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London Mathematical Society Lecture Note Series. 254
Galois Representations in Arithmetic Algebraic Geometry Edited by
A. J. Scholl University of Durham R. L. Taylor Harvard University
CAMBRIDGE UNIVERSITY PRESS
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© Cambridge University Press 1998
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CONTENTS Preface List of participants Lecture programme
The Eigencurve
vii viii ix
1
R. COLEMAN AND B. MAZUR
Geometric trends in Galois module theory
115
BOAS EREZ
Mixed elliptic motives
147
ALEXANDER GONCHAROV
On the Satake isomorphism
223
BENEDICT H. GROSS
Open problems regarding rational points on curves and varieties
239
B. MAZUR
Models of Shimura varieties in mixed characteristics
267
BEN MOONEN
Euler systems and modular elliptic curves
351
KARL RUBIN
Basic notions of rigid analytic geometry
369
PETER SCHNEIDER
An introduction to Kato's Euler systems
379
A. J. SCHOLL
La distribution d'Euler-Poincare d'un groupe profini JEAN-PIERRE SERRE
461
PREFACE This volume grew out of the London Mathematical Society symposium on "Galois representations in arithmetic algebraic geometry" held in Durham from the 9th to the 18th of July 1996. We understood our title rather loosely and the symposium considered many recent developments on the interface between algebraic number theory and arithmetic algebraic geometry. There were six expository courses on 1. Galois module structure 2. Shimura varieties in mixed characteristic 3. p-adic comparison theorems 4. the work of Kato on the Birch-Swinnerton-Dyer conjecture 5. polylogarithms 6. rigid analysis and modular forms We are very grateful to the organisers of each of theses courses (Chinburg, Oort, Fontaine, Kato, Goncharov and Coleman) as well as all the other lecturers who worked hard to make these courses highly successful. In addition to the short courses there were 14 research seminars. We would also like to thank these lecturers and particularly those who have contributed to this volume. The symposium received generous financial support from the EPSRC and from the EU (through the network on "automorphic forms and arithmetic algebraic geometry"). We were particularly pleased that this enabled a large number of young European researchers to attend. Finally we would like to thank Steve Wilson and the department of mathematics at Durham University for their help with the organisation of the meeting. This volume contains both expository and research articles. We are particularly grateful to the authors (Erez, Mazur, Moonen and Schneider) who have put a lot of time into preparing what we feel will be a useful collection of expositions. (Schneider's introductory lectures on rigid analysis were particularly well received at the symposium and we hope that we did right in pressuring him to write them up despite their relatively elementary nature.) The rest of the articles are research papers mostly based either on one of the short courses or on one of the individual lectures at the symposium. We would like to thank all the authors for allowing us to publish their work here. Tony Scholl Richard Taylor
PARTICIPANTS B. Agboola (Princeton) F. Andreatta (Utrecht) P. Balister (London) P.R. Bending (Oxford) P. Berthelot (Rennes) A. Besser (UCLA) B. Birch (Oxford) D. Blasius (UCLA) S. Bloch (Chicago) G. Boeckle (Strasbourg) C. Breuil (Ecole Polytechnique) K. Buecker (Grenoble) 0. Bultel (Oxford) D. Burns (KCL) C. Bushnell (KCL) K. Buzzard (Berkeley) N. Byott (Exeter) J.W.S. Cassels (Cambridge) R. Chapman (Exeter) 1. Chen (Oxford) F. Cherbonnier (Orsay) T. Chinburg (Pennsylvania) J. Coates (Cambridge) R. Coleman (Berkeley) P. Colmez (ENS Paris) D. Delbourgo (Cambridge) R. de Jeu (Durham) J. de Jong (Harvard) F. Diamond (Cambridge) M.E.T. Dickinson (Oxford) T. Dokshitzer (Utrecht) B. Edixhoven (Rennes) B. Erez (Bordeaux) G. Faltings (Max-Planck Institute) I. Fesenko (Nottingham) L. Figueiredo (Rio de Janeiro) M. Flach (Caltech) J. Fontaine (Orsay) A. Frohlich (Cambridge) A. Goncharov (MIT) R. Greenberg (Washington) B. Gross (Harvard) M. Harris (Paris 7) S. Howson (Cambridge) C. Huyghe (Rennes) L. Illusie (Orsay) F. Jarvis (Oxford) K. Kato (Tokyo)
G. Kings (Munster) V. Kolyvagin (Johns Hopkins) H. Knosper (Munster) S. Kudla (Maryland) L. Lafforgue (Orsay) E. Landvogt (Munster) S. Lichtenbaum (Brown) A. Macintyre (Oxford) J. Manorhamayum (Cambridge) D. Mauger (Paris-Nord) B. Mazur (Harvard) L. Merel (Paris) J. Merriman (Kent) W. Messing (Minnesota) B.J.J. Moonen (Munster) H. Nakamura (IAS) J. Nekovar (Cambridge) R. Noot (Rennes) L. Nyssen (Strasbourg) Y. Ochi (Cambridge) R. Odoni (Glasgow) F. Oort (Utrecht) G. Pappas (Princeton) A. Plater (Bordeaux 1) F. Pop (Heidelberg) M. Rapoport (Wuppertal) K. Ribet (Berkeley) K. Rubin (Ohio State) N. Schlappacher (Strasbourg) A. Schmidt (Heidelberg) P. Schneider (Munster) A.J. Scholl (Durham) J-P. Serre (Paris) N. Shepherd-Barron (Cambridge) V. Snaith (McMaster) R. Taylor (Oxford) M.J. Taylor (UMIST) J. Tilouine (Paris-Nord) T. Tsuji (Tokyo) A. Vasiu (ETH Zurich) M. Volkov (Orsay) R. Weissauer (Mannheim) A. Werner (Munster) J. Wilson (Oxford) S.M.J. Wilson (Durham) J.P. Wintenberger (Strasbourg) S. Wortmann (Cologne) M.A. Young (Durham)
Lecture programme 9:30-10:30
11:15-12:15
2:30-3:30
4:00-5:00
5:15-6:15
Tuesday 9th July
Al
Cl
D1
Il
12
Wednesday 10th July
B1
C2
El
13
14
Thursday 11th July
A2
C3
E2
15
Friday 12th July
B2
C4
Saturday 13th July Monday 15th July
17
Fl
D3
18
A3
F2
B3
19
Tuesday 16th July
B4
F3
D4
I10
Ill
Wednesday 17th July
A4
E3
112
113
114
Thursday 18th July
I15
F4
116
117
118
16
D2
A: GALOIS MODULE STRUCTURE 1) Ted Chinburg (Pennsylvania): Geometric group actions and Galois structure. 2) George Pappas (Princeton): The generalised Frohlich conjecture in any dimension. 3) Martin Taylor (UMIST, Manchester): Hermitian Euler characteristics and c-constants. 4) David Burns (Kings, London): Motivic Galois structure invariants. B: SHIMURA VARIETIES IN MIXED CHARACTERISTIC 1) Bas Edixhoven (Rennes): Introduction. Moduli schemes of abelian varieties. 2) Ben Moonen (Munster): Models of Shimura varieties. 3) Johan de Jong (Harvard; Princeton University): Integral crystalline cohomology. 4) Adrian Vasiu (ETH Zurich; Berkeley): Points of integral canonical models of Shimura varieties of preabelian type.
C: P-ADIC COMPARISON THEOREMS 1) Pierre Colmez (Paris VI): Fontaine's rings and the conjectures. 2) Jean-Marc Fontaine (Orsay): Abelian varieties. 3) Bill Messing (Minnesota): Syntomic and crystalline cohomology. 4) Takeshi Tsuji (Kyoto): p-adic vanishing cycles. D: THE WORK OF KATO ON THE BIRCH-SWINNERTON-DYER CONJECTURE 1) Tony Scholl (Durham): Kato's Euler system. 2/3) Kazuya Kato (Tokyo) 4) Karl Rubin (Ohio State): Euler systems and Selmer groups.
E: POLYLOGARITHMS 1) Sasha Goncharov (MIT): Geometry of polylogarithms and regulators. 2) Sasha Goncharov (MIT): L-functions of elliptic curves at s = 2. 3) Sasha Goncharov (MIT): Multiple polylogarithms and motivic Galois groups. F: RIGID ANALYSIS AND MODULAR FORMS 1/2) Peter Schneider (Munster): Rigid Analysis. 3) Robert Coleman (Berkeley): Serre's p-adic Fredholm theory in families. 4) Robert Coleman (Berkeley): The curve of q-expansions.
I: INDIVIDUAL LECTURES 1) Florian Pop (Heidelberg): An introduction to anabelian geometry I. 2) Jean-Pierre Serre (Paris): Euler-Poincare characteristics of profinite groups. 3) Barry Mazur (Harvard): Open questions about rational points on curves and varieties. 4) Fred Diamond (Cambridge): The Taylor- Wiles construction and multiplicity one. 5) Hiroaki Nakamura (IAS): An introduction to anabelian geometry H. 6) Dick Gross (Harvard): Modular forms for groups over Q whose real points are compact. 7) Pierre Colmez (Paris): Iwasawa theory of de Rham representations. 8) Gerd Faltings (Bonn): Fundamental groups of algebraic curves. 9) Gerd Faltings (MPI): Almost etale extensions. 10) Steve Kudla (Maryland): Height pairings and derivatives of Eisenstein series. 11) Michael Rapoport (Wuppertal): Special cycles on Siegel threefolds. 12) Victor Kolyvagin (Johns Hopkins): On the arithmetic of cyclotomic fields. 13) Michael Harris (Paris): p-adic uniformization and local Galois correspondences. 14) Ken Ribet (Berkeley): Torsion points on Xo(37). 15) Takeshi Tsuji (Kyoto): CBt. 16) Rainer Weissauer (Mannheim): Siegel modular forms mod p. 17) Loic Merel (Berkeley): Arithmetic of elliptic curves and diophantine equations. 18) Johan de Jong (Harvard/Princeton): Alterations.
The Eigencurve R. COLEMAN AND B. MAZUR in memory of Bernard Dwork
Let p be a prime number and CP the completion of an algebraic closure of the field Qp of p-adic numbers. Let N be an integer relatively prime to p. To describe our main object, we assume that p > 2, that the group of units in the ring Z/NZ is of order prime to p, and we restrict our attention (at least in this introduction) to classical modular cuspidal eigenforms f = E,°°,_1 anq" on ro(pN) of weight k > 2, with Fourier coefficients in CP, and normalized so that a1 = 1. By "eigenform" let us mean eigenform for the Hecke operators Tt for primes $ not dividing pN, for the Atkin-Lehner operators Uq for primes q dividing pN and for the diamond operators (d) for integers d prime to Np. By the slope of such an eigenform we mean the non-negative rational number or = ordp(ap), where ap is (both) the p-th Fourier coefficient of f and the Up-eigenvalue of f. Assume that the newform associated to f is either a newform for Fo(N) or ro(pN) (the latter case can occur only if Q = (k - 2)/2).
By the work of Hida (cf. [H-ET] for an exposition of this theory, and for further bibliography given there) one knows that any such eigenform f of weight k > 2 and slope 0 is a member of a p-adic analytic family f" of (overconvergent) p-adic modular eigenforms of slope 0 parameterized by their p-adic weights is (and, such that fk = f) for is ranging through a small p-adic neighborhood of k in (p-adic) weight space. Let us call this result p-adic analytic variation of slope 0 eigenforms. But Hida's results are, in fact, much more precise: If AN := ZP[[(Z/NZ)* x ZP]], Hida constructs a finite flat AN-algebra (let us call it T'P ,N) which is universal, in a certain sense, for slope 0 (overconvergent) eigenforms of tame level N, and such
that the associated rigid analytic space to To N (let us call it
is
the rigid-analytic space parameterizing p-adic analytic families of slope 0 eigenforms. If WN is weight space, i.e., the rigid analytic space associated to AN, then the p-adic families f,, alluded to above are obtained from the finite flat projection CC N -* WN, Hida having proved that this mapping is Reprinted from `Galois Representations in Arithmetic Algebraic Geometry', edited by A. J. Scholl & It. L. Taylor. ©Cambridge University Press 1998
R. Coleman & B. Mazur
2
etale at any classical modular point of weight > 2 in C° N.
The natural question arising from this work of Hida for eigenforms of slope 0 is to find the appropriate generalization of that theory valid for arbitrary finite slope eigenforms. The slope 0 theory has a significant simplifying advantage over the general theory in that there is available a clean idempotent operator (call it a°) projecting to the slope 0 part of the theory. This idempotent e°, a p-adic idempotent reminding one a bit of the "holomorphic projector" in classical analysis, is a bounded operator on the Banach spaces involved, and in fact it has no denominators and therefore acts as an idempotent on all aspects of the theory (e.g., parabolic cohomology, as well as spaces of modular forms); moreover the image of this idempotent is of finite type over whatever is the natural base ring.
In [C-BMF] a satisfactory analogue of Hida's p-adic analytic variation theorem for slope 0 eigenforms was established for finite slope classical eigenforms (at least for those satisfying a mild condition; cf. Cor. B5.7.1 of [C-BMF]).
The ultimate aim of the theory developed in this article is to provide a more global counterpart to the work done in [C-BMF] and construct a rigid analytic curve Cp,N (analogous to CP N) which parameterizes all finite slope overconvergent p-adic eigenforms of tame level N, and to study its detailed geometry: in particular, we wish to understand the nature of the projection of Cp,N to weight space WN.
For reasons of space, and time, we do this only in the case of p > 2, and tame level N = 1 in the present article. We do, however, treat noncuspidal eigenforms as well as eigenforms on I'1(pm), as opposed to the more restricted class of eigenforms delineated at the beginning of this introduction. For more precise, yet still introductory, statements concerning our main results, the reader might turn to section 1.3 (and in particular to the Theorems A,B,C,D,E,F,G formulated there). For the rest of this introduction, we suppose that p > 2, and discuss the case of tame level
N=1.
We construct a rigid analytic curve Cp = Cp,N=1 over Qp whose Cpvalued points parameterize all finite slope overconvergent p-adic eigenforms of tame level N = 1 with Fourier coefficients in C. We call Cp the (p-adic) eigencurve (of tame level N = 1). Hida's rigid space Cp = CC,N=1 which
parameterizes slope 0 eigenforms of tame level 1 occurs as a component part (cf. section 1.2) of our eigencurve Cp, but in contrast to Hida's theory, the natural projection of Cp to weight space is not of finite degree. The
The Eigencurve
3
eigencurve Cp has a natural embedding c H (pa,1/uc)
into the rigid analytic space Xp x Al where Xp is the rigid analytic space attached to the universal deformation ring Rp of certain Galois (pseudo-) representations and Al is the affine line. For a discussion of pseudorepresentations, see Chapter 5 below. The particular residual pseudo-representations for which Rp is the universal deformation ring we call p-modular residual representations (of the Galois group GQ,{p,,,.}). These are the residual pseudo-representations coming from the classical modular eigenforms of
finite slope and level a power of p (See section 5.1 for their definition.); there are only finitely many such residual pseudo-representations, and Rp is a complete semi-local noetherian ring. If c E Cp corresponds to the overconvergent eigenform ff, the first coordinate of c with respect to this embedding is the Galois pseudo-representa-
tion r. E Xp(Cp) attached to the eigenform ff and the second coordinate (1/uc) is the inverse of u, := the Up-eigenvalue of the eigenform ff. We define the eigencurve Cp C Xp x Al to be the rigid analytic subspace cut out by an ideal generated by certain specific rigid analytic functions, these functions being Fredholm series over Rp (cf. 1.2) obtained by pullback to Xp x Al of the characteristic series of certain completely continuous systems of operators (cf. 4.3). In general, a Fredholm series over a complete local
noetherian ring R is an entire power series (with constant term 1) in one variable T over R, and we call a Fredholm variety over R (see section 1.2 below) a rigid analytic subspace of Xp x A' which is cut out by an ideal generated by a collection of Fredholm series over R (here T is the variable parameterizing the affine line A'). Let us refer to the nilreduced rigid analytic space subjacent to the eigen-
curve as the reduced eigencurve CPed. For a brief discussion of the process of passing to the nilreduction of a rigid analytic space, see section 1.2 below. We define irreducible components there and prove under sufficiently general hypotheses that every point in such a space lies on one.
We show that each irreducible component of the reduced eigencurve Cped is isomorphic (at least outside of a discrete set of points) to some reduced irreducible Fredholm hypersurface over the Iwasawa ring A (i.e., it is a Fredholm variety defined by a single (irreducible) Fredholm series with coefficients in the Iwasawa algebra A) via an isomorphism that preserves projection to weight space. Each irreducible component of a Fredholm hypersurface over A is again a Fredholm hypersurface over A, and it is seen (Cor. 1.3.12 below) to either be isomorphic to the open complement of a
R. Coleman & B. Mazur
4
finite set of points in the rigid analytic space attached to a finite flat Aalgebra, or else to be of infinite degree. When (and if) the first case holds, we would be in a situation very analogous to Hida's theory for slope 0 eigen-
forms. But so far we have not yet been able to determine for any positive slope irreducible component of Cped which of these two alternatives hold! It follows from our results that the natural projection of each irreducible component of the reduced eigencurve to weight space is componentwise almost surjective in the sense that its image avoids at most a finite number of C. -valued points in the component of weight space within which it lands; in particular, each irreducible component of Cped has eigenforms of all (but a finite number of) weights. For a more precise statement, see Theorem B of 1.5.
It also follows from our results that any convergent eigenform of finite slope and integral weight (of tame level 1) is overconvergent if and only if its q-expansion is approximable p-adically by the q-expansions of classical eigenforms (of tame level 1). (Note that the q-expansions of Serre's p-adic modular forms are the limits of the q-expansions of classical modular (but not necessarily eigen) forms.) For a more precise statement, see Theorem G of 1.5.
In Chapter 7, we construct a second rigid space D, by means of the Banach module theory of [C-BMF], which is evidently a curve. We prove the above assertions (in particular, that it is a curve) about Cped by proving them about D and showing that D rv cared
From the its construction, one sees that the projection of D to weight space is locally in-the-domain finite flat, meaning that D is covered by admissible affinoid domains U such that the restriction of projection to weight space to U is a finite flat mapping of U onto its image in W.
The image of a rigid analytic morphism can be quite intricate, and, in particular, the projection Cped -4 Xp has an infinite number of double points (e.g., those coming from the classical modular eigenforms of level 1). The image of Cped in XP contains the "infinite ferns" studied in the articles [GM-FM], [GM 3] and [M-IF].
We may pull back the universal pseudo-representation on Xr to Cped via the natural projection, to obtain a rigid analytically varying pseudorepresentation on Cped which is realizable, at least on the complement of a certain discrete set of points on Cped as an 0cyea-linear continuous Gal (Q/Q)-representation unramified outside p with the property that the
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restriction of this representation to a Cp-valued point c E Cp yields the p-adic Galois representation attached to the eigenform ff.
Open questions. Is Cp reduced? (I.e., does Cp = D?) Is Cp smooth? Does Cped have a finite or an infinite number of components? Do any of these components have infinite genus? Is every component of Cped of slope > 0 of infinite degree over weight space, or are there components of positive
slope that are of finite degree over weight space? Do there exist p-adic analytic families of overconvergent eigenforms of finite slope parameterized by a punctured disc, and converging, at the puncture, to an overconvergent eigenform of infinite slope? (If so, one would want to complete the eigencurve a bit by including these missing points of infinite slope.) Having constructed
the (global) eigencurve allows us to ask more global questions along the lines of conjectures made by one of the authors of the present article, with Fernando Gouvea, and bears on recent work of Daqing Wan [Wa]: Are the ramification points of the natural projection from Cp to weight space infinite in number? We would like to know where those ramification points are; specifically given a point c E Cp of weight ic E W let us say that an affinoid subdomain U C W containing ic is a weight-parameter space for c if there is an affinoid neighborhood V C Cp of c whose natural projection to U is finite etale. Can one find, given any point c corresponding to a classical eigenform of weight k of slope strictly less than k - 1, a weight-parameter
space for c of radius greater than the inverse of a linear function of the slope of c? One can prove, using Wan's results [Wa], that there are such weight-parameter spaces of radius greater than the inverse of a quadratic function of the slope if the slope is strictly less than k - 1 and not equal to (k - 1)/2. We mentioned above that, excluding a discrete set of points (call this set A C Cped), the reduced eigencurve parametrizes a rigid-analytically varying family of Galois representations, i.e. we have a continuous representation
p : Gal (Q/Q) -* GL2(Oged_A) Can this family of Galois representations p be extended over the excluded set A? Is there a continuous representation of Gal (Q/Q) into the group of units in a rigid Azumaya algebra of rank 4 over Oc,, which, in the appropriate sense, extends p? Can one construct p (or at least p restricted to an appropriate Cped - A) from the cohomology of modular curves in the following sense: Let 9L be the polynomial algebra over A in the countable set of generators denoted Te for prime numbers t # p and an extra generator denoted Up. For a more complete discussion of this A-algebra ?-l see Chapter 6. There is a natural homomorphism of ?d to the ring Oc, of rigid
6
R. Coleman & B. Mazur
analytic functions on the eigencurve; in particular, in the discussion below we view Oct, as f-algebra. Define the discrete abelian group
M:= limH'(X1(p'); Qp/Zp), the direct limit of etale cohomology groups of the modular curves X1(pf), this limit being compiled via the mappings on cohomology induced from
the natural projections Xi(pn+l) -> X1(p') for all n > 1 (in the present brief description we omit saying which natural projections these are). The Galois group GQ,{p,,,.} acts on this direct limit in the natural way, as does the algebra f (each of the generators acting as the corresponding Hecke or Atkin-Lehner operator) and the actions of ?-l and Gq,{p,,,} commute, allowing us to view M as an f[[Gq,{p,,,,,}]]-module. Let M* := Hom(M; Qp/Zp) be the pontrjagin dual of M, viewed as compact f[[GQ,{p,,,,}]]-module. There are various (possibly equivalent) ways of spreading the cohomology f-module M* over the eigencurve. The simplest way to do this is to form V := the completed tensor product of the f-module M * with the f-algebra OcP. Viewing V as quasi-coherent sheaf over the eigencurve, the Gq,{p,_}action on Nl*, which commutes with the action of ?-l, induces a Oc,,-linear action on V. Is it the case that, over Cp, or possibly just over some large portion of Cp, the quasi-coherent sheaf V is locally free of rank 2, and the GQ,{p,,,}-representation on it is equivalent to the family of representations p discussed above? On the converse side it would be good to find an intrinsic operation which cuts out of M* precisely those Galois representations which are attached to overconvergent eigenforms. All these questions have analogues when the cohomology group
M := limH1(X1(p');Qp/Zp) is replaced by (direct limits of) parabolic cohomology of higher weights, and one might ask for a theory, following Hida's work, which deals with all weights. Relevant to this are the results of Hida [H-HA] and Gouvea [G-ApM].
Does Glenn Stevens' construction of the p-adic L-function of a (p-adic) eigenform on ro (pN) (cf. [St] ) work well over the eigencurve, and, in particular, does it give an L-function (with the coefficients of its Taylor expansion rigid analytic functions on Cp) which interpolates all the classical p-adic Lfunctions? We hope to show in a later publication, that at least a weak version of this is true. The locus of zeroes of this L-function, once constructed,
would itself be a rigid-analytic curve (call it Gp) admitting a natural rigid analytic projection to Cp. This deserves study. particularly intriguing is the nature of this projection in the neighborhood of a double-zero of the L-function.
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It would be interesting to construct a local version of the eigencurve using the theory of crystalline Galois representations, in the following sense. Let X now refer to the rigid-analytic universal deformation space of a fixed absolutely irreducible representation of the Galois group of Qp into GL2(Fr).
It is known that X is a smooth 5-dimensional rigid-analytic ball. By a crystalline point in (the 6-dimensional) rigid-analytic space X x A' - {0},
let us mean a pair (x, u) where x E X classifies a crystalline representation, and u E Al - {0} is one of its the Frobenius eigenvalues in the Fontaine-Dieudonne module attached to the crystalline representation x. (For a general overview of crystalline, semi-stable, and potentially semistable representations, see [F].) Is it the case that the rigid-analytic closure
of the set of crystalline points in X x A' - {0} is a 3-dimensional rigidanalytic subvariety ? Is it a rigid analytic subspace of affine three-space?
There are also certain foundational questions which deserve much better understanding than we have, at present. For example, one still lacks a satisfactory conceptual definition of what it means for a modular form (of general p-adic weight) to be overconvergent. This notion is clear for integral weight, but at present, for general weight, we have only an ad hoc procedure based on the availability of families of Eisenstein series of general weight (See Chapter 2 below: you multiply your modular form by such an
Eisenstein series to get the product to be of weight 0 and then ask that this be overconvergent as a rigid analytic function on a suitable affinoid in the modular curve). As a consequence of this awkward strategy, treatment of any serious property about overconvergent modular forms depends upon our detailed understanding of the family of Eisenstein series. It would be good to a. have a more direct definition of overconvergent modular forms and of families of overconvergent modular forms, and at the same time b. have a closer understanding of the properties of families of Eisenstein series (including knowledge of their zero-free regions).
An improvement in our state of knowledge of b would help in our understanding of the detailed geometry of the eigencurve. It would be important, as well, to have some detailed computations of specific affinoid subdomains of CP. In this regard, see forthcoming work of Matthew Emerton who (augmenting earlier calculations of Coleman and Teitelbaum) gives a complete description of the geometry of that part of the eigencurve for p = 2 and tame level 1 having minimal slope for their weight (cf. [Em]). See also [CTS] where some results on the low slope part of the 3adic eigencurve are proven. We might mention here that in these examples
(p = 2, 3), the slope tends to zero as one approaches the "boundary" of
R. Coleman & B. Mazur
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weight space. Are there components of the p-adic eigencurve (for some p) where this phenomenon does not occur? To be sure, a satisfactory general theory of the eigencurve must deal with
all tame levels N (and as a prelude for this, one necessary task, which we carry out, is to set up the deformation theory of pseudo-representations with fixed tame level). In the present paper, although we make the construction
of the eigencurve only for N = 1, in some sections (where it is easy to do so) we work with more general level N in preparation for the general theory. Our conventions concerning level will be signaled at the start of each section.
Is there an a priori deformation-theoretical approach to the eigencurve, and to the Galois representations that the eigencurve parameterizes? (We explain more precisely what we mean by this at the end of Chapter 1.) Our lack of such an approach accounts for some of the difficulty we have in analyzing local properties of the eigencurve. Is there a natural formal scheme over ZP whose associated rigid analytic space is the eigencurve? One is, in any event, guaranteed (by the preprint [LvP]) that any connected onedimensional, separated, rigid analytic space over QP is the generic fiber of some formal scheme which is flat over Zr,. It might be interesting to study irreducible components of the closed fiber of a formal scheme whose generic fiber is (a piece of) the eigencurve. In this connection, J. Teitelbaum has a computer program that produces such irreducible components in a range of slopes (for p = 2, 3).
In a later publication we hope to present more foundational material regarding the connection between Katz modular function and convergent eigenforms (including the proof of the compatibility of the action of the diamond operators, i.e., the proof of Prop. 3.4.2 below which we omitted from this article). We are deeply indebted to Kevin Buzzard for his extremely helpful comments throughout the preparation of this article. We also wish to thank Brian Conrad for his helpful suggestions on Chapter 1 and Matthew Emerton for his close reading of an early draft.
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Table of Contents.
1. Rigid analytic varieties .......................................
10
1.1 Rigid analytic spaces attached to complete local noetherian rings. 1.2 Irreducible components and component parts. 1.3 Fredholm varieties. 1.4 Weight space. 1.5 The eigencurve as the Fredholm closure of the classical modular locus. (Statement of the main theorems.) 2. Modular forms ................................................
35
2.1 Affinoid sub-domains of modular curves. 2.2 Eisenstein series. 2.3 Katz p-adic modular functions. 2.4 Convergent modular forms and Katz modular functions. 3. Hecke algebras ................................................
47
3.1 Hecke eigenvectors and generalized eigenvectors.
3.2 Action on Mk (Np', v; K). 3.3 Action on Katz Modular Functions. 3.4 Action on Mt(N). 3.5 Action on weight is forms. 3.6 Remarks about cusp forms and Eisenstein series.
4. Fredholm determinants .......................................
55
4.1 Completely continuous operators and Fredholm determinants. 4.2. Factoring characteristic series. 4.3. Analytic variation of the Fredholm determinant. 4.4. The Spectral Curves.
5. Galois representations and pseudo-representations
......
68
5.1. Deforming representations and pseudo-representations. 5.2. Pseudo-representations attached to Katz modular functions.
6. The eigencurve ................................................
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6.1 The definition of the eigencurve. 6.2. The points of the eigencurve are overconvergent eigenforms. 6.3. The projection of the eigencurve to the spectral curves. 6.4 The Eisenstein curve.
7. The eigencurve constructed as finite cover of spectral curves .........................................................
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7.1. Local pieces. 7.2. Gluing. 7.3. The relationships among the curves D,,. 7.4. D is reduced. 7.5. Equality of D and Cred 7.6. Consequences of the relationship between D and C. References
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Chapter 1. Rigid analytic varieties.
1.1 Rigid analytic spaces attached to complete local noetherian rings. Let p be a prime number, and CP the completion of a fixed algebraic closure of Qp,. Let v be the continuous valuation on Cp such that v(p) = 1. Define the absolute value I by Jal = p v(a), for a E C. By the ring of integers OK in a complete subfield K of Cp, we mean the set of elements of absolute value at most 1. For a rigid analytic variety Y (see §9.3 of [BGR]), A(Y) will denote the ring of rigid functions on Y, for a E A(Y), Jal will denote the spectral semi-norm of a (which may be infinite valued) and A°(Y) will denote the sub-OK-algebra of rigid functions with spectral semi-norm at most 1. For an affinoid algebra B, we let Max(B) denote the corresponding rigid variety over K. Now let R = OK be the ring of integers in a fixed finite extension K of Qp, in Cp. Let k denote the residue field of R. Let A be a complete local noetherian R-algebra with maximal ideal mA and residue field A/mA = I
k. Consider the functor A -* XA which attaches to each such complete noetherian local ring, its associated rigid analytic space over K. We refer to section 7 of [de J] for the construction, and for its basic properties (cf. loc. cit. Definition 7.1.3, 7.1.4a, and 7.1.5). Briefly, the construction may be given as follows: For each real number r > 0, let Ar be the p-adic completion of the quotient by its p-power torsion, of the ring A[{Yr,(a)}]
where (a) ranges over unordered tuples of elements in mA subject to the relations p[k(a)r)yr,(a) = fJ(a),
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where k(a) is the length of (a) and the product is over the entries of (a). (Note: A, = 0 if r is sufficiently large.) If r > s > 0 there is a natural Ahomomorphism from A3 to Ar determined by y, (a) H p[k(a)rl -[k(a)sl ys (a) If r is rational, this ring is finitely generated over A. Indeed, if {t,, ... , tn} generate mA and r = g/h with g, h positive integers, then A, is generated by y(a) where the entries of (a) are in {t,, ... , t,a} and the length of (a) is at most h, because in this case, mA C p9Ar. It follows that the admissible formal schemes, in the sense of [BL-FRI&II], SpfAr, where r is rational, glue together into an admissible formal scheme over R and we can make a rigid space XA out of this formal scheme as in [BL-FRI&II] covered by the affinoids X,.:= Max(Ar. 0 K). In a word, XA is the union of the affinoids
Xr's for all positive rational r. In particular, the Cr-valued points of XA are given by (continuous) homomorphisms of the R-algebra A into C1, (or, equivalently, into 0cp): XA(CP) = HomR(A, Cr),
and we have a natural ring homomorphism A -* A°(XA). To a complete Noetherian semi-local ring, we attach the union of the rigid spaces attached to each of its localizations at maximal ideals. An important special case
for us is A = ZP[[T]] (here R = Zr,, and K = QP). For r < 1, Ar is the ring of rigid functions defined over Q, and bounded by 1 on the disk B[0,p-r] = {x E C, I IxI < p-''} in CP, and XA is the open unit disk, B(0,1), viewed as rigid analytic space over Q. More generally, if one takes A to be the ring of rigid analytic functions bounded by 1 on a residue class (formal fiber) in an affinoid (see [Bo] ), XA will be canonically isomorphic to that residue class. All of the rigid spaces we shall be considering in this article are separated and have a countable admissible covering by affinoid subsets. Following the terminology of [JP] one says that a rigid space X over a complete non-archimedean valued field K is paracompact if it admits an admissible covering by affinoid subsets, any one of which meets only finitely many others. If a rigid space X over a complete non-archimedean valued field K admits an admissible covering by a countable increasing sequence
of affinoid subdomains X0 C X, c
C Xi c
C X, let us call X
nested. For any complete noetherian ring R, its associated rigid space XR described above is nested, as is XR x A', the product of XR with the affine line, which will be introduced in the next section. According to [LvP], any connected, separated, one-dimensional rigid space X over a complete nonarchimedean valued field K is paracompact (this is no longer the case in higher dimensions), and is the generic fiber of a formal scheme which is flat over Spf(OK), where OK is the ring of integers of K. As a general reference for the relationship between formal schemes and rigid analytic spaces, the reader might consult [L], [BL-FRI&II], [JP], [LvP], and pp. 429-436 of [Ra].
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1.2. Irreducible components and component parts. If t: S -+ Z is a morphism of rigid spaces, t is said to be a closed immer-
sion and Z is said to be a Zariski-closed analytic subvariety (via t) if there exists an admissible affinoid covering {U2}jEI of Z such that for all U2 is a closed immersion of affinoid i E I the induced map t: t-1(UZ) varieties [BGR §9.5.3]. It follows that for any open affinoid subvariety U of X, t: t-1(U) -4 U is a closed immersion of affinoid varieties [BGR Prop. 9.5.3/2]. If S is a Zariski-closed analytic subvariety of Z, we shall refer to S, for short, as Zariski-closed and note that (since affinoids are noetherian) the category of Zariski-closed subspaces of a given rigid space Z enjoys the local descending chain condition: any infinite descending chain of Zariskiclosed subspaces of Z stabilizes finitely when restricted to any given affinoid in Z. If t: S -a X is a closed analytic subvariety of X we let the interior of S be the rigid subspace which is the union of affinoids of the form t-1 (U) where U is an open affinoid subvariety of X and t: t-1(U) -+ U is an isomorphism. The rigid Zariski-closure of a morphism t: S -4 Z is the minimal element in the category of closed immersions W -* Z through which t factors. The fact that the rigid Zariski-closure of S exists follows from the fact
that if S -* Z factors through W -* Z and W' -+ Z it factors through W x Z W' -* Z, and from the local descending chain condition described above. If S C_ XA xK A' (C.), consider the rigid space over Cp which is the disjoint union of the points S (we denote this space by the same letter S). We define the the rigid Zariski closure of the set of points S to be the rigid Zariski closure of the morphism i : S -* X. If X is a rigid analytic space over K, by the nilreduction of X, denoted
Xred (cf. page 389 of [BGR]) we mean the unique rigid analytic subspace of
X over K (constructed in loc. cit.) whose admissible affinoid subdomains urea = Max(Oured) are in one:one correspondence with the admissible affinoid subdomains U = Max(Ou) of X, where Oured is the quotient of Ou by its nilradical. The natural closed immersion Xred C X induces a one:one correspondence on sets of Cp valued points. One says that X is nilreduced (or reduced) if Xred = X Definition. Suppose Z is a rigid space. Then by a component part of Z we mean a (non-empty) rigid subspace of Z which is the rigid Zariski-closure
in Z of an admissible open of Z. We will say that Z is irreducible if it has only one component part (i.e., itself). An irreducible component of Z is a component part which is itself irreducible. Notes. It may be worth explicitly pointing out that the definition of irreducible above ignores nilpotent elements, and in particular (as in standard algebraic geometric usage) our definition allows the possibility that irreducible components be nonreduced; moreover, embedded components are
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simply not counted. As we'll see, in Corollary 1.2.1/1, if Z is an affinoid, Z possesses only a finite number of distinct irreducible component parts, and any component part of Z may be viewed at least set-theoretically as a (finite, of course) union of irreducible component parts. Let fi(Z) denote the set of irreducible component parts of Z. One should note that 4D(-) is not in general functorial in Z (e.g., consider the case of Z = a single point mapping to the intersection of two irreducible components of a rigid analytic space Z'). Nevertheless, rigid analytic morphisms t : Z -+ Z' which embed Z as an open subdomain in Z' induce natural mappings 4D (t) : 4(Z) - D (Z') and we shall make use of this in what follows. Brian Conrad and Ofer Gabber have independently proposed another definition of irreducible components of X when X is reduced. One first passes to the normalization of X (which one has to prove is a well defined rigid space), takes connected components and then defines the irreducible components of X to be the images of these components. Using this definition one can show that irreducible components have the right localization properties. Conrad has established that, in this case, these irreducible components are the same as ours.
Lemma 1.2.1. Suppose X is an affinoid. Then X is irreducible if and only if the map A(X) -a A(X ),, is an injection for all closed points x of X. (Here A(X)X denotes the localization of A(X) at x.)
Proof. Suppose 0 # f E A(X) such that f H 0 in A(X )X for some x E MaxA(X). Then there exists an admissible open neighborhood U of x in X such that f H 0 in A(U). It follows that f maps to 0 in the ring of functions on the Zariski-closure Z of U and hence Z # X. Now suppose that A(X) injects into A(X)E for all x and Z is a component part of X equal to the Zariski-closure of an admissible open U of X. Then by Proposition 9.5.3/1 of [BGR] Z is an affinoid of the form MaxA(X)/I for some ideal I of A(X). It follows that I maps to zero in A(X)x for any closed point of U
andsol=OandZ=X.
Corollary 1.2.1/1. (i) If X is an affinoid, the irreducible components of X are afinoids ZI := MaxA(X)/I where I is a maximal ideal among the set S of ideals which map to zero in A(X),, for some closed point x. (ii) (In particular:) If A(X) is an integral domain X is irreducible. (iii) The number of irreducible components X is equal the number of minimal prime ideals of A(X).
Proof. (i) It follows from the previous lemma that if I satisfies the above hypotheses then ZI is irreducible. Conversely, suppose ZI is an irreducible component part of X for some ideal I of A(X) and U is an admissible open of X contained in ZI. Then I maps to zero in A(X)E for all closed points
x of U so I E S. If I C J where J E S, then Zj would be a component
R. Coleman & B. Mazur
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part of ZI equal to ZI if and only if I = J. (ii) follows from (i). (iii) It is clear that if I E S then I is contained in some minimal prime ideal of A(X). Now suppose P is a minimal prime ideal of A(X). Let Ip be the set of f E A(X) such that f 0 in A(X)X for some closed point x of Zp. Then Ip is the ideal of all f E A(X) such that there exists a t ¢ P such that t f = 0. The ideal Ip is contained in P and contains a power of P. It follows that Ip is not contained in any other minimal prime ideal. From its definition we see that Ip is a maximal element of S contained in P. It follows that Ip is a maximal element of S and the map P -* ZIP is a one to one correspondence between the minimal prime ideals of A(X) and the irreducible components of X.
We say an affinoid X is equidimensional if the Krull dimension of OX,x, where x is a closed point of X is independent of x. If this happens, this number equals the Krull dimension of A(X). Lemma 1.2.2. Suppose X is a reduced equidimensional affinoid. Then the singular locus of X has codimension at least 1 in X. Proof. The proof of this follows the lines of the classical proof in algebraic geometry (see [Ha] Chapter 1, section 5). First, we can express A(X) as the quotient of K (T1i . . . , T11) by an ideal generated by elements fl, . . . , fm
for some m and n. If X has dimension r, the singular locus of X is the set of points where the rank of the corresponding Jacobian matrix has rank < n - r so is a Zariski closed subaffinoid. To show it has codimension at least 1 one show that the non-singular locus on each irreducible component Z of X is non-empty. By rigid Noether normalization, Corollary 6.1.3/2 of [BGR], we can find a find a finite map 0 from Z onto B''. Then the singular locus of Z is contained in the set of ramification points of 0. Let F be the fraction field of A := A(B''). Then since A(Z) is an integral domain and Z --+ Br is finite and surjective, A(Z) injects into A(Z) OA F which is a finite extension of F. It follows that ZU is flat over the complement U of the zeroes of a non-zero element in A(Br) and hence the discimininant ideal D in A(U) of ZU/U is defined and non-zero. Moreover, the ramification points of 0 in ZU lie over the zeroes of D. Thus the non-sinular locus of Z maps onto a non-empty open of Br.
Proposition 1.2.3. Suppose X is a reduced affinoid of equidimension n. If I is an ideal of A(X) and V = Max {A(X)/I}, then V is a component part of X if and only if V is reduced and the Krull dimension of OV,,, equals n for all closed points of V.
Proof. If V is a component part, I = n.IES J where S is a set of minimal prime ideals. It follows that V is reduced and the Krull dimension of OV,,,
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equals n for all closed points v of V. Suppose V satisfies the latter hypotheses. Let Y be the maximal component part of X contained in V. Suppose Y : V. Let J be the ideal corresponding to Y. Since V is reduced it follows
that there exist an f E J and a vo E V such that f doesn't vanish at vo. We can assume f (vo) = 1. Let B be the subdomain of V defined by {v E V: I f(v) -11 < Cpl}.
Then B fl Y = 0. Now If v is point of V which is smooth in X then V contains an affinoid neighborhood of v in X and hence V contains the component part of X which is the Zariski closure of this neighborhood. Hence B is contained in the singular locus of X. Since this is a subaffinoid of codimension at least one, it follows that the Krull dimension of V at vo which equals the Krull dimension of B at vo is at most dimension n - 1. 1
Proposition 1.2.4. Suppose X is a reduced affinoid and Y is an affinoid subdomain. Then if Z is a component part of X, Z fl Y is a component part of Y. Proof. This follows in the equidimensional case, from the fact that Y is reduced by Corollary 7.3.2/8 of [BGR] and Prop. 1.2.3. The general case of the lemma follows from the equidimensional case and the fact that X is the union of equidimensional component parts. Indeed, if Z is a component part of X and V is the equidimensional part of X of dimension n, the part
of Z of equidimension n is a component part of V and we-can apply the equidimensional part of the proposition to V, Y fl V and Z fl V. Ofer Gabber, using the normalization definition of irreducible component mentioned above has proved that arbitrary rigid spaces can be covered by irreducible components. However, the following result is sufficient for our purposes:
Proposition 1.2.5. Suppose X is a reduced rigid analytic space which has an admissible open cover by of affinoilts {Xi: i E S2}, where S2 is some index
set such that Xi.7 := Xi fl Xj is an affinoid for each i and j. Then every closed point of X lies on an irreducible component of X.
Proof. This proposition is true if X itself is a reduced affinoid (e.g., as can be seen from Cor. 1.2.1/1). In particular (fixing on some index 0 E Sl) Proposition 1.2.5 is true for the affinoid subdomain Xo C X. It will then follow from
Lemma 1.2.6. Let AO be an irreducible component part of X0. Then AO is contained in a (unique) irreducible component of X. Proof. We will define, inductively, a collection of component parts, A,,,i of the affinoid subdomain Xi for pairs of integers i > 0 and n > 0 . We start
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R. Coleman & B. Mazur
the induction with A0,0 := A0 which is an irreducible component part of
X0 and A0,i = 0 for i # 0. Suppose, for a given integer n > 0 we have defined a component part An,i of Xi for all i E Z. We proceed to define the component parts An+1,i of X. Put Yn,j,k := Xj,k n An,k, which is a component part of Xj,k by Proposition 1.2.4. Let An+1,j be the union over k of the Zariski closures in Xj of Yn,j,k (there are only a finite number of these, as each of them is a component part of Xj). For each j these An, j stabilize for large n since they are all component parts and An+1,i 2 A. Let Z be the union of the An,i over all n and i. Then (using Prop. 1.2.4) Z is a rigid subspace of X and a component part by definition. We claim that Z is irreducible. Suppose U is any non-empty affinoid subdomain contained
in Z. We will show that if U is the rigid Zariski closure of U in X, then U = Z. We may suppose that U C An,i for some n and i. Let V be the rigid Zariski closure in Xi of U. Then V C An,i. It follows by construction
that if n > 0 there exists a j so that An-1j n V contains a non-empty affinoid subdomain of Xi. Since this is also contained in Z we can repeat this process inductively and conclude that U contains a non-empty affinoid subdomain of A0j for some j > 0. Therefore U contains A0,0 = A0. In particular, U contains the union of the A0,,; for all j > 0. Now, working up the inductive definition of An,i over all n, one sees that Yn,j,k C U for all n, j, k and consequently, U = Z.
1.3. Fredholm varieties. If A is a topological ring, recall that A(T) C A[[T]] is the set of power series whose coefficients tend to 0 in A. Let R = OK C K where K is a complete subfield of Ci,, as in the previous section. For any integer m, the closed disk
of radius p"° about the origin; that is, the rigid analytic subspace of AK whose CP valued points consist of {x E Cp I Jxj < pm} will be denoted, as usual, B[0, pm']K = Max(OK(pm'T)). We have A' = U'=1B is a rigid space over K and F(T) is a power series with coefficients in A(X) we will say F(T) is entire over X if it is the series associated to a rigid and
analytic function on X xK A}f. If X is a reduced affinoid, an equivalent formulation is: F(T) is entire over X if and only if the power series F(T) lies in (the intersection of) the subrings A(X)(pmT) C A(X)[[T]] for all natural numbers m. This is so because X xK A}{ = U,,=1 X xK B[0, p"'']K. The set of entire power series over X, then, forms an A(X)-subalgebra of the ring of power series A(X)[[T]], denoted A(X){{T}} C A[[T]].
If A is a local ring with maximal ideal mA, and P(T) _ E°O_1 anT" is a power series with coefficients an E A, we will say that P(T) is entire over the ring A if the coefficient an is in MI n' where cn are integers such than the sequence of real numbers {cn/n} tends to oo. The set of entire power series (in T) over a local ring A forms an A-subalgebra of the ring of
The Eigencurve
17
power series A[[T]] which we denote A{{T}} C A[[T]]. For example, if A is a field, then A{{T}} = A[T], the ring of polynomials in T over A. If A is a complete noetherian local R-algebra with finite residue field, where R is the
ring of integers in a finite extension K of Qp, and if XA is the rigid space over K associated to A, then we have a natural A-algebra homomorphism
A{{T}} -> A(XA){{T}}. Definition. If A is a local ring, a Fredholm series over A is an entire power series over A with constant term 1. If X is a rigid space, a Fredholm series over X is an entire power series over A(X) with constant term 1. By
the Fredholm function associated to a Fredholm series P(T) over A(X) we mean the rigid analytic function on X x Al determined by P(T) (i.e., the image of P(T) under the natural homomorphism A{{T}} -4 A(XA){{T}}). If A is a complete noetherian local R-algebra with finite residue field, where R is the ring of integers in a finite extension K of Qp, and if XA is the rigid space over K associated to A, an A-Fredholm hypersurface in XA x KAK is the rigid analytic subvariety cut out by a Fredholm series over A. An AFredholm subvariety in XA x Al is a rigid analytic subvariety of XA x Al cut out by a non-empty (possibly infinite) collection of A-Fredholm series. If S is a subset of the CP valued points of (XA x A') (Cr), by the Fredholm closure of S we mean the smallest Fredholm subvariety of XA x A' whose Cp-valued points contain S. Note. We extend the above notions to complete semi-local noetherian rings A in the evident manner. For example, if A is a complete semi-local noetherian ring, a power series P(T) E A[[T]] is a Fredholm series over A if the image of P(T) in Am[[T]] is a Fredholm series, where m ranges through the maximal ideals of A, and Am is the completion of A at m. Fredholm subvarieties are contained in the complement of XA X K (0, 1) in XA X K AK{. Inversion of the second coordinate in XA x Al therefore sends Fredholm sub-
varieties to closed, rigid analytic subvarieties of XA xK B[0,1]\{0}, where B[0,1]\{0} is the punctured closed unit disk. The rigid Zariski-closure of a set of CP valued points S C XA xK AK(Cr) is a reduced rigid analytic space and therefore the rigid Zariski-closure of S factors through the nilre-
duction of the the Fredholm closure of S. We do not, at present, know whether it is generally true that the rigid Zariski-closure of S is equal to this nilreduction.
Lemma 1.3.2. Suppose X is a reduced affinoid. a. The only invertible Fredholm series H(T) over X is the constant series H(T) - 1.
R. Coleman & B. Mazur
18
b. For any positive integer n, put Yn = X xK B[0, pn] C Y = X xK AK. Let G(T), F(T) be two Fredholm series over X with the property that for each n, if Gn(T), F,, (T) denote the restrictions of these Fredholm series to Yn, we have that Gn (T) divides Fn (T) in the ring of rigid analytic functions
on Yn. Then there is a Fledholm series H(T) over X such that
F(T) = G(T)H(T). Proof. a. We may assume X 0. Let H(T) = 1 + a1T + + anT + be an entire series over A and suppose there exists a k > 0 such that ak # 0. Then since X is reduced there exists a closed point x of X (L) for some finite
extension L of K such that ak(x) 0 0. Thus we can assume that {x} = X and so A(X) = L in which case the lemma is obvious. b. For each n > 1 let H,,(T) denote the rigid analytic function on Yn
such that Fn(T) = G(T)H(T), and note that Hn(T) is uniquely determined by this property since Gn(T), having constant term 1, cannot be a zero divisor. This system of functions {Hn}n is compatible under restriction
and patches together to give us a Fredholm series H(T) with the required properties.
Lemma 1.3.3. Suppose X is a reduced affinoid over K. For any positive integer n, put Yn = X xKB[0, pn], as before. Let Z -+ X XKAK be a closed immersion such that for each n its intersection with Yn, Zn := Z XK Yn, is the affinoid cut out by some rigid analytic function gn(T) E A(X) (pnT) on Yn with gn(0) = 1. Then there is a Fredholm series G(T) over X such that
for n> 1, lim gni(T)IY = G(T)IYn, n < nl-roo
where Z is the subvariety cut out by G(T). Proof. Our hypotheses imply that we may write 9m+i (T) I ym = 9m (T) hm (T)
where hm(T) E A(Y,,,,)* and hm(0) = 1. Write h,n (T) _ >00am,kTk k-o
where a.,,,,,k E A(X). The fact that h.,,,.(T) is a unit on Y,,,, and h(0) = 1 implies
Iam,kl < p-,k
for
k > 0.
The Eigencurve
19
It follows, since X is reduced and so the spectral norm determines the topology on A(X), that the product
11 hm(T)
m>n
converges to a unit on Yn for all n > 1. This concludes the proof.
Definition. We say that an affinoid algebra A over K is relatively factorial if every F(T) E A(T) with constant term 1 (i.e., F(O) = 1) can be factored into a (finite) product of primes (i.e., elements which generate prime ideals). We call an affinoid X over K relatively factorial if the affinoid algebra A(X) is. It follows that such a factorization is unique up to units. (The point is that no element in A(T) with constant term 1 is a zero divisor.) The Tate algebra ( T 1 , . . .. , Tn) is an example of a relatively factorial (in fact, factorial) algebra by Theorem 5.2.6/1 of [BGR]. Proposition 1.3.4. Suppose X is a relatively factorial affinoid over K such that A(X) is an integral domain and F(T) is a Fredholm series over A(X) which cuts out the (Fredholm) hypersurface Z. Then every component part W of Z is a Fredholm hypersurface whose defining equation is a divisor of F(T) in the ring A(X){{T}}.
Proof. Let V = X X K AK and, as before, for positive integers n let Y,, = X xK B[0,pn] and Zn = Z xv Yn. Suppose U is an admissible open in Z whose rigid Zariski-closure is W. Let Cn be the Zariski-closure of U xv Yn in Yn.
Because X is relatively factorial, we can write F(T)IY,. =
11
sz(T)
ZEP(Z,.)
where the product is taken over all the irreducible component parts Z of Zn, and sz (T) is a rigid analytic function on Yn (with the property that sz (0) = 1) defining the irreducible component part Z. Because U xVYY is an admissible open of Zn = Z xVYn and A(X) is an integral domain, there of irreducible reduced affinoid subdomains of Yn exists a collection such that U xv Vn,,n is the affinoid cut out by F(T) restricted to Vn,m and the collection {U xv Vn,,,,,} is an admissible open cover of U xv Yn (for m ranging through an appropriate index set, call it M).
Put hn,m(T) _ fl sz(T), Z
R. Coleman & B. Mazur
20
where the product is taken over those irreducible component parts Z E (D(Z,,) which are in the image of the natural mapping 4(U XV Vn,.n) - -D(Z,,),
there being such a natural mapping (as discussed in the note in section 1.2) since U x v Vn,,,,, is a subdomain of Z. Then U x v V,,,,,, is the affinoid in Vn,,,,. cut out by the restriction of hn,,n(T) to Vn,,,,,. An easy argument gives that the map A(Yn)/hn,,n(T)A(Yn) to A(Vn,,,,,)/hn,,n(T)A(Vn,,n) is an injection from which it follows that the Zariski-closure of U x v Vn,,n in Y, is the affinoid cut out by hn,n,,(T). Therefore Cn is the affinoid in Yn cut out by kn(T)
fjsz(T),
z
where the product is now taken over those irreducible component parts Z E (D(Z,,) which are in the union of the images of the natural mappings 4D(U xv Vn,,,,,) - 4)(Zn), for m ranging through the full index set M. So, for n' > n, the restriction of kni (T) to Yn may be expressed as follows,
kn'(T)IY = fJSz(T), where Z runs through those elements of which have the property that they are contained in an element of (D(Zn,) which is in the image of (D(U XV Vn',m') - - (Zn')
for some m'. Since these collections of irreducible component parts of Zn are increasing as n' grows and contained the finite set 4)(Zn) they stabilize for large n'. Putting fin (T) := knI (T)Iyn for n' large enough so that the resulting function fin (T) is unchanged if we augment n' any further, we see that the rigid analytic function gn (T) defines the subspace W x v Yn C Zn and (of
course) divides Fn(T) := F(T)1Yn. By Lemma 1.3.3, there is a Fredholm series G(T) over X which defines W. By Lemma 1.3.2 (b) we have that G(T) divides F(T), i.e., we have a factorization F(T) = G(T)H(T) where H(T) is a (well-defined) Fredholm series over X. We note that it follows from the proof of this proposition that UnCn = W and
W xvYn= U Cn' xvYn. n'>n
Corollary 1.3.5. With notation as above, if Z is the rigid Zariski-closure in V of the afEnoid B := Bz cut out by sz(T) in Yn, then Z is irreducible. Proof. Suppose Z is not irreducible. Then it follows from the proposition that F(T) = G(T)H(T) for two non-trivial entire series G(T) and H(T)
The Eigencurve
21
such that sz(T) divides the restriction of one and only one of G(T) or H(T) to Y,,. It follows that the rigid Zariski-closure of B is contained in the component part of Z corresponding to G(T) or H(T) which is a contradiction.
