ0 is independent of the choice of s. Lemma 7.29. The map ev0 is P− -equivariant, the map σ0 is P+ -equivariant, the A[P+ ]-modules σ0 (M) and M are isomorphic. Proof. It is clear on the definition of M P that ev0 is P− -equivariant. We show that σ0 is L + -equivariant using the s-model. Let t ∈ L + . We choose t ∈ L + , r ∈ N with t t = s r . Then ϕt ϕt = ϕ r and ϕt = ψt ϕ r . We obtain for tσ0 (m)(s k ) = σ0 (m)(s k t) the following expression σ0 (m)(t −1 s k+r ) = ψt (σ0 (m)(s k+r )) = ψt ϕ r+k (m) = ϕt ϕ k (m) = ϕ k ϕt (m) = σ0 (tm)(s k ). Hence tσ0 (m) = σ0 (tm). We show that σ0 is N0 -equivariant using the N -model. Let n 0 ∈ N0 and m ∈ M. Then n 0 σ0 (m) = σ0 (n 0 m), because for k ∈ N, u ∈ N0 , n 0 σ0 (m)(s −k us k ) = σ0 (m)(s −k us k n 0 ) = σ0 (m)(s −k us k n 0 s −k s k ) = (ψ k ◦ us k n 0 s −k ◦ ϕ k )(m) = (ψ k ◦ u ◦ ϕ k )(n 0 m) = σ0 (n 0 m)(s −k us k ).

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275

The compact induction of M from P− to P is defined to be the A[P]submodule c-Ind PP− (M) := McP of M P generated by σ0 (M). The space McP is the subspace of functions f ∈ M P with compact restriction to N , equivalently such that f (s k+r ) = ϕ k ( f (s r )) for all k ∈ N and some r ∈ N. The restriction to s Z is an s Z -isomorphism (Proposition 7.25) McP

→

Z

c-Indss −N ,ψ (M).

By Proposition 7.11, the map A[P] ⊗ A[P+ ] M → c-Ind PP− (M) [s −k ] ⊗ m → (ϕ −k ◦ σ0 )(m) is an isomorphism. Lemma 7.30. The compact induction functor from P− to P is isomorphic to c-Ind PP− A[P] ⊗ A[P+ ] M A (P+ )et → M A (P),

(7.18)

and is exact. Proof. For the exactness see Corollary 7.12.

3.4. P-equivariant map Res : C c∞ (N, A) → End A (M P ) Let Cc∞ (N , A) be the A-module of locally constant compactly supported functions on N with values in A, with the usual product of functions and with the natural action of P, P × Cc∞ (N, A) → Cc∞ (N , A),

(b, f ) → (b f )(x) = f (b−1 .x).

For any open compact subgroup U ⊂ N , the subring C ∞ (U, A) ⊂ Cc∞ (N , A) of functions f supported in U , has a unit equal to the characteristic function 1U of U , and is stable by the P-stabilizer PU of U . We have b1U = 1b.U . The A[PU ]-module C ∞ (U, A) and the A[P]-module C c∞ (N, A) are cyclic generated by 1U . The monoid P+ = N0 L + acts on End A (M) by P+ × End A (M) → End A (M) (b, F) → ϕb ◦ F ◦ ψb . Note that we have ψut = ψt ◦ u −1 .

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Proposition 7.31. There exists a unique P+ -equivariant A-linear map res C ∞ (N0 , A)

→

End A (M)

respecting the unit. It is a homomorphism of A-algebras. Proof. If the map res exists, it is unique because the A[P+ ]-module C ∞ (N0 , A) is generated by the unit 1 N0 . The existence of res is equivalent to Lemma 7.17 as we will show below. For b ∈ P+ we have the idempotent res(1b.N0 ) := ϕb ◦ ψb ∈ End A (M). (7.19) + We claim that for any finite disjoint union b.N0 = i∈I bi .N0 with bi ∈ P+ , the idempotents res(1bi .N0 ) are orthogonal of sum res(1b.N0 ). We may assume that b = 1, since the inclusion bi .N0 ⊂ b.N0 yields b−1 bi ∈ P+ . Write bi = u i ti with u i ∈ N0 and ti ∈ L + , and choose t ∈ L + such that t ∈ ti L + , say t = ti li (with li ∈ L + ). Let (n i j ) j be a system of representatives for N0 /li .N0 . Since M is étale, Lemma 7.17 shows that, for each i, the idempotents (ϕn i j li ◦ ψn i j li ) j are orthogonal, with sum id M . Note that (vi j := u i ti n i j ti−1 )(i, j) form a system of representatives for N0 /t .N0 , so again by Lemma 7.17 the idempotents (ϕvi j t ◦ ψvi j t )(i, j) are orthogonal with sum id M . The claim follows, since vi j t = bi (n i j li ), so ϕvi j t ◦ ψvi j t = ϕbi ◦ ϕn i j li ◦ ψn i j li ◦ ψbi . The claim being proved, we get an A-linear map res : C ∞ (N0 , A) → End A (M) which is clearly P+ -equivariant and respects the unit. It respects the product because, for f 1 , f 2 ∈ C ∞ (N0 , A), there exists t ∈ L + such that f 1 and f 2 are constant on each coset ut N0 t −1 ⊂ N0 . Hence res( f 1 f 2 ) = v∈J (N0 /t N0 t −1 ) f 1 (v) f 2 (v) res(1vt.N0 ) = res( f 1 ) ◦ res( f 2 ). The group P = N L acts on End A (M P ) by conjugation. We have the canonical injective algebra map F → σ0 ◦ F ◦ ev0

:

End A M → End A (M P ).

(7.20)

It is P+ -equivariant since, by Lemma 7.29 for b ∈ P+ , we have b ◦ σ0 ◦ F ◦ ev0 ◦b−1 = σ0 ◦ ϕb ◦ F ◦ ψb ◦ ev0 .

(7.21)

We consider the composite P+ -equivariant algebra homomorphism res

C ∞ (N0 , A) −−→ End A (M) −→ End A (M P ), sending 1 N0 to R0 := σ0 ◦ ev0 and, more generally, 1b.N0 to b ◦ R0 ◦ b−1 for b ∈ P+ .

G-equivariant sheaves on G/P For f ∈ M P , R0 ( f ) ∈ M P vanishes on N − N0 and R0 ( f )(s k ) In the N -model, R0 is the restriction to N0 . We show now that the composite morphism extends to Cc∞ (N , A).

ϕ k ( f (1)).

277 =

Proposition 7.32. There exists a unique P-equivariant A-linear map Res Cc∞ (N , A) → End A (M P ) such that Res(1 N0 ) = R0 . The map Res is an algebra homomorphism. Proof. If the map Res exists, it is unique because the A[P]-module Cc∞ (N , A) is generated by 1 N0 . For b ∈ P we define Res(1b.N0 ) := b ◦ R0 ◦ b−1 . We prove that b ◦ R0 ◦ b−1 depends only on the subset b.N0 ⊂ N , and that for any finite disjoint decomposition of b.N0 = 2i∈I bi .N0 with bi ∈ P, the idempotents bi ◦ R0 ◦ bi−1 are orthogonal of sum b ◦ R0 ◦ b−1 . The equivalence relation b.N0 = b .N0 for b, b ∈ P is equivalent to b P0 = b P0 because the normalizer of N0 in P is P0 . We have b ◦ R0 ◦ b−1 = R0 when b ∈ P0 because res(1b.N0 ) = res(1 N0 ) = id (Proposition 7.31). Hence b ◦ R0 ◦ b−1 depends only on b.N0 . By conjugation by b−1 , we reduce to prove that the idempotents bi ◦ R0 ◦ bi−1 are orthogonal of sum R0 for any disjoint decomposition of N0 = 2i ∈I bi .N0 and bi ∈ P. The bi belong to P+ , and Proposition 7.31 implies the equality. To prove that the A-linear map Res respects the product it suffices to check that, for any t ∈ L + , k ∈ N, the endomorphisms Res(1vt N0 t −1 ) ∈ End A (M P ) are orthogonal idempotents, for v ∈ J (N−k /t N0 t −1 ). We already proved this for k = 0 and for all t ∈ L + , and s k J (N−k /t N0 t −1 )s −k = J (N0 /s k t N0 t −1 s −k ). Hence we know that (s k ◦ Res(1vt N0 t −1 ) ◦ s −k )v∈J (N−k /t N0 t −1 ) are orthogonal idempotents. This implies that (Res(1vt N0 t −1 ))v∈J (N−k /t N0 t −1 ) are orthogonal idempotents. Remark 7.33. (i) The map Res is the restriction of an algebra homomorphism C ∞ (N, A) → End A (M P ), where C ∞ (N , A) is the algebra of all locally constant functions on N . For this we observe

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Peter Schneider, Marie-France Vigneras, and Gergely Zabradi

1. The A[P+ ]-module C ∞ (N0 , A) is étale. For t ∈ L + , the corresponding ψt satisfies (ψt f )(x) = f (t xt −1 ). 2. The map ( f, m) → res( f )(m) C ∞ (N0 , A) × M → M is ψt equivariant, hence induces to a pairing C ∞ (N0 , A) P × M P → M P . 3. The A[P]-module C ∞ (N0 , A) P is canonically isomorphic to C ∞ (N, A). (ii) The monoid P+ × P+ acts on End A (M) by ϕ(b1 ,b2 ) F := ϕb1 ◦ F ◦ ψb2 . For this action End A (M) is an étale A[P+ × P+ ]-module, and we have ψ(b1 ,b2 ) F = ψb1 ◦ F ◦ ϕb2 . Definition 7.34. For any compact open subsets V ⊂ U ⊂ N0 and m ∈ M, we denote resU := res(1U ), MU := resU (M), m U := resU (m), resU V := resV | MU : MU → M V . For any compact open subsets V ⊂ U ⊂ N and f ∈ M P ResU := Res(1U ), MU := ResU (M P ), fU := ResU ( f ), ResU V := ResV | MU : MU → M V . Remark 7.35. The notation is coherent for U ⊂ N0 , as follows from the following properties. For b ∈ P+ we have – resb.U = ϕb ◦ resU ◦ψb (Proposition 7.31); – b ◦ ResU = σ0 ◦ ϕb ◦ resU ◦ ev0 and ResU ◦b−1 = σ0 ◦ resU ◦ψb ◦ ev0 ; – (ResU f )(1) = resU ( f (1)). We note also that Proposition 7.32 implies: Corollary 7.36. For any compact open subset U ⊂ N equal to a finite disjoint union U = 2i∈I Ui of compact open subsets Ui ⊂ N , the idempotents ResUi are orthogonal of sum ResU . Corollary 7.37. For u ∈ N , the projector Resu N0 is the restriction to N0 u −1 in the N -model. Proof. We have Resu N0 = u ◦ Res N0 ◦u −1 and Res N0 is the restriction to N0 in the N -model. Hence for x ∈ N , (Resu N0 f )(x) = (Res N0 u −1 f )(xu) vanishes for x ∈ N − N0 u −1 and for v ∈ N0 , (Resu N0 f )(vu −1 ) = (u −1 f )(v) = f (vu −1 ). The constructions are functorial. A morphism f : M → M of A[P+ ]modules, being also A[P− ]-equivariant induces a morphism Ind PP− ( f ) :

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279

M P → M P of A[P]-modules. On the other hand, M P is a module over the non unital ring Cc∞ (N , A) through the map Res. The morphism Ind PP− ( f ) is Cc∞ (N , A)-equivariant. Since Res is P-equivariant , it suffices to prove that Ind PP− ( f ) respects R0 = σ0 ◦ ev0 which is obvious.

3.5. P-equivariant sheaf on N We formulate now Proposition 7.32 in the language of sheaves. Theorem 7.38. One can associate to an étale A[P+ ]-module M, a Pequivariant sheaf S M of A-modules on the compact open subsets U ⊂ N , with – sections MU on U , – restrictions ResU V for any open compact subset V ⊂ U , – action f → b f = Resb.U (b f ) MU → Mb.U of b ∈ P. Proof. (a) ResU U is the identity on MU = ResU (M) because ResU is an idempotent. V U (b) ResW ◦ ResU V = Res W for compact open subsets W ⊂ V ⊂ U ⊂ N . Indeed, we have ResW ◦ ResV = ResW on MU . (c) If U is the union of compact open subsets Ui ⊂ U for i ∈ I , and U U f i ∈ MUi satisfying ResUii ∩U j ( f i ) = ResUij∩U j ( f j ) for i, j ∈ I , there exists a unique f ∈ MU such that ResU Ui ( f ) = f i for all i ∈ I . (c1) True when (Ui )i∈I is a partition of U because I is finite and ResU is the sum of the orthogonal idempotents ResUi . (c2) True when I is finite because the finite covering defines a finite partition of U by open compact subsets V j for j ∈ J , such that V j ∩ Ui is empty or equal to V j for all i ∈ I, j ∈ J . By hypothesis on the f i , if V j ⊂ Ui , then the restriction of f i to V j does not depend on the choice of i, and is denoted by φ j . Applying (c1), there is a unique f ∈ MU such that ResV j ( f ) = φ j for all j ∈ J . Note also that the V j contained in Ui form a finite partition of Ui and that fi is the unique element of MUi such that ResV j ( f i ) = φ j for those j. We deduce that f is the unique element of MU such that ResUi ( f ) = f i for all i ∈ I . (c3) In general, U being compact, there exists a finite subset I ⊂ I such that U is covered by Ui for i ∈ I . By (c2), there exists a unique f I ∈ MU such that f i = ResUi ( f I ) for all i ∈ I . Let i ∈ I not belonging to I . Then the nonempty intersections Ui ∩U j for j ∈ I form a finite covering of Ui by compact open subsets. By (c2), f i is the unique element of MUi such that ResUi ∩U j ( f j ) = ResUi ∩U j ( f i ) for all nonempty Ui ∩ U j . The

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element ResUi ( f I ) has the same property, we deduce by uniqueness that f i = ResUi ( f I ). (d) Let f ∈ MU . When b = 1 we have clearly 1( f ) = f . For b, b ∈ P, we have (bb )( f ) = Res(bb ).U ((bb ) f ) = Resb.(b .U ) (b(b f )) = b(b f ). For a compact open subset V ⊂ U , we have b ◦ ResV ◦ ResU = ResbV ◦b ◦ ResU in End A M P hence b ResU V = Resb.V b. Proposition 7.39. Let H be a topological group acting continuously on a locally compact totally disconnected space X . Any H -equivariant sheaf F (of A-modules) on the compact open subsets of X extends uniquely to a H -equivariant sheaf on the open subsets of X . Proof. This is well known. See [4] §9.2.3 Prop. 1. Remark 7.40. The space of sections on an open subset U ⊂ X is the projective limit of the sections F (V ) on the compact open subsets V of U for the restriction maps F(V ) → F (V ) for V ⊂ V . By this general result, the P-equivariant sheaf defined by M on the compact open subsets of N (Theorem 7.38), extends uniquely to a P-equivariant sheaf S M on (arbitrary open subsets of) N . We extend the definitions 7.34 to arbitrary open subsets U ⊂ N . We denote by ResU V the restriction maps for open subsets V ⊂ U of N , by ResU = ResUN and by MU = ResU (M P ). In this way we obtain an exact functor M → (MU )U from M A (P+ )et to the category of P-equivariant sheaves of A-modules on N . Note that for a compact open subset U even the functor M → MU is exact. Proposition 7.41. The representation of P on the global sections of the sheaf S M is canonically isomorphic to M P . Proof. We have the obvious P-equivariant homomorphism (ResU )U

M P −−−−−→ M N = lim MU . ← − U

The group N is the union of s −k .N0 = s −k N0 s k for k ∈ N. Hence M N = limk M N−k . In the s-model of M P we have Ress −k .N0 = R−k and by the lemma ← − 7.13 the morphism f → (Ress −k .N0 ( f ))k∈N M P is bijective.

→

MN

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281

Corollary 7.42. The restriction ResUN : M N → MU from the global sections to the sections on an open compact subset U ⊂ N is surjective with a natural splitting. Proof. It corresponds to an idempotent ResU = Res(1U ) ∈ End A (M P ).

3.6. Independence of N0 Let U ⊂ N be a compact open subgroup. For n ∈ N and t ∈ L, the inclusion ntU t −1 ⊂ U is obviously equivalent to n ∈ U and tU t −1 ⊂ U . Hence the P-stabilizer PU := {b ∈ P | b.U ⊂ U } of U is the semi-direct product of U by the L-stabilizer L U of U . As the decreasing sequence (Nk = s k N0 s −k )k∈N forms a basis of neighbourhoods of 1 in N and N = ∪r∈Z N−r , the compact open subgroup U ⊂ N contains some Nk and is contained in some N−r . This implies that the intersection L U ∩ s N is not empty hence is equal to sUN where sU = s kU for some kU ≥ 1. The monoid PU = U L U and the central element sU of L satisfy the same conditions as (P+ = N0 L + , s), given at the beginning of Section 3.2. Our theory associates to each étale A[PU ]-module a P-equivariant sheaf on N . The subspace MU ⊂ M P (Definition 7.34) is stable by PU because b ◦ ResU = Resb.U ◦b for b ∈ P and Mb.U = Resb.U (M) ⊂ ResU (M) = MU . As MU = ⊕u∈J (U/t.U ) u Mt.U for t ∈ L U the A[PU ]-module MU is étale. Proposition 7.43. The P-equivariant sheaf S M on N associated to the étale A[P+ ]-module M is equal to the P-equivariant sheaf on N associated to the étale A[PU ]-module MU . Proof. For b ∈ PU we denote by ϕU,b the action of b on MU and by ψU,b the left inverse of ϕU,b with kernel MU −b.U . We have MU = Mb.U ⊕ MU −b.U and for fU ∈ MU , ϕU,b ( f U ) = b fU , ψU,b ( fU ) = b−1 Resb.U ( fU ), (ϕU,b ◦ ψU,b )( f U ) = Resb.U ( f U ).

(7.22)

By the last formula and Remark 7.35, the sections on b.U and the restriction maps from MU to Mb.U in the two sheaves are the same for any b ∈ PU . This implies that the two sheaves are equal on (the open subsets of) U . By symmetry they are also equal on (the open subsets of) N0 . The same arguments for arbitrary compact open subgroups U, U ⊂ N imply that the P-equivariant sheaves on N associated to the étale A[PU ]-module MU and to the étale A[PU ]-module MU are equal on (the open subsets of) U and on (the open

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subsets of) U . Hence all these sheaves are equal on (the open subsets of) the compact open subsets of N and also on (the open subsets of) N .

3.7. Etale A[ P+ ]-module and P-equivariant sheaf on N Proposition 7.44. Let M be an A[P+ ]-module such that the action ϕ of s on M is étale. Then M is an étale A[P+ ]-module. Proof. Let t ∈ L + . We have to show that the action ϕt of t on M is étale. As L = L + s −N with s central in L, there exists k ∈ N such that s k t −1 ∈ L + . This implies ϕ k = ϕs k t −1 ◦ ϕt in End A (M) and s k N0 s −k ⊂ t N0 t −1 . As ϕ is injective, ϕt is also injective. For any representative system J (t N0 t −1 /s k N0 s −k ) of t N0 t −1 /s k N0 s −k and any representative system J (N0 /t N0 t −1 ) of N0 /t N0 t −1 , the set of uv for u ∈ J (N0 /t N0 t −1 ) and v ∈ J (t N0 t −1 /s k N0 s −k ) is a representative system J (N0 /s k N0 s −k ) of N0 /s k N0 s −k . Let ψ be the canonical left inverse of ϕ. We have id = u◦ v ◦ ϕ k ◦ ψ k ◦ v −1 ◦ u −1 u∈J (N0 /t N0 t −1 )

=

u∈J (N0 /t N0 t −1 )

u◦

u∈J (N0 /t N0 t −1 )

=

v∈J (t N0 t −1 /s k N0 s −k )

v ◦ ϕt ◦ ϕt −1 s k ◦ ψ k ◦ v −1 ◦ u −1

v∈J (t N0 t −1 /s k N0 s −k )

u ◦ ϕt ◦ (

v ◦ ϕt −1 s k ◦ ψ k ◦ v −1 ) ◦ u −1 .

v∈J (N0 /t −1 s k N0 s −k t)

We deduce that ϕt is étale of canonical left inverse ψt the expression between parentheses. Corollary 7.45. An A[P+ ]-submodule M ⊂ M of an étale A[P+ ]-module M is étale if and only if it is stable by the canonical inverse ψ of ϕ. Proof. If M is ψ-stable, for m ∈ M every m u,s belongs to M in (7.12). Hence the action of s on M is étale, and M is étale by Proposition 7.44. Corollary 7.46. The space S(N0 ) of global sections of a P+ -equivariant sheaf S on N0 is an étale representation of P+ , when the action ϕ of s on S(N0 ) is injective. Proof. By Proposition 7.44 it suffices to show that S(N0 ) = ⊕u∈J (N0 /s N0 s −1 ) us(S(N0 )). But this equality is true because N0 is the disjoint sum of the open subsets us N0 s −1 = us.N0 and S(us.N0 ) = us(S(N0 )).

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The canonical left inverse ψ of the action ϕ of s on S(N0 ) vanishes on S(us N0 s −1 ) for u = 1 and on S(s N0 s −1 ) is equal to the isomorphism S(s N0 s −1 ) → S(N0 ) induced by s −1 . Theorem 7.47. The functor M → S M is an equivalence of categories from the abelian category of étale A[P+ ]-modules to the abelian category of Pequivariant sheaves of A-modules on N , of inverse the functor S → S(N0 ) of sections over N0 . Proof. Let S be a P-equivariant sheaf on N . By Corollary 7.46, the space S(N0 ) of sections on N0 is an étale representation of P+ because the action ϕ of s on S(N0 ) is injective. We show now that the representation of P on the space S(N )c of compact sections on N depends uniquely the representation of P+ on S(N0 ). The representation of N on S(N )c is defined by the representation of N0 on S(N0 ), because S(N )c = ⊕u∈J (N/N0 ) S(u N0 ) and S(u N0 ) = uS(N0 ) for u ∈ N . The group P is generated by N and L + . For t ∈ L + , the action of t on S(N )c is defined by the action of N on S(N )c and by the action of t on S(N0 ), because tS(u N0 ) = tut −1 tS(N0 ) with tut −1 ∈ N for u ∈ N . We deduce that the A[P]-module S(N )c is equal to the compact induced representation S(N0 )cP , and that the sheaves S and SS (N0 ) are equal. Conversely, let M be an étale A[P+ ]-module. The A[P+ ]-module S M (N0 ) of sections on N0 of the sheaf S M is equal to M (Theorem 7.38).

4. Topology 4.1. Topologically étale A[ P+ ]-modules We add to the hypothesis of Section 3.2 the following (a) A is a linearly topological commutative ring (the open ideals form a basis of neighbourhoods of 0). (b) M is a linearly topological A-module (the open A-submodules form a basis of neighbourhoods of 0), with a continuous action of P+ P+ × M → M (b, x) → ϕb (x). We call such an M a continuous A[P+ ]-module. If M is also étale in the algebraic sense (Definition 7.16) and the maps ψt , for t ∈ L + , are continuous we call M a topologically étale A[P+ ]-module.

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Lemma 7.48. Let M be a continuous A[P+ ]-module which is algebraically étale, then: (i) The maps ψt for t ∈ L + are open. (ii) If ψ = ψs is continuous then M is topologically étale. Proof. (i) The projection of M = M0 ⊕ M1 onto the algebraic direct summand M0 (with the submodule topology) is open. Indeed let V ⊂ M be an open subset, then M0 ∩ (V + M1 ) is open in M0 and is equal to the projection of V . We apply this to M = ϕt (M) ⊕ Ker ψt and to the projection ϕt ◦ ψt . Then we note that ψt (V ) = ϕt−1 ((ϕt ◦ ψt )(V )). (ii) Given any t ∈ L + we find t ∈ L + and n ∈ N such that t t = s n . Hence ψt t = ψt ◦ ψt = ψ n is continuous by assumption. As ψt is surjective and open, for any open subset V ⊂ M we have ψt−1 (V ) = ψt ((ψt ◦ ψt )−1 (V )) which is open. Lemma 7.49. (i) A compact algebraically étale A[P+ ]-module is topologically étale. (ii) Let M be a topologically étale A[P+ ]-module. The P− -action (b−1 , m) → ψb (m) P− × M → M on M is continuous. Proof. (i) The compactness of M implies that #

M = ϕt (M) ⊕

uϕt (M)

u∈(N0 −t N0 t −1 )

is a topological decomposition of M as the direct sum of finitely many closed submodules. It suffices to check that the restriction of ψt to each summand is continuous. On all summands except the first one ψt is zero. By compactness of M the map ϕt is a homeomorphism between M and the closed submodule ϕt (M). We see that ψt |ϕt (M) is the inverse of this homeomorphism and hence is continuous. (ii) Since P0 is open in P− = L −1 + P0 we only need to show that the restriction of the P− -action to t −1 P0 × M → M, for any t ∈ L + , is continuous. We contemplate the commutative diagram t −1 P0 × M t·× id

P0 × M

/M O ψt

/M

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where the horizontal arrows are given by the P− -action. The P0 -action on M induced by P− coincides with the one induced by the P+ -action. Therefore the bottom horizontal arrow is continuous. The left vertical arrow is trivially continuous, and ψt is continuous by assumption. Lemma 7.50. For any compact subgroup C ⊂ P+ , the open C-stable Asubmodules of M form a basis of neighbourhoods of 0. Proof. We have to show that any open A-submodule M of M contains an open C-stable A-submodule. By continuity of the action of P+ on M, there exists for each c ∈ C, an open A-submodule Mc of M and an open neighbourhood Hc ⊂ P+ of c such that ϕx (Mc ) ⊂ M for all x ∈ Hc . By the compactness of C, there exists a finite subset I ⊂ C such that C = ∪c∈I (Hc ∩C). By finiteness of I , the intersection M := ∩c∈I Mc ⊂ M is an open A-submodule such that M := c∈C ϕc (M ) ⊂ M. The A-submodule M is C-stable and, since M ⊂ M ⊂ M, also open. Let M be a topologically étale A[P+ ]-module. Since P0 is open in P the A-module M P is a submodule of the A-module C(P, M) of all continuous maps from P to M. We equip C(P, M) with the compact-open topology which makes it a linear-topological A-module. A basis of neighbourhoods of zero is given by the submodules C(C, M) := { f ∈ C(P, M) | f (C) ⊂ M} with C and M running over all compact subsets in P and over all open submodules in M, respectively. With M also C(P, M) is Hausdorff. It is well known that the regular action of P on C(P, M) is continuous (see for instance Proposition 7.52(ii) for a proof). Therefore M P is characterized inside C(P, M) by Z closed conditions and hence is a closed submodule. Similarly, Indss −N (M) and Z Ind N N0 (M) are closed submodules of C(s , M) and C(N , M), respectively, for the compact-open topologies. Clearly the homomorphisms of restricting maps Z (Proposition 7.25) M P → Indss −N (M) and M P → Ind N N0 (M) are continuous. Z

Lemma 7.51. The restriction maps M P → Indss −N (M) and M P → Ind N N0 (M) are topological isomorphisms. Proof. The topology on M P induced by the compact-open topology on the Z s-model Indss −N M is the topology with basis of neighbourhoods of zero Bk,M = { f ∈ M P | f (s m ) ∈ M for all − k ≤ m ≤ k}, for all k ∈ N and all open A-submodules M of M. One can replace Bk,M by Ck,M = { f ∈ M P | f (s k ) ∈ M},

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because Bk,M ⊂ Ck,M and conversely given (k, M) there exists an open Asubmodule M ⊂ M such that ψ m (M ) ⊂ M for all 0 ≤ m ≤ 2k as ψ is continuous (Lemma 7.49), hence C k,M ⊂ Bk,M . The topology on M P induced by the compact-open topology on the N -model Ind N N0 M is the topology with basis of neighbourhoods of zero Dk,M = { f ∈ M P | f (N−k ) ⊂ M}, for all (k, M) as above. We fix an auxiliary compact open subgroup P0 ⊂ P0 . It then suffices, by Lemma 7.50, to let M run, in the above families, over the open A[P0 ]-submodules M of M. Let C ⊂ P be any compact subset and let M be an open A[P0 ]-submodule of M. We choose k ∈ N large enough so that Cs −k ⊂ P− . Since Cs −k is compact and P0 is an open subgroup of P we find finitely many b1 , . . . , bm ∈ P+ −1 P . The continuity of the maps ψ implies such that Cs −k ⊂ b1−1 P0 ∪. . .∪bm bi 0 the existence of an open A[P0 ]-submodule M of M such that ψbi (M ) ⊂ M for any 1 ≤ i ≤ m. We deduce that Ck,M ⊂ C bi−1 P0 s k , M ⊂ C(C, M). i

Furthermore, by the continuity of the action of P+ on M, there exists an open submodule M such that v∈J (N0 /Nk ) vϕ k (M ) ⊂ M . The second part of the formula (7.16) then implies that Dk,M ⊂ Ck,M .

The maps ev0 : M P → M and σ0 : M → M P are continuous (Section 3.3). cont ⊂ E := End (M P ) the We denote by Endcont A A (M) ⊂ End A (M) and E subalgebra of continuous endomorphisms. We have the canonical injective algebra map (7.20) f → σ0 ◦ f ◦ ev0

:

cont Endcont . A (M) → E

Proposition 7.52. Let M be a topologically étale A[P+ ]-module. (i) If M is complete, resp. compact, the A-module M P is complete, resp. compact. (ii) The natural map P × M P → M P is continuous. (iii) Res( f ) ∈ E cont for each f ∈ Cc∞ (N , A) (Proposition 7.32).