Proposition 1.3.6. Suppose X is a relatively factorial affinoid over K such that A(X) is an integral domain and F(T) is an entire series over X such that F(0) = 1 and the hypersurface Z cut out by F(T) is irreducible. Then there exists a prime element G(T) in A(X){{T}} such that G(T)m = F(T) and G(0) = 1 for some positive integer m.
Proof. Set V = X x A1. Let W be the induced reduced affinoid corresponding to Z xv Y,,. If rn
H k=1
7rn,k(T)Mn,k
is the prime factorization of F(T)lyy. ((7rn,k): (7rn,j) if k # j) then Wn is the affinoid determined by rn
7rn,k(T). k=1
Now, if f < n, the affinoid determined by the restriction of 7rn,k(T) to Yt is a product of the 7rej(T) to some exponents but, by Corollary 7.3.2/10 of [BGR], it is reduced since the fiber product of the affinoid B determined by 7rn,k(T) in Yn and YE is an affinoid subdomain of B. Hence all these exponents are at most 1. Since (by the dimension argument used above) every 7rtj (T) divides the restriction of one and only one of the 7r,,,k (T) we see
that the fiber product of Wn and Y1 equals Wt. It follows that W := UnWn is a Zariski closed subspace of V that satisfies the conditions of Lemma 1.3.3 and hence is the hypersurface cut out by a Fredholm series G(T). Moreover, for each n we know there exists an element Hn(T) E A(Y,,) such that
G(T)HH(T) = F(T). By Lemma 1.3.2.b, these Hn(T) glue together into a Fredholm series H(T) such that F(T) = G(T)H(T).
Let U be the closed subvariety of Z determined by H(T). On Yn it is the affinoid determined by rn
11 7rn,k (T)mn,k-1 k=1
Suppose there exists n and k such that mn,k - 1 # 0, then since the rigid Zariski-closure in U of the affinoid cut out by 7rn,k in Yn is an irreducible
R. Coleman & B. Mazur
22
component of W, and since (we claim) W is irreducible, it must be equal to W and hence G(T) divides H(T). Continuing in this way we conclude F(T) = G(T)m where m is equal to any of the non-zero mn,k. Proof of claim. Suppose that W is not irreducible. Then we see that G(T) has a non-trivial factorization into Fredholm series. G(T) = R(T)S(T). It follows that there exist integers n and k, j such that 1 < k, j < n, 1rn,k(T)
divides R(T)Iy , and 7rn,j(T) does not. In particular, it follows that the rigid Zariski-closure of the affinoid in Yn determined by 7rn,k(T)mn,k is an irreducible component of the hypersurface cut out by R(T)mn,k but the rigid Zariski-closure of the affinoid determined by 7rn,j(T)mnj is not. Hence the rigid Zariski-closure of the first affinoid is not Z, a contradiction. Since W is both reduced and irreducible, it follows that G(T) is prime.
Theorem 1.3.7. Suppose X is a relatively factorial afhnoid over K such that A(X) is an integral domain. Then, if F(T) is an entire series over X such that F(O) = 1, there exist distinct prime entire series P2(T) and positive integers n2 such that Pi (0) = 1 and F(T) = P1(T)nl . P2(T)nz ...Pk(7')nk ... Moreover, this factorization is unique up to order. Proof. Let notation be as in Proposition 1.3.6. For each n and 1 < k < n, we know from Corollary 1.3.5 that the rigid Zariski-closure in Z of the affinoid Bn,k in Yn cut out by 7rn,k(T) is an irreducible component of Z, Zn,k, which after Propositions 1.3.4 and 1.3.6 is cut out by the mn,k-th power of a prime entire series Pn,k(T) such that Pn,k(0) = 1 for some positive integer mn,k.
T o hide these unsightly double indices, let .t(T) (for 'y = 1, 2, 3, ...) be the list of these Pn,k's, putting the indices (n, k) in lexographic order; and correspondingly, denote mn,k by my. Then the product 11 P7(T)
y>1
converges by the argument in Lemma 1.3.3 since P7' (T) is a unit on Yn for large y and P7"(T)(0) = 1. The product converges to F(T) by Lemma 1.3.2. This yields the existence of the factorization. Now suppose Q(T) and H(T) are entire series with constant term 1 in A(X){{T}} such that Q(T) is prime and Q(T)H(T) = F(T). Then for every 'y, it follows that .y(T) divides H(T) or Q(T). One of them, say Pk(T), must divide Q(T)
since Q(T) is not a unit and A(X){{T}} is an integral domain. Hence
The Eigencurve
23
Q(T) = P,,(T) since both series are prime and there are no non-trivial units with constant term 1 in A(X){{T}} by Lemma 1.3.2. From this, we may deduce that the factorization of F(T) is unique. In the following, if U C X is an affinoid subdomain, and Z -> X a rigid space (over X) we let ZU = Z x x U denote the fiber product.
Lemma 1.3.8. Suppose X is a relatively factorial affinoid such that A(X) is an integral domain. Let F(T) = G(T)H(T) be an equation in Fredholm
series over X such that G(T) 0 1. Suppose Z(G) C Y = X x A', the hypersurface cut out by G(T) is a non-empty component part of Z(F) C Y, the hypersurface cut out by F(T). Then if U is a non-empty affinoid subdomain of X, Z(G)u is a non-empty component part of Z(F)U.
Proof. The rigid space Z(G)U is non-empty, for if it were empty then all the coefficients of G would have to vanish on U C X but then Lemma 1.3.1 implies G(T) = 1. The only way for Z(G)u not to be a component part of Z(F)u is if there were an irreducible component V C Z(G) C Y such that the following holds. If Yn = X x B[0, pn], Vn = V xy Yn. But from Lemma 1.3.1, it follows that H(T) vanishes identically when restricted to Vnred for all n contradicting the assertion that Z(G) is a component part of Z(F).
Lemma 1.3.9. Suppose X is an affinoid over K such that A(X) is an integral domain, L is a finite Galois extension of K with Galois group r and P(T) is a prime entire series over XL := X x L such that P(O) = 1. Then if P,(T), P2 (T), . . . , P, (T) is the orbit of P(T) under the action of r,
Q(T) := 11Pi(T) is a prime entire series over X.
Proof. Suppose there exist entire series F(T) and G(T) over X such that Q(T) divides F(T)G(T) in A(X){{T}}. Then in A(XL){{T}}, P(T) divides F(T) or G(T). Suppose
P(T)H(T) = F(T). It follows that P°(T)H°(T) = F(T) for all o, c F. Since P°(T) doesn't divide P(T) unless it equals P(T) by Lemma 1.3.2, we conclude that there exists an entire series R(T) over XL such that
Q(T)R(T) = F(T). Since Q and F have coefficients in A(X) and A(XL){{T}}
is an integral domain, it follows that R(T) is an entire series over X.
R. Coleman & B. Mazur
24
Lemma 1.3.10. Suppose that A is a complete local R algebra as in §1.1 and A = A°(XA). Suppose F(T) E A{{T}} and we have a factorization
F(T) = G(T)H(T)
with G(T), H(T) E A(XA){{T}} and G(O) = H(O) = 1. Then G(T), H(T) E A{{T}}.
Proof. Suppose G(T) _ n>0 anTn and there exists an n such that a 0 A. Then, since A = A°(XA), there exists an x E XA(K) such that Ian(x)I > 1. Denoting G,,(T) := En>0an(x)Tn we see that GE(T) E K{{T}}, has a zero in k of absolute value strictly less than 1. The same must be true about FA(T) which is patently false. Hence, G(T) E A{{T}} and we reach the same conclusion for H(T). The proof of the next theorem follows the general format of the proof of Theorem 1.3.7.
Theorem 1.3.11. Let A = 0[[T,,..., Tm]] be a power series ring in m variables over 0, the ring of integers in a field K which is finite over Qp. Denote its associated rigid space over K by XA. Let F(T) be a non-trivial Fredholm series over A. Then there exist unique distinct prime Fredholm series over A, Pi(T), and positive integers ni such that
F(T) = P1(T)ni ...Pk(T)nk ... Proof. Recall the notation introduced in the construction of the rigid analytic space XA in section 1.1. In particular, for each rational number, 1 > r > 0, we have an affinoid subdomain of X,. C XA. We may identify XA with the open unit n dimensional polydisk v(ti) > 0 and X, with the affinoid subpolydisk defined over Qp, v(ti) > r. Let B,n := A(Xi/m) ®Qp K where K = Qp(pm, p1/m) Then B,n = K(X1, ... , X,) and so is relatively factorial. It follows from Theorem 1.3.7, Lemma 1.3.9 and Lemma 1.3.3 that there exists a countable set In such that F(T)I x,l. = 11 P,,i(T)1-n,i, iEI,
where P,,,i(T) E A(Xl/n){{T}} are prime entire series on X11, and the MO are positive integers. Now if n > .£, arguing as in the proof of Proposition 1.3.6 we see that for each i E It, there exists a unique j (n, .£, i) E In such that PP,i (T) divides Pnj (n,P,i) (T) I x,/, and it divides it exactly once. In fact, mt,i = mn,j(n,P,i) Let S(n, 2, j) be the subset of It such that Pn,j(T)Ixl1e
The Eigencurve
25
is divisible by Pe,i(T). Because of Lemma 1.3.8, this set is non-empty and because of Lemma 1.3.3, for j E In,
II
Pn,j(T)Ixj, =
PL,i(T)
iES(n,e,j)
Now let
Qe,i (T) _ Pn,j (n,e,i) (T ). This equals
II
Pe,k(T),
kET(e,i)
where T (.t, i) = UnS(n, .£, j (n, t, i)). If Jt is a subset of Ie such that U T (B, i) = Ie
and T (.t, i) fl T (B, j) = O
iEJe
ifi,jEJeii0j,then F(T) I x11 = 11
Qe,i(T)n`
iEJe
Now it follows from the definitions that Qn,AnAi) (T) I x11 = Qt, i (T ),
and hence for each i E It there exists a unique entire series Qi(T) over XA such that Qi (T) I x111 = Qe,i (T) and so
F(T) = fi Qi(T)m','. iEJe
It follows from Lemma 1.3.10 that Qi(T) E A{{T}} and it follows by an argument similar to that used in Theorem 1.3.7 that Q2(T) is prime and that the factorization is unique. ,
Comments about degree and the geometry of Fredholm hypersurfaces. Suppose, for concreteness, that A = O[[T1, ... , T,,,,]] is a power series ring in m variables over 0, the ring of integers in a field K which is finite over Q Let X = XA be the open unit ball of dimension m over K. Any Fredholm series over A (which is different from the constant function 1) is either a polynomial of (finite) degree d > 0 over A, P.
26
R. Coleman & B. Mazur
P(T) = 1 + a1T + a2T2 + + adTd, with ad nonzero, or else it is an entire power series P(T) = 1 + a1T + a2T2 + , where there are arbitrarily large integers n for which the corresponding coefficient an doesn't vanish. Consider the Fredholm hypersurface Z C X x Al whose defining equation is given by P(T) = 0. We view Z as rigid space over K. In the first case, i.e., when P(T) is a polynomial of degree d, consider the monic polynomial
P* (U) := UdP(1/U) = Ud + a,Ud-1 + a2Ud-2 + ... ad. Form the finite flat A-algebra E := A[U]/P*(U), and consider the quotient
algebra E -+ E := E/(U) = A/(ad). Both E and E are complete semilocal noetherian 0-algebras. Let XE and XE denote their respective rigid analytic spaces over K. We have a canonical immersion XE " XE, and the rigid space Z is the complement of XE in XE. The natural projection
XE -+ X is finite flat of degree d, and therefore the projection Z -* X is a mapping of generic degree d (with a decrease in the degree of fibers occurring over the zero-locus of the function ad on X). If the image of ad is a unit in A ®n K, then XE is empty, and Z = XE. In the case where P(T) has an infinite number of nonvanishing coefficients we shall say that the Fredholm hypersurface Z is of infinite degree over X; this is a natural terminology, for, in this case, there are infinitely many points in the fiber of Z above each of the points x lying outside a proper subspace of X. This discussion, together with Theorem 1.3.11 above, immediately gives:
Corollary 1.3.12. If Z is the Fredholm hypersurface defined by a Fredholm series over A, then each irreducible component Z,, of Z is either 1. a rigid space of the form XE \ XE where XE -* X is the rigid space associated to a finite flat A-algebra, and where XE C XE is the rigid space associated to a closed subscheme of Spec E which projects isomorphically onto a proper closed subscheme of Spec A. or else is 2. of infinite degree over X.
Corollary 1.3.13. When A is isomorphic to 0[[T]], any irreducible component Zo of a Fredholm hypersurface defined by a Fredholm series over A has the property that its image in X omits at most a finite number of points. Note that if A is the completion at any maximal ideal of the Iwasawa algebra AN of the next section, where cb(N) is prime to p, then A satisfies the hypotheses of Cor. 1.3.13.
27
The Eigencurve
1.4 Weight space. Let p be a prime number, and fix CP as in 1.1. Put q = p if p > 2 and q = 4 if p = 2. Let N be a positive integer prime to p. Put Zp N := lim (Z/Np'Z)*, and
AN = limZp[(Z/Np"Z)*] = Zp[[(Z/NZ)* x Z;]], the projective limits being taken as n tends to infinity. The ring AN (sometimes referred to as the Iwasawa algebra) is the completed topological group algebra with ZP coefficients of the profinite group Zp N. The group structure of the finite group (Z/Np"Z)* induces, in the standard way, a commutative (associative) Hopf algebra structure on the group ring Zp[(Z/Np"Z)*] (which one can view as the affine Hopf Algebra associated to the Cartier dual of the constant group scheme (Z/Npf1Z)* over Spec Zp). Passing to the limit as n tends to oo, we get a (completed) associative and commutative Hopf algebra structure AN -+ AN®ZpAN. We have a canonical factorization, Z*AN =
(Z/NqZ)* x (1 + qZp),
the first factor being identified with the torsion subgroup of ZP N and the second factor being the wild part of ZP N, i.e., the multiplicative group of elements congruent to 1 mod qN. If a E ZP N, we let [a] denote the corresponding element in AN and ((a)) to avoid confusion with the diamond operators the projection of a to 1 + qZP with respect to the above factorization. We have a corresponding factorization of the Iwasawa algebra, AN
Zp[(Z/NqZ)*] 0 Z,[[(1 + qZp)]],
where the second factor admits a (not very natural) isomorphism with a power series ring Zp[[t]] (in one variable t over Zp by sending [1 + q] E Zp[[(1 +gNZp)]] to 1 + t E Zp[[t]]. If the finite group (Z/NqZ)* is of order prime to p (equivalently, if p > 2 and no prime divisor of N is congruent to 1 mod p) then AN is a regular (noetherian) semi-local ring of Krull dimension two, and, conversely, if AN is regular, then N has the property just stated.
Denote by W = WN the rigid analytic space over Qp, XAN associated to the formal Spf (Zp)-scheme Spf (AN) as in §1.1. The completed Hopf algebra structure on AN endows WN with the structure of (one-parameter) commutative rigid analytic group where multiplication corresponds to the usual multiplication of characters. We refer to WN, with its structure as
R. Coleman & B. Mazur
28
rigid analytic group over QP, as weight space of tame level N. The CP-valued points of WN are the continuous CP valued characters on ZP,N. If N divides M, the natural projection Zp,M -> Zp,N induces an injective homomorphism WN y WM which allows us to identify the rigid analytic
group WN with an open and closed rigid analytic subgroup of WM. In particular, for any N we identify the connected component of the identity in the rigid analytic group WN with the connected component of the identity, call it 13, in W1, which can be identified, by restriction, with the continuous characters on 1+gZp. The group of connected components, i.e., the quotient group DN := WN/13 may be canonically identified with the Cp dual of the
finite group (Z/NqZ)* and since we have a natural restriction map from WN to B we may regard WN as DN x 13. The rigid analytic group B is isomorphic to the open disk of radius 1 about the identity element in Cp with its induced multiplicative group structure, and the Cr-valued points of B are identified with continuous homomorphisms Zp[[(1 + gNZp)]] Cp. Setting AM = Zp[[(1 + qZp)]], we have that AM°), the ring of rigid analytic functions over Qp and bounded by 1 on 13, is the classical Iwasawa algebra, and the standard completed Hopf algebra structure on AM induces the rigid
analytic group structure on B. If is E WN, we let (ic) E 8 C WN be the character a H ic(((a))). It is trivial on (Z/NqZ)*.
Definition. Ifp > 2 and {s E Cp: v(s) > -1+1/(p-1)}, let 77s : AN -> Cp denote the continuous character a H ((a))8, and denote by 13* the space of all these characters rls; i.e., 13* := {rls : a H ((a))3 1 s E Cp, v(s) > -1 + 1/(p - 1)}.
If p = 2, we let 13* :_ {r19 : a H ((a))3
1
s E Cp, v(s) > -1}.
By an accessible weight-character (of tame level N) we shall mean a Cp-valued point of WN of the form X71k where k E ZP and x is a character of finite order. We sometimes refer to such a point rc by the coordinates (x, k).
Let 1 be the trivial character and T E W be the the unique character which is trivial on (Z/NZ)* x (1 + qZp) and the identity on p(Zp). If k is an integer, the k-th power map (a H ak) on Zp N composed with projection to C* is an accessible weight-character in the sense just defined, and has coordinates (Tk, k). We refer to this homomorphism as the character of integral weight k (and trivial nebentypus). More generally, if e : Zr N -+ C* is any continuous character of finite order, the point e Tk Rk E WN (Cr)
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will be said to be an arithmetic point of weight k and nebentypus character e. Most of the rigid analytic spaces we will deal with in this article will come along with a natural (rigid analytic) morphism to WN (for some appropriate tame level N). We shall refer to this morphism by the phrase projection to weight space. All the rigid analytic morphisms we encounter will preserve projections to weight space.
1.5. The eigencurve as the Fredholm closure of the classical modular locus. (Statement of the main theorems) For a prime number p and N (a tame level) an integer prime to p , let S denote the finite set of places of Q consisting of the infinite place, p, and the prime divisors of N. Let GQ,S be the Galois group of the maximal algebraic extension of Q in C) which is unramified outside S. One would eventually
wish to treat all primes p and all tame levels N but in the present article we prove our main theorems only for p > 2 and tame level N = 1. For the rest of this section, then, we let p be an odd prime number and N = 1. In section 5.1 below we shall define a certain collection of residual GQ,s-
representations into finite fields of characteristic p which include, up to equivalence, (the semi-simplification of) all characteristic p residual Galois representations coming from classical modular forms of p-power level. These we call p-modular residual representations of GQ,S of tame level 1. We also recall in detail in 5.1 the notion of pseudo-representation, following [Wi], [T], [Ny], [Ro] and [H-NO]. Briefly, if D is a topological ring, to any continuous representation p : GQ,S -4 GL2(D)
H g
a(g) b(g) c(g) d(g)
which sends complex conjugation in GQ,S to the diagonal matrix
k1 01 0 -1
'
its associated 2-dimensional pseudo-representation r with values in D is taken to be the pair of (continuous) functions a, 6 : G -* D given by the formulas, a(g) = a(g); 5(g) = d(g), noting that a and 5, so defined, are invariants of the equivalence class of p. It is convenient to define, as well, the function : G x G -+ D by the formula (g, h) = b(g)c(h), and although the function is determined by either function a or 6 because e(g, h) = a(gh) - a(g)a(h) = 5(hg) - S(h)b(g),
R. Coleman & B. Mazur
30
we refer to the triple of functions r := a, 8), as the pseudo-representation associated to p. A (general) pseudo-representation is roughly a triple of functions satisfying the same relations as an arbitrary a, 8) obtained from a representation in the above way. For any 2-dimensional pseudorepresentation r = (, a, (3) on GQ,S with values in D and any g E GQ,S we set
trace(r(g)) = a(g) + /3(g)
and
det(r(g)) = a(g),3(g) - fi(g).
In 5.1 we shall consider the (complete noetherian, semi-local) ring Rp defined as the universal (p-complete, noetherian, semi-local) deformation ring for (degree two) modular pseudo-representations of GQ,S of tame level 1. Being universal, there is a pseudo-representation r"i' from GQ,S with values in Rp which specializes to any deformation of a p-modular pseudorepresentation. Here let us just note that Rp includes among its factor rings the universal Galois deformation rings of all absolutely irreducible modular residual representations (i.e., residual representations over finite fields of characteristic p) attached to newforms of levels dividing powers of p (see 5.1). As we show in section 5.1, Rp may viewed in a natural way as a A-algebra
(with its weight A-algebra structure).
Let XRp = X, be the rigid analytic space over Qp associated to the complete semi-local ring Rp regarded as a Zp-algebra, as in §1.1, endowed with the projection to weight space, ir : Xp -* W coming from the weight A-algebra structure of Rp. Using [Wi], one can see that any two-dimensional continuous representation p : GQ,s -4 GL2(Cp) of tame level 1 whose associated residual representation is p-modular corresponds to a (unique) point XP E Xp(Cp) (given by its associated pseudo-representation).
To a normalized (classical) modular eigenform f on F1(M) over Cp, where M is a positive multiple of p, of weight k and character X with qexpansion
f(q)
=co+q+c2g2+c3g3+...
one can attach an odd degree 2 representation p f over Cp (up to equivalence) whose associated pseudo-representation r f :_ a, 8) satisfies
trace(r f(Frobl)) = c`
and
det(r f(Frob`)) = X(1)1'-1
for primes 1 not dividing M induced from the universal pseudo-representation via a unique homomorphism Rp -* Cp, i.e., a unique Cp-valued point X f E Xp(Cp).
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31
If, further, M is a power of p and f is of finite slope, i.e., if its Upeigenvalue u := cp is nonzero, denote by x f the point (xpf,1/u) E (Xp X
A')(CP) Definition 1. The classical modular locus of tame level 1, M C (Xp x A1)(CP)
is the set of points x f = (xf, 1/u) where f runs through all (classical) modular eigenforms of finite slope, as above, on I71(pm) for m > 1.
In this article we shall give two different constructions of rigid analytic curves which parameterize the collection of all overconvergent eigenforms of tame level N = 1 and of finite slope (i.e., having non-zero Up eigenvalue).
Definition 2.
The eigencurve C = C, of tame level N = 1 is the
Fredholm closure (see Section 1.2) in Xl, x Al of the classical modular locus (of tame level 1).
Note: We have jumped the gun in our terminology for the above definition
does not immediately allow us to see that C, is a curve. In this paper we shall not define the eigencurve of general tame level, and only concern ourselves with tame level N = 1. Thus, in what follows, we will drop the tag "of tame level N = 1" and refer to the eigencurve of tame level 1
simply as the eigencurve. It follows from the above definition that Cp and its nilreduction CPed are nested in the sense defined in section 1.1, and therefore Proposition 1.2.5 implies that Cped is the union of its irreducible components.
Here is a statement of some of the main results of this paper. By the reduced eigencurve, we mean the nilreduction of the eigencurve Cn
Theorem A. Let Co C Xp x Al be an irreducible component part of the reduced eigencurve. Then there exists an element -yo E Rp such that the mapping
XpxA1- W x A induced by the A-algebra homomorphism A{{T}} -* Rp{{T}} (T H when restricted to C,, is generically an isomorphism of Co onto a Fredholm hypersurface in W x A'. By "generically" we simply mean, since we are dealing with a rigid analytic curve, after the exclusion of a discrete set of points (i.e., a 0-dimensional Zariski-closed subspace). Theorem A follows directly from Corollary 7.6.2, and its proof. 1
32
R. Coleman & B. Mazur
Theorem B. The natural projection of any irreducible component of the reduced eigencurve to weight space is component-wise almost surjective in the sense that given any irreducible component of the reduced eigencurve, the complement of its image in the unique irreducible component of weight space containing that image is (empty, or) consists of at most a finite number of weights.
Proof. This is Corollary 7.4.2 below. The argument for it uses Theorem A together with the Corollary 1.3.13.
Theorem C. The projection of the reduced eigencurve to weight space
is locally in-the-domain finite flat in the sense that C7d is covered by admissible affinoid domains U such that the restriction of projection to weight space to U is a finite flat mapping of U onto its image in W. Proof. This follows directly from the construction of D in Chapter 7 (Prop. 7.2.2) together with the fact that D '-' Cpr d (Theorem. 7.5.1).
It follows from Theorem C that Cp is a curve in the sense that it is an equidimensional rigid analytic space of dimension 1. Since the universal pseudo-representation ring Rp is a complete semilocal ring whose maximal ideals are in one-one correspondence with (certain) equivalence classes of semi-simple residual representations GQ,S -p GL2(Fp) (see section 5.1 below) we have:
Theorem D. If two classical modular points of tame level N, if, if, E (Xp x A') (C,,) lie on the same irreducible component of the reduced eigen-
curve, then f - f modulo the maximal ideal of OcP in the sense that their Fourier expansions are congruent modulo that maximal ideal (and their associated residual representations have equivalent semi-simplifications).
In view of Theorem D, given p : Gq,s -* GL2 (Fp), a semi-simple representation, we will say that a component C; of the eigencurve is of type p if one (equivalently: all) of its classical modular forms have the semisimplifications of their associated residual representations equivalent to p. Define Cp to be the rigid analytic subspace of Cp which is the union of all components of the eigencurve of type p. We say a rigid subspace V of a rigid space U is admissibly closed-open if there exists a rigid subspace W of U disjoint from V such that {V, W} is an admissible open cover of U.
Corollary 1.5.1. The subspaces Cp are admissibly closed-open in Cp.
Proof. Indeed, the proof we sketched for theorem D understates the case a bit: Since the ambient rigid-analytic space Xp x Al breaks up unto the disjoint union of rigid analytic spaces Xp x A' where Xp is the universal deformation space of the pseudo-representation (of GQ,S) associated to p,
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33
for p running through the semi-simple residual' representations described above and since Cp = Cp fl XP x Al , Corollary 1.5.1 follows. The classical modular forms of finite slope correspond to (certain) points
on the eigencurve. What interpretation is there for the remaining points? For this we have Theorem 6.2.1 below, a paraphrase of which is given by the following theorem.
Theorem E. There is a natural one-one correspondence
CHf, between the Cp-valued points c on the eigencurve and overconvergent (see section 5.2) modular eigenforms fc of finite slope of tame level 1. This oneone correspondence is characterized by the conditions that ir(c), the natural projection of c to Xp, when viewed as a pseudo-representation, is the pseudorepresentation associated to the overconvergent modular eigenform ff by the Gouvea-Hida Theorem (see section 5.2), and the natural projection of c to AQp is the reciprocal of the Up-eigenvalue of f,.
We note that the notion of an overconvergent eigenform of integral weight was introduced in [K-PMF] and its generalization to more general arithmetic weight characters was made in [C-CO] and [C-COHL] and to non-arithmetic weight in [C-BMF] (see also [G-ApM]). We might note that the compatibility of these two definitions was not formally established previously but is now, as a consequence of Corollary 2.2.6.
Theorem F. The reduced eigencurve is the rigid Zariski-closure of the classical modular locus.
Proof. By Theorem E, the points of the eigencurve correspond to overconvergent eigenforms. By Theorem B, every irreducible component of the eigencurve has points of all but a finite number of possible weights in the component of weight space to which it projects. In particular, for each irreducible component C,, of the eigencurve, there is a positive integer ko such that C,, has points of all integral weights > k,,. One immediately sees from this that every irreducible component Co has points of positive integral weight k and of slope < k -1 (and different from k 21). By Corollary B5.7.1 of [C-BMF] there is an affinoid disk in Co whose projection to weight space is an isomorphism C,, -* W,, C W onto an affinoid W,, in W, and such that for a topological dense set Do in the set of arithmetic weights in W0(Qp) the points of C,, mapping to Do are given by classical eigenforms. This proves Theorem F. A similar argument to that used above also gives the following result (which is Proposition 7.6.5 below).
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34
Theorem G. If { f,a} is a sequence of normalized eigenforms of tame level
1 with Fourier coefficients in Cp such that fn has weight in E W(Cp), v(ap(fn)) is bounded independently of n, the sequence in converges in (Z/(p - 1)Z) x Zp to some weight is E W(Cp) and the sequence {fn(q) E Cp[[q]](1/q)} converges coefficientwise to a series f (q). Then, f (q) is the qexpansion of an overconvergent modular eigenform f of tame level 1, weight i, and finite slope. Let f be a Katz modular eigenfunction (see section 2.2 below) of tame level 1, of finite slope, with (normalized) Fourier expansion
f
=q+a2Q2+a3g3+...
with coefficients aj E Cp. Suppose that f is of accessible weight-character (cf section 1.4) (x, s) with s E Zp. Then f is overconvergent if and only if f is the limit in the q-expansion topology of a sequence of classical, normalized, cuspidal-overconvergent (cf section 3.6) modular eigenforms of tame level 1.
In section 5.2 we show that the eigencurve is universal for certain rigid analytic families of (overconvergent, finite slope) modular eigenforms. We lack, however, a completely satisfactory theory here, for we do not have an a priori concept of infinitesimal deformation of (overconvergent, finite slope) modular eigenforms. By "a priori" we mean prior to constructing the eigencurve: it would be good, for example, when given an overconvergent eigenform of finite slope f, to have a theory which produces the Zariski tangent space of the eigencurve at the point f, and produces it in a reasonably computable manner (perhaps as an appropriate cohomology group) . We will now summarize some of the key properties of the eigencurve in a purely rigid-geometric way without mentioning deformations. The p-adic eigencurve of tame level 1 is a rigid analytic curve C with the following
properties: There is a rigid analytic morphism n from C to W and rigid analytic functions tn, n > 1 with the following properties: Suppose .f (q) =
angn n>O
is a normalized eigenform on X1(p"), r > 0, of weight k, such that ap # 0 whose character on (Z/p''Z)* is cXr-k, where x E B. Then there exists a unique point x f on C such that ic(xf) = exrlk and tn(xf) = an. Call these points the classical points. On the other hand, if x E C(Cp) such that n(x) = ex77k where k is an integer, x E B is a character of finite order and v(tp(x)) < k - 1 then x is classical. Moreover, as we stated above, C is the rigid Zariski-closure of the classical locus. This is not, however, enough to characterize C.
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Chapter 2. Modular Forms. 2.1 Affinoid sub-domains in modular curves. If M > 1 is an integer, and S is a scheme, µM /S denotes the finite flat group scheme over S given by as the kernel of multiplication by M in the multiplicative group (over S). Let p be a prime number. Fix an integer N prime to p, and m > 1 and let us begin our discussion in level Npm. Our running assumption in this section is that, unless otherwise explicitly mentioned, Npm > 5. Let Y1(N pm) be the (uncompactified) modular curve classifying isomorphism classes of pairs (6, a) where £ is an elliptic curve over a Qp-scheme S, a : AN.pm * £ is an injection (over the base S). This is equivalent to the [F1[(Npm)] problem (over Qp) as discussed in [KM]. Since Npm > 5, the above problem is rigid and representable in the terminology of 4.7 of [KM] (cf. pp. 327, 328 of [KM]) and we have a universal family
(u : Ei(Np') -*
Y1(Npm'),
a).
Let w be the invertible line bundle on Yi(Npm)l QP given by
w= By the discussion of (10.13) (cf. the summarizing table (10.13.9.1)) of [KM],
since Npm > 5 we have an extension of the line bundle w to a line bundle (which we denote by the same letter) on the compactification X1(Npm)/QP, and a canonical Kodaira-Spencer isomorphism w®2
uXl (Npm )i QP (log
cusps),
of line bundles on X1(Npm). Assuming now that N > 5, the reader is invited to consider, as well, a noncompact model (over Zp) of this moduli scheme given by the analysis in [KM] of the [I'1[(Npm)] problem (over Zp) whose closed fiber is the (geometrically irreducible) incomplete Igusa curve of level Npm (cf. Chapter 12 of [KM]). It is sometimes useful to keep in mind the entire tower of modular curves (*)
as n tends to oo (m < n). Consider, for example, a triple (£, aN, ate) where
£ is an elliptic curve over Zp with ordinary reduction mod p (i.e., with either good ordinary reduction, or multiplicative type reduction), where a : AN y 9 is an injective homomorphism (over Spec Zp) and where a,,,
R. Coleman & B. Mazur
36
is an embedding G,,,,[p°°] " E[p°°] of the p-divisible group attached to G",L (over Spec Zp) into the p-divisible group attached WE. Such a triple (£, aN, ate) (which we will sometimes refer to as a p-trivialized elliptic curve with tame I71(N)-structure) gives rise to a cofinal system of Spec Zp-valued points of (**)
as follows. For each n restrict a,, to get an embedding apn : µpn -4 E, and define aNpn : ILNpn '- £ to be the unique embedding which restricts to aN and apn respectively on µN and µpn. We then associate to the triple (£, aN, ate) the cofinal system n H (E, aNpn ). Conversely, a cofinal system of Spec Zp-valued points of (**) comes from such a triple (£, aN, ate). Given such a triple, the embedding a,,,, induces an isomorphism G. = E of formal completions at the origin. The image of the standard differential
dt/t on G. under this isomorphism then gives us a specific differential w generating the fiber of the line bundle w over the point (E, aNpn) of Y1(N p") (Qp), for any n > 1. (Although we will not use this point of view in the present paper, one way of thinking of this, is to invoke the projective
limit Y1(N p°O) of the projective system (*) of curves and to view the process of stipulating a generating section w E w over Y1 (N p°O) as giving us a canonical trivialization of the line bundle w over Y1 (N p'). See 1. 1 of [G] for slightly more discussion of this.) Let us now return to the modular curve X1(N p""), taken over the base Qp.
First, let Z1 (N p'n) denote the inverse image under reduction to Spec(Fp) of the complement of the supersingular points on the irreducible component containing the cusp oo in the above model of X1 (N p"") over Spec(Zp). We have that Z1(N p"n) is an affinoid subdomain of the rigid analytic space X1(N p"") over Qp (see [C-RLC]). In §B2 of [C-BMF] a system of affinoid neighborhoods Z1 (N p"°) (v) (called there X1 (N p"n) (v) )
of Z1 (N p') in X1 (N ptm) is described, where the parameter v (which we sometimes refer to as the radius) is a rational number allowed to vary in
the range 0 < v < p(2--)/(p + 1). Let Im denote the set of such v. In particular, Z1(N p'n) (0) = Z1(N p'n) and when v > 0 these neighborhoods are strict (cf. the definition of strict neighborhood below). We shall briefly sketch the definition of Z1 (N p) (v) here for p > 5 (for the full story, see [C-BMF], section B2). Let A be the level 1, characteristic p, Hasse invariant modular form. For a non-negative rational number v E Q, Z1(Np) (v) is- the affinoid subdomain of X1(Np) whose points over Cp correspond to pairs (£, a) over S = Spec(R), where R is the ring of integers in Cp, as above such that v(A(E, rl)) < v where 71 generates Q61 /s and (E, )
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37
is the reduction of (E, 77) modulo p, and a(pp) is the canonical subgroup (see Chapter 3 of [K-PMF] where the canonical subgroup is defined for
all N and p > 5). We will also have occasion to consider the modular curves Xl (Np'', B), where £ is a prime not dividing Np, which classify triples
(E, a, C) where E and a are as above and C is a subgroup scheme of E of order t. Then we have an affinoid subdomain Z1(Npm,t) with affinoid neighborhoods Z1(Npm, .£) (v) for v E Im defined in a similar way.
In what follows let us fix a rational number v and focus attention on the affinoid sub-domain Z1 (N pm) (v) of that width. We also reinstitute the
hypothesis N pm > 5. Definition. Let V be a rigid analytic space. By an overconvergent family of rigid analytic functions on Z1 (N pm) parameterized by V we mean a rigid analytic function F on Z1 (N pm) x V with the property that there is an admissible open covering of V by affinoids Vj (j E J) such that there are positive numbers vj E Im (j E J) for which the restriction of F to Vi extends to a rigid analytic function on Z1(N pm) (vj) x Vj. More generally,
Definition. If X -4 Y is a morphism of rigid spaces over K, we say that X is affinoid over Y if for each affinoid subdomain Z in Y, XZ is an affinoid. Suppose W -4 Y is a map of rigid spaces and X is a rigid subpace of W which is affinoid over Y, then we say that an admissible open subspace V of W is a strict neighborhood of X over Y in W if there is an admissible covering of Y by affinoids Vj such that for each j there exists an affinoid neighborhood Uj of Xv3 in Vv; such that the morphism Xvs -+ Uv, factors through a subscheme of Uv, finite over Y. Finally, if X, W and Y are as above, we say that a rigid function f on X is overconvergent in W over Y if f extends to some strict neighborhood of X in W over Y. When Y is Max(K), we just say f is overconvergent on X in W.
We denote the ring of these functions on X which are overconvergent in W over Y by At (X/Y, W) and the subring of those functions bounded by 1 we denote At(X/Y, W)°. We drop Y from the notation when it is a point and we drop W when it is understood from the context. In particular, the ring of overconvergent families of rigid analytic functions on Z1(Nq) parameterized by V is denoted At(Z1(Nq)v/V). If BK[0,1] and BK(0,1) denote the affinoid and wide open unit disks over K, At(BQp[0,1]BQp(° 1)/Bq (0,1))° may be identified as the ring of power series in the parameter variable T over AM which are of the form E000 A,,Tn Ilan] where Ii is the maxsuch that for some positive real number a, An E imal ideal of AM. (For more discussion of this notion see §A5 of [C-BMF] as well as §1 of [C-CCS].) It is less straightforward to define the notion of
R. Coleman & B. Mazur
38
overconvergent modular form or family of overconvergent modular forms. Cf. section 2.4 below for the general definition but, for now, we define overconvergence only for integral weights: Let wk for k E Z denote the k-th tensor power of w, and consider the vector space of rigid analytic sections of wk over the affinoid Z, (N p-) (v). More explicitly, letting K be a complete subfield of Cp, and the affinoid over K induced from Zl (N pm) (v) by base change, define the K-vector space Mk (Npm, v; K) := wk (Zl (N pm) (V) 1K),
which has a natural K-Banach space structure, as described in section B2 of [C-BMF].
If v > 0, an element of Mk(Npm, v; K) is called a v-overconvergent
modular form (of weight k and level N pm). By definition an overconvergent modular form (of tame level N with coefficients in K) is a v-overconvergent modular form of level N pm for some v > 0, and some m > 0, i.e., it is an element in the p-adic Frechet space given by the union of these p-adic Banach spaces. (Compare this with the definition of overconvergent function, and form in A5 and B4 of [C-BMF] as well as in [C-COHL]). The elements of Mk(Npm, v; K) for v = 0, i.e., the space of rigid analytic
sections of wk over Zl (N pm) will be referred to simply as convergent; convergent modular forms on X1(Npm) of weight k coincide with Katz padic modular functions of that tame level and weight-character (see section 2.4 below). The standard automorphisms and correspondences on the curve X1(N pm) (which we shall list below) induce natural continuous operators on the Banach space Mk(Npm, v; K). We explain this in detail in the next chapter. For now, we will discuss the diamond operators.
For any positive integer m, the group (Z/NpmZ)* acts on the modular curve Xi(Npm) over Qp as follows. For r E (Z/NpmZ)*, the operator (r) sends the pair (£, a) to (£, r a). This gives us a compatible action of the profinite group ZP,N = lim(Z/NpmZ)* on the tower (*) and more specifically, for every positive integer m, we have compatible actions of the finite quotient group (Z/NpmZ)* of ZP N on the pair (**) (X1(Npm) , WX1(Npm)). ry1
Lemma 2.1.1. The action described above induces a continuous homomorphism from the topological group ZP N to the group Autrlg.an. (Zl (Npm) (v),
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39
WZ1(Npm)(,)) of rigid analytic automorphisms of the pair consisting of the affinoid Zi(Npm)(v) in Xi(Npm) and the restriction of the line bundle w to this affinoid. This induces a natural action of the Iwasawa algebra AN as an algebra of continuous operators on the Banach space Mk (Npm, v; K).
2.2 Eisenstein series. In this section we work in tame level N = 1. We recall and continue the development of the theory of the fundamental family of modular forms, i.e., the Eisenstein family. Some of this theory was discussed and/or proved on pages 446-448 of [C-BMF] as well as in [C-CCS]. See also section 7.1 of Hida's book [H-ET] for further proofs and discussion.
Let W = WN for N = 1 and W+ denote that part of weight space W consisting of characters i such that K(-1) = 1. We have the p-adic (-function (*(rc) (notation as in section B1 of [C-BMF]) which is a rigid
analytic function on W+ away from the point it = 1 (at which it has a simple pole). If rc = (x, s) is an accessible weight, (*(r.) = Lp(x,1 - s),
where Lp(x, s) is the Kubota-Leopoldt p-adic L-function. For n > 1 set djn,(d,p)=1
which we view as an Iwasawa function on W+. If rc 2
1 and (*(r.) 0 0, set
°O
vk(n)g
EK(q) = 1 + S*pj n=1
and for rc = 1, put Erjq) = 1. Compare the discussion on page 447 of loc. cit. Let us refer to this family of q-expansions Ek(q), parameterized by the rigid analytic subspace Weis = jr, E W+ and (*(n)
0}
as the basic Eisenstein family. Note that 5 C Wis. We also use the notation E°(q) :=
1+clq+e2g2+...,
where the coefficients en are the rigid analytic functions in A(Weis) with the property that en(rc) = 2Q*(n)/(*(rc). The power series E°(q) E A(Weis)[[q]]
has the corresponding property that rc(E°(q)) = E,(q).
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40
One immediately checks that the power series E° (q) is fixed under the action of the U-operator on A(Weis) [[q]] which takes power series E0 0 to E°O_0 ap q". Recall that if rc = (x, k) E W+ is arithmetic and k > 0 and (i) # 0, then E,. (q) is the q-expansion of a weight k classical (see Miyake [Mi] Chapter 7) (hence v-overconvergent, for any v) modular eigenform on Xi(l.c.m.(q, condx)) with character xT-k. Here condX is the smallest power
of p such that x is trivial on 1 + condX ZP. More generally, if rc = (x, k) with k any integer we have that (*(i)E,c(q) is the q-expansion of a weight k overconvergent modular eigenform on Xi(l.c.m.(q, condX)) with character X7--k as we will show in Corollary 2.2.6 below when x trivial on µ(Zp). (The general case follows from this and Theorem 2.1 of [C-CCS] applied to C*(i)E,c(q)/E(w)(q)). (See also Proposition 11.3.22 of [G-ApM].) As discussed in [C-BMF, §B1] and elsewhere, if a is trivial on the sub-
group of roots of unity, u(Qp), in Q,, then JE,c(q) - 11 < 1.
Definition. The restricted Eisenstein family E is the restriction of the basic Eisenstein family of tame level 1 to the subspace B C W (cf. notation of section 1.4) .
Remark 1. For any tame level N, as we saw in section 1.4, the rigid analytic group WN is the disjoint union of a finite number of 13-cosets of the form x B where x is a character of finite order in DN. This is important in that it will allow us to transform modular forms of arbitrary weights to functions on modular curves by division by Eisenstein series in the restricted Eisenstein family. But here we considered only tame level N = 1, and now
restrict to the special case where x = 1 is the trivial character. Some important facts about the restricted Eisenstein family are given below.
Proposition 2.2.1. The q-expansion, E(q) of the restricted Eisenstein family has coefficients in A(°):
E(q) = 1 + clq + E2g2 +---
E A(°)[[q]],
and the coefficients ej for j > 1 lie in the maximal ideal of AM.
Proof. This is well known. For example, it is a consequence of Iwasawa's theorem on p-adic L-functions [I] (cf. Theorem 16 of [S-FMZ]). Theorem 2.2.2 (additivity). Let Z = Zl(q). There exists a rigid analytic function on ZBxs bounded by one and overconvergent over 13 x 13 (i.e. an element ofA°(ZBxB/13 x B))t) with q-expansion Ea(q)EQ(g) Ea/3 (q)
The Eigencurve
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Proof. Let E = E(TO 1). This is the classical weight one Eisenstein series on
X1(q) used in Chapter B of [C-BMF]. For v(s) > 1 - 1/(p - 1) we know, by Corollary B4.5.2 of [C-BMF], that E(TO 3)(q)/E(q)8 is the q-expansion of
a function on Z. overconvergent over 13*. It is without zeroes on Z1(q), since its q-expansion is congruent to 1, hence on Zl (q) (v) for some v > 0.
It follows that if v(s) and v(t) are greater that 1 - 1/(p - 1), a = (T°, s) and ,3 = (,r°, t) then (E«(q)/E(g)s) . (Ep(g)/E(g)t) E.,3(q)/E(q)s+1
is the q-expansion of a function on ZB.xB. overconvergent over 13* x 13*. Hence, we obtain the conclusion of the theorem with 5 replaced by 13*. We also note that the q-expansion coefficients of this function lie in A®A. We want to apply a generalization of Theorem 2.1 of [C-CCS] (i.e., Theorem 2.2.4 below) which we must now prepare to prove. Let In be the maximal ideal of An := Zp[[S1i ... , S.]]. For a E An let vA(a) = min{m: a E I,}.
Suppose t = (t1,. .. , tn) is an n-tuple of real numbers < 1. We define a norm II IIt on An by setting for I11: btL1SMIIt = max{IbMItM}. M
Here M is a multi-index. The norm 111 It is the spectral norm on the disk B[t], I Si I < ti, which vt(a) = -log I IaI It.
is an affinoid when ti E pQ for 1 < i < n. Set
Lemma 2.2.3. There exist positive constants A(t) and B(t) such that for all non-zero a E An, A(t)vA(a) < vt(a) :5 B (t) VA (a).
Proof. Let a = pmo SM where M = (ml, ... , Mn) and m° + E M = m. Then, I IaI It = I pI tm° . tM, so
vt(a) = m° logp+
mi (- log ti).
Thus the lemma follows if we take B(t) = max{log p, - log t1i ... , - log tn}
and A(t) = min{logp,-logt1i...,-logtn}. As a corollary we get that a series over An which is convergent on 13n x B[0,1] is overconvergent over the open unit n-polydisk l3n if and only if it is overconvergent over any single affinoid subpolydisk around the origin. We are ready to prove the generalization of Theorem 2.1 of [C-CCS]
anticipated above. Let l3n be the n-dimensional unit polydisk. Then An may be regarded as the ring of rigid analytic functions on Bn defined over Qp and bounded by 1.
R. Coleman & B. Mazur
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Theorem 2.2.4 (q-expansion Principle). Suppose, t E Rn and 0 < ti < 1. Then, if G E At(ZB[t]IB[t]) and G(q) E A,,[[q]], G uniquely analytically continues to an element ofAt(ZB, Proof. The proof goes exactly like that of Theorem 2.1 in [C-CCS] and we employ the notation of that paper. In particular if Y is an affinoid, A°(Y)[X]t denotes the ring of power series with coefficients in A(Y) which overconverge on B[0,1] x Y over Y and are bounded by 1 there. In particular, if Y is the origin consider as a rigid space over Qp, A°(Y)[X]t = Zp[X]t. We apply Lemma 2.2 of [C-CCS] which asserts, that there exists a finite morphism f from Zt onto B[0,1]t such that f -1(0) = oo and f is separable. We base change f to a finite morphism from (ZB[t]/B[t])t to (B[O,1]B[t]/B[t])t and let Tr f be the corresponding trace map. We let X be the standard parameter on Al and conclude that for r E At(Z)°, Tr f(rG) is in both A,a[[X]] and At(B[0,1]B[t]/B[t]). It follows from this that if D generates the discrim-
inant ideal in Zp[X]t of At(Z)°/Zp[X]t, then DG E At(Zgn/13n). Since f is separable, p XD. Then Lemma 2.3 of [C-CCS] generalizes immediately to
Lemma 2.2.5. Let t E Rn, 0 < to < 1. Suppose a(X) E A0(B[t])[X]t and suppose that there exists a D(X) E Zp[X]t such that p XD(X) and D(X)a(X) E An[X]t, then a(X) E An[X]t. However, we feel that one of the assertions made in the proof of Lemma 2.3 of [C-CCS] requires more justification. Namely, we will now justify the
convergence of the sum (in the notation of [C-CCS]) S := En Anhn(X). Recall, hn (X) is an element of Z, (X) satisfying
Xn - rn(X) = D(X)hn(X), where D(X) E Zp[X]t and rn(X) is either 0 or a polynomial of degree strictly less than d over Zr,. This is already enough to show that hn (X) is overconvergent but we need more. Let S-1 > s > 1 so that D(X) converges on B[0, s] and has no zeroes on the half open annulus A(1, s]. It follows that hn(X) converges on B[0, s]. Let C = Il/D(X)I A[,,,]- It follows that for large n, I hn (X) I is at most Csn on the circle A[s, s] and hence (by the maximum modulus principle) on the disk B[0, s]. It follows, since IAnIt < on, that the sum S converges to an element of A° (B [t]) [XI t as claimed. We conclude the proof of the Theorem 2.2.4 the same way as that of Theorem 2.1 of [C-CCS]:
Suppose b,,. .. , bd is a basis for A°(Zt) over Zp[X]t. We may write, G = a,(X )b1 + a2(X)b2 + ... + ad(X)bd
where ai(X) E A°(B[t])[X]t. Then as DG E At(ZB, /13n)°, it follows, since b1i... , bn is also a basis for At(Z/3n/Bn)° over Bn, that D(X)a$(X) E A[X]t. Thus we may apply the previous lemma and deduce Theorem 2.2.4. 1 We now give a complete proof of a claim made in §B1 of [C-BMF].
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Corollary 2.2.6. If ic E 13 is arithmetic of weight k and nebentypus e, then E,c(q) is the q-expansion of an overconvergent modular form in Mk(pm, v, Qp(µpm )) for some v > 0 where p"` is LCM(q, cond6).
Proof. This is automatically true for k > 0 since then Ek is classical of the appropriate weight and level. Moreover, in this case, knowledge about the field of definition follows from the q-expansion principle. In general, we
proceed by induction on -k. Suppose the corollary is true when k > -n. Let a E 13 be the arithmetic character with coordinates (T°, 1). Suppose k = -n. Then Theorem 2.2.4 implies &(q) equals E«,(q) F(q), E. (q) where F is an overconvergent function on Zl (Nq). Since E0, reduces modulo
p to the (p - 1)-st root of the Hasse invariant on Z1(q), it has no zeroes there and hence no zeroes in a strict neighborhood of this affinoid. The corollary now follows as the induction hypothesis applies to Ea,c (q) which
has weight -n + 1.
1
Proposition 2.2.7. Let .£ be a prime number. The ratio of E(q) and E(qe),
El(q) := E(q)/E(ql), is the q-expansion of a rigid analytic function on Zl (q, 2) x 13 if $ # p, and on Zl(q) x 13 if f = p, overconvergent over B. The function El (q) has Fourier coefficients in AM, its constant term being 1, and all other coefficients lying
in the maximal ideal of 0). Proof. The proof of existence of the overconvergent rigid analytic function runs along the same lines as the proof of Corollary 2.1.1 of [C-CCS] which is this proposition with .£ = p, as did that of the additivity theorem, with Z replaced by Z1 (q, 2) if £ p. The statement about the coefficients follows from Proposition 2.2.1. 1
Remarks. For fuller control of the geometry of the eigencurve it would be useful to know explicit affinoid regions of the type X x Z1(Npt)(v) on which some, or all of the functions Et(q) converge. (For a preliminary result along these lines, in the case of p = 2 see forthcoming work of M. Emerton.)
We also feel that it is generally a good idea for anyone who is just beginning the study of p-adic modular forms to spend time specifically concentrating on, and understanding the nature of, the Eisenstein family. It is at the root of much of the theory. For example:
R. Coleman & B. Mazur
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1. (It is the ur-example of a p-adic family.) The Eisenstein family provides a very explicit example of a p-adically varying Fourier expansion (of eigenforms) parameterized by p-adic weight which interpolates to give classical modular eigenforms at positive integral weights. To our knowledge,
Serre was the first person to signal the p-adic analytic variation of the Fourier coefficients of the Eisenstein family, and the fact that this could be used to provide an alternate construction of p-adic L-functions. 2. (It provides a way of understanding Hida families.) Given a classical
parabolic modular eigenform f of tame level N and weight k > 2 which is (p-)ordinary (i.e., which has slope zero), the restricted Eisenstein family can be used in conjunction with Hida's projection operator to the subspace of ordinary modular forms, to enable one to see, a bit more concretely than is offered by the general theory, the analytically varying family of ordinary eigenforms of which f is a member. Briefly, just multiply f by E,,(q) where ic ranges through B and then project this family to a family of ordinary modular forms using Hida's projection operator. In the special case where the rank of the space of ordinary modular eigenforms of that tame level is 1 then things are particularly easy: the projection would then be family of ordinary modular eigenforms (for all Hecke operators) parameterized by an open subspace of weight space. But the general case is hardly more difficult. 3. (Multiplying by Eisenstein series provides us with a convenient way of passing from weight 0 to general weight. In this way we get a convenient definition of overconvergence for modular forms of arbitrary p-adic weight,
and for families of modular forms by requiring the corresponding form of weight 0, or family of forms of weight 0, to be overconvergent. See, for example, the definitions given in B4 of [C-BMF], and repeated in section 2.4 below. Moreover, the multiplier function EP enables one to compare the operation of the Atkin-Lehner operator UP on weight 0 modular functions to the same operator on modular forms of general weight. See [C-BMF] and section 2.4. The fact that the Fourier coefficients of Ep are nice is one of the key ingredients for a good p-adic analytic dependence (on weight) of Up eigenvalues.
2.3. Katz p-adic Modular Functions. For the beginning of this section fix any prime number p and an arbitrary tame level N. Denote the Iwasawa algebra AN by A, for short. By a Aadic ring, we mean an algebra over A which is complete with respect to the A-adic topology, i.e., B is such a ring if and only if B is a A-algebra and B lim B/rad(A)'B where rad(A) is the radical of A.