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Proof. (i) If M is complete, by [3] TG X.9 Cor. 3 and TG X.25 Th. 2, the compact-open topology on C(P, M) is complete because P is locally compact. Hence, M P as a closed submodule is complete as well. If M is compact, the s-model of M P is compact as a closed subset of the compact space M N . Hence by Lemma 7.51, M P is compact. (ii) It suffices to show that the right translation action of P on C(P, M) is continuous. This is well known: the map in question is the composite of the following three continuous maps P × C(P, M) −→ P × C(P × P, M) (b, f ) −→ (b, (x, y) → f (yx)), P × C(P × P, M) −→ P × C(P, C(P, M)) (b, F) −→ (b, x → [y → F(x, y)]), and P × C(P, C(P, M)) −→ C(P, M) (b, ) −→ (b), where the continuity of the latter relies on the fact that P is locally compact. (iii) It suffices to consider functions of the form f = 1b.N0 for some b ∈ P. But then Res( f ) = b ◦ σ0 ◦ ev0 ◦b−1 is the composite of continuous endomorphisms.

P 4.2. Integration on N with value in Endcont A (M )

We suppose that M is a complete topologically étale A[P+ ]-module. We denote by E cont the ring of continuous A-endomorphisms of the complete A-module M P with the topology defined by the right ideals cont P EL = Homcont A (M , L)

for all open A-submodules L ⊂ M P . Lemma 7.53. E cont is a complete topological ring. Proof. It is clear that the maps (x, y) → x − y and (x, y) → x ◦ y from E cont × E cont to E cont are continuous, i.e. that E cont is a topological ring. The composite of the natural morphisms cont P P E cont → lim E cont /E L → lim Homcont A (M , M /L) ← − ← −

L

L

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is an isomorphism (the natural map M P → limL M P /L is an isomor← − phism), hence the two morphisms are isomorphisms since the kernel of the cont P P cont is map E cont → Homcont A (M , M /L) is E L . We deduce that E complete. Definition 7.54. An A-linear map Cc∞ (N, A) → E cont is called a measure on N with values in E cont . The map Res is a measure on N with values in E cont (Proposition 7.52). Let Cc (N, E cont ) be the space of compactly supported continuous maps from N to E cont . We will prove that one can “integrate” a function in Cc (N , E cont ) with respect to a measure on N with values in E cont . Proposition 7.55. There is a natural bilinear map Cc (N , E cont ) × Hom A (Cc∞ (N , A), E cont ) → E cont / ( f, λ) → f dλ. N

Proof. (a) Every compact subset of N is contained in a compact open subset. It follows that Cc (N , E cont ) is the union of its subspaces C(U, E cont ) of functions with support contained in U , for all compact open subsets U ⊂ N . cont (b) For any open A-submodule L of M P , a function in C(U, E cont /E L ) is cont cont locally constant because E /E L is discrete. An upper index ∞ means that we consider locally constant functions hence cont cont cont C(U, E cont /E L ) = C ∞ (U, E cont /E L ) = C ∞ (U, A)⊗ A E cont /E L .

There is a natural linear pairing cont cont (C ∞ (U, A) ⊗ A E cont /E L )× Hom A (C ∞ (U, A), E cont ) → E cont /E L

( f ⊗ x, λ) → xλ( f ). cont Note that E cont /E L is a right E cont -module. cont (c) Let f ∈ Cc (N, E ) and let λ ∈ Hom A (Cc∞ (N , A), E cont ). Let U ⊂ N be an open compact subset containing the support of f . For any open Acont submodule L of M P let f L ∈ Cc∞ (U, E cont /E L ) be the map induced by f . Let / cont f L dλ ∈ E cont /E L U

be of (b). The elements . the image of ( f L , λ) by the natural pairing cont = lim E cont /E cont to f dλ combine in the projective limit E U L L ← −L

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. . give an element U f dλ ∈ E cont . One checks easily that U f dλ does not depend on the choice of U . We define / / f dλ := f dλ. N

U

We recall that J (N /V ) is a system of representatives of N / V when V ⊂ N is a compact open subgroup. Corollary 7.56. Let f ∈ Cc (N , E cont ) and let λ be a measure on N with values in E cont . Then / lim f (v) λ(1vV ) = f dλ V →{1}

N

v∈J (N /V )

with the limit over compact open subgroups V ⊂ N shrinking to {1}. Proof. We choose an open compact subset U ⊂ N containing the support of f . Let L be an open o-submodule of M P and a compact open subgroup V ⊂ N such that uV .⊂ U and f L (proof of Proposition 7.55) is constant on uV for all u ∈ U . Then U f L dλ is the image of f (v) λ(1vV ) v∈J (N/V ) cont by the quotient map E cont → E cont /E L .

Lemma 7.57. Let f ∈ Cc (N , E cont ) be a continuous map with support in the compact open subset U ⊂ N , let λ be a measure on N with values in E cont , and let L ⊂ M P be any open A-submodule. There is a compact open subgroup VL ⊂ N such that U VL = U and / cont f 1uV dλ − f (u)λ(1uV ) ∈ E L N

for any open subgroup V ⊂ VL and any u ∈ U . Proof. The integral in question is the limit (with respect to open subgroups V ⊂ V ) of the net ( f (uv) − f (u))λ(1uvV ). v∈J (V /V ) cont is a right ideal it therefore suffices to find a compact open subgroup Since E L VL ⊂ N such that U VL = U and cont f (uv) − f (u) ∈ E L

for any u ∈ U and v ∈ VL .

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We certainly find a compact open subgroup V˜ ⊂ N such that U V˜ = U . The map U × V˜ → E cont (u, v) → f (uv) − f (u) is continuous and maps any (u, 1) to zero. Hence, for any u ∈ U , there is an open neighbourhood Uu ⊂ U of u and a compact open subgroup Vu ⊂ V˜ cont . Since U is compact we have U = such that Uu × Vu is mapped to E L Uu 1 ∪ . . . ∪ Uu s for finitely many appropriate u i ∈ U . The compact open subgroup VL := Vu 1 ∩ . . . ∩ Vu s then is such that U × VL is mapped to cont EL . Let C(N , E cont ) be the space of continuous functions from N to E cont . For any continuous function f ∈ C(N , E cont ), for any compact open subset U ⊂ N and for any measure λ on N with values in E cont we denote / / f dλ = f 1U dλ U

N

where 1U ∈ C ∞ (U, A) is the characteristic function of U hence f 1U ∈ Cc (N , E cont ) is the restriction of f to U . The “integral of f on U ” (with respect to the measure λ) is equal to the “integral of the restriction of f to U ”. Remark 7.58. For f ∈ Cc (N , E cont ) and φ ∈ Cc∞ (N , A) we have / / / f φd Res = φ f d Res = f d Res ◦ Res(φ). N

N

N

Proof. This is immediate from the construction of the integral and the multiplicativity of Res.

5. G-equivariant sheaf on G/ P Let G be a locally profinite group containing P = N L as a closed subgroup satisfying the assumptions of Section 3.2 such that (a) G/P is compact. (b) There is a subset W in the G-normalizer NG (L) of L such that – the image of W in NG (L)/L is a subgroup, – G is the disjoint union of Pw P for w ∈ W . We note that Pw P = N w P = PwN .

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(c) There exists w0 ∈ W such that N w0 P is an open dense subset of G. We call C := N w0 P/P the open cell of G/P. (d) The map (n, b) → nw0 b from N × P onto N w0 P is a homeomorphism. Remark 7.59. These conditions imply that G = P P P = C(w0 )C(w0−1 ) where P := w0 Pw0−1 and C(g) = Pg P for g ∈ G. Proof. The intersection of the two dense open subsets gC and C in G/P is open and not empty, for any g ∈ G. The group G acts continuously on the topological space G/P, G × G/P → G/P (g, x P) → gx P. For n, x ∈ N and t ∈ L we have nt xw0 P = nt xt −1 w0 P = (nt.x)w0 P hence the action of P on the open cell corresponds to the action of P on N introduced before Proposition 7.32, i.e. the homeomorphism N → C,

u → xu := uw0 P

is P-equivariant. When M is an étale A[P+ ]-module, this allows us to systematically view the map Res in the following as a P-equivariant homomorphism of A-algebras Res : Cc∞ (C, A) → End A (M P ) and the corresponding sheaf (Theorem 7.38) as a sheaf on C. Our purpose is to show that this sheaf extends naturally to a G-equivariant sheaf on G/P for certain étale A[P+ ]-modules. When M is a complete topologically étale A[P+ ]-module we note that also integration with respect to the measure Res (Proposition 7.55) will be viewed in the following as a map Cc (C, E cont ) → E cont / f → f d Res C

on the space Cc (C, to E cont .

E cont )

of compactly supported continuous maps from C

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5.1. Topological G-space G/ P and the map α Definition 7.60. An open subset U of G/P is called standard if there is a g ∈ G such that gU is contained in the open cell C. The inclusion gU ⊂ N w0 P/P is equivalent to U = g −1 U w0 P/P for a unique open subset U ⊂ N . An open subset of a standard open subset is standard. The translates by G of N0 w0 P/P form a basis of the topology of G/P. Proposition 7.61. A compact open subset U ⊂ G/P is a disjoint union 5 U= g −1 V w0 P/P g∈I

where V ⊂ N is a compact open subgroup and I ⊂ G a finite subset. Proof. We first observe that any open covering of U can be refined into a disjoint open covering. In our case, this implies that U has a finite disjoint covering by standard compact open subsets. Let g −1 U w0 P/P ⊂ G/P be a standard compact open subset. Then U = 2u∈J uV (disjoint union) with a finite set J ⊂ U and V ⊂ N is a compact open subgroup. Then g −1 U w0 P/P = 2h∈I h −1 V w0 P/P (disjoint union) where I ={u −1 g | u∈J }. For g ∈ G and x in the nonempty open subset g −1 C ∩ C of G/P (Remark 7.59), there is a unique element α(g, x) ∈ P such that, if x = uw0 P/P with u ∈ N , then guw0 N = α(g, x)uw0 N . We give some properties of the map α. Lemma 7.62. Let g ∈ G. Then (i) g −1 C ∩ C = C if and only if g ∈ P. (ii) The map α(g, .) : g −1 C ∩ C → P is continuous. (iii) We have gx = α(g, x)x for x ∈ g −1 C ∩ C and we have α(b, x) = b for b ∈ P and x ∈ C. Proof. (i) We have g −1 C ∩ C = C if and only if g N w0 P ⊂ N w0 P if and only if g ∈ P. Indeed, the condition h Pw0 P ⊂ Pw0 P on h ∈ G depends only on Ph P and for w ∈ W , the condition w Pw0 P ⊂ Pw0 P implies ww0 ∈ Pw0 P hence ww0 ∈ w0 L by the hypothesis (b) hence w ∈ L.

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(ii) Let N g ⊂ N be such that N g w0 P/P = g −1 C ∩ C. It suffices to show that the map u → α(g, uw0 P)u : Ng → P is continuous. This follows from the continuity of the maps u → guw0 N : N g → Pw0 P/N = Pw0 N/N and bw0 N → b : Pw0 N /N → P. (iii) Obvious. Lemma 7.63. Let g, h ∈ G and x ∈ (gh)−1 C ∩h −1 C ∩C. Then hx ∈ g −1 C ∩C and we have α(gh, x) = α(g, hx)α(h, x). Proof. The first part of the assertion is obvious. Let x = uw0 P and hx = vw0 P with u, v ∈ N . We have huw0 N = α(h, x)uw0 N , gvw0 N = α(g, hx)vw0 N, and α(gh, x)uw0 N = ghuw0 N . The first identity implies α(h, x)uw0 P = vw0 P, hence v −1 α(h, x)u ∈ P ∩ w0 Pw0−1 . Hypothesis (d) easily yields P ∩ w0 Pw0−1 = L, hence α(h, x)u = vt for some t ∈ L. Multiplying the second identity on the right by w0−1 tw0 we obtain gvtw0 N = α(g, hx)vtw0 N = α(g, hx)α(h, x)uw0 N . Finally, by inserting the first identity into the right-hand side of the third identity we get α(gh, x)uw0 N = gα(h, x)uw0 N = gvtw0 N = α(g, hx)α(h, x)uw0 N which is the assertion. It will be technically convenient later to work on N instead of C. For g ∈ G let therefore N g be the open subset of N such that C ∩ g −1 C = Ng w0 P/P. We have N g = N if and only if g ∈ P (Lemma 7.62 (i)). We have the ∼ homeomorphism u → xu := uw0 P/P : N − → C and the continuous map (Lemma 7.62 (ii)) N g −→ P u −→ α(g, xu ) such that gu = α(g, xu )u n(g, ¯ u) α(g, xu )u = n(g, u)t (g, u)

for some n(g, ¯ u) ∈ N := w0 N w0−1 , for some n(g, u) ∈ N, t (g, u) ∈ L .

(7.23)

Lemma 7.64. Fix g ∈ G and let V ⊂ g −1 C ∩ C be any compact open subset. ˙ . . . ∪˙ Vm by compact open subsets There exists a disjoint covering V = V1 ∪ Vi and points xi ∈ Vi such that

294

Peter Schneider, Marie-France Vigneras, and Gergely Zabradi α(g, xi )Vi ⊂ gV

for any 1 ≤ i ≤ m. ∼

Proof. We denote the inverse of the homeomorphism u → xu : N − → C by x → u x . The image C ⊂ P of V under the continuous map x → α(g, x)u x V → P is compact. As (Lemma 7.62 (iii)) α(g, x)x = gx ∈ gV for any x ∈ V , under the continuous action of P on C, every element in the compact set C maps the point w0 P into gV . It follows that there is an open neighbourhood V0 ⊂ C of w0 P such that C V0 ⊂ gV . This means that α(g, x)u x V0 ⊂ gV

for any x ∈ V .

Using Proposition 7.61 we find, by appropriately shrinking V0 , a disjoint covering of V of the form V = u 1 V0 ∪˙ . . . ∪˙ u m V0 with u i ∈ N . We put xi := u i w0 P. We denote by G X := {x ∈ G | x X ⊂ X } the G-stabilizer of a subset X ⊂ G/P and by G †X := {g ∈ G | g ∈ G X , g −1 ∈ G X } = {x ∈ G | x X = X } the subgroup of invertible elements of G X . If G X is open then its inverse monoid is open hence G †X is open (and conversely). Lemma 7.65. The G-stabilizers G U and G †U are open in G, for any compact open subset U ⊂ G/P. Proof. By Proposition 7.61 it suffices to consider the case where U = U w0 P/P for some compact open subgroup U ⊂ N . As U w0 P ⊂ G is an open subset containing w0 there exists an open subgroup K ⊂ G such that K w0 ⊂ U w0 P. The set U/(K ∩ U ) is finite because U is compact and (K ∩ U ) ⊂ U is an open subgroup. The finite intersection K := 6 6 −1 = −1 is an open subgroup of K which is u∈U/(U ∩K ) u K u u∈U u K u normalized by U . But K U = U K implies that K U w0 P = U K w0 P ⊂ U (U w0 P)P = U w0 P, and hence that K ⊂ G U . We deduce that G U is open. Hence G †U is open. Remark 7.66. The G-stabilizer of the open cell C is the group P. Proof. Lemma 7.62 (i). For U ⊂ C the map GU × U → P

,

(g, x) → α(g, x)

(7.24)

is continuous because, if U = U w0 P/P with U open in N , then the map (g, u) → guw0 N : G U × U → Pw0 P/N = Pw0 N /N is continuous (cf. the proof of Lemma 7.62 (ii)).

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5.2. Equivariant sheaves and modules over skew group rings Our construction of the sheaf on G/P will proceed through a module theoretic interpretation of equivariant sheaves. The ring Cc∞ (C, A) has no unit element. But it has sufficiently many idempotents (the characteristic functions 1V of the compact open subsets V ⊂ C). A (left) module Z over Cc∞ (C, A) is called nondegenerate if for any z ∈ Z there is an idempotent e ∈ Cc∞ (C, A) such that ez = z. It is well known that the functor sheaves of A-modules on C → nondegenerate Cc∞ (C, A)-modules which sends a sheaf S to the A-module of global sections with compact support 0 Sc (C) := V S(V ), with V running over all compact open subsets in C, is an equivalence of categories. In fact, as we have discussed in the proof of Theorem 7.38 a quasi-inverse functor is given by sending the module Z to the sheaf whose sections on the compact open subset V ⊂ C are equal to 1V Z . In order to extend this equivalence to equivariant sheaves we note that the group P acts, by left translations, from the right on Cc∞ (C, A) which we write as ( f, b) → f b (.) := f (b.). This allows to introduce the skew group ring AC := Cc∞ (C, A)#P = ⊕b∈P bCc∞ (C, A) in which the multiplication is determined by the rule (b1 f 1 )(b2 f 2 ) = b1 b2 f 1b2 f 2

for bi ∈ P and f i ∈ Cc∞ (C, A).

It is easy to see that the above functor extends to an equivalence of categories

P-equivariant sheaves of A-modules on C − → nondegenerate AC -modules. We have the completely analogous formalism for the G-space G/P. The only small difference is that, since G/P is assumed to be compact, the ring C ∞ (G/P, A) of locally constant A-valued functions on G/P is unital. The skew group ring AG/P := C ∞ (G/P, A)#G = ⊕g∈G gC ∞ (G/P, A) therefore is unital as well, and the equivalence of categories reads

G-equivariant sheaves of A-modules on G/P − → unital AG/P -modules. For any open subset U ⊂ G/P the A-algebra Cc∞ (U, A) of A-valued locally constant and compactly supported functions on U is, by extending functions by zero, a subalgebra of C ∞ (G/P, A). It follows in particular that AC is a subring

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of AG/P . There is for our purposes a very important ring in between these two rings which is defined to be A := AC ⊂G/P := ⊕g∈G gCc∞ (g −1 C ∩ C, A). That A indeed is multiplicatively closed is immediate from the following observation. If supp( f ) denotes the support of a function f ∈ C ∞ (G/P, A) then we have the formula supp( f 1 f 2 ) = g −1 supp( f 1 ) ∩ supp( f 2 ) g

for g ∈ G and f 1 , f 2 ∈ C ∞ (G/P, A).

(7.25)

In particular, if f i ∈ Cc∞ (gi−1 C ∩ C, A) then supp( f 1 2 f 2 ) ⊂ g2−1 (g1−1 C ∩ C) ∩ (g2−1 C ∩ C) ⊂ (g1 g2 )−1 C ∩ C. g

We also have the A-submodule Z := ⊕g∈G gCc∞ (C, A) of AG/P . Using (7.25) again one sees that Z actually is a left ideal in AG/P which at the same time is a right A-submodule. This means that we have the well defined functor nondegenerate A-modules → unital AG/P -modules Z → Z ⊗A Z. Remark 7.67. The functor of restricting G-equivariant sheaves on G/P to the open cell C is faithful and detects isomorphisms. Proof. Any sheaf homomorphism which is the zero map, resp. an isomorphism, on sections on any compact open subset of C has, by G-equivariance, the same property on any standard compact open subset and hence, by Proposition 7.61, on any compact open subset of G/P. Proposition 7.68. The above functor Z → Z ⊗A Z is an equivalence of categories; a quasi-inverse functor is given by sending the AG/P -module Y to 0 V ⊂C 1V Y where V runs over all compact open subsets in C. Proof. We abbreviate the asserted candidate for the quasi-inverse functor by 0 R(Y ) := V ⊂C 1V Y . It immediately follows from Remark 7.67 that the functor R, which in terms of sheaves is the functor of restriction, is faithful and detects isomorphisms. By a slight abuse of notation we identify in the following a function f ∈ C ∞ (G/P, A) with the element 1 f ∈ AG/P , where 1 ∈ G denotes the unit

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element. Let V ⊂ C be a compact open subset. Then 1V AG/P 1V is a subring of AG/P (with the unit element 1V ), which we compute as follows: 1V AG/P 1V = 1V gC ∞ (V , A) = g1g −1 V C ∞ (V , A) g∈G

=

g∈G

gC ∞ (g −1 V ∩ V, A).

g∈G

We note: – If U ⊂ V ⊂ C are two compact open subsets then 1V AG/P 1V ⊃ 1U AG/P 1U . – Let f ∈ C c∞ (g −1 C ∩ C, A) be supported on the compact open subset U ⊂ g −1 C ∩C. Then V := U ∪ gU is compact open in C as well, and U ⊂ 0 g −1 V ∩V . This shows that Cc∞ (g −1 C∩C, A) = V ⊂C C ∞ (g −1 V ∩V , A). We deduce that

1V AG/P 1V = AC ⊂G/P = A.

V ⊂C

A completely analogous computation shows that 1V Z = 1V A. Given a nondegenerate A-module Z the map 1V (Z ⊗A Z ) = (1V Z) ⊗A Z = (1V A) ⊗A Z → 1V Z 1V a ⊗ z = 1V ⊗ 1V az → 1V az therefore is visibly an isomorphism of 1V AG/P 1V -modules. In the limit with respect to V we obtain a natural isomorphism of A-modules ∼ =

R(Z ⊗A Z ) − → Z. On the other hand, for any unital AG/P -module Y there is the obvious natural homomorphism of AG/P -modules Z ⊗A R(Y ) → Y a ⊗ z → az. It is an isomorphism because applying the functor R, which detects isomorphisms, to it gives the identity map. Remark 7.69. Let Z be a nondegenerate A-module. Viewed as an AC -module it corresponds to a P-equivariant sheaf Z on C. On the other hand, the AG/P on G/P. We module Y := Z ⊗A Z corresponds to a G-equivariant sheaf Y |C = extends the sheaf have Y Z , i.e., the sheaf Y Z.

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We have now seen that the step of going from A to AG/P is completely formal. On the other hand, for any topologically étale A[P+ ]-module M, the P-equivariance of Res together with Proposition 7.52 imply that Res extends to the A-algebra homomorphism Res :

P AC → Endcont A (M ) b fb → b ◦ Res( f b ).

b∈P

b∈P

When M is compact it is relatively easy, as we will show in the next section, to further extend this map from AC to A. This makes crucial use of the full topological module M P and not only its submodule McP of sections with compact support. When M is not compact this extension problem is much more subtle and requires more facts about the ring A. We introduce the compact open subset C0 := N0 w0 P/P of C, and we consider the unital subrings A0 := 1C0 AG/P 1C0 = gC ∞ (g −1 C0 ∩ C0 , A) g∈G

and AC 0 := 1C0 AC 1C0 =

bC ∞ (b−1 C0 ∩ C0 , A)

b∈P

of A and AC , respectively. Obviously AC 0 ⊆ A0 with the same unit element 1C0 . Since g −1 C0 ∩ C0 is nonempty if and only if g ∈ N0 P N0 we in fact have A0 = gC ∞ (g −1 C0 ∩ C0 , A). g∈N0 P N0

The map A[G] −→ AG/P sending g to g1G/P is a unital ring homomorphism. Hence we may view AG/P as an A[G]-module for the adjoint action G × AG/P −→ AG/P (g, y) −→ (g1G/P )y(g1G/P )−1 . One checks that AC ⊆ A are A[P]-submodules, that AC 0 ⊆ A0 are A[P+ ]submodules, and that the map Res : AC −→ E cont is a homomorphism of A[P]-modules. Proposition 7.70. The homomorphism of A[P]-modules ∼ =

A[P] ⊗ A[P+ ] A0 −→ A b ⊗ y −→ (b1G/P )y(b1G/P )−1

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is bijective; it restricts to an isomorphism A[P] ⊗ A[P+ ] AC 0 −→ AC . Proof. Since P = s −N P+ the assertion amounts to the claim that

A= (s −n 1G/P )A0 (s n 1G/P ) n≥0

and correspondingly for AC . But we have (s −n 1G/P ) gC ∞ (g −1 C0 ∩ C0 , A) (s n 1G/P ) = s −n gs n C ∞ ((s −n g −1 s n )s −n C0 ∩ s −n C0 , A) for any n ≥ 0 and any g ∈ G. P Suppose that we may extend the map Res : AC 0 −→ Endcont A (M ) to an A[P+ ]-equivariant (unital) A-algebra homomorphism

R0 : A0 −→ End A (M P ). By the above Proposition 7.70 it further extends uniquely to an A[P]equivariant map R : A −→ End A (M P ). Lemma 7.71. The map R is multiplicative. Proof. Using Proposition 7.70 we have that two arbitrary elements y, z ∈ A are of the form y = (s −m 1G/P )y0 (s m 1G/P ), z = (s −n 1G/P )z 0 (s n 1G/P ) with m, n ∈ N and y0 , z 0 ∈ A0 . We can choose m = n. It follows that yz = (s −m 1G/P )y0 z 0 (s m 1G/P ) = (s −m 1G/P )x0 (s m 1G/P ) with x0 := y0 z 0 ∈ A0 , and that R(yz) = R((s −m 1G/P )x0 (s m 1G/P )) = s −m ◦ R0 (x0 ) ◦ s m = s −m ◦ R0 (y0 ) ◦ R0 (z 0 ) ◦ s m = (s −m ◦ R0 (y0 ) ◦ s m ) ◦ (s −m ◦ R0 (z 0 ) ◦ s m ) = R(y) ◦ R(z).

Note that the images Res(AC 0 ) and R0 (A0 ) necessarily lie in the image of End A (M) = End A (Res(1C0 )(M P )) by the natural embedding into End A (M P ). This reduces us to search for an A[P+ ]-equivariant (unital) A-algebra homomorphism R0 : A0 −→ End A (M)

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which extends Res |AC 0 . In fact, since for g ∈ N0 P N0 and f ∈ C ∞ (g −1 C0 ∩ C0 , A) we have g f = (g1g −1 C0 ∩C0 )(1 f ) with 1 f ∈ AC 0 it suffices to find the elements Hg = R0 (g1g −1 C0 ∩C0 ) ∈ End A (M)

for g ∈ N0 P N0 .

Note that P+ = N0 L + is contained in N0 P N0 = N0 L N N0 . Proposition 7.72. We suppose given, for any g ∈ N0 P N0 , an element Hg ∈ End A (M). Then the map R0 :

g∈N0 P N0

A0 −→ End A (M) g f g −→ Hg ◦ res( f g ) g∈N0 P N0

is an A[P+ ]-equivariant (unital) A-algebra homomorphism which extends Res |AC 0 if and only if, for all g, h ∈ N0 P N0 , b ∈ P ∩ N0 P N0 , and all compact open subsets V ⊂ C0 , the relations H1. res(1V ) ◦ Hg = Hg ◦ res(1g −1 V ∩C0 ), H2. Hg ◦ Hh = Hgh ◦ res(1(gh)−1 C0 ∩h −1 C0 ∩C0 ), H3. Hb = b ◦ res(1b−1 C0 ∩C0 ). hold true. When H1 is true, H2 can equivalently be replaced by Hg ◦ Hh = Hgh ◦ res(1h −1 C0 ∩C0 ). Proof. Necessity of the relations is easily checked. Vice versa, the first two relations imply that R0 is multiplicative. The third relation says that R0 extends Res |AC 0 . The last sentence of the assertion derives from the fact that we have Hgh ◦ res(1(gh)−1 C0 ∩h −1 C0 ∩C0 ) = Hgh ◦ res(1(gh)−1 C0 ∩C0 ) ◦ res(1h −1 C0 ∩C0 ) = Hgh ◦ res(1h −1 C0 ∩C0 ) since Hgh ◦ res(1(gh)−1 C0 ∩C0 ) = Hgh by the first relation. The P+ -equivariance is equivalent to the identity R0 ((c1G/P )g f g (c1G/P )−1 ) = ϕc ◦ R0 (g f g ) ◦ ψc where c ∈ P+ and f g is any function in C ∞ (g −1 C0 ∩ C0 ). By the definition of R0 and the P+ -equivariance of res the left-hand side is equal to Hcgc−1 ◦ ϕc ◦ res( f g ) ◦ ψc whereas the right-hand side is ϕc ◦ Hg ◦ res( f g ) ◦ ψc .

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Since ψc is surjective and res( f g ) = res(1g −1 C0 ∩C0 ) ◦ res( f g ) we see that the P+ -equivariance of R0 is equivalent to the identity Hcgc−1 ◦ ϕc ◦ res(1g−1 C0 ∩C0 ) = ϕc ◦ Hg ◦ res(1g −1 C0 ∩C0 ). But as a special case of the first relation we have Hg ◦ res(1g−1 C0 ∩C0 ) = Hg . Hence the latter identity coincides with the relation Hcgc−1 ◦ ϕc ◦ res(1g −1 C0 ∩C0 ) = ϕc ◦ Hg . This relation holds true because ϕc = Hc and by the second relation Hcgc−1 ◦ Hc = Hcg and Hc ◦ Hg = Hcg ◦ res(1g −1 C0 ∩C0 ).

5.3. Integrating α when M is compact Let M be a compact topologically étale A[P+ ]-module. Then M P is compact, hence the continuous action of P on M P (Proposition 7.52) induces a continuous map P → E cont . We will construct an extension R es of Res to AC ⊂G/P by integration. For any g ∈ G, we consider the continuous map α(g,.)

αg g −1 C ∩ C −−−→ P → E cont . We introduce the A-linear maps ρ A = AC ⊂G/P → Cc (C, E cont ) g f g → αg f g g∈G

g∈G

and R es A = AC ⊂G/P → E cont / a → ρ(a)d Res . C

For b ∈ P the map αb is the constant map on C with value b (Lemma 5.3 iii). It follows that R es | AC = Res is an extension as we want it.

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Theorem 7.73. R es is a homomorphism of A-algebras. Proof. Let g, h ∈ G and let Vg and Vh be compact open subsets of g −1 C ∩ C and h −1 C ∩ C, respectively. We have to show that R es((g1Vg )(h1Vh )) = R es(g1Vg ) ◦ R es(h1Vh ) holds true. This is, by definition of R es, equivalent to the identity / / / αgh 1h −1 Vg ∩Vh dRes = αg 1Vg dRes ◦ αh 1Vh dRes. C

C

C

Let Ug , Uh be the open compact subsets of N corresponding to Vg , Vh and let f be the map αg 1Vg , seen as a map on N with support on Ug . Let L ⊂ M P be an open A-submodule and let VL be chosen as in Lemma 7.57, with λ = Res. If we let Nk,v = α(h, xv ).(v Nk ), then the P-equivariance of Res combined with Remark 7.58 yield, for v ∈ Uh and k ≥ 1 / / αg 1Vg dRes ◦ α(h, xv ) ◦ Res(1v Nk ) = f dRes ◦ Res(1 Nk,v ) ◦ α(h, xv ) C

N

/ = N

( f 1 Nk,v )dRes ◦ α(h, xv ).

Writing α(h, xv ) = n v tv with n v ∈ N and tv ∈ L, for k large enough we have Ug tv Nk tv−1 ⊂ Ug and tv Nk tv−1 ⊂ VL for all v ∈ Uh (by compactness of (tv )v∈Uh ). Since Nk,v = (α(h, xv ).v)tv Nk tv−1 , we deduce that Nk,v ∩ Ug = ∅ ⇔ α(h, x v ).v ∈ Ug ⇔ hx v ∈ Vg and hence, by Lemma 7.57 for all sufficiently large k we have, uniformly in v ∈ Uh , / f 1 Nk,v dRes ≡ 1xv ∈h −1 Vg ∩Vh f (α(h, xv ).v) ◦ Res(1 Nk,v ) = N

cont = 1xv ∈h −1 Vg ∩Vh α(g, hx v ) ◦ Res(1 Nk,v ) (mod E L ).

Combining the last two relations with Lemma 7.63, and using again the P-equivariance of Res, we obtain for k large enough and for all v ∈ Uh / αg 1Vg dRes ◦ α(h, xv ) ◦ Res(1v Nk ) C

cont ≡ 1xv ∈h −1 Vg ∩Vh α(gh, xv ) ◦ Res(1v Nk ) (mod E L ).

The result follows by summing over v and letting k → ∞ (Corollary 7.56).

5.4. G-equivariant sheaf on G/ P Let M be a compact topologically étale A[P+ ]-module. We briefly survey our construction of a G-equivariant sheaf on G/P functorially associated with M.

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From Proposition 7.32 we obtain an A-algebra homomorphism Res Cc∞ (C, A)#P → E cont which gives rise to a P-equivariant sheaf on C as described in detail in Theorem 7.38. By Theorem 7.73, it extends to an A-algebra homomorphism R es AC ⊂G/P → E cont . This homomorphism defines on the global sections with compact support McP of the sheaf on C the structure of a nondegenerate AC ⊂G/P -module. The latter leads, by Proposition 7.68, to the unital Cc∞ (G/P, A)#G-module Z ⊗A McP which corresponds to a G-equivariant sheaf on G/P extending the earlier sheaf on C (Remark 7.39). We will denote the sections of this latter sheaf on an open subset U ⊂ G/P by M U . The restriction maps in this sheaf, for open subsets V ⊂ U ⊂ G/P, will simply be written as ResU V M U → M V. We observe that for a standard compact open subset U ⊂ G/P with g ∈ G such that gU ⊂ C the action of the element g on the sheaf induces an iso∼ =

morphism of A-modules M U − → M gU = Mg U . Being the image of a continuous projector on M P (Proposition 7.52), Mg U is naturally a compact topological A-module. We use the above isomorphism to transport this topology to M U. The result is independent of the choice of g since, if gU = hU for some other h ∈ G, then hU ⊂ (gh −1 )−1 C ∩ C and, by construction, the endomorphism gh −1 of M hU is given by the continuous map R es(gh −1 1h U ). A general compact open subset U ⊂ G/P is the disjoint union U = ˙ ...∪ ˙ Um of standard compact open subsets Ui (Proposition 7.61). We U1 ∪ equip M U = M U1 ⊕ . . . ⊕ M Um with the direct product topology. One easily verifies that this is independent of the choice of the covering. Finally, for an arbitrary open subset U ⊂ G/P we have M U = lim M V, ← − where V runs over all compact open subsets V ⊂ U, and we equip M U with the corresponding projective limit topology. It is straightforward to check that all restriction maps are continuous and that any g ∈ G acts by continuous homomorphisms. We see that (M U)U is a G-equivariant sheaf of compact topological A-modules. Lemma 7.74. For any compact open subset U ⊂ G/P the action G †U × (M U) → M U of the open subgroup G †U (Lemma 7.65) on the sections on U is continuous. Proof. Using Proposition 7.61, it suffices to consider the case that U ⊂ C. Note that G †U acts by continuous automorphisms on M U = MU . By (7.24) the map

304

Peter Schneider, Marie-France Vigneras, and Gergely Zabradi G †U × U → E cont (g, x) → αg (x)

is continuous. Hence ([3] TG X.28 Th. 3) the corresponding map G †U → C(U, E cont ) is continuous, where we always equip the module C(U, E cont ) of E cont -valued continuous maps on U with the compact-open topology. On the other hand it is easy to see that, for any measure λ on C with values in E cont , the map / . dλ C(U, E cont ) → E cont U

is continuous. It follows that the map G †U → E cont g → R es(g1U ) is continuous. The direct decomposition M P = MU ⊕ MC −U gives a natcont through which the above map ural inclusion map Endcont A (MU ) → E factorizes. The resulting map G †U → Endcont A (MU ) is continuous and coincides with the G †U -action on MU . As MU is compact this continuity implies the continuity of the action G †U × MU → MU . The same construction can be done, starting from the compact topologically étale A[PU ]-module MU , for any compact open subgroup U ⊂ N . Proposition 7.75. Let U ⊂ N be a compact open subgroup. The Gequivariant sheaves on G/P associated to (N0 , M) and to (U, MU ) are equal. Proof. As the P-equivariant sheaves on the open cell associated to (N0 , M) and to (U, MU ) are equal by Proposition 7.43, and as the function αg depends only on the open cell, our formal construction gives the same G-equivariant sheaf.

6. Integrating α when M is non compact Recall that we have chosen a certain element s ∈ Z(L) such that L = L − s Z and (Nk = s k N0 s −k )k∈Z is a decreasing sequence with union N and trivial

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intersection. We now suppose in addition that (N k := s −k w0 N0 w0−1 s k )k∈Z is a decreasing sequence with union N = w0 N w0−1 and trivial intersection. We have chosen A and M in Section 4.1. We suppose now in addition that M is a topologically étale A[P+ ]-module which is Hausdorff and complete. Definition 7.76. A special family of compact sets in M is a family C of compact subsets of M satisfying: C(1) C(2) C(3) C(4)

Any compact subset of a compact set in C also lies in C. 0 If C1 , C2 , . . . , C n ∈ C then in=1 Ci is in C, as well. For all C ∈ C we have N0 C ∈ C. 0 M(C) := C∈C C is an étale A[P+ ]-submodule of M.

Note that M is the union of its compact subsets, and that the family of all compact subsets of M satisfies these four properties. Let C be a special family of compact sets in M. A map from M(C) to M is called C-continuous if its restriction to any C ∈ C is continuous. We equip the A-module HomCont A (M(C), M) of C-continuous A-linear homomorphisms from M(C) to M with the C-open topology. The A-submodules E(C, M) := { f ∈ HomCont (M(C), M) : f (C) ⊆ M}, A for any C ∈ C and any open A-submodule M ⊆ M, form a fundamental system of open neighbourhoods of zero in HomCont (M(C), M). Indeed, A this system is closed for finite intersection by C(2). Since N0 is compact the E(C, M) for C such that N0 C ⊆ C and M an A[N0 ]-submodule still form a fundamental system of open neighbourhoods of zero (Lemma 7.50 and C(3)). We have: – HomCont (M(C), M) is a topological A-module. A – HomCont (M(C), M) is Hausdorff, since C covers M(C) by C(4) and M is A Hausdorff. – HomCont (M(C), M) is complete ([3] TG X.9 Cor.2). A

6.1. (s, res, C)-integrals We have the P+ -equivariant measure res : C ∞ (N0 , A) −→ Endcont A (M) on N0 . If M is not compact then it is no longer possible to integrate any map in the A-module C(N0 , Endcont A (M)) of all continuous maps on N0 with values in Endcont (M) against this measure. This forces us to introduce a notion of A integrability with respect to a special family of compact sets in M.

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Definition 7.77. A map F : N0 → HomCont (M(C), M) is called integrable A with respect to (s, res, C) if the limit / Fd res := lim F(u) ◦ res(1u Nk ), k→∞

N0

u∈J (N0 /Nk )

where J (N0 /Nk ) ⊆ N0 , for any k ∈ N, is a set of representatives for the cosets in N0 /Nk , exists in HomCont A (M(C), M) and does not depend on the choice of the sets J (N0 /Nk ). We suppress C from the notation when C is the family of all compact subsets of M. Note that we regard res(1u Nk ) as an element of Endcont A (M(C)). This makes sense as the algebraically étale submodule M(C) of the topologically étale module M is topologically étale. One easily sees that the set C int (N0 , HomCont (M(C), M)) of integrable A maps is an A-module. The A-linear map / .d res : C int (N0 , HomCont (M(C), M)) −→ HomCont A A (M(C), M) N0

will be called the (s, res, C)-integral. We give now a general integrability criterion. Proposition 7.78. A map F : N0 −→ HomCont (M(C), M) is (s, res, C)A integrable if, for any C ∈ C and any open A-submodule M ⊆ M, there exists an integer kC,M ≥ 0 such that (F(u) − F(uv)) ◦ res(1u Nk+1 ) ∈ E(C, M) for any k ≥ kC,M , u ∈ N0 , and v ∈ Nk . Proof. Let J (N0 /Nk ) and J (N0 /Nk ), for k ≥ 0, be two choices of sets of representatives. We put sk (F) := F(u) ◦ res(1u Nk ) and sk (F) u∈J (N0 /Nk )

:=

F(u ) ◦ res(1u Nk ).

u ∈J (N0 /Nk )

Since HomCont (M(C), M) is Hausdorff and complete it suffices to show that, A given any neighbourhood of zero E(C, M), there exists an integer k0 ≥ 0 such that sk (F) − sk+1 (F), sk (F) − sk (F) ∈ E(C, M)

for any k ≥ k0 .

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For u ∈ J (N0 /Nk+1 ) let u¯ ∈ J (N0 /Nk ) and u ∈ J (N0 /Nk+1 ) be the unique elements such that u Nk = u¯ Nk and u Nk+1 = u Nk+1 , respectively. Then sk (F) = F(u) ¯ ◦ res(1u Nk+1 ) u∈J (N0 /Nk+1 )

and hence sk (F) − sk+1 (F) =

(F(u(u −1 u)) ¯ − F(u)) ◦ res(1u Nk+1 ). (7.26)

u∈J (N0 /Nk+1 )

Since u −1 u¯ ∈ Nk it follows from our assumption that the right-hand side lies in E(C, M) for k ≥ kC,M . Similarly sk+1 (F) − sk+1 (F) = (F(u) − F(u(u −1 u ))) ◦ res(1u Nk+1 ); u∈J (N0 /Nk+1 )

again, as u −1 u ∈ Nk+1 ⊆ Nk , the right-hand sum is contained in E(C, M) for k ≥ kC,M .

6.2. Integrability criterion for α Let Ug ⊆ N0 be the compact open subset such that Ug w0 P/P = g −1 C0 ∩ C0 . This intersection is nonempty if and only if g ∈ N0 P N0 , which we therefore assume in the following. We consider the map αg,0 : N0 −→ Endcont A (M) $ Res(1 N0 ) ◦ αg (xu ) ◦ Res(1 N0 ) if u ∈ Ug , u −→ 0 otherwise cont under the natural (where we identify Endcont A (M) with its image in E embedding (7.20) using that Res(1 N0 ) = σ0 ◦ ev0 ). Restricting αg,0 (u) ∈ Endcont A (M) to M(C) for any u ∈ N0 we may view αg,0 as a map from N0 to Endcont A (M(C)) since M(C) is an étale A[P+ ]-submodule of M. However, as we do not assume M(C) to be complete, it will be more convenient for the purpose of integration to regard αg,0 as a map into HomCont A (M(C), M). We want to establish a criterion for the (s, res, C)-integrability of the map αg,0 . By the argument in the proof of Lemma 7.64 (applied to V = g −1 C0 ∩ C0 ) we may choose an integer k g(0) ≥ 0 such that, for any k ≥ k g(0) , we have Ug Nk ⊆ Ug and

α(g, xu ).u Nk ⊆ gUg

for any u ∈ Ug .

(7.27)

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Lemma 7.79. For u ∈ Ug and k ≥ k g we have αg,0 (u) ◦ res(1u Nk ) = α(g, xu ) ◦ Res(1u Nk ). Proof. Using the P-equivariance of Res we have α(g, xu ) ◦ Res(1u Nk ) = Res(1α(g,xu ).u Nk ) ◦ α(g, xu ) ◦ Res(1u Nk ) = Res(1 N0 ) ◦ Res(1α(g,xu ).u Nk ) ◦ α(g, xu ) ◦ Res(1u Nk ) = Res(1 N0 ) ◦ α(g, xu ) ◦ Res(1 N0 ) ◦ Res(1u Nk ) = αg,0 (u) ◦ res(1u Nk ) where the second identity follows from (7.27). (0)

For u ∈ U g and k ≥ k g we put Hg,J (N0 /Nk ) :=

α(g, xu ) ◦ Res(1u Nk ).

(7.28)

u∈Ug ∩J (N0 /Nk )

By Lemma 7.79, each summand on the right-hand side belongs to End A (M(C)). If αg,0 is (s, res, C)-integrable, the limit Hg :=

lim

(0)

k≥k g ,k→∞

Hg,J (N0 /Nk )

(7.29)

exists in HomCont (M(C), M) and is equal to the (s, res, C)-integral of αg,0 A / αg,0 d res = Hg . (7.30) N0

We investigate the integrability criterion of Proposition 7.78 for the function αg,0 . We have to consider the elements g (u, k, v) := (αg,0 (u) − αg,0 (uv)) ◦ res(1u Nk+1 ),

(7.31)

(0)

for u ∈ Ug , k ≥ k g , and v ∈ Nk . By Lemma 7.79, we have g (u, k, v) = (αg,0 (u) ◦ res(1u Nk ) − αg,0 (uv) ◦ res(1uv Nk )) ◦ res(1u Nk+1 ) = (α(g, xu ) ◦ Res(1u Nk ) − α(g, xuv ) ◦ Res(1uv Nk )) ◦ Res(1u Nk+1 ) = (α(g, xu ) − α(g, xuv )) ◦ Res(1u Nk+1 ) = (α(g, xu ) − α(g, xuv )) ◦ u ◦ Res(1 Nk+1 ) ◦ u −1 . Recall that N g ⊂ N is the subset such that N g w0 P/P = g −1 C ∩ C. Lemma 7.80. For u ∈ Ng and v ∈ N such that uv ∈ N g we have: i. v ∈ Nn(g,u) ; ¯ ii. α(g, xuv ) = α(g, xu )uα(n(g, ¯ u), xv )u −1 .

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Proof. i. Because of gu = α(g, xu )u n(g, ¯ u) we have α(g, xu )u n(g, ¯ u)v = guv ∈ α(g, xuv )uv N and hence n(g, ¯ u)vw0 P = u −1 α(g, xu )−1 α(g, xuv )uvw0 P ∈ Pw0 P. ii. By i. the equation n(g, ¯ u)vw0 N = α(n(g, ¯ u), xv )vw0 N holds. Hence guvw0 N = α(g, xu )u n(g, ¯ u)vw0 N = α(g, x u )uα(n(g, ¯ u), xv )vw0 N and therefore α(g, xuv )uv = α(g, xu )uα(n(g, ¯ u), xv )v. Let f : Ug → P be the map u → α(g, xu )u. The previous computation shows that for all u ∈ Ug and v ∈ Nk we have g (u, k, v) = ( f (u) − f (uv)v −1 ) ◦ Res(1 Nk+1 ) ◦ u −1 .

(7.32)

Let f (u) = n(g, u)t (g, u), with n(g, u) ∈ N0 and t (g, u) ∈ L. Also, write gu = f (u)n(g, u) with n(g, u) ∈ N . Since t (g, Ug ) ⊂ L and n(g, Ug ) ⊂ N (1) (0) are compact subsets, there is k g ≥ k g such that (1)

g := t (g, Ug )s k g ⊂ L + ,

n(g, Ug ) ⊂ N −k (1) . g

(7.33)

Proposition 7.81. For any compact open subgroup P1 of P0 there is (2) (1) (2) k g (P1 ) ≥ k g such that for all k ≥ k g (P1 ), u ∈ Ug and v ∈ Nk (1)

f (u) − f (uv)v −1 ∈ N0 s k−k g (1 − P1 ) g s −k . Proof. We abbreviate n(u) = n(g, u) and similarly for t (u) and n(u). Since f (u)n(u)v = guv = f (uv)n(uv), we have f (u) − f (uv)v −1 = f (u)(1 − n(u)vn(uv)−1 v −1 ) = n(u)(1 − t (u)n(u)vn(uv)−1 v −1 t (u)−1 )t (u). (1)

Since n(u) ∈ N0 , t (u) ∈ s −k g g and (t (u))u∈Ug is compact, it suffices to prove that for any compact open subgroup P2 of P0 we have n(u)vn(uv)−1 v −1 ∈ s k P2 s −k for sufficiently large k. But if v = s k n 0 s −k , we can write −k n(u)vn(uv)−1 v −1 = s k (s −k n(u)s k )n 0 (s −k n(uv)−1 s k )n −1 0 s

∈ s k N k−k (1) ( g

n 0 ∈N0

−k n 0 N k−k (1) n −1 0 )s . g

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The result follows from the compactness of N0 and the fact that the N k ’s shrink to {1} as k → ∞. (2)

Corollary 7.82. For any compact open subgroup P1 of P0 and k ≥ k g (P1 ) (1)

g (Ug , k, Nk ) ⊂ N0 s k−kg (1 − P1 ) g sψ k+1 N0 . Proof. Proposition 7.81 and relation (7.32) show that (1)

g (Ug , k, Nk ) ⊂ N0 s k−k g (1 − P1 ) g s ◦ s −(k+1) ◦ Res(1 Nk+1 ) ◦ N0 . The P-equivariance of Res yields s −(k+1) ◦ Res(1 Nk+1 ) = Res(1 N0 ) ◦ s −k−1 , cont P and this is the image of ψ k+1 ∈ Endcont A (M) in End A (M ). The result follows. This leads to an integrability criterion for αg,0 , which depends only on (s, M, C). Proposition 7.83. We suppose that (s, M, C) satisfies: C(5) For any C ∈ C the compact subset ψ(C) ⊆ M also lies in C. T(1) For any C ∈ C such that C = N0 C, any open A[N0 ]-submodule M of M, and any compact subset C+ ⊆ L + there exists a compact open subgroup P1 = P1 (C, M, C+ ) ⊆ P0 and an integer k(C, M, C+ ) ≥ 0 such that s k (1 − P1 )C+ ψ k (C) ⊆ M

for any k ≥ k(C, M, C+ ).

(7.34)

Then the map αg,0 : N0 → HomCont (M(C), M) is (s, res, C)-integrable for all A g ∈ N0 P N0 . Proof. By the general integrability criterion of Proposition 7.78, the map αg,0 is integrable if for any (C, M) as above, there exists kC,M,g ≥ 0 such that g (u, k, v) ∈ E(C, M)

for any k ≥ kC,M,g , u ∈ Ug , and v ∈ Nk . (7.35) (2) By Corollary 7.82, this is true if kC,M,g ≥ k g (P1 ) and (1)

s k−kg (1 − P1 ) g sψ k+1 (C) ⊂ M,

(7.36)

because N0 M = M and N0 C = C. We note that the set C+ = g s is contained in L + by (7.33) and is compact, (1)

that the set C = ψ kg +1 (C) ⊂ M is compact and N0 C = C because the map ψ is continuous and N0 ψ(C) = ψ(s N0 s −1 C) = ψ(C), and that (7.36) is equivalent to (1)

(1)

s k−k g (1 − P1 )C+ ψ k−kg ⊂ E(C , M).

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By our hypothesis, there exists an open subgroup P1 ⊂ P0 such that this (1) inclusion is satisfied when k ≥ k g + k(C , M, C + ). For kC,M,g := max(k g(1) + k(C , M, C + ), k g(2) (P1 )) (1)

(7.35) is satisfied. By construction, P1 depends on ψ k g only on C, M, g.

+1 (C), M,

(7.37) g s, hence

Later, under the assumptions of Proposition 7.83, we will use the argument in the previous proof in the following slightly more general form: for C, M, C+ as in the proposition and an integer k ≥ 0 we have

s k−k (1 − P1 (ψ k (C), M, C+ ))C+ ψ k ⊆ E(C, M)

(7.38)

k

for any k ≥ k + k(ψ (C), M, C + ).

6.3. Extension of Res Proposition 7.84. Suppose that (s, M, C) satisfies the assumptions of Proposition 7.83 and that the (s, res, C)-integral Hg of αg,0 is contained in End A (M(C)) for all g ∈ N0 P N0 . In addition we assume that: C(6) For any C ∈ C the compact subset ϕ(C) ⊆ M also lies in C. T(2) Given a set J (N0 /Nk ) ⊂ N0 of representatives for cosets in N0 /Nk , for k ≥ 1, for any x ∈ M(C) and g ∈ N0 P N0 there exists a compact A-submodule C x,g ∈ C and a positive integer k x,g such that Hg,J (N0 /Nk ) (x) ⊆ C x,g for any k ≥ k x,g . Then the Hg satisfy the relations H1, H2, H3 of Proposition 7.72. Remark 7.85. The properties C(3), C(5), C(6) imply that for any u ∈ N0 , k ≥ 1, and C ∈ C also res(1u Nk )(C) lies in C. Indeed, res(1u Nk ) = u ◦ ϕ k ◦ ψ k ◦ u −1 . We prove now H1 and H3, which do not use the last assumption. The proof of H2 is postponed to the next subsection. Proof. For the proof of H1 let V ⊂ C0 be a compact open subset and let U1 , U2 be the compact open subsets of N0 corresponding to V and g −1 V ∩C0 . To prove that res(1V ) ◦ Hg = Hg ◦ res(1g −1 V ∩C0 ), it suffices to verify that if k is large enough, then for all u ∈ U g we have Res(1U1 ) ◦ α(g, xu ) ◦ Res(1u Nk ) = α(g, xu ) ◦ Res(1u Nk ) ◦ Res(1U2 ). (7.39) If Nk,u = α(g, xu ).(u Nk ), then by P-equivariance of Res, (7.39) is equivalent to

312

Peter Schneider, Marie-France Vigneras, and Gergely Zabradi Res(1U1 ∩Nk,u ) ◦ α(g, xu ) = α(g, xu ) ◦ Res(1u Nk ∩U2 ).

(7.40)

Write α(g, xu ) = n u tu with n u ∈ N and tu ∈ L. If k is large enough, then for all u ∈ U g we have U2 Nk ⊂ U2 and U1 tu Nk tu−1 ⊂ U1 . Since Nk,u = (α(g, xu ).u)tu Nk tu−1 , we obtain U1 ∩ Nk,u = ∅ ⇔ α(g, xu ).u ∈ U1 ⇔ gx u ∈ V ⇔ xu ∈ g −1 V ∩ C0 ⇔ u ∈ U2 ⇔ u Nk ⊂ U2 . Hence (7.40) is equivalent to 0 = 0 or to Res(1 Nk,u ) ◦ α(g, xu ) = α(g, xu ) ◦ Res(1u Nk ), which is true as Res is P-equivariant. H3. For b ∈ P ∩ N0 P N0 we have αb,0 = constant map on N0 with value res(1C0 ) ◦ b ◦ res(1C0 ) and hence Hb = res(1C0 ) ◦ b ◦ res(1C0 ) = b ◦ res(1b−1 C0 ∩C0 ).

6.4. Proof of the product formula We invoke now the full set of assumptions of Proposition 7.84 and we prove the product formula Hg ◦ Hh = Hgh ◦ res(1h −1 C0 ∩C0 ) for g, h ∈ N0 P N0 . This suffices by Proposition 7.72. (0) (1) (0) Let k0 := max(k g , kh , k gh ) + 1 and let k ≥ k0 . (0)

(1)

(0)

As k ≥ kh (because kh ≥ kh (7.33)), the set Uh is a disjoint union of cosets u Nk . We choose a set J (N0 /Nk ) ⊂ N0 of representatives of the cosets in N0 /Nk and for each u ∈ J (N0 /Nk ) ∩ Uh a set Ju (N0 /Nk−k0 ) ⊂ N0 of representatives of the cosets in N0 /Nk−k0 with n(g, u) ∈ Ju (N0 /Nk−k0 ) (see (7.23)). We write Hg ◦ Hh − Hgh ◦ res(1h −1 C0 ∩C0 ) as the sum over u ∈ J (N0 /Nk ) ∩Uh of (Hg ◦ Hh − Hgh ◦ Res(1Uh )) ◦ Res(1u Nk ) = ak,u + bk,u + ck,u ,

(7.41)

where ak,u :=(Hg ◦ Hh − Hg,Ju (N0 /Nk−k0 ) ◦ Hh,J (N0 /Nk ) ) ◦ Res(1u Nk ) bk,u :=(Hg, Ju (N0 /Nk−k0 ) ◦ Hh,J (N0 /Nk ) −Hgh,J (N0 /Nk ) ) ◦ Res(1Uh )◦ Res(1u Nk ) ck,u :=(Hgh,J (N0 /Nk ) − Hgh ) ◦ Res(1Uh ) ◦ Res(1u Nk ).

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The product formula follows from the claim that bk,u = 0 and that for an arbitrary compact subset C ∈ C such that N0 C = C, and an arbitrary open A[N0 ]-module M ⊂ M, ak,u and ck,u lie in E(C, M) when k is very large, independently of u. The claim results from the following three propositions. Because (s, M, C) satisfies Proposition 7.83, we associate to (C, M, g) the integer kC,M,g defined in (7.37) which is independent of the choice of the J (N0 /Nk ). For the sake of simplicity, we write (k) (k+1) H(k) − H(k) g := Hg, J (N0 /Nk ) , sg := Hg g .

(7.42)

(0)

By (7.26), we have, for k ≥ k g , sg(k) =

g (u, k, vu )

u∈Ug ∩J (N0 /Nk+1 )

for some vu ∈ Nk . It follows from Corollary 7.82 that, for any given compact open subgroup P1 ⊂ P0 , we have (1)

sg(k) ∈ < N0 s k−k g (1 − P1 ) g sψ k+1 N0 > A

for k ≥ k g(2) (P1 ),

(7.43)

where we use the notation < X > A for the A-submodule in End A (M) gener(k) ated by X . We deduce from the proof of Proposition 7.83, that sg ∈ E(C, M) for any k ≥ kC,M,g . Proposition 7.86. (Hg − Hg,J (N0 /Nk ) ) ◦ Res(1u Nk ) ∈ E(C, M) for any k ≥ kC,M,g . (0)

Proof. When k ≥ 0, k2 ≥ max(k − 1, k g ), u ∈ Ug , v ∈ Nk we have that g (u , k2 , v) ◦ Res(1u Nk ) is equal either to g (u , k2 , v) or to 0. If follows that sg(k2 ) ◦ Res(1u Nk ) ⊆ E(C, M)

for any k2 ≥ max(k − 1, kC,M,g ) and k ≥ 0.

Now we fix k ≥ kC,M,g . Note that Res(1u Nk )(C) is contained in C by the (k ) stability of C by ψ, ϕ, and u ±1 . Therefore the sequence (Hg 2 ◦ Res(1u Nk ))k2 converges to Hg ◦ Res(1u Nk ) in HomCont (M(C), M). In particular, we have A 2) (Hg − H(k g ) ◦ Res(1u Nk ) ⊆ E(C, M)

for any k2 ≥ max(k − 1, kC,M,g ) and k ≥ 0. The statement follows by taking k2 = k. This establishes that ck,u lies in E(C, M) when k ≥ kC,M,gh . Note that the proposition is true also for any other system J (N0 /Nk ) ⊂ N0 of representatives for the cosets in N0 /Nk for the same integer kC,M,g . We (k) write H(k) g and sg for the elements defined in (7.42) for J (N0 /Nk ).

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Proposition 7.87. There exists an integer kC,M,g,h,k0 ∈ N, independent of the choices of J (N0 /Nk ) and J (N0 /Nk ), such that: (k+1−k )

(k−k )

0 i. Hg ◦ Hh(k+1) − Hg 0 ◦ Hh(k) ∈ E(C, M), for all k ≥ (k−k ) (k) kC,M,g,h,k0 , and the sequence (Hg 0 ◦ Hh ) converges to Hg ◦ Hh in Cont Hom A (M(C), M). (k−k ) (k) ii. (Hg ◦ Hh − Hg 0 ◦ Hh ) ◦ Res(1u Nk ) ∈ E(C, M), for all k ≥ kC,M,g,h,k0 .

Proof. i. To prove the first assertion, we write (k+1)

(k)

0) ◦ H H(k+1−k g h

0) ◦ H − H(k−k g h

(k)

(k)

(k−k 0 ) 0) ◦ s = H(k+1−k ◦ Hh . g h + sg (1)

(k)

(k)

Note that, when k ≥ k g , the endomorphisms Hg in the A-module < N0 s

(1) k−k g

(7.44)

and Hg are contained

g ψ k N0 > A , because

α(g, xu ) ◦ Res(1u Nk ) (1)

= n(g, u)t (g, u)u −1 us k ψ k u −1 ⊂ N0 s k−k g g ψ k N0

for u ∈ Ug .

We consider any compact open subgroup P1 ⊂ P0 and we assume k ≥ (2) (2) max(k g (P1 )+k0 , kh (P1 )). With (7.43) we obtain that (7.44) is contained in (1)

(1)

< N0 s k+1−k0 −kg g ψ k+1−k0 N0 s k−kh (1 − P1 ) h sψ k+1 N0 > A (1)

(1)

+ < N0 s k−k0 −kg (1 − P1 ) g sψ k−k0 +1 N0 s k−kh h ψ k N0 > A . Recalling that ψ a (N0 ϕ a+b (m)) = ψ a (N0 )ϕ b (m) = N0 ϕ b (m) for a, b ∈ N and m ∈ M, we see that this is contained in (1)

(1)

< N0 s k+1−k0 −kg g N0 s k0 −kh

−1

(1)

(1 − P1 ) h sψ k+1 N0 > A (1)

+ < N0 s k−k0 −kg (1 − P1 ) g N0 s k0 −kh h ψ k N0 > A . As k + 1 − k0 − k g(1) ≥ k g(2) (P1 ) + 1 − k g(1) ≥ 1 and as g ⊂ L + , we have (1)

(1)

N0 s k+1−k0 −kg g N0 ⊂ N0 s k+1−k0 −k g g , and this is contained in (1)

(1)

< N0 s k+1−k0 −k g g s k0 −kh (1)

−1

(1 − P1 ) h sψ k+1 N0 > A (1)

+ < N0 s k−k0 −k g (1 − P1 ) g s k0 −kh h ψ k N0 > A .