Definition. Let B be a A-adic ring. By a Katz p-adic modular function over B (of tame level N) we mean a function F which assigns to
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any isomorphism class (E, a.., v) of trivialized elliptic curves with level Nstructure (v) over a A-adic B-algebra D an element F(E, ate, v) E D, and whose formation commutes with base change. For a discussion of this notion and its variants (e.g., cuspidal Katz p-adic modular functions) see Chapter 4 of [K-PMF], section 1.3.1 of [G-ApM], and all of [K-2], especially the appendix there. If F is a Katz modular function
of tame level N over the A-adic ring B and a E ZP N, a = (b, c) where b E (Z/NZ)* and c E Z*, we set Fj(a)(E, ate, v) = F(E, c-laoo, bv), for all isomorphism classes of triples (E, ate, v) of trivialized elliptic curves with level N structure over a B-algebra D. If 0 : Zp N B* is a con-
tinuous homomorphism we will say that F has weight-character 0 if FI(a) = cD(a)F for all a E ZP N where 'OD is the composition of 0 with the homomorphism B* D*, induced from the structure homomorphism of the B-algebra D. For any p-adically complete Zp algebra B, let Vn,m (B) be as in [G, §1.3.1]. Let Npt > 5. The ring Vn,m(B) is the ring of functions on Z1(Npm) x B/pnB where Z1(Npm) is the underlying formal scheme over Zz, which is in the affinoid connected component Z1(Np'n) of oo in the ordinary locus of X1(Npm). Then the ring of Katz holomorphic p-adic modular functions over B (of tame level N) is lim lim Vn,m
n m For this, see ibid. and [K-2].
Proposition 2.3.1. The q-expansion E(q) is the q-expansion of a Katz p-adic modular function 6 over A(°) with the identity weight-character.
Proof. Fix an integer n > 0. Let Rn = A(°)/([1 + q]Pn - (1 + q)2Pn). Then A = limRn. Let En(q) be the restriction of E(q) to Rn. Let x(a) _ ((a))2V,(a) where 0 is a character of finite order on 1 + qZr. Then since x(E(q)) is the q-expansion of a classical weight 2 modular form with character OT-2, and fJ ([1 + q] - (1 + q)2?p(l + q)), [l + q]rn (l
-
+ q)2Pn =
0
where the product is over all characters 0 on 1 + qZp trivial on 1 + qpn Zp, En (q) is the q-expansion of a Katz p-adic modular function over the normal-
ization of R. It follows from the q-expansion principle that it is, in fact, the q-expansion of a Katz p-adic modular function over Rn and hence, passing to the inverse limit, E(q) is the q-expansion of a Katz p-adic modular function over AM. ,
R. Coleman & B. Mazur
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2.4 Convergent modular forms and Katz modular functions. Let p be a prime number, N a tame level, and m > 1, and assume, for this
section, that Np't > 5.
Definition. We say F(q) _ ° angt1, an E K, is the q-expansion of a convergent (resp. overconvergent) form of tame level N and weightcharacter rc E W over K if F(q)/E(,,) (q) is the q-expansion of a rigid (resp. overconvergent) function on Z1(Nq) in X1(Nq) of character rc/(rc) for the
action of (Z/NqZ)*. The K-vector space of these q-expansions with the property that the corresponding function converges on Zl (Nq) (v) will be denoted M,, (K, N, v).
If U is an admissible open subspace of B we also say that 00
F(q) = E angn, n=o
a E A(U), is the q-expansion of a family of convergent (resp. overconvergent) forms over U of tame level N if F(q)/E(q) is the q-expansion
of a rigid (resp. overconvergent) function on U x Zi(Nq). We say this family has type 8 E DN if this function has character 8 for the action of (Z/NqZ)*. We call the A(U) module of these forms Mt (N), and for v > 0, we call Mu (N) (v) the submodule consisting of those families over U whose respective functions converge on U x Zl (Nq) (v).
Thus, in particular, it follows from the above definition that E(qP) is the q-expansion of a family of overconvergent forms over B of tame level 1 and type 1. Note that when , is arithmetic of weight k, M,, (K, N, v) is contained in Mk(Np', v, K) for ptm = LCM{q, condx} where x is the wild part of ic for small enough v > 0 because of Corollary 2.2.6. Also, as an immediate consequence of Theorem 2.2.2 we get
Proposition 2.4.1. If F(q) and G(q) are the q-expansions of overconvergent forms of weight-characters a and ,Q, then F(q)G(q) is the q-expansion of an overconvergent form of weight character a,d.
Theorem 2.4.2. Let F be a convergent family of modular forms over an affinoid subspace X of W, with q-expansion coefficients in A°(X). Then the q-expansion of F is the q-expansion of a Katz modular function F over A°(X). Proof. What we have is that F(q)/E(q) is the q-expansion of a rigid analytic
function on X x Z1(q). But A°(X x Z1(q)) = A°(X)®A°(Z1(q))
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is just V00,i(A°(X)), where i = 2 if p = 2 and i = 1 otherwise, so F(q)/E(q) is the q-expansion of a Katz modular function over A° (X ). The result now follows from 2.3.1 since A° (X) is naturally a A-adic ring.
Chapter 3. Hecke Algebras Fix a prime number p and a tame level N relatively prime to p. The precise conditions on these will be given at the head of each section below. Whenever we deal with a level written as Np'n we assume that Npm > 5.
3.1 Hecke eigenvectors and generalized eigenvectors. In this section we work with arbitrary p and level N. By 9 l = W N let us mean the (commutative) polynomial algebra over the topological ring AN in the infinitely many variables labelled Te for prime numbers £ not dividing
N p and Uq for primes q dividing N p. We view 9-l as a topological AN-algebra given its weak topology. More explicitly, if for any finite set of monomials S in the variables Te and Uq we let ?is denote the free ANsubmodule of 9-l of finite rank generated by the monomials in S, given the product topology it inherits from the topology on AN, a subset of the topological ring 9-l is open if and only if its intersection with every 7-Is is open. In particular, continuous ring homomorphisms from 9-l have the property that their restriction to AN are continuous. Purely formally, we may define the universal q-expansion 00
E Tn qn
E
9-l[[q]]
n=1
by the following rules: For prime numbers £ which do not divide Np, we put Te = TE; for prime numbers 2 which divide Np, we put Te = Ut; and we define Tn for any positive integer n by the finite recursive relations summarized by the equality of formal Dirichlet series 00
=[J(1-Tt-t-s+A .j-2s)-1
ETn ns n=1
P
where the product is taken over all prime numbers £, and where [e] is defined to be 0 if t divides pN and to be the image of .£ E lim(Z/NpnZ)* C AN if
doesn't divide pN. If 1 : 71 -+ Cp is a Cp-valued character (i.e., a continuous ring homo-
morphism) by the weight-character of ' we mean the homomorphism
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w = mp : AN -+ CP which is the restriction of T to AN. It is sometimes useful to refer to the q-expansion 00
E T (Tn) - qn
E
Cp[[q]]
n=1
as the Fourier expansion of the character T. For any such T : f -+ CP1 it is convenient to define the following ideal: First consider the ring 7-l ®AN C p
where the tensor product is made via the weight-character w : AN -* C. Define mw C 7-l ®AN Cp
to be the ideal generated by all elements of the form r ®1® T (T) for T E 7-l.
Definition 1. Given a continuous homomorphism w : AN -+ C, and a power series 00
an . qn
.f (q) :=
E
Cp[[q]],
n=1
say that f (q) is a normalized Hecke eigenvector of weight w if there is a character I : 7-l -4 CP of weight-character w whose Fourier expansion is f (q).
There are two other equivalent (and more standard) ways of formulating this notion of f (q) being a (normalized) Hecke eigenvector of weight w. To give these definitions, let us fix a weight-character w, and define the weight w action of 7{ on Cp[[q]] as follows: we let the coefficient ring AN act on Cp[[q]] via scalars through the homomorphism w, and define the action of Te by the standard rule: For primes £, 00
f
bn . qn
Tj
E
Cp[[q]],
n=1
where bn = an/,+t`w([t]) ant, where we set anal = 0 if 1 In and w([t]) = 0 if tjNp. The weight w action of 7-l on C,[[q]] extends naturally to an action of the tensor product 7-l ®AN CP considered above ( this tensor product being made via the weight-character w : AN -* Cr).
Definition 2. The power series f (q) := E°O 1 an qn E Cp[[q]] is a normalized Hecke eigenvector of weight w if and only if the coefficient a1, of q1 in its Fourier expansion is equal to 1, and if there is a character T : 7-l -* CP of weight-character w such that f (q) is an eigenvector for the weight w action of T on C,[[q]] with eigenvalue 'Y(T), for all prime numbers t. Equivalently we may ask that a1 = 1, and that the power series
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49
f (q) E Cp[[q]] be annihilated by the ideal mp C 9i®A,,, Cp acting on Cp[[q]].
If this is the case, then we refer to 1F as the Hecke eigenvalue character of f (q).
Definition 3. We will say that a nonzero power series f (q) := E°_1 a, q"
in C,[[q]], is a generalized Hecke eigenvector or just a generalized eigenvector of weight w if there is a character IT as above, of weightcharacter w, and some positive integer v such that f (q) E Cp[[q]] is annihilated by the ideal (m,p)" C W ®AN Cp. If f (q) is such a generalized eigenvector, we refer to W as the Hecke eigenvalue character underlying f(q).
3.2. Action on Mk(Npm, v; K). In this section we work with arbitrary p and square-free tame level N, and as usual, when a level Np' is invoked, we assume that Np' > 5. For prime numbers £ not dividing N p we have the Hecke operators TI, and for primes q dividing N p, we have the Atkin-Lehner operators Uq. These operators come from correspondences, all defined, when Np' > 5, on the modular curve Xi(Npt) (cf. [M-W]) all commute with each other, and also commute with the diamond correspondences (d) for d E (Z/NprZ)*.
Lemma 3.2.1. For q 0 p, the Hecke, Atkin-Lehner and diamond correspondences T1, Uq and (d) induce rigid analytic correspondences on the affinoid Z1(Npm)(v) and compatible correspondences on the restriction of w to this affinoid. These operators induce continuous AN-linear operators on the Banach space Mk (Npm, v; K) which commute with each other.
Proof. The main point is that if E is an elliptic curve over a scheme S of characteristic p and Ver: E() -* E is Verschiebung, then if v is a generator of SZE1s,
Ver*v = A(E, v)v(p). This is equivalent to the dual formula of Katz on page 97 of [K-PMF]. From this one can deduce that if h: F -* E is an etale isogeny, then
A(F, h*v) = A(E, v).
(1)
This means one can define operators on Mk (Npm, v; K) in the standard way. Eg., suppose m = 1 and f is a prime, such that (E, Np) = 1. Then if (E, a, C) is a triple corresponding to an R-valued point on Zl (Npm, l) (v), where R is the ring of integers in a finite extension of Qp and v E I,, and 17 is a generator of QE11R, then
v(A(E,i)) =v,
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where (E, ) is the reduction of (E, 77) modulo p. Then the image of a(lip) in E/C is the canonical subgroup and (1) implies
v = v(A(E/C, 6)) where
is any generator of Q(E/C)/R Thus the two maps from X1(Np,2)
to Xl(Np) take Zl(Np,t)(v) onto Zl(Np)(v) for V E I. We also have a completely continuous operator corresponding to U. acting on Mk(Npm, v; K) for v E Im (see section B2 of [C-BMF]). Note: One must stipulate, specifically, whether one means to take these operators as the ones induced from direct image or the inverse image of the geometric automorphisms or correspondences listed. Our convention will be to take the direct image, i.e.,
(r)=(r)*, Te=Te*, Uq=Uq*. Had we made the other choice, i.e., had we taken the inverse image, we would have had
(r-1)*= (r)*, etc.; cf. the discussion and formulas in [M-W] Ch. 2, section 5. We view ?I as acting on the (Banach) AN-modules Mk (Np71, v; K) in
the evident manner. We will also have use for notation for the sub-ANalgebra Ii' C ? generated by all the Te's, for prime numbers B not dividing N p. Thus when N = 1, 7.1 is the polynomial algebra over 9-l' generated by the single symbol Up, i.e., by the Atkin-Lehner operator at p. If 4) is a Cp-valued character on 71, and if a modular form f E Mk(Npm, v; Cp) is an eigenvector for the action of 71 with Hecke eigenvalue character equal to P then if f (q) E Cp is the Fourier expansion of f and if the coefficient of q in f (q) is 1, then f (q) is a normalized eigenvector in the sense of §3.1, of weight
w4,. We refer to f and f (q) simply as normalized (overconvergent, modular) eigenforms; if f (q) is a generalized eigenvector in the sense §3.1, we refer to f and f (q) as generalized (overconvergent, modular) eigenforms. 3.3 Action on Katz Modular Functions. Let N be any square-free tame level. By the Katz-Hecke Algebra Tp(N) we shall mean the completion of the AN-algebra generated by the Hecke operators and diamond operators (as described in §2.3) over Zp acting (faith-
fully) on the space V of all Katz p-adic modular functions of tame level N with respect to the compact open topology on the ring of Zp endomorphisms of V (see section II.1 of [G-ApM]). As Hida shows, in Theorems 3.1 and 3.2 of [H-HA],
Tp(N) = limhk(Np') V
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for any weight k > 2 where hk(Np") is the Hecke algebra acting on weight k modular forms of level Np" over Zr,.
As we will show in section 5.2 (for p odd and tame level N = 1) there is a natural homomorphism Rp -+ Tp(1) =: Tp, where Rp is the universal deformation ring of two dimensional p-adic pseudo-representations of the Galois group of Q of tame level N.
Proposition 3.3.1. If p
2, the ring Tp is a complete semi-local noetherian ring with finite residue fields.
Remark. We will give a proof of Noetherian-ness, motivated by the proof of Corollary 111.5.7 [G-ApM], in Corollary 5.2.3 below. We will prove the rest of the proposition now (for any tame level N). Proof.. As remarked, for any fixed weight k > 2, Tp,N is the projective limit of the Hecke algebras hk(Np") acting (faithfully) on weight k modular forms of level Np" over Zp. This is enough to show it is complete, since hk(Np") is. By duality, any homomorphism of hk(Np") into a field of characteristic p comes from an eigenform of this level modulo p. Since there are only finitely many, say n, systems of eigenvalues of tame level N (cf. the Proposition 5.1.1), for large v, there exist maximal ideals ml (v), ... , Mn (v) of hk(Np") such that mi(v + 1) is the inverse image of mi(v) and hk(Np") = i
Hence, mi := limmi(v) is a maximal ideal of Tp,N, (Tp,N)m, is a complete " local ring and
Tp,N = ®(Tp,N)mi. I
3.4. Action on Mt(N). In this section we work with arbitrary p, square-free tame level N, and we assume that Np' > 5. Let E(q) E A(°) [[q]] be the q-expansion of the restricted Eisenstein family, so that
i(E(q)) = Ek(q)
for all is E B. By Proposition 2.2.7, for any prime number $ the ("multiplier") function EQ whose q-expansion is E(q)/E(qt) is an overconvergent
function on Zl (q f) if e # p, on Zl (p) if t = p > 5, and when p = 2 or p = 3 and 2 = p we work, rather, on the affinoid Zl (5q), making use of the strategy explained in section 2.5.
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We set Mt (N) = Mat (N) where the latter was defined in §2.4. This is an alteration of that given in section B.4 of [C-BMF] where the definition was the same except that B was replaced by B*. For prime numbers e, we will define operators Te copying Lemma B5.1 of [C-BMF] mutatis mutandis. If F E Mt(N) and e E (Z/NZ)* x Z* define
FI(f)* _ ([((t))] . E)
(E I(e)
I .
For prime numbers e let fe be the operator on A(B)[[q]] given by
Ot(E angn) = E anegn. n
n
Lemma 3.4.1. For each prime number e there is a unique continuous operator T(e) on Mt(N) such that FIT(e)(q) = ''e(F(q)) when £INp, and
(FIT(e))(q) =Ot(F(q)) +e-1(FI (e)*)(qe) when e does not divide Np. In fact, for each affinoid X of 8, and each prime
e there is a vt (X) > 0 such that the operator T(i) preserves the subspace M(N, X) (v) for all v in the range 0 < v < vt(X). The proof of the existence of the operators goes exactly the same as that of Lemma B5.1 of ibid., to obtain the assertion about T(t) and vt(X), one must use Proposition 2.2.7 and the property of the Hasse invariant used in the proof of Lemma 3.2.1. The nature of the vt (X) depends on the analytic properties of Et(q).
Proposition 3.4.2. The mapping F H F from overconvergent modular forms to Katz modular functions respects the Hecke module structure.
Proof. We omit the proof of this proposition in this article, noting firstly that the main thing to check here is the action of the diamond operators since the other operators can be checked by looking at q-expansions, and noting secondly that this check on the action of the diamond operators is not trivial: it will be done in a subsequent article. If we could show that the vt (X) could be chosen independently of e we would be in a position to put a natural topology on the algebra generated by the above operators (but see Chapter 7).
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3.5. Action on weight n forms. We use the same formulas as in the lemma of the last section. The existence of the operators follows as a corollary of that lemma by specialization. It follows from Proposition 3.4.2 that action of each of the Katz Hecke operators corresponding to T(() and (d) preserve M,, (K, N, v) (see section 2.4) for v > 0 and sufficiently small.
3.6. Remarks about cusp forms and Eisenstein series. For this discussion, let p > 2 and let us assume the tame level N is equal to 1. Let f be a (p-adic) overconvergent modular eigenform (of tame level 1) of finite slope. Consider its Fourier expansion f = E' o angTb (with coefficients an = an (f) E Cp).
Definition. We say that f is a cuspidal-overconvergent eigenform if ao = 0; that is, if the constant term of its Fourier expansion vanishes. We have included the hyphen between cuspidal and overconvergent because our condition only requires the parabolicity condition ao = 0 at the cusp oo and therefore, for example, even certain classical Eisenstein series whose Fourier expansion at the cusp oo have no constant term, but whose Fourier expansion at the cusp 0 have nonzero constant term, would be counted by the above definition as cuspidal-overconvergent, even though when considered as a classical eigenform they would not be cuspidal. This is specifically the case for the eigenform on Xo(p) which was called in [GMFM,CPS] the "evil twin" of the Eisenstein series of weight k and level 1.
Let us examine the properties of a non-cuspidal-overconvergent eigen-
form f of weight ic, of tame level 1, and of finite slope, whose Fourier expansion we may normalize to be in the form: f = 1 + alq + . Let Al denote the eigenvalue of the operator T1 on f (for primes t eigenvalue of Up, we have
p) and Al the
Al = Al ao(f) = ao(f ITe) = ao(f)(1 +£-1K([e])) = (1 +£ 'ic([t])), and similarly AP = AP ao (f) = ao (f I Up) = ao (f) = 1. It follows that the eigenvalues of the operators Te and Up on such an f are determined solely by the weight ic, and the eigenvalues for the operators Tn for n > 1 are given by o*(n) as in section 2.2. These eigenvalues are equal to the corresponding eigenvalues of the operators in question on E,, , the member of weight is of the basic Eisenstein family, as discussed in section 2.2 (for the weights rc for which E,, is defined; i.e., when (*(,%) # 0). In particular, all these eigenforms f are of slope 0. Now make a similar computation on the coefficient al, using the above evaluation of the eigenvalues, to give a
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formula for all of the coefficients a,,(f) (n > 2) as specific multiples of al(f) where the multiples depend only up n and the weight K. For example :
al(f) = a1(fITi) = At al(f) = (1 +e-1ic([Q])) . a1(f). Proposition 3.6.1. The only normalized (in the sense of the previous paragraph) non-cuspidal-overconvergent eigenforms of tame level 1 are Eisenstein series. Specifically, for any weight ic, if (* (ic) # 0 then the unique normalized non-cuspidal-overconvergent eigenform of weight s; of tame level 1 is Ek, while if (* (rc) = 0 there are no normalized non-cuspidal-overconvergent eigenforms of weight Ic of tame level 1.
Proof. Step 1. We first show that if i # 0, and if f and g are two normalized non-cuspidal-overconvergent eigenforms (of tame level 1) of weight rc, then al (f) = a1(g). Suppose that we are given two such eigenforms with al (f) : a1(g); we may assume with no loss of generality that a1 (g) 0. We already know that these eigenforms have the same eigenvalues for all the Ti's and for U,,. Now consider the modular form
f
al(f)
al(g) 9,
which is an eigenform of weight is and has Fourier expansion identically equal to the non-zero constant 1 - al(f)/al(g). But an overconvergent eigenform whose Fourier expansion is a constant is necessarily of weight 0 (for a discussion of this fact, see, e.g. sections 4.4, 4.5 of [K-PMF]).
Step 2. We next show that if (*(K) = 0, there are no normalized noncuspidal-overconvergent eigenforms (of tame level 1) of weight K. For in this case, despite appearances, E,, is cuspidal-overconvergent, and also is 0 (for
at ,c = 0, (* has a pole and not a zero). If there were a normalized noncuspidal-overconvergent eigenform f (of tame level 1) of weight r., then by the discussion above, for an appropriate constant c, f - c E,c would have its Fourier expansion equal to the constant 1, which is impossible since is 0.
Step 3. It remains to deal with the case of weight #c = 0. Here, the Eisenstein series E,, has its Fourier expansion equal to the constant 1, and if there were another normalized non-cuspidal-overconvergent eigenform f (of tame level 1) of weight 0, then f - E,, would have its Fourier expansion equal to a nonzero constant times 1 v* 1(n)q'. But this latter Fourier series is not the Fourier series of an overconvergent eigenform, by Lemma 4 of [CGJ], and therefore such an f does not exist.
En
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Chapter 4. Fredholm determinants. 4.1.
Completely continuous operators and Fredholm determi-
nants. Let p be any prime number, N a square-free tame level. The levels Npm and qN which occur below are assumed to be > 5. The results employed in this section are mainly due to Serre [S 1]. A special role is played in this theory by the Atkin-Lehner operator Up E ?-l for the important reason that it acts completely continuously (cf. section Al of [C-BMF] for the definition)
on Mk (Npm, v; K) for k E Z (cf. 3.11 and 3.12 of [K-PMF], B3 of [CBMF], and Proposition 11.3.15 of [G-ApM]), and we'll show it acts essentially completely continuously on the p-adic Frechet space of overconvergent forms of any weight ic E W over CP
lim M,,(Cp, v). v-+0+
Since the composition of a completely continuous operator and a continuous operator is again completely continuous, it follows that any operator in the ideal U C IL generated by Up is completely continuous in its action on the Banach space Mk (Npm, v; K). Let U E U be any element in this ideal, and
denote by U,, the operator of MK that it induces. When k E Z, the p-adic Fredholm theory applies, giving us well-defined Fredholm determinants
Pu(Npm,k,X,T) := det(1 - T T. UkIMk(Np'v; K; X)).
Here, for X a character on (Z/Np'nZ)*, let Mk(Np'n, v; K; X) be the subspace of Mk (Np', v; K) on which the diamond operators (d), d E (Z/qNZ)* act via the character X. These Fredholm determinants are independent of the choice of radius v (provided that v is in the range stipulated above) as one can see using proposition 4.3.2 below (Cor. H.3.18 of Gouv6a [G-ApM] and section A3 of [C-BMF]). As we'll see, in section 4.3, the operator U. acting on MK has a Fredholm determinant PC. (is, T). The Fredholm determinants are power series in the variable T (indeed they are entire functions of T with coefficients in the ring of integers of K, or Cp, with constant term 1, and hence are Fredholm functions in the sense of section 1.2). For the basics regarding these p-adic Fredholm determinants, see Chapter A of [C-BMF]. The key fact about them is given in Theorem 4.1.1 below. For any U as above, and any element is E CP define
MK,ty-w} C MK
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to be the CP vector subspace consisting of vectors in the kernel of some positive power of the operator U - u I (where I is the identity operator) acting on M,c. We shall refer to an element of M,, {U-u} C M,c as a generalized eigenform for U (with eigenvalue u). The subspace Mk {U-u} is stable under the action of W.
Theorem 4.1.1. Suppose Ic E WN. Over CP, the power series PU(n,T) may be written as a product
i PU(n,T) _ fl (1 - iij .T)e.i j=1
for J either +oo or a positive finite integer. Here the uj's are distinct elements in Cp - {0}; the ej's are positive integers (ej = the multiplicity of uj in the product expansion above) and the ordering of the uj is such that the rational numbers oj = ordp(uj) are increasing (but not necessarily strictly, of course). If J = +oo then vj tends towards oo. For any (nonzero) element u E Cp - {0} the Cp vector subspace,
M,c,{U-u} C M, of generalized eigenforms for U with eigenvalue u is ofCr-dimension equal to the multiplicity of u in the product expansion of the Fredholm determinant PU (k, x, T). That is, the dimension of M,. {U-u} is zero if ii is not equal to
one of the uj's and is equal to ej if u = uj. Proof cf. Propositions 11 and 12 of section 7 in [S-ECC].
Definition. Let M be a K-vector space and L : M -+ M a K-linear operator. A nonzero vector m E M will be said to have L-slope o E Q if there is a polynomial Q(T) E K[T] whose Newton polygon has a single side
which is slope -v such that m is in the kernel of Q(L). For v a positive rational number, and U as above, denote by Mrc,{slope(U)=o} C M,-
the Cp-vector subspace of vectors of slope v. The space Mr. {slope(U)=o} where uj ranges through the is generated by the vector spaces inverses of the zeroes of PU(tc,T) such that ordp(uj) = v. The subspace M,c,{slope(U)=o} is stable under the action of W.
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Proposition 4.1.2. Let a E Id be of the form 1 + p r for r E -l. Put U = a Up. The Cp-vector subspace M-,{slope(U)=a}
depends only on r, and a. It is independent of the choice of a.
Remark. More generally, Proposition 4.1.2 holds when a is a unit in the p-completion, Id,, := lim f/p i-l of the ring 71. Proof. Let a, a' E 9d be two elements of the form required in our proposition, and put U = a Up, and U' = a'. Up. For any U-eigenvalue ii of slope owe shall show that any element in Mk,{U-u} is generated by U'-generalized eigenforms with U'-slope equal to 0. This will establish the inclusion M.c,{slope(U)=a} C M,c,{slope(UI)=a}
and, by symmetry, the opposite inclusion as well. Fix, then, a u as above. Note that the operators a, a' E W induce nonsingular linear transformations of the Cp-vector space
M:=M.{U-u}, and more stringently they induce automorphisms of an Ocp-lattice in M. Using the same symbols a, a', U, etc. for the endomorphisms of M that they induce, let -y = a' a-' : M -+ M. The eigenvalues of ry are all units in Oc,,. Decompose the finite dimensional Cp-vector space M as a direct sum of (generalized) eigenspaces for the operator ry. We work on each of these (generalized) eigenspaces in turn. Let g E Cp be an eigenvalue of ry and consider now the CP subspace M. C M consisting of generalized eigenforms for y with eigenvalue g. We have the following formula involving (commuting) elements in the algebra Endc,,(M9):
yU - giiI = Since ry - g I and U - is I are nilpotent operators on M9 (and since all the operators in the above formula commute with each other) it follows that 'yU - gu I = U' - gv, I is nilpotent as well on Mg, concluding the proof of our Proposition.
Remark. We denote this Cp-vector space M,c {slope(U)=o} simply as M.,o
to reflect the fact that it depends only on o and not on the choice of a (provided that a is chosen as stipulated in the above proposition).
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4.2. Factoring Characteristic Series. Suppose A is a Banach algebra, (M, 11) is an orthonormizable Banach Amodule, and V is a completely continuous operator on M such that its norm IV I is < 1. In A2 of [C-BMF] the characteristic series of V operating on M, denoted det(l - TV) E A°{{T}}, was defined. By Corollary A2.6.1 of [C-BMF], if A is semi-simple, det(l - TV) only depends on (V and) the topology of M. Note that the definition of semisimple used here is that of [C-BMF]. Specifically, a normed ring A is semisimple if the intersection of its maximal ideals is the zero-ideal, and if, for every maximal ideal m c A, the residual norm on A/m is multiplicative. Note (cf. "Example (i)" on page 427 of [C-BMF]) that if A is a reduced affinoid algebra over a complete subfield of Cr and A is viewed as Banach algebra with norm equal to the supremum norm, then A is semi-simple. It follows from Corollary A2.6.1 and Lemma A1.4 of [C-BMF] that
Proposition 4.2.1. If A is semi-simple and Max(A) is a rigid space which has an admissible covering by affinoid subdomains Xi such that there exists an orthonormizable norm on MX; equivalent to the induced norm then any completely continuous operator on M over A has a characteristic series in A{T} which depends only on the topology on M.
Theorem 4.2.2. Suppose A is a semi-simple Banach algebra, (M, is an orthonormizable Banach module and N is a free submodule of M of finite rank such that there exists a continuous projector from M onto N. Then, locally, there exists a norm on M equivalent to I ( I such that both N and M/N with their induced norms are orthonormizable. I
Proof. Suppose {ei}iEI, for some index set I, is an orthonormal basis for M and {ni}iES is a basis for N where S is a finite subset of I. Write
ni = E ai,j ej jEI
It follows that we may cover Spec(A) with affine opens such that for each such open U, there exist {ji}iES such that det (a ,jk )i,kES
is invertible on U. We may suppose U = Spec(A) and ji = i. We may now change our basis for N so that aij = 6ij for j E S. Now {ni, ej}iES,jVs is a basis for M. We define a new norm I" on M by requiring this set to is equivalent to I Suppose be an orthonormal basis. We claim that I
mEMand
m=Ebini+1] bjej. iES
j0s
II
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Then IImll N is a continuous projector. Then l I7fII is defined. Let I
F = Ker 7r and 7r' = 1 - 7r. Clearly lei, 7r'(ej): i E S, j S} is a basis for M. Let I I I' be the norm on M determined by making this an orthonormal I
basis. W e claim that I I I ' is equivalent to 1 I
1
1
1
.
Let
M = E biei + E bi7r'(ei)
iES
ivs
Then,
biei - 7r(Ebiei) +>biei.
m= iES
iqS
ivS
It follows that IImII < Max{1, Ilirll}llmll'-
Also if m =
diei, then bi = di if i V S and
bi = di - E djcji, ivs if
7r(ej) = Y, cjkek. kES
We see that IImll' a,,(U) - T"
E AN{{T}}
uniquely determined by the property that for every weight , E WN, the image of PU(T) E AN{{T}} C AN[[T]]
under the homomorphism AN[[T]] -* CP[[T]] induced by rc : AN -+ Cr is equal to the Fredholm determinant PU(ic;T) E Cr[[T]]. Here PU(ic; T) refers to the Fredholm determinant of the completely continuous system of operators U on the system of Banach modules M,tt(N,1/i) for i large enough, where Mr ,t (v) is the Banach module over Cp consisting of v-overconvergent modular forms of tame level N. Before we begin the proof we need to talk about completely continuous systems of operators. First suppose (M, I 11) is a Banach module over a I
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Banach algebra (A, 11). By an orthogonal basis of M over A we mean a set {ei}iEl such that every element m of M is the limit of a unique series of the form T, aiei, iEI
where ai E A and limi,(,. llaieill = 0 such that 11mMI = maxiEJ{fail IIm1Il}. If JM* l = IA* 1, where M* = M- {0}, an orthogonal basis can be converted to an orthonormal basis. The proof of Proposition 3.1 of [C-CCS] yields:
Proposition 4.3.2. Suppose M and N are Banach A-modules, f : M -+ N is a continuous map of Banach A-modules, and UM: M -4 M and UN: N -* N are completely continuous (A-linear) maps of Banach A-modules such
that
M
M
N E4 N commutes. Then if there exists an orthogonal basis B of M such that f (B) is an orthogonal basis of N,
det(1 - TUM) = det(1 - TUN) E A{{T}}. Now suppose given a sequence of Banach A-modules Mn for n > 1 and continuous A-linear mappings for each n > 1, fn : Mn -* Mn+l with the property that there is an orthogonal basis Bn of Mn such that Bn+l :_ fn(B,) is an orthogonal basis of Mn+1. Let M = limMn. Suppose further that IM I = lA*l for all n > 1, and that there exists an orthogonal basis of MO whose image in Mn is orthogonal for each n. Then if U is an operator on M which restricts to a completely continuous operator on Mn for each n, we set
det(1 -TU) = det(1
E A{{T}},
for any n. Here we have used Proposition 4.3.2 to guarantee that det(1 is independent of n. We say that U is a completely continuous
operator on the sequence {Mn}n > 1. If the restriction of the operator U to Mn is of norm < 1 for all n > 1, then det(1 - TU) E A°{{T}}. More generally, we must allow the Banach algebra A to vary as well. So, suppose we have:
i. a sequence of Banach algebras An for integers n > 1, and contractive (this means norm decreasing) ring homomorphisms A.,,, -4 An for
m>n>1,
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ii. a function n H sn from the set of non-negative integers to nonnegative integers, and for each n > 1, a sequence of Banach An-modules {Mn,i}i>s such that IMn it = JA*J (for integers i > sn), iii. continuous An-module homomorphisms fn,i : Mn,i --4 Mn,i+i for all n > 1 and i > sn such that there is an orthogonal basis Bn,i of Mn,i with the property that fn,i(Bn,i) C Mn,i+l is an orthogonal basis of M; i+1, and for all m > n > 1 and i sufficiently large, is the completed tensor product of A,,,, and Mn,i.
Definition. We call the above structure a system of Banach modules. Given such a system of Banach modules M, to give a completely continuous system of operators V on M one must give a completely continuous operator Vn,i on Mn,i over An, for each n > 1 and i > sn, such that all the obvious diagrams commute and such that V,,,,,i = Vn,i 0 1 if m > n > 1 and
i>sm
Given a system of Banach modules M as above, we will say that M is a system of Banach modules over the sequence of Banach algebras {An}nForm the projective limit,
A:= lim An, of the sequence of Banach algebras and contractive mappings, An,, -4 An, given in i. above, to obtain a topological ring which we will call the limit ring A of the system M. Consider the topological ring A° := limA° C A (it is, in fact, a subring of A since projective limit over the ordered set of natural numbers is left-exact).
Proposition 4.3.3. Let V be a completely continuous system of operators on the system of Banach modules M. Let A = lim A,,,, be the limit ring of M and A° C A as above. There is a unique formal power series which we denote det(1 - TV) E A[[T]]
such that for each n > 1, the natural projection of det(1 - TV) to An[[T]] is the entire power series det(1 - TVn,il Mn,i)
for any i _> Sn. If the operators Vn,i of the system V all have operator norm < 1, (the norm of an operator on a Banach module was defined in section Al of [C--BFM]) then det(1 - TV) E A°[[T]]. Proof. This follows from Proposition 4.3.2 together with Theorem A2.1 and Lemma A2.5 of [C-BMF].
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We call det(1 - TV) E A[[T]] the Fredholm determinant of the completely continuous system of operators V.
Proof of Theorem 4.3.1. We now describe a specific system of completely continuous operators to which Proposition 4.3.2 applies.
The sequence of Banach algebras.
To give the Banach algebras, let us refer to the terminology of section 1.1 where to any complete noetherian local ring R is attached a rigid space X = XR given as a union of a nested sequence of affinoid subdomains, denoted X,. = XR,r C XR for positive rational numbers r, where X,. C X,.' if r > r'. Let us use the same notation, when R is a complete noetherian semi-local ring, to obtain XR as a union of a nested sequence of affinoid subdomains, XR,r C XR, by taking X,. = XR,r to be the disjoint union of the correspondingly denoted affinoids for each of the (finite) connected components of XR. Now put R = AN and, for every integer n > 1 define the Banach algebra An to be the affinoid algebra of XAN,1/n C XAN.
Note that for each n > 1 there is a natural continuous ring homomorphism AN -* An C An, and the mappings An -+ An+1 are AN-algebra homomorphisms, as well as contractive homomorphism of Banach algebras.
Proposition 4.3.4. For the above sequence of Banach algebras, the natural continuous ring homomorphism AN
is an isomorphism of topological rings. Any power series in AN[[T]] whose
projection to An[[T]] is an entire power series (over An) for all n > 1 is entire over AN, Proof. . The first sentence of the proposition is (essentially) Proposition 1.1 and Corollary 1.1.1 of [C-CCS]. For the convenience of the reader we sketch the outlines of its proof. Given our assumptions on the level, the ring AN is a product of regular local rings of the form O[[S]]; i.e., power series rings in one variable over complete discrete valuation rings 0 which are finite over Z. It suffices to prove the analogous statement with R = 0[[S]]. Let, then, R be such a ring, let I I be the multiplicative norm on the discrete valuation ring 0 whose value on any uniformizer is 1/p E R. Let K denote the field of fractions of 0 which we can take to be a (finite degree) field extension of Qp contained in Cp. Denote the maximal ideal of R by MR = (ir, S) C R (here
it is a uniformizer of 0). As discussed (at least for the discrete valuation ring Zp) in section 1.1, X = XR is the open unit disk, and one sees that for n > 1, An is the ring of rigid functions defined over K and bounded by 1 on the disk B[O,p-1/nl = {x E Cp xl < p-1/'n} in Cp.
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For every real number t in the range 0 < t < 1 define the t-norm I It on a power series E°_0 bnS' E R by the formula 00
E bnS"It = Maxn{
Ibnltn
}
n=0
which (if t E I C7, I) is the norm obtained by mapping an element of R into A°(B[0, t]) and then taking the supremum norm of its image. One checks
that for f E R, If it 8n,
for integers i > sn = 1. We have that JMM*jj _ IA*J. Denote by fn,i : Mn,i -4 Mn,i+1
(for all n, i > 1) the continuous An-module homomorphisms induced by the natural inclusion mappings of the corresponding affinoids. Moreover, for all m > n > 1 and i > 1 we have that Mnti i is the completed tensor product of Ana and Mn Note also that each Banach An-module Mn,i is orthonormizable, by Lemma A5.1 and the remark immediately following that lemma, in [C-BMF]. What remains to be done is to check that for each Banach An-module Mn i, we may find an orthogonal basis Bn i C Mn i such that fn,i(Bn,i) C Mn,i+l is an orthogonal basis of Mn,i+i. This follows from
i
Proposition 3.1 and Corollary 4.2.1 of [C-CCS]. Furthermore, by Proposition A5.2 of [C-BMF], combined with what has been said above, we see that the operator U (cf. section 3.4) on Mn',i is a completely continuous operator (on
the Banach An-module Mn i)
Remark. By the discussion in section 2.4, one has that for each n > 1 there is an integer, call it an, which is large enough so that if i > vn, the mapping
defines an isomorphism of the Banach An-module, Mn,i of rigid analytic functions on XAN, l/n x Zl (Nq) (v) onto the Banach An-module
Mt(N)xnN,l/ n (1/i) of v-overconvergent modular forms over XAN,i/n of tame level N, where v = 1/i. If i > Qn, Put Mn,i := Mt(N)xAN 11n(1/i), and let us concentrate on the system of Banach modules given by the Mn,i (i > an). We note that the Banach An-module Mn,i inherits an orthogonal An-basis from Mn,i which is brought (under fn,i) to an orthogonal An-basis of Mn,i+i.
The sequence of completely continuous operators. We now return to U = a Up given in the hypothesis of Theorem 4.3.1. This operator is the product of an element a E ?i which may be written as a polynomial in a finite number of Hecke operators TI, (j = 1, . . . , µ) and the operator Up, the coefficients of this polynomial being elements of AN. The operator Up, when viewed as acting on overconvergent modular forms F E Mn,i (for i large enough) is given by the formula
FlUp = EU(F/E).
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Recalling Lemma 3.4.1 (and the notation ve (X) described in that Lemma) let us put sn := Max{ an, vUP (XAN,1/n), ve; (XA,,11n); 7 = 1, ... , µ }
Define (for n > 1 and i > Sn) the transformations Vn,i : Mn,i -* Mn,i
to be the An-linear operator on (1/i)-overconvergent modular forms given by U = a Up. Since Ul, is completely continuous on Mn,i, so are all the operators Vn,i. By the formulas given for the operators T; and for Up in section 3.4, we see that the operators Vn,i are all of norm < 1. We have established
Lemma 4.3.5. The above system V := {Vn,i}n,i is a system of completely continuous operators of norm < 1 on the system of Banach modules M {Mn,i}n,i.
It follows from Propositions 4.3.3 and 4.3.4, that the system of operators V has a Fredholm determinant,
Pv(T) := det(1 - TV) E AN[[T]]
such that for each n > 1, the natural projection of PP(T) to An[[T]] is the entire power series det(1 - TVn,ilMn,i) for any i > sn. Moreover, by comparing the definitions one sees directly that for every weight r. E WN, the image of PU(T) under the homomorphism AN[[T}] -* C,[[T]] induced by is : AN -3 C, is equal to the Fredholm determinant PU(ic;T) E CP[[T]]. This concludes the proof of Theorem 4.3.1.
Remark. In the special instance where the element U is the Atkin-Lehner operator U,, itself, the coefficients of the Fredholm determinant may be given in closed form as in formulae of Appendix I of [C-BMF].
4.4 The Spectral Curves. In this section we again suppose that (q5(Nq), p) = 1 and Nq > 5.
Definition. For each U E U, define ZU C WN x A' to be the rigid space cut out by the Fredholm determinant, ZU : PC, (T) = 0,
viewed as rigid analytic (Fredholm) closed hypersurface (over Qp) in the (smooth) rigid analytic surface WN x Al (cf. section 1.2; recall that WN is
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weight space of tame level N and Al is the affine line parameterized by the variable T). From its definition we see that ZU C WN x A' is a rigid analytic Zariskiclosed subvariety which is (equi-dimensional) and of dimension equal to 1 (i.e., it is a rigid analytic curve). Since ZU is rigid-Zariski-closed in the nested (cf. 1.1) rigid analytic space WN x A' it follows that ZU is nested.
We will refer to the curve ZU as the Spectral curve attached to U. It is easy to find examples where ZU is disconnected. Can Zv have an infinite number of connected components? As in Appendix I of [C-BMF] one sees that the natural projection mapping
7r: Zv - WN can be of infinite degree.
Theorem 4.4.1. For any n E WN(Cp) and any non-zero element u E Cp, the Cp-valued point (rc,1/u) E WN x Al lies on the curve ZU if and only if there exists an overconvergent eigenform (eigenform for all Hecke and Atkin-Lehner operators) f with Fourier coefficients in Cp with the following characteristics: The overconvergent form f is of tame level N, of weightcharacter rc, and has U-eigenvalue equal to u.
Addendum. We also have that the Cp vector space of overconvergent (generalized) eigenforms with the above characteristics is of dimension equal
to the ramification index of Zv over WN at (rc,1/u). Proof. . This follows immediately from Theorems 4.1.1 and 4.3.1 (compare also: Lemma A2.5 of [C-BMF] and Proposition 12 of [S-ECC] or Theorem A4.5 of [C-BMF]).
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Chapter 5. Galois representations and pseudo-representations. 5.1. Deforming representations and pseudo-representations. Let p > 2 be a prime number, and let our tame level N be an arbitrary positive integer not divisible by p. Let S be the finite set of places of Q consisting of oo and the prime divisors of Np. Let GQ,s denote the Galois group of the maximal algebraic extension of Q in C which is unramified outside the set S. Let c E GQ,s denote complex conjugation. We begin by describing the residual representations of interest to us. By the catch-phrase residual representation we shall mean a two-dimensional vector space V over a finite field F of characteristic p endowed with an odd F-linear continuous GQ,s-action. By odd we mean that the determinant
of the operation c on V is -1 E F. In particular, we will be considering residual representations that come from modular eigenforms, by which we mean the following: If m is an integer and f is a (classical, cuspidal) modular eigenform on ['1(Npt) with ring of Fourier coefficients Of C Oc, where Of is a (finite) complete local ring extension of ZP with field of fractions of Of denoted by K f and with residue field isomorphic to F f, let Vf denote the associated two dimensional (irreducible) continuous representation of GQ,s over K f attached to f in the usual sense (cf. Section 3 of [D-D-T]). Let Of C Vf be a GQ,s-stable Of-lattice in the Kf-vector space Vf. Let us allow ourselves the abuse of notation
Vf=Of®ofF1, but note that in the case where Vf is not absolutely irreducible, the isomorphism class of the F f[[GQ,s]]-module Vf is not uniquely determined by f; it depends, as well, on the lattice Of C Vf chosen.
Definition. Let us say that a F[[GQ,s]]-module V comes from a modular eigenform (of tame level N) if its extension of scalars to a finite field extension field F C F' is isomorphic as module over the ring F'[[GQ,s]] to a Galois representation Vf OF f F', where f is a (classical) modular eigenform of tame level N, and F f -* F' is a homomorphism of fields.
By a p-modular, tame level N, residual representation (a pmodular representation, for short) let us mean a two-dimensional vector space V over a finite field F of characteristic p endowed with an F-linear continuous GQ,s-action and satisfying the following properties. 1. The F[[GQ,s]]-module V descends to no proper subfield of F. That is, there is no proper subfield F' C F and sub- F'[[GQ,s]]-module V' C V whose
change of scalars to F is isomorphic to V. Equivalently (since the Schur
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index of these representations is 1) the character of the GQ,S-representation V maps surjectively to F. 2. The F[[GQ,s]]-module V comes from some classical modular eigen-
form f on I'1(Npm) for some positive integer m. We call the pseudorepresentation associated to the semi-simplification of a p-modular repre-
sentation a p-modular pseudo-representation. Proposition 5.1.1. For given N and p > 2 there are only a finite number of isomorphism classes ofp-modular, tame level N, residual representations. Proof.
Step 1. Let us first show that there are only a finite number of such representations which are absolutely irreducible. Denote by p" an absolutely irreducible p-modular residual representation satisfying the further requirements in our proposition above. By basic representation-theoretic results over finite fields, one has that p is defined over the field generated over Fr by the traces of the p(g)'s for g E GQ,S (for example, see the Corollary of section 6 of [M-DTR] so we have from hypothesis 1. that F is generated by traces, as described above. Moreover, the isomorphism class (over F) of the F-representation p is determined by the character
g is Tracep (p(y)). . So we must show that there are only a finite number of character-functions occurring as characters of absolutely irreducible representations p satisfying our hypotheses. A quick way of doing this is to make use of the work surrounding Serre's Conjecture (see [S-RD2] and the discussion given in [Ri]). Specifically, note, that if p is odd, and p arises from a modular form on ri(Npm) (with N prime to p) then it arises from a modular form on F1(N). This is a well-known fact ( Remarque on p. 195 of [S-RD2]) and a proof is given in [Ri] (Theorem 2.1). Now it is also known (cf. Theorem 4.3 of [Ed]; see also the discussion in [Ri]) that any modular residual Galois representation p with values in a finite field of characteristic p and tame level N arises from an eigenform on r1(N) and having weight k(p) defined by Serre ([S-RD2] section 2). Since one easily computes from the formulas given in section 2 of [S-RD2]
For this, cf. the Corollary of section 5 of [M3]
that k(p) < p2 since it follows that if p is a p-modular residual Galois representation of fixed tame level N, then p is associated to one of a finite number of modular eigenforms, from which the proposition follows. Step 2. We are left with the task of showing finiteness of the set of isomorphism classes of (absolutely) reducible p-modular, tame level N, residual representations (p > 2). Here we first show that the characters of such representations (with values in the algebraic closure FP ) are finite in number.
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But any such character is the sum of two degree 1 characters 01+02 of GQ,S where S is the collection consisting of the prime at oo and the prime divisors
of pN. Moreover the conductors of the characters 0 which can occur are bounded. It follows (by the Kronecker-Weber theorem if nothing else) that only a finite number of O's can occur. Fix two such characters 01, 02, let K{y 1, 2}/Q denote the splitting field of the two characters (meaning the fixed field in Q of the subgroup of GQ,S given by the intersection of the kernels of the homomorphisms 01, 02 : Q -+ Fr). Now consider the set of isomorphism classes of two-dimensional Fr,-vector spaces V equipped with
continuous representations p : GQ,S -4 AutF(V) with character equal to 01 + 02 , and admitting a line (i.e., a one-dimensional Fr-vector subspace) stabilized by the action of GQ,s with character 01. The corresponding quotient space (which is also a line) has GQ,S character equal to 02. Consider such a {V, p}. The subgroup H = ker(p) C GQ,S has, as fixed field, an Abelian, exponent p, extension-field of which is unramified outside the set of primes of K{,p1,yp2} lying above S. Since the maximal Abelian, exponent p, extension-field of K{,p1,,p2} which is unramified outside primes
lying above S is finite, we see that the collection of all IV, p}'s we are presently considering have the property that there is a fixed finite quotient group 4P of GQ,S through which the p's all factor. It follows that there are only a finite number of {V, p}'s up to isomorphism. 1 We shall now study continuous, two-dimensional, odd GQ,s-representations over commutative topological rings A in which 2 is invertible; that is, continuous A-linear representations of GQ,S into the topological group of automorphisms of free modules M of rank two over such rings A, for which the complex conjugation c has determinant (over A) equal to -1 E A. Since 2 is invertible in A, by the symmetrizing and anti-symmetrizing with respect to c, we have a canonical decomposition of A-modules, M = M+ ® Mwhere
M':={mEMI c(m)=±m}. By the c-normal matrix description for the G-action on M we mean the matrix which describes the G-action in terms of the above direct sum decomposition: a(g) b(g)
g c(g) d(g)
where a(g) E HomA(M+, M+), b(g) E HomA(M-, M+), c(g) HomA(M+, M-), and d(g) E HomA(M-, M-).
E
By a c-normal A-basis for such an A[Gq,s]-module M is meant a Abasis {x, y} such that x is an A- generator of M+ and y is an A-generator of M-. Equivalently, it is a basis for which c(x) = x and c(y) = -y.
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Remark Let A = D, a complete noetherian semi-local (commutative) ring in which 2 is invertible. Then any (odd) two-dimensional representationmodule M over D admits a c-normal D-basis . To see this, consider the quotient of D by its Jacobson radical rad(D) which may be decomposed
as a product of fields D/rad(D) = fl' =1 kj, and let ej : D -4 kj be the projection to the j-th factor. Now form the direct sum decomposition of D-modules, M = M+ ® M- and the analogous direct sum decomposition of D/rad(D)-modules
M/rad(D) M = (M/rad(D) M)+ ® (M/rad(D) M)-. By noting that the dimensions of the vector spaces (M/rad(D) M)+ OD kj
over kj are all 1 one sees that (M/rad(D) M)} is free of rank one over D/rad(D). Choosing generators x and y of the free rank one modules (M/rad(D) M)+ and (M/rad(D) M)- respectively, and lifting these to elements x and y of the D- modules M+ and M-, we see by Nakayama's lemma that x and y are generators of M+ and M-, respectively, giving us a surjective D-module homomorphism
h:D®D-4 M ax ® by. Since M and D ® D are both free of rank 2 over D the surjection h is immediately seen to be an isomorphism. That is, {x, y} constitutes a c-normal D-basis of M. Now let us return to A, a commutative topological ring in which 2 is invertible and M a free rank defined by (a, b)
two A-module endowed with an odd, continuous, A-linear representation of
G = GQ,S. Let p : G -3 AutA(M) denote this G-action. Following [Wi], [T] (cf. also [H-NO]), and a suggestion of Buzzard, define the (continuous) functions
a=ap:G-#A; S=Sp:G-*A
by the rules:
a(g) a(g)
traceA(p(g)) + traceA(p(cg))
2
traceA(p(g)) - traceA(p(cg)) 2
Define the continuous function _ p : G x G -a A to be
(g, h) =:= a(gh) - a(g)a(h). If M admits a c-normal basis for the G-action on M and if we consider the matrix representation with respect to such a basis: a(g) b(g)
g c(g) d(g)
we have: a(g) = a(g), 5(g) = d(g), and (g, h) = b(g)c(h).
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Lemma. The triple of continuous functions
r :=
a, S)
satisfy the following relations (for all g, h, k, f E G = GQ,s):
a(g . h) = a(g)a(h) + 6(g, h) S(g h) = S(g)S(h) + e(h, g)
(gh, k) = a(g)6(h, k) + 6(h)6(g, k) 6(g, hk) = a(k)6(g, h) + S(h)6(g, k)
(1)
(2)
(3)
and a(1) = 6(1) = 1; a(c) = 1; S(c) = -1
(4)
(g, h) = 0 if either g or h is 1 or c. Proof. We leave this as an exercise to the reader, with the suggestion that each of the formula-checks can be made to follow from a single computation in an an appropriate universal situation. For example, one can check formula (1), the second equation being the only one that needs checking, by replacing
G by the group 9 generated by elements denoted c, g, h with the single relation c2 = 1, and replacing A by the Z[1/2]-algebra A which is obtained from the polynomial ring in twelve variables over Z[1/2] (four variables making up the matrix coefficients of a 2 x 2-matrix C, G, H for each of the three elements c, g, h) by inverting the determinants of G, H, and insisting that the determinant of C be 1 and its trace be 0. To check formula (1) in the context of the lemma, it suffices to check the analogue of formula (1)
for the representation c -+ C, g N G, h -* H of G on the free module M := A ® A or, since A is an integral domain, it suffices to check these formulas after extension of scalars to the field of fractions A, K. But, since K is a field, M ®A K admits a c-normal basis, with respect to which it is easy to compute everything.
G x G -+ A and a, S Recall that a triple r = a, S), where G -> A are continuous functions, is called a (continuous) pseudorepresentation of G = GQ,S with values in A if formulas (1)-(4) are satisfied. From (1) we see that r is determined by the pair of functions (a, S). If M is a free A-module of rank two equipped with a continuous, odd, Alinear G-action, and if p : G -* GL2 (A) is the corresponding representation, the A-valued pseudo-representation r = (Cp, ap, Sp) as defined above will be referred to as the pseudo-representation attached to the G-module M, or to
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the representation p. If r =
a, 8) is a pseudo-representation with values
in A, we define
Trace(r) = a +8:G-A, and, for g E G, put Det(r)(g) = a(g) . 8(g) - e(g, g)
One checks, using (1)-(4), that Det(r) : G -> A* is a (continuous) homomorphism. If r is the pseudo-representation attached to p, we have that Trace(r) = Trace(p) and Det(r) = Det(p). It follows from a result of Carayol and Serre (cf. the Corollary of section 5 of [M-DTR] ) that if A = D is a a complete noetherian local ring, and if p is absolutely irreducible, then the pseudo-representation r determines p up to equivalence.
The theory we have discussed, requiring that M be free of rank two over
A, generalizes immediately to the situation where M is locally free. But let us focus on this in the special case of continuous pseudo-representations of G = GQ,s with values in the ring of rigid analytic functions A = A(Y), where Y is a rigid analytic space. If Y is a rigid analytic space, consider pairs (V, p) where V is now a rank two vector bundle over the rigid analytic Y and p is a continuous, odd, Oy-linear representation of G = GQ,S on V,
p:G-iAuto,, (V), and where the action of complex conjugation, p(c) induces a decomposition of V into the direct sum of two line bundles, i.e. invertible Oy-modules,
V =V+ED V-, where p(c) fixes V+, and acts as multiplication by -1 on V-. We leave to the reader the exercise of defining the pseudo-representation r with values in A(Y) associated to such a representation p.
One of the aspects of the usefulness of pseudo-representations is that a, 8) with values in a field K which given any pseudo-representation r = is not of characteristic 2, a theorem of Wiles [Wi] guarantees that r is indeed the pseudo-representation attached to a representation p : GQ,s -+ GL2 (K). Moreover, the representation p is irreducible if and only if there exists a pair of elements (so, to) E GxG such that (so, to) 0. Compare Proposition 1.1 of [H-NO]). A simple version of this theorem over rigid analytic spaces can be proved using the same method, and we now prepare for its formulation.
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Definition. Let Y be a rigid analytic space over Qp and r =
a, S)
a continuous pseudo-representation of GQ,S into the ring A(Y) of rigid
analytic functions on Y. By the ideal of reducibility of the pseudorepresentation I, C Oy let us mean the sheaf of Oy-ideals generated by all the functions e(s, t) E A(Y) for (all pairs) (s, t) E G x G. For each point y E Y(Cp) evaluation of the pseudo-representation r at the point y gives a Cp-valued pseudo-representation ry to which (by the theorem of Wiles referred to above) one can attach an equivalence class of representations py : GQ,s -4 GL2(Cp). We have that py is irreducible if and only y is not contained in the support of the ideal I,.
Theorem 5.1.2. Let Y be a rigid analytic space over Qp and r =
a, 8)
a continuous pseudo-representation of GQ,S with values in A(Y). Suppose that there is a pair of elements so, to E GQ,S such that e(so, to) E A(Y) is a non-zero-divisor, and is a generator of the Oy-module I,. Then there is a continuous representation p : GQ,S -- GL2(A(Y))
whose associated pseudo-representation is r. Proof. This is just an exercise in globalizing Wiles' theorem; one can use the argument of Proposition 1.1 in [H-NO], with hardly any change, to prove the above theorem. Explicitly, for g E G, we repeat the formulas of [H-NO] (after adjusting for the changes in powers of 2). A representation p is given by
gH
4
(so, g)
09)
>
fI
.
Remark. In particular, if Y is a smooth rigid analytic curve, and r = (1;, a, 8) a continuous pseudo-representation of GQ,S with values in A(Y) then Y can be covered by open rigid analytic subspaces U over each of which the restriction of the pseudo-representation r satisfies the hypothesis of the above theorem, or else has the property that the function is identically
0. If the open U falls into the first category, Theorem 5.1.2 applies to the restriction of r to U giving us a rigid analytic GQ,s-representation over U whose corresponding pseudo-representation is r restricted to U ; if it fall in the second category, the corresponding functions a and 8 restricted to U are continuous characters of G with values in OO, and r restricted to U is the pseudo-representation associated to direct sum of the two degree one GQ,s-representations given by the characters a and 6. Can one get a finer, more global theorem along the above lines by considering Oy-linear representations of GQ,S into the groups of units in Azumaya algebras (of rank 4) over Oy?
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Deformations of pseudo-representations compile well forming universal deformation rings as we demonstrate below in the two dimensional case. For the general case, under the assumption that the pseudo-representation is irreducible, see [Ny] or [Ro].
Theorem 5.1.3. Let k be a finite field of characteristic p
2. Let denote a pseudo-representation of G = GQ,S with values in k. Then there is a universal lifting of 1 to a pseudo-representation I/i"" of G with values in a complete noetherian local ring Runiv(v') with residue field k which is universal in the sense that given any pseudo-representation b with values in a complete noetherian local ring D with residue field k, such that 0 is a lifting of , there is a unique continuous homomorphism of local rings
7ro : Runiv
D
reducing to the identity on k, such that 0 is the composition of ,/,univ and 70 .
Proof. One can prove this easily using Schlessinger's criterion [Sch]; compare [Ny]. Briefly, fix a pseudo-representation = of G = GQ,s
into k. We consider deformations of . Namely, let d be the category of complete noetherian local rings with residue field k. For A E C, let F(A) be the set of pseudo-representations of G into A lifting. Suppose we have a Cartesian diagram of elements of d:
Al
A2
The natural map b: F(A3) -3 F(A1) XF(Ao) F(A2)
is an isomorphism and we will write down its inverse. Suppose, ?Pi = (ai, Si, 4i) are pseudo-representations with values in Ai for i = 1, 2 and the images of these in A0 are the same. Then e(bi, /2) =: ((ai, a2), (S1, S2),
e2))
is a pseudo-representation with values in A3. Clearly, e is the inverse of b. Thus conditions H1, H2 and H4 of [Sch] are automatic. The only thing we have to check is H3. That is, we must show that for A = k [E] , the dual
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numbers over k, that F(A) = F(k[e]), which is a k-vector space, is finite dimensional, or equivalently, is finite. is irreducible (which, by definition, means that there is a pair of Case 1.
elements s and tin G such that e(s, t) # 0). Then as mentioned above (and compare any of our basic references; e.g., [H-NO,§1]) we know that for any b E F(k[c]) there /exists an irreducible representation p: G -+ GL2(F(k[e]))
such that p(c) = I 0 11 I whose associated pseudo-representation equals . Because any\ two representations that have the same character, take the value I
01
the form (
0
)
on c, and are irreducible, must be conjugate by a matrix of
0 ) where l and k are in F(k[e])*, if we fix one representation,
into GL2(k) whose associated pseudo-representation is ', then there is a unique strict equivalence class of liftings of it with pseudo-representation ,0. By [M-DGR], the vector space of strict equivalence classes of liftings of p is isomorphic to H1(G,Ad(p)) which is of finite dimension over k. Case 2. /i is reducible. That is, (s, t) = 0 for all s and tin G. It follows from
(1) and (4) that, a =: a and d =: S are characters on G with values in V. Let 0 = (a, /3, ) E F(k[e]). Write a(s) = a(s) + eR(s), ,3(s) = b(s) + eS(s) and (s, t) = EX (s, t). Then (3) is trivial and if R' = R/a and S' = S/d, (1) implies that R'(st) - (R'(s) + R'(t)) = X (s, t)/a(st) S'(st) - (S'(s) + S(t)) = X (s, t)/d(st).
(5)
(6)
Moreover, if y(s, t) = X (s, t)ld(st) and X(s) = a(s)/d(s), (2) implies: y(st, r) = X(s)y(t, r) + y(s, r) y(s, ru) = y(s, r)X(u) + y(s, u).
(7)
In other words, y is a continuous bi-cocycle of G with values in k where the action of G on k is via the character X. That is, r y(., r) is a one-cocycle
hr and if V is the k vector space of one-cocycles, the map r H (hr-1) is a one-cocycle with values in V where G now acts via the character X-1 Since, G satisfies the finiteness condition 4P of [M-DGR], we see first that V is a finite dimensional and then that the space of these cocycles is finite dimensional. Finally, given y and hence C, we see that the difference of any two R"s satisfying (5) or of any two S"s satisfying (6) is a homomorphism from G into k. Since there are only finitely many such homomorphisms, the set of liftings of & to k[e] is finite.
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Remark. In the case where the pseudo-representation ' ' 1comes from a continuous, odd, two-dimensional representation p of G = GQ,S with values
in the finite field k with the property that p is absolutely irreducible (or more generally, such that the only endomorphisms commuting with p are scalars) then there exists a universal deformation ring R = Runiv(p) as in [M-DGR] and a universal deformation puniv : GQ,s --f GL2(R)
of the residual representation p. In this case, we have that the universal deformation ring Runiv (p) is canonically isomorphic to the universal deformation ring Runiv() of the pseudo-representation 1 attached to p, and (making the canonical identification of these two rings) the pseudorepresentation attached to puniv is ,/,univ 4 From now on, in this article, we suppose that N = 1. By Proposition 5.1.1, there are only a finite number of non-isomorphic pmodular (modular, tame level 1, residual, characteristic p) representations. Let i,b1i ?2i ... , z%i. be the set of the distinct pseudo-representations corresponding to these p-modular (modular, tame level 1, residual, characteristic p) representations. So the set of Oj's is in natural one:one correspondence with the set of (isomorphism classes of) semi-simplifications of p-modular (modular, tame level 1, residual, characteristic p) representations. We will refer to any one of these pseudo-representations 0j (j = 1, ... m) simply as a p-modular pseudo-representation. For each p-modular Eli, let R1, denote its universal deformation ring as constructed in Theorem 5.13. Put j=m
Rp = 11 Rpj. j=1
We think of Rp as the universal p-adic deformation ring of residual representations which are p-modular, tame level 1, (pseudo-)representations,
and will refer to Rp as the universal deformation ring (of tame level 1) for short. Of course, to be more precise we should perhaps qualify the adjective universal to emphasize that the only residual representations we consider are the p-modular ones, defined at the beginning of this section. The ring Rp is a semi-local complete noetherian ring, because of the proposition proved at the beginning of this section, and it comes along with a universal pseudo-representation, runiv = ({univ G,univ, funiv)
with values in Rp.
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For each p-modular pseudo-representation Vi, let X,p denote the rigid analytic space over Qp associated to Rp, and let XP = fl 1 X,p be the rigid analytic space over Qp associated to Rp. We will refer to Xp as the universal deformation (rigid analytic) space (of tame level 1). As hinted above, there is a well-behaved part of Rp and a not-so-well-behaved part. Specifically, when 0 is an irreducible p-modular pseudo-representation
(in the sense that its associated e-function does not vanish identically), then 0 is the pseudo-representation of an irreducible (odd, degree two) Grepresentation. Any such representation is absolutely irreducible (because an irreducible, but absolutely reducible, degree two representation is seen to have Abelian image, and therefore it stabilizes the two eigenvectors of the image of c). Let V denote a k-vector space of dimension two, with klinear G-action having O as corresponding pseudo-representation. Then V , being an absolutely irreducible residual G-representation, has an associated universal deformation space (cf. [M-DGR]) which we denote Rp equipped with a continuous G-action on a (free rank two) RV-module V (the universal deformation of the G-module V) which satisfies the universal lifting property for deformations of the G-module V (or equivalently, the G-representation to AutR V is universal for strict equivalence classes of liftings of the Grepresentation on V). Passage to the corresponding pseudo-representation of the G action on V induces a natural isomorphism (cf. [Ny]) R p ^_' Rp.
Let Oj (j = 1.... p) be the subset of the set of p-modular pseudorepresentations which are irreducible, and Oj (j = µ + 1,...m) the pmodular pseudo-representations which are not. For each j = 1,...,u fix a degree two G-representation Vj with pseudo-representation equal to Oj. Define RV,, j1=11
j=1
and
m RPred
:_ 11 R
V)i'
Then R = Rsrr x Rred p
p
p
and corresponding to this product decomposition, we have a decomposition of associated rigid analytic spaces, XP = Xirrp U XPred
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The Eigencurve
The essential difference between the two factors Rp r and Reed for us is that over the first factor we have a continuous representation puniv-irr : G -* GL2(RP r)
satisfying a universal property vis a vis irreducible p-modular residual representations (of tame level 1) whereas over the second factor we must make do with a continuous pseudo-representation (which nevertheless does indeed satisfy a corresponding universal property for pseudo-representations). The representation puniv-irr induces a rigid analytic, OX,rr-linear G-represenP tation on the free OXirr-bundle of rank two over Xirr In contrast, we do not show the existence of an analogous rigid analytic, U` XPed-linear Grepresentation over Xped. Nevertheless, Theorem 5.1.2 will provide partial results along these lines, applicable to open smooth neighborhoods of the modular curve.
The ring Rp has "a" natural A-algebra structure. This is defined by noting that the (determinant) mapping p : G -4 Rp of the universal pseudorepresentation, given by the formula p(g) := auniv(g)Suniv(g) _ univ(g)
is, in fact, a determinant: it is a continuous homomorphism to RP* and factors through the Abelianization X : Gab = Z;, where we have identified Gab with ZP via the cyclotomic character X; that is, if C is a p-power root
of unity, g(c) = X(g)
C. This mapping ZP -4 Rp extends uniquely to a continuous ring homomorphism which we will denote here pdet : A = Zp[[Zp]] -+ Rp and which gives a A-algebra structure to Rp that might be referred to as the determinant A-algebra structure on Rp (for, tensoring Rp via this A-algebra structure with a character
K:G-*Z; gives the universal ring of pseudo-deformations of possessing determinant character ic; for more discussion about this see section 24 of Chapter 5 of [M-DTR]). One should note that when one works with modular forms, as we are doing, it is more conventional to twist this A-algebra structure p = pdet
to get a A-algebra structure on Rp that gives directly the weight of the modular form, rather than the determinant of its (pseudo-)representation. The conversion formula is P.t([T]) = -'' p-det(['Y]),
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where for ry E Zp, we denote its image in the completed group ring A by [7], and the multiplication-law in the right-hand side of the above formula is via the natural Zp-algebra structure of Rp. We will view Rp from now on with its weight A-algebra structure, i.e., the one given by pwt structure. The rigid analytic space Xp is thereby endowed with a canonical projection to weight space Wl = W.
5.2. Pseudo-representations attached to Katz modular functions. Keep the notation of the previous section. Let TP := Tp(1) denote the completion of the algebra generated by Hecke operators acting on Katz modular functions of tame level 1 as described in section 3.3. Then as proven in §3 of [H2],
Theorem 5.2.1 (Gouvea-Hida).
There exists a continuous pseudo-
representation rr from Gq,{p} to Tp such that, for primes l
"trace"(ir(Frobl)) =T(l)
and
,{'p,
"det"(x(Frobt)) = (l)*/l.
In particular, as TP is the inverse limit of Noetherian rings (see section 3.3), we get a natural (unique) ring homomorphism Rp
Tp
bringing the universal pseudo-representation of tame level 1 to ir.
Corollary 5.2.2. Suppose m is a maximal ideal ofTP such that the corresponding residual representation p is absolutely irreducible (cf. section 5.1). Then, there exists a representationp,,,.: Gq,s 4 such that, for 1 ,{'Np,
trace(p(Frobz)) = T(l)
and
det(p(Frobt)) _ (l)*/l.
Corollary 5.2.3. If p > 2, the ring TP of Hecke operators on Katz modular functions of tame level 1 is noetherian. Proof. We know that TP is a complete semi-local p-adic ring, so all we have to show is that the completion (Tp),,,, is noetherian for any maximal ideal m. Let T , denote the subring of Tp which is the completion of the A-algebra
generated by the Hecke operators T(l) for (1,p) = 1. Then it follows that T'' = li_m hk (p k) where hk (pk) is the subalgebra of hk (pk) generated by the diamond operator and the Hecke operators T (I) for (1,p) = 1. The pseudo-
representation ir above actually takes values in Ti,. Let m' = m fl TIP and
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81
let Tn denote the reduction of ir: GQ,{p} T, modulo m'.* It follows that there exists a homomorphism of R(im) into (Tp),,,, whose image contains the image of T(l) and (l)* for all 1 # p. Since R(2ti,n) is compact, using the Tchebotarev density theorem, we see that this homomorphism is surjective. Hence this complete local ring (T'),,,, is Noetherian. (When m corresponds to an absolutely irreducible representation this is the conclusion of Corollary 111.5.7 of [G-ApMF].) We need the following general proposition to conclude the proof:
Proposition 5.2.4. Suppose Rn are complete p-adic local rings and Rn+l -+ Rn are surjections. Then R = lim Rn is a complete local ring. n
Let m be its maximal ideal and k its residue field. The ring R is Noetherian if and only if the dimensions of the Zariski tangent spaces dimk mRn/m2Rn are bounded independently of n. Proof. The Noetherianness of R implies the uniform boundedness of the dimensions of the Zariski tangent spaces. If now dimk mRn/m2Rn is bounded
independently of n, it follows that d =: dimk m/m2 is finite. Suppose t1, ... , td E m is a basis modulo m2. Then there is a natural map from W(k) to R and the map W(k)[[Tl,...,Td]] -+ R, which sends T$ to ti is a surjection. Since W (k) [[T1, ... , Td]] is Noetherian we may conclude the proof.
We can now complete the proof of the corollary. Let Tn = h2 (p') and Tn = h2(pn). Then Tn = T,*a[un], where U = limun. Let k = T/m and k' = T'/m'. Since, (T')m' is Noetherian, there exists a1,. .. , ad in m' which reduce to a basis of m'/(m')2 over V. Let g(x) E k'[x] be the minimal monic polynomial for U mod m over k', which exists since k is finite. Suppose f
is its degree. Now suppose n is such that k' = Tn' /M'T' := m' and T/mT := Mn- We identify these isomorphic fields. Let g be a lifting of g to a monic polynomial in T' [x]. It follows that ao =: g(un) E M' We can write any element a of Tn in the form k
-
a=
bj(un)9(unA
* We note that, T'p is also a complete semi-local ring with finite residue fields. Moreover, the map Max(Tp)->Max(T'p) is surjective by Atkin-Lehner theory and bijective away from the ordinary points.
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where bj (x) E T' [x] and deg bj (x) < f
.
Then a E m,, if and only if
bo(x) E m' T' [x]. This means that, in this case, we can write d bo(x)
Y' ck(x)ak k=1
where ck (x) E Tn [x]. We conclude,
a = bi(un)ao + c1(un)ai +
. cd(un)ad mod mTn.
In particular, dimk mn/mn < dimkl m;,/(m'n)2 + 1,
which concludes the proof i Corollary. dimk mlm2 < dimk' m'/(m')2 + 1.
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83
Chapter 6. The Eigencurve. 6.1. The definition of the eigencurve. Keep the notation of Chapter 5. We assume that p > 2, and recall that we are working with tame level N = 1. Let Rp be the universal deformation space for (p-modular) pseudo-representations of tame level 1, as before, and X, is the associated rigid analytic (universal deformation) space of tame level 1. Recall that RP has a natural V-algebra structure, and is the receiving ring for the universal pseudo-deformation r""' of GQ,S with the notational conventions of Chapter 5. Let Yp denote the rigid analytic variety given as the product
Y=Yp=XpxA', where the parameter for A' will be given by the variable T. We prepare to apply the constructions of section 1.2 to Y. In particular, putting R = Rp for short, we have a natural homomorphism of the ring of entire power series R{{T}} into the ring A(Y) of rigid analytic functions on Y.
Let a E ?-l' be any element such that the image, t a E R, is a unit in R (where t : W -+ R denotes the canonical homomorphism). Let U E U be the product of a and Up. Let PU(T) E A{{T}} be the Fredholm determinant of the completely continuous system of operators associated to U as discussed in Chapters 3 and 4 (and was also denoted Pc.(T) in Chapter 5). Put OU := PU(T/(t' a))
which, by Theorem 4.3.1 and our hypothesis about a, is a Fredholm series in the ring R{{T}}, and consequently determines a Fredholm function on Yp, which we will denote by the same letter. Consider the ideal I in A(Y) generated by the Fredholm functions OU for all a's which satisfy our hypothesis.
Definition 1. The eigencurve of tame level 1 is the closed, rigid analytic QP subvariety,
C=Cp C Yp = XpxA'
whose ideal of definition is I E A(Y).
Remark. We shall be proving that C is a curve (cf. below) but this fact is not evident from the definition. Equivalently, we can think of C as follows: For every a E 9-l' such that t a is a unit in R we define the mapping ra: Xp x A' -* W x A' by t ra(x,t)7r (x), t a(x) )
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where it : Xp -4 W is the canonical projection to weight space (see section 5.1). Then Cp
= flr-1(Z,,,.u),
a where the spectral curve Z,,.U is the locus of zeroes of the Fredholm function
P«.U(T) on W x A'. By its definition, the curve Cp comes along with a rigid analytic projection to the universal deformation space Cp -* Xp and therefore by composition of this morphism with the canonical projection, 7r : X -+ W, we have a canonical projection, Cp - W, of Cp to weight space. There are also
natural morphisms of Cp to ZU for every U = a Up with t a E R*, and these natural morphisms are compatible with projection to weight space. Explicitly, we have the ring homomorphism
tU : Al{t}} -+ R{{T}} which extends the natural ring homomorphism t : A -+ R by sending the
variable t to T/(t a). The ring homomorphism tU brings the Fredholm determinant PU(T) E A{{T}} to ¢U E R{{T}}. Since PU(T) generates the defining ideal of the rigid analytic curve (the Fredholm hypersurface)
ZUCWxA' and since 0U is in the defining ideal of Cp C Xp x A', tU induces a rigid analytic morphism, call it aU : Cp -* Z. Given a Cp-valued point c E Cp let us refer to its two coordinates c =
(r, 1/u) E Xp x Al as its associated pseudo-representation r = r. and inverse Up-eigenvalue respectively, and we will refer to the quantity u = u, as the Up-eigenvalue of c. The point c induces a ring homomorphism
T,:'1->Cp as follows. The associated pseudo-representation r = r, induces a homomorphism ',.: fl' -+ Cp as in the previous discussion and IC is the unique (ring homomorphism) extension of IQr to the polynomial ring fl = fl'[Up] which sends Up to u E Cp. As in the discussion in Section 3.1 we may attach to the ring homomorphism T, a Fourier expansion F, which we will
refer to as the Fourier expansion of the point c E Cp: n=oo
E W.,(T)gn E Cp[[q]],
n=1
where the conventions regarding the symbols Tn are as in Chapter 3.
The point c E Cp is determined by its Fourier expansion F,.
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6.2. The points of the eigencurve are overconvergent eigenforms. Recall that an eigenform f for (the full A-algebra of operators) 9d, with Fourier expansion f = 1:n00 o anq', is called normalized if either al = 1, or else we are in the specific case of weight 0 where it is possible for al to vanish, in which case the standard recurrence relations that hold for the Fourier coefficients of a f -eigenform force all the nonconstant coefficients of f to vanish; in this idiosyncratic case, let us call f normalized if f is the constant power series 1. For a positive integer n this normalized eigenvector has TT-eigenvalue o*,(n). Any eigenform for f is a multiple of a normalized eigenform.
Theorem 6.2.1. There is a one-one correspondence between the set of normalized overconvergent modular eigenforms with Fourier coefficients in CP, of tame level 1, weight w E W(Cp), having nonzero UP-eigenvalue, and the set of Cp-valued points of weight w on the eigencurve C = CP. If f is a normalized overconvergent modular eigenform (of tame level 1) with Fourier coefficients in CP and with nonzero UP-eigenvalue with Fourier expansion n=oo
f=
angn E Cp[[q]], n=0
with al = 1, then the point c E C to which f corresponds has its Fourier expansion equal to f - ao (i.e., the Fourier expansion of c is that of f deprived of its constant term). Proof. Starting with an overconvergent modular eigenform f with the characteristics formulated in the statement of the above theorem, one sees (by Theorem 4.4.1) that there is a point zU E ZU for every U E U corresponding to f and therefore, one immediately sees from the construction of the rigid analytic space Cp that there is a (CP valued) point c E Cp corresponding to f as asserted in the statement of our theorem. It is somewhat more delicate to go the other way. For this, let us start with a Cp-valued point c of the eigencurve C = Cp. We shall fix c in the discussion below, and define a list of objects dependent upon c. Note that we are working systematically, in the proof below, over CP. Denote the weight of c by w (i.e., w is the CP valued point of W which is the image of c under the canonical projection Cp -* W). AU (c) E ZU such that to is in R*, let zU For U = a Up with a E
The points zU = (w, 1/w) E W x A' are, of course, of fixed weight w. Moreover, we have that ordp(u) is equal to v := the slope of c, i.e. Q = ordp (u) where u is the Up-eigenvalue of c (Note that ordp (u) = ordp (u)
since to E R*). Applying Theorem 4.1.1 to the points zU E ZU, we have
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that, for every U as above, there is at least one overconvergent modular form of weight w which is an eigenform (not just-a generalized eigenform) for U = aUP with eigenvalue u. Moreover, the U-eigenvalue of each such eigenform is equal to u = the U-eigenvalue of c. The CP-vector space VU of these overconvergent U- eigenforms (of weight w, tame level 1, and Ueigenvalue equal to u) is finite dimensional, and is stabilized by the action of the commutative A-algebra W. Therefore VU contains eigenforms for the action of the entire algebra W. It follows that VU contains at least one normalized I.1 -eigenform. Let F& denote the set of all normalized eigenforms in V. That is, FU is the set of normalized overconvergent modular forms of tame level 1 which are I.1 -eigenforms and such that their
U-eigenvalue is equal to u. Then for every U = a . U, FU is a finite nonempty set.
Lemma 6.2.2. The intersection of the sets FCT (running over all U's as above) is nonempty and, in fact, consists in precisely a single eigenform f. Remarks and Proof of Lemma 6.2.2. Assume, for a second, that there is an eigenform
fEn.TU. U
It follows that f has the same Fourier expansion as c. To see this, first note that f has the same UP-eigenvalue as c (because f E Fu,,). Moreover, for T E Ii' and U = (1 + p T) . UP, since f E FU, we have that f has the same U-eigenvalue as c. Since the operators UP and U commute, it follows that f also has the same r-eigenvalue as c. This argument yields two things. First, we see that it suffices to show existence of f, for then its uniqueness follows. Second, we see that it suffices to show that
n U
is nonempty where the intersection is taken over a special list of U's. Specif-
ically, let fli £2, ... be an enumeration of all the prime numbers different from p so that UP, TE1, Tee ... is an enumeration of the full list of Hecke and Atkin-Lehner operators.
Sublemma 6.2.3. If for a fixed infinite sequence of positive integers al, a2, ... and for the list Uo = UP
U1 = (1 + pa1TE1)Uo
U2 = (1 +pa2Te2)Ui
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of operators, the intersection no F& is nonempty as U runs through Uk for k > 0, then this intersection is nonempty as U runs through all (1 +p - T) - Up. In the proof of Lemma 6.2.2 that we give below, we shall assume given an enumeration of the primes (different from p) as above, and we will be generating inductively a particular sequence of positive integers a1, a2i .. . as required by sublemma 6.2.3 to prove lemma 6.2.2.
Step 1. Consider all the U's of the form U = a Up where a = 1 + p T for ,r E H. For these U's, the eigenforms of fixed weight w and eigenvalue of fixed slope a all lie in the vector space M,c,o
defined at the end of section 4.1 (here, ic = w) which we denote simply M,
for short; recall that M comes along with a natural f-module structure. By Proposition 4.1.2, this (finite dimensional) M is generated by all overconvergent generalized U-eigenforms of weight w and U-slope equal to or, where U is any element of U as above. In particular, it is generated by all overconvergent generalized UP eigenforms of weight w and UP slope equal to or. Since there are only a finite number of eigenvalues for the operator Up acting on the (finite dimensional) space M we may find an integer a1 large enough to distinguish Up-eigenvalues in the sense that if A and A' are distinct Up- eigenvalues in its action on M, then ordp(A - A') < a1.
Step 2. Put a1 := 1 +pal Tel, and U1 = a1 Up. Here, £1 is the first prime given in our enumeration above, and a1 is the integer obtained in Step 1. Let
fl E so that the U1-eigenvalue of fl is the same as that of c. We claim that it then follows that both the Up- and the Ttl Ceigenvalues of fl are the same as those of c. To see this, let us establish notation for the UP and the Ttleigenvalues of fl and of c by the formulas: Up
fl = tti,(f) fl
;
Tt, c = tt"(c)
c.
Denoting by ul the common Ul-eigenvalue of fl and c, we have two expressions for ul; namely: ill = (1 +pal ttl,(f)) up,(f) = (1 +pal . tt1,(o) up,(c)
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Since all the elements of the above formula lie in the ring of integers of Cp, it follows that ordp(up (f) - up,(,)) a1 which implies, by Step 1, that up (f) = up (c), giving also that ttl (f) = tel (c). Since there only a finite number of eigenvalues for the operator Ul acting on the (finite dimensional)
space M we may find an integer a2 > al large enough so that if A and A' are distinct U1- eigenvalues in the space M, then ordp(A - A') < a2 - a1.
Step 3. We follow the pattern of Step 2. Put a2 := 1 + p12 TI,, and U2= a2 - U1. Let f2 E 'FU2
so that the U2-eigenvalue of f2 is the same as that of c. With the analogous notational conventions as in Step 2 (i.e., the eigenvalue of an operator on f2 or on c is denoted by the symbol denoting the operator, only in lower case, and with an (f) or a (c) in the subscript) we have:
i = (1 + pat . 42,(f)) . ul,(f) = (1 +p¢2 - te2,(C)) - "L(C)'
By an analogous argument to that of Step 2, since a2 is large enough to distinguish U1-eigenvalues, we get from the above equation first that the Ul-eigenvalue of f2 and of c are equal, and then that the Tee- eigenvalues of f2 and of c are equal as well. Since the Ul-eigenvalue of f2 and of c are equal, a direct application of the argument of Step 2 then gives that f2 and c agree on their Tel- and UP eigenvalues as well. Let a3 > a2 be an integer large enough, now, to distinguish U2-eigenvalues in the action of U2 on M.
Step 4. Proceeding by induction as in Steps 2 and 3 we may obtain a sequence of integers a1 < a2 < ... ak ... and corresponding operators Uk for k = 1, 2, ... and overconvergent modular forms fk E TUk C M which have the property that fk and c agree on their Up-eigenvalues as well as their Tt3 -eigenvalues for j = 1, . . . , k. Since the sets 'FU,, are finite, it follows that
there is at least one element f E nk 1.FUk C M which, by the previous discussion, is then an eigenform for Up and the Hecke and Atkin-Lehner operators TI for all prime numbers .£ # p. This f is then an overconvergent (normalized) modular eigenform of tame level 1 whose eigenvalues for
the action of any element in f is the same as that of c. This establishes Theorem 6.2.1.
6.3. The projections of the eigencurve to the spectral curves. But the above proof gives more, and to record what it gives, let us consider multiplicities of various spaces of generalized eigenforms. Define: µ(c) := the dimension over Cp of the space of all overconvergent generalized eigenforms (for f) in the w-isotypic component corresponding to c in M = Mk(Cp, )a.
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89
Similarly, for U E U of the appropriate form (i.e., U = (1 + p T) Up with r E 7-l) let z E ZU be the image of c under AU, and put µ(z) := the dimension over Cp of the space, call it MU c M, of all overconvergent modular forms in M = Mk(Cp)o which have the same weight as c, and which are generalized U-eigenforms with the same U-eigenvalue as c. We have the following geometric properties of the projections of the ZU's to weight space, and the morphisms A : C -> ZU (using the Reisz Theory developed in A4 of [C-BMF]).
Proposition 6.3.1. The structure of ZU over weight space. Let U = (1 + p r) Up with r E W. Then ZU is a Fredholm hypersurface
in W x A'. Proof. This follows from Theorems 4.3.1 and 4.4.1.
Proposition 6.3.2. The structure of A : C -3 ZU. a. The morphism AU : C -+ ZU is surjective.
b. If z is point in Zv(Cp) then tt(z) = Y'A(c) where the sum ranges over the points c in C(Cp) which map to z. Proof. Part a follows from the fact that every point z in ZU (Cp) corresponds to a finite dimensional space of eigenforms for U with fixed weight and slope by theorems 4.1.1 and 4.3.1 which is stable under the action of W. It follows that this space contains an eigenform and if c is the coresponding point of
C, c maps to z. Part b follows from the fact that the space of generalized eigenforms corresponding to a point z E ZUis (Cr)of dimension µ(z) and is the direct sum isotypic components for the action of W. Moreover each such isotypic component corresponds to a point c E C(Cp) which maps to z and has dimension µ(c). We now state some consequences of the above Propositions and of The-
orem 6.2.1 (and its proof). We will have occasion to refer below to the sequence of operators Uk for k = 1, 2.... given in that proof. Corollary 6.3.3. For c E C as in Theorem 6.2.1 with weight w, we have: 1. If k is sufficiently large, 'FN = 1 f' }.
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2. Let zk E ZUk denote the image of c under AUk . If k is sufficiently large, then A(c) = lz(zk).
Proof. The first statement was proved in the course of the proof of Theorem
6.2.1. But the second comes along with it for if k is sufficiently large to guarantee that FUk = If,}, then any Uk-generalized eigenform in MUk is an f-generalized eigenform in the isotypic component corresponding to c. 0
6.4. The Eisenstein curve. Define the Eisenstein pseudo-representation r,E.is =
a.18, aEis)
of weight is by the formulas ,Ei8(g, h) = 0, akis(g) = 1, and let denote the continuous Cp valued character of GQ,{p,,.} which is the determinantcharacter of representations attached to modular eigenforms of weight ic, as discussed at the end of section 5.1. Specifically, consider the composition of the homomorphisms 77r
-i zp -+ A* -4 Cp,
where the first homomorphism is x, the canonical homomorphism arising from action on p-power roots of 1 (so for C E Qp a p-power root of unity, g(C) _ (X(9)) the second is the natural injection, and the third is the restriction of r. to multiplicative groups of units. Calling x again the Cp valued character induced by x (i.e., the composition x : GQ,{p,,,,,} -+ Z* C Cp, we put Eis -1
8 =x
-7r"
More systematically, one can simply define a single pseudo-representation rEis = (tEis, aEis, 5Eis) of GQ,{p,oo} with values in A by taking Eis
0 and aEis = 0 and 8Eis the evident character GQ,{p,oo} -+ A* so that ic(rEis) = rEis for all Cp-valued weights r.. By universality for p-modular pseudo-representations, we then get that rEis is the pseudo-representation induced from a continuous ring-homomorphism irEis : Rp -+ A (which sends the radical of Rp to the maximal ideal in A). By XEis C Xp let us mean the rigid analytic subspace of Xp associated to the closed subscheme Spec (A) C
Spec (Rp) and define the Eisenstein curve CE" to be the rigid subspace of Xp x Al defined by:
CEisXEisx1cx pxA1 P
o
that is, the Up-eigenvalue ((in the sense discussed in section 6.1) of the points of CP is is taken to be identically equal to 1. By Propositions 3.6.1,
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91
and 6.2.1, we see that the set of Cp-valued points of CP ig is contained in the set of Cp-valued points of the eigencurve Cp, and, moreover, any point of Cp which is not in C 19 is represented by a normalized cuspidal-overconvergent eigenform (of tame level 1 and finite slope).
Chapter 7. The eigencurve as a finite cover of a spectral curve. In this chapter, we will construct a curve which we will call D, which we will prove is isomorphic to the reduced rigid space associated to the eigencurve. We perform this construction by patching together generalizations of the pieces made in the proof of Theorem B5.7 of [C-BMF]. 7.1. Local pieces.
We will need the following generalization of Theorem B5.7 of [C-BMF] (see also Theorem 1 of [CST]): We let p be an odd prime number, and work with tame level 1. We keep the notation of the previous chapters except for the
following changes: For a E f let P0,(T) denote the Fredholm determinant of aUp, Za the spectral curve of aUp and Aa the natural map from Cp to Za. (These were called PaUp (T), ZaUp and Aoup, respectively, in previous sections.) We view Za as a rigid analytic Fredholm curve over Qp. Let Ira : Za -+ W be the projection of Za to weight space. Now fix a E W. If V be an irreducible affinoid subdomain in Za, then by Lemma A5.6 of [CBMF], its image Y := Ira (V) in W is an affinoid subdomain of W. Denote by Zy C Zr,, the inverse image of Y under the morphism Ira. Let Co, be the collection of affinoid subdomains V in Za such that V is finite over its image Y := 7ra(V) and which are admissibly closed-open in Zy. Let Ca denote the set of irreducible members of Co,. We know using Proposition A5.8 of [C-BMF] that Co, is an admissible covering of Za.*
Fixing a E Ii, for each affinoid subdomain Y E W, there is a one:one correspondence
V +* { P, (T) = Qv(T)H(T) }, where V runs through the subset of Ca with image Y, and where the equa-
tions Pa(T) = Qv(T)H(T) run through all factorizations of Pa(T) over A := A(Y), with Qv (T) a polynomial in A[T] whose constant term is one * It was said in [C-BMF] that the proof of Proposition A5.8 given there only works over Cp, but it actually works more generally. First the phrase "xEV" should be replaced by "closed points x of V." Then the affinoid W found in the proof of Lemma A5.9 can be defined, in the same way, for arbitrary K.
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and whose leading coefficient is a unit in A, and with H(T) an entire function in A{{T}} such that (Qv, H) = 1. If V and Qv(T) are connected via the above one:one correspondence, we have that
V = Max A(Y)(T)/(Q*,(T)), recalling the notation TdQv(T-1), where d = degree(Qv(T)). Let M,(v) be the Banach A-module of v-overconvergent modular forms of tame level 1 over the affinoid domain Y, as in section 2.4, for sufficiently small v > 0, and Mt = the A-module of overconvergent modular forms of tame level 1 over the affinoid domain Y. For each v > 0 we obtain a direct sum decomposition MY (v)
= N(V; v) ®F(V; v)
of closed A(Y)-modules for which annihilates N(V; v) and is invertible on F(V; v). Using the results of section 4.3 we see that N(V; v) is independent of v for v close to 1. The submodule N(V) C Mt is then defined to be N(V; v) for v sufficiently close to 1. Remark. In the notation of section A4 of [C-BMF], we may write
N(V) = N&, (Qv), where UY is the restriction of U = aU to MY. By Theorem A4.5 of [C-BMF], N(V) is locally free over A = A(Y) of
constant rank dv := degQv(T). Define T(V), the overconvergent finite slope Hecke algebra over V, to be the image of fl ®A in EndA(N(V)). The A-algebra T(V) is complete since EndA(N(V)) is finite over A = A(Y). We have that T(V) is the ring of rigid analytic functions on an affinoid D(V) over Qr finite flat over Y of degree d. The argument for this can be found
on pp. 437-438 of [C-BMF]. In fact, A is a PID and so N(V) is free. In summary, so far, we have associated to every element V of Ca with image Y C W, an affinoid D(V) which is finite and flat over Y.
V/Y i D(V)/Y
.
Since any member of Ca is a finite disjoint union of elements in C', we may extend the above association (by taking disjoint unions) to any member V
of Ca. Now let p denote the projection of W x Al to A'. We will apply the q-expansion principle to show:
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93
Theorem 7.1.1. Suppose V E Ca such that Y:= ia(V). Let L C Cn be a finite extension of Qp. For an L-valued point x E D(V)(L), let fix: Tv -4 L denote the corresponding homomorphism. Now suppose is is a weightcharacter in Y(L), and let (D(V))r denote the fiber in D(V) over is . For each x E (D(V)),c(L) the vector space W. over L of weight-character s; overconvergent eigenforms F over L such that
FITn=?lx(Tn)F (recalling the notation of 3.1) is one dimensional. Moreover, the correspondence x H Wx is a bijection between (D(V)),,(L) and the one dimensional spaces of eigenforms of weight-character is whose inverse (aU),ti-eigenvalue lies in p(Vk(L)).
The proof runs along the same lines as that of Theorem B5.7 of [CBMF]. We construct a pairing between T(V) and N(V) using q-expansions in the same way. Specifically, if T E T(V) and f E N(V), define
(T, f) := a1(T f), where a1 denotes the coefficient of q1 in the Fourier expansion. The main difference between the present setting and Theorem B5.7 of [C-BMF] is that the pairing here is not perfect, in general, because of the possible existence of a non-zero constant term in the q-expansions. Our pairing leads to an exact sequence of A-modules
0 - T(V) -4 HomA(N(V), A) 4M -+ 0,
(7.1.1)
where M is the residue field of 13 at the trivial character 1 if 1 E N(V)1 and p(h) is then h1 (1) in this case, otherwise M = 0. The proof of this requires our comparison of convergent modular forms and Katz modular functions, Theorem 2.4.2, and the injectivity of the q-expansion map on Katz modular functions by the second lemma of section XI of [K-LME]. Keeping this in mind, we can (almost) reformulate this result in terms of q-expansions. In particular, we obtain,
Corollary 7.1.2. Let notation be as in the theorem and set F. (q) = E77x(Tn)gn n>1
Then if is is not trivial or if is is trivial and % is not the homomorphism which takes Tn to v* 1(n), F. (q) is the q-expansion of a normalized overconvergent eigenform over L (minus its constant term) of weight-character Ic.
Here o* 1(n) is, as in 2.2, the sum of the reciprocals of the squares of the positive divisors of n which are relatively prime to p. We have a natural finite
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map of D(V) to V since the image of where Qv(T) = in T(V) is zero, by Theorem A4.5 of [BMF] (see also Corollary 4.2.3 above).
TdQV(T-1),
This map is not necessarily an injection but it is when V is reduced (this follows from the Cayley-Hamilton theorem). Since both V and D(V) have degree d over Y we deduce:
Proposition 7.1.3. If Qv(T) is square free, then the map from D(V) to V is generically an isomorphism and moreover D(V) is reduced.
The last assertion follows from the following lemma, taking X to be D(V) and W to be ira(V). Lemma 7.1.4. Suppose X -+ W is a morphism of affinoids of dimension one such that W is reduced and irreducible and A(X) is finite free over A(W). Then if X is generically reduced, it is reduced. Proof. Indeed, since X is generically reduced, of dimension one and finite over Y there exists an open Y in W such that Xy is reduced (just take Y to be the complement of the image of the finitely many points where X is not reduced). Thus if e E A(X) and e2 = 0, the image of e in A(Xy) must be zero. But, the other hypotheses imply that the map A(X) -* A(Xy) is an injection. Thus e = 0.
7.2. Gluing. Fix a E 9{, and let Z = Za. C := Ca.
Lemma 7.2.1. If V1 and V2 belong to C so does V1 fl V2. Moreover, if V1 C V2, D(V1) is naturally an affinoid subdomain of D(V2).
Proof. Let M = Mt. Let Yi be the image of V in W and Y12 = Y1 fl Y2. Then, it is easy to see that Vi fl Zy1 2 E C so we may assume Vi = V fl Zy12. The hypotheses imply that there exists functions el and e2 on Zy12 such that ei l Vi = Sij. It follows that V1 is the disjoint union of two subdomains Ul and U2, disconnected from each other, where e2 equals 1 and 0 respectively. Hence each of these must be finite over Y1. But U1 = V1 fl V2. Now suppose V1 C_ V2. First as we argued above V3 := V2 fl Zy1 E C. Moreover, it is not hard to see that D(V3) is naturally isomorphic to D(V2)y1 which is an affinoid subdomain of D(V2). Thus we may suppose Y1 = Y2. Let Qi(T) be the factor of Pa(T) over Y1, such that Qi(0) = 1, corresponding to V. Then, again arguing as above, V2 = V1 U V1' where V1 and V1 are disconnected and finite over Y1. It follows that
Q2(T) = Q1(T)Qi(T),
where Q1(T) and Q2(T) are relatively prime. We may apply the theory of Riesz decomposition (as in section A4 of [C-BMF]) because aU is associated
The Eigencurve
95
to a completely continuous system (cf. section 4.3) of operators on My1 = (v). We have NQ2 = NQ, ® NQI,.
From this we see that
D(V2) = D(Vi) lD(V1') and so D(V1) is an affinoid subdomain of D(V2) as asserted.
Construction of the curve Da. Now we will glue all the D(V)'s, for V E Ca, together to make a rigid analytic space Da using Proposition 9.3.2/1 of [BGR]. For U, V E Ca, let D(U, V) denote the image of D(UnV) in D(U). We know this is an affinoid subdomain. Let q5u v be the natural map from D(U, V) to D(V, U). Clearly, ¢U v o ¢v U = id, D(U, U) = D(U) and cbU U = id. Also OU v induces an isomorphism ¢U v w: D(U, v) n D(U, W) -4 D(V, u) n D(V, W) such that
OUVW=cWVU°cUWV because D(U, v) n D(U, W) is the image of D(U n v n w) in D(U). Thus the triple ({D(U)}, {D(U, V)}, {qU,v})U,vEC satisfies all the hypotheses of
the aforementioned proposition and the D(U)'s glue together into a rigid curve Da Note that Do is empty. Let za denote the projection from Da onto Z. By construction,
Proposition 7.2.2. The morphism za: Da -* Za is finite, and the projection to weight space it : Da -+ W has the property that it is locally in-the-domain finite flat in the sense that Da is covered by affinoid subdomains Da(V) C Da which have the property that their images Y := ir (Da (V)) C W are affinoid subdomains of W and moreover Da (V) is finite flat over Y. The rigid analytic spaces Da are curves; that is, they are equidimensional of dimension 1.
7.3. The relationships among the curves D. Consider the set P of all pairs S = (X, N) such that: a. X is an affinoid in W. b. N is a Tx-submodule of MX (here TX = 7-l 0 A(X)),
c. N is locally free of finite rank over A(X) and there exists a Txmodule projector from MX onto N which comes from a continuous projector
from MX(v) on N, where v > 0 is sufficiently close to 0 so that all the
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elements on N are the images of elements in MX (v). (Note that the module N, being of finite rank, is contained in the image of the Banach space M, (v) for v > 0 and sufficiently small, and the results of section 4.2 apply.)
d. The operator T (p) is invertible on N. (In this way the elements of N correspond to families of forms of finite slope).
One source of such pairs is the following: Let notation be as above. Suppose V E C,, and Qv(T) E A(ir,,(V))[T] the corresponding polynomial.
Let Nv be the kernel of Qt,(aUP) in M. Then, using Theorem A4.5 of [C-BMF], we see that Sv := (ira(V), Nv) E P. Even though MX is not naturally a Banach-module over A(X), writing MX = we may view MX as being given by a system of Banach modules, as in the terminology of section 4.3, and using Proposition 4.3.3 an operator U in the ideal U has a characteristic series det(1 - TUIMX) in A(X){{T}}. It follows from Corollary 4.2.3 that if a E W and U = aUP,
det(1 - TUIMX) = det(1 - TURN) det(l - TUIMX/N).
Suppose S = (X, N) is a pair in P. Let T(S) denote the ring generated over A(X) by the image of TX in EndX (N). It is the algebra of rigid functions on a one dimensional affinoid D(S). If V E Ca, D(Sv) = D(V) and T (Sv) = T (V). We also see that if S = (X, N) and S' = (X', N') are elements of P such that X' C X and N' C NX, := N ®A(X) A(X') then we have an induced natural ring homomorphism from the image of T(S) to T(S') which in turn induces a natural map
D(S') -4 D(S), which we will refer to below as the "functorial map". We also set
Tp,oF = limT(S) (the "OF" stands for overconvergent of finite slope). Using Proposition 3.4.2 and Theorem 2.4.2 we obtain compatible maps of TP into T(S) for S E P and hence a map from Tp into TP,OF
Construction of a natural mapping from D(S) into Da. Suppose a E 9l, S = (X, N) E P and a is invertible on N. Let U = aU. By Corollary A2.6.2 of [C-BMF] and Theorem 4.2.2 (since U is stabilizes N and A(X) is semi-simple) we may factor the Fredholm characteristic series Pa (T) over X,
P.(T) = Qs(T)Hs(T)
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97
where Qs(T) = det(1 - TUIN) is a polynomial over A(X) whose degree is the rank of N and whose leading coefficient is a unit in A(X), and where Hs (T) E A(X) { {T } }, and in contrast to members of Ca, here Hs (T) is not necessarily relatively prime to Qs(T) . Define
Zs := Za,s := Max(A(X)[T]/Q(T)) which is an affinoid and note that we have natural mappings D(S) -+ Zs -4 Z:= Z,,,.
Now because C := Ca is an admissible cover of Z, there exists a finite collection Z1,. .. , Zn of members of C which covers the image of Zs, i.e.,
such that the collection {Zs xZ Zi}i is an admissible open cover of Z. Let Xi = 1fa(Zi) and Xi = Xi fl X. Corresponding to Zi there is a Riesz theory idempotent ei which acts on The idempotent ei also may be thought of as a rigid analytic function on Zxj which is 1 on Zi and 0 on its complement. As ei E TX, (actually in the completion of the subring generated by U) ei acts on Nx;. Also, over Xi fl Xj, ei and ej commute. MX:.
Let fi = 1 - ei and Ei = eiNx; and F$ =
fiNxi. Then
Nx' =E2®Fi. (X" Ei) and Ei Consider the pairs Si (Xi', Fi) in P. Then, we have a factorization Q(T) = det(1 - TUIE;) det(1 - T(JFF;), which, given our notational conventions, may be written
Q(T)=Qs,(T)-QE;(T), the polynomial factors Qs, and QE; being relatively prime because of the Riesz theory. Moreover, we have the affinoids ZS, := Zc,s, := Max(A(X%)[T]/Qs;(T)), and
Zr, := Z,,,Ei := Max(A(X,)[T]/QE;(T)). It is easy to see that Si := (Xi', Ei) and
S23 :_ (Xi j = X' fl Xj, E1 := eiejNx'nx: )
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all lie in P. We will have later use for the natural mapping D(S1) -* D(Zi). This map is obtained as follows. First, consider the pair Zi := (Xi, e2Nx;) E P. Since X' C Xi and eiNx! C eiNx; ®A(x;) A(X'), we obtain a functorial
map (see above) D(Si) -3 D(Zi). Next note that D(Zi) = D(Zi).
Lemma 7.3.2. The natural mapping Zs; -3 Zs X Z Zi is an isomorphism. Proof.
Zs X z Zi = (Zs)x, X Z Zi and
(Zs)x; = Zs, I ZEi because Qs, (T) and QE: (T) are relatively prime. As ZE, x z Zi = 0 (again using the R.iesz theory) Zs X Z Zi = Zs; X Z Zi.
The lemma then follows since the natural map from Zs; to Z is contained in Zi.
Lemma 7.3.3. The natural map D(Si) -+ D(S) xz Zi is an isomorphism.
Proof. As in Lemma 7.3.2,
D(S) XZ Zi = D(Si) xz Zi and because the map from D(SS) to Z factors through Zs; we see that this equals D(Si).
Lemma 7.3.4. The natural map D(Sij) -+ D(Si) XD(S) D(Sj) is an isomorphism. Moreover, the restriction to D(Sij) of either the map from D(Si) to D xz Zi or D(Sj) to D xz Zj is the same map to D xz Zij (where Zij = Zi XZ Z,). Proof. This comes from
D(Si) XD(S) D(Sj) = (D(S) Xz Zi) XD(S) (D(S) Xz Zj)
=D(S)xzZZj
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99
We saw above that Zap E C, but we can be more precise. Over Xtij Xg fl Xj, we have the decomposition
M=eiejMED eifjMED ejffM®fifjM. This corresponds to a factorization, over X$ fl Xj,
PQ(T) = A(T)B(T)C(T)D(T)
where D(T) E A(XXj){{T}} and A(T), B(T), C(T) are polynomials in A(Xij)[T], and Qz: (T) = A(T)B(T) QZ3 (T) = A(T)C(T).
We see that Qz;, (T) = A(T) and if Zii # 0, the image of Z$j in 13 equals Xij and the projector corresponding to Zap is eaej. By Lemma 7.3.3, we see
that D(S2j) = D(S) XD D(Z2j) and the map from D(SSj) factors through the projection to D(Z3). From this the lemma follows. 1 The Lemmas 7.3.3 and 7.3.4 allow us to patch together mappings.
Proposition 7.3.5. Let S E P. There is a unique map from D(S) to D0, characterized by the following property: Suppose V E Ca, with image
Y = ira(V). Let e be the corresponding idempotent acting on MY and S' = (Y fl X, eNy) E P. Then the following diagram commutes: D(S)
t
D,,
t
D(S') -4 D(V) where the left and the bottom arrows are the natural ones coming from the
inclusions S'(N) C N(S)Ynx and S'(N) C N(V)Ynx Proof. It follows from Lemma 7.3.3 that the D(Si) cover D(S) and from Lemma 7.3.4 that all the morphisms D(S$) -+ D(Z2) (constructed above) glue together to give a morphism from D(S) to Da. To check that one gets the same map from a different covering of the image of ZS by a collection of elements of C,, it is enough to check this when the second covering is a refinement of the first in which case it follows easily from Lemmas 7.3.2-4. We may deduce from the above that we have a natural maps pp,a: DO - Da for all 0 E a71 as follows: Suppose V1, V2 E Cp then, as /3 is invertible on N(V$), by the above we have morphisms D(V1) -+ D,, and D(V2) -+ D,,. We must show that they agree on D(V1) XD,, D(V2) which is D(V1 xZp V2). Hence, it is enough to prove this when V2 C_ Vl (i.e. V1 xZp V2 =
V2). Suppose Zl,... , Z,, cover the image of Zs(vl). Then as N(V2) is a
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submodule of N(Vl),r,(v2), Zl, ... , Zn also cover the image of Zs,,2 . We can therefore use Z1, ... , Z to make our maps. Let in the above notation, with S = Sv,, S$ (Vj) = Si, Xti (Vj) = X. We must check that the following diagram commutes:
D(SS(V1)) - D(ZZ)
t
II
D(Si(V2))
-*
D(Z$),
but this follows from ea(N(V2)x,(V2)) c eti((N(V1)x(v2))X.(V2)) _ (ea(N(V1)X,(V1))x,(v2).
Lemma 7.3.6. If ,Q E aTp,OF and -y E /3Tp,0F Pp,« 0 Py,p = P7,a
Proof Let UECy,VECp,WEC,,,X=iry(U),Y=ip(V)and Z= ira(W). Let ev and eu be the idempotents corresponding to V and W (We suppress mentioning in the notation where they are defined here and in the following). Then if S' = (X fl Y, evN(U)) and S" = (Y fl Z, ewN(V)), using Proposition 7.3.5, we see that the following diagram commutes: D(U)
P4
Dp
P
Da
t
fi
D(S') -+ D(V)
t
D(S")
l
-* D( W).
Now if s", = (X fl Y fl Z, ewevN(U)), it is easy to see that
D(S')
-+
D(V)
D(S"')
--#
D(S")
t
t
commutes, where the right and bottom arrows are the natural ones. Now using the previous corollary and the fact that the D(S"')'s constructed in this way cover D(U) we see that the composition of pp,c, and py,p restricted to D(U) is py,c,. This completes the proof of the lemma. Since p,,,,, is the identity,
Corollary 7.3.7. If /3 and a generate the same ideal in T p,OF, then pp,c, is an isomorphism. Let D = D1. The above gives us maps from D to Zc, for all a E TP,oF The curve D will be shown to be isomorphic to the reduced eigencurve.
The Eigencurve
101
If 0 E 7-l such that t /3 is invertible, then the image of ,3 in TP,OF is invertible because the homomorphism from 7 into TP,OF is the composition of the homomorphisms
W-+ Rp-+ Tp*Tp,oF (see sections 3.3, 5.2 and 7.2).
Remark. If a and /3 are non-associate elements of TOF then Da and Dp may be very different. For example, if a = 0, Dc, will be empty although as we'll see D := D, is quite large. If a is an idempotent, Dc, is an admissibly closed-open subspace of D.