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We assume, as we may, that the compact open subgroup P1 of P0 satisfies (1) t P1 t −1 ⊆ P1 for all t in the compact set g s k0 −kh −1 of L + . Then we finally obtain that (7.44) is contained in (1)

(1)

< N0 s k+1−k0 −k g (1 − P1 ) g s k0 −kh h ψ k+1 N0 > A (1)

(1)

+ < N0 s k−k0 −k g (1 − P1 ) g s k0 −kh h ψ k N0 > A . This subset of End A (M) is contained in E(C, M) when (1)

(1)

s k+1−k0 −k g (1 − P1 ) g s k0 −kh h ψ k+1 (C) (1)

(1)

and s k−k0 −k g (1 − P1 ) g s k0 −kh h ψ k (C) are contained in E(C, M) because N0 C = C and M is an A[N0 ]-module. By (7.38), this is true when P1 is contained in (1) (1) P1 (ψ k0 +kg (C), M, g s k0 −kh h ) and k ≥ kC,M,g,h,k0 where (2)

(1)

kC,M,g,h,k0 := max(k g(2) (P1 ) + k0 , kh (P1 ), k(ψ k0 +k g (C), (1)

M, g s k0 −kh h )).

(7.45)

The first assertion of i. is proved. We deduce the second assertion from the following claim and the last assumption of Proposition 7.84: Let ( An )n∈N and (Bn )n∈N be two convergent sequences in HomCont A (M(C), M) with limits A and B, respectively; assume that (Bn )n∈N and B are in End A (M(C)) and that, for any x ∈ C there exists an A-submodule C ∈ C such that Bn (x) ∈ C for any large n. Then, if the sequence (An ◦ Bn )n∈N is convergent, its limit is A ◦ B. Let D be the limit of the sequence (An ◦ Bn )n . It suffices to show that, for any open A-submodule M ⊆ M and any element x ∈ M(C) we have (D − A ◦ B)(x) ∈ M. We write D − A ◦ B = (D − An ◦ Bn ) − (A − An ) ◦ Bn − A ◦ (B − Bn ). Obviously (D − An ◦ Bn )(x) ∈ M for large n. Secondly, the elements Bn (x) for any large n are contained in some compact A-submodule C ∈ C, hence also (B − Bn )(x). Moreover A − An ∈ E(C, M) for large n. Hence (A − An ) ◦ Bn (x) ∈ M for large n. Finally, A being C-continuous there is an open A-submodule M ⊆ M such that A(M ∩ C) ⊆ M. Furthermore (B − Bn )(x) ∈ M ∩C for large n. Hence A ◦(B − Bn )(x) ∈ M for large n. ii. This follows from the second assertion in i. together with Remark 7.85. We have now proved that ak,u ∈ E(C, M) when k ≥ kC,M,g,h,k0 .

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Proposition 7.88. For u ∈ J (N0 /Nk ) ∩ Uh , we have Hg,Ju (N0 /Nk−k0 ) ◦Hh,J (N0 /Nk ) ◦Res(1u Nk ) = Hgh, J (N0 /Nk ) ◦Res(1u Nk ). (7.46) Proof. The left side of (7.46) is α(g, xv ) ◦ Res(1v Nk−k0 ) ◦ α(h, xu ) ◦ Res(1u Nk ). v∈Ug ∩Ju (N0 /Nk−k0 )

The right side of (7.46) is α(gh, x u ) ◦ Res(1u Nk ) if u ∈ J (N0 /Nk ) ∩ Uh ∩ Ugh and is 0 if u does not belong to Ugh . We recall that α(h, xu )u = n(h, u)t (h, u)

(1)

with n(h, u) ∈ N0 and t (h, u) ∈ L + s −kh .

It follows that α(h, xu )u Nk w0 P ⊆ n(h, u)Nk−k (1) w0 P ⊂ n(h, u)Nk−k0 w0 P. h

We obtain Res(1v Nk−k0 ) ◦ α(h, xu ) ◦ Res(1u Nk ) $ α(h, xu ) ◦ Res(1u Nk ) if v Nk−k0 = n(h, u)Nk−k0 , = 0 otherwise. We check now that u ∈ U gh ∩ Uh if and only if n(h, u) ∈ Ug . Indeed xu = uw0 P/P belongs to h −1 C0 ∩ C0 = Uh w0 P/P, xu ∈ (gh)−1 C0 ∩ h −1 C0 ∩ C0

if and only if

hx u ∈ g −1 C0 ∩ C0

and hxu = α(h, x u )xu = n(h, u)w0 P/P. It follows that u ∈ Ugh ∩ Uh if and only if n(h, u) ∈ Ug . As Ju (N0 /Nk−k0 ) contains n(h, u), we have v = n(h, u) when v Nk−k0 = n(h, u)Nk−k0 . We deduce that the left side of (7.46) is 0 when u does not belong to Ugh and otherwise is equal to α(g, hx u ) ◦ α(h, xu ) ◦ Res(1u Nk ) = α(gh, xu ) ◦ Res(1u Nk ), where the last equality follows from the product formula for α (Lemma 7.63). We have proved that bk,u = 0, therefore ending the proof of the product formula.

6.5. Reduction modulo pn We investigate now the situation that will appear for generalized (ϕ, )modules M, where the reduction modulo a power of p allows us to reduce to the simpler case where M is killed by a power of p. We will use later

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this section to get a special family Cs in M such that the (s, res, Cs )-integrals Hg exist for all g ∈ N0 P N0 and satisfy the relations H1, H2, H3 of Proposition 7.72. We assume now that (A, M) satisfies: a. A is a commutative ring with the p-adic topology (the ideals pn A for n ≥ 1 form a basis of neighbourhoods of 0) and is Hausdorff. b. M is a linearly topological A-module with a topology weaker than the p-adic topology (a neighbourhood of 0 contains some p n M) and M is a Hausdorff and topological A[P+ ]-module as in Section 6 (we do not suppose that M is complete). c. The submodules pn M, for n ≥ 1, are closed in M. d. M is p-adically complete: the linear map M → limn≥1 (M/ p n M) is ← − bijective. For all n ≥ 1, we equip M/ p n M with the quotient topology so that the quotient map pn : M → M/ p n M is continuous. The natural homomorphism ∼ =

M −→ lim(M/ p n M) ← − n≥1

is a homeomorphism, and the natural homomorphism ∼ =

n Endcont Endcont A (M) −→ lim A (M/ p M) ← − n≥1

is bijective. We have: – For a subset C of M, let C be the closure of C. Then C = lim p (C) ← −n≥1 n and if C is closed, C = limn≥1 pn (C). If C is p-compact (i.e. pn (C) are ← − compact for all n ≥ 1), then C is compact, and conversely ([2] I.29 Cor. and I.64 Prop.8). – An endomorphism f of M which is p-continuous (i.e. the endomorphism f n induced by f on M/ p n M is continuous for all n ≥ 1) is continuous, and conversely. – An action of a topological group H on M which is p-continuous (i.e. the induced action of H on M/ p n M is continuous for all n ≥ 1) is continuous, and conversely. – If the M/ p n M are complete for all n ≥ 1, then M is complete. – The image Cn in M/ p n M, for all n ≥ 1, of a special family C of compact subsets in M such that, for all positive integers n, p n M ∩ M(C) = p n M(C) is a special family. In this case, one has M(Cn ) = M(C)/ p n M(C).

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– M is a topologically étale A[P+ ]-module if and only if M/ p n M is a topologically étale A[P+ ]-module, for all n ≥ 1. If we replace “topologically” by “algebraically”, this is the same proof as for classical (ϕ, )-modules (see Subsection 7.3). The canonical inverse ψs of the action ϕs of s is continuous if and only if it is p-continuous. We introduce now our setting which will be discussed in this section. We suppose that: – M is a topologically étale A[P+ ]-module, and M/ p n M is complete for all n ≥ 1. – We are given, for n ≥ 1, a special family Cn of compact subsets in Mn = M/ p n M such that Cn contains the image of Cn+1 in Mn for all n ≥ 1. Let C be the set of compact subsets C ⊂ M such that pn (C) ∈ Cn for all n ≥ 1. Lemma 7.89. C is a special family in M and M(C) = limn≥1 M(Cn ). ← − Proof. C(1) It is obvious that a compact subset C of C ∈ C is in C because pn is continuous and pn (C ) is compact. C(2) pn commutes with finite union hence C is stable by finite union. C(3) pn commutes with the action of N0 hence C ∈ C implies N0 C ∈ C. C(4) By definition x ∈ M(C) if and only if pn (x) ∈ M(Cn ) for all n > 1. The compatibility of the Cn implies that the M(Cn ) form a projective system. We deduce M(C) = limn≥1 M(Cn ). As the latter ones are topolog← − ically étale, the topological A[P+ ]-module M(C) is topologically étale by Remark 7.24. We have the natural map lim Hom A (M(Cn ), M/ p n M) → Hom A (lim M(Cn ), lim M/ p n M) ← − ← − ← − n

n

n

= Hom A (M(C), M). Lemma 7.90. The above map induces a continuous map lim HomCAn cont (M(Cn ), M/ p n M) → HomCcont (M(C), M), A ← −

(7.47)

n

for the projective limit of the Cn -open topologies on the left-hand side. Proof. Let f = lim f n be a map in the image, and let C ∈ C. Then f |C is the ← − projective limit of the f n | pn (C) hence is continuous. This means that the map

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319

in the assertion is well defined. For the continuity, let C ∈ C and M ⊂ M be an open A-submodule. The preimage of E(C, M) is equal to lim HomCAn cont (M(Cn ), M/ p n M) ∩ E( pn (C), M + p n M/ p n M) . ← − n

n

Since M contains some

p n o M, this intersection is equal to the open submodule

{( f n ) ∈ lim HomCAn cont (M(Cn ), M/ pn M) : f n ∈ E( pn (C), M + p n M/ p n M) ← − n

for n ≤ n 0 }.

Proposition 7.91. In the above setting assume that all the assumptions of Proposition 7.84 are satisfied for (s, M/ p n M, Cn ) and for all n ≥ 1. Then, for all g ∈ N0 P N0 , the functions αg,0 : N0 → HomCont (M(C), M) A are (s, r es, C)-integrable, their (s, r es, C)-integrals Hg belong to End A (M(C)) and satisfy the relations H1, H2, H3 of Proposition 7.72. Proof. In the following we indicate with an extra index n that the corresponding notation is meant for the module M/ p n M with the special family Cn . Then αg,0 (u) is the image of (αg,0,n (u))n by the map (7.47), for u ∈ N0 . It follows that Hg,J (N0 /Nk ) is the image of (Hg,J (N0 /Nk ),n )n for g ∈ N0 P N0 . By assumption the integral Hg,n = limk→∞ Hg,J (N0 /Nk ),n exists, lies in HomCAn ont (M(Cn ), M/ p n M), and satisfies the relations H1, H2, H3 of Proposition 7.72. The continuity of the map (7.47) implies that the image of (Hg,n )n is equal to the limit limk→∞ Hg,J (N0 /Nk ) , therefore is the integral Hg of αg,0 . The additional properties for Hg are inherited from the corresponding properties of the Hg,n . Under the assumptions of Proposition 7.91, we associate to (s, M, C), an A-algebra homomorphism R es AC ⊂G/P → End A (M(C) P ) via Propositions 7.72, 7.70, which extends the A-algebra homomorphism Res Cc∞ (C, A)#P → End A (M(C) P ) constructed in Proposition 7.32. The homomorphism Res gives rise to a P-equivariant sheaf on C as described in detail in Theorem 7.38. The homo morphism R es defines on the global sections with compact support M(C)cP

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of the sheaf on C the structure of a nondegenerate AC ⊂G/P -module. The latter leads, by Proposition 7.68, to the unital Cc∞ (G/P, A)#G-module Z ⊗A M(C)cP which corresponds to a G-equivariant sheaf on G/P extending the earlier sheaf on C (Remark 7.69).

7. Classical (ϕ, )-modules on OE 7.1. The Fontaine ring OE Let K /Q p be a finite extension of ring of integers o, of uniformizer p K and residue field k. By definition the Fontaine ring OE over o is the p-adic completion of the localization of the Iwasawa o-algebra (Z p ) := o[[Z p ]] with respect to the multiplicative set of elements which are not divisible by p. We choose a generator γ of Z p of image [γ ] in OE and we denote X = [γ ] − 1 ∈ OE . The Iwasawa o-algebra (Z p ) is a local noetherian ring of maximal ideal M(Z p ) generated by p K , X . It is a compact ring for the M(Z p )-adic topology. The ring OE can be viewed as the ring of infinite Laurent series n∈Z an X n over o in the variable X with limn→−∞ an = 0, and (Z p ) as the subring o[[X ]] of Taylor series. The Fontaine ring OE is a local noetherian ring of maximal ideal p K OE and residue field isomorphic to k((X )); it is a pseudocompact ring for the p-adic (= strong) topology and a complete ring (with continuous multiplication) for the weak topology. A fundamental system of open neighbourhoods of 0 for the weak topology of OE is given by (On,k = p n OE + M(Z p )k )n,k∈N or by (Bn,k = p n OE + X k (Z p )n,k∈N . Other fundamental systems of neighbourhoods of 0 for the weak topology are (On := On,n )n≥1

or

(Bn := Bn,n )n≥1 .

7.2. The group G L(2, Q p ) We consider the group G = G L(2, Q p ) and , ∗ Z∗p Zp 0 1 Zp N0 := , := , L 0 := 0 1 0 0 1 Z p − {0} 0 L ∗ := , 0 1

0 , Z∗p

G-equivariant sheaves on G/P

Nk :=

1 0

pk Z p 1 + pk Z p , L k := 1 0

0 1 + pk Z p

321 for k ≥ 1,

Pk = L k Nk for k ∈ N, the upper triangular subgroup P, the diagonal subgroup L, the upper unipotent subgroup N , the center Z , the mirabolic monoid P∗ = N0 L ∗ , and the monoids L + = L ∗ Z , P+ = N0 L + . The subset of non invertible elements in the monoid L ∗ is a 0 N−{0} s p = {sa := for a ∈ pZ p − {0}}. 0 1 N−{0}

An element s ∈ s p Z is called strictly dominant. In the following we identify the group Z p with N0 . The action of P+ on N0 induces an étale ring action of P+ (trivial on Z ) on (N0 ) which respects the ideal generated by p. This action extends first to the localization and then to the completion to give an étale ring action of P+ on OE determined by its restriction to P∗ . For the weak topology (and not for the p-adic topology), the action P+ × OE → OE of the monoid P+ on OE is continuous (see Lemma 8.24.i in [12]). For t ∈ L + the canonical left inverse ψt of the action ϕt of t is continuous (this is proved in a more general setting later in Proposition 7.119).

7.3. Classical étale (ϕ, )-modules N−{0} s p Z.

Let s ∈ A finitely generated étale ϕs -module D over OE is a finitely generated OE -module with an étale semilinear endomorphism ϕs . These modules form an abelian category Met OE (ϕs ). We fix such a module D. In the following, the topology of D is its weak topology. For any surjective OE -linear map f : ⊕d OE → D, the image in D of a fundamental system of neighbourhoods of 0 in ⊕d OE for the weak topology is a fundamental system of neighbourhoods of 0 in D. Finitely generated (N0 )-submodules of D generating the OE -module D will be called lattices. The map f sends ⊕d (Z p ) onto a lattice D 0 of D. We note On,k := p n D + M(Z p )k D 0 and Bn,k := p n D + X k D 0 . Writing On := On,n and Bn := Bn,n , (On )n and (Bn )n are two fundamental systems of neighbourhoods of 0 in D. The topological OE -module D is Hausdorff and complete. A treillis D0 in D is a compact (N0 )-submodule D0 such that the image of D0 in the finite dimensional k((X ))-vector space D/ p K D is a k[[X ]]-lattice ([5] Déf. I.1.1). A lattice is a treillis and a treillis contains a lattice. For n ≥ 1, the reduction modulo p n of D is the finitely generated OE module D/ p n D with the induced action of ϕs . The action remains étale, because the multiplication by p n being a morphism in Met OE (ϕs ) its cokernel

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belongs to the category. The reduction modulo p n of ψs is the canonical left inverse of the reduction modulo p n of ϕs . The reduction modulo p n of a treillis of D is a treillis of D/ p n D. Conversely, if the reduction modulo p n of a finitely generated ϕs -module D over OE is étale for all n ≥ 1, then D is an étale ϕs -module over OE because D = lim D/ p n D. ← −n The weak topology of D is the projective limit of the weak topologies of D/ p n D. When D is killed by a power of p and D0 is a treillis of D, we have: 1. D0 is open and closed in D. 2. (M(Z p )n D0 )n∈N and (X n D0 )n∈N form two fundamental systems of open neighbourhoods of zero in D. 3. Any treillis of D is contained in X −n D0 for some n ∈ N. 0 4. D = k∈N X −k D0 . 5. D0 is a lattice. The first four properties are easy; a reference is [5] Prop. I.1.2. To show that D0 is a lattice, we pick some lattice D 0 then D0 is contained in the lattice X −n D 0 for some n ∈ N by property 3. Since the ring (N0 ) is noetherian the assertion follows. When D is killed by a power of p, the weak topology of D is locally compact (by properties 2 and 5). Proposition 7.92. Let D be a finitely generated étale ϕs -module over OE . Then ϕs and its canonical inverse ψs are continuous. Proof. (a) The above OE -linear surjective map f : ⊕d OE → D sends (ai )i to i ai di for some elements di ∈ D. As ϕs is étale, the map (ai )i → i ai ϕs (di ) also gives an OE -linear surjective map ⊕d OE → D. Both surjections are topological quotient maps by the definition of the topology on D, and the morphism ϕs of OE is continuous. We deduce that the morphism ϕs of D is continuous. (b) The image of ⊕d (N0 ) by f is a lattice D0 of D. For any k ∈ N the

(N0 )-submodule D0,k of D generated by (ϕs (X k ei ))1≤i≤d also is a treillis of D because ϕs is étale. Here {ei | 1 ≤ i ≤ d} is a generating family of D0 . We have ψs (D0,k ) = X k D0 (cf. Lemma 7.19). (c) When D is killed by a power of p, we deduce that ψs is continuous by the properties 1 and 2 of the treillis. When D is not killed by a power of p, we deduce that the reduction modulo p n of ψs is continuous for all n; this implies that ψs is continuous because (A = o, D) satisfy the properties a,

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b, c, d of Section 6.5, and D/ p n D is a (finitely generated) étale ϕs -module over OE . We put D + := {x ∈ D : the sequence (ϕsk (x))k∈N is bounded in D} and D ++ := {x ∈ D | lim ϕsk (x) = 0}. k→∞

(7.48)

Proposition 7.93. (i) If D is killed by a power of p, then D + and D ++ are lattices in D. (ii) There exists a unique maximal treillis D ' such that ψs (D ' ) = D ' . (iii) The set of ψs -stable treillis in D has a unique minimal element D & ; it satisfies ψs (D & ) = D & . (iv) X −k D ' is a treillis stable by ψs for all k ∈ N. Proof. The references given in the following are stated for étale (ϕs p , )modules but the proofs never use that there exists an action of and they are valid for étale ϕs p -modules. (i) For s = s p this is [5] Prop. II.2.2(iii) and Lemma II.2.3. The properties of s p which are needed for the argument are still satisfied for general s in the following form: – ϕs (X ) ∈ ϕsmp (X ) (Z p )× where s = s0 s m p z with s0 ∈ , m ≥ 1, and z ∈ Z. k k – (ϕs (X )X −1 ) p ∈ p k+1 (Z p ) + X ( p−1) p (Z p ) for any k ∈ N. (ii) and (iii) For any finitely generated OE -torsion module M we denote its Pontrjagin dual of continuous o-linear maps from M to K /o by M ∨ := ∨ Homcont o (M, K /o). Obviously, M again is an OE -module by (λ f )(x) := ∨ f (λx) for λ ∈ OE , f ∈ M , and x ∈ M. It is shown in [5] Lemma I.2.4 that: – M ∨ is a finitely generated OE -torsion module, – the topology of pointwise convergence on M ∨ coincides with its weak topology as an OE -module, and – M ∨∨ = M. Now let D be as in the assertion but killed by a power of p. One checks that D ∨ also belongs to Met OE (ϕs ) with respect to the semilinear map ϕs ( f ) := f ◦ ψs for f ∈ D ∨ ; moreover, the canonical left inverse is ψs ( f ) = f ◦ ϕs . Next, [5] Lemma I.2.8 shows that:

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– If D0 ⊂ D is a lattice then D0⊥ := {d ∈ D ∨ : f (D0 ) = 0} is a lattice in D ∨ , and D0∨∨ = D0 . We now define D & := (D ∨ )+ and D ' := (D ∨ )++ . The purely formal arguments in the proofs of [5] Prop. II.6.1, Lemma II.6.2, and Prop. II.6.3 show that D & and D ' have the asserted properties. For a general D in Met OE (ϕs ) the (formal) arguments in the proof of [5] Prop. II.6.5 show that ((D/ p n D)& )n∈N and ((D/ p n D)' )n∈N are well defined projective systems of compact (Z p )-modules (with surjective transition maps). Hence D & := lim(D/ p n D)& ← −

and

D ' := lim(D/ p n D)' ← −

have the asserted properties. (iv) X −k D ' is clearly a treillis. As X divides ϕs (X ) = (1 + X )a − 1 in

(Z p ) = o[[X ]] (when s ∈ sa Z for some a ∈ pZ p \ {0}), there exists f (X ) ∈ o[[X ]] such that ϕs (X k ) = X k f (X )k . So we have ψs (X −k D ' ) = ψs (ϕs (X −k ) f (X )k D ' ) = X −k ψs ( f (X )k D ' ) ⊂ X −k ψs (D ' ) ⊂ X −k D ' since D ' is ψs -stable by definition. Proposition 7.94. Let D be a finitely generated étale ϕs -module over OE killed by a power of p. For any compact subset C ⊆ D, there exists an r ∈ N such that

ψsk (N0 C) ⊆ X −r D ++ . k≥0

Proof. Since N0 C is compact and D ++ and D ' are treillis, there exists l ∈ N such that N0 C ⊂ X −l D ' ⊂ X −2l D ++ . By (iv) of Proposition 7.93 we obtain for all k ≥ 0 ψsk (N0 C) ⊂ ψsk (X −l D ' ) ⊂ X −l D ' ⊂ X −2l D ++ and we can take r = 2l. Corollary 7.95. Let D be a finitely generated étale ϕs -module over OE . For any compact subset C ⊆ D and any n ∈ N, there exists k0 ∈ N such that

ψsk (N0 C) ⊆ D ' + p n D. k≥k0

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325

Proof. We may assume that D is killed by a power of p. In view of (the proof of) Proposition 7.94 it suffices to show that for all l > 0 there exists a k0 such k that ψs 0 (X −l D ' ) = D ' . By Proposition 7.93(ii) and (iv) we have D ' = ψsk+1 (D ' ) ⊆ ψsk+1 (X −l D ' ) ⊆ ψsk (X −l D ' ) ⊆ X −l D ' 6 for any k ≥ 0. Hence k ψsk (X −l D ' ) is a treillis in D on which ψs is surjective. Therefore it coincides with D ' by the maximality of D ' (Proposition 7.93(ii)). On the other hand, the Z p [[X ]]-module (X −l D ' )/D ' is killed by both X l and p n and hence is finite. So there exists a k0 such that ψsk (X −l D ' ) = D ' for all k ≥ k0 . For any submonoid L ⊂ L + containing a strictly dominant element, an étale L -module over OE is a finitely generated OE -module with an étale semilinear action of L . A topologically étale L -module over OE will be an étale L -module D over OE such that the action L × D → D of L on D is continuous. This terminology is provisional since we will show later on (Corollary 7.125) in a more general context that any étale L -module over OE in fact is topologically étale and, in particular, is a complete topologically étale o[N0 L ]-module in our previous sense. a b Let D be a topologically étale L + -module over OE and let g = ∈ c d GL2 (Q p ). Denote 0 1 1 b w0 = , u(b) = . 1 0 0 1 Using the formula

ar + b gu(r )w0 = cr + d

a , c

one checks that the set U g defined by g −1 C0 ∩ C0 = Ug w0 P/P is Ug = u(X g ) where

ar + b ∈ Z p }. cr + d

X g = {r ∈ Z p |cr + d = 0,

For each r ∈ X g we can write

gu(r )w0 = u(g[r ])t (g, r )w0 u where ar + b g[r ] = ∈ Zp, cr + d

, t (g, r ) =

c cr + d

det g cr +d

0

,

0 . cr + d

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We deduce that for u(r ) ∈ Ug we have α(g, xu(r) ) = u(g[r ])t (g, r). Let now s = sa z ∈ L + be strictly dominant, with z ∈ Z and a ∈ pZ p − {0}. There exists a positive integer k g,s such that for all k ≥ k g,s we have t (g, r )s k ∈ L + . Note that Nk = s k N0 s −k = u(a k Z p ). We deduce that for k ≥ k g,s the operators Hg, J (N0 /Nk ) introduced in (7.28) are equal to the operators Hg,s,J (Z p /a k Z p ) = (1 + X )g[r] ϕt (g,r)s k ψsk ((1 + X )−r ). r∈X g ∩J (Z p /a k Z p )

Proposition 7.96. Let D be a topologically étale L + -module over OE . For the cont compact open topology in Endcont o (D), the maps αg,0 : N0 −→ Endo (D), for g ∈ N0 P N0 , are integrable with respect to s and res, for all s ∈ L + strictly dominant, i.e. s = sa z with a ∈ pZ p − {0} and z ∈ Z , their integrals / Hg = αg,0 d res = lim Hg,s,J (Z p /a k Z p ) N0

k→∞

for any choices of J (Z p /a k Z p ) ⊂ Z p , do not depend on the choice of s and satisfy the relations H1, H2, H3 of Proposition 7.72. Proof. By Proposition 7.91, we reduce to the case that D is killed by a power of p and to showing the assumptions of Proposition 7.84 for the family of all compact subsets of D. The axioms Ci , for 1 ≤ i ≤ 6, are obviously satisfied by continuity of ϕs , ψs , and of the action of n ∈ N0 on D. i. We check first the convergence criterion of Proposition 7.83, using the theory of treillis, i.e. of lattices, in D. Given a lattice M ⊆ D, a compact subset C ⊆ D such that N0 C ⊆ C, and a compact subset C+ ⊆ L + , we want to find a compact open subgroup P1 ⊂ P0 and an integer k0 ∈ N such that s k (1 − P1 )C+ ψsk (C) ⊆ M

(7.49)

for all k ≥ k0 . We choose r0 ∈ N with ϕsk (D ++ ) ⊂ M for all k ≥ r0 , as we may by the definition of D ++ . We choose r, k0 ∈ N such that k0 ≥ r0 and

ψsk (C) ⊆ X −r D ++ , k≥k0

as we may by Proposition 7.94. Applying C+ we obtain

C+ ψsk (C) ⊆ C+ (X −r D ++ ). k≥k0

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The continuity of the action of P+ on D implies that C+ (X −r D ++ ) is com pact. Hence we can choose r ∈ N such that C+ (X −r D ++ ) ⊆ X −r D ++ and we obtain

C+ ψsk (C) ⊆ X −r D ++ . k≥k0

As X −r D ++ is compact and D ++ an open neighbourhood of 0, the continuity of the action of P+ on D there exists a compact open subgroup P1 ⊆ P+ such that

(1 − P1 )X −r D ++ ⊆ D ++ . Hence we have s k (1 − P1 )C+ ψsk (C) ⊂ ϕsk (D ++ ) ⊂ M for all k ≥ k0 . ii. To obtain all the assumptions of Proposition 7.84 for the family of all compact subsets of D, it remains to prove that, given x ∈ D and g ∈ N0 P N0 , s = sa z with a ∈ pZ p − {0} and z ∈ Z, and (J (Z p /a k Z p ))k , there exists a compact C x,g,s ⊂ D and a positive integer k x,g,s such that Hg,s,J (Z p /a k Z p ) (x) ∈ C x,g,s for any k ≥ k x,g,s . This is clear because D is locally compact (by hypothesis D is killed by a power of p) and the sequence (Hg,s,J (Z p /a k Z p ) (x))k converges. iii. The independence of the choice of s ∈ L + strictly dominant results from the fact that, for z ∈ Z , e ∈ Z∗p , and a positive integer r , we have kr k k r k kr (zs pr e )k N0 (zs pr e )−k = s kr p N0 s p and ϕzs pr e ψzs pr e = ϕs p ψs p as ψzse is the right and left inverse of ϕzse . Remark 7.97. Let D be a topologically étale L + -module over OE , on which .Z acts through a character ω. The pointwise convergence of the integrals N0 αg,0 dres is a basic theorem of Colmez, allowing him the construction of the representation of GL2 (Q p ) that he denotes D ω P1 . Our construction coincides with Colmez’s construction because our Hg ∈ Endcont o (D) are the same as the Hg of Colmez given in [6] lemma II.1.2 (ii). Indeed, α(g, xu(r) ) = u(g[r])t (g, r ) = , ω(cr + d)u(g[r ]) where g [r] =

det g . (cr+d)2

det g (cr +d)2

0

0 g [r ] = ω(cr + d) 0 1

g[r ] , 1

This coincides with Colmez’s formula.

The major goal of the paper is to generalize Proposition 7.96. See Proposition 7.141.