7.4. D is reduced. In this section, we will prove D is nested (cf. 1.1), reduced, and each irreducible component of D maps surjectively and generic isomorphically onto a Fredholm hypersurface (see Prop. 7.4.5 below). By construction we have a map w: D -* W. By the results of §7.3, D = Da for all a E TP OF. This means that we have finite maps za: D -* Za, and again by construction the following diagrams commute:
where the mappings to weight space are the natural ones; i.e., the mapping
D -* W is w and the mapping Zc, -* W is Ira. Let va: Z0, -* A' be the natural projection to the affine line. If a E Tp OF, the map s: P -* -v(va(z«(P)) is independent of a and we call it v. Indeed, if we think of a closed point on D as a one dimensional space of eigenforms, as we can by Theorem 7.1.1, s is just the slope of these forms. We know that every point on the nilreduction Dred of D lies on an irreducible component part by Proposition 1.2.5. Let Z = Z,.
Lemma 7.4.1. Suppose a E T*, OF. If Y is an irreducible component of Dred then the map Y -* factors through a surjective map onto an irreducible component part of Zred.
Proof. Let V E Ca. Let Dl, ... , Dn be the irreducible components in D(V)red in Yv = Y fl D(V). These correspond, by Corollary 1.2.1, to minimal prime ideals P,,. .. , P of Tv. Since the map from D(V) to V is surjective on closed points, it follows that the pullback Qi of Pi to V is a minimal prime ideal of A(V). and let Zi = MaxA(V)/Qi be the irreducible component of of V''ed corresponding to Qi. Since D(V) to V is finite, it
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follows from the going-up theorem, Theorem 5.11 of [A-M], that the map from Di to Zi is surjective. Let Zv be the component part of Vred which is the union of the irreducible components ZZ for i = 1 n. The natural
map from Yv to Zv is finite and surjective. The Zv's for V E Ca glue together into a component part Z of Zaed and the maps glue together into a finite surjective map from Y onto Z. Since Zaed is Fredholm hypersurface it follows that Z is as well. Since Y is irreducible so is Z. Write A = rjP-i AU) where each of the p-1 factor rings AU) are isomorphic to a power series ring in one variable over Zp, and where the product decomposition of rings corresponds to the decomposition of the rigid space W as a (disjoint) union of its p-1 irreducible components, W = [J WU), where AU) is the algebra over Qp of rigid analytic functions bounded by 1 on the rigid analytic space W(j).
Corollary 7.4.2. Let w : D -* W be the projection mapping to weight space. If D is an irreducible component of Dred then w maps D into a unique irreducible component, Will), of W (where 1 < jv < p-1) and the mapping
w:D-4 W(jD) is almost surjective on Cp-valued points in the sense that the complement of D(Cp) in WU-) (Cp) is finite.
Proof. Let D be an irreducible component of Dred Then D projects onto an irreducible component of Za d by Lemma 7.4.1. To prove Corollary 7.4.2 it suffices, then, to prove for irreducible components of Za d. By Theorem 4.3.1, Za is cut out by a Fredholm series PU(T) over A. Consider the corresponding factorization p-1
PU(T) _ 11 PU(T)U) j=1
where P(T)U) E AU) { {T } } for j = 1 . . p- 1. Since AU) is isomorphic to a power series ring in one variable over Zr,, Theorem 1.3.11 applies, allowing
us to identify the irreducible components of Ze d (and also of Za) with factors of the Fredholm series PU(T)U) for j = 1... p - 1. Corollary 1.3.13 then applies, which concludes the proof of our corollary. If W is an affinoid open in W and t E Q, the conditions
w(P) E W
v(P) < t determine an affinoid open D(W, t) in D.
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103
This follows because the conditions Ira (Q) E W
v(oa(Q)) < t determine an affinoid open Z,, (W, t) in Z. for any a of which D(W, t) is the pullback so is an affinoid by the rigid Chevalley lemma Proposition 9.4.4/1 of [BGR].
Lemma 7.4.3. For W an affinoid open in W and t E Q. There exists an a E Tp of such that Za(W, t) is reduced. Proof. We know D(W, t)red has finitely many irreducible components by Corollary 1.2.1/1. Using Proposition 1.2.5 we see that these are contained in a finite number of irreducible components of D each map almost surjectively (in the sense described in Corollary 7.4.2) to some component WW) of W. In particular, for each
v, there is a point 8, on D of some arithmetic weight k and slope s, with 0 < sv < (k - 1)/2. Now choose a E Tp of as in Chapter 6 so as to distinguish all the classical forms of weight k and slope s for all v = 1,
, n.
Claim. Za(W, t) is reduced. Let Z be an irreducible component part of Za containing the image < ,
of 8 under za : Da -* Za. Then by the arguments in Chapter 6 (Props. 6.3.1- 6.3.3) Z is reduced (There is only one modular form corresponding to (,,. This implies Z is generically reduced, but since it is an irreducible Fredholm hypersurface it must be reduced.) Now the claim and our lemma follow because Za (W, t) is the union over v of Z fl Za (W, t).
Lemma 7.4.4. D(W, t) is reduced. Proof. If we knew Za (W, t) was an element of Ca we could apply Proposition
7.1.3 to conclude that D(W, t) is reduced. But it may not be so we must work harder. We go to the proof of Proposition A5.8 of [BFM]. We must generalize it slightly and replace BK with W. (We also use the proof of Lemma A5.9 in loc. cit. which works even when d = 0). Translating what was proved there, we know that for u E Q, u > t, and sufficiently close to t, there exists a finite covering {Uo, Ul,... , Ud} of W by affinoids in W such that the the affinoid Vi in Za, whose points in Za consist of all Q such that 7ra(Q) E Ui
and
v(Qa(Q)) < u
lie in Ca and is finite over Ui. Moreover, S,, := Ui>,1 Ui contains the affinoid
W in W consisting of all P E W such that the fiber of Ira restricted to
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Za (W, t) at P has degree at least n. We can also assume such that Un is the rigid Zariski closure of W fl U,,. Since Un is the Zariski closure of Wn fl Un and Za(W,t) is reduced, it follows that the Zariski closure of (Va)wnnvn is Vn. But (Vn)wnnun = (Z. (W, t))un which is reduced by Corollary 7.3.3/10 of [BGR] since Z0, (W, t) is reduced. Thus Vn is generically reduced. Now we can apply Proposition 7.1.3 to conclude that D(Vn) is reduced. Since the D(V,,) cover D(W, t) our lemma follows.
Proposition 7.4.5. The rigid analytic curve D is reduced, nested, and and each irreducible component of D maps surjectively and generic isomorphically onto a Fredholm hypersurface.
Proof. The D(W, t)'s form an admissible cover of D, so D is reduced by Lemma 7.4.4. Now suppose D is an irreducible component of D. We may suppose D is the Dl mentioned above. We claim it is generically isomorphic to Zl. We know D(W, t) is generically isomorphic to Za(W, t) (here a is as in the previous lemmas). It follows that V, fl D(W, t) is generically isomorphic to Zl fl Za (W, t) since these affinoids are open and Zariski dense in their respective spaces. Since Z, is nested, so is D. Our proposition follows.
7.5. Equality of D and Creel.
Set C = Cp. Recall C''ed denotes the nilreduction of C. We maintain the notation of the previous section so in particular, we have morphisms za: D -+ Za (which is contained in W x Al - {0}) for all a E Tp,OF. We also have a morphism 5: D -* Xp x (A' - {0}),
given by c H (p(c), 1/u(c)) where p(c) is the pseudo-representation and u(c) is the Up eigenvalue attached to c, or more precisely: The morphism from D to Xp arises from the map Tp -+ T p,OF described in section 7.3 and the pseudo-representation described in section 5.2. The map from D to A' - {0} is the composition of the morphism from D to the spectral curve
Z =: Z, and the natural projection of or,: Z -> A' - {0}. We constructed the eigencurve C, in section 6.1, as follows: For every a E f' such that
a is a unit in Rp we have a map ra from Y := Xp x (A' - {0}) to W x (A' - {0}) such that t
ra(x, t) = (w(x),1/t (t a(x))). Then C is the rigid analytic subspace of Y which is the intersection of the subspaces
r,l(Za) :=Y xwx(A'-10)) Za,
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105
where a ranges over the elements in 9-l' as above. The key point is that
rao8=za. Thus 8 factors through C. It follows that we have a natural map w: D -* C such that the diagrams
commute for all a E TOC'
Theorem 7.5.1. The map w factors through a map to Cred and induces an isomorphism (which we continue to denote by the same letter)
w: D^ Cred Proof. That w factors through Cred follows from the fact that D is reduced. That w is a bijection on closed points follows from Theorems 6.2.1 and 7.1.1. To prove it is an isomorphism of rigid spaces, we will first prove that it is a generic isomorphism and then prove it is a local isomorphism. For the first assertion we need the following lemma. For the second assertion we will show below that the morphism from D to X1, x A' is locally a closed immersion.
Lemma 7.5.2. Suppose Eo and El are reduced rigid spaces (over K a complete subfield of Cr), and ho : Eo -4 El and h, : El -* Eo are rigid analytic mappings which induce mappings on K-valued points which are two-sided inverses of one another; i.e., the compositions of ho : E0(K) El (k) and hl : El (k) -+ Eo(K) in either order yield the identity mapping on the relevant set of K-valued points. Then ho : Eo = El and h, : El Eo are rigid analytic isomorphisms between Eo and El and are two-sided inverses of one another as rigid analytic mappings. Proof. It will suffice to show that if E is a reduced rigid space over K, and h: E -+ E is a morphism of rigid spaces over K such that h is the identity on K-valued points, then h is the identity. Let S be an admissible cover of E by affinoid subdomains. Let V E S. Then there exists an admissible cover
U. of V by affinoid opens and V E S such that hju factors through the inclusion Vi -+ E. Now since, h(V(K)) = V(K), h(U1(K)) C_ (V n V)(K) and since V is an affinoid subdomain of E, V n V is an affine subdomain of V. It follows from Proposition 7.2.2/1 of [BGR] that hju, actually factors through V n V2 and so hl v factors through the inclusion V -* E. Thus we may suppose E is an affinoid. Let A = A(E). We need to show h*: A -4 A is the identity. Suppose a E A. Let m be a maximal ideal of A and (extending
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scalars, if necessary) suppose c E K such that a - c E m. Then h*a - c E m
and so h*a - a E m for all maximal ideals m. Since A is reduced (and semi-simple), it follows that h*a = a. Now we apply this to our eigencurve. Let a E f whose image in Tp,OF
is invertible and suppose V is a reduced component part of Z«. Let Dv be the fiber above V in D and Cv the fiber above V in C''ed (Thus, by definition, Cv is reduced.) Then we have a commutative diagram
Dv -* Cv V and we know w is one:one on closed points and the restriction of za to Dv is generically an isomorphism onto V using the fact that D is reduced and Lemma 7.4.1. Let S be the set of points of V where za is not an
isomorphism, V' = V - S, D' = Dv - z«'(S). C' = Cv - A;1(S), w' the restriction of w to D' etc. Let p: V' -* D' be the inverse of za: D' 4 V1. Then
(poA')ow'=id. Applying Lemma 7.5.2 with Eo = D' , El = C' and ho = w', h, = p o A' that WIDE: Dv -* Cv is generically an isomorphism. Since every component part of D composed of finitely many irreducible components is generically isomorphic to a reduced component part of Z. for some a E T p OF we may conclude that w is a generic isomorphism. Now we will show w is a local
isomorphism. Let V be a open in C := C, and Y its image in W. Let X be an affinoid open in Xp and U a affinoid in Al such that the image of X in W is Y and such that the image of D(V) in Xp x Al is contained in X x U (more precisely, such that the morphism D(V) -+ Xp x A' factors through X x U for after we get this second property, we can get the first by taking fiber products). Since A(D(V)) is generated over A(Y) by the image of the Hecke operators which can be identified as functions on Xp x A' and A(Y) maps into A(X x U) the map from D(V) to X x U is a closed immersion.
Claim. For any point on D there exists a V E C such that P E D(V) and an X and U as above such that the image of 6(D(V)) equals 8(D) n (X x U). Suppose for the moment that we have this. Let E be the intersection of C with X x U. It is a subaffinoid and its points are the same as those of D(V). It follows that D(V) maps via a closed immersion onto the nilreduction of E. Since it is a bijection on closed points and D(V) is reduced we conclude that it is an isomorphism. Thus the claim implies that w is a local isomorphism. To prove the claim, suppose v(P) = t. Then using the analysis of Lemma A5.9 of [BGR], we see there exists an affinoid open neighborhood W of w(P)
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in W such that the affinoid Z(W, t) E C (this is the affinoid called Z1 (W, t) in the last section). The choices V = Z(W, t), X = W and U = B[0, pt] will fulfill the requirements of the claim. We now prove the theorem. Suppose X is an affinoid open in Xr whose image in W is an affinoid open W. The map from D(W, t) to Cred factors
through C(X, t) := Cred n (X x B[0, pt]). Moreover, both D(Wt) and C(X, t) are affinoids and the map f: D(W, t) C(X, t) is a bijection on closed points. It follows from the above that f is an isomorphism after the removal of finitely many points S in D(W, t) and is a local isomorphism. Let A = D(W, t) - S, B = C(X, t) - w(S) and let 7p: B -a A be the inverse of wIA. Now choose affinoid neighborhoods U of S in D(W, t) and V of w(S)
in C(X, t) such that w restricts to an isomorphism U -4 V. We can do this because S is finite and w is a injection on closed points. Let ¢: V -> U be the inverse of wIv. It is clear that 0 and 0 glue together into a morphism p(X, t): C(X, t) -+ D(W, t) inverse to W I D(w,t) As X and t vary the affinoids C(X, t) for an admissible open covering of Cred and the p(X, t) glue together into an inverse of w. 1
7.6. Consequences of the relationship between D and C. The first consequence of section 7.5 is that we now know that the eigencurve
is a curve since D is by construction. Moreover, Theorems A and B of Chapter 1, section 1.5, follow from the results of sections 7.4 and 7.5. We also established
Corollary 7.6.1. Every irreducible component of the reduced eigencurve C contains a classical point.
Corollary 7.6.2. Each irreducible component of Cred is generically isomorphic to a Fredholm hypersurface over W.
For the record, let us extract a slightly more precise statement that is useful in understanding the geometry of the rigid-analytic curve C. In the special case of arithmetic weight a = (x, k) where the slope v of c is less than k - 1 and different from (k - 2)/2 it follows from [C-BMF] that the CP vector space Mk(Cr; x)o consists entirely of classical modular forms, and therefore it follows from standard facts that f acts semi-simply (with multiplicity one) on M = M,,,Q. In particular, µ(c) = 1. We therefore get:
Corollary 7.6.3. Let c be a Cp-valued point of the eigencurve C = Cp of weight-character w = x 07]k E W and of slope v < k -1, and o # (k - 2) /2. Then c is a smooth point of the curve Cred and the projection Cred -+ w is locally an isomorphism in a neighborhood of c.
One should note that if or > k - 1 the point c may indeed be ramified in the projection mapping of C to weight space. We do not have to go far
R. Coleman & B. Mazur
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to see this phenomenon. Say that a point c E C of weight w = X ® T7k is of
critical slope if the slope of c is equal to k - 1. There are examples (cf. [C-OC]) of points c of critical slope ramified under the projection mapping of C to weight space. From the previous discussion, we deduce:
Corollary 7.6.4. If Q E Tp OF, and dp E Zp, then the map D -+ Zp is finite flat of degree at most 7i(dp) in a neighborhood of dp. Moreover, if any of the points d E D above dp satisfy the conditions of the previous corollary, they all do and this degree equals p(dp).
We may now prove Theorem G stated in section 1.5:
Proposition 7.6.5. If If,,} is a sequence of normalized eigenforms of tame level 1 with Fourier coefficients in CP such that fn has weight nn E W(Cp), v(ap(fn)) is bounded independently of n and the sequence {fn(q) E Cp[[q]](1/q)} converges coefficientwise to a series f(q). Then the sequence ,n converges in (Z/(p - 1)Z) x Zp to some weight i E W(Cp) and f (q) is the q-expansion of an overconvergent modular eigenform f of tame level 1, weight is, and finite slope. Let f be a Katz modular eigenfunction (see section 2.2) of tame level 1, of finite slope, with (normalized) Fourier expansion 2 f = q + a2q + a3g3+...
with coefficients aj E C. Suppose that f is of accessible weight-character (cf. section 1.4) (x, s) with s E Zp. Then f is overconvergent if and only if f is the limit in the q-expansion topology of a sequence of classical, normalized, cuspidal-overconvergent (cf. section 3.6) modular eigenforms of tame level 1.
Proof. That the in converge to a n E W(Cp) follows from Theorem 2 of [S 3]. We will give two proofs of the rest of the first part of this proposition.
For the first proof note that because the slopes are bounded and the weights converge, eventually all the fn correspond to points Pn on an affinoid of the form D(V) for some V E C1. Moreover in the notation of Theorem 7.1.1, 7gp,,(T),, is the n-coefficient of fn(q). Since the images of Hecke operators generate the ring of functions on D(V) over the weight
space, it follows that the points Pn converge to a point P on D(V) such that qp(Tn) is the n-coefficient of f (q). By Theorem 7.1.1, P correspond to an eigenform of weight ic with q-expansion f (q) and finite slope.
For the second proof and the proof of the rest of this theorem, let us first consider the various natural topologies on sets of eigenforms. For every prime number f E P we have a natural continuous mapping TE : (XP x (A1 \ {0})) (Cp) -+ Cp
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by putting Te(x,1/u) := Tt(x) if f # p, where we view the Hecke operator TE as rigid analytic function on Xp, and putting Tp(x,1/u) := u. On defines
the q-expansion topology on the set Xp x (A' \ {0})(Cp) by means of these functions. Explicitly, let P be the set of all prime numbers, and Cp the product of CP with itself countably many times, indexed by P and given the natural product topology. Consider the mapping
T : Xp x (A' \ {0})(Cp) -> CP
,
where for any f E P, the .£-th entry of T(c) is Tt(c). By definition, the q-expansion topology on the set Xp x (A' \ {0}) (Cp) is the topology this set inherits via the mapping T to the topological space CP P.
Lemma 7.6.6. The natural topology that the set Xp x (A' \ {0})(Cp) inherits as the Cp-valued points of the rigid analytic space Xp x (A' \ {0})
(which we will call the rigid analytic topology) is the same as its qexpansion topology.
Proof. This follows from the fact that if 1-l' is the A-algebra generated by the T-,'s (for £ # p) (cf. Chapter 2), the natural ring homomorphism of R' to Rp is topologically dense. This fact can be regarded as given by a "standard argument" at least for those components of Rp which actually correspond to representations rather than pseudo-representations (for if you take such a component , e.g., a completion of Rp with respect to a maximal ideal m (call it R,,,,) and if you consider R911 C Rm which is the closure of the image
of 9d' in Rm, one can see that (by continuity and Cebotarev) the image of the entire Galois group under the trace mapping is contained in R. We may assume this is a local ring with the same residue field as Rm and then by standard Schur-type arguments you see that the universal representation over R,,,, descends to a representation r' over R. Universality of R7, allows us to conclude that R'm = Rm (for if not, there would be two distinct ringinducing the same representation: the identity homomorphisms Rm and the projection to R'). Now this argument works word-for-word (only is easier) for pseudorepresentations, because we don't need "standard Schur-type arguments." The fact that the trace descends immediately implies (from the definition of pseudo-representation) that the entire pseudo-representation descends. , The first part of our proposition is a consequence of this lemma. For Proposition 6.2.1 allows us to identify our sequence { fa} with a sequence of CP valued points of (the nilreduction of) the eigencurve. But since the reduced eigencurve is a Zariski-closed rigid analytic subspace, its set of Cvalued points is a closed subset of Xp x (A' \ {0}) (Cp) in the rigid analytic topology. By the lemma, this set is also closed in the q-expansion topology.
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As for the second part, let K be the set of all Katz modular eigenfunctions of tame level 1, of finite slope, of arbitrary weight, but with Fourier expansion as hypothesized in our proposition. We view K as a topological space, given the q-expansion topology. Let c : IC -* X p x A' (Cr) be the mapping which assigns to a Katz modular eigenform f E IC the point c(f) E X, x Al (Cp) whose coordinate in Xp(Cp) corresponds to the associated pseudo-representation of f, as given by the Gouvea-Hida Theorem (see section 5.2) and whose coordinate in A'(Cp) is the reciprocal of its Up-eigenvalue. The mapping c is a homeomorphism of the topological space IC onto its image in Xp x (A1 \ {0})(Cp). This may be seen from the above
lemma by noting that the Te-eigenvalues (for all £ # p) together with the UP eigenvalue of f determine the q-expansion of f (see the discussion about this in section 3.1 ). Now take a Katz modular eigenform f E IC satisfying all the hypotheses of our proposition, and in particular which is of accessible weight-character. Suppose that f is overconvergent. The point c(f) is then, by Theorem 6.2.1, a Cp-valued point of the reduced eigencurve, and since f is of finite slope and accessible weight, using the argument in the proof of Corollary 7.6.3
one sees that f is the limit of classical eigenforms of tame level 1 (with normalized q-expansions).
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Gal(Q/Q). Duke Math. J. 54 (1987) 179-230. Serre, J. P., Formes modulaires et fonctiones zeta p-adiques, Modular Functions of one Variable III, SLN 350, (1972) 69-190. Stevens, G., Overconvergent Modular Symbols and a Conjecture of Mazur, Tate, and Teitelbaum, to appear. Taylor, R.: Galois representations associated to Siegel modular forms of low weight. Duke Math. J. 63 (1991) 281-332 Wan, D.: Dimension Variation of classical and p-adic modular forms, to appear in Inventiones Math. Wiles, A.: On ordinary A-adic representations associated to modular forms, Inventiones math. 94 (1988) 529-573.
DEPARTMENT OF MATHEMATICS, EVANS HALL, UNIVERSITY OF CALIFORNIA AT
BERKELEY, BERKELEY, CA 94720, USA
[email protected] DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY, ONE OXFORD STREET,
CAMBRIDGE, MA 02138, USA
[email protected] Geometric trends in Galois module theory B. EREZ to A. Frohlich on his 80th birthday
Abstract We review recent contributions by various authors to the study of group actions in arithmetic geometry related to L-functions. The results we present considerably extend previous work on the Galois module structure of rings of integers and units in algebraic number fields. In particular we present the solution of a generalized Frohlich Conjecture relating the module structure of de Rham cohomology to epsilon
constants on arithmetic schemes of arbitrary dimension, and we discuss new invariants attached to equivariant motives, which generalize the omega invariants introduced by Chinburg in connection with the Stark Conjectures.
Contents Introduction 1
Refined Euler characteristics and analytic classes l.a Analytic classes . . . . . . . . .. . . . . . . . . . . 1.b 1.c 1.d
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Tame actions ............................... 119 Perfect complexes in etale cohomology and omega invariants . . . . . . . . . . . . . . . . . Nearly perfect complexes . .
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6 The equivariant arithmetic genus
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7 Equivariant motives
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Introduction Four talks-by D. Burns, T. Chinburg, G. Pappas and M.J. Taylor-were given
at the Durham conference, which concerned results in what one may call geometric Galois module theory. In this survey article we attempt to explain the main results presented at the conference. We try to show how various lines of investigation, which originated in the work on the Galois module structure in number fields and which were inspired by recent advances in arithmetic geometry have converged to give a coherent body of results. One result we will focus on is the proof of a generalized Frohlich Conjecture. Recall that for a tamely ramified Galois extension of number fields N/K, with Galois group G, the ring of integers ON in N is a projective ZGmodule. The original Frohlich Conjecture, proved by Taylor in [T1], asserts
that the obstruction to ON being stably free is given by the signs of the constants in the functional equation of the Artin L-functions of symplectic representations of G. The generalized Frohlich Conjecture concerns the relation between the epsilon constants and the Galois modules attached to a group action on an arithmetic scheme. Chinburg made a first fundamental advance towards finding the right formulation of the conjecture, by proving a result analogous to Taylor's theorem for varieties over a finite field in [C3]. Let (X, G) be a tame action of a finite group G on a projective scheme X over a commutative ring R. Chinburg showed how to define a homomorphism from the Grothendieck group Go (G, X) of coherent G-sheaves on X to the Grothendieck group CT(RG) of finitely generated RG-modules which are cohomologically trivial for G: the refined Euler characteristic. If R is a field and X is a projective variety over R, such Euler characteristics were considered by Nakajima [Na2]; in this case CT(RG) may be identified with the Grothendieck group K0(RG) of all finitely generated projective RG-modules. Suppose now that R = Fp is a field of prime order p and that X is a smooth variety over FP. Chinburg proved that a suitable combination T (X, G) of refined Euler characteristics of sheaves of differentials is characterised by the epsilon factors attached to the action (X, G). Then he translated this result into an equation inside the projective classgroup Cl(ZG), which is a strict analogue of Taylor's theorem. Indeed, mimicking the constructions performed in the case of number fields, he defined a root-number class W 7y in Cl(ZG)
Geometric trends in Galois module theory
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which is determined by the signs at infinity of the epsilon constants e(Y,V), for V ranging over the irreducible symplectic CG-modules. He also introduced a ramification class Rx/y, which depends on the epsilon constants of the branch locus of the covering X/Y, and showed the equality
Resz("(X, G)) = Wx/y + Rx/y
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where Resz sends a projective FpG-module to the (stable equivalence) class in Cl(ZG) of the ZG-module obtained by restricting scalars from Fp to Z. The corresponding conjecture for flat X over Z formulated in [CEPT2] involves a different class than IF (X, G). This conjecture was proved in [CEPT2] for tame actions on arithmetic surfaces and for free actions in arbitrary dimensions. Again a suitable combination of refined Euler characteristics of sheaves of differentials was related to the generalization of the root-number class appearing in the original Frohlich Conjecture. The case of tame actions in arbitrary dimensions is now also dealt with, see [CPT]. Another important set of results concerns some new invariants for equivariant motives which have been defined using recent work on L-values. The prototype of such invariants is the omega invariant introduced by Chinburg in [Cl], in relation with Tate's work on the Stark Conjectures. For N/K a (not necessarily tamely ramified) Galois extension of number fields, with Galois group G, this is an invariant 52(N/K, 3) in Cl(ZG), which encodes information on the Galois structure of the group of S-units in N, for S a large enough (finite, G-stable) set of places of N. The new invariants defined in [BFI] and [CKPS1] for Tate motives, encode information about the Galois structure of higher K-groups of S-integers. Moreover the very general approach of Burns and Flach sheds a new light on the work of Chinburg and Frohlich, which had emphasized the analogies that exist between additive and multiplicative Galois module structure. The contents of this paper can be divided into four parts. The first part consists of Sect. 1, in which we review various ways to obtain refined Euler characteristics attached to group actions. Such Euler characteristics arise from bounded complexes of finitely generated G-modules, which compute various kinds of sheaf cohomology and whose terms are of a restricted typesay projective or cohomologically trivial. If, for instance, the action (X, G) is tame, then the Cech complexes computing the cohomology of coherent G-sheaves are quasi-isomorphic to bounded complexes of finitely generated cohomologically trivial G-modules. The refined Euler characteristics arising from such complexes are one of our main objects of study in the second part of the paper, which consists of Sect. 2 to 4. There we present results whose common denominator is that they relate various refined Euler characteristics to classes defined in terms of L-values and epsilon constants. Sect. 2 contains results for tame actions on varieties over finite fields. Here, following
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[CEPT5], we begin by indicating a new approach to Chinburg's results of [C3], in which Lefschetz-Riemann-Roch Theorems are used to compute the refined Euler characteristics. Such results are also used in the proof of the generalized Frohlich Conjecture. In the second half of Sect. 2 we present a recent result from [CKPS2], which builds on work of Lichtenbaum to compute an equivariant Euler characteristic for the multiplicative group G,,,, in terms of L-values. Sect. 3 reports on the generalized Frohlich conjecture, and Sect. 4 shows how one can characterize epsilon factors in terms of Galois modules, following [CEPT3] and [CEPT4]. Thus, as was explained above, in the second part we mainly deal with combinations of Euler characteristics for various sheaves of differentials. In contrast, in the third part, we describe results about the Euler characteristic of an arbitrary sheaf on certain
surfaces equipped with a free action. Sect. 5 deals with a topic which has been initiated in [T2], where Taylor formulated the conjecture that certain orders defined using torsion points on elliptic curves admit a normal basis (see loc.
cit. p. 433). We present Agboola's and Pappas' geometric approach to
this problem, which generalizes previous work by Srivastav-Taylor and which comes from [P1]. In Sect 6, again following Pappas, we show how Deligne's Riemann-Roch theorem can be used to deduce that the refined Euler characteristic of the structure sheaf of a surface has order 1 or 2. In the last part we discuss the work of Burns-Flach and Burns on the omega invariants attached to equivariant motives mentioned at the beginning of the introduction. Acknowledgements. This is a good time for reviewing our subject and I thank the organizers for having given me the oppurtunity to do this. (For other aspects of Galois module theory see the reviews [C-N,C,F,T] and [C-N,T].) This presentation is largely independent of the four talks given at the conference and has actually been influenced by results obtained after the conference. However I have greatly benefited from discussions with the speakers and I thank them for having shared their insights with me. Many
thanks also go to A. Agboola and Ph. Cassou-Nogues for fruitful discussions, and to the referee, whose careful reading of the manuscript lead to the correction of many imprecisions.
1
Refined Euler characteristics and analytic classes
We are interested in classes arising from an action (X, G) of a finite group G on an arithmetic scheme X. For the main part of the paper we will be dealing with classes inside the projective classgroup Cl(ZG). Recall that this is the quotient of the Grothendieck group K0(ZG) of projective ZG-modules by the
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subgroup generated by the class of ZG. The classgroup can also be identified with the subgroup of rank 0 elements in Ko(ZG). Moreover, Schanuel's lemma implies that one can identify Ko(ZG) with the Grothendieck group CT(ZG) of cohomologically trivial G-modules.
l.a Analytic classes There are two kinds of classes in which we are interested. The first arises from considering homomorphisms from the group RG of (virtual) complex representations to the multiplicative group of complex numbers, such as those defined by epsilon constants or L-value/regulator quotients. These define classes in the classgroup via Frohlich's so-called Hom-description of the classgroup. Let
E be a large enough (finite) Galois extension of the rationals. In particular assume that all representations of G are realized over E. Let J(E) denote the group of ideles of E. Then RG and J(E) are modules for the Galois group of E/Q. It can be shown that the group of Galois equivariant homomorphisms between these two modules surjects onto the classgroup: cl : HorGal(E/Q)(RG, J(E)) + CTZG)
.
(For this one uses Swan's result that every projective ZG-module is locally free, see e.g. [Fl]; it is sometimes more convenient to work with an algebraic closure Q of the rationals in place of E.) Two examples of analytic classes: (a) Cassou-Nogues and Frohlich have shown how to define a Galois equivariant homomorphism by using the root numbers attached to a Galois extension of number fields (see [F2]); (b) Tate's formulation of the Stark conjecture says that a certain L-value/regulator quotient defines a Galois equivariant homomorphism (see [Ta] Chap. I, 5.1). These two constructions are fundamental in the Galois module theory of number fields and have been greatly generalized. For instance, root number classes have been defined for tame actions on arithmetic schemes in [C3] 5.8, [CEPT2] 3.2 and also for motives endowed with a group action arising from base change by a Galois extension in [B1] Sect. 4. One can usually control the order of these analytic classes and locate them inside the classgroup. Thus they give a good measure for the triviality or otherwise of classes defined in other ways and that are related to them.
1.b
Tame actions
The second kind of classes arise from taking Euler characteristics of perfect complexes, that is (complexes quasi-isomorphic to) bounded complexes with finitely generated and projective terms. Let X be a projective scheme over a regular Noetherian commutative ring R, equipped with an action by a finite group G. We will say that the action
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of G on X is tame, if the order of the inertia group of each point is prime to the residue characteristic of the point. From [C3], [CE] and [CEPT1], Sect. 8 one has the Euler characteristic homomorphism x = xCT : Go (G, X) - CT (RG)
from the Grothendieck group of coherent G-sheaves on X to that of cohomologically trivial RG-modules. Indeed one shows that for any coherent Gsheaf F on X there is a perfect complex (Mi)i of RG-modules, well-defined up to quasi-isomorphism, which computes the cohomology of F and one puts x(F) = E1(-1)i[Mi] (see [CEPT1] Sect. 8, where the case of actions by not necessarily constant finite affine group schemes is also considered; [Nal] contains a special case also noted independently by Kani). Recall that Go (G, X) (resp. Ko (G, X)) denotes the Grothendieck group of all coherent (resp. locally free) G-sheaves on X. For regular X the forgetful map defines an isomorphism from Ko (G, X) to Go (G, X) and we will identify these two groups when X is regular.
1.c
Perfect complexes in etale cohomology and omega invariants
Other kinds of perfect complexes of arithmetic interest arise as follows. The motivating example is Tate's construction of a four term exact sequence involving S-units and which represents a fundamental class coming from class field theory. Let N/K be a Galois extension of number fields with Galois group G and let S be a finite G-stable set of places of N. Denote by U the group of S-units in N, and by X the kernel of the augmentation/degree map on the free abelian group over S. Recall from [Ta] 11.5 that if S is large enough (see loc. cit.), then there is an exact sequence of ZG-modules
0-4U-4A-+ B -+X -+ O, which represents Tate's fundamental class a in Ext2 (X, U) and where A and B are finitely generated and cohomologically trivial. By a homological lemma proved in-say-[Wa] 1.3, the class [A] - [B] in Ko(ZG) only depends on U, X and a. In [Cl] Chinburg showed that this class is independent from S as long as S is large enough, so the class only depends on the extension N/K.
It is denoted 1(N/K, 3). Note that it actually lies in the rank 0 subgroup of K0(ZG) by the Dirichlet S-unit Theorem. Recall that this subgroup is naturally isomorphic to the classgroup Cl(ZG). Independently, Pappas and Snaith saw how to use the results of Kahn in [K] to construct a four term sequence relating the third and second K-group
of the S-integers in N, in the case that N/K does not ramify at infinity
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(see [Sn1] Ch. 7). A direct approach, using only K-theory, to get classes related to the Galois structure of higher K-groups does not seem possible. However one gets four term sequences for all extensions N/K which relate K2z_1 to K2i_2i for any i > 1, by using etale cohomology and the Chern character homomorphisms (see [CKPS1] and the account in [BF2] 4.1). One of the basic ingredients in the construction is a lemma which we state in an imprecise form (see [Kal] 4.17 or [BF1] 1.20).
Lemma 1.1 For any prime p such that no place over p lies in S, the etale cohomology with compact support of a smooth constructible Zp-sheaf F on the ring of S-integers can be computed by a perfect complex of ZPG-modules.
When G is abelian, this lemma was used by Burns and Flach in [BF1] to assign Galois structure invariants to the base change to N of a motive over K provided certain standard conjectures hold. These conjectures hold for the Tate motives Z(i), for example. The restriction to abelian G resulted from the use of determinants over the group ring of G. The case of arbitrary G and Tate motives was considered unconditionally by Chinburg, Pappas, Kolster and Snaith in [CKPS1]; they used a different method which involves the construction of certain four-term sequences of G-modules. The four-term sequence method was generalized to motives by Burns [B1]. An approach developed in [CKPS2] which does not rely on four-term sequences will be described in Sect. l.d. The four-term sequence approach of [CKPS1] for the Tate motive Z(i) proceeds in the following way. Let Cp(i) be the cokernel of the injection ZP(i) -> Eve. i,,.ivZP(i), where i, : SpecN -+ SpecON,s is the natural map. By Artin-Verdier duality, this sheaf has cohomology concentrated in degrees 0 and 1, thus one obtains an exact sequence of ZPG-modules
0-*H°(Cp(i))-*M-M'-*H1(Cp(i))-+ 0, with M and M' cohomologically trivial ZPG-modules. This four term sequence defines an element in ExtZpG(H'(Cp(i)), H°(Cp(i))). By taking prod-
ucts over p one gets a class E in Ext4G(jjp H1(Cp(i)), rJP H°(Cp(i))). By using results in [DF] on the Chern character homomorphisms, one is then lead to consider the following approximations to the K-groups of S-integers. One puts K2z_2 = !lp H1(C,,(i)) and shows that there is a finitely generated submodule K2i_1 of !l, H°(C,,(i)), which becomes equal to it after extending scalars from Z to Z. Hence we can view E as an element in ExtZG(K2z-2®zZ, K2Z-1®zZ) = Extzc(K2z-2, K _1)
Because the groups K2z_j are finitely generated for j = 1 and 2 and cup product with E induces cohomology isomorphisms, E is represented by an
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exact sequence
0 -4 K2i_1 -3 Ai - Bi -+ K2i_2 -> 0 in which Ai and Bi are finitely generated cohomologically trivial ZG-modules. The classes [Ai] - [Bi] of K0(ZG) are not of rank 0, hence one subtracts from
them the free ZG-module ZED, which is the free abelian group on the set E.,, of complex embeddings of N. The resulting classes are the generalized omega invariants of [CKPS1]:
cl (N/K) = [An+1] - [B.+,] - [ZEW] .
The omega invariants of Burns and Flach are discussed briefly in the last section.
1.d
Nearly perfect complexes
In his work on the value at 1 of the zeta function of a surface X over a finite field, Lichtenbaum was lead to define a (numerical) Euler characteristic for the multiplicative group Gm, viewed as a sheaf for the etale topology. Assume that the Brauer group H2(X, Gm) is finite. Even under this assumption, one cannot expect H3(X, Gm) to be finitely generated. However, this group is dual to a finitely generated group. Using this fact, Lichtenbaum was able define a numerical Euler characteristic X(X, Gm). The Euler characteristic X(X, Ga) of the additive sheaf Ga in the etale topology equals the usual numerical Euler characteristic of the coherent sheaf Ox. Lichtenbaum proved that (under the assumption that H2(X, Gm) is finite), the quotient X(X, Ga)/X(X, Gm) equals ±1 times the leading term of the zeta function of X at s = 1; [Li2] 4.1 and [Lil] Sect. 3. Now let (X, G) be a free action of the finite group G on the integral scheme X. Suppose R is a ring and that F' is a complex of sheaves of R-modules for the etale topology on X/G, which is bounded below. Suppose that for each subgroup H of G, only finitely many of the cohomology groups Hi(X/H, F') are non-trivial. By arguments similar to those mentioned in Sect. 1.b, one can show that the etale hypercohomology H'(X, F') is isomorphic in the derived category to a bounded complex of RG-modules which are cohomologically trivial for G. For X a regular surface over a finite field, let C' be the complex computing the cohomology of Gm on X. Assume still that the Brauer group H2(X, Gm) is finite. For i = 3 or 4, let Li = H4-i(X, Gm)/H4-i(X, Gm)tar Lichtenbaum's work on the cohomology of Gm in [Li2] Sect. 3.4 and 4, shows that for i = 3 or 4, there are isomorphisms Ti : Homz(Li, Q/Z) -4 H2(X, Gm)div ,
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where Adiv is the subgroup of divisible elements of A. The same result is obtained by S. Saito in [SS] when X is a regular surface which is flat over Z with finite Brauer group. Put Li = 0 and Ti = 0 for i # 3 or 4. The triple (C', (Li)i, (Ti)i) defines what is called a nearly perfect complex in [CKPS2]. In loc. cit. the authors show how to define an Euler characteristic class in Ko(ZG) for any nearly perfect complex. The class only depends on the quasi-isomorphism class of the complex. If C' happens to have finitely generated cohomology, then Li and ri are trivial and this new Euler characteristic coincides with the Euler characteristic of the perfect complex obtained from C' by the usual approximating procedure.
2
Varieties over finite fields
In this section we review two ways of giving a module theoretic interpretation of results on zeta and L-functions for varieties over finite fields. The material presented in the first half of this section comes from [C3] (the presentation follows [CEPT5]). It relates the Galois structure of de Rham cohomology to e-factors. In the second half of the section, following [CKPS1], we relate the Euler characteristics X(X, Gm), introduced at the end of the previous section, to L-values.
2.a
Galois structure of de Rham cohomology
The work of Chinburg presented here represents the coming together of two lines of research, which take their origin in the Galois theory of the thirties. Namely Noether's work on the Normal Integral Basis problem and the work of Chevalley-Weil on the Galois structure of differentials (see [CE] for the references). The Galois structure of differentials for varieties over fields has been studied by various authors and, as we mentioned above, in his study of the question Nakajima already considered refined Euler characteristics. In his set-up, one knew how to calculate the character of the Euler characteristics. Projectivity then showed this was sufficient for their determination as modules. Following the approach suggested by the Frohlich Conjecture, Chinburg shifted the emphasis from the explicit determination of the module structure of differentials to comparing Euler characteristics to analytic invariants associated to L-functions. Let p be a prime number. In this subsection we consider a smooth projective variety X which is equidimensional of dimension d over the finite field k with q = pf elements. We assume X is equipped with an action by a finite group G, which is tame in the sense of Grothendieck and Murre. (This is
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stronger than the numerical tameness discussed in Sect. 1.b, see [C3], [CE] and [GM] for details; if one has resolutions of singularities, some of the results can be generalized to apply to the weaker notion of tameness in which one requires only that the order the inertia group of each point is relatively prime to the residue characteristic of the point.) The zeta function Z(X, t) satisfies the functional equation Z(X, t) = ±(gdt2)-ex/2Z(X, q-dt-1)
Here ex is the topological (or 2-adic) Euler characteristic, also equal to the self-intersection number of the diagonal Ax in X x X. The valuation at p of the constant e(X) := can be interpreted in terms of the de Rham cohomology of X. Namely, let aX/k denote the sheaf of i-differentials on X and let x(Qix/k) denote its (numerical) Euler characteristic, then d
2
ex = (-1)`(d - i)X(SZX/k) i=O
(see e.g. [CEPT5]; this is a consequence of the Hirzebruch-Riemann-Roch theorem and Serre duality).
Write Y = X/G for the quotient of the action, then for any complex representation V of G there is defined an algebraic number e(Y, V), which is the constant in the functional equation of the Artin L-function attached to the action (X, G) and V. Actually e(Y, V) only depends on the character Xv of V and we shall therefore write e(Y, Xv) = e(Y, V). Let vp(-) denote the valuation at p. For any g in G, let S(g)
E vp(e(Y, X))X(g) x
where the sum runs over the set of irreducible complex characters of G. Then,
by the above, -S(1) = vp(e(X)) equals [k Fp]d ex/2 and thus has an expression in terms of de Rham cohomology. The first main result is an :
expression for S(g) in terms of the Brauer traces of g acting on a virtual kGmodule, which is a combination of refined Euler characteristics of the sheaves SZX/k, as defined in Sect.l.b. Consider the class in K0(FpG) d
IF (X, G) _ (-1)'(d - i)ResFp(X°1'(Qix/k)) i=0
where Res denotes restricting scalars from k to Fp. For N an FpG-module and g in G of order prime to p, let BTr9(N) denote the Brauer trace of g on N. Then we have (see [C3] Thm. 5.2)
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Theorem 2.1 For any g in G of order prime to p, S(g) = BTr9(W(X, G))
.
In [C3] Chinburg deduces the result from work of Milne and Illusie on the slopes of Frobenius in crystalline cohomology. A more direct approach is used in [CEPT5] to deal with curves and surfaces. It goes as follows. Once the c-factors are connected to congruence Gauss sums by using the results of T. Saito in [ST], one can use Stickelberger's Theorem to determine the valuations vp(c(Y, x)). The computation of the Brauer trace of g on W (X, G) is carried out by use of a Lefschetz-Riemann-Roch (LRR) theorem going back to Donovan (see [Do] and [BFQ]). (To use LRR one has to analyze in detail the geometry of a tame action. This analysis becomes combinatorially involved for varieties of dimension higher than two, which explains the restriction on
dimension imposed in [CEPT5], but see [CPT] for a way to organize the computation in higher dimension.) Let us indicate how the proof works for a curve X. The group K0(FpG) injects into the group Go(FpG) of all finitely generated FpG-modules and the Brauer trace is defined on the latter group (see [Se]). For simplicity of notation assume X is defined over k = Fp and assume that k is large enough. Let f : X -* Spec k
denote the structure morphism. There is a direct image homomorphism f* : Go (G, X) -+ Go(kG)
,
which factors over the refined Euler characteristic map. Also, for g 0 1
in G let < g > denote the subgroup of G generated by g, then there is a commutative diagram
Ko(G,X) - Ko(,X) f* d-
Go(kG)
-f
. f*
Go(k < g >)
Assume that g has order prime to p, then the fixed point scheme X9 is regular and the inclusion i : X9 -+ X is a regular embedding. One has a restriction map i* : Ko(< g >,X) --* Ko(< g >,X9), and an isomorphism Ko(< g >, X9) =' Ko(X9) 0 Go(kG). It follows that the conormal sheaf N = Nx9lx is locally free ([FuL]). Consider the class a_1 (N) := [Ox] - [N] in Ko(< g >, X9) (recall that X is a curve; in general one needs to consider the exterior powers of N up to dim(X)). Let W = W(k) denote the ring of Witt vectors of k (here W = Zp). One has a "localization" map Ko(< g >, X9) = Ko(X9) ® Go(kG) --+ Ko(X9) O W
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induced by the Brauer trace BTrg : Go(kG) -> W. The image of A_1(N) under this map is invertible. The Lefschetz-Riemann-Roch Theorem asserts that the following diagram is commutative Ko(X9) ® W
Ko(< g >, X) f*
f9®W
B9
G0(kG)
W = Go(k) ® W
This gives a way to compute the Brauer trace appearing in the theorem. Let us calculate the Brauer trace BTrg(f*(OX)). By LRR every closed point x which is ramified in the covering X/Y contributes to the computation. The conormal sheaf Al at x is a one dimensional representation of G over k (large enough!)-say k Xx. So the contribution of x to a_1(N) is 1 - Xx. Hence:
BTr(f (0X )I= v' g
+
xEX9
1
1 - Xx(g)
The expression S(g) can be evaluated by using Deligne's expression for the 6factors of curves as a product of local terms, which are essentially congruence Gauss sums by tameness (see [De2]). The valuation of such Gauss sums can be computed by Stickelberger's Theorem (see [Fl] Thm. 27 or [La] IV.4). The theorem then results from the observation that if for an integer e the complex number w 1 is such that we = 1, then there holds 1
1-w
_ l
e-1
en=1
nwn
Now, following Chinburg, we want to reformulate the above theorem as a result on classes in the classgroup Cl(ZG). Consider the map
Rest : Ko (FpG) -* Cl(ZG) obtained by composing restriction of scalars and considering the stable equivalence class. The result we want to state is that Resz('I(X, G)) only depends on root number data at infinity for symplectic characters and on root number data over the branch locus b of the covering X/Y. We first factor Rest over the map cl giving the Hom-description (see Sect. 1.b). Let 12 (resp. 52p) denote the absolute Galois group of Q (resp. Qp) and let RG,p be the group of representations of G over the algebraic closure QP of Qp. There is an injection 0 : Ko (FpG) -* Homnp (RG,p, QP )
obtained by writing a projective FPG-module P as the quotient of a projective
module P by pP, and letting
PH(XHP-'X)
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where mx denotes the multiplicity of X in P 0 QP. Next, let (Q)P' = (Q (9q QP)" and choose an embedding h of Q in QP. This gives an isomorphism h*-1 : Homon(RG>p, QP)
HomSe(RG, (Q) P )
The last group maps into Homn(RG, J(Q)), the group on which cl is defined. Then Rest = cl o h*-1 o A .
For any place v of 4 denote by the image of e(Y,V) under the injection of Q" into the group of ideles J(Q), through (Q)v . Note that for V symplectic e(Y, V) is totally real and so has a well defined sign at each archimedean place of Q. These signs give an element sign (e,,. (Y) (V)) in (Q)x _ (Q ®Q R)". The root number class attached to the covering X/Y is then the analytic class Wx/r = cl(Xv H hoo(Xv)) where the p-component of the idele
for V irreducible, is given by
Xv not symplectic p finite hoo(Xv)p = I sign(e.(Y)(V)) p = oo , Xv symplectic 1
1
Let as before b be the ramification divisor of X/Y. There exists a p-primary idele lep(b)lp(Xv) such that JEp(b)IP(Xv) ep(b)(Xv) is a p-unit idele. Then the ramification class is the analytic class Rx/Y = cl(Xv H Eoo(b) - lep(b)Ip1(Xv)) .
The main result in [C3] for X of arbitrary dimension can now be stated (see loc. cit. Thm. 5.10). Theorem 2.2 With the above notation and assumptions, the following equality holds in the classgroup Cl(ZG), Resz(T (X, G)) = Wxly + Rx1 v
.
We make some remarks on the proof, in particular we indicate why only 6factors for symplectic representations and those supported on the branch locus appear in this result. Using the above factorization of Resz, Theorem 2.1 can be restated as the equality Resz(W(X,G)) = -cl(Iep(Y)Ip)
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Here all characters and all of Y play a role. Let U = Y \ b and let e(U) be the product of the e(U) over of Q. 0 in Moreover of the of Q. In loc. cit. Thm. 5.12 Chinburg uses the connection between epsilon factors and the determinant of the (opposite of) Frobenius on v-adic etale cohomology to show that the right hand side of the last displayed equation equals (up to sign) the analytic class defined by I CP(Y) I P -
(EY(U)eoo(U))
Using the additivity of epsilon factors under disjoint unions, this can be written as (Y) . I ep(b)IpI ep(U)Ip - ep(U) - e,,. (b) In loc. cit. Thm. 5.13 Chinburg shows how to deduce that cl (I ep (U) I pep (U)) _
0 from the work of T. Saito in [ST] and Taylor's Fixed-Point Theorem for group determinants (see [Fl] Chap. 11.6, Thm.10A). Thus U does not appear anymore. Moreover for any V, all finite components of the idele (e... (Y) ham) (Xv) are trivial, and the infinite components are positive if V is symplectic. From this one deduces that cl(e,,.(Y) ham) = 0 and thus only symplectic characters come into the definition of W l y. This then concludes our sketch of the proof of the theorem.
2.b
L-values and the cohomology of G,n, for surfaces
Let X be a smooth projective geometrically connected surface over a finite field and assume that its Brauer group H2 (X, Gm) is finite. Let the finite group G act freely on X so that the quotient morphism X -+ Y = X/G is etale. We describe a result from [CKPS2] on the image of the Euler characteristic X(X, Gm) defined in the previous section, in the Grothendieck group of finitely generated ZG-modules, under the forgetful map f : Ko(ZG)
Go(ZG)
.
Let GoT(ZG) be the Grothendieck group of all finite ZG-modules, then we also have the forgetful map z : GoT(ZG) -* Go(ZG) .
By work of Queyrut, the group GOT(ZG) admits a Hom-description, so that one can define analytic classes in it as well. We shall define such a class by using the leading terms at 1 of the L-functions attached to (X, G) and show that it is equal to a class whose image under z is f (X(X, Gm)).
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Let cv denote the leading term at s = 1 of the L-function L(q-s, V). Then the function CX,G : Xv H CV
is Galois equivariant and thus defines a class in G°T (ZG) (see [CKPS2] Thm. 2.8). Let X : H1(X, Gm) -i H'(X, Gm)D = Homz(H'(X, Gm), Z)
denote the map induced by the intersection pairing on divisors. It has finite kernel and cokernel. Define classes in G°T (ZG) by
XT(-, Gm) = [H°(X, Gm)] +[coker(.)]
- [H1(X, Gm)tor] + [H2(X, G,,,,)]
- [H3(X,
[H4(X, Gm)]
and
XT(X, Ga) _ [H°(X, Ox)] - [H1(X, Ox)] + [H2(X, Ox)] Then, z(XT(X, Gm)) = f (X(X, G,,,,)) and a slight generalization of the work in [Li2] allows one to show that cx,G = XT(X, Gm) -XT(X, Ga). But because we assume the action is free, by work of Nakajima [Na2], z(XT(X, Ga)) = 0, so f (X(X, Gm)) = Z(CX,G). The final part concerns identifying the right
hand side in this last equality with a class only depending on the signs of the leading terms for symplectic characters. Let Lx,l be the class defined in G°(ZG) via the Hom-description by the element h of Homo(RG, J(Q)) such that for irreducible X v finite v = oo, X not symplectic I sign(cx,G) v = oo, X symplectic 1
h(X),, =
1
Then Lx,l has order at most two and z(cx,G) = Lx,i. In summary
Theorem 2.3 f (X(X, Gm)) = Lx,l .
3
The generalized Frohlich Conjecture
Once results analogous to Taylor's theorem, for varieties over finite fields were known (in any dimension!), one started wondering about the possibility of formulating a generalized Frohlich Conjecture for arithmetic schemes. Such a conjecture was formulated in [CEPT2] and proven there in many cases. A proof of that conjecture in general is contained in [CPT]. To arrive at a reasonable conjecture, the first thing to try was, of course, to see if one could define classes analogous to those defined by Chinburg in
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130
the case of-say-arithmetic surfaces. One had to find a combination of refined Euler characteristics, which was related to epsilon factors and possibly obtain a relation like Chinburg's T = W + R. Classes analogous to the root number class and the ramification class could be defined by using Deligne's theory of e-constants of representations of Weil-Deligne groups ([De2]). But, as we will see, the strict analogue of IF does not seem to be related nicely to e-constants. Let X be a regular projective scheme flat over Spec Z, equidimensional of dimension d+1, which admits an action by the finite group G. Let Y = X/G. Assume that the action is tame in the sense that, for every closed point x of X, the inertia subgroup at x has order relatively prime to the characteristic
of the residue field of x. Then the branch locus b in Y is fibral and every coherent sheaf on X admits a refined Euler characteristic. Which combination of sheaves is related to a root number class? A root number class would have order at most two and it would lie inside the so-called kernel group D(ZG),
that is the group of classes in the classgroup, which become trivial after extending scalars to one (hence any) maximal order in QG containing ZG. One can show that, even for surfaces, the class ' does not lie in the kernel group (see [C3] Rem. 3.9). The correct class in the present framework is the following. Note that in general the sheaf Qx' lz of absolute differentials on X is not locally free, however by the regularity of X we can identify the two groups Go(G, X) and Ko(G, X). Thus it make sense to take exterior powers of classes in Go(G, X). We define the Euler-de Rham class in Cl(ZG), attached to the tame action (X, G), to be
=
d E(-1)iXCT(AZ([QX]))
X(X,G) i=0
It is shown in [CEPT2] Sect. 1.4.5 that if (2, G) is a tame action on a variety of dimension d + 1 over a finite field, which is fibered over a curve C, then the Euler-de Rham class defined using the relative differentials for ,i/C equals the class W (2, G). To conclude the comparison of X and' over a finite field, we note that (up to sign) X is the Euler characteristic of the top Chern class of Q1 whereas ' is the Euler characteristic of the penultimate Chern class (see [CPT]). The root number class and the ramification class are defined by analogy with the case of varieties over finite fields and the case of extensions of number fields. Let X = X ® Q be the generic fibre of X. The local epsilon factors that come into the definition of the root number class are those attached, via a recipe due to Deligne, to the motives X ®G V whose realization are given by the cohomology groups (H*(X) 0 V)G (see [De2]). In fact the epsilon One must factor at v is modified by the epsilon factor of the fiber also choose a fixed auxilary prime f such that Xe is smooth together with an embedding Qp -* C (see [CEPT2]). One can then define a root number class
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Wx,e and a ramification class Rx. The ramification class is defined in terms of the epsilon factors of the fibers of the reduced branch locus of the quotient
X- y=X/G.
Theorem 3.1 In addition to the above assumptions, also assume that the quotient Y is regular, with special fibers Yp that are divisors with strict normal crossings and for each p assume that the multiplicities of the irreducible components of Y, are prime to p. Then
x(X,G)=Wx,e+Rx. This was proved in [CEPT2] for surfaces and for free actions in arbitrary dimensions. The general case is the main theorem of [CPT]. Below we indicate a proof for free actions and outline the strategy for the general case. One of the reasons for the restrictions imposed on y is the use of the results in [ST] on epsilon factors. Another reason is that they imply that a sheaf of relative log-differentials, which is used in the proof, is locally free. It is conjectured that the result holds without these restrictions. For the fact that Taylor's theorem is a special case of this result see [CEPT2] Thm. 4.13 (and also [P2]).
Assume now the action (X, G) is free. Then the quotient morphism it X -+ y is etale and the ramification class is trivial. Moreover, every coherent G-sheaf on X is of the form lr*F for some coherent sheaf F on Y. The idea is to reduce to the one dimensional case, where one can apply Taylor's result. Let cy :_ (-1)dcd(SZy/Q) denote the fundamental class of the generic fiber Y of Y. This is an element in the Chow group CHd(Y) represented by a 0-cycle on Y. Write Cy as a linear combination of closed points of Y
cy=Eej[Pj] Let Pj denote the Zariski closure of Pj in Y. Then OP,,,yx is a G-stable order inside the G-Galois algebra whose spectrum is Pj Xy X. Let S; be the normalization of Pj xy X. By comparing Ej ej [Os3] to E" o(-1)i)i(1,ly) in Ko(G, X) = Go(G, X) and by using results of Thomason [Th] and Nakajima [Nal], [Na2], one can show that the Euler-de Rham class can be represented as
x(X, G) = (-1)d
ej . xcT(S
G)
Thus by applying Taylor's Theorem, one is able to relate the Euler-de Rham class to a combination of root number classes coming from number field extensions. The relation to the root number class Wx,l comes from a product formula for the symplectic root numbers at infinity. This formula shows that the root numbers of the motives X ®G V over R for symplectic V can be
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obtained as the product of the symplectic root numbers of the points in the canonical cycle of Y. In the proof of the product formula, the root numbers of the real motive are calculated using a formula of Deligne, which expresses them in terms of Hodge numbers (see [Del]). The product of root numbers of the points in the canonical cycle is expressed in terms of the Euler numbers of the connected components of the real locus of Y. This uses the topological Lefschetz fixed point theorem and a result on cycles and characteristic classes due to Borel-Haefliger. The formula then results from a (long) calculation. For the details see [CEPT2] Sect. 5 and 6. Let us now go back to the general case. Then not every G-sheaf on X is the pull-back of a sheaf on Y. Still, as in the case of a free action, the strategy will be to study the pull-back of a locally free sheaf whose refined Euler characteristic is closely related to the Euler-de Rham class. Note that this class can be viewed as the Euler characteristic of the top Chern class of
i. x
X(X,G) _ (-l)dxCT(cd(QX)) Because of our assumptions on y, the sheaf Sly (log) = Sly/z (log ys d/ log S)
of relative logarithmic differentials is locally free. Here S denotes the set consisting of those primes p at which the fibre Yp is not smooth and those
supporting the branch locus of X/y (recall that X/y is generically etale because the action is tame), and Sly(log) is defined with respect to the morphisms of log-schemes (y, ys d) -4 (Spec Z, S). We put Xi(X,G) _ (-1)dXC1(c1(1r*Qy(log)))
and we let X2 (X, G) = X(X, G) - Xi (X, G). By using a Moving Lemma for Chern classes, X, (X, G) can be written in terms of the refined Euler characteristics of one dimensional schemes flat over Z and hence can be related to symplectic root number classes by using Taylor's theorem. The relation to
the root number class is achieved by using the product formula for the root numbers of real motives mentioned above and the product formula for epsilon factors of tame sheaves proved by Saito in [ST]. To deal with X2 (X, G) one notes that because it is the Euler characteristic of the difference of two classes in KO (G, X) which give the same class in Ko (G, X), using [Th] Thm. 2.7, it can be written as X2 (X, G) = E Resz(XCT (Fp)) PES
where the Fp are elements of G0(G, Xped). Then one defines a class 'p by using the T-classes of Chinburg attached to the strata of the reduced fiber XrPed, such that Resz(XCT(Fp))
= (-l)dResz('p)
.
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This equality is proved by showing, using LRR, that the Brauer traces of XCT(Fp) and Tp (for elements different from the identity) are the same. The
relation to root numbers is achieved by using the results of Chinburg on varieties over finite fields. For the details see [CEPT2] and [CPT].
4
Characterizing epsilon factors
So far we have seen how to determine (combinations of) refined Euler characteristics in terms of epsilon factors. Proving a second conjecture of Frohlich, Cassou-Nogues and Taylor have shown how to characterize the signs of symplectic root numbers attached to tame Galois extensions of number fields, in terms of the hermitian Galois module structure of rings of integers supplied by the bilinear trace form. Here we describe analogous results for the cases of varieties over finite fields and of schemes over Zp. We also state a theorem generalizing a result of Serre, which asserts that for orthogonal representations the epsilon factors of curves are positive.
4.a Varieties over finite fields Recall the notation of Sect. 2.a. In particular let e(Y, V) denote the epsilon constant attached to an action (X, G) with quotient Y, and a complex representation V of G. In this subsection we consider real characters of even degree with trivial determinant, not just symplectic characters. Let RG denote the subgroup of RG which is generated by these characters. If the character of the representation V is real valued, then e(Y, V) is real. The following is an easy generalization, using Poincare duality, of a result of Serre (proved in [FQ], see also [CEPT3] Thm. A).
Theorem 4.1 If the dimension of X is even (resp. odd) and if V is a symplectic (resp. orthogonal) representation, then e(Y, V) is positive.
By the results of Chinburg described in Sect. 2.a, the p-adic valuations of the epsilon constants e(Y, V) are integers, which are determined by the class %F (X, G). To determine the sign of e(Y, V) for Xv in RG, we introduce the (generalized) adelic hermitian classgroup A(ZG). Let E be a large enough Galois extension of the rationals and let Jf (E) denote the group of finite ideles of E. The properties of A(ZG) that we need are: (a) there is a surjection p : Homaai(E/q)(RG, Jf(E))) -4 A(ZG) , analogous to the map giving the Hom-description of Sect. 1.a;
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(b) A(ZG) contains a subgroup isomorphic to HomGa,`(E/Q)(RG, Q"), called the group of rational classes.
(This last fact is a generalization of work of Cassou-Nogues and Taylor, see [CEPT3] Thm. 3.2 and [F3] Cor. 3 to Thm. 17.) By composing the injection A of Sect. 2.a with the map induced on the Hom-groups by restricting homomorphism from RG to RG and with the surjection µ, we obtain a map
a : Ko(FPG) -* A(ZG) . We shall use a stratification (Ui, Gi)i of (X, G) and classes q/(Ui, Gi) whose image under the maps ai, defined in the same way as a with G replaced by Gi, are rational classes. These classes combine to determine the epsilon constants. For simplicity assume that all irreducible components of the branch locus B
in X are in the same G-orbit. Then one can define locally closed Ui in X with a number of desirable properties. Namely:
(1) all points of Ui have the same inertia group Ii, which is normal in the stabilizer Di of Ui in G; (2) Ui carries an action of Gi = D/Ii; T (3) X = Lh LIgEG/Di 9Ui
The epsilon factor of X can be expressed in terms of the epsilon factors of (Ui, Gi). Thus, if we could characterize these epsilon factors in terms of IFinvariants, we would be done. However, Chinburg's T-invariants are only defined for smooth projective varieties with a tame action. The idea then is to define 'I/(Ui, Gi)_by inclusion-exclusion. For instance, if the difference between the closure Ui of Ui and Ui is the union of two closed varieties Zl and Z2, which intersect in P, then we would put
'F (Ui) _'FA) - 'F(ZO - T(Z2) +'F (P) . The surjection Di -> Gi and the inclusion Di -4 G induce maps on the level of the Hom-groups denoted Infi and Indi respectively. Denote by er (X, G) the restriction to R' of the map sending XV in RG to e(Y, V). The following is essentially Thm. C of [CEPT3].
Theorem 4.2 With the above notation (i) The classes ai(T (Ui, Gi)) are rational. (ii)
er(X, G) = 11 Indi o Infi(ai(`I'(Ui, Gi)))
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Let us explain this result for curves. The Ui above are defined in terms of a stratification of the branch locus B of X/Y. If X is of dimension one, the branch locus is a collection of points, and epsilon factors for points are roots of unity. From this and the theorem one deduces the following.
Corollary 4.3 Let X be of dimension one, then Er(X,G) = a([H°(X,Ox)] - [H1(X,Ox)])
4.b
Arithmetic schemes
Here, following [CEPT4], we indicate how one can characterize symplectic epsilon constants, in terms of the intersection numbers of certain Pfaffian divisors with a sheaf constructed from logarithmic differentials. Although this does not transpire from our summary, the results we present rely heavily on the work of T. Saito in [ST]. Let X be a regular projective scheme flat over Spec Zp, which is equidimensional of dimension d + 1. We assume that X is equipped with an action by a finite group G, which is such that the quotient y is regular connected and such that the pair (y, Y x Qp) is tame in the sense of [Ka2] (2.2) (e.g. Y semistable). Then the sheaf fly (log) = ci ,1 (log yped/ log Fp)
of relative logarithmic differentials is locally free of Oy-rank d. Let d
.\-1(QY(log)) = E(-1)'Az(Qy(log)) i=o
Assume that the action (X, G) is tame over a vertical divisor with strictly normal crossings. Then, for any symplectic representation V of G, imitating Frohlich's construction in [F3] 11. 3, we can define a divisor P f (X, G) (V) on y by using the trace form at the local level. This divisor is called the Pfaffian divisor. Under the tameness assumption, for any symplectic representation V of degree 0, the constant eo (X ®G V) is (up to sign) an integral power of p. This power can be determined in terms of the Pfaffian divisor as
vp(EO(X ®c V)) = d(V) := deg(\-1(9y(log)) Pf(X,G)(V)) (The right-hand side can be thought of as an intersection number; see loc. cit.)
This equality can be refined to get a statement about classes inside a classgroup. Let 0 denote the ring of integers in the cyclotomic field
Q(e2i'r/p).
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As in the previous subsection, we consider the adelic hermitian classgroup A(OG), which comes with a surjection a : HomGai(E/Q)(RG, Jf(E)) + A(OG)
and which contains a subgroup of rational classes. One shows that the image by a of the homomorphism sending Xv to the idele with p-component (-P)d(V)
and all other components 1, is a rational class c and thus corresponds to an element of HomGal(E/Q) (RG, Q") again denoted by c.
Theorem 4.4 For all symplectic representations V of G c(V) = eo(X ®G (V - (dim(V) 1)))
.
Corollary 4.5 If Y has semistable reduction, then for any symplectic representation sign(c(V)) = sign(eo(X ®G V))
5
.
Normal bases for elliptic division orders
Let us begin by recalling what a normal basis for a Galois extension of rings is. Let F be a finite group and let R be a ring. A F-Galois extension of R is an R-algebra S on which r acts by ring automorphisms in such a way that R = S' and the S-algebra S OR S is isomorphic to the algebra Map(F, s) of all maps from r to S, under the map x ® y i-+ (ry H xry(y)). In other words, Spec S is a principal homogeneous space/torsor for the constant group scheme FR associated to the algebra Map(F, R). Then S has a normal basis, if S is isomorphic to RF as an RF-module. Note that RF is the dual R-algebra of Map (17, R).
We will be interested in the situation where S is replaced by an order whose spectrum is a torsor for a non-necessarily constant group scheme obtained as the kernel of an isogeny between abelian varieties. Let R be the ring of
integers in a number field and let A be an abelian scheme on T = SpecR. Consider an isogeny f from A to A, and let G denote its kernel. For any R-valued point of A, say Q : T -> A, we can form the pull-back XQ = A XAT
of f along Q. This is an affine T-scheme which is a torsor for G. Let A (resp. 8Q) denote the ring of G (resp. XQ). These are both modules over the linear dual B = HomR(A, R). It can be shown that BQ is a locally free module of finite rank over B. Assume for simplicity that A is free over B. Then the map from the group of R-valued points A(R) to the classgroup of locally free B-modules, which sends Q to the class of BQ, can be shown to be a homomorphism
0: A(R) -* Cl(8)
.
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This is called the class invariant homomorphism. It has been first introduced by Waterhouse in [Wat] and it has been studied in relation with Galois module theory by Taylor and then Agboola, Cassou-Nogues, Srivastav et al. (see [T2] and the surveys [C-N,T], [BT] and [C-N,C,F,T]). It is clear that a point Q defines an element in the kernel if and only the order l3Q is free over B, i. e. if and only if it admits a normal basis. One of the main results about the class invariant homomorphism is that for A = E of dimension 1 and f multiplication by the power of a prime p > 3, then any torsion point is in the kernel of ': E(R)t,,,. C ker(z/))
.
Srivastav-Taylor proved this for elliptic curves with complex multiplication, by finding explicit normal basis generators using modular functions (see [SrT]).
Then, in [Ag2] Agboola was able to prove the result without assuming the curve has complex multiplication. In [Agl] Agboola showed the result to be equivalent to the following geometric result on the restriction of torsion line bundles, which was proved by a new method and in complete generality by Pappas in [P1].
Theorem 5.1 Let E be an abelian scheme of relative dimension 1 over a commutative ring R and let m be an integer coprime to 6. Let G be a torsion line bundle on E with trivial restriction to the 0-section. Then the restriction of G to the subscheme of m-torsion points En is a trivial line bundle. In [P1], Pappas shows how to construct examples of abelian schemes fibered over an affine curve defined over a finite field, for which the analogous result does not hold. The restriction to integers prime to 6 is necessary as shown by Bley-Klebel in [B1K1] and by Cassou-Nogues-Jehanne in [C-N,J].
6
The equivariant arithmetic genus
In the preceding sections we have insisted on the fact that the main results about refined Euler characteritics concern combinations of such. Not much is known about the Euler characteristics of individual sheaves. In this section we will see how, following [P2], one can obtain results on the refined Euler characteristic of one sheaf alone. Let X be an arithmetic surface, that is a projective scheme of dimension 2 flat over Spec Z. Assume X supports a free action by a finite abelian group G and, as before, denote the quotient by -7r : X -+ Y = X /G. For simplicity we assume that Y is a local complete intersection. We want to show how to use Deligne's Riemann-Roch theorem for relative curves of [De3] to deduce that under the stated hypotheses
2xCT(OX)=0.
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Let GD denote the Cartier dual of G (viewed as a constant group scheme). Then because the action is free, and hence tame, Ox is locally free over OyG (see e.g. [CEPT1]). So we get a line bundle C = G[7r] on y = y x GD. Let h denote the structure map of y over GD. Then, xCT(Ox) =
det : Ko(GD) -+ Pic(GD) is the determinant homomorphism, and we identify Pic(GD) with the quotient Cl(ZG) of CT(ZG). Recall from [De3], that to any two line bundles C and M on y one can associate a line bundle < G, M > on GD. Also let w = wy/GD denote the relative dualizing sheaf for h. Then Deligne's Riemann-Roch theorem (loc. cit. (7.5.1)) gives detRh,(,C[ir])02 =< G[ir],,C[7r] ®w-1 > ®detRh,(Oy)®2
So we deduce that 2
XCT(Ox) =< L, L > < G, w-1 > .2 . xCT(Oy)
The last term in the right hand side is an induced class. Up to sign, the middle term equals the refined Euler characteristic of the structure sheaf of the inverse image by 7r of a divisor representing the relative dualizing sheaf. One can show that it is trivial by using the results of Taylor [T1] and Nakajima [Na2] mentioned in Sect. 3. So there remains
In order to prove that the right hand side is zero one writes it in terms of a G,,,-biextension of (GD, GD) and one uses the fact that there are no nontrivial biextensions on finite multiplicative group schemes over Spec Z. The latter fact can be obtained using Herbrand's Theorem (see [P2] or [Maz]). It is interesting to note that in the approach of [P2], the "2-dimensional, absolute" Galois structure problem of determining XCT(Ox) is controlled by the "1-dimensional, relative" problem of determining the structure of a torsor over GD. Actually Pappas' results are more general, here we only considered the simplest interesting case. [P2] also contains results connecting XCT(Ox) to values of the class invariant homomorphism considered in the previous section.
7
Equivariant motives
In this section I review the results on generalized omega invariants which I am aware of. These results fall under three headings: (i) Tate motives, (ii)
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equivariant motives obtained by base change by abelian extensions and (iii) motives obtained by base change by an arbitrary finite Galois extension.
Let N/K be a finite Galois extension of number fields, and let G = Gal(N/K). Chinburg's invariants St(N/K, i) for i = 1,2,3 (see [Cl], [C2]) arose from Tate's work on Stark's conjecture and on the cohomology of onedimensional class field theory. The goal of extending Galois structure theory to the context of the generalizations of Stark's conjecture proposed by Lichtenbaum, Beilinson, Bloch-Kato, and others is mentioned at the end of the introduction of [C3]. The first major step towards this goal was achieved by Burns and Flach in [BF1]. Let M be a motive defined over the number field K and let MN
denote the base change of M by the finite Galois extension N/K. Burns and Flach assumed that the extension N/K is abelian, and used work of Bloch, Kato, Fontaine and Perrin-Riou. (See [Fo] for a presentation of the non-equivariant theory.) They defined a "fundamental line" 8(MN), which is a QG-module, which over the completions of Q, conjecturally is isomorphic to some standard modules: RG over R, and the determinant of a complex RI',(Mp) computing the cohomology with supports of the p-adic realization of MN, over Qp. The definition of E(MN) uses determinants of QG-modules.
This is why one needs to assume G is abelian. The invariant cI(N/K, M) defined by Burns and Flach is the class in Pic(ZG) of an invertible module E(MN)z in the fundamental line. This module is defined as the intersection over all primes p of invertible ZpG-modules which are obtained from perfect complexes, via Lemma 1.1. For a general motive, the construction only works under some assumptions. However these are all satisfied for Tate motives and for the motive hl (X) (1) of an abelian variety X defined over K, such that the Tate-Shafarevich group of XN is finite. The problem of removing the restriction that N/K be abelian was studied by Chinburg, Kolster, Pappas and Snaith for the case of Tate motives. They developed a different approach from Burns and Flach based on the idea of using Chern character maps from K-theory to etale cohomology to construct four-term sequences of G-modules analogous to the Tate S-unit sequence.
As described in Sect. 1.c, they obtained invariants 52,,(N/K) for all finite Galois N/K and all Tate motives Z(n) for n > 1 which generalize Chinburg's 11(N/K, 3) = Sto(N/K) (multiplicative) invariant. In [CKPS1] the S2"(N/K) invariants are related to the Galois structure of K-groups under the condition that the Quillen-Lichtenbaum conjecture on the Chern character maps is true. A generalization of Stark's conjecture, namely the Lichtenbaum-Gross conjecture, implies that SZn(N/K) equals the root number class modulo the kernel group D(ZG). (This implication can in fact be proved in the function field case for n > 2). The relation between 111(N/K) and the root number class for quaternionic extensions of the rationals is considered in [CKPS3].
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In [BF2], Burns and Flach have compared their invariant for Tate motives, with the invariants of [CKPS1]: they show under very general assump-
tions that for n > 0 the image of their 11(N/K, Q(-n)) by the involution on Pic(ZG) induced by g g-1, equals Sln(N/K). Also Sl(N/K, Q(1)) becomes equal to the inverse of the image of Chinburg's Sl(N/K, 1) inside Pic(Z[1/(21G1)]G). (The invariant studied there is S21(N/K) by [CKPS4].) Burns has applied the four-term sequence approach of [CKPS1] to more general motives than Tate motives (see [B2]), and has related these generalizations to non-commutative generalizations of Iwasawa's Main Conjecture
(see Kato's papers [Kal] and [Ka3]). Burns has also applied the four-term sequence method to define analogues of Chinburg's 1(N/K, 2) invariant for equivariant motives of the form MN (see [B2]). Geometric class field theory leads naturally to complexes that are problematic from the point of view of the four-term sequence technique; this motivated the work of Chinburg, Kolster Pappas and Snaith on nearly perfect complexes which has been described in Sect. 1.d. The depth and breadth of current research on values of L-functions associated to motives seems certain to be reflected in future work on the Galois module structure theory of motives.
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der Math., 3. Folge, Bd. 1, Springer-Verlag, Heidelberg-New York-Tokyo, 1983. [F2]
Frohlich, A.: Some problems of Galois module structure for wild extensions, Proc. London Math. Soc. 37(1978), 193-212.
[F3]
Frohlich, A.: Classgroups and Hermitian Modules, Progress in Math. 48, Birkhauser, Boston-Basel-Stuttgart, 1984.
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Frohlich, A., Queyrut, J.: On the functional equation of the Artin L-function for characters of real representations, Inventiones math. 20(1973), 125-138.
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Fulton, W., Lang, S.: Riemann-Roch Algebra, Grund. math. Wiss. 277, Springer-Verlag, New York-Berlin-Heidelberg, 1985.
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Grothendieck, A., Murre, J. P.: The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme, Lect. Notes in Math. 208, Springer-Verlag, Heidelberg, 1971.
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Kahn, B.: Descente galoisienne et K2 des corps de nombres, Ktheory 7(1993), 55-100.
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Kato, K.: Iwasawa theory and p-adic Hodge theory, Kodai Math. J. 16(1993), 1-31.
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Kato, K.: Class field theory, D-modules, and ramification on higher dimensional schemes, part 1, Amer. Jour, Math. 116(1994), 757-784.
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Kato, K.: Lectures on the approach to Iwasawa theory for HasseWeil L-functions via BdR, Part I, 50-163, in "Arithmetic Algebraic Geometry", LNM 1553, Springer-Verlag, 1992.
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Lang S.: Algebraic number theory, Addison-Wesley, Reading, 1970.
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Lichtenbaum, S.: Zeta-functions of varieties over finite fields
at s = 1, in "Arithmetic and geometry-papers dedicated to I.R. Shafarevich...", Vol. I, Progress in Math. 35, Birkhauser, Boston-Basel-Stuttgart, 1983. [Li2]
Lichtenbaum, S.: Behavior of the zeta-function of open surfaces at s = 1, in "Algebraic Number Theory-in honor of K. Iwasawa", Adv. Stud. in Pure Math. 17(1989), 271-287.
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Mazur, B.: Modular curves and the Eisenstein ideal, Publ. Math. IHES 47(1977), 33-186.
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Pappas, G.: On torsion line bundles and torsion points on abelian varieties, to appear in Duke Math. J.
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Pappas, G.: Galois modules and the theorem of the cube, preprint Princeton University, 1996.
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Saito, S: Arithmetic theory of arithmetic surfaces, Ann. of Math. 129(1989), 547-589.
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Serre, J.-P. Representations lineaires des groupes finis, 3eme ed., Hermann Paris, 1978.
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Snaith, V.P.: Local fundamental classes derived from higher dimensional K-groups, to appear in Proc. London Math. Soc.
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99(1990), 165-184. [Ta]
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en s = 0, Progress in Math. 47, Birkhauser, Boston-BaselStuttgart, 1984. [T1]
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LABORATOIRE DE MATHEMATIQUES PURES, UNIVERSITE BORDEAUX 1, 351, COURS DE LA LIBERATION, F-33405 TALENCE CEDEX, FRANCE
[email protected] Mixed elliptic motives ALEXANDER GONCHAROV
Contents 1
Introduction
147
2 Two basic examples: a survey
156
3 Mixed motives and motivic Lie algebras
163
4 Conjectures on the motivic Galois group
168
5 Towards the Lie coalgebra G*1(E)
179
6
Reflections on elliptic motivic complexes
7 The complexes B(E, n) * and B*(E, n) *
185 190
8 The regulator integrals, Eisenstein-Kronecker series and a conjecture on L(Sym'E, n + 1)
197
9 The complexes B(E; n)* and motivic elliptic polylogarithms 206 10 Elliptic Chow polylogarithms and generalized EisensteinKronecker series 1
215
Introduction
Summary. Let E be an elliptic curve over an arbitrary field k and 9d the motive H'(E)(1). We define complexes B(E; n + 2)* and conjecture that they are quasiisomorphic to RHomMMk (Q, Sym'N(1)). If k is a number field this together with Beilinson's conjecture on regulators lead to a precise conjecture expressing the special values L(Sym'nE, n + 1) via the classical Reprinted from `Galois Representations in Arithmetic Algebraic Geometry', edited by A. J. Scholl & R. L. Taylor. ©Cambridge University Press 1998
148
Alexander Goncharov
Eisenstein-Kronecker series. It can be considered as an elliptic analog of Zagier's conjecture. We give a simple motivic interpretation of the elliptic polylogarithms and show how it together with the motivic formalism implies that the complexes B(E; n + 2)' should map naturally to RHomMMk (Q, Sym'nR(1)). When E degenerates to the nodal curve our complexes lead to the motivic complexes from [G1-2] reflecting the properties of the classical polylogarithms.
The complex B(E; 3)' was constructed in [GL]. The groups similar to H1B(E; n + 2)' were also discussed in [W], [W2], where it was conjectured that they inject into We formulate several conjectures about the category of mixed elliptic motives and its motivic Galois group. Perhaps the most unexpected are conjecture 1.1 about a small category of mixed elliptic motives and its generalization to all mixed motives, conjecture 4.12. The others generalize some conjectures about the mixed Tate motives ([G1-2]). In the end we define the generalized Eisenstein-Kronecker series which should be related to L(SymnE, n+m). For n = 1, m = 2 this was conjectured in [D3] and proved in [G3].
1. A triptych: special values of L-functions, motivic complexes and motivic Galois groups. Let E be an elliptic curve over a number field k and L(SymnE, s) the L-function of the n-th symmetric power of h1(E). The seminal Beilinson conjecture relates the special values of the L-function of a motive X over a number field, considered up to a Q* -factor, with the volume of the image of certain pieces of the algebraic K-theory of X under the regulator maps. However for symmetric powers of elliptic curves one should be able to say more about the special values. It is natural to adress the problem in the language of motives. Let M.Mk (resp. MMx) be the (hypothetical) abelian Q-category of all mixed motives over a field k (resp. all mixed motivic sheaves over a regular scheme X, [Be]).
Let Q(-1) := h2(P') be the Tate motive, Q(n) := Q(1)®n for any integer n and M(n) := M 0 Q(n). Let E be an elliptic curve over a field k. Then h1(E) is a pure motive of weight 1. The cup product defines an isomorphism A2h1(E) -> Q(-1). Set 7d := hl(E)(1). It is a simple object of weight -1. The elliptic motives are the direct summands of the motives 9L®n(m). They form a rigid abelian tensor category PE.
Example. If E has no complex multiplication then PE is equivalent to the category of finite dimensional rational GL2-modules over Q. The objects Sn?d(m) are simple and mutually non isomorphic. Any simple object in PE is isomorphic to one of them. The category ME of mixed elliptic motives is the smallest abelian tensor subcategory of MMk which contains the elliptic motives and is closed under extensions.
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For an integer a let L* (Sym'E, a) be the leading coefficient of the Taylor expansion of the L-function at s = a. For a given pair of integers n, m there are the following intimately related problems: Problem A. Find explicit formulas for special values of the L-functions of pure elliptic motives, i.e. L* (SymnE, n + m)
Elk,
k
is a number field
Problem B. Construct explicitly elliptic motivic complexes
RHomMMk(Q(0), S'il(m)),
k
is an arbitrary field
(1)
They may be non zero only if -n - 2m, the weight of the motive Symn9-l(m), is negative. Problem C. Find a precise description of the Galois group of the category of mixed elliptic motives.
A weaker version of the Problem A concerns the special values of Lfunctions up to a Q*-factor. We will refer to it as Problem A*. We are making sense of the motivic Ext's in a usual way. Namely, let E(n) be the kernel of the sum map
E n+1
-- E,
(x1 7 ...7
xn+1)Hxl+...+xn+1
(2)
For a group A living on En+i the notation As9n means the part which is alternating under the action of the group Sn+i. Beilinson's description of Ext'MMX (Q(0), Q(n)) implies (see lemma 3.4) that one should have ExtMMk (Q(O), S"H(m)) =
Q
(3)
(Here y is the -y-filtration on the K-groups). If k is a number field these groups are expected to be zero for i > 1. Beilinson's regulator map ExtNtMk (Q(0), SnI (m)) -> Extl-Mxs(R(0), SnH(m)) is provided by the realization functor from MMk to the category IR- MILS of mixed Hodge structures over Problem A is the deepest one. It is of arithmetic nature, while Problems B and C are geometric. We expect such a solution of Problem C that gives the desired answer for Problem B. This answer in the case of number fields should resolve Problem A*. The regulator map should be constructed first over C, brining analysis into the picture and providing a key for Problem B. In these problems one can consider the category of mixed motives generated by powers of any simple pure motive X. However the case of mixed elliptic motives seems to be especially interesting.
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To smell the flavor of the problems let us look at similar questions for the category of mixed Tate motives over a field k (i.e. the rigid tensor subcategory
of MMk generated by the Tate motive Q(1) ). Then Problem A is about special values Ck(n) of the Dedekind zeta function of a number field k. The ideal answer to Problem A* is given by Zagier's conjecture [Z2]. In Problem B we are looking for complexes RHomMMk (Q(0), Q(n)) for an arbitrary field k. In [G1],[G2] we have constructed complexes B(Q(n);
,tan(k) -* Bn-1(k) ® k* -->
... -+
tan(k) ®
An-2k*
Ank*
(4)
which reflect the properties of classical n-logarithm function Lin(z). Here ,tan(k) is the quotient of Z[k*] along a certain subgroup Rn, which in the case k = C is the subgroup of all functional equations for the classical n-logarithm function. It is placed in degree 1. These complexes ®Q were conjectured to be quasiisomorphic to RHomMMk (Q, Q(n)). A detailed exposition of this philosophy in the two simplest cases, for the motives Q(2) and W(1), is given in chapter 2. According to the Tannakian formalism the category of mixed Tate motives a field k is equivalent to the category of finite dimensional representations of its motivic Galois group. The motivic Galois group is a semidirect product
of G,,,, and a prounipotent algebraic group scheme U(k) over Q. Let L(k) be the Lie algebra of U(k). The action of G,,,, provides a grading L(k). _ ®,,>1L(k)_n (it is negatively graded thanks to a weight argument). The answer to the Problem C for mixed Tate motives which we have in
mind is this. Let I(k).:= ®n>2L(k)_n. It is an ideal, and L(k)./I(k). = (k*)v. (Here V -* Vv is a duality between the inductive and projective limits of finite dimensional Q-vector space.)
The Freeness Conjecture. ([G1],[G2]) I(k). is a free graded pro-Lie algebra generated by the groups Bn(k)v, n > 2, sitting in degree n. For a more precise version see Conjecture 1.20 in [G2], where we proved that it is equivalent to the description (4) of the motivic complexes. There are several other candidates for the motivic complexes, for example Bloch's beautiful complexes of higher Chow groups [B1] and their versions. However the complexes (4) are the smallest possible and the only ones directly related just to the classical polylogarithms. For a "cycle" construction of the motivic Lie algebra see [BK]. In this paper I will formulate an elliptic analog of the freeness conjecture, see conjecture 1.1 below.
2. A small category of mixed elliptic motives and the freeness conjecture for its motivic Galois group. The Tannakian formalism tells us that there exists a pro-Lie algebra L(E) in the tensor category PE such that the category of mixed elliptic motives is equivalent to the category of
Mixed elliptic motives
151
modules over L(E) in the category PE. There is a different tensor structure 0' on the category of pure elliptic motives. Assume that E has no complex multiplication. Then Sild(m) 0' S,'W(m') = S"+n'?-l(m + m'). Denote by PP the category of pure elliptic motives with this tensor structure. Let J be the set of k-points of the jacobian of E. Let V be a Q-vector space and M an object of a category C. Then if V is finite dimensional there is an object V®M of C such that Homc(N, V ®M) = V 0 Homc (N, M). If V is an inductive limit of finite dimensional Q-vector spaces then we get an object of the category of Ind-objects in C.
Let L, L* be objects in PE and p : L --* L* is a projection. Suppose L has a structure of a Lie algebra in PE. Say that the quotient L* is a Lie algebra with respect of the both tensor structures on the category of pure elliptic motives if there is a structure of a Lie algebra in PP on L* given by the commutator map [, ]* : A'. L -+ L* and the following natural diagram is commutative: APE L
L
4 A2p
p
A2 L
L*
Then there are canonical maps of cohomology groups H* (L*) -+ H;_ ,(L).
Conjecture 1.1 Assume that k = k. Then a) L(E) has a unique quotient L*(E) which is a pro-Lie algebra with respect to the both tensor structures on the category of pure elliptic motives and such
that one has H . (L*(E)) = HPE(L(E)) b) There is an ideal I*(E) C L*(E) with an abelian quotient
L*(E)/I*(E) = (k*)" ®9-l ®J" ®9-l The pro-Lie algebra I*(E) should be free in the tensor category PP.
The Lie algebra L*(E), considered as a Lie algebra in PE, is the central theme of our story. The tensor category PE at first seems to be only a tool which allows us to express the properties of the Lie algebra L*(E) in a neat form. See nevertheless the proof of the theorem 1.2 in s.4.4. The category .ME of finite dimensional modules over L* (E) in the category PE is a subcategory of ME which we call the small category of mixed elliptic motives. More about
the Lie algebra L*(E) in s. 4.8. The conjectures about L*(E) imply a very specific structure for the elliptic motivic complexes. So far the existence of such a Lie algebra L* (E) is a conjecture. However we proved
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152
Theorem 1.2 Let us assume for simplicity that E is not a CM curve. Then there exists a differential graded pro-Lie algebra DL*(E) in the category PE such that for (m, n) (0, 0) one has HPE (DL*(E))Snn(m)v
=
grn+mKn+2m-i(E(n))sgn
Here VM is the M-isotypical component of a simple object V in PE. The DG pro-Lie algebra DL* (E) is constructed in s.4.4. This result does not assume any conjecture on mixed motives. Assuming the following vanishing conjecture (a version of the BeilinsonSoule conjecture; the left hand side is defined via (3)) ExtMMk (Q(0), SnW (m)) = 0
for i < 0
one can show that HH(DL*(E)) = 0 for j < 0. Here a is the differential in L*(E).
Conjecture 1.3 Assume k = k. a) H3(DL*(E)) = 0 for j :0. b) H80(DL*(E)) is isomorphic to the pro-Lie algebra L*(E) anticipated in conjecture 1.1.
Remark. Let G be a reductive group. Then there is a new tensor structure the category of finite dimensional G-modules given by VA 0 V -- Va+µ, where V\ is the G-module with the highest weight A. Notice that the category of all pure motives is equivalent to the category of finite dimensional modules over a pro-reductive group. The Lie coalgebra G* (E) has the following canonical filtration. Set CC1RHomMMk (Q(0), Q(2))
inducing the isomorphism on H1 and H2. The right hand side can be understood, for example, as the complex of Bloch's higher Chow groups. The existence of such a morphism follows also from the motivic philosophy and the motivic construction of the dilogarithm in [BGSV]. According to the Beilinson-Soule vanishing conjecture one should have gr2K4+i(k) 0 Q = 0
for i > 0
One can reformulate this by saying that the canonical morphism RHomMMk (Q(0), Q(2)) - * T>1RHomMMk (Q(0), Q(2))
is a quasiisomorphism The Bloch-Suslin complex is definitely "the smallest possible" representative of the motivic complex RHomMMk (Q(0), Q(2)). I think it is also "the best possible", but this is of cource matter of taste. It gives a solution of the Problem B for the motive Q(2) which we have in mind. ii) Now let k = C. Recall that the dilogarithm is the following multivalued analytic function of on CP'\{0, 1, oo}): Lie (z)
J
z log(1
- t) dt
It has a single-valued version, the Bloch-Wigner function:
L2(z) := ImLi2(z) + arg(1 - z)
log JzJ
Let
L2 : Z[C'] -+ IR,
{z} t-) L2(z)
Then one can show that G2(Rq 2) (C)) = 0, so we get a well defined homomorphism L2 : Bq(2i (C) -f
Theorem 2.1 The restriction of the homomorphism L2 to the subspace H'B(Q(2); k)' 0 Q = Kind(C) ® Q coincides with the Borel regulator on Kind (C) 0 Q.
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158
Here "coincides" means coincides up to a known rational factor depending on a normalization of the Borel class we choose. A simple direct proof of this result see in [G1]. Finally, if k = F is a number field combining the two theorems above with the Borel theorem we get the solution of the Problem A* for (F(2):
Theorem 2.2 a) There exists elements yi, ..., yr E H'B(Q(2); F) ® Q such that q - (F(2) = 7.2(ri+r2)dF1/2det(G2(aa(yj)))
for certain q E Q*. b) For any yi, ..., yr E H1(B(Q(2); F)') ® Q one has the formula above
with gEQ. To solve the Problem A one should be able to determine effectively the constant q(yl A ... A yr). I do not know how to do this. So we see that the analysis (the dilogarithm) indeed suggested a key for the solution of the Problem B, which leads to a solution of the Problem A*. The relation with the Problem C was explained in details in [G2] and will be scketched briefly in section 4. Remark. The standard notation for the groups BQ(n) (k) is B,, (k).
2. The N(1)-story: the elliptic deformation of the Bloch-Suslin complex and L(E, 2) ([GL]) . Let us suppose that k is an algebraically closed field. Let IE be the augmentaion ideal of the group algebra Z[E(k)], and IE its fourth power. Let B* be the quotient of IE by the subgroup generated by the elements (f) * (1 - f)-, where * is the convolution in the group algebra Z[E], f E k(E)*, and g-(t) := g(-t). Then there is a map 8W(1) : B3 (E) -4 k* ® J
(9)
whose construction will be outlined below. The complex we get putting the left group in the degree 1 is called the complex B*(E, 3)'. This is our candidate for RHomMMk(Q(0), N(1)). If k = C we can "define" Sa(l) as follows: E ni{ai}
E ni9(ai) 0 ai
where
9(z) = 11(1 - qnz) := q'/12z 2 H (1 - qjz) nEZ
j>0
qjz-1) j>0
To make sense out of this formula for arbitrary field k we proceed as follows (see also s. 4.6 of [GL]).
Mixed elliptic motives
159
i. The group B2(E). Let E be an elliptic curve over an arbitrary field k. In s.2.1-2.2 of [GL] we defined a group B2(E/k) = B2(E) which fits in the
following diagram 0
t 0 -> Gm(k) _+ B2(E/k) - S2J(k)
+
0
to Z[E(k)\0]
a a. here the horisontal sequence is exact, and the such that p o 0 {a} vertical is exact at least if k = k. The map 0 is defined modulo 6-torsion. Moreover, if K is a local field there exists a canonical homomorphism
h : B2(E/K) -+ whose restriction to the subgroup K* C B2(E/K)) is given by x --> log JxJ, ( see s. 2.3). Let hK be the canonical local height. Then -hK is given by the composition
Z[E(K)\0] - B2(E/K) -11 The group B2(E) appears naturally from the theory of biextensions. Namely, let P be the rigidified Poincare line bundle over J x J: its fiber over (0, 0) is identified with k*. The restriction of P to x x J (resp. J x y) minus the zero section has canonical structure of a commutative algebraic group over k. It is the extension of J by Gm corresponding to x (resp y). This means that for every point (x, y) E J x J we have a k* torsor T(x,y) together with morphisms of k*-torsors T(x1,y) ®k* T(X2,y) -- T(x1+2.2,y)
T(X,y1) ®k* T(=,y2)
T(x,y1+y2)
providing the group structure "in horizontal and vertical directions" such that the diagram T(.1,y,) X T(-2,y,) X T(X1,y2) X T(X2,y2) -4 T(x1,y1+y2) X T(X2,y1+y2)
T(X1+x2,y1) X T(,1+22,y2)
T(X1+2i2,y1+y2)
is commutative. Then by definition the k*-torsor p-1(x y) is given by T(x,y). The horizontal sequence in the diagram above is just a different way to spell
Alexander Goncharov
160
these properties of the torsors T(,,,y). For the defenition of the map 0 see [GL]. Roughly speacking it comes from a section of the restriction of the Poincare line bundle to the diagonal deleted at zero. Set {a}2 := 0({a}) E B2(E).
ii) The group B3(E). Consider the homomorphism (J := J(k)) 6W(1) : Z[E(k)]
B2(E) ® J,
{a}
-2{a}2 ® a
Let R3 be the subgroup of I4 given by the elements (f) * (1 - f) when f E k(E)*. Let R3 C Z[E(k)] be the subgroup generated by R3 and the distribution relations
-,m - E {b}),
a E E(k), m= -1, 2
mb=a
Then 6W(1) (R3(E)) = 0 by theorem (3.3) in [GL]. Setting
B3(E) :_
R3(E)
we get a homomorphism 5-H(l) : B3(E) -* B2(E)®J. We define B3 (E) as the quotient of IE by the subgroup R3* (E). There is a map i : B3* (E) --+ B3(E)
provided by the inclusion i : IE -+ Z[E]. We define Sx(1) as a morphism making commutative the diagram B3 (E)
B3(E)
65xc1
k* ® J
B2(E) ® J
Let us consider the following complex
B(E;3)*:
B2(E)0 J-- J®A2J-*A3J
B3(E)b
(10)
Here the middle arrow is {a}20b F--3 a®anb and the last one is the canonical projection. The complex is placed in degrees [1, 4]. It is acyclic in the last two terms. It was proved in [GL] that we get the commutative diagram: 0
0
B3 (E)
k* ® J
B3 (E)
S3J
a
B2 (E) ®J -3 J ® A2J -> A3J
-> S2J®J - J®A2J - A3J 4-
0
0
Mixed elliptic motives
161
where the vertical sequences are exact, and the bottom one is the Koszul complex, and thus also exact. So B* (E; 3)' is canonically quasiisomorphic to B(E; 3)'. If k k we postulate the Galois descent:
B*(E/k; 3); := B*(E/k; 3). Gal(k/k) iii) The elliptic dilogarithm, algebraic K-theory and regulator on K2(E/C).
Let E(C) = C*/qz be the complex points of an elliptic curve E. Here q exp(21riT), Imr > 0. £2,q(Z) := E'C2(gnz),
'C2,q(Z-1) = -'C2,q(Z)
nEZ
(The series converges since G2 (z) has a singularity of type I zI log I z I near z = 0
and G2(z) = -G2(z-1)). When k = C the group R3(E) is the subgroup of functional equations for the elliptic dilogarithm. (The rigidity conjecutre tells us that R3(E) should give all of them). In particular the homomorphism ,C2,q : Z[E(C)] -4 R,
{a} H 'C2,q(a)
annihilates the subgroup R3(E/C).
Let K(E)- be the minus part of the action of the involution x -+ -x,
xEE.
Theorem 2.3 [GL] a) Let k = k. Then there is a sequence
0 -+ Tor(k*, J) -* gr2K2(E)- -+ B3(E) -+ k* ® J -+ gr2K1(E)- -3 0 It is exact modulo 2-torsion (and exact in k* 0 J). b) Now let k = C. Then the composition
9r'
K2(E)- -+ B3(E)
c41[t
coincides (up to a known multiple) with the Bloch-Beilinson regulator map.
Since we have the Galois descent on gr2Ki(E)-, we immediately get
Corollary 2.4 Let E be a curve over an arbitrary field k. Then H1(B*(E; 3)') = gr2K2(E)-,
H2(B*(E; 3)') = grzKi(E)Q
Using the complex B (E, 3) instead of B* (E, 3) the following lemma makes
the conditions on the divisor P effectively computable (see a numerical example in s.1.3 of [GL]) .
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162
Lemma 2.5 Let E be an elliptic curve over a number field F. Then an Frational divisor P = E nj (Pj) over 0 belongs to Ker(B3(E) --* B2(E) ®J) if and only if it satisfy the following conditions: a)
>2
®P, ®P,= O in S3J((Q)
(11)
b) For any valuation v of the field F(P) generated by the coordinates of the points P3
E
Pj = O
in J(O)OR
(12)
In particular for any field k the group of k-rational divisors on E(c) satisfying the conditions a)-b) above maps surjectively to gr2K2(Elk)Q, and when
k is a number field this map is compatible with the regulator map in the obvious way.
Combining lemma 2.5 with Beilinson's conjecture on regulators we come to an explicit formula for L(E, 2), which we formulate only for k = Q leaving the general case to the reader.
Conjecture 2.6 Let E be an elliptic curve over Q. Then for any Q-rational divisor P on E(Q) satisfying the conditions (11) and (12), and an integrality condition at each prime p where E has a split multiplicative reduction (see 57) one has 7rr2,q(P) = q L(E, 2) for a certain q E Q Adding to the game Beilinson's results on regulators for modular curves we proved this formula for modular elliptic curves over Q ([GL]):
Theorem 2.7 Let E be a modular elliptic curve over Q. Then there exists a Q-rational divisor P on E(Q) satisfying the conditions of conjecture above such that 7rG2,q(P) = q L(E, 2) for a certain q E So for the motive ?l(1) we see the same kind of relationship between the Problems A* and B. It remains to see the role of the motivic Galois group of the category of mixed elliptic motives, i.e. to understand Why the motivic complex RHommm,k(Q(0),?-l(1)) has the shape (9), how it reflects the structure of the motivic Galois group, and what it tells us about the motivic Galois group? In particular, how to define the group B3 (E) it terms of the motivic Lie algebra of the category ME? The answers are given in chapter 4 below.
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163
Mixed motives and motivic Lie algebras
1. Categories of mixed motives. Let MMx be the (hypothetical) abelian category of all mixed motivic sheaves over a regular scheme X. When X = Spec(k), k is an arbitrary field, we get the category MMk. We will assume that it satisfies all the expected properties conjectured by Beilinson [Be]. In particular any object of MMx has a canonical increasing weight filtration W.; morphisms are strictly compatible with W.. We will ignore the fact that existence of such an abelian category is not known yet. Let 7r : X --a Spec(k) be the structure morphism. There are the Tate
sheaves Q(n)x := x`Q(n)Spec(k) which we usually denote simply by Q(n). The basic conjecture is Beilinson's description of Ext's between them:
Conjecture 3.1 ExtMM(X) (Q(O)x, Q(n)x) = grnK2n-i(X) ® Q
Consider the category of pure motives over a field k. One can have in mind Grothendieck's category of motives with morphisms given in terms of the Chow correspondences modulo numerical equivalence. We will assume that it is a semisimple abelian category. Now let P be a rigid tensor subcategory of the category of pure motives, and Mp the tensor category of mixed motives generated by P. This means that Mp is closed under extensions, contains P as a full subcategory and the weight graded quotients of any object of Mp belong to P. Examples. 1. P is the category of pure Tate motives. Mp is the category of mixed Tate motives. 2. P is the category of pure elliptic motives. Then .Mp is the category of mixed elliptic motives. 3. P is the category of all pure motives over k, so Mp is the category of all mixed motives. There is a canonical fiber functor
T: .Mp -+ P, M
®nEzgrnM
Let us axiomatise this situation. Namely, let P be an abelian semisimple rigid tensor Q-category . We will say that Mp is a mixed category over P if it is an abelian rigid tensor Q-category containing P as a full subcategory with the following properties:
1. Any object M of Mp carries a canonical finite filtration W.M (the weight filtration). 2. Morphisms are strictly compatible with W.. 3. The graded objects grWM belong to P. 4. Homm, (M, N) are finite dimensional.
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Remark. We can assume also that Mp is an F-category where F is an arbitrary field of characteristic zero. This is essential when E is a curve with complex multiplication by OF, so ME is an F-category. However to simplify a bit notations we will assume F = Q below. Below we will assume that Mp is such a category, not necessarily of motivic origin. 2. The fundamental Lie algebra of a mixed category over P.
Set H(Mp) := End('). It is a cocommutative Hopf algebra in the tensor category P. Let L(Mp) be the Lie algebra of all derivations of the functor ':
L(Mr) = Der(v) = {F E End(T)JFx®y = Fx 0 idy + idx ® Fy} It is a Lie algebra in the tensor category ProP. Its universal enveloping algebra is isomorphic to the Hopf algebra H(.Mp). The Tannakian formalism shows that the functor T provides an equivalence between the category Mp and the category of L(Mp)-modules in the category P. A Lie coalgebra in a tensor category C is an object G together with a
cobracket 5 : G -* A'G such that the composition G -* AC 2,C
a®s
®a Ac 3,C
is zero. The standard complex of C is defined as follows: CC (G) :=,C -L AC 2,C
a®=-®b
AC 3,C -+ A4,C --f ...
Here G placed in degree 1, and the differential has degree +1. We define the cohomology groups of a Lie coalgebra C setting H' (,C) := H' (C' (G) ).
There is a duality V -3 V", (V")" = V between the inductive and projective limits of objects in the cateogory P. Set .C(Mp) := L(Mp)". It is a Lie coalgebra in the tensor category IndP. Recall that for an Q-vector space V which is an inductive limit of finite dimensional Q-vector spaces and an object X of a category C there is an object V ® X of IndC . Any pure motive W can be canonically decomposed into the isotypical components:
W = OMHomM(M,W) ® M where the sum is over the isomorphism classes of simple objects in Mp. The standard complex of C(Mp) splits into a direct sum of isotypical components
(G(Mp))M - (A2c(Mp))M
(A3G(Mp))M -
...
(13)
Notice that C(Mp)M is zero unless the weight M is > 0. Therefore each of the complexes (13) has finite length.
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3. The Galois group of a mixed category. Let
:.Mp -+ VectQ
be composition of the fiber functor T with a fiber functor cp from P to the category of finite dimensional Q-vector spaces. Then G(.Mp) := Aut®4) is a proalgebraic group scheme over Q. It is the Galois group of the category Mp. There are two canonical functors between the tensor categories .Mp and
P: the inclusion functor P y .Mp, and the functor
gr' : Mp -* P
gru' : X
®nezgrnX
Their composition is the identity functor on P. These functors obviously respect the fiber functor, and so lead to homomorphisms of groups G(Mp) ->
G(P) and G(P) -+ G(Mp). Thus the group G(Mp) is a semidirect product:
0 -* U(Mp) -* G(Mp) -f G(P) -* 0 Passing to Lie algebras we get:
0 -* LieU(Mp) -* LieG(Mp) -* LieG(P) -* 0 So LieU(Mp) is a pronilpotent Lie algebra in the category of G(P)-modules. The category of finite dimensional G(Mp)-modules is equivalent to the category of U(.Mp)-modules in the category of finite dimensional G(P)-modules. Since the group scheme U(Mp) is prounipotent, it is equivalent to the category of LieU(.Mp)-modules in the category of G(P)-modules. One can think about it as of the category G(P)-modules equipped with an action of
the Lie algebra LieU(Mp) such that the action LieU(Mp) ® V -* V is a morphism of G(P)-modules.
Lemma 3.2 Let M be a pure object. Then RHomLieu(Mp)-mod(cc(Q(0)), FP(M)) Z Mv = ((A C(M7))M, a)
4. A description of the fundamental Lie coalgebra. (Compare with [BMS] and [BGSV]). Let A, B be simple objects of the category P. Let us say that an object M of Mp is n-framed by A, B if we are given nonzero homomorphisms
vA:A-*gro M,
.fB:gr wM-*B
Consider the finest equivalence relation on the set of all objects n-framed by A, B such that (M; VA; fB) and (M'; vA; fB) are equivalent if there is a morphism M -4 M' respecting the frames. Denote by A(A, B) the set of equivalence classes of mixed motives n-framed by A, B.
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Let us define an addition law setting (M; VA) fB) + (M'; VA'-) fB) := (M ® M'; (VA, VA); fB + fa)
The neutral element is A ® B with the obviuos framing. Indeed, the equivalence between (M ® A ® B; (VA, idA); (fB + idB)) and (M; VA; fB) is provided
by the natural morphisms M - M ® A y M ® A ® B. The inversion is given by -(M; VA; fB) (M; VA; - fB), and one also has (M; VA; -fB) = (M; -VA; fB). Let us prove the first claim. The (A, B)framed object object W 0 and p < -n and groM = H(Mp) '(Q(0)). Q(0), gr nM = B. Consider the cyclic submodule M'
Then its weights are bigger then -n, since otherwise cp((M; vqo); fB)®Bv) # 0. So we get a morphism M'@ B -+ M which obviously respects the frames. On the other hand there is canonical projection M'®B -+ Q(0)®B providing
by the projection M' -* gro = Q(0). Thus (M; vQ(o); fB) is equivalent to Q(0) ® B (with the natural frame). Now let us show that cp is surjective. Let f E H(Mp)B be a subobject isomorphic to By. Denote by Kerf the corresponding subobject of H(Mp)B such that H(Mp)B/ker f = B. Consider the H(Mp)-module: Q(0) ®H(.Mp)B/Kerf whith the action of H(Mp) provided by f, the nontrivial part of the action is a morphism H(Mp)B®Q(0) -- H(Mp)B/Kerf, and the obvious framing. The map cp sends it to f . The part b) follows easily from the definitions and the part a). The theorem is proved. Remark. A coalgebra similar to H'(Mp) was considered in [BMS]. However that coalgebra is not isomorphic to ours, it is bigger then H'(Mp).
4
Conjectures on the motivic Galois group
1. A conjecture on Ext's in the category of mixed elliptic motives. Lemma 4.1 RHomMMk(Q(0), Sym'IL(m)) = RHomMME(n) (Q(0), Q(n + m))[n]sgn (15)
Proof. Let p : E(n) --f Spec(k) be the canonical projection. Then we should have the motivic Leray spectral sequence E2,a = ExtmMk (Q(0), R9p,Q(n + m))
degenerating at E2 and abbuting to ExtMMEcn> (Q(0), Q(n + m)). Since H'(E (n))egn = 0
for
i # n;
H" (E (n))sgn =
Symnh'(1' )
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we get (15). Conjecture (3.1) tells us that the cohomology of the elliptic motivic complexes are given by the formula IT HomMMk (Q(0), Sym'it (m)) ®Q = grn+mKn+2m-i(E(n))sgn ®Q (16)
Let us choose a fiber functor cp : PE -a VectQ. Set H := cp(9-1) Then V(Q(1)) corresponds to the one-dimensional representation det : GL2(H) -G,n,
and Sm9d(n) to the GL2-module SmH(n), both with the trivial
LieU(ME)-action. The following conjecture plays a crucial role. Conjecture 4.2 Let E be an elliptic curve over an algebraically closed field k The canonical morphism RHomME(Q(0), Sn9-l(m)) -p RHomMMk(Q(0), Snf(m)) is a quasiisomorphism. In particular the Ext groups in the category ME should be isomorphic to the Ext groups (between the same objects) in the category MMk.