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8. A generalization of (ϕ, )-modules We return to a general group G. We denote G (2) := G L(2, Q p ) and the objects relative to G (2) will be affected with an upper index (2) . (a) We suppose that N0 has the structure of a p-adic Lie group and that we have a continuous surjective homomorphism : N0 → N0(2) . (2)

We choose a continuous homomorphism ι : N0 → N0 which is a section of (2) (which is possible because N0 Z p ). (2) We have N0 = N ι(N0 ) where N is the kernel of . We denote by L ,+ := {t ∈ L | t N t −1 ⊂ N , t N0 t −1 ⊂ N0 } the stabilizer of N in the L-stabilizer of N0 , and by L ,ι := {t ∈ L | t N t −1 ⊂ (2) (2) (2) N , tι(N0 )t −1 ⊂ ι(N0 )} the stabilizer of N in the L-stabilizer of ι(N0 ). We have L ,ι ⊂ L ,+ . (b) We suppose given a submonoid L ∗ of L ,ι containing s and a continuous (2) homomorphism : L ∗ → L + such that (, ι) satisfies (tut −1 ) = (t)(u)(t)−1 , tι(y)t −1 = ι((t)y(t)−1 ), (2)

for u ∈ N0 , y ∈ N0 , t ∈ L ∗ . (2)

The sequence (s n N0 s −n ) = (s)n N0 (s)−n in N (2) is decreasing with trivial intersection. The maps in (a) and (b) combine to a unique continuous homomorphism (2)

P∗ := N0 L ∗ → P+ .

8.1. The microlocalized ring (N0 ) The ring (N0 ), denoted by N (N0 ) in [12], is a generalization of the ring (2) OE , which corresponds to id (N0 ). We refer the reader to [12] for the proofs of some claims in this section. The maximal ideal M(N ) of the completed group o-algebra (N ) = o[[N ]] is generated by p K and by the kernel of the augmentation map o[N ] → o. The ring (N0 ) is the M(N )-adic completion of the localization of

(N0 ) with respect to the Ore subset S (N0 ) of elements which are not in M(N ) (N0 ). The ring (N0 ) can be viewed as the ring (N )[[X ]] of skew Taylor series over (N ) in the variable X = [γ ] − 1 where γ ∈ N0 and

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(γ ) is a topological generator of (N0 ). Then (N0 ) is viewed as the ring of infinite skew Laurent series n∈Z an X n over (N ) in the variable X with limn→−∞ an = 0 for the pseudocompact topology of (N ). The ring (N0 ) is strict-local noetherian of maximal ideal M (N0 ) generated by M(N ). It is a pseudocompact ring for the M(N )-adic topology (called the strong topology). It is a complete Hausdorff ring for the weak topology ([12] Lemma 8.2) with fundamental system of open neighbourhoods of 0 given by On,k := M (N0 )n + M(N0 )k for n ∈ N, k ∈ N. In the computations it is sometimes better to use the fundamental systems of open neighbourhoods of 0 defined by Bn,k := M (N0 )n + X k (N0 ) for n ∈ N, k ∈ N, and Cn,k := M (N0 )n + (N0 )X k for n ∈ N, k ∈ N, which are equivalent due to the two formulas X k (N0 ) ⊆ (N0 )X k +M(N0 )k

and (N0 )X k ⊆ X k (N0 )+M(N0 )k ,

We write On := On,n , Bn := Bn,n , and Cn = Cn,n . Then (On )n , (Bn )n , and (Cn )n are also fundamental system of open neighbourhoods of 0 in (N0 ). The action (b = ut, n 0 ) → b.n 0 = utn 0 t −1 of the monoid P,+ = N0 L ,+ on N0 induces a ring action (t, x) → ϕt (x) of L ,+ on the oalgebra (N0 ) respecting the ideal (N0 )M(N ), and the Ore set S (N0 ) hence defines a ring action of L ,+ on the o-algebra (N0 ). This action respects the maximal ideals M(N0 ) and M (N0 ) of the rings (N0 ) and

(N0 ) and hence the open neighbourhoods of zero On,k . Lemma 7.98. For t ∈ L ,+ , a fundamental system of open neighbourhoods of 0 in (N0 ) is given by (ϕt (On,k ) (N0 ))n,k∈N . Proof. We have just seen ϕt (On,k ) (N0 ) ⊂ On,k . Conversely, given n, k ∈ N, we have to find n , k ∈ N such that On ,k ⊂ ϕt (On,k ) (N0 ). This can be deduced from the following fact. Let H ⊂ H be an open subgroup. Then given k ∈ N, there is k ∈ N such that

M(H )k (H ) ⊃ M(H )k .

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Indeed by taking a smaller H we can suppose that H ⊂ H is open normal. Then M(H )k (H ) is a two-sided ideal in (H ) and the factor ring (H )/M(H ) (H ) is an artinian local ring with maximal ideal M(H )/M(H ) (H ). It remains to observe that in any artinian local ring the maximal ideal is nilpotent. Proposition 7.99. The action of L ,+ on (N0 ) is étale: for any t ∈ L ,+ , the map (λ, x) → λϕt (x) : (N0 ) ⊗ (N0 ),ϕt (N0 ) → (N0 ) is bijective. Proof. We follow ([12] Prop. 9.6, Proof, Step 1). (1) The conjugation by t gives a natural isomorphism

(N0 ) → t N t −1 (t N0 t −1 ). (2) Obviously t N t −1 (t N0 t −1 ) = (t N0 t −1 ) ⊗ (t N0 t −1 ) t N t −1 (t N0 t −1 ), and the map

(t N0 t −1 ) ⊗ (t N0 t −1 ) t N t −1 (t N0 t −1 ) → (N0 )⊗ (t N0 t −1 )

t N t −1 (t N0 t −1 )

is injective because t N t −1 (t N0 t −1 ) is flat on (t N0 t −1 ). (3) The natural map

(N0 ) ⊗ (t N0 t −1 ) t N t −1 (t N0 t −1 ) → (N0 ) is bijective. (4) The ring action ϕt : (N0 ) → (N0 ) of t on (N0 ) is the composite of the maps of (1), (2), (3), hence is injective. (5) The proposition is equivalent to (3) and ϕt injective.

Remark 7.100. The proposition is equivalent to: for any t ∈ L ,+ , the map (u, x) → uϕt (x) : o[N0 ] ⊗o[N0 ],ϕt (N0 ) → (N0 ) is bijective. et et 8.2. The categories M (L ∗ ) and MO (L ∗ ) (N0 ) E ,

By the universal properties of localization and adic completion the continuous homomorphisms and ι between N0 and N0(2) extend to continuous o-linear homomorphisms of pseudocompact rings,

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: (N0 ) → OE , ι : OE → (N0 ), ◦ ι = id .

(7.50)

If we view the rings as rings of Laurent series, (X ) = X (2) , ι(X (2) ) = X , and is the augmentation map (N ) → o and ι is the natural injection o →

(N ), on the coefficients. We have for n, k ∈ N, (2) (M (N0 )) = p K OE , (Bn,k ) = Bn,k ,

(7.51)

(2)

ι( p K OE ) = M (N0 ) ∩ ι(OE ), ι(Bn,k ) = Bn,k ∩ ι(OE ). (2)

We denote by J (N0 ) the kernel of : (N0 ) → (N0 ) and by J (N0 ) the kernel of : (N0 ) → OE . They are the closed two-sided ideals generated (as left or right ideals) by the kernel of the augmentation map o[N ] → o. We have

(N0 ) = ι(OE ) ⊕ J (N0 ), M (N0 ) = p K ι(OE ) ⊕ J (N0 ), (2)

X k (N0 ) = ι((X (2) )k (N0 ) ⊕ X k J (N0 ),

(7.52)

(2) Bn,k = ι(Bn,k ) ⊕ (J (N0 ) ∩ Bn,k ).

The maps and ι are L ∗ -equivariant: for t ∈ L ∗ , ◦ ϕt = ϕ(t) ◦ ,

ι ◦ ϕ(t) = ϕt ◦ ι,

(7.53)

thanks to the hypothesis (b) made at the beginning of this chapter. The map ι is equivariant for the canonical action of the inverse monoid L −1 ∗ , but not the map as the following lemma shows. Lemma 7.101. For t ∈ L ∗ , we have ι◦ψ(t) = ψt ◦ι. We have ◦ψt = ψ(t) ◦ if and only if N = t N t −1 . (2)

(2)

Proof. Clearly N0 = N ι(N0 ) and t N0 t −1 = t N t −1 tι(N0 )t −1 for t ∈ L. We choose, as we may, for t ∈ L ,ι , a system J (N0 /t N0 t −1 ) of representatives of N0 /t N0 t −1 containing 1 such that (2)

(2)

J (N0 /t N0 t −1 ) = {uι(v) |u ∈ J (N /t N t −1 ), v ∈ J (N0 /(t)N0 (t −1 ))}. (7.54) We have ι ◦ ψ(t) = ψt ◦ ι because, for λ ∈ OE , we have on one hand (7.12) λ= vϕ(t) (λv,(t) ), λv,(t) = ψ(t) (v −1 λ), (2)

(2)

v∈J (N0 /(t)N0 (t)−1 )

ι(λ) = (2)

ι(v)ϕt (ι(λv,(t) )), (2)

v∈J (N0 /(t)N0 (t)−1 )

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and on the other hand (7.12) ι(λ) =

uι(v)ϕt (ι(λ)uι(v),t ),

(2) (2) u∈J (N /t N t −1 ),v∈J (N0 /(t)N0 )(t)−1 )

where ι(λ)uι(v),t = ψt (ι(v)−1 u −1 ι(λ)). By the uniqueness of the decomposition, ι(λ)ι(v),t = ι(λv,(t) ), ι(λ)uι(v),t = 0 if u = 1. Taking u = 1, v = 1, we get ψt (ι(λ)) = ι(ψ(t) (λ)). A similar argument shows that ◦ ψt = ψ(t) ◦ if and only if N = t N t −1 . For λ ∈ (N0 ), λ= uϕt (λu,t ), λu,t = ψt (u −1 λ), u∈J (N0 /t N0 t −1 )

(λ) =

u∈ J (N0 /t N0

(u)ϕ(t) ((λu,t )) = t −1 )

vϕ(t) ((λ)v,(t) ).

(2) (2) v∈J (N0 /(t)N0 )

By the uniqueness of the decomposition, (λ)v,(t) =

(λuι(v),t ).

u∈J (N /t N t −1 )

We deduce that ◦ ψt = ψ(t) ◦ if and only if N = t N t −1 . Remark 7.102. ◦ ψs = ψ(s) ◦ , except in the trivial case where : N0 → (2) N0 is an isomorphism, because s N s −1 = N as the intersection of the decreasing sequence s k N s −k for k ∈ N is trivial. For future use, we note: Lemma 7.103. The left or right o[N0 ]-submodule generated by ι(OE ) in

(N0 ) is dense. Proof. As o[N0 ] is dense in (N0 ) it suffices to show that the left or right

(N0 )-submodule generated by ι(OE ) in (N0 ) is dense. This will be shown even with respect to the M (N0 )-adic topology. View λ ∈ (N0 ) as an infinite Laurent series λ = n∈Z λn X n with λn ∈

(N ) and limn→−∞ λn = 0 in the M(N )-adic topology of (N ). Further, note that the left, resp. right, (N0 )-submodule of (N0 ) generated by ι(OE ) contains (N0 )X −m , resp. X −m (N0 ), for any positive integer m. Finally, for each n ∈ N there exists μn in (N0 )X −m , resp. X −m (N0 ), for some large m, such that λ − μn ∈ M (N0 )n .

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n Let M be a finitely generated (N0 )-module and let f : ⊕i=1

(N0 ) → M be a (N0 )-linear surjective map. We put on M the quotient topology of the weak topology on ⊕in=1 (N0 ); this is independent of the choice of f . Then M is a Hausdorff and complete topological (N0 )-module and every submodule is closed ([12] Lemma 8.22). In the same way we can equip M with the pseudocompact topology. Again M is Hausdorff and complete and every submodule is closed in the pseudocompact topology, because (N0 ) is noetherian. The weak topology on M is weaker than the pseudocompact topology which is weaker than the p-adic topology. In particular the intersection of the submodules p n M for n ∈ N is 0. By [9] IV.3.Prop. 10, M is p-adically complete, i.e., the natural map M → limn M/ p n M is bijective. ← − Unless otherwise indicated, M is always understood to carry the weak topology.

Lemma 7.104. The properties a,b,c,d of Section 6.5 are satisfied by (o, M) and M is complete. Definition 7.105. A finitely generated module M over (N0 ) with an étale semilinear action of a submonoid L of L ,+ is called an étale L -module over

(N0 ). We denote by Met

(N0 ) (L ) the category of étale L -modules on (N0 ). Lemma 7.106. The category Met

(N0 ) (L ) is abelian.

Proof. As in the proof of Proposition 7.23 and using that the ring (N0 ) is noetherian. (2)

The continuous homomorphism : L ∗ → L + defines an étale semilinear (2) action of L ∗ on the ring id (N0 ) isomorphic to OE . Definition 7.107. A finitely generated module D over OE with an étale semilinear action of L ∗ is called an étale L ∗ -module over OE . An element t ∈ L ∗ in the kernel L =1 of acts trivially on OE hence ∗ bijectively on an étale L ∗ -module over OE . Remark 7.108. The action of L =1 on D extends to an action of the subgroup ∗ =1 is commutative or if we assume that for each of L generated by L =1 if L ∗ ∗ t ∈ L =1 there exists an integer k > 0 such that s k t −1 ∈ L ∗ . The assumption ∗ is trivially satisfied whenever L ∗ = H ∩ L + for some subgroup H ⊂ L. Indeed, the subgroup generated by L =1 is the set of words of the form ∗ ±1 x1 . . . xn±1 with xi ∈ L =1 for i = 1, . . . , n. So if we have an action of all ∗ the elements and all the inverses, then we can take the products of these, as

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well. We need to show that this action is well defined, i.e., whenever we have a relation x1±1 . . . xn±1 = y1±1 . . . yr±1

(7.55)

in the group then the action we just defined is the same using the x’s or the y’s. If L =1 is commutative, this is easily checked. In the second case, we can ∗ n r ki −1 ∈ L and choose a big enough k = ∗ i=1 ki + j =1 k j such that s x i −1 k k j s y j ∈ L ∗ . Then multiplying the relation (7.55) by s we obtain a relation in L ∗ so the two sides will define the same action on D. This shows that the actions defined using the two sides of (7.55) are equal on ϕsk (D) ⊂ D. (2) However, they are also equal on group elements u ∈ N0 hence on the whole ) k D = u∈J (N (2) /ϕ k (N (2) )) uϕs (D). 0

s

0

We denote by MOE, (L ∗ ) the category of étale L ∗ -modules on OE . et

Lemma 7.109. The category Met O

E,

(L ∗ ) is abelian.

Proof. As in the proof of Proposition 7.23 and using that the ring OE is noetherian. et We will prove later that the categories Met OE, (L ∗ ) and M (N0 ) (L ∗ ) are equivalent.

8.3. Base change functors We recall a general argument of semilinear algebra (see [12]). Let A be a ring with a ring endomorphism ϕ A , let B be another ring with a ring endomorphism ϕ B , and let f : A → B be a ring homomorphism such that f ◦ ϕ A = ϕ B ◦ f . When M is an A-module with a semilinear endomorphism ϕ M , its image by base change is the B-module B ⊗ A, f M with the semilinear endomorphism ϕ B ⊗ ϕ M . The endomorphism ϕ M of M is called étale if the natural map a ⊗ m → aϕ M (m) : A ⊗ A,ϕ A M → M is bijective. Lemma 7.110. When ϕ M is étale, then ϕ B ⊗ ϕ M is étale. Proof. We have B ⊗ B,ϕ B (B ⊗ A, f M) = B ⊗ A,ϕ B ◦ f M = B ⊗ f ◦ϕ A M ∼ B ⊗ A, f M. = B ⊗ A, f (A ⊗ A,ϕ A M) =

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Applying these general considerations to the L ∗ -equivariant maps :

(N0 ) → OE and ι : OE → (N0 ) satisfying ◦ ι = id (see (7.50), (7.53)), we have the base change functors M → D(M) := OE ⊗ (N0 ), M from the category of (N0 )-modules to the category of OE -modules, and D → M(D) := (N0 ) ⊗OE ,ι D in the opposite direction. Obviously these base change functors respect the property of being finitely generated. By the general lemma we obtain: Proposition 7.111. The above functors restrict to functors et D : Met

(N0 ) (L ∗ ) → MOE, (L ∗ ) and

et M : Met OE, (L ∗ ) → M (N0 ) (L ∗ ).

When M ∈ Met

(N0 ) (L ∗ ), the diagonal action of L ∗ on D(M) is: ϕt (μ ⊗ m) = ϕ(t) (μ) ⊗ ϕt (m) for t ∈ L ∗ , μ ∈ OE , m ∈ M.

(7.56)

When D ∈ Met OE, (L ∗ ), the diagonal action of L ∗ on M(D) is: ϕt (λ ⊗ d) = ϕt (λ) ⊗ ϕt (d) for t ∈ L ∗ , λ ∈ (N0 ), d ∈ D.

(7.57)

The natural map M : M → D(M), M (m) = 1 ⊗ m is surjective, L ∗ -equivariant, with a P∗ -stable kernel M := J (N0 )M. The injective L ∗ -equivariant map ι D : D → M(D), ι D (d) = 1 ⊗ d is ψt -equivariant for t ∈ L ∗ (same proof as Lemma 7.101). For future use we note the following property. Lemma 7.112. Let d ∈ D and t ∈ L ∗ . We have $ (2) ι D (ψt (v −1 d)) if u = ι(v) with v ∈ N0 , ψt (u −1 ι D (d)) = 0 if u ∈ N0 \ ι(N0(2) )t N0 t −1 . Proof. We choose a set J ⊂ N0(2) of representatives for the cosets in (2) (2) N0 /(t)N0 (t)−1 . The semilinear endomorphism ϕt of D is étale hence d= vϕt (dv,t ) where dv,t = ψt (v −1 d). v∈J

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Applying ι D we obtain ι D (d) = ι(v)ι D (ϕt (dv,t )) = ι(v)ϕt (ι D (dv,t )) v

=

v

ι(v)ϕt (ψt (ι D (v

−1

d))).

v

The map ι induces an injective map from J into N0 /t N0 t −1 with image included in a set J (N0 /t N0 t −1 ) ⊂ N0 of representatives for the cosets in N0 /t N0 t −1 . As the action ϕt of t in M(D) is étale, we have (7.12) m= uϕt (m u,t )) where m u,t = ψt (u −1 m) u∈J (N0 /t N0 t −1 )

for any m ∈ M(D). We deduce that ψt (ι(v −1 )ι D (d)) = ι D (dv,t ) when v ∈ J and ψt (u −1 ι D (d)) = 0 when u ∈ J (N0 /t N0 t −1 ) \ ι( J ). As any element of N0(2) can belong to a set of representatives of N0(2) /(t)N0(2) (t)−1 , we (2) deduce that ψt (ι(v −1 )ι D (d)) = ι D (dv,t ) for any v ∈ N0 . For the same −1 reason ψt (ι(u )ι D (d)) = 0 for any u ∈ N0 which does not belong to (2) ι(N0 )t N0 t −1 .

8.4. Equivalence of categories Let D ∈ Met OE , (L ∗ ). By definition D(M(D)) = OE ⊗ (N0 ), ( (N0 ) ⊗OE ,ι D), and we have a natural map μ ⊗ (λ ⊗ d) → μ(λ)d : OE ⊗ (N0 ), ( (N0 ) ⊗OE ,ι D) → D. Proposition 7.113. The natural map D(M(D)) → D is an isomorphism in Met OE , (L ∗ ). Proof. The natural map is bijective because ◦ι = id : OE → (N0 ) → OE , and L ∗ -equivariant because the action of t ∈ L ∗ satisfies ϕt (μ ⊗ (λ ⊗ d)) = ϕ(t) (μ) ⊗ ϕt (λ ⊗ d) = ϕ(t) (μ) ⊗ (ϕt (λ) ⊗ ϕt (d)), ϕt (μ(λ)d) = ϕ(t) (μ((λ))ϕt (d) = ϕ(t) (μ)(ϕt (λ))ϕt (d), by (7.56), (7.57). The kernel N of : N0 → Z p being a closed subgroup of N0 is also a padic Lie group, hence contains an open pro- p-subgroup H with the following property ([11] Remark 26.9 and Thm. 27.1): n For any integer n ≥ 1, the map h → h p is an homeomorphism of H onto an open subgroup Hn ⊆ H , and (Hn )n≥1 is a fundamental system of open neighbourhoods of 1 in H .

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The groups s k N s −k for k ≥ 1 are open and form a fundamental system of neighbourhoods of 1 in N . For any integer n ≥ 1 there exists a positive integer k such that any element in s k N s −k is contained in Hn , hence is a p n -th power of some element in N . We denote by kn the smallest positive integer such that any element in s kn N s −kn is a p n -th power of some element in N . Lemma 7.114. For any positive integers n and k ≥ kn , we have ϕ k (J (N0 )) ⊂ M (N0 )n+1 . Proof. For u ∈ N , and j ∈ N, the value at u of the p j -th cyclotomic polynomial p j (u) lies in M (N0 ) and n

up − 1 =

n

p j (u)

j=0

lies in M (N0 )n+1 . An element v ∈ s k N s −k is a p n -th power of some element in N hence v −1 lies in M (N0 )n+1 . The ideal J (N0 ) of (N0 ) is generated by u − 1 for u ∈ N and ϕ k (J (N0 )) is contained in the ideal generated by v − 1 for v ∈ s k N s −k . As M (N0 ) is an ideal of (N0 ) we deduce that ϕ k (J (N0 )) ⊂ M (N0 )n+1 . Lemma 7.115. i. The functor D is faithful. ii. The functor M is fully faithful. Proof. Obviously ii. follows from i. by Proposition 7.113. To prove i. let f : M1 → M2 be a morphism in Met

(N0 ) (L ∗ ) such that D( f ) = 0, i.e., such that f (M1 ) ⊆ J (N0 )M2 . Since M1 is étale we deduce that f (M1 ) ⊆ 6 k 6 (N )n M2 . Since the k ϕ (J (N0 ))M2 and hence, by Lemma 7.114, in n M 6 0 pseudocompact topology on M2 is Hausdorff we have n M (N0 )n M2 = 0. It follows that f = 0. Let M ∈ Met

(N0 ) (L ∗ ). By definition, MD(M) = (N0 ) ⊗OE ,ι (OE ⊗ (N0 ), M) = (N0 ) ⊗ (N0 ),ι◦ M. In the particular case where L ∗ = s N is the monoid generated by s, we et et denote the category Met

(N0 ) (L ∗ ) (resp. MOE, (L ∗ )), by M (N0 ) (ϕ) (resp. et et Met OE, (ϕ)). The category M (N0 ) (L ∗ ) (resp. MOE, (L ∗ )) is a subcategory of et Met

(N0 ) (ϕ) (resp. MOE, (ϕ)). Proposition 7.116. For any M ∈ Met

(N0 ) (ϕ) there is a unique morphism % M : M → MD(M)

in Met

(N0 ) (ϕ)

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D(% M ) such that the composed map D (% M ) : D(M) −−−−→ DMD(M) ∼ = D(M) is the identity. The morphism % M , in fact, is an isomorphism.

Proof. The uniqueness follows immediately from Lemma 7.115.i. The construction of such an isomorphism % M will be done in three steps. Step 1: We assume that M is free over (N0 ), and we start with an arbitrary finite (N0 )-basis (i )i∈I of M. By (8.2), we have M = (⊕i ∈I ι(OE )i ) ⊕ (⊕i ∈I J (N0 )i ). The (N0 )-linear map from M to MD(M) sending i to 1 ⊗ (1 ⊗ i ) is bijective. If ⊕i ∈I ι(OE )i is ϕ-stable, the map is also ϕ-equivariant and is an isomorphism in the category Met

(N0 ) (ϕ). We will construct a (N0 )-basis (ηi )i∈I of M such that ⊕i ∈I ι(OE )ηi is ϕ-stable. We have ϕ(i ) = (ai, j + bi, j ) j where ai, j ∈ ι(OE ), bi, j ∈ J (N0 ). j∈I

If the bi, j are not all 0, we will show that there exist elements xi, j ∈ J (N0 ) such that (ηi )i∈I defined by ηi := i + xi, j j ,

j∈I

satisfies ϕ(ηi ) = j∈I ai, j η j for i ∈ I . By the Nakayama lemma ([1] II §3.2 Prop. 5), the set (ηi )i ∈I is a (N0 )-basis of M, and we obtain an isomorphism in Met

(N0 ) (ϕ), % M M → MD(M), %(ηi ) = 1 ⊗ (1 ⊗ ηi ) for i ∈ I, such that D (% M ) is the identity morphism of D(M). The conditions on the matrix X := (xi, j )i, j ∈I are: ϕ(Id + X )(A + B) = A(Id + X ) for the matrices A := (ai, j )i, j∈I , B := (bi, j )i, j ∈I . The coefficients of A belong to the commutative ring ι(OE ). The matrix A is invertible because the

(N0 )-endomorphism f of M defined by f (i ) = ϕ(i ) for i ∈ I is an automorphism of M as ϕ is étale. We have to solve the equation A−1 B + A−1 ϕ(X )(A + B) = X. For any k ≥ 0 define Uk = A−1 ϕ(A−1 ) . . . ϕ k−1 (A−1 ) ϕ k (A−1 B) ϕ k−1 (A+B) . . . ϕ( A+B)(A+B).

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We have A−1 ϕ(Uk )(A + B) = Uk+1 . Hence X := k≥0 Uk is a solution of our equation provided this series converges with respect to the pseudocompact topology of (N0 ). The coefficients of A−1 B belong to the two-sided ideal J (N0 ) of (N0 ). Therefore the coefficients of Uk belong to the two-sided ideal generated by ϕ k (J (N0 )). Hence the series converges (Lemma 7.114). The coefficients of every term in the series belong to J (N0 ) and J (N0 ) is closed in (N0 ), hence xi, j ∈ J (N0 ) for i, j ∈ I . Step 2: We show that any module M in Met

(N0 ) (ϕ) is the quotient of another module M1 in Met (ϕ) which is free over

(N0 ).

(N0 ) Let (m i )i ∈I be a minimal finite system of generators of the (N0 )-module M. As ϕ is étale, (ϕ(m i ))i∈I is also a minimal system of generators. We denote by (ei )i∈I the canonical (N0 )-basis of ⊕i∈I (N0 ), and we consider the two surjective (N0 )-linear maps f, g : ⊕i ∈I (N0 ) → M, f (ei ) = m i , g(ei ) = ϕ(m i ). In particular, we find elements m i ∈ M, for i ∈ I , such that g(m i ) = ϕ(m i ). By the Nakayama lemma ([1] II §3.2 Prop. 5) the (m i )i∈I form another

(N0 )-basis of ⊕i∈I (N0 ). The ϕ-linear map ⊕i ∈I (N0 ) → ⊕i∈I (N0 ), ϕ( λi ei ) := ϕ(λi )m i i∈I

i∈I

therefore is étale. With this map, M1 := ⊕i∈I (N0 ) is a module in Met

(N0 ) (ϕ) which is free over (N0 ), and the surjective map f is a morphism in Met

(N0 ) (ϕ). Step 3: As (N0 ) is noetherian, we deduce from Step 2 that for any module M in Met

(N0 ) (ϕ) we have an exact sequence f

f

M2 − → M1 − →M →0 in Met

(N0 ) (ϕ) such that M1 and M2 are free over (N0 ). We now consider the diagram MD(M O 2)

MD( f )

% M2 ∼ =

M2

f

/ MD(M1 ) MD( f )/ MD(M) O O % M1 ∼ % = M f / M1 /M

/0. /0

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Since the functors M and D are right exact both rows of the diagram are exact. By Step 1 the left two vertical maps exist and are isomorphisms. Since D(MD( f ) ◦ % M2 − % M1 ◦ f ) = D( f ) ◦ D (% M2 ) − D (% M1 ) ◦ D( f ) = 0 it follows from Lemma 7.115.i that the left square of the diagram commutes. Hence we obtain an induced isomorphism % M as indicated, which moreover by construction satisfies D (% M ) = idD(M) . Theorem 7.117. The functors et et et M : Met OE , (L ∗ ) → M (N0 ) (L ∗ ), D : M (N0 ) (L ∗ ) → MOE , (L ∗ ),

are quasi-inverse equivalences of categories. Proof. By Proposition 7.113 and Lemma 7.115.ii it remains to show that the functor M is essentially surjective. Let M ∈ Met

(N0 ) (L ∗ ). We have to find a D ∈ Met (L ) together with an isomorphism M ∼ = M(D) in OE, ∗ et M (N0 ) (L ∗ ). It suffices to show that the morphism % M in Proposition 7.116 is L ∗ -equivariant. We want to prove that (% M ◦ ϕt − ϕt ◦ % M )(m) = 0 for any m ∈ M and t ∈ L ∗ . Since D (% M ) = idD(M) we certainly have (% ◦ ϕt − ϕt ◦ %)(m) ∈ J (N0 )MD(M) for any m ∈ M and t ∈ L ∗ . We choose for any positive integer r a set J (N0 /Nr ) ⊆ N0 of representatives for the cosets in N0 /Nr . Writing (7.12) m= uϕ r (m u,s r ), m u,s r = ψ r (u −1 m) u∈J (N0 /Nr )

and using that st = ts we see that (% M ◦ ϕt − ϕt ◦ % M )(m) =

ϕt (u)ϕr ((% M ◦ ϕt − ϕt ◦ % M )(m u,sr ))

u∈J (N0 /Nr )

lies, for any r , in the (N0 )-submodule of MD(M) generated by ϕ r (J (N0 ))MD(M). As in the proof of Lemma 7.115.ii we obtain 6 r r >0 ϕ (J (N0 ))MD(M) = 0. Since the functors M and D are right exact they commute with the reduction modulo p n , for any integer n ≥ 1.

8.5. Continuity In this section we assume that L ∗ contains a subgroup L 1 which is open in L ∗ and is a topologically finitely generated pro- p-group.