Remark. This conjecture should not be true if k is not an algebraically closed field. A similar conjecture for the category of mixed Tate motives is expected to be valid for all fields k. Question. Consider the mixed category M{x,} generated by given simple pure motives Xi. When could we expect that the Ext's in the category Mx coinside with the Ext's (between the same objects) in the category .MMk? 2. The main hero: a Lie coalgebra L* (E). Let us assume for simplicity that E is an ellliptic curve over a field k without complex multiplication. Then any simple object of the category PE of pure elliptic motives is isomorphic to just one of the objects S' t(m). PE is a rigid tensor category. Let us define a new tensor structure 0' on this category setting Sn9-l(m) ®' Sn'fl(m) -3 Sn+n'f(m + m') where the morphism is given by the usual tensor product followed by the projection to the Sn+n''d(m+m') component. We get an abelian tensor category PP which will be called the reduced tensor category of elliptic motives.
Remark. PP is not a rigid tensor category since there is no dual object for Sn9-t (m) if n > 0.
The mixed Tate motives are contained in the category of mixed elliptic motives, and we have a canonical functor MT(k) -* ME. Let lr* : G(k) -4 G(E) be the corresponding morphism of Lie coalgebras. Let L be a Lie coalgebra in the tensor category PE. We say that L* is a Lie subcoalgebra of L with respect to the both tensor structures, PE and
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PE, if C* has a structure of a Lie coalgebra in PE given by the coproduct P : C* -r APEC* and the diagram APEC
G
ti
TA2i
C* -+ APEG* is commutative. Then we get a morphism of complexes
G - APEG
tA 2i Ti C* . + APEG*
APEC T A3i APEG*
and thus a morphism of cohomology groups. i*
: H* (C*)
HPE(C)
Conjecture 4.3 There exists a Lie subcoalgebra G*(E) C C(E) which enjoys the following properties: 1. G*(E) is also a Lie coalgebra in the reduced tensor category PP. 2. The natural morphism of cohomology groups
HPE(G*(E)) -+ Hv. (G(E)) is an isomorphism. 3. 7r* maps C(k)Q(n)v isomorphically onto G*(E)Q(n)v.
A construction of a Lie coalgebra in the tensor category PE which hypothetically satisfies all these properties is given in s. 4.5. Remarks. 1. A Lie subcoalgebra of C(E) is usually not a Lie coalgebra in the tensor category PE. 2. HP. (C*(E)) are quite different from H,E(C*(E)). The reason is clear from the following example:
(V®fc)®(V®7c)
=
(V(& V)®s2l ® (V®v)®s2l
(v ®l) ®' (v ®l) =
(V (& V) ®s2?c
Consider G*(E) as a Lie coalgebra in PE. Let ME be the category of comodules over it in the category PE. If G*(E) is a Lie subcoalgebra of C(E), then ME is a subcategory of ME. We will call it the small category of mixed elliptic motives.
Warning. The natural functor PP -3 ME is not a tensor functor.
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The weight consideration shows that the picture of nonzero isotypical components of C* (E) looks as follows:
n T 7 6 5
4 3 2 1
0
-3 -2 -1 0
1
2
3
4 -+ m
A boldface point with coordinates (m, n) corresponds to nonzero isotypical component G*(E)S. (,n)v. For example G*(E)Q(O) = 0, G*(E)S2g{(_1)v = 0. 3.
Corollaries. According to conjectures (4.3) and (4.2) one has the
following quasiisomorphisms: (A,,c*(E), 0)Sn?{(m)v 4=3 RHom,,,EQ (0), S"fl(m)) 1_2
RHommm,k (Q(0), Si-(m)) So we get the basic formula
(17)
(18)
H'(APEG*(F'))Sn-l(m)v = grn+mKn+2m-i(E''(n))sgn ®Q
(19)
Using this we immediately conclude that G*(E)Q(1)v = k* 0 Q(1)v,
.C*(E)Hv
= J ®9-lv
More generally, let JS2m+lx(_m) be the image of the subgroup of homologically trivial cycles in CHm(E(2m-1))sgn under the Abel-Jacobi map. Then 1C*(E)S2n+1W(_n)v = JS2n+lg{(_n) ® S2n+1H(-n)v,
n>0
Indeed, if H is a pure motive of weight 1 then C*(G*(E))H = G*(E)H =
H,,(E)(G*(E))H a Hl(G*(E))H
Recall that B2(k) (denoted by Bq(1)(k) in chapter 2) is the Bloch group of a field k. Let B3(k) be its analog for the classical trilogarithm introduced in [G1-2]).
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172
Lemma 4.4 (17) together with the results of [G2], [GL] implies that the lower left corner of the diagram looks as folows (all groups are tensored by Q). Js3w(-1)
J B3(E) B2(k)
k*
0
B3(k)
Proof. The structure of the bottom row follows from 3) and the main results in [G2]. The 9{(1)v-isotypical component of the standard complex of G* (E) is
G*(E)-H(1)v -+ £*(E)Q(l)v ®.C*(E)wv = k* ® J Z -H (1)v
The motivic interpretation of the group B3 *(E) as a group of 9{(1)-framed mixed elliptic motives (see theorem 9.15 below) leads to a morphism of complexes
B3 (E) ®l(1)v - k* ®J ®l(1)v -* k* ®J 0 f(1)v
G* (E)x(i)v
Thanks to the main result of [GL] and property 2) this map is a quasiisomorphism. So G*(E)-H(1)v = B3* (E) ®9-l(1)v.
The Lie coalgebra G(E) has a much more complicated structure, see s.4.8 below for the simplest example: description of C(E)Q(1)v. Zero cycles on E(n). The only way how the object S 'N could appear as a direct summand of a tensor product of n simple objects of negative weights in the small tensor category PE is this: Sn9{
=
(n times)
9-l 0' ... 0' 9{
Set
G*(E)s2xII :=
(20)
Bs29{ ® S29-C
The S29-C'- isotypic component of the cochain complex APEG*(E) looks as follows:
(,C*(E) -+ APEG*(E))S,N,
=
(Bs2,(E)
a
A2J) ® S291v
Similarly the Snfl"-isotypic component of APEG*(E) is ( Sn9{v. Therefore we get Extn(Q(0), Sn97l)
=
CHn(E(n))S9n
=
... -*
a quotient of AnJ
AnJ) 0 (21)
The last equality indeed follows from the Bloch's theorem on zero cycles on abelian varieties [B15].
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Similarly we deduce that for m > 0 one should have
ExtM (Q(0), S? ?j (m))
=
a quotient of A"J (g. Km (k)
4. The duality between DG Lie coalgebras and commutative DG algebras. Let G' be a DG Lie coalgebra in a tensor category C, which is not supposed to be a rigid tensor category. This means that we have a complex (L', a) and a cobracket
8:G'-4A'(G') which is a morphism of complexes satisfying the co Jacobi identity. Let
C(L') := L'[-1] ® SS(.C'[-i]) ® SS(L'[-1]) ®... be the free super commutative algebra (without unit) generated by L'[-1]. There are two cohomological differentials on C(L'): the first is provided by the differential in C', and the second comes via the Leibniz rule from the cobracket 8. Their sum is a differential providing a structure of a commutative
DG algebra on C(L'). For example, if L' = C is concentrated in degree 0, then SS (L[-1]) = A'C[-n] and we get the standard cochain complex of the Lie coalgebra:
C(L) := C[-1] ® A2(L)[-2] ® A3(L)[-3] ®...
Let L' be a DG Lie coalgebra and A' is a DG commutative algebra. Define MC(Homk(L6[-1], A')) as the set of all degree zero elements in w E Homk(L'[-1], A') satisfying the Maurer-Cartan equation dw + 1w A w = 0. Here w A w is the composition
L'[-1] -+ Sc(L'[-1]) -* ScA' -* A' If we think of w as of element in (L'[-1])'' 0 A', we get the usual formula for w A w. Then one has
MC(Homk(L'[-1], A'))
=
HomDGC.(C(L), A)
We get a functor
C : DGCoLie -* DGCom There is a functor in the opposite direction:
F : DGCom -+ DGCoLie satisfying
HomDG.Lie(L', F(A'))
=
MC(Homk(L', A'[l]))
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Namely, F(A') is the cofree Lie supercoalgebra (A'[l]) generated by the complex A'[1]. There is a canonical projection p :.F(A*[l]) -3 A'[1]. According to the universality property of the cofree Lie coalgebras to define the differential on .F(A*[l]) it is sufficient to define its projection .2 (A'[1]) -* A'[1]. By definition it is the sum of the differential in A'[1] and the product A2(A'[1]) = S2(A')[2] -+ A'[1]. The functors C and F are obviously adjoint. So there are canonical morphisms
A' -+ CoF(A')
G' -> FoC(G')
Theorem 4.5 These morphisms are quasiisomorphisms. This theorem was proved by Quillen [Q] when C is the category of Qvector
spaces, but it is true for an arbitrary tensor category C. The functor F has another description via the bar construction. Namely, let be the reduced bar complex of a DG commutative algebra A'. It
has a structure of a Hopf algebra. Let Bo(A') be the augmentation ideal of B(A'). Then the space BO(A')/(Bo(A') Bo(A')) of the indecomposable elements has a natural structure of a DG Lie coalgebra (a good reference for the bar construction is ch. 2 of [BK]).
5. A cycle construction of G*(E) (Compare with [1314)). Let G,, be the semidirect product of the symmetric group S,,, and (Z/2Z)'. Let sgn be the one dimensional alternating representation of Gn where a generator of each factor Z/2Z acts by -1 and the restriction to Sn is the alternating representation. The group Gn acts naturally on En. The idea of the construction. Let rx (n) be a motivic complex on a regular scheme X, i.e., it is a complex of Q-vector spaces quasiisomorphic to RHomMMX (Q(0), Q(n)). We will need also a canonical morphism of com-
plexes rx(m) x ry(n) -* rx,,Y(m + n). We will take below the complex of Bloch's Higher Chow groups on X as a concrete version of rx (n) to work with. Then .V* (E) := ®2m+n>O,n>OrEn(m + n)sgn ® SnN(m)v
has a natural structure of a commutative DG algebra (without unit) in the reduced tensor category of pure elliptic motives. Namely, the product is provided by the natural morphism of complexes rEn (m + n)sgn x rEn' (m + n )sgn
-+
rEn+n' (m + 7n + n + n')sgn
(take the external product of the complexes on En+n' and project it onto the
signum part with respect to the action of the group Gn+n'). It remains to apply the functor F!
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Now let us spell out the details for the complexes of Bloch's Higher Chow groups. Let Zn(Em, k) be the Q-vector space of codimension n cycles with Q coefficients on Em x (F1\{1})k which are skewsymmetric with respect to the action of Gk on (P1\{1})'k and meet the faces of the coordinate cube defined by setting some of the coordinates equal to 0 or oo properly. Set Zm+n(En c)sgn := (Zm+n(En, c) (9 sgn)Gn. We define a complex Zm+n(En *)sgn placing Zm+n(En c)sgn in degree
2n + m - c
The differential is the alternating sum of the face maps: Zm+n(En,
C)sgn
Zm+n(En, C - 1)sgn
A ®'-product of RHom's
l
RHomMMk (Q(0), Sni(m)) ®' RHomMMk (Q(0), Sn'f-l(m')) -+
(22)
RHomMMk (Q(0), Sn+n'Ij(m + m'))
is provided by the usual tensor product
RHom(Q, A) 0 RHom(Q, B) -> RHom(Q, A (9 B) and the canonical projection S' 1-l(m) 0 Sn',H(m') _+ Sn+n'H(m + m').
Lemma 4.6 a) One has a quasiisomorphism Zm+n(En,
*)sgn
=
RHomMMk(Q(0), Snf(m))
b) The product of cycles followed by the projection to the Gn+nl -alternating part provides a morphism of complexes
Zm+n(En *)sgn ® Zm'+n'(En' *')sgn
zm+n+m'+n'(En+n' * + *')sgn
which coincides in the derived category with the ®'-product of RHom's (22). Proof. Follows from lemma (4.1) and the results of Bloch [B12], [BK]. Set
N*(I'+) :_
®2m+n>oZm+n(En
*)sgn ® Sni(m)''
There is a commutative product on N* (E) given by (Zm+n(En, *)sgn
® S'W(m)V) ® (Zm'+n' `En,, *')sgn Z Sn'W(m )v)
(zm+n+m'+n'
(En+n' * + *')sgn ®
Sn+n'W (m
+ MI) V )
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176
Proposition 4.7 N* (E) is a commutative DG algebra (without unit) in the reduced tensor category of pure elliptic motives.
Thus setting G*(E) := FN*(E) and using theorem 4.5 we get a proof of theorem 1.2.
Conjecture 4.8 Hi(FN*(E)) = 0 if i # 0. H°(FH*(E)) is a Lie coalgebra in the tensor category PE which is our candidate for G*(E). Proposition 4.9 Assuming the conjecture (4.8) one has for (n, m) # (0, 0): (Hv.(H°(G*(E))))s^x(m)" = RHomMM,k(Q(0), Sn'(m)) 0 SnH(m)v Proof. This folows immediately from theorem 4.5 and lemma 4.6.
6. The Freeness conjecture for mixed elliptic motives. The Lie coalgebras in this paper are Ind-objects in the category of pure motives. Denote by L(E), L*(E), I*(E) the Lie algebras dual to the Lie coalgebras G(E), G* (E), .T* (E). Set I*(E) :_ ®n+m>1L*(E)Snf(m)
It is clear from the picture above that I*(E) is an ideal in L*(E) and L*(E)/I*(E) = (k*)" ® Q(1) ® Jv ®H
(23)
is an abelian Lie algebra.
Conjecture 4.10 I*(E) is a free Lie algebra in the tensor category PP. A Lie algebra L in PE is free if and only if Hy. (L) = 0 for i > 1. Remark. According to the property 3) this conjecture implies the freeness conjecture for mixed Tate motives.
7. The vector space C(E)Q(1)v. It follows from (19) that one should have G(E)g2n+17{(_n)v = JS2n+1?[(-n) = CH n+1 (E 2n+1)sgn
Let M be a simple object of PE.
According to H.Weyl's theorem
HompE (M, ®mn) is an irreducible Gm-module. Denote it by pm() (we will omit (m) sometimes). Set
-
CH 2n+2 (E 4n+2) PQ(l)v := (CH2n+2(E4n+2) l 0 Pq(1)") G4n+2 C S2(Jg2n+lf(-n))
For any integer n > 0 one can define an abelian group Bpi) together with the following exact sequence:
0 -* k* _+ Bpi) -+ CH2n+2(E4n+2)PQ(1)v __+ 0
(24)
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177
For n = 0 this is the group BQ(1) (E) we discussed in chapter 2. In general the extension 24 comes in a similar way from the biextension related to codimension n + 1 cycles in E2n+1 (see [B13]).
Let us take the sum of all these extensions:
0 -4 k* 0 Q[t] - ®n>oB
®n>oCH2n+2(E4n+2)PQ(1)v ®Q(1)v
There is a homomorphism a : k* 0 Q[t] -+ k*,
--+ 0
a 0 to H a.
Theorem 4.11 ®n>oBQ(1)
kera Using the identification N ®n>o(A2L(E)s2n+1W(_n)v)Q(1)V = ®n>oCH2n+2(E4n+2)PQ(J)v
we get a canonical homomorphism, provided by (24)
Q(1)V
l
,C(E)q1)v -> $n>o(A2G(E)s2n}1%(_n)v)Q(1)V
This homomorphism gives the cobracket S. Its kernel is isomorphic to k*. Notice how simple is G*(E)q(1)v = k& and how complicated is L(E)Q(1)v!
8. The small motivic Galois group ? Let Mk be the abelian category of all pure motives over a field k. It should have a natural small tensor structure. Namely, if chark = 0 then Mk should be a neutral tannakian tensor category, so one should have a canonical equivalence
0: Mk -* G(Mk) - mod between the category of all pure motives and the category of finite dimensional
Q-rational representations of a pro-reductive group scheme G(Mk) over Q. The 1-adic cohomology fiber functor on Mk is given by b 0 Q. (The case chark # 0 we left to the reader). Let G(Mk)s be the maximal split over Q quotient of the proreductive
group G(Mk). According to E. Cartan's theory the irreducible G(Mk)smodules are the highest weight modules VA, where )( is the highest weight.
We define the new tensor structure 0' on the category of G(Mk)s - mod by VA ®' Vp := VA+p. The category G(Mk)s - mod is a subcategory of G(Mk) - mod. We conjecture that there exists a small tensor structure on the category G(Mk) - mod which coincides with the small tensor structure 0' defined above on the subcategory G(Mk)s - mod. Combining it with the equivalence
0 we would get a small tensor structure on the category of pure motives. Denote this new tensor category by M.
Alexander Goncharov
178
Conjecture 4.12 Assume k = k.
Then there exists a Lie subcoalgebra G*(Mk) C L(Mk) which is a Lie coalgebra for both tensor structures on the category of pure motives over k such that the canonical morphism
H J;'C*(Mk) -* H,,,tk (L(.Mk) is an isomorphism.
9. Some evidence for conjecture 4.12. Let X be a regular projective curve over an algebraically closed field k. Set hl (X) := hl (X) (1) and F := k(X). The Gersten complex K2(F) -+ k* 0 Z[X(k)] provides an sequence
K2(k) " K2(F) --+ k* 0 Z[X(k)]
Z
k* -+ 0
Here s : Z[X(k)] -* Z is the augmentation map, the left arrow can be proved to be an inclusion and the right one is surjective by the Weil reciprocity law. Let Z[X (k)]o be the subgroup of the degree zero divisors.
Lemma 4.13 Let us assume the rigidity conjecture for K3"d(F) ® Q. Then the complex
K2(k) -+ k* ® Z[X(k)]o placed in degrees [1, 2] and tensored RHomMMk (Q(0), hi (X)).
by Q is
(25)
quasiisomorphic to
Proof. The standard motivic Leray spectral sequence argument shows that one expects to have an exact sequence RHomMMk (Q(0), Q(2)) 0 there exists an abelian group Bsnfl(m) together with homomorphisms BSn3l(m) ---+ BSn-l3((m) ®J,
111
n + 2m > 0
82 : Bsnx(m) -1 Bsnfl(m-1) 0 k* m > 1 ::
For m > 1 one should have BS 2m}17{(-m) = JS2,n+lx(-m)
and
Bu(m) = Bm(k)
where Bm(k) are the groups introduced in [G1-2]. For instance B = J and 13 (1 = . b) One should get a bicomplex l sn'H(m) of the following shape.
Form>0: BSn (m) BS*.n_l J{/m) A
-?
Bsnfl(m-1) A k*
BsnW(1) /\ Am_lk*
J -* BS*.n-lj(m_l) A J A k* - ...
Bsn-l,H(l) A J A Am-lk*
A AnJ -*
A AnJ A k* -+ ...
->
AnJ A Amk*
Alexander Goncharov
186
Here the horizontal differentials are b A x are b A x i- 62(b) A x. For m = 0:
61(b) A x and the vertical ones
Bs2x A An-2J -* A" J
Bsnx -> Bs*.-1.n A J
For m < 0, set m' := -m. Then BSn7{(-mI) --+ BS'n-1x(-mW) 0 J - ... --- B;2m'+13{(-m,) ®
An-2"`'-1J
c) The total complex associated with the bicomplex rsnx(m) (abusing notations I will denote it also by 1 S*' (m)) after tensoring with Q should be quasiisomorphic to RHomMMk (Q(0), Snf(m)). Example. rfl(m) is the total complex associated with the bicomplex
Bu(m)
1
-+
B
[(m-,) Ak*
-> ...
1
BW(2) A
...
Am-Zk*
1
nJnAm-2k*
Am-ik*
B3 (E)
1
JAAmk*
d) There should be a variation Pn,m of mixed elliptic motives framed by Q(0) and Snjt(m) over a certain finite dimensional variety X (n, m) over k. The groups BsnW(m) should come from it as follows. The variation Pm,n provides a homomorphism ln,m : Q[X (n, m)(k)] -+ A(Q(0), Stf(m)) where ln,m({x}) for x E Xn,m, is the class of the framed mixed motive Pn,m(x). (Pn,m(x) is the fiber at x of the variation Pn,m). Passing to the coalgebra Lie we get a homomorphism ln,m : Q[X (n, m) (k)I --* G(E)Sn-R(m)v
Set BSn-H(m) := Im(ln m). The group Bsnjj(m) is the largest Q-subspace of BSnx(m) such that the restriction of the coproduct b to the group BsnW(m) has non zero components only in GSn-1W(m)v A J ®llv
®
GSnx(m-1)v A k* ® Q(1)v,
if m > 1
and in GSn-1f(m)v A J ® IRV,
if m < 1
By definition Sl and 62 are the components of the restriction of 6 to Bsnfl(m) SnH(m)v. The restriction of 6 to Bs-w(m) ® Sn9i(m)v may be more complicated.
Mixed elliptic motives
187
The periods of the R-Hodge realization of the variation Pn,,n are the new transcendental functions denoted Pn,m which is needed to get the special values L(S'i-t, m + n). Example. Pn,l is the (motivic) elliptic polylogarithm sheaf on E\0, and Pn,1 are the Eisenstein-Kronecker series from 1.5. The group Bsn(1) (resp. Bsnln(l)) should coincide with the group //
I2n+2
/
Bn+2(E) _ SE(E) ,
Bn+2(E) _
7n+2(E)
which will be defined later on. Here IE is the augmentation ideal of the group algebra Q[E(k)].
2. The structure of L* (E) and elliptic motivic complexes. Let us define the Q-vector spaces Csnj(m) by setting H1(I*(E)) = ®Csnfl(m) ® Sn9(m)v
Let b* be the the cobracket in the Lie coalgebra L*(E). Conjecture 4.10 means that 6* : Csn7i(m) -+Csnx(,n_1) A k*
®
Csn-1n(+n) A J
(35)
Lemma 6.1 The Sn9-t(m)v-isotypical component of the Serre-Hochshild spectral sequence for the ideal I*(E) C L*(E) computing cohomology of L*(E) collapses to the total complex associated with a bicomplex (which should be
tensored ®Snf(m)v): Csnn(m)
Csn-1.(m) ® J
-*
Csnx(m-1) ® k* CSn-iW(m) ® J ® k*
...
®S"`+[
Csnn(_[nz'])
1]k*
-a ... -> Csn-1v(-[n22]) ®J ®Sm+[°a'1 k* 1
1 1 Ca(m) 0 An-1J -> C4l(m_1) (9 An-1j® k* 1
1
... -4
1
Ca(m) ® AnJ _+ %m-1) ® AnJ ® k* -3 ... -*
Cx(1) ® An-1J ® Am-lk* 1
AnJ ® Amk*
The length of the rows decreases when we are going down. Here is how it
Alexander Goncharov
188
looks for n = 4, m = 1:
CSs3{(1) 0 J
CS4,J(_1) ® A2k*
CS4,h ® k*
CS4W(1)
-+
-> CC3W(_1) ® J ® A2k*
CS37i ® J ® k*
1-
CS2,J(1) ® A2J -4 CS21H (9 A2J 0 k*
%1) ® A3J k* ®A2J
Proof. The E1 term of this spectral sequence is Ep,q
=
CP
(k* ®J Hh (r
(Ei ))) sn,(.),,
Since H9,, (Z* (E)) is zero for q > 1 the only non zero rows are E1'0 and E1'1 Moreover E11'0 is zero unless p = n + m, and
El+n,° = AnJ ® Amk* Further, El11 is non zero only if 0 < p < m + n. If so, then EP,1
=
®a+b=p
CS -3{(m-b) 0 AaJ 0 Abk*
So the E1 term of the spectral sequence gives us precisely the groups in the bicomplex. The differential d1 : El -* El provides all the differentials exept the last two in the right bottom corner targeting to AnJ ® A91k*. These two we get from the differential d2. The lemma is proved. The structure of the quotient L*(E)/[I*(E),I*(E)]. There is an exact sequence of Lie algebras
- [I*(E),I*(E)]
0-+
1*
'
L*(E) [I*(E,),I*(E,)]
L*(E)
I*(E)
(
36
)
The action of L*(E) on I * (E) leads to the action of L* (E) /I * (E) on I*(E)/[I*(E),I*(E)]. The Lie algebra structure on L*(E)/[I*(E),I*(E)] is determined by this action. The inclusion L*(E)Q(1)v ®L*(E),h(1)v
* L*(E) provides a canonical split-
ting s : L*(E)/I*(E) L*(E)/[I*(E),I*(E)] as an extension of Q-vector spaces. Recall that L*(E)/I*(E) = (k*)' ® Q(1) ® Jv ® 9-l is an abelian Lie algebra. So to define the Lie algebra structure on L*(E)/[I*(E),I*(E)] we need to know the Q-vector spaces Csnh(..) and the homomorphisms (35). That is precisely what the following two conjectures are doing.
Mixed elliptic motives
189
Conjecture 6.2 i) There is a canonical map Bsn (m) " CS'nli(m) such that i(m) = 61 + 82. ii) BS*n (m) = C,*gn (m) if n = 0, 1. Otherwise one has an exact sequence
6* I BS'
0 -+ B*nW(m) -+ CS*nx(m) -+ BsnW(-[-]) ®Sm+[n2']k* --+ 0
(37)
Set Csn.(m) := CS*ny{(m)/B*n (m). Then 8* induces a map Csnji(m) --+ CS*nx(m_l) ® k*
®
Csn-i3{(m) ® J
(38)
Now we can formulate the second part of conjecture (4.10).
Conjecture 6.3 The second component of the map (38) is zero, and the first component coinsides with the homomorphism BSnW(-[z']) ®Sm+[nz']k* --+ BSnn(-[n2i]) ®
Sm+[n2']_1k ®k
(39)
given by the identity x the Koszul differential.
Theorem 6.4 Assume all the conjectures of this section. Then the complex FSnW(m) is quasiisomorphic to RHomME(Q(0), S'
(m))
Proof. The complex RHomME(Q(0), S"3l(m)) is quasiisomorphic to the
standard complex C'(G(E)) of the Lie coalgebra G(E). According to the property 1) there is a morphism of complexes
CPE(G*(E)) -+ C. (G(E)) which is a quasiisomorphism thanks to the property 2). The part ii) of conjecture (6.2) just means that there is a canonical embedding of complexes
: r*. x(m) `' CP. (,C*(E)) We are going to show that it is a quasiisomorphism. Using the part i) of conjecture (6.2) we see that the quotient of the bicomplex we got from the spectral sequence along the bicomplex r (m) looks as follows. The two bottom rows become zero, and each of the remaining rows
is B* times a Koszul complex. For instance in the case n = 4, m = 1 the quotient is B547i(_1) ® S2k*
01 Bs3 (_1) ®J ® S2k* o4.
Bs*, ®n 2J ®k*
Bs4 (-1) ® k* ® k* BS3'H(-1) ® J0
-+
BS 3w(_1) ® n2k*
oJ.
0
k* ® k*
o
Bs2w ® n2J ®k*
BS3x(_1) ®J
®n2k*
Alexander Goncharov
190
7
The complexes B(E, n)* and B*(E, n)'
1. An auxiliary following complex Q[A]
complex. Let A be an abelian group. Consider the
Q[A] ® AQ - Q[A] ® A2AQ -L Q[A] ® A3AQ
(40)
The differential is defined by the formula
{a}® aAbiAb2A...Ab(41) It is infinite if AQ := A ® Q is infinite dimensional. Let IA be the k-th degree of the augmentation ideal IA C Z[A].
Lemma 7.1 b(IA) c
IAk
1) ®A.
Proof. IA is generated by the elements
({al} - {0}) * ({al} - {0}) * ... * ({ak} - {0}) Clearly
6({al} - {0}) * ... * ({ak} - {0}) = > rj({aj} - {0}) * fail (9 ai j#i So the complex (40) has the "diagonal" filtration by subcomplexes i
IA
IA 3®A3AQ--*...
IA--l
Each graded quotient is isomorphic to the Koszul complex
S"AQ -f Sn-'AQ ® AQ -f ... -* AQ ® An-1AQ -* AmAQ 2. The groups B,,,(E). Recall that Bo(E) := 7G and B1(E) = J(k). We will usually write J for J(k). The group B2(E) is the group discussed in s. 2.1. In particular if k = k it is a quotient of Z[E(k)]. Recall that E(n-1) = {(xi, ..., xn) C En! E xi = 0} The group Sn acts naturally on E(n-1). Let pi : E(n-1) -4 E be the projection to the i-th factor. We will use the "coordinate notations" denoting p* f by f (xi) etc. Let us define the following diagram: KM (k(E(n-1))) An
/
Sn (k `E)*) On
I2n
E
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191
by setting µn : fl o ... o fn H E (-1)l al {p*a(1)f1, ..., p*a(n)fn} = E {p1fa(1), ..., pnfa(n)} aES" aES
For n> 2 set
pn : Snk(E)* -f I(/)
fl O ... o fn ~+ (fl) * (f2) * ... * (fn)
Definition 7.2 Let k = k and n > 3. Then Rn(E/k) is the subgroup of Z[E] generated by,Qn-l(Kerµn_1) and "distribution relations" :
{a}n - mn-2 E {b}n,
a E E(k),
m = -1,2
(42)
mb=a
Example R3(E) is generated by the elements (1- f)*(f)-, where {x}{-x}, and (42). Remark. It would be more natural to add to the subgroup Rn(E/k) the distribution relations for all m E Z, m 54 0. However we will get the same group, and for our purposes the definition above is technically more convenient.
Definition 7.3 Let k = k. Bn(E)
Z[E(k)] Rn.(E/k)
Theorem 7.4 8(Rn(E)) C Rn_1(E) ® J. The proof consists of two independent parts. For a more dificult one see proof of the theorem (7.9) below. The easy one follows from
Lemma 7.5 For any m E Z, m
0, one has
(Sn ({a}n - mn-2 . > {b}n) = 0 in the group Bn_1(E) ® J
(43)
mb=a
Proof. For n = 2 this is done in [GL]. The general case follows by induction: 8({a}n - mn-2 .
{b},,)
= {a}n-1 ® a - mn-3 E {b}n_3 ® mb = mb=a
mb=a
({a}n-1
_ mn-3 E
{b}n_1) ® a = 0
mb=a
So we get a homomorphism b : Bn(E) -* Bn_1(E) ® J and thus the following complex
Bn(E) -L Bn-1(E) ®J a * ... a- B2(E)
®An-2J -- J ®An-lJ a4 AnJ (44)
Alexander Goncharov
192
Here the very left group sits in degree one. The differential is defined by the formula (41) and has degree +1. If k is not an algebraically closed we postulate the Galois descent:
B(E/k, n + 2); := (B(E/k, n + 2).)Gal(k/k) Let us also define the groups Bn* (E) for k = k:
B* (E) := Im(I2n-2 -* Bn(E)) Here the map is induced by the natural inclusion I2n-2 " Z[E]. It follows from the lemma (7.1) that 6n(Bn*(E)) C Bn_1(E) ® J. So we can consider the following subcomplex B* (E, n)' of the complex (44).
Bn* (E) b>Bn_1(E)®J-
k*®An-2J
...
Proposition 7.6 The canonical morphism of complexes
B*(E, n + 2); - B(E, n + 2); is a quasiisomorphism. Proof. This morphism is injective by the definition of the groups Bn* (E). It follows immediately from the lemma below that the quotient is isomorphic to the Koszul complex
SnJQ -* Sn-1JQ®JQ -* ... -* S2JQ®An-2JQ -p JQ®An-1JQ -* A"JQ
Lemma 7.7 Bn(E)/Bn*(E) ® Q = SnJQ
Proof. We may assume that k is algebraically closed. We need to study the quotient of the group Q[E(k)]/I2n-2 by the subgroup generated by the distribution relations. Notice that Q[E(k)]/I2n-2 =Q$JQ®S2JQ®...®S2n-3JQ
where the isomorphism is given by i = (io, ..., i2n_3) where i,n
: Q[E(k)] -* SmJQ;
{a} H a'n
Let us denote by DRn the subgroup generated by the distribution relations (42). Then the homomorphism in = (io, ..., 2n, ..., 22n-3) : DRn
®JQ ®...
DS"J ®...
®S2n_3 J
Mixed elliptic motives
193
is surjective. Indeed, ik({a}n - mn-2
{b}n)
= (1 -
mn-k) . ak
mb=a
In particular in(DRn) = 0. The lemma and hence the proposition are proved.
3. The complex B* (E/k, n + 1); is an elliptic deformation of the complex B(Spec(k), n);
.
Let me recall that the complex B(Spec(k), n)'
looks as follows:
Bn(k) -a 13n-1(k) ®k* -> ... -_* B2 (k) 0
An-2k*
Ank*
where 13n(k) is the quotient of Z[k*] along a certain subgroup Rn. There is an important difference between these two complexes. The complex B(Spec(k), n) is defined directly in terms of a field k, while to define
the complex B(E/k, n + 1) we have to go to the algebraic closure of k and then postulate the Galois descent property. So in general they can only be quasiisomorphic.
Suppose k = k. When E degenerates to (Pl, {0} U {oo}) the complex B(E/k, n + 1% degenerates to a complex quasiisomorphic to B(Spec(k), n% (for n = 2 see s.3.4 in [GL]). In a sense the elliptic situation is simpler. For example our definition of the group Rn(E) is not inductive and more explicit then the definition of the
group Rn(k) in [G2]. In fact when E degenerates to (Pl, {0} U tool) our definition suggests a new way of defining of the groups 3n(k) from [G1]-[G2].
4. Proof of the theorem (7.4). Let a E Sn be a permutation and ai E E. Consider the following codimension p cycle in E(n-1): X (a; a1i ..., ap) := {(x1, ..., xn) C E(n-1) I xo(1) = a1, ..., xa(p) = ap}
It is a product of elliptic curves E. Let us define a homomorphism /,In;p
: APZ[E] ® Sn-Pk(E)* -i
Ij XE(E(n-1))p
An-p(k(X))*
by setting
fall A...Alap} ®fp+1o...o fnH E (-1))a1fp+l(xo(p+l)) A ... A fn(xa(n))IX(a;a1,...,ap) aESn
Denote by Din) the image of the homomorphism µn;p-1
The elements of type fl(xl) A ... A f n(x,n) in Amk(E(m-1))* and their linear combinations may be called the decomposable elements; this suggests the notation.
Alexander Goncharov
194
Let F be an arbitrary field with a discrete valuation v and the residue class
u H u.
F. The group of units U has a natural homomorphism U ---3
1. There is canonical homomorphism
Choose a uniformizer 7r E F*,
An(F*) - An-1(F'") uniquely defined by the properties av (u1 A ... A un) = 0;
a,,
(ir A U2 A ...nun) = (u2, ..., iin)
Consider the following complex on
An(k(E(n-1))) - II
E(n-1):
An-1(k(X)) -
...
XE(E(n-1))1
-
JJ
k(X)*
XE(E(n-1))n-1
(45)
Here a := Ex av(X) where v(X) is the valuation corresponding to the irreducible divisor X. Lemma 7.8 The groups Din) form a subcomplex in the complex (45)
Proof. Clear from the definitions. Let us define for p < n a homomorphism RAP+1) D()1
-+
2(n-P+1)
®APJ
We define )t+1) first on space of decomposable elements on the subvariety xl = a1, ..., x, = aP by the formula
n
(P+1)
fP+1(xP+1) A ... A fn(xn)Ix1=41>...,2p=ap
(fP+1)*...*(fn)*(a,+...+aa)0a,A...Aaa It extends uniquely to the space of all decomposable elements assuming the skewsymmetry with respect to the action Sn. In particulary it is defined on ER,n)
If p = n - 1 then D(n) = k* ® %3(n) n
:
k*®Z[E(n-1)]sgn
- k*®An-1J'
Z[E(n-1)]sgn and
we have a homomorphism
x®
an) -3 x®a1 A...nan-1
Finally, one can define a homomorphism
IE ®An-2J * k* ®An-1J Namely, there is a homomorphism
B3(E) ®An-2J
B3(E) ® Ai-1J {a}3 ® b1 A ... '-- {a}2 0 a n b1 A ...
Mixed elliptic motives
195
Let Bnk)(E) be the subgroup of Bn(E) generated by k-th degree of the aug-
mentation ideal I. The restriction of this homomorphism to B34) (E) An-2J C B3(E) 0 An-2J lands in k® ® An-1J. Indeed, the composition
Bq1) (E) ®An-1J -> S2J ®An-1J
IE ®An-2J is equal to zero.
Theorem 7.9 The maps Nn) provide a homomorphism of complexes D1(n)
-24
ant)
IE
-4
,b2
...
e4
D (n)
D n_1) l (nn-1)
$(2)
-* IE-2®J
...
IE
(9
An-2j
$(n) k*
®An-1 J
Proof. We will do in details the commutativity of the left square, at the same time proving the theorem (7.4). The commutativity of the other squares except the very right one is completely similar.
The commutativity of the right square is a more subtle statement. For n = 2 it was already proved in [GL], s.3. The general case n > 2 is similar. Consider an element f1(x1) A ... A fn(xn) E Ank(E(n-1))
(46)
Let va(f) be the order of zero of the function f (t) at t = a. The part of the coboundary (in the complex D(*n)) of the element (46) sitting on the divisor x1 = a1 is equal to E vat (f1) ' f2(x2) A ... A fn(xn),
x2 + ... + xn = -a1
(47)
at EE(k)
Let ta be the shift by a on the group E(k), so to{b} = {a + b} and to f (x) _ f (x - a) (sic). Then (ta f) = (f) * (a). Then it can be written as
J
Val (fl) ' f2(y2) A ... A tat fn(yn),
Y2 + ... + yn = 0
al EE(k)
Applying the homomorphism a(,2) to the element (47) we get >
vat (f1) ' (f2) * ... * (fn) * (al) ®a,
at EE(k)
On the other hand writing
(fl) * ... * (fn) = E vat (fl) ' ... ' van (fn) ' {a1 + ... + an} a;EE(k)
(48)
Alexander Goncharov
196
we get ((fl)*...*(fn))
_ E val(fl) ... vnn(fn)-jai +...+a,n}®(al+...+an) (49) aiEE(k)
Collecting the terms with a1 in the right factor we get just what needed: the formula (48). Taking into consideration the coboundaries of the element (46) sitting on the divisors xp = ap we will get the other terms (with ap) in (49). To prove the commutativity of the right square we replace it by a bigger diagram D( n-1)
nn-1)
nn)
B3(E) 0 An-2J a BQ(1) (E) ®An-1J and then prove its commutativity in a way similar to the proof of theorem 4.5 in [GL]. The theorem is proved. Consider the Gersten complex for the Milnor K-theory on
Kn (k(E(n-1))) - II
E(n-1):
JI
Kn 1(k(X)) -!+... -4
k(X)*
XE(E(n-1))n-1
XE(E(n-1))1
The Gersten complex is obtained from (45) by factorisation along the subgroup generated by the Steinberg relations. Let D(n) :
D(n) ---+ D(n)
... -> D(n)
be the image of the complex D(n) in the Gersten complex. In other words ,DP
(n) = D(n)/St(n)
where St(n) is the intersection of f)(-'n) with the subgroup generated by the Steinberg relations in IIXE(E(n-1))y_1 An-p+lk(X)*. Lemma 7.10 Qnp)(St(n)) = 0 in Bn*+2_p(E) ® AP-1J.
Proof. Consider the subvariety X (id; a1, ..., ap_1) in E(n-1) and its projection p : X (id; a1, ..., ap_1) -+
En-p
p : (x1i ..., Xn) --+ (xp, ..., xn-1)
The subgroup of Steinberg relations is generated by { f (xp, ..., xn-1), (1
- f) (xp, ..., xn-1), 91(xp, ..., xn_1), ...}
Mixed elliptic motives
197
Notice that xn = -al - ... - aa_1 - xp - ... - xn_1. This means that an element Ej { f U) (xp), ..., f,) (x,,) } of the subgroup of Steinberg relations can
be written as p*
{ fP7)(yp) ...
tal-...-apfn( )(yn)}
J
on
E(n-p) = { (yp, ..., yn) l yp + ... + yn = 0}
Now the lemma follows immediately.
Theorem 7.11 There is a canonical homomorphism of complexes D('n)
Al)
-*
V(n) &(2) +'
B,,+1(E) - Ba(E) ®J -
...
k* ®An-lJ
Proof. Follows immediately from lemma (7.10) and theorem (7.9).
8
The regulator integrals, Eisenstein-Kronecker series and a conjecture on L(SymnE, n + 1)
1. Beilinson's conjecture in the case of L(Symnhl(E),n+1). Let E be an elliptic curve over a number field k. According to the general Beilinson conjecture on regulators the special value L(Symnhl(E), n + 1) should be equal, up to standard factors, to the (co)volume of a certain lattice obtained as follows.
The n-th Deligne complex R(n)D on a regular variety X can be defined as the total complex associated with the following bicomplex: A (n
- 1)
1 .4 (n-1)
AX 1(n-1) t7rn
T 7rn
Qx
d
-
cmn+1
x
(50)
Here (AX, d) is the C°°-De Rham complex, irn : AX ® C -* AX (n - 1) is the projection induced by irn : C -* R(n - 1), z z + (-1)(n-1)2, the group A°x(n - 1) is placed in degree 1 and (SZX, 9) is the De Rham complex of holomorphic forms with logarithmic singularities at infinity.
Alexander Goncharov
198
One has the regulator map rBe :
(E('- 1), Q(n)) sgn -* HD
(E(n-1) ®Q R, R(n)) sgn
(51)
The right hand side is a group of purely topological origin: H. (E(n-1) ®Q R, R(n)) sgn =
HB 1(Ein-1> ®Q C, IR(n - 1)) sgn =
(®o:kticHsn-1(Eo(n-1)
(®o:k4C
(C), l[8(n - 1))sgn)
Symn-1Ha(Eo(C),l1 (1)))
Here + means invariants of the complex conjugation acting on a's and on the coefficients IR(n - 1).
The image of the regulator map (51) is conjectured to be a lattice. gives another lattice in HD ®. R, IR(n)) sgn The covolume of the lattice ImreBe measured with respect to the second latHD
(E(n-1)
(9Q IR, Q(n))
(E(n-1)
sgn
tice should coincide (up to standard factors) with the special value of our L-function at s = n + 1. 2. The Eisenstein-Kronecker series. Let me recall their definition:
For the relation with the elliptic polylogarithms see [BL], [Z].
Lemma 8.1 a) For any lattice r one has Ki,j(z; T) _ (-1)i+jKj,i(z; T) b) Suppose that the lattice r and a divisor P on C/I' are invariant under the complex conjugation. Then Ki,3 (P; T) E R.
Proof. Clear. Consider for each n > 2 a homomorphism
Kn : Z[E(C)] -+ Symn-2H1(E(C), C)
{z} H
Ka,b(z;T)(dz)a-1(dz)b-1
a+b=n
Theorem 8.2 Kn(R,,(E(C))) = 0. We will prove it in s 6.3 below. So we get a homomorphism
Kn : B,,[E(C)] -; Symn-2H1(L' (C), C)
This means that Rn(E(C)) is a subgroup of functional equations for the Eisenstein-Kronecker series.
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199
Lemma 8.3 Suppose that E is defined over R and the lattice r was defined using a real differential w. Then Symn-2H1(E(C),R(1))+
Kn : Bn[E(C)]+ ___+
Here + means invariants of the complex involution acting on both R(1) and E(C). Proof. Follows from lemma (8.1).
3. Computation of the regulator integral. The main result of this section is due to Deninger (see [D1], s.6). Our presentation is technically simpler since working with distributions we avoid the convergence problems. For any functions fl,..., fn on a manifold X consider the following (n - 1)form with values in R(n - 1) :_ (27ri)n-1R
rn(fi,..., fn) :_
(52)
Altn E Cj log I fl I d log I f2I A ... A d log I f2j+1 I A di arg f 2j+2 A ... A di arg fn j>0
Here Cj :=
(2j+1)!(n-2j-1)!
and Altn is the operation of alternation of fi's.
Then
drn(fi, ..., fn) = 7rn(dlog fl A ... Adlogfn) This just means that the pair (rn(f1,..., fn),
dlog f1 A ... A dlogfn)
is an n-cycle in the Deligne complex (50). It is the product in the real Deligne cohomology of the 1-cocycles (log I fi 1, d log fi). Set (-)
(-)
dz1 A...A dzp+q)
wp,q
(p,q)
(53)
(-) where dzi means either dzi or dzi and the sum is taken over all possible terms. For example w2,o = dz1 A dz2,
w1,1 = dzl A d22 + dz1 A dz2,
WO,2 = dz1 A dz2
The forms 7rnwp,q for -p > q, p + q = n - 1, form a basis over R in (E(n-1) , R(n - 1))sgn. We can represent elements in HB
HBn-1
1An-1))syn by their cup prod-
uct with forms 7rnwp,q: 1 (27ri)(n-1)
fE(r
1)(Q)
of
(- 1)"'rn(pa(1)fl, ..., po(n) fn) A7rnwp,q
Alexander Goncharov
200
Theorem 8.4 (-1)lalrn(pa(1)fl,.,PQ(n)fn)AWP,9
E(n-1)( oESn
I
= Cn7r .KP+1,9+1 (fl*...* fn)
where enEQ*
The constant Cn can be obtained from the proof below. Proof. It consists of several reductions of the integral. Step 1. The form Wp,q is skew invarant under the action of the group Sn. So the integral is equal to n.(
.
E(n-1)(C)
rn(pifi, ...,pnfn) ^ WP,9
Step 2. Let n
an(fl, ..., fn)
E(-1)'1oglfildloglfl I A ...dlogjfil... A dloglfnl
(54)
i=1
Lemma 8.5 For any functions fl,..., fn r E(n-1)(C)
rn(fl, ..., fn) AWP,9 = bn JE(n-1)(C) an(fl, ..., fn) AWP,9
where bn E Q" is a (computable) constant. Proof. One always have either d log fi A Wp,q = 0 or d log fi A wp,q = 0. So we can replace everywhere di arg f by ±d log If 1. Step 3.
fEfr1)(C)&', ...,fn)AWP,9 = n JE(n
1)
log IfnIdlog
If1IA...AdlOg I fn-1I AWP,9
(55) Indeed, log I fj l d log I f2l + log I f2I d log Ifl I = d(log Ifl l . log I f2 1) and so by the
Stokes theorem
Ln-i)(C)
d(log Ifi l log If2I)d log l f3l A ... A d log Ifn l AWp,q = 0
Step 4. Let us compute the right integral in (55).
Lemma 8.6 log If (z)(_
-ImT
where Cf is a certain constant.
27r
E va(f) (z
7Er\o
Iryl2,'Y)
+ Cf
(56)
Mixed elliptic motives
201
Proof. One can get a proof applying as to the both parts of (55). The constant C f can be computed from the decomposition of f on the product of theta functions using the formula in s. 18 ch. VIII of [We]. According to step 5 it does not play any role in our considerations. Step 5. By the Stokes formula JE(n-1)(q
we see that one can neglect the constants Cf. Step 6. We may suppose that in (55) the form Wp,q is written in variables z1, ..., z,,- 1. Then for each summand in Wp,q one can replace d log IfiI by 1/2
L9 log fi or 1/2 a log fi depending whether in this summand appeared dzi or dzi.
For example for the summand dz1 A ... A dzp A dzp+1 A ... A d2 _1 we will
have the integral
n 2n_1
E(n_1)(C)
log Ifn(zn)Ialog f1(zl) A ... A clog fp(zp)A
a log fp+l (zp+l) A ... A a log fn-1(zn-1) A dz1 A ... A dzp A dzp+l A ... A dzn_1
Differentiating the distributions we get
alogf(z)_ E Va(f) (z-a,y). tiEr\o
a log f (z) = -
I
Va (f )
y I2
(z-a,y)'y
7Er\o
The condition z1 + ... + zn = 0 just mean that we compute the value of the convolution of one variable distributions: log I f(z)I *
a 1ogf2(2) 192
logfp(z) a2
*
clogfp(z) az
alogf_1(z) *
*
az
at 0. Using the fact that the Fourier transform sends the convolution to the product and the formulas above we get the theorem. Recall we have a homomorphism Kn+1 : Z[E(C)] -> Symn-1HB(E(C), C) which is constructed as follows n-1
{z} H E Ka,b(Z) (dz)a-1(dz)b-1 E Symn-1Ha(E(C), C) i=o
Theorem (8.2) claims that it sends the subgroup Rn+1(E(C)) to zero.
Alexander Goncharov
202
4. Proof of the theorem (8.2). For any function f on a manifold one has d log If I A d log 11 - f I. So a non zero term in the integral (8.5) could be Ifldlog11-fl-log 11-fldlogIfl)ndloglf3ln...AdlogIfnIAWp,e
(log
One always has
(log Ifldlog11- fI-log11- fldloglfl)AWp,a= ±i (log If I darg(1- f) - log 11 - f I darg f) A wp,e
Further one has, even in the sence of the distribution, dr2 (f) = log If I darg (1 - f) - log 11 - f I darg f
So we can rewrite the integral as
fi
d(G2(f)Adlog lfslA...Adlog IfnlAwp,9)=0
It is zero by the Stokes formula. Now using theorem (8.4) relating Eisenstein-Kronecker series to the regulator integral we come to the proof of the theorem. Theorem (8.2) and lemma (8.1) imply
Theorem 8.7 The Eisenstein-Kronecker map Kn+1 provides a homomorphism Kn+1 :
Bn+1[E(Q)]Gat(0/F) _+ (e
:FvcSymn-1HB(Ea(C),R(1)))+
Combining these results with conjecture (1.4), our construction of the elliptic motivic complexes presented in s. 4.1 and Beilinson's conjecture on regulators we come to conjecture (8.8) about L-function of Symn-'h'(E) at s = n for an arbitrary elliptic curve E over a number field. 5. A conjecture on L(SymnE, n + 1) In this section we will assume for
simplicity that E be an elliptic curve over Q. We will left to reader as an easy exercise to generalise all the discussion to the case of an elliptic curve over an arbitrary field F. Conjecture (1.4) together with Beilinson's conjecture on regulators imply a precise conjecture on L(SymnE, n + 1)
For any divisor P = En,(P,) on E(C) put KZj(P) :_ En,KKj(P,). The integrality condition . Suppose E has a split multiplicative reduction
at p with N-gon as a special fibre. Let L be a finite extention of Qp of degree n = e f and OL the ring of integers in L. Let E° be the connected component of the Neron model of E over °L. Let us fix an isomorphism
Mixed elliptic motives
203
E pf = Qom/Fp1. It provides a bijection between 7G/eNZ and the components of EFp f . For a divisor P such that all its points are defined over L denote by d(P; v) the degree of the restriction of the flat extension of a divisor P to the v'th component of the (eN)-gon. Let Bn+1(x) be the (n + 1)-th Bernoulli polynomial. The integrality condition at p is the following condition on a divisor P, provided by the work of Schappaher and Scholl ([SS]). For a certain (and hence for any, see s. 3.3 in [GL]) extention L of Q such that all points of the divisor P are defined over
Lonehas([L:Q]=ef): E d(P; v)Bn+1(-N) = 0
(57)
vEZ/(eN)Z
Let Cn be the conductor of the system of the l-adic representations related
to Symnhl(E). Set 21+1 (ImT)(l+1)(I+2)'
N21+1 = C2l+11)
-
021 = C21
2
7f-21(ImT)(l+1)
z
Conjecture 8.8 a) For any elliptic curve over Q there exist [21 +1 Q-rational divisors Pa on E(Q) such that L(Symnh1(E), n + 1) -Q. /3ndetl Kb,n+2-b(Pa; T)1
(58)
(1 < a, b < [2] + 1), and the divisors Pa satisfy the following two conditions: i)
6(Pa)=O in Bn+1(E) 0 J(Q)Q
(59)
ii) the integrality condition: at each prime p where E has a split multiplicative reduction with special fibre a Neron N-gon
E d(P; v)Bn+1(-N) = 0
(60)
vEZ/(eN)Z
b) For any [2]+1 Q-rational divisors Pa on E(Q), satisfying the conditions above the right hand side of (58) is equal to q L(Snh1(E), n + 1) where q is a rational number, perhaps equal to 0. In [W] J.Wildeshaus, assuming standard conjectures about mixed motives,
formulated a conjecture similar to the part b) of the conjecture (8.8) ( an elliptic analog of the weak version of Zagier's conjecture). For n = 2 the formula (58) was proved for modular elliptic curves over Q in [GL]. Formula (58), even without precise conditions on the divisors Pa, is the most nontrivial part of the conjecture for n > 2 ( see also s.8 in [G4]). An
Alexander Goncharov
204
efficient way to formulate the conditions on the divisors P. without referring to the definition of the subgroups R,(E) is given in the chapter 7. When E degenerates to the nodal curve, the conjecture on L(Sym2E, 3) leads to Zagier's conjecture [Za2] at s = 3, which was proved in [G1]-[G2]. This gives a credit for the conjecture (8.8). The key condition (59) is obviously satisfied if Pa are (multiples of) torsion divisors. The determinants from (58) for torsion divisors where considered by Deninger ([De2], s.5) (and inspired by the Eisenstein symbol of Beilinson [Be]). They work well for CM elliptic curves. However Mestre and Schappacher [SM] deduced from a result of Serre that for a given non CM elliptic curve over Q for all n > no the determinant is always zero for any Q-rational torsion divisors Pa. So to get the L-values one has to consider the non torsion divisors.