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We will show that the L ∗ -action on any étale L ∗ -module over (N0 ) is automatically continuous. Our proof is highly indirect so that we temporarily will have to make some definitions. But first a few partial results can be established directly. Let M be a finitely generated (N0 )-module. Definition 7.118. A lattice in M is a (N0 )-submodule of M generated by a finite system of generators of the (N0 )-module M. The lattices of M are of the form M 0 = ri=1 (N0 )m i for a set (m i )1≤i≤r of generators of the (N0 )-module M. We have the three fundamental systems of neighbourhoods of 0 in M: (

r i =1 r

(

i =1 r

(

On,k m i = M (N0 )n M + M(N0 )k M 0 )n,k∈N ,

(7.58)

Bn,k m i = M (N0 )n M + X k M 0 )n,k∈N ,

(7.59)

Cn,k m i = M (N0 )n M + Mk0 )n,k∈N ,

(7.60)

i =1

where Mk0 is the lattice ri=1 (N0 )X k m i , and is different from the set X k M0 when N0 is not commutative. If M is an étale L ∗ -module over (N0 ), for any fixed t ∈ L ,+ we have a fourth fundamental system of neighbourhoods of 0 in M: r ( ϕt (On,k ) (N0 )ϕt (m i ))n,k∈N , i=1

given by Lemma 7.98, because (ϕt (m i )1≤i ≤r is also a system of generators of the (N0 )-module M. Proposition 7.119. Let L be a submonoid of L ,+ . Let M be an étale L module over (N0 ). Then the maps ϕt and ψt , for any t ∈ L , are continuous on M. Proof. The ring endomorphisms ϕt of (N0 ) are continuous since they preserve M(N0 ) and M(N ). The continuity of the ϕt on M follows as in part (a) of the proof of Proposition 7.92. The continuity of the ψt follows from ψt (

r i=1

ϕt (On,k ) (N0 )ϕt (m i )) =

r i=1

On,k ψt ( (N0 ))m i =

r i=1

On,k m i .

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The same proof shows that, for any D ∈ Met OE , (L ∗ ), the maps ϕt and ψt , for any t ∈ L ∗ , are continuous on D. Proposition 7.120. The L ∗ -action L ∗ × D → D on an étale L ∗ -module D over OE is continuous. Proof. Let D be in Met OE , (L ∗ ). Since we already know from Proposition 7.119 that each individual ϕt , for t ∈ L ∗ , is a continuous map on D and since L 1 is open in L ∗ it suffices to show that the action L 1 × D → D of L 1 on D is continuous. As D is p-adically complete with its weak topology being the projective limit of the weak topologies on the D/ p n D we may further assume that D is killed by a power of p. In this situation the weak topology on D is locally compact. By Ellis’ theorem ([8] Thm. 1) we therefore are reduced to showing that the map L 1 × D → D is separately continuous. Because of Proposition 7.119 it, in fact, remains to prove that, for any d ∈ D, the map L 1 −→ D, g −→ gd is continuous at 1 ∈ L 1 . This amounts to finding, for any d ∈ D and any lattice D0 ⊂ D, an open subgroup H ⊂ L 1 such that (H − 1)d ⊂ D0 . We observe that (X m D++ )m∈Z is a fundamental system of L 1 -stable open 0 neighbourhoods of zero in D such that m X m D++ = D. We now choose an m ≥ 0 large enough such that d ∈ X −m D++ and X m D++ ⊂ D0 . The L 1 action on D induces an L 1 -action on X −m D++ / X m D++ which is o-linear hence given by a group homomorphism L 1 → Auto (X −m D++ / X m D++ ). Since D++ is a finitely generated o[[X ]]-module which is killed by a power of p we see that X −m D++ / X m D++ is finite. It follows that the kernel H of the above homomorphism is of finite index in L 1 . Our assumption that L 1 is a topologically finitely generated pro- p-group finally implies, by a theorem of Serre ([7] Thm. 1.17), that H is open in L 1 . We obtain (H − 1)d ⊂ (H − 1)X −m D++ ⊂ X m D++ ⊂ D0 .

In the special case of classical (ϕ, )-modules on OE the proposition is stated as Exercise 2.4.6 in [10] (with the indication of a totally different proof). Proposition 7.121. Let L be a submonoid of L ,+ containing an open subgroup L 2 which is a topologically finitely generated pro- p-group. Then the L -action L × (N0 ) → (N0 ) on (N0 ) is continuous. Proof. Since we know already from Propositions 7.99 and 7.119 that each individual ϕt , for t ∈ L , is a continuous map on (N0 ) and since L 2 is open

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in L it suffices to show that the action L 2 × (N0 ) → (N0 ) of L 2 on

(N0 ) is continuous. The ring (N0 ) is M (N0 )-adically complete with its weak topology being the projective limit of the weak topologies on the

(N0 )/M (N0 )n (N0 ). It suffices to prove that the induced action of L 2 on = (N0 )/M (N0 )n is continuous. The weak topology on is locally compact since (Bk = (X k (N0 ) + M (N0 )n )/M (N0 )n )k∈Z forms a fundamental system of compact neighbourhoods of 0. By Ellis’ theorem ([8] Thm. 1) we therefore are reduced to showing that the map L 2 × → is separately continuous. Because of Proposition 7.119 it, in fact, remains to prove that, for any x ∈ , the map L 2 −→ , g −→ gx is continuous at 1 ∈ L 2 . This amounts to finding, for any x ∈ and any large k ≥ 1, an open subgroup H ⊂ L 2 such that (H − 1)x ⊂ Bk . We observe that the Bk , for k ∈ Z, are L 2 -stable of union . We now choose an . The L -action on induces an L m ≥ k large enough such that x ∈ B−m 2 2 action on B−m /Bm which is o-linear hence given by a group homomorphism /B ). Since B is isomorphic to o[[X ]] ⊗ (N )/M(N )n L 2 → Auto (B−m o m 0 /B is as an o[[X]]-module, and (N )/M(N )n is finite, we see that B−m m finite. It follows that the kernel H of the above homomorphism is of finite index in L 2 . Our assumption that L 2 is a topologically finitely generated prop-group finally implies, by a theorem of Serre ([7] Thm. 1.17), that H is open in L 2 . We obtain (H − 1)x ⊂ (H − 1)B−m ⊂ Bm ⊂ Bk .

Lemma 7.122. i. For any M ∈ Met

(N0 ) (L ∗ ) the weak topology on D(M) is the quotient topology, via the surjection M : M → D(M), of the weak topology on M. ii. For any D ∈ Met OE , (L ∗ ) the weak topology on M(D) induces, via the injection ι D : D → M(D), the weak topology on D. Proof. i. If we write M as a quotient of a finitely generated free (N0 )module then we obtain an exact commutative diagram of surjective maps of the form // M . ⊕n (N0 ) i=1

⊕i

⊕ni=1 OE

M

/ / D(M)

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The horizontal maps are continuous and open by the definition of the weak topology. The left vertical map is continuous and open by direct inspection of the open zero neighbourhoods Bn,k (see (7.51)). Hence the right vertical map M is continuous and open. ii. An analogous argument as for i. shows that ι D is continuous. Moreover ι D has the continuous left inverse M(D) . Any continuous map with a continuous left inverse is a topological inclusion. An étale L ∗ -module M over (N0 ), resp. over OE , will be called topologically étale if the L ∗ -action L ∗ × M → M is continuous. Let Met,c

(N0 ) (L ∗ ) et,c and MOE , (L ∗ ) denote the corresponding full subcategories of Met

(N0 ) (L ∗ ) and Met (L ), respectively. Note that, by construction, all morphisms in ∗ OE , et Met (L ) and in M (L ) are automatically continuous. Also note that ∗ ∗

(N0 ) OE , by Proposition 7.119 any object in these categories is a complete topologically étale o[N0 L ∗ ]-module in our earlier sense. Proposition 7.123. The functors M and D restrict to quasi-inverse equivalences of categories et,c et,c et,c M : Met,c OE , (L ∗ ) → M (N0 ) (L ∗ ), D : M (N0 ) (L ∗ ) → MOE , (L ∗ ).

Proof. It is immediate from Lemma 7.122.i that if L ∗ acts continuously on M ∈ Met

(N0 ) (L ∗ ) then it also acts continuously on D(M). On the other hand, let D ∈ Met OE , (L ∗ ) such that the action of L ∗ on D is continuous. We choose a lattice D0 in D with a finite system (di ) of generators. (2) Given t ∈ L ∗ we introduce Dt := i (N0 )t.di which is a lattice in D since the action of t on D is étale. Also D0 + Dt is a lattice in D. The (N0 )-module M(D) is generated by ι D (D0 ) as well as by ι D (D0 + Dt ) and both (Cn ι D (D0 ))n∈N

and (Cn ι D (D0 + Dt ))n∈N

are fundamental systems of neighbourhoods of 0 in M(D) for the weak topology. To show that the action of L ∗ on M(D) is continuous, it suffices to find for any t ∈ L ∗ , λ0 ∈ (N0 ), d0 ∈ D0 , n ∈ N a neighbourhood L t ⊂ L ∗ of t and n ∈ N such that L t .(λ0 ι D (d0 ) + Cn ι D (D0 )) ⊂ t.λ0 ι D (d0 ) + Cn ι D (D0 + Dt ). The three maps λ → λι D (d0 ) : (N0 ) → M(D) d → λ0 ι D (d) : D → M(D) (λ, d) → λι D (d) : (N0 × D) → M(D)

(7.61)

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345

are continuous because ι D is continuous. The action of L ∗ on D and on (N0 ) is continuous (Proposition 7.121). Altogether this implies that we can find a small L t such that L t .λ0 ι D (d0 ) ⊂ t.λ0 ι D (d0 ) + Cn ι D (D0 + Dt ). Since ι D is L ∗ -equivariant we have, for any n ∈ N, L t .Cn ι D (D0 ) = (L t .Cn ) ι D (L t .D0 ). The continuity of the action of L ∗ on (N0 ) shows that L t .Cn ⊂ Cn when L t is small enough and n is large enough. (2) (2) For d ∈ D0 we have L t . (N0 )d ⊂ (N0 )(L t .d). The action of L ∗ on D is continuous hence, for any n , we can choose a small L t such that (2) L t .d ⊂ t.d + Cn D0 . We can choose the same L t for each di and we obtain L t .D0 ⊂

(2)

(2)

(N0 )t.di + Cn D0 .

i

Applying ι D , we obtain ι D (L t .D0 ) ⊂ ι D (Dt ) + Cn ι D (D0 ) and then (L t .Cn ) ι D (L t .D0 ) ⊂ Cn ι D (Dt ) + Cn C n ι D (D0 ). We check that C n C n ⊂ Cn,n+n ⊂ Cn when n ≥ n. Hence when n is large enough, L t .(Cn ι D (D0 )) ⊂ Cn ι D (Dt + D0 ). This ends the proof of (7.61). et,c et Proposition 7.124. We have Met,c OE , (L ∗ ) = MOE , (L ∗ ) and M (N0 ) (L ∗ ) = Met

(N0 ) (L ∗ ).

Proof. The first identity was shown in Proposition 7.120 and is equivalent to the second identity by Theorem 7.117 and Proposition 7.123. Corollary 7.125. Any étale L ∗ -module over (N0 ), resp. over OE , is a complete topologically étale o[N0 L ∗ ]-module in our sense. Proof. Use Propositions 7.119 and 7.124.

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9. Convergence in L + -modules on (N0 ) In this section, we use the notation of Section 8 where we assume that N is a padic Lie group. We assume that and ι are continuous group homomorphisms : P → P (2) , ι : N (2) → N , ◦ ι = id, (2) such that (L + ) ⊂ L (2) + , (N ) = N , (ι ◦ )(N0 ) ⊂ N0 , and

tι(y)t −1 = ι((t)y(t)−1 ) for y ∈ N (2) , t ∈ L.

(7.62)

The assumptions of Section 8 are naturally satisfied with L ∗ = L + . Indeed, the compact open subgroup N0 of N is a compact p-adic Lie group, the group (N0 ) is a compact non-trivial subgroup N0(2) of N (2) Q p hence N0(2) is isomorphic to Z p and is open in N (2) , the kernel of | N0 is normalized by L ,+ . (2) (2) Note that L + normalizes ι(N0 ) since (L + ) normalises N0 and (7.62). et et Let M ∈ M (N0 ) (L + ) and D ∈ MOE , (L + ) be related by the equivalence of categories (Theorem 7.117), M = (N0 ) ⊗OE ,ι D = (N0 )ι D (D). We will exhibit in this section a special family Cs of compact subsets in M such that M(Cs ) is a dense o-submodule of M, and such that the P-equivariant sheaf on C associated to the étale o[P+ ]-module M(Cs ) by Theorem 7.38 extends to a G-equivariant sheaf on G/P. We will follow the method explained in Subsection 6.5 which reduces the most technical part to the easier case where M is killed by a power of p.

9.1. Bounded sets Definition 7.126. A subset A of M is called bounded if for any open neighbourhood B of 0 in M there exists an open neighbourhood B of 0 in (N0 ) such that B A ⊂ B. Compare with [12] Def. 8.5. The properties satisfied by bounded subsets of M can be proved directly or deduced from the properties of bounded subsets of

(N0 ) ([15] §12). Using the fundamental system (7.59) of neighbourhoods of 0, the set A is bounded if and only if for any large n there exists n > n such that

(M (N0 )n + X n (N0 )) A ⊂ M (N0 )n M + X n M 0 ,

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equivalently X n −n A ⊂ M (N0 )n M + M 0 . We obtain (compare with [12] Lemma 8.8): Lemma 7.127. A subset A of M is bounded if and only if for any large positive n there exists a positive integer n such that

A ⊂ M (N0 )n M + X −n M 0 . The following properties of bounded subsets will be used in the construction of a special family Cs in the next subsection. – Let f : ⊕ri=1 (N0 ) → M be a surjective homomorphism of (N0 )modules. The image by f of a bounded subset of ⊕ri=1 (N0 ) is a bounded subset of M. For 1 ≤ i ≤ r , the i-th projections Ai ⊂ (N0 ) of a subset A of ⊕ri=1 (N0 ) are all bounded if and only if A is bounded. – A compact subset is bounded. – The (N0 )-module generated by a bounded subset is bounded. – The closure of a bounded subset is bounded. – Given a compact subset C in (N0 ) and a bounded subset A of M, the subset C A of M is bounded. – The image of a bounded subset by f ∈ Endcont o (M) is bounded. The image by M of a bounded subset in M is bounded in D. – A subset A of D is bounded if and only if the image An of A in D/ p n D is bounded for all large n. – When D is killed by a power of p, a subset A of D is bounded if and only if A is contained in a lattice, i.e. if A is contained in a compact subset (by the properties of lattices given in Section 7.3). Lemma 7.128. The image by ι D of a bounded subset in D is bounded in M. Proof. Let A ⊂ D be a bounded subset and let D 0 be a fixed lattice in D. For all n ∈ N there exists n ∈ N such that A ⊂ p n D + (X (2) )−n D 0 by Lemma 7.127. Applying ι D we obtain

ι D (A) ⊂ p n ι D (D) + X −n ι D (D 0 ) ⊂ M (N0 )n M + X −n M 0 where M 0 = (N0 )ι D (D 0 ) is a lattice in M. By the same lemma, this means that ι D (A) is bounded in M.

9.2. The module Msbd Definition 7.129. Msbd is the set of m ∈ M such that the set of M (ψ k (u −1 m)) for k ∈ N, u ∈ N0 is bounded in D.

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The definition of Msbd depends on s because ψ is the canonical left inverse of the action ϕ of s on M. We recognize m u,s k = ψ k (u −1 m) appearing in the expansion (7.12). Proposition 7.130. Msbd is an étale o[P+ ]-submodule of M. Proof. (a) We check first that Msbd is P+ -stable. As Msbd is N0 -stable and P+ = N0 L + , it suffices to show that tm = ϕt (m) ∈ Msbd when t ∈ L + and m ∈ Msbd . Using the expansion (7.12) of m and st = ts, for k ∈ N and n 0 ∈ N0 , we write ψ k (n −1 0 tm) as the sum over u ∈ J (N0 /Nk ) of k k −1 −1 k ψ k (n −1 0 tuϕ (m u,s k )) = ψ (n 0 tut ϕ (ϕt (m u,s k ))) −1 = ψ k (n −1 0 tut )ϕt (m u,s k ),

and M (ψ k (n −1 0 ϕt (m))) as the sum over u ∈ J (N0 /Nk ) of −1 M (ψ k (n −1 0 tut )ϕt (m u,s k )) = vk,n 0 M (ϕt (m u,s k )) = vk,n 0 ϕt ( M (m u,s k )), (2)

−1 bd where vk,n 0 := (ψ k (n −1 0 tut )) belongs to N0 or is 0. As m ∈ Ms , the set of M (m u,s k ) for k ∈ N and u ∈ N0 is bounded in D. Its image (2) by the continuous map ϕt is bounded and generates a bounded o[N0 ]bd submodule of D. Hence ϕt (m) ∈ Ms . (b) The o[P+ ]-module Msbd is ψ-stable (hence Msbd is étale by Corollary 7.45) because we have, for m ∈ Msbd , u ∈ N0 , k ∈ N,

ψ k (u −1 ψ(m)) = ψ k+1 (ϕ(u −1 )m).

(7.63)

The goal of this section is to show that the P-equivariant sheaf on C associated to the étale o[P+ ]-module Msbd extends to a G-equivariant sheaf on G/P. We will follow the method explained in Subsection 6.5. Let pn : M → M/ p n M be the reduction modulo p n for a positive integer n. Recall that M is p-adically complete. Lemma 7.131. The o-submodule Msbd ⊂ M is closed for the p-adic topology, in particular Msbd = lim(Msbd / pn Msbd ). ← − n

Moreover is the set of m ∈ M such that pn (m) belongs to (M/ p n M)bd s for all n ∈ N, and we have Msbd

Msbd = lim(M/ p n M)bd s . ← − n

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Proof. (a) Let m be an element in the closure of Msbd in M for the p-adic topology. For any r ∈ N, we choose m r ∈ Msbd with m − m r ∈ pr M. For each r , we choose r ≥ 1 such that M (ψ k (u −1 m r )) ∈ pr D + X −r D 0 for all k ∈ N, u ∈ N0 , applying Lemma 7.127. We have M (ψ k (u −1 m)) ∈ M (ψ k (u −1 m r ) + pr M)

= M (ψ k (u −1 m r )) + pr D ⊂ pr D + X −r D 0 . By the same lemma, m ∈ Msbd . This proves that Msbd is closed in M hence p-adically complete. (b) The reduction modulo p n commutes with M , ψ, and the action of N0 . The following properties are equivalent: – m ∈ Msbd , – { M (ψ k (u −1 m)) for k ∈ N, u ∈ N0 } ⊂ D is bounded, – { M/ pn M (ψ k (u −1 pn (m))) for k ∈ N, u ∈ N0 } ⊂ D/ p n D is bounded for all positive integers n, a. pn (m) ∈ (M/ p n M)bd s for all positive integers n. We deduce that m → ( pn (m))n : Msbd → limn (M/ p n M)bd s is an ← − isomorphism. Proposition 7.132. D = Dsbd and Msbd contains ι D (D). Proof. (i) We show that D = Dsbd . By Lemma 7.131, we can suppose that D is killed by a power of p. Let d ∈ D. By Corollary 7.95, for n ∈ N, there (2) exists k0 ∈ N such that ψ k (v −1 d) ∈ D ' for k ≥ k0 , v ∈ N0 . As D ' ⊂ D (2) is bounded, and as the set of ψ k (v −1 d) for all 0 ≤ k < k0 , v ∈ N0 , is (2) also bounded because the set of v −1 d for v ∈ N0 is bounded and ψ k is bd continuous, we deduce that d ∈ Ds . (ii) We show that Msbd contains ι D (D) by showing { M (ψ k (u −1 ι D (d))) for k ∈ N, u ∈ N0 } (2)

= {ψ k (v −1 d) for k ∈ N, v ∈ N0 } when d ∈ D (the right-hand side is bounded in D by (i)). We write an (2) element of N0 as ι(v)u for u in N and v ∈ N0 . By Lemma 7.112, ψ k (u −1 ι(v)−1 ι D (d)) = ψ k (u −1 ι D (v −1 d)) = s −k u −1 s k ψ k (ι D (v −1 d)) when u ∈ s k N s −k and is 0 when u is not in s k N s −k . When u ∈ s k N s −k we have M (s −k u −1 s k ψ k (ι D (v −1 d))) = ψ k (v −1 d) as ι D is ψ-equivariant.

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Proposition 7.133. Msbd is dense in M. Proof. Msbd ⊂ M is an o[N0 ]-submodule, which by Proposition 7.132 contains ι D (D). The o[N0 ]-submodule of M generated by ι D (D) is dense by Lemma 7.103. We summarize: we proved that Msbd ⊂ M is a dense o[N0 ]-submodule, stable by L + , and the action of L + on Msbd is étale. Remark 7.134. It follows from Lemma 7.131 and the subsequent Proposition 7.135 that Msbd is a (N0 )-submodule of M.

9.3. The special family Cs when M is killed by a power of p We suppose that M is killed by a power of p. Proposition 7.135. 1. For any lattice D0 in D, the o-submodule Msbd (D0 ) := {m ∈ M | M (ψ k (u −1 m)) ∈ D0 for all u ∈ N0 and k ∈ N} of M is compact, and is a ψ-stable (N0 )-submodule. 2. The family Cs of compact subsets of M contained in Msbd (D0 ) for some lattice D0 of D, is special (Definition 7.76), satisfies C(5) (Proposition 7.83) and C(6) (Proposition 7.84), and M(Cs ) = Msbd is a

(N0 )-submodule of M. Proof. 1. (a) As and ψ are continuous (Proposition 7.119) and D0 ⊂ D is closed, it follows that Msbd (D0 ) is an intersection of closed subsets in M, hence Msbd (D0 ) is closed in M. As Msbd (D0 ) is an o[N0 ]submodule of M and o[N0 ] is dense in (N0 ) we deduce that Msbd (D0 ) is a (N0 )-submodule. It is ψ-stable by (7.63). The weak topology on M is the projective limit of the weak topologies on M/M (N0 )n M, and we have ([2] I.29 Corollary) Msbd (D0 ) = lim(Msbd (D0 ) + M (N0 )n M)/M (N0 )n M. ← − n≥1

Therefore it suffices to show that (Msbd (D0 ) + M (N0 )n M)/M (N0 )n M is compact for each large n. We will show the stronger property that it is a finitely generated (N0 )-module. (b) We prove first that Msbd (D0 ) is the intersection of the (N0 )-modules generated by the image by ϕ k of the inverse image −1 M (D0 ) of D0 in M, for k ∈ N,

G-equivariant sheaves on G/P Msbd (D0 ) =

(N0 )ϕ k (−1 M (D0 )).

351

(7.64)

k∈N

The inclusion from left to right follows from the expansion (7.12), as m ∈ Msbd (D0 ) is equivalent to m u,s k = ψ k (u −1 m) ∈ −1 M (D0 ) for all u ∈ N0 and k ∈ N. The inclusion from right to left follows from M ψ k u −1 ( (N0 )ϕ k (−1 M (D0 ))) = D0 . (c) We pick a lattice M0 of M such that −1 M (D0 ) = M0 + J (N0 )M, as J (N0 )M is the kernel of M . By Lemma 7.114 we can choose for each n ∈ N a large integer r such that ϕ r (J (N0 )M) ⊆ M (N0 )n M. Therefore we have Msbd (D0 ) ⊆ (N0 )ϕ r (M0 + J (N0 )M) ⊆ (N0 )ϕ r (M0 ) + M (N0 )n M. We deduce (Msbd (D0 ) + M (N0 )n M)/M (N0 )n M ⊆ ( (N0 )ϕr (M0 ) + M (N0 )n M)/M (N0 )n M. The right term is a finitely generated (N0 )-module hence the left term is finitely generated as a (N0 )-module since (N0 ) is noetherian. 2. The family is stable by finite union because a finite sum of lattices is a lattice. If C ∈ Cs then N0 C ∈ Cs because Msbd (D0 ) is a (N0 )-module. We have M(Cs ) = ∪ D0 Msbd (D0 ) = Msbd , when D0 runs over the lattices of D, the last follows from the fact that a bounded subset of D is contained in a lattice (this is the only part in the proof where the assumption that M is killed by a power of p is used). Apply Proposition 7.130. Property C(5) is immediate because Msbd (D0 ) is ψ-stable. Property C(6) follows from ϕ(Msbd (D0 )) ⊂ Msbd (Ds ) where Ds is the lattice of D generated by ϕ(D0 ) (this uses part (a) of the proof of Proposition 7.130). Consider the lattice M ++ = (N0 )i D (D ++ ) of M. Since (N0 ) and D ++ are ϕ-stable and since ϕ and ι D commute, M ++ is ϕ-stable and M (M ++ ) = D ++ . Hence for a subset S ⊂ M we have S ⊂ X r M ++ + J (N0 )M ⇔ M (S) ⊂ (X (2) )r D ++ .

(7.65)

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Proposition 7.136. Let r ∈ N, C+ ⊂ L + a compact subset and let D0 be a lattice in D. There is a compact open subgroup P1 ⊂ P0 and k0 ≥ 0 such that for all k ≥ k0 s k (1 − P1 )C+ Msbd (D0 ) ⊂ X r M ++ + M (N0 )r M. Proof. Denote for simplicity S = Msbd (D0 ). By definition M (S) ⊂ D0 . Since (2) P+ acts continuously on D, (C+ )D0 is compact and (X (2) )r D ++ is open, (2) (2) (2) there is a compact open subgroup P1 ⊂ P+ such that (1 − P1 )(C+ )D0 ⊂ (X (2) )r D ++ . We may choose a compact open subgroup P1 of P0 such that (2) (P1 ) ⊂ P1 , hence M ((1 − P1 )C+ S) ⊂ (X (2) )r D ++ . Relation (7.65) yields (1 − P1 )C+ S ⊂ J (N0 )M + X r M ++ . Choosing k0 such that ϕ k (J (N0 )) ⊂ M (N0 )r for k ≥ k0 (as we may by Lemma 7.114), the result follows from the ϕ-stability of X r M ++ (which follows from the fact that ϕ(X r ) ∈ X r (N0 ) and ϕ(M ++ ) ⊂ M ++ ). Corollary 7.137. Property T(1) in Proposition 7.83 is satisfied. Proof. Let C, C+ , M as in Proposition 7.83 and choose r such that M (N0 )r M + X r M ++ ⊂ M. Choose a lattice D0 such that C ⊂ Msbd (D0 ). As Msbd (D0 ) is ψ-stable (Proposition 7.135), we can choose the subgroup P1 and k(C, M, C+ ) = k0 given by Proposition 7.136. bd Recall that we defined (7.42) operators sg(k) = H(k+1) − H(k) g g on Ms = M(Cs ). From now on we fix a lattice D0 in D and g ∈ N0 P N0 . We denote S = Msbd (D0 ).

Corollary 7.138. There is k g ≥ 0 such that for all x ∈ N and k ≥ x + k g ++ ψ x ◦ N0 ◦ sg(k) (S) ⊂ −1 ). M (D

Proof. Let r = 1, C+ = g s and choose P1 and k0 as in Proposition 7.136, so ++ ) for k ≥ k . Since S is (N )[ψ]-stable and that s k (1 − P1 )C+ S ⊂ −1 0 0 M (D ++ ) is o[N ]-stable, relation (7.43) and the inequality k (2) (P ) ≥ k (1) −1 (D 0 1 g g M ++ ) for k ≥ x + k + k (2) (P ). Applying ψ x ◦ N yield sg(k) (S) ⊂ N0 ϕ x −1 (D 0 1 0 g M (2) yields the desired result, with k g = k0 + k g (P1 ). Lemma 7.139. There is a lattice D1 in D such that for all u ∈ Ug , k ≥ k g ≥ (1) k g and x ≥ k − k g ψ x ◦ N0 ◦ α(g, xu ) ◦ Res(1u Nk )(S) ⊂ −1 M (D1 ).

(7.66)

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= s k g t (g, U ). This is a compact subset of L , since k ≥ k . Proof. Let C+ g + g g Since S is (N0 )[ψ]-stable, we have Res(1u Nk )(S) ⊂ u ◦ ϕ k (S), hence α(g, xu ) ◦ Res(1u Nk )(S) ⊂ N0 ◦ t (g, u) ◦ ϕ k (S) ⊂ N0 ϕ k−k g (C+ S). S), Hence the left-hand-side of (7.66) is contained in ψ x−k+kg ( (N0 )C+ −1 −1 (D ), because S ⊂ (N )ϕ x−k+k g ( (D )) which is a subset of (N0 )C+ 0 0 0 M M and C+ (N0 ) ⊂ (N0 )C+ . Thus (2)

M (ψ x ◦ N0 ◦ α(g, xu ) ◦ Res(1u Nk )(S)) ⊂ (N0 )(C+ )(D0 )

and the last subset of D is compact, hence contained in some lattice D1 . bd ++ ). Corollary 7.140. For all k ≥ k g we have H(k) g (S) ⊂ Ms (D1 + D bd ++ Moreover, Hg (S) ⊂ Ms (D1 + D ).

Proof. The second assertion follows from the first by letting k → ∞, since Msbd (D1 + D ++ ) is closed in M. For the first assertion, we need to prove that (k) ++ ) for all x ≥ 0 and k ≥ k . Fix x ≥ 0. If ψ x (N0 Hg (S)) ⊂ −1 g M (D1 + D k ≤ k g + x, simply add all relations (7.66) for u ∈ J (Ug /Nk ). If k > k g + x, (x+k ) ( j) (k) the equation Hg = Hg g + k−1 j =x+k g sg and Corollary 7.138 show that (x+k g )

x ψ x (N0 H(k) g (S)) ⊂ ψ (N0 Hg

(x+k g )

But we have already seen that ψ x (N0 Hg

++ (S)) + −1 ). M (D

++ ). (S)) ⊂ −1 M (D1 + D

Proposition 7.141. All the assumptions of Proposition 7.84 are satisfied. Proof. Property T(1) was checked in Corollary 7.137. Property T(2) and the fact that Hg preserves Msbd (for g ∈ N0 P N0 ) follow from Corollary 7.140 and the fact that any m ∈ Msbd is in S = Msbd (D0 ) for some lattice D0 in D.

9.4. Functoriality and dependence on s Let Z (L)†† ⊂ Z (L) be the subset of elements s such that L = L − s N and (s k N0 s −k )k∈Z and (s −k w0 N0 w0−1 s k )k∈Z are decreasing sequences of trivial intersection and union N and w0 N w0−1 , respectively (see Section 6). Let M be a topologically etale L + -module over (N0 ) and let D := D(M). We have D/ p n D = D(M/ p n M) for n ≥ 1. By Lemma 7.104, M satisfies the properties a,b,c,d of Subsection 6.5 and is complete (the same is true for M/ p n M). The image D0,n in D/ p n D of any lattice D0,n+1 in D/ pn+1 D is a lattice and the maps and ψ commute with the reduction modulo p n , hence n bd (M/ p n+1 M)bd s (D0,n+1 ) maps into (M/ p M)s (D0,n ). Therefore the special n+1 family Cs,n+1 in M/ p M maps to the special family Cs,n in M/ p n M. As

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in Lemma 7.89 we define the special family Cs in M to consist of all compact subsets C ⊂ M such that pn (C) ∈ Cs,n for all n ≥ 1. By Proposition 7.135 and Lemma 7.131 we have M(Cs ) = Msbd . Theorem 7.142. Let s ∈ Z (L)†† and M ∈ Met

(N0 ) (L + ). (i) The (s, res, Cs )-integrals Hg,s of the functions αg,0 | Msbd for g ∈ N0 P N0 exist, lie in Endo (Msbd ), and satisfy the relations H1, H2, H3 of Proposition 7.72. (ii) The map M → (Msbd , (Hg,s )g∈N0 P N0 ) is functorial. Proof. (i) By Proposition 7.141 the assumptions of Proposition 7.91 are satisfied. (ii) Let f : M → M be a morphism in Met

(N0 ) (L + ). For m ∈ M we denote E s (m) = { M (ψsk u −1 m) for u ∈ N0 , k ∈ N}. We have D( f )(E s (m)) = E s ( f (m)) when m ∈ M,

(7.67)

because the maps M : M → D and M : M → D sending x to 1⊗x for x ∈ M or x ∈ M satisfy M ◦ f = D( f )◦ M , and f is P − -equivariant by Lemma 7.22. Any morphism between finitely generated modules on OE is continuous for the weak topology (cf. [12] Lemma 8.22). The image of a bounded subset by a continuous map is bounded. We deduce from (7.67) that E s (m) bounded implies E s ( f (m)) bounded, equivalently m ∈ Msbd implies f (m) ∈ Ms bd . For m ∈ Msbd we have f (Hg,s (m)) = Hg,s ( f (m)) where Hg,s (.) = lim n(g, u)ϕt (g,u)s k ψsk u −1 (.), k→∞

u∈J (N0 /s k N0 s −k )

because f is P+ and P− -equivariant by Lemma 7.22. We investigate now the dependence on s ∈ Z (L)†† of the dense subset Msbd ⊆ M and of the (s, res, Cs )-integrals Hg,s . Lemma 7.143. Z (L)†† is stable by product. Proof. Let s, s ∈ Z(L)†† . Clearly L − s n = L − s −n s n s n ⊂ L − (ss )n because L − is a monoid and s −1 ∈ Z(L)− = Z (L) ∩ L − . Therefore L = L − (ss )N . The sequence ((ss )k N0 (ss )−k )k∈Z is decreasing because s k+1 s k+1 N0 s −k−1 s −k−1 ⊂ s k s k+1 N0 s −k−1 s −k ⊂ s k s k N0 s −k s −k .

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The intersection is trivial and the union is N because s k s k N0 s −k s −k ⊂ s k N0 s −k when k ∈ N and s k s k N0 s −k s −k ⊃ s k N0 s −k when −k ∈ N. One makes the same argument with w0 N0 w0−1 . (2)

Lemma 7.144. (i) The action of t0 ∈ −1 (L 0 ) ∩ L + on D is invertible. (ii) There exists a treillis D0 in D which is stable by −1 (L (2) 0 ) ∩ L +. Proof. (i) is true because the action of t0 on D is étale and N0(2) = (2) (t0 )N0 (t0 )−1 . (ii) Let s ∈ Z (L)†† and let ψs be the canonical inverse of the étale action ϕs of s on D. We show that the minimal ψs -stable treillis D & of D (2) (Proposition 7.93(iii)) is stable by −1 (L 0 ) ∩ L + . (2) For t0 ∈ −1 (L 0 ) ∩ L + we claim that ϕt0 (D & ) is also a ψs -stable treillis in D. We have ψs ψt0 = ψt0 ψs as t0 ∈ Z (L). Multiplying by ϕt0 on both sides, one gets ϕt0 ψs ψt0 ϕt0 = ϕt0 ψt0 ψs ϕt0 . Since ψt0 is the two-sided inverse of ϕt0 by (i) we get that ϕt0 and ψs commute. Hence ϕt0 (D & ) is (2) a compact o-module which is ψs -stable. It is a (N0 )-module because (2) (2) any λ ∈ (N0 ) is of the form λ = ϕ(t0 ) (μ) for some μ ∈ (N0 ) and & λϕt0 (d) = ϕt0 (μd) for all d ∈ D. As D contains a lattice and ϕt0 is étale, we deduce that ϕt0 (D & ) contains a lattice and therefore is a treillis. By the minimality of D & we must have D & ⊂ ϕt0 (D & ). Similarly one checks that ψt0 (D & ) is a treillis. It is ψs -stable because ψs and ψt0 commute. Hence D & ⊂ ψt0 (D & ). Applying ϕt0 which is the two-sided inverse of ψt0 we obtain ϕt0 (D & ) ⊂ D & hence D & = ϕt0 (D & ). We denote by Z(L)† ⊂ Z(L) the monoid of z ∈ Z (L)+ = Z (L) ∩ L + such that z −1 w0 N0 w0−1 z ⊂ w0 N0 w0−1 . We have Z (L)†† Z (L)† ⊂ Z (L)†† . (2) (2) Note that L (2) 0 contains the center of G L(2, Q p ) and that Z(L )† = L + . For m ∈ M, t ∈ L + , u ∈ U, and a system of representatives J (N0 /t N0 t −1 ) ⊂ N0 for the cosets in N0 /t N0 t −1 we have (7.12) m= uμt,u , μt,u := ϕt ψt (u −1 m). (7.68) u∈J (N0 /t N0 t −1 )

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For g ∈ N0 P N0 and s ∈ Z (L)†† , we have the smallest positive integer k g,s as (0) in (7.27). For k ≥ k g,s , we have Hg,s,J (N0 /Nk ) ∈ Endcont o (M) where (compare with (7.28)) Hg,s,J (N0 /Nk ) (m) = n(g, u)t (g, u)μs k ,u . (7.69) u∈J (U g /Nk )

When m ∈ Msbd , the integral Hg,s (m) is the limit of Hg,s,J (N0 /Nk ) (m) by Theorem 7.142 and (7.29). Proposition 7.145. Let s ∈ Z (L)†† , t0 ∈ −1 (L (2) 0 ) ∩ Z (L)† and r a positive integer. (i) We have Mstbd0 ⊆ Msbd = Msbd r . (ii) For g ∈ N0 P N0 we have Hg,s = Hg,st0 on Mstbd0 and Hg,s = Hg,sr on Msbd . Proof. (a) Note that st0 and s r in proposition belong also to Z(L)†† . (2) For a treillis D0 in D which is stable by −1 (L 0 ) ∩ L + (Lemma 7.144), (2) (X (2) )−r D0 is a treillis in D; it is also stable by t0 ∈ −1 (L 0 ) ∩ L + because (2)

(2)

ϕ(t0 ) ((X (2) )−r (N0 )) = ϕ(t0 ) ((X (2) )−r )ϕ(t0 ) ( (N0 )) (2)

= (X (2) )−r (N0 ). When M is killed by a power of p, this implies with Proposition 7.135 that Msbd is the union of Msbd (D0 ) when D0 runs over the lattices of D which (2) are stable by −1 (L 0 ) ∩ L + . (b) We suppose from now on, as we can by Lemma 7.131, that M is killed by bd a power of p to prove Mstbd0 ⊂ Msbd = Msbd r . Let m ∈ Mst (D0 ) where D0 0 (2)

is a −1 (L 0 )∩ L + -stable lattice of D. For u ∈ N0 and k ∈ N, using (7.12) for t = t0k we obtain that M (ψsk (u −1 m)) = M ( v ◦ ϕtk0 ◦ ψtk0 ◦ v −1 ◦ ψsk (u −1 m)) v∈J (N0 /t0k N0 t0−k )

=

(v)ϕtk0 ( M (ψstk 0 (ϕsk (v −1 )u −1 m)))

v∈J (N0 /t0k N0 t0−k )

lies in D0 , since D0 is both N0(2) - and ϕt0 -invariant and M (ψstk 0 (u m)) ∈ D0 for u ∈ N0 . Therefore Mstbd0 (D0 ) ⊂ Msbd (D0 ) and by (a) we deduce Mstbd0 ⊂ Msbd .

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For any m ∈ M we observe that { M (ψskr (u −1 m)) for k ∈ N, u ∈ N0 } ⊂ { M (ψsk (u −1 m)) for k ∈ N, u ∈ N0 }, as ψskr = ψsr k . We deduce that Msbd (D0 ) ⊂ Msbd r (D0 ) for any lattice D0 of D hence Msbd ⊂ Msbd . Conversely, for k ∈ N we write k1 = r k − k2 with r 1 k ∈ N and 0 ≤ k2 < r and we observe that M (ψsk1 (u −1 m)) = M ( v ◦ ϕsk2 ◦ ψsr k (ϕsk1 (v −1 )u −1 m)) v∈J (N0 /s k2 N0 s −k2 )

=

(v)ϕsk2 ( M (ψskr (ϕsk1 (v −1 )u −1 m))).

v∈J (N0 /s k2 N0 s −k2 )

r−1 i (2) The (N0 )-submodule Dr generated by i=1 ϕs (D0 ) is a lattice because the action ϕs of s on D is étale. We deduce that Msbd r (D0 ) ⊂ Msbd (Dr ) since M (ψskr (u m)) ∈ D0 for u ∈ N0 , m ∈ Msbd r (D0 ). bd bd bd Therefore Msbd r (D0 ) ⊂ Ms (Dr ) hence Ms r ⊂ Ms . It is obvious that Hg,s = Hg,s r on Msbd . (0) (2) (c) Let g ∈ N0 P N0 , k ≥ k g,s , t0 ∈ −1 (L 0 ) ∩ Z (L)† and r ≥ 1. We have (0)

(0) k g,st0 ≤ k g,s ,

(0)

(0) k g,s r ≤ k g,s

because (st0 )k N0 (st0 )−k ⊂ Nk and (s r )k N0 (s r )−k = Nkr ⊂ Nk . Let d in D and v ∈ N0 . By (7.12) we have k d= uϕst ◦ ψstk 0 (u −1 d) 0 (2)

(2)

u∈J (N0 /(st0 )k N0 (st0 )−k )

= (2)

uϕsk ◦ ψsk (u −1 d), (2)

u∈J (N0 /(s)k N0 (s)−k )

with the second equality holding true summand per summand, because ψt0 is the left and right inverse of ϕt0 on D (Lemma 7.144 (i)) and (2) (2) (t0 )N0 (t0 )−1 = N0 . Since ι D commutes with ϕt and ψt for t ∈ L + , this implies k vι D (d) = vι(u)ϕst ◦ ψstk 0 (ι(u)−1 ι D (d)) 0 (2)

(2)

u∈J (N0 /(st0 )k N0 (st0 )−k )

=

vι(u)ϕsk ◦ ψsk (ι(u)−1 ι D (d)),

(2) (2) u∈J (N0 /(s)k N0 (s)−k )

again with the second equality holding true summand per summand. We choose, as we can, systems of representatives J (N0 /(st0 )k N0 (st0 )−k ) and

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(2)

(0)

J (N0 /s k N0 s −k ) containing ι( J (N0 /(s)k N0 (s)−k )). For k ≥ k g,s ≥ (0) k g,st0 , we obtain Hg,st0 ,v J (N0 /(st0 )k N0 (st0 )−k ) (vι D (d)) = Hg,s,v J (N0 /s k N0 s −k ) (vι D (d)). Passing to the limit when k goes to infinity, and using linearity we deduce that Hg,st0 = Hg,s on the o[N0 ]-submodule < N0 ι D (D) >o generated by ι D (D) in Mstbd0 . (d) Let m ∈ Msbd (D1 ) with D1 ⊂ D a ψs -stable lattice (Proposition 7.93 (iv)). For a positive integer k, and a set of representatives J (N0 /s k N0 s −k ), we write m in the form (7.12) m= uϕsk (ι D (d(s, u)) + m(s, u)) u∈J (N0 /s k N0 s −k )

with m(s, u) in J (N0 )M and d(s, u) = M (ψsk (u −1 m)) in D1 . Then m(s) := uϕsk (ι D (d(s, u))) lies in < N0 ι D (D) >o u∈J (N0 /s k N0 s −k )

because ι D is L + -equivariant. Moreover m − m(s) is contained in the o[N0 ]-submodule N0 ϕsk (J (N0 )M) generated by ϕsk (J (N0 )M). We show that m(s) ∈ Msbd (D1 ). For v ∈ N0 and r ≤ k we have

ψsr (v −1 (m − m(s))) = ψsr (v −1

(7.70)

uϕsk (m(s, u)))

u∈J (N0 /s k N0 s −k )

=

ψsr (v −1 u)ϕsk−r (m(s, u))

u∈J (N0 /s k N0 s −k )

which lies in J (N0 )M since m(s, u) is in J (N0 )M and J (N0 )M is N0 and ϕs -stable. This shows that M (ψsr (v −1 m(s))) = M (ψsr (v −1 m)) lies in D1 . On the other hand, for r > k we have M (ψsr (v −1 m(s))) = M (ψsr (v −1 uϕsk (ι D (d(s, u))))) =

u∈J (N0 /s k N0 s −k )

M (ψsr−k (ψsk (v −1 u)ι D (d(s, u))))

u∈J (N0 /s k N0 s −k )

which lies in D1 . Indeed, since D1 is ψs -stable the formula in part (ii) of the proof of Proposition 7.132 implies that ι D (D1 ) ⊆ Msbd (D1 ); hence the ι D (d(s, u)) lie in the ψs - and N0 -invariant subspace Msbd (D1 ). We conclude that m(s) ∈ Msbd (D1 ).

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Therefore, for any ψst0 -stable lattice D1 ⊂ D, any k ≥ 1, and any set of representatives J (N0 /(st0 )k N0 (st0 )−k ), we have defined an o-linear homomorphism m → m(st0 )

Mstbd0 (D1 ) → Mstbd0 (D1 ) ∩ < N0 ι D (D) >o

such that k m − m(st0 ) ∈ Mstbd0 (D1 ) ∩ ϕst (J (N0 )M). 0

(7.71)

By (c) we have Hg,st0 (m(st0 )) = Hg,s (m(st0 )) for m ∈ Mstbd0 (D1 ). (e) To end the proof that Hg,st0 = Hg,s on Mstbd0 (D1 ) we use the Cs -uniform convergence of (Hg,s,J (N0 /s k N s −k ) )k . We fix systems of representatives J (N0 /(st0 )k N0 (st0 )−k ) and J (N0 /s k N s −k ), for any k ≥ 1. We also (2) choose a lattice D0 ⊂ D which is stable by −1 (L 0 ) ∩ L + and such that D1 ⊂ D0 . We recall that Mstbd0 (D1 ) is compact (Proposition 7.135 i)) and that Mstbd0 (D1 ) ⊂ Mstbd0 (D0 ) ⊂ Msbd (D0 ) by (b). For any open (N0 )submodule in the weak topology M0 ⊂ M, there exists a common constant (0) (0) k0 ≥ k g,s ≥ k g,st (by (c)) such that for k ≥ k0 , 0 Hg,st0 , J (N0 /(st0 )k N (st0 )−k ) ∈ Hg,st0 + E(Mstbd0 (D1 ), M0 ) Hg,s,J (N0 /s k N s −k ) ∈ Hg,s +

E(Mstbd0 (D1 ),

M0 ).

(7.72) (7.73)

On the left-hand side of (7.72), (7.73), we have continuous endomorphisms of M. By Lemma 7.114, there exists an integer k1 ≥ k0 such that they send k N0 ϕst10 (J (N0 )M) into M0 . Therefore, for m ∈ Mstbd0 (D1 ), they send the element m − m(st0 ) associated to k1 and J (N0 /((st0 )k1 N0 (st0 )−k1 ) as in (d) (7.71) into M0 hence Hg,st0 (m − m(st0 )) and Hg,s (m − m(st0 )) lie in M0 . By (d) we obtain that Hg,st0 (m) − Hg,s2 (m) lies in M0 for m ∈ Mstbd0 (D1 ). The statement follows since we chose M0 to be an arbitrary open neighbourhood of zero in the weak topology of M. Definition 7.146. We define the transitive relation s1 ≤ s2 on Z(L)†† generated by (2)

s1 = s2 t0 for t0 ∈ −1 (L 0 ) ∩ Z (L)† or s1r1 = s2r2 for positive integers r1 , r2 . Proposition 7.145 admits the following corollary. Corollary 7.147. Let s1 , s2 ∈ Z (L)†† . (i) When s1 ≤ s2 we have Msbd ⊆ Msbd and Hg,s1 = Hg,s2 on Msbd . 1 2 1

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(ii) When the relation ≤ on Z (L)†† is right filtered, we have Hg,s1 = Hg,s2 on Msbd ∩ Msbd . 1 2 Proof. (i) If s1 ≤ s2 then there exists, by definition, a sequence s1 = s1 ≤ s2 ≤ . . . ≤ sm = s2 in Z (L)†† such that each pair si , si+1 satisfies one of the two conditions in Definition 7.146. Hence we may assume, by induction, that the pair s1 , s2 satisfies one of these conditions, and we apply Proposition 7.145. (ii) When there exists s3 ∈ Z (L)†† such that s1 ≤ s3 and s2 ≤ s3 , by (i) Msbd 1 bd and H bd ∩ M bd . and Msbd are contained in M = H = H on M g,s g,s g,s s s s 1 2 3 2 3 1 2 Proposition 7.148. We assume that the relation ≤ on Z(L)†† is right filtered. Then, the intersection and the union

M∩bd := Msbd ⊂ M∪bd := Msbd s∈Z (L)††

s∈Z (L)††

are dense étale L + -submodules of M over (N0 ). For g ∈ N0 P N0 the endomorphisms Hg ∈ Endo (M∪bd ) equal to Hg,s on bd Ms for each s ∈ Z(L)†† , are well defined, stabilize M∩bd and satisfy the relations H1, H2, H3 of Proposition 7.72. Proof. M∩bd is an L + -submodule of M over (N0 ) by Proposition 7.130 and Remark 7.134. It is dense in M by Proposition 7.132 and Lemma 7.103. The action of L + on M∩bd is étale because M∩bd is L − -stable. When ≤ is right filtered, M∪bd is a (N0 )-module by Corollary 7.147(i). For the same reasons as for M∩bd , it is an étale L + -submodule of M over (N0 ). By Corollary 7.147 the Hg are well defined and stabilize M∩bd . They satisfy the relations H1, H2, H3 of Proposition 7.72 because the Hg,s satisfy them (Theorem 7.142). We summarize our results and give our main theorem. Theorem 7.149. For any s ∈ Z(L)†† , we have a faithful functor Ys : Met

(N0 ) (L + )

→

G-equivariant sheaves on G/P,

which associates to M ∈ Met

(N0 ) (L + ) the G-equivariant sheaf Ys on G/P such that Ys (C0 ) = Msbd . When the relation ≤ on Z (L)†† is right filtered, we have faithful functors Y∩ , Y∪ : Met

(N0 ) (L + )

→

G-equivariant sheaves on G/P,

which associate to M ∈ Met

(N0 ) (L + ) the G-equivariant sheaves Y∩ and Y∪ on G/P with sections on C0 equal to Y∩ (C0 ) = M∩bd and Y∪ (C0 ) = M∪bd .

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361

Proof. The existence of the functors results from Proposition 7.148, Theorem 7.142, Proposition 7.72, and Remark 7.69. We show the faithfulness of the functors. For a non zero morphism f : M → bd M in Met

(N0 ) (L + ), we have f (M∩ ) = 0 because f is continuous ([12] Lemma 8.22) and M∩bd containing (N0 )ι D (D) is dense (Proposition 7.148). We deduce Y∩ ( f ) = 0 since it is nonzero on sections on C0 . A fortiori Ys ( f ) = 0, and Y∪ ( f ) = 0.

10. Connected reductive split group We explain how our results apply to connected reductive groups. (a) Let F be a locally compact non archimedean field of ring of integers o F and uniformizer p F . Let G be a connected reductive F-group, let S be a maximal F-split subtorus of G and let P be a parabolic F-subgroup of G with Levi component L containing S and unipotent radical N . Let X ∗ (S) be the group of characters of S, let L , resp. , be the subset of roots of S in L, resp. G, and let +,N be the subset of roots of S in N (we suppress the index N if P is a minimal parabolic F-subgroup of G). Let s be any element of S(F) such that α(s) = 1 for α ∈ L and the p-valuation of α(s) ∈ F ∗ is positive for all roots α ∈ +,N . For any compact open subgroup N0 of N (F), the data (P(F), L(F), N (F), N0 , s) satisfy all the conditions introduced in the section on étale P+ -modules, Section 3 and Subsection 3.2, the assumptions introduced in Section 6, and in Section 9. (b) We suppose that P is a minimal parabolic F-subgroup. Let W ⊂ NG (L) be a system of representatives of the Weyl group N G (L)/L and let w0 be the longest element of the Weyl group. The data (G(F), P(F), W ) satisfy the assumptions of Section 5 on G-equivariant sheaves on G/P. (c) We suppose until the end of this chapter that F = Q p , G is Q p -split and P is a Borel Q p -subgroup. The Levi subgroup L = T of P is a split Q p -torus. The monoid of dominant elements and the submonoid without unit of strictly dominant elements are T (Q p )+ = {t ∈ T (Q p ), α(t) ∈ Z p for all α ∈ }, T (Q p )++ = {t ∈ T (Q p ), α(t) ∈ pZ p − {0} for all α ∈ }.

362

Peter Schneider, Marie-France Vigneras, and Gergely Zabradi With our former notation Z(L) = T (Q p ), Z (L)†† = T (Q p )++ . For each root α ∈ , let u α : Q p → Nα (Q p ), tu α (x)t −1 = u α (α(t)x) for x ∈ Q p , t ∈ T (Q p ), (7.74) be a continuous isomorphism from Q p onto the root subgroup Nα (Q p ) of N (Q p ) normalized by T (Q p ). We can write an element u ∈ N (Q p ) in the form u= u α (x α ) α∈+

for any ordering of + . The coordinates x α = xα (u) ∈ Q p of u are determined by the ordering of the roots, but for a simple root α, the coordinate xα : N (Q p ) → Q p

(7.75)

is independent of the choice of the ordering, and satisfies u α ◦ xα = 1. We suppose, as we can, that the u α have been chosen such that the product N0 = u α (Z p ) α∈+

is a group for some ordering of + . Then N0 is the product of the u α (Z p ) = Nα (Z p ) for any ordering of + . We choose a simple root α. We consider the continuous homomorphisms α : P(Q p ) → P (2) (Q p ), ια : N (Q p )(2) → N (Q p ), α ◦ ια = 1, defined by α(t) α (ut) := 0

xα (u) 1 (2) (2) , ια (u (x)) := u α (x) for u (x) := 1 0

x , 1

for t ∈ T (Q p ), u ∈ N (Q p ), x ∈ Q p . They satisfy the functional equation tια (y)t −1 = ια (α (t)yα (t)−1 ) for y ∈ N (Q p )(2) and t ∈ T (Q p ). The data (N0 , α , ια ) satisfies the assumptions introduced in Sections 8 and 9. We consider the binary relation s1 ≤ s2 on T (Q p )++ generated by s1 = s2 s0 with s0 ∈ T (Q p )+ , α(s0 ) ∈ Z∗p , or s1n = s2m with n, m ≥ 1. Lemma 7.150. The relation s1 ≤ s2 on T (Q p )++ is right filtered.

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Proof. Let = {α = α1 , . . . , αn }. The image of T (Q p )++ by A = (val p (αi (.))αi ∈ is contained in (N − {0})n and s1 ≤ s2 depends only on the cosets s1 T (Q p )0 and s1 T (Q p )0 , where T (Q p )0 = {t ∈ T (Q p ), α(t) ∈ Z∗p for all α ∈ }. (a) First we assume that, for any positive integer k, there exists s[k] ∈ T (Q p ) such that A(s[k] ) = (k, 1, . . . , 1). Then we have s[k] ≤ s[k+1] , and s ≤ s[k(s)] for s ∈ T (Q p )++ with k(s) = val p (α(s)). For any s1 , s2 in T (Q p )++ we deduce that s1 ≤ s[k(s1 )+k(s2 )] and s2 ≤ s[k(s1 )+k(s2 )] . Hence the relation ≤ on T (Q p )++ is right filtered. (b) When G is semi-simple and adjoint the dominant coweights ωα1 , . . . , ωαn for = {α = α1 , . . . , αn } form a basis of Y = Hom(Gm , T ), and A(T (Q p )++ ) = (N − {0})n . Hence s[k] exists for any k ≥ 1. (c) When G is semi-simple we consider the isogeny π : G → G ad from G onto the adjoint group G ad ([13] 16.3.5). The image Tad of T is a maximal split Q p -torus in G ad . The isogeny gives a homomorphism T (Q p ) → Tad (Q p ), inducing an injective map between the cosets T (Q p )++ /T (Q p )0 → Tad (Q p )++ /Tad (Q p )0 respecting ≤, and such that for any tad ∈ Tad (Q p ) there exists an inten ger n ≥ 1 such that tad ∈ π(T (Q p )). Given s1 , s2 ∈ T (Q p )++ there exists sad ∈ Tad (Q p )++ such that π(s1 ), π(s2 ) ≤ sad by (b) and (a). Let n n n ≥ 1 such that sad = π(s3 ) for s3 ∈ T (Q p ). We have sad ≤ sad hence π(s1 ), π(s2 ) ≤ π(s3 ). This is equivalent to s1 , s2 ≤ s3 . (d) When G is reductive let π : G → G = G/Z 0 be the natural Q p homomorphism from G to the quotient of G by its maximal split central torus Z 0 . The group G is semi-simple, π(T ) = T is a maximal split Q p -torus in G , π |T gives an exact sequence 1 → Z 0 (Q p ) → T (Q p ) → T (Q p ) → 1, inducing a bijective map between the cosets T (Q p )++ /T (Q p )0 → T (Q p )++ /T (Q p )0 respecting ≤. By (c), ≤ is right filtered on T (Q p )++ . We deduce that ≤ is right filtered on T (Q p )++ . By Theorem 7.117 and Theorem 7.149, we can associate functorially to an étale T+ -module D over OE ,α different sheaves: • For any s ∈ T++ , a G(Q p )-equivariant sheaf Ys on G(Q p )/P(Q p ) with sections on C0 equal to M(D)bd s .

364

Peter Schneider, Marie-France Vigneras, and Gergely Zabradi

• The G(Q p )-equivariant sheaves Y∩ and Y∪ on G(Q p )/P(Q p ) with secbd tions on C0 equal to ∩s∈T++ M(D)bd s and ∪s∈T++ M(D)s . In general M(D) is different from ∪s∈T++ M(D)bd s , by the following proposition. Proposition 7.151. Let M be an étale T+ -module M over α (N0 ). When the root system of G is irreducible of positive rank r k(G), we have: (i) If r k(G) = 1, the G(Q p )-equivariant sheaf on G(Q p )/P(Q p ) with sections Msbd over C0 does not depend on the choice of s ∈ T++ , and M = Msbd . (ii) If r k(G) > 1, a G(Q p )-equivariant sheaf of o-modules Y on G(Q p )/P(Q p ) such that Y(C0 ) ⊂ M and (u α (1) − 1) is bijective on Y(C0 ), is zero. Proof. We prove (i). If r k(G) = 1, then OE = α (N0 ) and M = D is an étale T+ -module over OE . With the same proof as in Proposition 7.96, we have Msbd = M for any s ∈ T++ and the integrals Hg for g ∈ N0 P N0 do not depend on the choice of s. (ii) is equivalent to the property: an étale o[P+ ]-submodule M of M which is also a R = o[N0 ][(u α (1) − 1)−1 ]-submodule of M, and is endowed with endomorphisms Hg ∈ Endo (M), for all g ∈ N0 P(F)N0 , satisfying the relations H1, H2, H3 (Proposition 7.72), is 0. (a) Preliminaries. As r k(G) ≥ 2 and the root system is irreducible, there exists a simple root β such that α + β is a root. The elements n α := u α (1) and n β := u β (1) do not commute. By the commutation formulas, n α n β = n β n α h for some h = 1 in the group H = γ Nγ (Z p ) for all positive roots of the form γ = i α + jβ ∈ + with i, j > 0. Note that H is normalized by Nα (Z p ). Let s ∈ T++ . We have the expansion (7.12) (n α h − 1)−k = uϕs (ψs (u −1 (n α h − 1)−k )) (7.76) u∈J (Nα (Z p )H/s Nα (Z p )H s −1 )

in R. We choose, as we can, a lift wβ of sβ in the normalizer of T (Q p ) such that – wβ n β ∈ n β P(Q p ) – wβ normalizes the group N+ −β (Z p ) = γ Nγ (Z p ) for all positive roots γ = β. The subset Nβ (Z p ) ⊂ Nβ (Z p ) of u β (b) such that wβ u β (b) ∈ u β (Z p )P(Q p ), contains n β but does not contain 1. The subset Uwβ ⊂ N0 of u such that wβ u ∈ N0 P(Q p ) is equal to

G-equivariant sheaves on G/P

365

Uwβ = Nβ (Z p )N+ −β (Z p ) = N+ −β (Z p )Nβ (Z p ). Hence Uwβ = uUwβ , i.e. wβ−1 C0 ∩C0 = uwβ−1 C0 ∩C0 , for any u ∈ N+ −β (Z p ). (b) Let M be an R = o[N0 ][(n α − 1)−1 ]-module of M, which is also an étale o[P+ ]-submodule, and is endowed with endomorphisms Hg ∈ Endo (M), for all g ∈ N0 P(F)N0 , satisfying the relations H1, H2, H3 (Proposition 7.72), and let m ∈ M be an arbitrary element. We want to prove that m = 0. The idea of the proof is that, for s ∈ T++ , we have m = 0 if Hwβ (n β ϕs (m)) = 0 and that Hwβ (n β ϕs (m)) = 0 because it is infinitely divisible by n γ − 1, where γ = sβ (α). An element in M with this property is 0 because n γ − 1 lies in the maximal ideal of α (N0 ). Let a ∈ Z p . The product formula in Proposition 7.84ii implies Hwβ ◦ Hn aα ◦ res(1w−1 C β

0 ∩C0

) = Hwβ n aα ◦ res(1w−1 C β

0 ∩ C0

)=

Hn aγ wβ ◦ res(1w−1 C0 ∩C0 ) = Hn aγ ◦ Hwβ ◦ res(1w−1 C0 ∩C0 ) β

β

−1 −1 −1 −a since n −a α wβ C0 ∩ C0 = wβ C0 ∩ C0 = wβ n γ C0 ∩ C0 . For all k ∈ N, the elements

m k :=(n α − 1)−k n β ϕs (m) = n β (n α h − 1)−k ϕs (m)

(7.77)

lie in the image of the idempotent res(1w−1 C0 ∩C0 ) ∈ Endo (M), because β

mk =

n β uϕs (ψs (u −1 (n α h − 1)−k m))

u∈J (Nα (Z p )H/s Nα (Z p

(7.78)

)H s −1 )

by (7.76), (7.77). Therefore the product relations between Hwβ , Hn aα and Hn aγ imply Hwβ (n β ϕs (m)) = Hwβ ((n α − 1) m k ) = k

k a=0

=

k

k a=0

k Hwβ ◦ Hn aα (m k ) a

(−1)

k Hwβ ◦ Hn aα ◦ res(1w−1 C ∩C )(m k ) 0 0 β a

(−1)k−a

k Hn aγ ◦ Hwβ ◦ res(1w−1 C0 ∩C0 )(m k ) β a

k−a

a=0

=

(−1)

k−a

= (n γ − 1)k Hwβ (m k ). Hence Hwβ (n β ϕs (m)) = 0 since it is infinitely divisible by n γ − 1 which lies in the maximal ideal of α (N0 ). We also have

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Peter Schneider, Marie-France Vigneras, and Gergely Zabradi

n β ϕs (m) = H1 ◦ res(1w−1 C0 ∩C0 )(n β ϕs (m)) = Hwβ ◦ Hwβ (n β ϕs (m)) = 0. β

As n β ◦ ϕs ∈ Endo

(M )

is injective, we deduce m = 0.

Corollary 7.152. There exists a G(Q p )-equivariant sheaf on G(Q p )/P(Q p ) with sections M on C0 if and only if rk(G) = 1.

References [1] [2] [3] [4] [5] [6] [7]

[8] [9] [10] [11] [12]

[13] [14] [15] [16]

Bourbaki N., Algèbre commutative. Ch. 1 à 4. Masson 1985. Bourbaki N., Topologie générale. Ch. 1 à 4. Hermann 1971. Bourbaki N., Topologie générale. Ch. 5 à 10. Hermann 1974. Bosch S., Güntzer U., Remmert R., Non-Archimedean analysis. Springer 1984. Colmez P., (ϕ, )-modules et représentations du mirabolique de G L 2 (Q p ). Astérisque 330, 2010, 61–153. Colmez P., Représentations de G L 2 (Q p ) et (ϕ, )-modules. Astérisque 330, 2010, 281–509. Dixon J. D., du Sautoy M. P. F., Mann A., Segal D., Analytic pro- p groups. Second edition. Cambridge Studies in Advanced Mathematics, 61. Cambridge University Press, 1999. Ellis R., Locally compact transformation groups. Duke Math. J. 24, 1957 119–125. Gabriel P., Des catégories abéliennes. Bull. Soc. Math. France 90, 1962, 323–448. Kedlaya K., New methods for (ϕ, )-modules, preprint (2011), http://math.mit. edu/kedlaya/papers/new-phigamma.pdf Schneider P., p-Adic Lie groups. Springer Grundlehren, Vol. 344, Springer, 2011. Schneider P., Vigneras M.-F., A functor from smooth o-torsion representations to (ϕ, )-modules. Volume in honour of F. Shahidi. Clay Mathematics Proceedings, Volume 13, 525–601, 2011. Springer T. A., Linear Algebraic Groups. Second edition. Birkhäuser, 2009. Vigneras M.-F., Représentations -modulaires d’un groupe réductif p-adique avec = p. PM 137, Birkhäuser, 1996. Warner S.: Topological rings. Elsevier, 1993. Zábrádi G., Exactness of the reduction of étale modules. J. Algebra 331, 2011, 400–415.

8 Intertwining of ramified and unramified zeros of Iwasawa modules Chandrashekhar Khare and Jean-Pierre Wintenberger

1. Introduction Let F be a totally real field, p > 2 a prime, F∞ ⊂ F(μ p∞ ) the cyclotomic Z p -extension of F with Galois group = Z p = γ . This short note is a sequel to [5]. The sole aim is to point out the ubiquity of a phenomenon dicussed in a particular case in our previous paper. Namely, the (arithmetic) eigenvalues of γ acting on Galois groups of maximal p-abelian unramified extensions of F(μ p∞ ) intertwine with the eigenvalues acting on inertia subgroups of ramified p-abelian extensions of F(μ p∞ ). We make this vague philosophy precise in the text below after alluding for mise-en-scène to the p-adic L-functions that lurk suggestively in the wings, but do not play an explicit role in the algebraic computations of this note. Let ψ be an even Dirichlet character of F. Consider the p-adic L function ζ F, p (s, ψ) = L p (s, ψ), s ∈ Z p , which is characterised by the interpolation property L p (1 − n, ψ) = L(1 − n, ψω−n )v| p (1 − ψω−n (v)N (v)n−1 ), for n ≥ 1 a positive integer. When ψ = 1, we denote the corresponding L-function by ζ F, p (s). It is known that L p (s, ψ) is holomorphic outside s = 1, is holomorphic everywhere when ψ = 1, and otherwise has at most a simple pole at s = 1. This pole is predicted to exist by the conjecture of Leopoldt, which asserts the non-vanishing of the the p-adic regulator of units of F. The residue of ζ F, p at s = 1 has been computed by Pierre Colmez: Ress=1 ζ F, p (s) =

2d R F, p h F , √ 2 DF

CK was partially supported by NSF grants. JPW is member of the Institut Universitaire de France. Automorphic Forms and Galois Representations, ed. Fred Diamond, Payman L. Kassaei and c Cambridge University Press 2014. Minhyong Kim. Published by Cambridge University Press.

367

368

Chandrashekhar Khare and Jean-Pierre Wintenberger

where R F, p is the p-adic regulator for F. The non-vanishing of R F, p is the Leopoldt conjecture for F and p. We recall a folklore conjecture that is a very particular case of the general conjectures of Jannsen ([4]) about the non-vanishing of higher regulators. Conjecture 8.1. (Non-vanishing of higher p-adic regulators) For an integer m = 0, L F, p (m, ψ) = 0 if either m = 1 or ψ = 1. Furthermore, ζ F, p (s) has a pole at s = 1. As a supplement in the case m = 0, the case of “trivial zeros”, the multiplicity of the zero at s = 0 of L p (s, ψ) is conjectured to be given by the number of v| p such that (1 − ψω−1 (v)) = 0. We call the zeros of ζ F, p (s) unramified zeros following a similar usage in [8]. The main conjecture of Iwasawa theory realises the zeros of the L-functions as roots of the characteristic polynomial of γ acting on “unramified arithmetic spaces”. Nevertheless it gives no direct information about the zeros, belying the Hilbert–Polya philosophy in this case! Our basic observation, coming from [5] that we reinforce here, is that the unramified zeros (of the p-adic L-function) always intertwine with (the eigenvalues of γ acting on) ramification at p. Further the Leopoldt zeros, i.e. eigenvalues of that correspond on a finite index subgroup to its action on p-power roots of unity, intertwine with ramification at Q, for finite sets of primes away from p, for a generic choice of Q. This shows that the non-vanishing of p-adic regulators is equivalent to splitting of ramification in naturally occuring exact sequences of Iwasawa modules. The reader is referred to Theorems 8.5, 8.9 and 8.10 for precise statements. The proofs of these theorems rely on numerical coincidences between – dimensions of certain Galois cohomology groups whose computation result from the work of Soulé and Poitou–Tate duality, – dimensions of Iwasawa modules that follow from theorems of Iwasawa describing the structure of inertia at p (resp. at a set Q of auxiliary primes) in the Galois group of the maximal odd abelian p-extension of F(μ p∞ ) that is unramified outside p (resp. Q).

2. Galois cohomology 2.1. In our previous paper [5] we paid attention to integral questions, while here we work over Q p exclusively. We consider F a totally real field, an odd prime p.

Ramified and unramified zeros

369

We consider a sufficiently large finite extension K of Q p with ring of integers O (that will contain values of the character under consideration). Let S p be the set of places of F above p and ∞ and let S be a finite subset of places of F containing S p . Let G S be a Galois group of the maximal algebraic extension of F unramified outside S. We consider a potentially crystalline, or arithmetic, character χ of G S , of (parallel) weight m and thus of the form ηχ pm where χ p is the p-adic cyclotomic character, and η a finite order character. We impose that χ (c) is independent of the choice of complex conjugation c ∈ G F , and call it odd or even according as this value is either −1 or 1. We denote by ω the Teichmüller character. We consider the cohomology subgroup H(1p) f (G S , K (ηχ pm )) of H 1 (G S , K (ηχ pm )) defined by imposing for v ∈ S, v ∈ / S p the condition to be unramified (although χ might be ramified at these v). We denote by H 1f (G S , K (ηχ pm )) their Bloch–Kato subgroups, where for v primes over p, we impose the Bloch–Kato finiteness condition ([2]). We denote the dimensions over K of H(1p) f (G S , K (ηχ pm )) and H 1f (G S , K (ηχ pm )) by h 1 (χ ) and h 1f (χ ). We use h 1split (χ ) to denote dimensions of cohomology groups where we ask that the classes are split locally at all places above p and unramified at other primes. We have the tables.

2.2. Odd χ χ = ηχ pm , χ odd h 1 (χ ) h 1f (χ )

m>1 d d

d+

v| p

m=1 −1 v , η ) − δχ ,χ p d − δχ ,χ p

h 0 (G

m≤0 d 0

2.3. Even χ χ = ηχ pm , χ even h 1 (χ ) h 1f (χ )

m>1 0 0

m=1 0 (G , η−1 ) h v v| p 0

m≤0 δχ ,id + h 1split (χ p χ −1 ) 0

Remark. The situation for even χ is thus not satisfactory as we do not have an explicit formula in all cases for h 1 (χ ). The conjectures in [4], concerning non-vanishing of higher p-adic regulators, predict the vanishing of h 1split (χ p χ −1 ) for χ an even arithmetic character of G F (see also [1] 5.2). This follows from the main conjecture of Iwasawa theory if the weight of χ is > 0.

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2.4. Ingredients of the computation We justify the values in the tables. We need the following ingredients: • (Bloch–Kato) h 1f (χ ) = dim K K (χ )G F + ords=1−m L(η−1 , s). • (global duality) χ even: h 1 (χ ) = h 1split (χ p χ −1 ) + δχ ,id +

h 0 (G v , χ p χ −1 ).

v| p

• (global duality) χ odd: h 1 (χ ) = d + h 1split (χ p χ −1 ) − δχ ,χ p +

h 0 (G v , χ p χ −1 ).

v| p

The (global duality) equalities follow from Theorem 8.7.9. of [6] and: • (local Euler Poincaré characteristic) h 1 (G v , V ) = h 0 (G v , V ) + h 2 (G v , V ) + [Fv : Q p ]dim K (V ) where v| p. • (local duality) h 0 (G v , V ∗ (1)) = h 2 (G v , V ). The Bloch–Kato formula, which directly implies the bottom row of both tables, follows from a theorem of Soulé (Theorem 1 of [7]) and duality as we justify. – The theorem of Soulé directly implies Bloch–Kato formula for h 1f (χ ) for m > 1. – The case of m = 1 for the bottom rows follows from Kummer theory. – We have the equality for m ≤ 0 and χ even: h 1f (χ ) = h 1f (χ p χ −1 ) − d + δχ ,id , and for m ≤ 0 and χ odd: h 1f (χ ) = h 1f (χ p χ −1 ). These equalities follow from Theorem 8.7.9 of [6] and the fact that for m ≤ 0, 1 (G , χ ). These two H 1f (G v , χ ) coincides with the unramified cohomology Hur v equalities allow us to deduce the case m ≤ 0 of the Bloch–Kato formula from the Theorem of Soulé (see also [1], §4.3.1). This checks the second rows of both the tables.

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371

Let us check the first row. For the first column in the case m > 1, we use the fact that h 1 (G v , χ ) = h 1f (G v , χ ). The case m ≤ 0 even χ follows from (global duality) as h 0 (G v , χ p χ −1 ) = 0 for all v above p. For the case m = 1, we use global duality and that h 1split (η−1 ) = 0 which follows easily from the fact that a Z p -extension of a number field has to be ramified at a place above p. For χ odd and m ≤ 0, we use global duality and the fact that h 1split (χ p χ −1 ) ≤ h 1f (χ p χ −1 ) = 0.

2.5. Galois groups We consider ε totally odd and ψ totally even finite order characters of G F , such that εψ = ω. We set ψ(n) = ψ(ω−1 χ p )n := ψκ n , and likewise for ε(n). We consider characters ψζ of that send a chosen generator γ of to a p-power root of unity ζ and consider ψ(n)ψζ , ε(n)ψζ . We assume that ψ is of type S, i.e. we assume that the field Fψ cut out by ψ is linearly disjoint from the cyclotomic Z p -extension F∞ of F. We denote by the Galois group of F∞ /F with choice of generator γ , consider

the completed group algebra Z p [[]] that is isomorphic to Z p [[T ]] via the homomorphism which sends γ → 1 + T . We let K = ⊗ K . We consider the Galois group Gψ = Gε = Gal(Fψ (μ p )/F) × Gal(F∞ /F) of Fψ (μ p∞ ) over F. A continuous character χ of Gψ with values in O∗ is the product of a finite character χψ of Gal(Fψ (μ p )/F) and a character χ of . The character χ induces a K -algebra map K → K . We denote by Pχ the corresponding prime ideal of K kernel of this map. It is generated by γ − χ (γ ). The character χψ induces a morphism f χ : Z p [Gal(Fψ (μ p )/F)] → O ⊂ K . The morphism Z p [[Gψ ]] → K induced by χ is the composite of the map Z p [[Gψ ]] → K induced by f χ and the morphism K → K . We consider the maximal, abelian pro- p extension L∞ of Fψ (μ p∞ ) unram and set X ified everywhere: we denote its Galois group by X ∞ ∞ = X∞ ⊗ K . We set X ∞,ε to be the maximal quotient of X ∞ on which Gal(Fψ (μ p )/F) acts by ε. We also consider the analogous extensions L∞,ε (Q) and corresponding Galois group X ∞,ε,Q when Q is any set of places of F and we allow ramification above Q. There are two natural cases to consider: • Q is all the places above p, and then we replace Q by p in the notation; • Q is a finite set of places disjoint from the places above p. Let Fψ,∞ be the cyclotomic Z p -extension of the totally real field Fψ . We the Galois group of the maximal abelian prop- p extension of denote by Y∞

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. As above, we denote Fψ,∞ that is unramified outside p, and Y∞ = Q p ⊗ Y∞ by Y∞,ψ the related quotient on which Gal(Fψ (μ p )/F) acs by ψ, and recall the perfect -equivariant Iwasawa pairing

Y∞,ψ × X ∞,ε → K (1). Lemma 8.2. Let ψ be an even character of G F of type S and n ∈ Z. (1) We have h 1 (ψζ ψ(n)) = dim K (Y∞,ψ /Pψζ ψ(n) ) if ψζ ψ(n) is not trivial, and h 1 (ψζ ε(n)) = dim K (X ∞,ε, p /Pψζ ε(n) ). (2) We have that dim K ((X ∞,ε ) Pψζ ε(−n) /Pψζ ε(−n) ) = dim K ((Y∞,ψ ) P

ψζ−1 ψ(n+1)

/Pψ −1 ψ(n+1) ). ζ

Proof. For (i) we use the inflation-restriction sequence relative to G S → Gψ . Recall that the unramified condition for η ∈ H(1p) f (G S , K (εχ pm )) at v not above p and such that ψ is ramified at v (Section 2.1). We check that this condition is equivalent to that the restriction of η to the kernel of the map G S → Gψ is unramified at v. For (ii), we invoke the pairing of Iwasawa. We use the cyclicity of to identify dimensions of twisted invariants and covariants for . We rederive a standard result about trivial zeros of p-adic L-functions, usually proved using genus theory, which is a corollary to the lemma. Proposition 8.3. Suppose ε is an odd character of Gal(Fψ (μ p )/F) as before. Then dim K ((X ∞,ε ) Pψζ ε /Pψζ ε ) = h 1 (χ p ψζ−1 ε−1 ) = h 0 (G v , ψζ−1 ε−1 ). v| p

Proof. We deduce the first equality using (2) of Lemma 8.2 for n = 0 and the second equality using the above table for m = 1 even χ. Note that v| p h 0 (G v , ψζ−1 ε−1 ) is the number of places v| p such that the

Euler factor (1−ψζ−1 ε−1 (v)) is 0. It is conjectured by Greenberg that the trivial zeros occur semisimply i.e. dim K (X ∞,ε /Pψζ ε ) = dim K (X ∞,ε ) Pψζ ε .

3. Main conjecture and higher regulators We consider the Deligne–Ribet p-adic L-function ζ F, p (s, ψ) for an even character ψ of F of type S. It is defined on Z p when ψ is non-trivial, and on

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Z p \{1} when ψ is trivial. It is characterised by the interpolation formula that for integers n ≥ 1, ζ F, p (1 − n, ψ) = L p (1 − n, ψω−n ), where the superscript denotes that we have dropped the Euler factors at p. There is a power series Wψ (T ) in K = Z p [[]] ⊗ K (the latter by the isomorphism that sends a chosen generator γ of = Gal(F∞ /F) to 1 + T and u := χ p (γ )), with the property that: Wψ (u s − 1) = ζ F, p (s, ψ) u 1−s − 1 when ψ is trivial, and Wψ (u s − 1) = ζ F, p (s, ψ) otherwise. Furthermore we have: Wψψζ (T ) = Wψ (ζ −1 (1 + T ) − 1); see the introduction of [8], where the notation is G ψ (T ) for Wψ (u (1 + T )−1 − 1). Then the main conjecture asserts for characters ψ of type S that (Wψ (T )) = char K (X ∞,ψ −1 ω ), i.e. the characteristic polynomial of the action of γ on the finite dimensional K -vector space X ∞,ε generates the same ideal as Wψ (T ). (We ignore μ-invariants via this formulation.) When m is an integer = 0, Conjecture 8.1 is equivalent via the main conjecture to the statement that the generalised ζ u m -eigenspace of X ∞,ψ −1 ω ⊗ Q p for the action of γ is trivial . When m = 0, via the main conjecture, we have that the generalised ζ -eigenspace is of dimension given by the number of v| p of F such that (1 − ψζ−1 ε−1 (v)) = 0.

4. Intertwining of ramified and unramified zeros 4.1. Ramification at p We consider the exact sequences: 0 → I Q → X ∞,ε,Q → X ∞,ε → 0, of finitely generated K -modules with Q = p (see 2.5).

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4.1.1. Ramification at p and intertwining with non-trivial unramified zeros Next theorem follows from Th. 25 of [3] : Theorem 8.4. We have an ismorphism of K -modules ) s { dK ⊕ j =1 IndG ℘ K (1)} j Ip = , K (1) where G ℘ are the decomposition groups of the places ℘ above p in . We deduce from this and the computations in Galois cohomology earlier: Theorem 8.5. Let χ = εψζ κ m be an odd arithmetic character of Gε of weight m, and let Pχ be the corresponding prime ideal of O for O such that χ is valued in O ∗ . Then the exact sequence 0 → (I p ) Pχ → (X ∞,ε, p ) Pχ → (X ∞,ε ) Pχ → 0, of finitely generated K -modules splits if and only if (X ∞,ε ) Pχ vanishes. This is equivalent to that ζ F, p (m, ψζ−1 ε−1 ω) = 0 when χ = χ p , and when χ = χ p , to that ζ F, p (s) has a pole at s = 1. Proof. One direction is trivial. For the other direction, one notes by the theorem of Iwasawa that dim K ((I p ) Pχ /Pχ ) is d if m = 1 and d + 0 −1 v| p h (G v , ε ) − δχ ,χ p if m = 1. We have the numerical coincidence: dim K ((I p ) Pχ /Pχ ) = h 1 (χ ). Further one notes from Lemma 8.2 that h 1 (χ ) = dim K (X ∞,ε, p ) Pχ /Pχ . Thus if the exact sequence in the theorem splits, we deduce that (X ∞,ε ) Pχ /Pχ = 0, which is equivalent to the vanishing of (X ∞,ε ) Pχ . By the main conjecture proved by Wiles, the vanishing of (X ∞,ε ) Pχ /Pχ is equivalent to the vanishing of Wψ at ζ u m − 1. It is equivalent to the vanishing of ζ p (m, ψψζ−1 ) = ζ F, p (m, ψζ−1 ε−1 ω). Remarks. 1. We also deduce that an equivalent formulation of the non-vanishing of the higher regulator conjecture is the conjecture that the exact sequence 0 → (I p ) Pχ → (X ∞,ε, p ) Pχ → (X ∞,ε ) Pχ → 0, of finitely generated K -modules splits for all odd arithmetic characters χ of G of non-zero weight. 2. As trivial zeros (in weight 0) do occur we get examples of Iwasawa modules in which ramification is allowed at p such that γ acts non-semisimply.

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We have a conditional result for any character χ = εκ s of Gε not necessarily arithmetic, and that follows by similar arguments. Proposition 8.6. Let χ = εκ s be any odd character of Gε . Assume further that 1 (G −1 Hsplit F, p , χ p χ ) is trivial. Then the exact sequence 0 → (I p ) Pχ → (X ∞,ε, p ) Pχ → (X ∞,ε ) Pχ → 0, of compact finitely generated K -modules splits if and only if (X ∞,ε ) Pχ = 0. 1 (G −1 Remark. Note that the vanishing of Hsplit F, p , χ p χ ) = 0 is predicted by Greenberg’s conjecture that the p-part of the class group of the cyclotomic Z p -extension F∞ of F is finite. The above proposition suggests that there may be a formulation of the main conjecture using Ext K (X ∞,ε , I p ).

4.2. Ramification away from p 4.2.1. ψ and ψζ trivial Consider the maximal abelian p-extension L ∞ of F , and denote its Z p -rank by 1 + δ. The Leopoldt conjecture asserts that δ = 0. Definition 8.7. (generic sets Q) We say that a finite set of primes Q of cardinality r away from p is generic if the rank r Q of the subgroup generated by the Frobq ’s for q ∈ Q in Gal(L ∞ /F) is min(r, 1 + δ). The terminology is meant to reflect the fact that when δ > 0, the Frobq1 , Frobq2 will be linearly independent in Z1+δ = Gal(L ∞ /F) for most p choices of q1 , q2 . If r = 2 and we choose a prime q1 freely, then for a density one set of primes q2 , the set Q = {q1 , q2 } is generic. We now show the intertwining of the Leopoldt zero u = u γ with the ramification at Q provided Q is a generic set of primes with |Q| > 1. Proposition 8.8. Let Q be a finite set of primes of F away from p. Then the subgroup I Q of X ∞,ω,Q generated by the conjugacy class of the inertia groups Iq for q ∈ Q is isomorphic as a Gal(F∞ /F) = Gal(F(μ p )/F) ×-module to (q∈Q IndG q K (1)) K (1) where G q is the decomposition subgroup at q in = Gal(F∞ /F), and where we declare that Gal(F(μ p )/F ) acts by ω. Proof. This follows from class field theory. Note that when the primes q are inert in F∞ /F then I Q = K (1)r−1 .

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Theorem 8.9. Let Q be generic set of primes Q of cardinality r ≥ 2. Then the exact sequence 0 → (I Q ) Pχ p → (X ∞,ω,Q ) Pχ p → (X ∞,ω ) Pχ p → 0 splits if and only if (X ∞,ω ) Pχ p = 0, i.e., if and only if the Leopoldt conjecture is true. Thus if there is a Leopoldt zero, then it intertwines with the ramified zeros at Q for a generic set of primes (away from p) with |Q| ≥ 2. Proof. If the Leopoldt conjecture is true, (X ∞,ω ) Pχ p = 0 and the exact sequence splits. Let us prove the converse. For a finite dimensional vector space V over K endowed with a continuous action of G F that is unramified outside a finite set of places, and a set of Selmer conditions L = {Lv } for Lv ⊂ H 1 (Fv , V ) where Lv is outside a finite set of places the unramified subgroup, we have the formula : h 1L (F, V ) − h 1L⊥ (F, V ∗ (1)) =h 0 (F, V ) − h 0 (F, V ∗ (1)) + (dim K Lv − h 0 (Fv , V )). v

We apply this formula for V = K (1), and with the Selmer conditions L to be unramified everywhere. In particular the Selmer condition is trivial at places v above p as V Iv = 0 for v above p. We get: h 1L (F, V ) − h 1L⊥ (F, V ∗ (1)) = −1. Furthermore, we have h 1L⊥ (F, K ) = 1 + δ. Consider the Selmer conditions L Q that arise when we allow ramification at Q, i.e., (L Q )v = Lv for v ∈ / Q and (L Q )v = H 1 (G v , K (1)) for v ∈ Q. We get: h 1L Q (F, V ) − h 1L⊥ (F, V ∗ (1)) = −1 + r. Q

Furthermore, we have h 1L⊥ (F, K ) = 1 + δ − r Q . We see that: Q

h 1L Q (F, K (1)) = h 1L (F, K (1)) + r − r Q . If the exact sequence splits, it remains exact after reduction modulo Pχ p , hence we have: h 1L Q (F, K (1)) = h 1L (F, K (1)) + r − 1.

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Thus we get r Q = 1. As Q is generic and | Q |≥ 2, i.e., r Q = min(r, 1 + δ) with r ≥ 2, we get that 1 + δ = 1, thus δ = 0 and the Leopoldt conjecture is true. 4.2.2. Weight 1, ψ or ψζ non-trivial Theorem 8.10. Consider the exact sequence 0 → (Iq ) Pψζ εκ → (X ∞,ε,q ) Pψζ εκ → (X ∞,ε ) Pψζ εκ → 0. Assume ε = ω or that ζ = 1. Then the sequence splits for all choices of primes q of F away from p, if and only if (X ∞,ε ) Pψζ εκ = 0. Proof. We only sketch the proof as its very similar to the proof of Theorem 8.9. By (2) of Lemma 8.2, if (X ∞,ε ) Pψζ εκ = 0, the maximal abelian p-extension L of Fε−1 ψ −1 ω = Fψψ −1 on which Gal(Fψψ −1 /F) acts by ψψζ−1 and which ζ

ζ

ζ

is unramified outside p, has Galois group that is of positive rank as a Z p module. Let q be a prime of F away from p, that splits in Fψψ −1 , such ζ

that the Frobenius at a prime above q of Fψψ −1 is a non-torsion element in ζ

Gal(L/Fψψ −1 ). With V = K (ψζ εκ) and the Selmer conditions as above, it ζ

follows that h 1L⊥ − h 1L⊥ = 1. q

Using the above formula, we get h 1Lq − h 1L⊥ − h 1L + h 1L⊥ = 0. If the exact q

sequence were to split, we would have h 1Lq = h 1L , which is a contradiction.

References [1] Joël Bellaiche. An introduction to the conjecture of Bloch and Kato. Lectures at the Clay Mathematical Institute summer School, Honolulu, Hawaii , 2009. [2] Spencer Bloch, Kazuya Kato. L-functions and Tamagawa numbers of motives. The Grotendieck Festschrift, vol. 1, 333-400, Prog. Math. 86, Birkhäuser Boston, Boston, MA, 1990. [3] Kenkichi Iwasawa. On Z -extensions of algebraic number fields. Ann. of Math. (2) 98 (1973), 246–326. [4] Uwe Jannsen. On the -adic cohomology of varieties over number fields and its Galois cohomology, in Galois groups over Q (Berkeley, CA, 1987), 315–360, Math. Sci. Res. Inst. Publ., 16, Springer, New York, 1989. [5] Chandrashekhar Khare, Jean-Pierre Wintenberger. Ramification in Iwasawa modules and splitting conjectures. To appear in International Mathematics Research Notices.

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[6] Jürgen Neukirch. Cohomology of number fields. 2nd edition. Grundlehren des mathematischen Wissenschaften, 323, Springer, 2008. [7] Christophe Soulé. On higher p-adic regulators, in Algebraic K -theory. Evanston 1980 Lecture Notes in Math., 854, Springer, 1981. [8] Andrew Wiles. The Iwasawa conjecture for totally real fields. Ann. of Math. (2) 131 (1990), no. 3, 493–540.

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