6. A more explicit form of the conditions on the divisors Pa. Let P = E niPi and k(P) be the field generated by the points Pi. Let us denote by h,, the canonical local height related to a valuation v of a number field K. Let K,, be the completion of a number field K corresponding to v. Recall the height homomorphism h,, : B2(E(K,,)) -+ it If v is a non archimedean valuation then the target of this homomorphism is (log p) Q. Let us consider a homomorphism
dm : Bn+1(E) --4 Bm(E) ®
Sn+l-m J,,
{a}n+1 --+ {a}n+1-m ® a ... a
We will need the following pairs of homomorphisms. If m = 2: i) A homomorphism
p2®id
:B2(E)®Sn-1JQ-4S2Jc (D
Sn-1JQ
where P2 : {a}2 F--? a a. For any valuation v of we have
ii) The height homomorphism
h ®id : B2 (E)
®Sn-1 JQ
-4 R (9Q
Sn-1 JQ
where v is any valuation of the field K(P). For m > 2: iii) The Bernoulli homomorphism, defined for any bad prime where E has a split multiplicative reduction with the Neron N-gon: Beret, : B n(E) 0 Sn+1-m JQ {a}m 0 bl
... bn+1-m E"+
-4
Sn+1-mJ,
d(a, v)Bm(N) . bl
...
bn+l-m
Mixed elliptic motives
205
iv) The Eisenstein-Kronecker homomorphism
Km ® id : Bm(E) 0
Sn+l-mJQ
Symn-2H1(E(C),R(1))+ 0
Sn+l-m JJ
Remarks. 1. In formulas above JQ means J(O)Q. However for a given divisor P we land in Gal(Q/Q)-invariant part of powers of J(k(P))Q. 2. Let us consider the Eisenstein-Kronecker homomorphism only on the kernel of the Bernoulli homomorphism. Then Beilinson's conjecture on regulators means that it should land in (the regulator lattice in Symn-2H1(E(C), R(1))+) ® Sn+1-mJQ So by the Mordell-Weil theorem if k(P) is a given number field then the target group is a finite dimensional Q-vector space. 3. If v is a p-adic valuation of the field k(P), then (logp)-1 E Q, and so the target of the height homomorphism is a finite dimensional Q-vector space. 4. If the a divisor P is in the kernel of the height homomorphisms for all archimedean valuations but one then thanks to the product formula it is
sent to zero by all of them. In particulary if k(P) = Q we can forget the archimedian valuation. Composing each of these homomorphism with the appropriate map dm we get homomorphisms (p2 ®id) o d2,
(h ®id) o d2,
Ber1n o d,,,,
(K,,, 0 id) o d,n
(61)
here m > 2.
Proposition 8.9 The condition b(P) = 0 in the group Bn(E) ® JQ implies that all of the homomorphisms (61) are equal to zero.
Proof. Clear. The height condition is the crucial one. If it is satisfied, then for a given field k(Pa) the other conditions should give only a finite number of conditions on the divisors Pa. If a Q-rational divisor P is sent to zero by all of the homomorphisms (61)
then this essentially means that 5(P) = 0 in the group Bn(E) ® J(Q)Q. To see this we write the homomorphism dm as a composition of homomorphisms
b®id Bn+1(E)_+Bn(E)®JQ->(Bn-1(E)®JQ)0JQ_+...-4Bm(E)®JOn+1-m
(62)
followed by the projection Bm(E) ® Jan+l-m
-, Bm(E) ®
Let us spell the details in the first interesting case: n = 3.
Alexander Goncharov
206
Proposition 8.10 Let us assume that the Bloch-Beilinson regulator rae K2(E)z -+ R is injective. Then if for n = 3 a Q-rational divisor P belongs to the kernel of homomorphisms (61) then 6(P) = 0 in the group B3(E) ® JJ
Proof. Consider the homomorphism
d2 : B4(E) -* B2(E) ® S2J,
{a}4' --* {a}2 0 a a
(63)
Suppose an element P E B4(E) is in the kernel of the homomorphism (p2 (9 id) o d2. Then P E B4* (E) and f (P) E ((0" 0 S2J(Q))Ga1(O/Q.
The image of the divisor P under this homomorphism belongs to the subgroup k(P)* 0 S2J(k(P)) . The hight condition is a way to say that it is equal to zero. Let us write the map d2 as a composition
B4 (E) -* B3 (E) ®J --* (B2(E) (9 J) ®J - B2 (E) ® S2J Notice that
Ker(B3(E)®J -+ (B2(E)®J)0J)OQ = Ker(B3(E) -* B2(E)®J)®J®Q Consider the homomorphism
B3(E(C)) 0 J(C) -* J(C) ® R,
b 0 {a}3 '--* K2,1(a) ® b
(64)
There is a homomorphism K2(E(C)) -> B3(E(C)) such that the following diagram is commutative (see [GL]):
K2(E(C))
4
R
Id
B3(E((C)) A So assuming the injectivity of the regulator K2(E)z -+ R we see that (64) should be injective on Ker(B3(E) -> B2(E) ® J) ® J ® Q.
9
The complexes 13(E; n)' and motivic elliptic polylogarithms
1. In this chapter k = k, and all abelian groups are tensored by Q, so we work with the corresponding Q-vector spaces. For instance J := J 0 Q etc. Let
Q[E(k)] -* Q[E(k)] 0 J,
{a} H {a} ® a
Mixed elliptic motives
207
Theorem 9.1 Let us assume standard conjectures on mixed motives. Then there exist canonical homomorphisms In : Q[E(k)] --> C(E)Sn-2,H(1)v such that the following digram is commutative:
-
Q[E(k)]
-1- In-2 0 id
In-1 -4-
C(E) sn-2W(1)v
1
Q[E(k)] ® J
-24 £(E) Sn-3W(1)V ® J
Proof. The proof is based on the existence and basic properties of the motivic elliptic polylogarithms of Beilinson and Levin [BL]. For any nonzero point a E E(k) let GQ1i be an element of (Q(0), 7-l) with corresponds to
a E J under the isomorphism ExtME(Q(0),7-1) = J. Set Gam) := Sm(Ga1)). The motivic elliptic (n -1)-logarithm at a is a mixed elliptic motive Ela_1(a) which provides a certain extension class in Ext E(7l,Gan-1)(1)). In particular its weight graded quotients are 71, Q(1), 7-1(1), ..., S(n-1)7-t(1)
Therefore it has canonical framing and so defines an element of A(7-1, Sin-1)l (t)). After tensoring it by 7-l and twisting by Q(-1) we can introduce a natural framing by Q and S(n-2) l(1) (since S(n-1)7{®7l = S(n)WE) S(n-2)7.1). Therefore we picked up an element In-i(a) E .A(Q, S(n-2)f(1)). The commutativity of the diagram follows from the properties of the elliptic polylogarithms ([BL]). The crucial point is this. Since W, The projection to C(E) of the elements in A(Sik>7-t(1),S(n-1)7-(1)) coming from the canonical framing by S(k)7{(1)
and S(n-1)7{(1)) of the motive Eln_1(a) are zero provided 0 < k < n - 2. So projecting the coproduct of Eln_1(a) to A2G(E) the only nonzero contribution we get is given by the component of the coproduct coming from .A(7-1, Sin-2>l(1)) 0 A(S(n-2)7.1(1), S(n-1)7.1(1)). The fact that it is equal to ln_2(a) A a follows immediately from the basic property of the elliptic polylogarithm motive (see [BL]).
Definition 9.2 Rn(E) = Kerln-1,
In(E) _ Q[E(k)] Rn (E)
Theorem (9.1) implies that b provides a well defined homomorphism 8 B,,(E) -> Bn_1(E) ® J. So we get a complex B(E; n)':
Bn(E) -> Bn_1(E) ®J
... -+ B2 (E) ®An-2J -, J 0 An-1J --+ AnJ
Alexander Goncharov
208 Set
r,, (J) := Ker(J ® An-1J -* A'J) The theorem (9.1) and this definition immediately imply that there exists canonical homomorphism of complexes
-#
8n(E) In-1 ,l.
Bn_1(E) ®J
-* ... -# B2(E) ®An-2J -> rn(J)
ln_2 (9 id J.
l l1 ®id
C(E)s
,C(E)sn-2x(1)
0 J -> ... --+ ,C(E)Q(1)v ® An-2J
11
-* rn(J)
Lemma 9.3 The bottom complex is a subcomplex of the Sn-231(1)"-isotypical component of the standard cochain complex of the Lie coalgebra L(E).
So if k = k, we get a canonical injective morphism of the complexes
,t3(E; n + 1)' -+ (A'L(E), a)Sn-lW(l)V
(65)
Let Km 1 be the sheaf of Milnor K-groups. Combining this morphism with the canonical morphism from the right hand side to RHommm,k(Q(0), Sn-131(1))
provided by the inclusion functor ME -4 MMk we get Corollary 9.4 Let us assume the formalism of mixed motives. Then a) there exists canonical homomorphisms
H'(B(E; n + 1);)
grnKn+1-i(E(n-1))99n
(9 Q
(66)
b) The homomorphism for i = 1 is injective.
Indeed, thanks to (65) this is true if k = k. The general case follows since we have the descent property both for rational K-theory and, (by definition), for the complexes B(E; n + 1);.
Remark. We do not expect a morphism of complexes (65) exist if k is not algebraically closed. The reason is this. If k is not algebraically closed we have postulated the Galois descent for the complexes 13(E; n + 1)'. On the other hand the standard complex of the Lie algebra L(Elk) should not satisfy the Galois decent. I hope a stronger result should be valid:
Conjecture 9.5 a) grnKn+l-i(E(n-1))sgn ® Q =
H2-1(E(n-1),
K' )89n ®Q
(67)
Mixed elliptic motives
209
b) There exists a canonical isomorpism in the derived category 13(E; n + 1) ; = RHomMMk (Q(0), Snl (1))
in particular H2(13(E, n + 1);) = 9rn7Kn+1-i(E("-1))sgn ®Q
(68)
c) Let k = k. Then the homomorphism of complexes (65) is a quasiisomorphism.
If the conjecture (4.2) is correct, then b) is equivalent to c). According to the lemma (9.3) this conjecture implies conjecture (1.4). Part a) of the conjecture is trivial for n = 2.
Lemma 9.6 For n = 3 and k = k the part a) of the conjecture (9.5) follows from the rigidity conjecture for K3nd(k)
Proof. The statement boils down to Ker,6 = Ima in the sgn-part of the diagram
a
A3k(E(2))* 1
lYE(E(2))1 B2(k(Y))
UYE(E(2))1 A2k(Y)*
- B2(k(E(2))) ® k(E(2))*
0
Let Q = LIYNY. Then Kerfy = K3nd(k(Y))Q. By the rigidity conjecture any point y E Y provides an isomorphism K3nd(k)Q = K3nd(k(Y))Q. So Ker,3/Ima is a subgroup of ®k(E(2))* _+
Coker(K3nd(k)Q
11
K3nd(k)Q) = CH1(E(2))s9n ®Q = 0
YE(E(2))1
The lemma is proved.
2. Motivic realization of elliptic polylogarithms. Let f = (fi,..., fn) be an n-tuple of rational functions on E. Motivation. Consider the following multivalued analytic function on {n - tuples of rational functions on
P(E (n-1); f;f'-Y) :=
7
E((C)} x Hn-1(E(n-1)(C)),gn
o(x1))dlog f2(x2)n...ndlog fn(xn) (69)
Alexander Goncharov
210
where -y is a cycle representing a nontrivial class in Hn_1(E(n-1)(C))sgn,,. The
subscript sgnn means the skewsymmetric part with respect to the permutations. (A better way to define this function is given by formula (74) below.) We will show that this function is a period of a mixed elliptic motive. Choose a coordinate z on P1. Let An be the coordinate cube in (P'; OUoo)n,
i.e. union of 2n hyperplanes zi = 0, zi = oo. Notice that (P1)n\An = (Gm)n. Let us define a codimension n cycle Z(E(n-1);f) C E(n-1) X (pl)" as follows:
Z(E(n-1); f) := Alt(yl..... a-){xl, ..., xn; fl(xl), ..., f"(xn)} U E(n-1) X {1, ..., 1}
Here we use the coordinate system zl,..., zn, i.e. zi = fi(xi). The group Sn acts on (Gn)n by permutations. So we get an action of the group Sn x Sn on E(n-1) X (Gm)n. In this chapter we mark by the subscript sgn the skewsymmetric part under this action. Consider the following mixed motive h(E(n-1); f) :=
Hn(E(n-1) X
(Gm)n, Z°(E (n-1); f )) (n)sgn
where Z°(E(n-1); f) := Z(E(n-1); f)\(Z(E(n-1); f) n An) More precisely, h(E(n-1); f) (-n) := Rnp!.F(E; f )sgn where F(E; f) is the E(n-1) x following mixed motivic sheaf on Take the constant sheaf on (lFl)n.
the complement to Z(E(n-1); f) U E(n-1) X An in j* to the divisor E(n-1) X An and then by j! to
E(n-1) x (pl)n; extend it by Z(E(n-1); f).
Lemma 9.7 h(E(n-1); f) is a mixed elliptic motive framed by Q(0) and Symn-l7-l(1).
Proof i) Q(0)-component of the frame. Let us prove that 9r2Hn(E(n-1) X (Gm)n, Zo(E(n-1); f))sgn = Q(-n)
Notice that Hn(Gm) gn = Hn(Gm)n and Hi(Cm) gn = 0 for i < n. So the projection E(n-1) x (Gm)n -* (Gm)n induces an isomorphism Q(-n) = gr nH"((Gm)n)sgn -- gr2 H"(E(n-1) X (Gm)n)sgn The canonical morphism Hn(E'(n-1) X
(Gm)"
ZO(E("-1);
f ))sgn -f Hn(E(n-1) X (Gm)"),gn
induces an isomorphism after taking gr2. Indeed, there is an exact sequence H"-1Zo(E("-1); f) -+ H"(E(n-1) X (Gm)", ZO(E("-1); f))sgn -+
Mixed elliptic motives
211 HnZ°(E(n-1); f)
Hn(E(n-1) x (Gm)n)sgn -+ and
gri Hn-1Z°(E(n-1); f) = gr2 -1Hn-1Z°(E(n-1); f) = 0 since Z°(E(n-1); f) is an open regular variety of dimension n - 1. ii) Symn-131(1)-component of the frame. One has
Hn(E(n-1) x (pl)n, Z(E(n-1); f))sgn = Symn-lhl(E) Indeed,
Hn(E(n-1) X (P1)n)gn = 0. So there is
(70)
an exact sequence
Hn-1(E(n-1) x (pl)n)sgn -? Hn-1(Z(E(n-1);f))sgn Hn(E(n-1) x (pl)n, Z(E(n-1); f))sgn -* 0 Further, Hn-1(E(n-1) X (1P1)n)sgn = Symn-lhl(E) Symn-1h1(E.,) ® Symn-1h1(E)
Hn-1(Z(E(n-1); f))sgn =
and a is injective. The restriction map induces an isomorphism
grn 1Hn(E(n-1) X
(F1)n,
Z(E(n-1);
f ))sgn +
(71)
grn-1Hn(E(n-1) x (Gm)n, Z°(E(n-1); f))sgn
because (Z(E(n-1); f) is regular of dimension n - 1). Wn-1Hn-1Z°(E(n-1); f) =
Wn_1Hn-1Z(E(n-1); f)
Combining (70) and (71) we get the Symn-131(1)-component of the frame. The lemma is proved. Finally, we show that h(E(n-1); f) is a mixed elliptic motive by induction using the following basic observation: the intersection of the cycle Z(E('a-1); fl'...' fn) with any codimension 1 face of the cube On is a sum of cycles of form Z(E(n-2); g1i ..., gn_1). For example
Z(E(n-1); fl,..., fn) n {zl = 0} =
m5(fl) .
Z(E(n-2); f2' ...j fn)
SEE
where m5(f) is the multiplicity of zero at x. The lemma is proved.
Remark 9.8 One can apply the same construction to n arbitrary functions f1(z1, ..., zn), ..., fn(zl, ..., zn) on E(n-1). However it is not clear whether the motive h(E(n-1); fl,..., fn) is a mixed elliptic motive in general.
Alexander Goncharov
212
The functions fl,..., fn on the E define a map
f:
E(n-1) _+
(x1, ..., xn) -+ (f1(x1), ..., fn(x1))
(Pl)n,
Efn-1) Let be the image of this map and Consider the following motive:
h(E(n-1);
f) :=
Efn-1)
:= Efn-1)\(Efn-1) n
Efn-1))(n)sgn
0 (72)
Lemma 9.9 Suppose that f* : Hn_1(E(n-1))sgn -4 Hn_1(Ef -1))sgn is a nonzero map. Then a) h(E(n-1); f) is a (Q(0),Symn-17d(1))-framed mixed elliptic motive. b) h(E(n-1); f) = h(E(n-1); f) as framed motives. c) If f* = 0 then h(E(n-1); f) = 0.
Proof a) is very similar to the proof of the lemma above. For instance the Symn-1hl(E) part of the framing comes from isomorphism Wn-1Hn((Pl)n\An, Ef -1)) -* Sym"`-lhl(E)
Namely, Hi(Pn)gn = 0, so there is an isomorphism Hn-l(E(f
n-1))sgn -4 Hn((p1)n, Ef _1))sgn Hn-1(Efn-1))sgn -j Hn-1(E(n-1))sgn
Combining it with f* :
we get a mor-
phism Hn((P1)n,
Hn_1(E(n-l))sgn
Ef
= Symn-lhl(E)
(73)
b) The projection E(n-1) X (P')n
(Pl)n
induces a morphism respecting the frames. c) is clear from the construction. The lemma is proved.
The period corresponding to this framing is exactly the function (69). Indeed, consider the differential form wo := d log(zl) n ... A d log(zn)
in (CPI)'\On. Let I' be a relative n-chain in CPn which bounds an (n - 1)cycle 'y, ['y] E Hn-1(((CPI )n, Ef
n-l))sgn Then Ln(E(n-1); f; _Y)
= Jrr
o
The Stokes formula shows that the integrals (74) and (69) coinside.
(74)
Mixed elliptic motives
Proposition 9.10
213
The R-valued period of the Hodge realization of
h(E('a-1); f) is given by £(E (n-1);
f)
Altx1i...Xnrn(fi, ..., fn) A Wp,q
This integral coincides with the one computed by means of the EisensteinKronecker series, as was explained before.
Recall that A(Q, Symn-17{(1)) is the group of mixed elliptic motives framed by Q(O) and Symn-19-1(1).
Lemma 9.11 For any )i E k* one has the equality of framed motives h(E(n-1); fl'...'
fn) =
h(E, (n-1);
Al
fl,..., An
fn)
Proof. The action of an element g = (.A1i ..., An) E (tv,nn on Pn\An = G,,, gives a morphism of motives h(E(n-1); Al fi,..., An fn) -+ h(E(n-1); fl'...' fn) which obviously preserves the framing. Theorem 9.12 There is a well defined homomorphism of abelian groups Mn : Snk(E)* -+ A(Q(O), Symn-17 (1))
fl o ... o fn H
h(E(n-1);
f)
It is zero if one of the functions fi is a constant; one has m* (Kerµn) = 0.
Proof. Below a generalization of the construction above is given. Let D° be the group of degree zero divisors on E. For any d := (d1i ..., dn) let us construct a mixed elliptic motive h(E(n-1); d) := h(E(n-1); d1i ..., dn) framed by Q(0) and Symn-17-1(1).
Let P be the rigidified Poincare line bundle over J x J. For any two degree zero divisors dl, d2 with disjoint support there is an element < d1i d2 >E p[d1],[d2]
where [di] E J is the class of a degree zero divisor di. Consider the following (n - 1)-cycle
Z(E(n-1); d) C E(n-1) x pn
(75)
c0n-1); d) := Alt(x1i...,xn)(xl) ..., xn; < d1, (x1) - (0) >; ...; < dn, (xn) - (0) >) E(n-1) Here x1 + ... + xn = 0 and (x1, ...) xn) belongs to Zariski open part of
where the divisors di and (xi) - (0) are disjoint. and set h(E(n-1); d) := H'
(E(n-1) X Pn,
Z(E(n-1); d))
(n)
Alexander Goncharov
214
Theorem 9.13 a) h(E(n-'); d) is a mixed elliptic motive framed by Q(O) and Symn-'7i(1). b) There is a well defined homomorphism of abelian groups di o ... o do H h(E(n-1); d)
SnD° -+ A(Q(O), Symn'-'?-l(1))
Proof. Restriction to a fiber of the Poincare line bundle provides an isomorphism
gr2 H'(P) 4 H'(Gm.) = Q(-1) and thus we get a first part of the framing:
Q(-n) -* gr2 Hn(Pn)
gr2
Hn(F'i(n-1) X
Pn)
where ir : E(n-') X Pn -+ Pn is the natural projection. The second part of the framing comes from the fundamental cycle of E(n-1)
just as before. The rest is straitforward.
3. Motivic construction of the complex B* (E; 3). Recall the convolution map 02 : S2k(E)* -f B*(E), fi o f2 - (fi) * (f2). We are going to show that the diagram S2k(E)* y,32
B* (E)
m2 -- G*(E)x(1)
provides a well defined homomorphism l2
: B*(E) -+G*(E)x(1)
(Here G*(E)W(1)v = G*(E)N(1) ®1. (1)v so G*(E)x(1) is a Q-vector space.)
Consider the map µ2 :
S2k(E)*
K2(K(E) {k*, k(E)*}_'
fl o f2 H {fl(x), f2(-x)} - { fl(-x), f2(x)}
According to theorem 3.9 in [GL] one has
Theorem 9.14 tc2(Ker/32) = 0 It remains to use that l2(f * (1 - f)) = 0 by theorem 9.12. Theorem 9.15 We get a commutative diagram B*(E)Q
(k* ®d)Q
l2 J G*(E)W(1)
G*(E)Q(1) ®,C*(E)x
To prove this theorem we need only to compute bh(E; fl, f2), which is left to the reader.
Mixed elliptic motives
10
215
Elliptic Chow polylogarithms and generalized Eisenstein-Kronecker series
1. Elliptic Chow polylogarithms. The single valued version. Let C be a codimension n cycle in Ek X (P1)t, k + l = 2n - 1, skewinvariant under the action of Gk x Gt.
Recall the forms 7rnWp,q (see (53)), which for p > q, p + q = n - 1 form a basis over R in HB 1(En 1, R(n - 1)),ggn,,. We represent elements in HB 1(En 1, R(n - 1))sgn by their cup product with the forms 7rnWp,q: The single valued elliptic Chow polylogarithm is a function
k+1 =2n- 1
Pk,l : Zn(Ek,1) -* HB 1(E1njj1, R(n - 1))sgn
defined as follows. Let it : C -+ (P')' and p : C --+ Ek. If k > 0: < Pk,t (C), Wp,q >:= fc
qr*rk-1(zl, ..., zk) A p*Wp,q
p ? q, p + q = n -
Here we integrate over the nonsingular part of the complex points of the cycle C. The integral is always convergent (see [G6]). For example P2,1(C) := Alt(x2,x3) furl log Izi 7r*w A ir3W
The multivalued elliptic Chow polylogarithm, denoted Pk,l(C), is defined as follows. Let (x1, ..., xk, zk+1, ...) zk+l) be the coordinates on Ek X (IN)'. Assume l 0. Let irz (resp. pj) be the projection of C to the i-th coordinate
IN (resp. j-th factor E) in Ek x (pl)t i). Assume k < n. Then Pk,l(C) :=
Alt(ck xG,) f
piryx...xpkryx1rk}16x...x1rk+,b
log zn+ld log zn+2 A ... A d log z2n-1
ii). Assume n < k < 2n - 1. Then Pk,l(C) := pn+lw A ... A p*w n (log zk+ld log zk+2 A ... A d log z2n-1)
A1t(Gk xG,) fl,ryx...xp p
,ry
Example 1. The multivalued elliptic Chow dilogarithms: Pa,3(C) := Alt(G3) f
is
log z2d log z3
P1,2(C) := Alt(G1xa2) f logz2dlogz3 pity
Alexander Goncharov
216
P2,1(C) := Alt(G2xG1) fip2W - logz3
p7 Example 2. The multivalued elliptic Chow trilogarithms: Po,5(C) := Alt(G5) 1116"2*6 logz3dlogz4 A dlogz5
P1,4 (C) := Alt(G1 xG4) f
log z3d log z4 A d log z5
P2,3(C) := Alt(G2 xG3) fi
log z3d log z4 A d log z5
pi7x1r2b
p 7 xp27
psW A log Z4 A dlogz5
P3,2(C) := Alt(G3xG2) f
pi-xp27
P4,1(C) := Alt(G4 xG1) f
p17xp27
paW A p4W . log z5
The multivalued elliptic Chow polylogarithms are periods of mixed motives, which are easy to write down.
2. Some interesting cycles. Let Ln (a) be the codimension n cycle in (p1)2n-1 responsible for the classical n-logarithm (see [B16] and [BK]): Ln (a) := {x1, ..., xk_1i 1-x1, 1-x2/x1, ..., 1-xk_1/xk_2, 1-a/xk_1} E Zn(2n-1)
Consider the following cycle in Zn(E(n-k-1), k + n): Alt(x1...,Xn-k) (x1, ...) xn-k, Lk (fl (x1)), f2(x2), ...,
Notice that we are using this
E(n-k-1),
not
fn-k(xn-k)
(76)
En-k-1.
Examples of cycles. in
Z2(E, 2) : in
Z2(3) :
in
Z3(E(2), 3) :
in
Z3(E, 4) : in
i3(5):
Alt(x1,x2){x1, fl (Xl), f2 (X2)}
{z1, 1 - z1,1 - a/zl}
Alt(x1,x2,x3){xl, x2, fl(xl), f2(x2), f (x3)}
Alt(x1,x2){x1, 1 - x1, 1 - fl(xl)/xl, f2(x2)}
{zl,z2i1-z1,1-z2/z1,1-a/z2}
I think the single valued elliptic Chow polylogarithm on these cycles should L(Symn-k-1E, n). provide the new transcendental functions needed for
3. The generalized Eisenstein-Kronecker series and L(SymnE, n +
m) for m > 1. Conjecture 2.1 in [G2] for the field k(E(n)) tells us that
Mixed elliptic motives
217
H 1(Speck(E(n)), Q(n + m)) is generated by the sums of the elements of form
E{f0(i)(x)}m ® fl(i)(x) A ... A ffi)(x)
in
Bm(k(E(n))) ® Ank(E(n))*
i
satisfying the condition
E{f0(Z)(x)}m-iAf0 )(x)Afiz)(x)A...Af,)(x) in
Bm-1(k(E(n)))®An+lk(E(n))*
(77)
Definition 10.1 Do,1m) is the subgroup of 13m(k(E(n))) 0 Ank(E(n))* generated by the elements Alt(x( ..... x,){f(xo)}m ® g1(x1) A ... A gn(Xn)
(78)
where f, gk E k(E)*, x0 + ... + xn = 0.
Denote by X the element (78). Set
0(X) :=
Alt(xo,...,xn){fo(xo)}m-l A fo(xo) A g1(x1) A ... A
It belongs to the group 13m_1(C(E(n))) 0
gn(xn)
An+1C(E(n))*.
n
r(X) :_i=1E(-1)' E
vagi-Alt(xo,...,x,.){fo(xo)}m®gl(xl)A...gi...Agn(xn)1xi=a
aEE(k)
Here vag is the valuation of g at the point a. Let dn,m 0 + r. r(X) lies in the sum of the groups 13m(C(Xi,a))* 0 An-1C(Xi,a)* where Xi,a is the divisor xi = a.
Conjecture 10.2 a) There exists a map Kerdn,m -+ H 1(E(n), Q(n + m))sgn which in the case k = C commutes with the regulator map. b) One might hope that this map is surjective.
Part a) of this conjecture can be deduced from standard conjectures. If m = 1 this is exactly conjecture discussed in chapter 5. If n = 1 the conjecture
follows from the conjecture 2.1 in [G2], see also conjecture 8 in [G4] and n = 1, m = 2 it is proved in [G3]. In general the main argument for the hope expressed in the part b) is simplicity of the ansatz used to define the elements (78).
Element (78) provides a cycle of type (76), as was explaned above. The value of the regulator on the element which lies in Kerdn,m should coinside
Alexander Goncharov
218
with the value of the elliptic Chow polylogarithm Pn,m on the corresponding cycle. Thus to get the generalized Eisenstein-Kronecker series responsible for
the special values L(SymnE, n + m) we evaluate the Chow polylogarithm Pn,m on the cycle (76) using the Fourier decomposition of ,Ck(f(x)) and then the same method as in chapter 6. This boils down to computation of the following regulator integral (p + q = n) r({f}m(D91n...Agn;Wp,q) :=
loglfI'-2,a2(1-f,f)Aan
JE'
(C)
(91,...,9n)Awp,q
Set
r({f}m(9 g1A...Agn):= E r({f }m (991 A ... A 9n, Wq,p) (dz)p(d2)q E SymnH1(E(C), C) p+q=n
Let
(Z[E] ® Z[E] 0 Z[E])E= (Z[E x E x E])E be the abelian group generated by the elements {x, y, z} where x, y, z E E(k), subject to the relations {x, y, z} = {x+c, y+c, z+c} for any c E E(k). Define On,m : { f }m ®g1 0 ... 0 gn H (Z [E] 0 Z[E] 0 Z[E])E
{.f}m0g10...0gn
*(1-f)®(.f)®(91)*...*(9n)
Consider the following functions where p + q = n, m > 2: z)Im..)mX
Kpq(x n,m l y,
7r
\'ym(yo - 71) + 7m (7o - ry1)) ''Ym
(x) -YO) (Y, ry1 + ... + 7m-1) (Z,'ym) 17012 '
...
1 _ x,gm 1
17m-11217m12"
I will call them the generalized Eisenstein-Kronecker series. For n = 1 this is the functions Km+1(x, y, z) defined in [G4], see also [G3]. For n = 1, m = 2 this function was considered by Deninger [D3].
Conjecture 10.3 There exists a variation of mixed elliptic motives over (ExExE)/E such that its real periods are given by the generalized EisensteinKronecker series. Define a homomorphism Kn,m : (Z[(E x E x E) (C)]) E-* Sym"H1(£' (C), C)
{x, y, z} H
y, z)(dz)p(dz)q p+q=n
Mixed elliptic motives
219
Theorem 10.4 Assume Alt(.(,._..) ({f
(xo)}m-1 0 f (xo) A 91(x1) A ... A gn(xn))
=0
in the group ,Bm_1(C(E(n))) 0 An+1C(E(n))*. Then
r({f}m(9 glA...Agn)
=Kn,m0On,m({f}m®910...ogn)
The proof is completely similar to the proof of theorems 8.4 and theorems 3.4 and 4.7 in [G3], and thus is omitted.
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[BL] Beilinson A.A., Levin A.M.: Elliptic polylogarithms, Symposium in pure mathematics, 1994, vol 55, part 2, 101-156. [BMS] Beilinson A.A., MacPherson R., Schechtman V.V.: Notes on motivic cohomology, Duke math. Journal, 54 (1987), 679-710. [BGSV] Beilinson A.A., Goncharov A.A., Schechtman V.V., Varchenko A.N.: Aomoto dilogarithms, mixed Hodge structures and motivic cohomology, the Grothiendieck Feschtrift, Birkhauser, vol 86, 1990, p. 135171.
[B11] Bloch S.: Higher regulators, algebraic K-theory and zeta functions of elliptic curves, Lect. Notes U.C. Irvine, 1977. [B12] Bloch S.: Algebraic cycles and higher K-theory, Advances in Math. vol. 61 (1986) 267-304.
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[B13] Bloch S.: Cycles and biextensions, Contemporary Math. 1989, vol 83, 19-31
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Amer. J. of Math., 4 (1991), 771-791.
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[De2] Deligne P.: A quoi servent les motifs?, in Symp. in Pure Math., v. 55, part 1, (1994), 143-163. [D1] Deninger, C.: Higher regulators and Hecke L-series of imaginary quadratic fields I, Invent. Math. 96, 1-96 (1989) [D2] Deninger, C.: Higher regulators and Hecke L-series of imaginary quadratic fields II, Ann. of Math, 132, N1 (1990), 131-158.
[D3] Deninger, C.: Higher order operations in Deligne cohomology. Inventiones Math. 122, N1, (1995), 289-316.
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[G2] Goncharov A.B.: Polylogarithms and motivic Galois groups, Symposium in pure mathematics, 1994, vol 55, part 2, 43-96. [G3] Goncharov A.B.: Deninger's conjecture on special values of L-functions of an elliptic curve at s = 3, Special volume dedicated to Manin's 60th
birthday, Plenum, 1997 (alg-geom e-preprint, Preprint MPI January 1996.)
[G4] Goncharov A.B.: Polylogarithms in arithmetic and geometry, Proc. ICM-94 in Zurich, vol 1, 374-387.
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[G5] Goncharov A.B.: The Eisenstein-Kronecker series and L(Sym2E, 3), Preprint September 1995. [G6] Goncharov A.B.: Chow polylogarithms and regulators. Math. Res. Letters, Ni (1995) 95-112
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[MS] Mestre J.-F., Schappacher, N.: Series de Kronecker et fonctions L des puissances symmetriques de courbes elliptiques sur Q, In: Arithmetic
algebraic geometry. Van de Geer. G., Oort, F., Steenbrink, J. (eds) (Prog. Math., vol. 89, 209-245). Birkhauser 1991. [Q] Quillen D. : Rational homotopy theory, Ann. of Math, 90 (1969) 204295.
[SS] Schappacher N., Scholl, A.: The boundary of the Eisenstein symbol, Math. Ann. 1991, 290, 303-321. [S] Suslin A.A.: K3 of a field and Bloch's group, Proceedings of the Steklov Institute of Mathematics 1991, Issue 4.
[We] Weil A.: Elliptic functions according to Eisenstein and Kronecker, Ergebnisse der Mathematik, 88, Springer 77.
[W] Wildeshaus J.: On an analog of Zagier's conjecture for elliptic curves, Duke Math. Journal, (1997) vol. 87 N2, 355-407. [W2] Wildeshaus J.: On the generalized Eisenstein symbol, Preprint, 1997. [Z] Zagier D.: The Bloch- Wigner-Ramakrishnan polylogarithm function, Math. Ann. 286, 613-624 (1990)
[Z2] Zagier D.: Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields, Arithmetic Algebraic Geometry (G.v.d.Geer, F.Oort, J.Steenbrink, eds.), Prog. Math., Vol 89, Birkhauser, Boston, 1991, pp. 391-430. DEPARTMENT OF MATHEMATICS, BROWN UNIVERSITY, PROVIDENCE, RI 02912,
USA
[email protected] On the Satake isomorphism BENEDICT H. GROSS
In this paper, we present an expository treatment of the Satake transform. This gives an isomorphism between the spherical Hecke algebra of a split reductive group G over a local field and the representation ring of the dual group G. If one wants to use the Satake isomorphism to convert information on eigenvalues for the Hecke algebra to local L-functions, it has to be made quite explicit. This was done for G = GL,, by Tamagawa, but the results in this case are deceptively simple, as all of the fundamental representations of the dual group are minuscule. Lusztig discovered that, in the general case, certain Kazhdan-Lusztig polynomials for the affine Weyl group appear naturally as matrix coefficients of the transform. His results were extended by S. Kato. We will explain some of these results below, with several examples. CONTENTS 2. 3. 4. 5. 6. 7. 8.
The algebraic group G The Gelfand pair (G, K) The Satake transform Kazhdan-Lusztig polynomials Examples L-functions The trivial representation Normalizing the Satake isomorphism Acknowledgements
223 225 227 229 231 232 234 235 237
References
237
1.
1. The algebraic group G Throughout the paper, we let F be a local field with ring of integers A. We fix a uniformizing parameter 7f for A, and let q be the cardinality of the residue field A/irA. Reprinted from `Galois Representations in Arithmetic Algebraic Geometry', edited by A. J. Scholl & R. L. Taylor. ©Cambridge University Press 1998
224
Benedict H. Gross
Let G be a connected, reductive algebraic group over F. We will assume
throughout that G is split over F. Then G is the general fibre of a group scheme (also denoted G) over A with reductive special fibre; in the semisimple case G is a Chevalley group scheme over A. We fix a maximal torus contained in a Borel subgroup T C B C G, all defined over A, and define the Weyl group of T by W = NG(T)/T. Define the characters and co-characters of T by
X' = X'(T) = Hom(T, G,,,) X. = X. (7:) = Hom(G., T)
These are free abelian group of rank 1 = dim(T), which are paired into Z. The first contains the roots of G : the characters Hom(G,,,,, of T occuring in the adjoint representation on Lie (G), and the second contains the coroots [1, Ch.6,§1]. The subset T+ of positive roots which occur in the representation on Lie (B) satisfies: = -D+ U -4D+. It determines a root basis A C of positive, indecomposible roots. When G is of adjoint type, the elements of A
give a Z-basis of X'. The root basis determines a positive Weyl chamber P+ in X.(T), defined by
P+={AEX.:(A,a)>OallaE4D+}
(1.1)
={AEX.: (A,a)>0allaEz } Let (1.2)
2p = > a
in X'(T)
Then, for all A in P+, the half-integer (A, p) is non-negative. There is a partial ordering on P+, written A > p, if the difference A - p
can be written as the sum of positive coroots. If a is a basic coroot in then (1.3)
(61P) = 1
Hence A > p implies that (A - p, p) is a non-negative integer.
Let G be the complex dual group of G. This is a connected, reductive group over C whose root datum is dual to G. If we fix a maximal torus in a Borel subgroup t C B C G, there is an isomorphism X' (T) -- X.(22)
which takes the positive roots corresponding to b to the positive coroots corresponding to B. The elements A in P+ C X'(T) index the irreducible representations VA of the group d : A is the highest weight for b on VA. Let
On the Satake isomorphism
225
xA = Trace(V,,) be the character of VA, viewed as an element of Z[X'(t)]. Then xa lies in the subring R(G) = 7G[X'(T)]w
fixed by the Weyl group. If we write
xa=Ema(µ)-[a], then ma(p) = m,(wa). Hence it suffices to determine the integers MA(p) for ,u in P+, as these weights represent the orbits of the Weyl group on A simple result is that the integer m,\ (p) = dim V\ (p) is non-zero if and only if A > µ in P+ [5, pg.202-203].
2. The Gelfand pair (G, K) We define compact and locally compact topological groups by taking the A- and F-rational points of the group scheme G:
K = G(A) c G = G(F)
(2.1)
Then K is a hyperspecial maximal compact subgroup of G [17, 3.8.1]. Similarly, we have the locally compact, closed subgroups
T = T(F) C B = B(F) C G = G(F). We let N = N(F), where N is the unipotent radical of B. Then (2.3) B = T < N, and (2.2)
det(ad(t)I Lie(N)) = 2p(t).
(2.4)
The Hecke ring Il = 'H(G, K) is by definition the ring of all locally constant, compactly supported functions f : G -* Z which are K-bi-invariant:
f (kx) = f (xk') = f (x) for all k, k' in K. The multiplication in 7t is by convolution (2.5)
f g (z) = JG f (X) . g(x-iz)dx
where dx is the unique Haar measure on G giving K volume 1. We will see below that the product function f g also takes values in Z. The characteristic function of K is the unit element of 1 . Each function f in f is constant on double cosets KxK; since it is also compactly supported it is a finite linear combination of the characteristic functions char(KxK) of double cosets. Hence these characteristic functions give a Z-basis for W. For any A E X.(T) = we have the element A(7r) in T(F). Since )(A*) C T(A) C K, the double coset K\(7r)K depends only on A, not on the choice of uniformizing element. Here we view A multiplicatively, so (A + p) (7r) _ A(ir) . µ(ir)
Benedict H. Gross
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Proposition 2.6. (cf. [17, p.51]) The group G is the disjoint union of the double cosets KA(ir)K, where A runs through the co-characters in P+.
This is a refinement of the Cartan decomposition: G = KTK; for G = GLn it is proved by the theory of elementary divisors [13, pg.56-57]. It follows
that the elements cA = char(KA(7r)K)
(2.7)
A E P+
give a Z-basis for f, and multiplication is determined by the products (2.8) cA cN, = Ena,,(v) c,,, with na,,(v) E Z. To obtain an explicit formula for the integers na,N,(v), we write v(7r) = t
KA(ir)K = ll xiK K,u(7r)K = 11 yjK Then na,µ(v) _ (ca cµ)(t)
= foca(x)c,(x-1t)dx
_
®iK
cµ(x-1t)dx
cµ(kxi
1t)dk
L i c (xi 1t)
_ #{(i, j) : v(7r) E xiyjK} Since we can take xi = A(7r) and yj =,u(ir), this shows that na,µ(A + µ) > 1. In fact, we will see later that na,,,(A + i.t) = 1 and that n,\,, (v) # 0 implies that v < (A + µ). Therefore (2.9)
Ca
Cµ = CA+µ + > na,µ(v) c u 0, the constant coefficient of q(a,P>P(p) is equal to 1. Let
2pv = E a"
(4.3)
be the sum of all positive coroots.
(W Proposition 4.4. The coefficient d,\(p) appearing in the Satake isomorphism is given by the formula d,\ (p) = P,,,\ (q) =
q('U-µ,P) E e(o)P(ti(\ w
+ pv) - (p + p')),
where e(a) = det(QIX.(T)) is the sign character on the Weyl group W. Kato shows that the polynomial P, ,,\(q) defined by the alternating sum in Proposition 4.4 is a Kazhdan-Lusztig polynomial for the affine Weyl group Wa of G. It is associated to the pair of elements w, < w,, in the extended affine Weyl group Wa = X' (T) >i W of maximal length in the double cosets WpW and WAW respectively. These elements have lengths: f(wµ) = (p, 2p) + dim(G/B) and 2(w,\) _ (A, 2p) + dim(G/B). The general theory of Kazhdan-Lusztig polynomials then implies that PN,,a(q) has non-negative integer coefficients, and has degree strictly less than (A- p, p) in q [9, pg.215].
If we set q = 1, P(p) becomes the partition function, and P,,A(1) = dim V\(p) by a formula of Kostant (cf.
[5, pg.421]). More generally, R. Brylinski [2] has shown how to obtain P,u,A(q) from the action of a principal
SL2 in d on the space V\(p).
Assume p < A in P+. Then P, ,\(q) has constant coefficient = 1. In particular, (4.5)
dim VA(p) = 1 = P,,,,\ (q) = 1.
A non-trivial case is due to Lusztig [9, p.226]. Assume G is simple and A is the highest coroot (= the highest weight of the adjoint representation of d).
On the Satake isomorphism
231
Then 0 < A in P+ and (4.6)
qm.-1
Po,A(q) _ 8=1
where m1, M2)) ... , ml are the exponents of G [1, ch.5,§6].
5. Examples
We first treat the case G = PGL2. Then O = SL2(C) and X.(T) _ X'(T) = Z X, where X is the character of the standard representation on C2. If A = nX and p. = mX are elements of P+ (m, n > 0), then A > p if and only if
n>m
(5.1)
{ n-m
(mod 2).
Since dim V\ (p) = 1, we have da (p) = 1 in this case. If we use the traditional notation (5.2)
T,m = char(K("ml)K)
for cµ in 1L we obtain the well-known formula [12, p.73] (5.3)
q2 (TraceSymn(C2)) =s(
T.Tm). 0
and (7.3)
dim(G/Pa) = #{a E 4D : (A, a) < 0}
If A = 0 we find P,, = G; if A is regular we find P,, = B. Let
£:Wa=X'(T)>iW-+ Z be the length function on the extended affine Weyl group, defined in [9, pg.209]. The following is a simple consequence of the Bruhat-Tits decomposition of G [17, 3.3.1].
On the Satake isomorphism
235
Proposition 7.4. For all A in P+, we have: deg(ca) _ E qe(y)/ qe(w) _ #(GIPa)(q) WAW
W
q(a,2n)
gdim(G/Pa
Moreover, A is a minuscule co-weight if and only if
deg(ca) = #(G/PA)(q) It is also known that the Satake parameter of the trivial representation is the conjugacy class s = p(q) = 2p(g1/2) in G(C). Equivalently, if s0 =
(7.5)
g
1/2
q_1/2
in SL2(C)
is the Satake parameter of the trivial representation of PGL2, then s is the image in d(C) of s° in a principal SL2. This gives a check on our various formulas. For example, when G = G2 we found g3X1(s) = a1 + 1
85X2 (s) = a2 + a1 + 1 + q4
On the trivial representation, we find a1 = deg(ca,) = q6 +q5 +q4 +q3 +q2 +q a2 = deg(cA2) = q10 + q9 + q8 + q7 + q6 + q5 Since
Vi = S6(C2) V2 = S10(C2) + S2(C2)
as representations of the principal SL2 in G2, we find g3X1(s) = g3(g3 + q2 + q + 1 + q-1 + q-2 + q-3)
= q6+q5+q4+q3+q2+q+1
1
g5X2(s) = q10+q9+q8+q7+2q6+2q5+2q4+q3+q2+q+1 which checks!
One consequence of Proposition 7.4 is that the degrees of Hecke operators are quite large. For example, if G = E8 and A # 0, deg(ca) > q58.
8. Normalizing the Satake isomorphism One unpleasant, but necessary, feature in the Satake isomorphism is the presence of the irrationalities q1/2. As already noted, these are not needed in the case when p E X* (T). When the derived group G' of G is simplyconnected, we will see how they can be removed by a choice of normalization.
Let Y be the quotient torus GIG'. The exact sequence: (8.1)
1-*G'-- G-*Y->1
Benedict H. Gross
236
induces an exact sequence (8.2)
0 -* X* (Y) -* X'(T) -* Hom(Z , Z) -* 0
Since (a, p) is integral for all coroots, there is a class py in aX' (Y) with
mod X'(T) The class py is well-determined in the quotient group (8.3)
p
choice of representative in
py
ZX'(Y)/X'(Y). A
2X'(Y) will be called a normalization.
Since X'(Y) = X.(Z), where Z is the connected center of G, a normalization py gives us a central element (8.4)
in 2(C) C G(C).
z = pY(q) = 2py(g1/2)
We adopt the convention that the normalized Satake parameter of an unramified representation 7r is given by:
s' = z s(ir) in G(C). Of course, this depends on the choice of py. It has the advantage that the (8.5)
relation between eigenvalues of the Hecke operators c,\ on 7rK and the traces X .\(S') is now algebraic, involving only integral powers of q. Indeed, xa(s') _ q(a,PY)X'\(s), so (8.6)
q(AP-PY)Xa(s)
=
(CAI1rK) + E da(p)(Cµl7rK). µ 1:
V where g(x) E K[x] is a polynomial of degree > 5 with no multiple roots.
B. Mazur
242
V
VT
I t
T
For each t E K-{0}, we get a smooth curve V over K of fixed genus > 1, and therefore by Falting's Theorem, I Vt(K) J is finite. Let us show, as illustration of our method, that (assuming Lang's Conjecture) there is a unform upper
bound B to the cardinalities of V(K) for all t E K - M. On the one hand, thinking of Faltings' Theorem, one might imagine that the rational points are fairly scarce in VT: there are only finitely many Krational points on each fiber. But, of course, VT is a rational variety over K (for any choice of coordinates x, y we may solve for t) and therefore the Krational points are, in fact, all over the place; they are dense in the total space VT. In a word, the rational points, fiber-by-fiber, are not "correlated" (I'm not using the term "correlated" in any rigorous sense). But, as we will see in a moment, Lang's Conjecture implies that these K-rational points satisfy a certain kind of "two-point correlation". To visualize this, consider what we might call the "two-point correlation space" for VT = V XT V, i.e., the family parametrized by T which for each given t E T is the product of V with itself: Vt X Vt
V2 T
T t
The variety VT2 is a threefold, whose equations are:
t y2 = g(x),
t v2 = g(u),
and we immediately see that we have a dominating mapping of VT2 onto the surface W given by the equation
W:z2=g(x).g(u) by sending x H x, u H u, and z H t
v y.
Open problems regarding rational points
243
Now the surface W is of general type, and therefore by Lang's Conjecture,
its K-rational points lie in a "proper" subvariety (in particular, a curve), whose inverse image in the threefold VT2 is therefore a subvariety of dimension _< 2 which contains all the K-rational points of VT2 . In other words, Lang's Conjecture tells us that there are some nontrivial algebraic relations satisfied
by the locus of couples of K-rational points on the same fiber. Let, then, S C VT2 be a surface containing the Zariski closure of the set of K-rational points of VT2 (we may assume that every component of S is two-dimensional, and for simplicity of discussion we will do so).
Step 1: Consider the points t E T for which the entire fiber Vt2 of VT2 is contained in S. There are at most a finite number of such points {t1, t2, ... , It suffices to find an upper bound B to the cardinalities of the sets V (K) for all t E K - {0} other than this finite set of points. Relabel the complement of the finite set {t1, t2, ... , in A' - {0} as T and restrict our family to this parameter space T so that we may now assume that S contains no component which is equal to a fiber of VT T. Step 2: Consider the projection VT2 -+ VT to the first coordinate, and let it : S -4 VT be the restriction of this projection to S. Now suppose that 7r is finite . Then our bound B can be taken to be the "generic" degree of 7r.
For, take a value t E T(K). Then either V has no K-rational points at
all (and then B would be a perfectly good upper bound for the cardinality of V (K)) or else there is a point u E V (K). Fix such a point u, and consider the fiber above u for the projection VT2 -> VT . This fiber is Jul x V and therefore is isomorphic over K to V t. But the set of its K- rational points is contained in S, and therefore B is an upper bound for its cardinality. We are now ready to consider the general situation. Step 3: Consider the locus C C VT consisting of those points for which the mapping 7r: S -3 VT is not finite, and let B1 denote the "generic" degree of 7r when restricted to the inverse image of the complement of C. Let ¢: C -4 T denote the restriction to C of the projection f : VT -4 T. Now the mapping 0 is finite, for if it weren't finite at some point then it would have the property that S contains the entire fiber above that point. Let B2 denote the generic degree of ¢. I claim that the uniform bound B that we seek may be taken to be the maximum of B1 and B2. For let t E T(K). If the cardinality of Vt(K)
is < B2, then we are done. If it is > B2 then there is a K-rational point u E V which does not lie in C, and hence the argument of Step 2 bounds the cardinality of V(K) by B1.
3. Finer uniformity? If we knew the Lang locus of the surface W of general type given by the equation z2 = g(x) g(u), finer information could be obtained for the example
B. Mazur
244
t y2 = g(x) of the previous section. There are, of course, the "evident" curves of genus 0 lying on W, i.e.,
x = u, z = g(x);
x=u,z=-g(x); x = x0,
for x0 E k such that g(xo) = 0;
u = uo,
for uo E K such that g(uo) = 0.
If VT -4 T is a family defined over K for which these curves comprise the full Lang locus of W, i.e., for which these are all the curves of genus < 1 lying on W, then the above argument gives the following more stringent uniformity result (conditional, of course, on Lang's Conjecture): For every number field extension L/K there is a finite set of curves defined over L,
C(L)={CC;i=1,...,m} such that for all t E L - {0} if V is not isomorphic over L to a curve in C(L) then there is at most one antipodal pair of non-Weierstrass L-rational points in V (L). Here "antipodal" means that the pair of points is stable under the hyperelliptic involution of V t.
But the above remark is of limited use at present, because to my knowledge, for no polynomial g(x) has the Lang locus of the corresponding surface W (of the construction above) been computed. Can one compute some example?
4. The general "correlation theorem" The illustrative case we have just discussed might suggest to us that we could put the following theorem to good use.
THE FIBERED POWER THEOREM. Let T be an irreducible variety over C, and let f : VT -* T be a smooth proper family of varieties of general type, and let fN:VTN -* T denote the N-th fiber power of VT over T. Then there exists a positive integer N such that VTN dominates a variety W of general type (by a rational mapping, 0: VTN- --> W).
ADDENDA. If the family VT/T is of "maximal variation" then VTN is itself of general type. If [the fibers are curves, and if] VT -4 T is defined over
Open problems regarding rational points
a number field K, then W and the rational mapping 0 : VTN taken to be defined over K.
245
W may be
Remarks. This theorem was proved in the case of fiber dimension one in [C-H-M 1] and was conjectured to be true in general, in that paper. Brendan Hassett proved this in case the fiber dimension is two [H] and Abramovich [A 2] has recently shown the theorem in full generality (see also his earlier note: [A 1]). We might mention in passing that the bracketed requirement in the second Addendum above (that the fibers are curves) is not necessary if one appeals to the proof of the theorem in [A 2]. Here are a few remarks which may evoke something of the flavor of the proof. First, although we have assumed that our family is smooth in the statement of the theorem, in the actual proof of the theorem we must compactify VT and therefore must deal with possible singularities. Second, let us loosely use the word "positive" to mean "having many pluricanonical sections" and
the word "negative" to mean "not necessarily having many pluricanonical sections". Our goal, of course, is to show that a desingularization of a compactification of some fiber power VTN is "positive". Now the base, T, is "negative" (which may hinder us in our goal). The fibers of VT -4 T are, of course, "positive" (which will help us) and therefore, the idea of increasing N sounds like a good idea to counteract the possibly "negativity" of the base. A problem which one encounters, however, is that as N gets larger the singularities of the compactification will also get more complicated, and this is a hindrance. In a word, it is a race (as N goes to infinity) between the good effect of the high power of the fiber and the bad effect of the complicating singularities. Happily, the good effect wins out. In the proof for fiber dimension one in [C-H-M 1] we make strong use of the stable compactification of families of curves, which have very mild controllable singularities. In the proof for families of surfaces due to Hassett, again it is the known structure of the moduli space for stable surfaces of general type that is relied on. In the proof of the full theorem due to Abramovich, first a general positivity result of Kollar is used, and the idea of de Jong's alterations [dJ] is adapted to this situation to control the problem due to singularities (Abramovich finds a Galois cover Y -+ VT where Y is a composition of curves with at worst nodal singularities).
5. Uniformity statements for the number of rational point on curves I hope the "illustrative example" of section 2 is sufficient to suggest that the theorem given in the previous section can be used to show that Lang's Conjecture implies the existence of uniform upper bounds for the numbers of rational points of certain varieties, as these varieties vary in algebraic families.
Restricting attention to curves, at the moment, Lang's Conjecture implies
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B. Mazur
the three statements which I have labelled A, B, and C below. In these statements, g will denote a number > 1, and "curve" will mean smooth proper curve.
A. There is a number N(g) with the following property. For each number field K, there is a finite set C(g, K) of curves of genus g defined over K such
that if C is any curve of genus g defined over K with more than N(g) Krational points, then C is isomorphic over K to one of the curves in C(g, K). Call the curves in C(g, K) "K- exceptional curves of genus g".
Questions. Assume A to be true, i.e., that there is such a number N(g). Let us take N(g) to be the smallest positive number for which A is true, and let us take C(g, K) to be the corresponding sets of "K- exceptional" curves.
1) Find lower bounds for N(g). To show N(g) > some number B you need only find some fixed number field K and an infinity of K- isomorphism classes of curves over K of genus g each of which possesses more than B points rational over K. In this direction, both Mestre and Brumer have shown (by independent constructions) that N(g) > 16(g+1) for all g > 1. For particular small g, i.e., g = 2, 3, one has slightly better lower estimates for N(g).
2) Does the number of K-exceptional curves of genus g depend only on (g and) the degree of K over Q? For the proof that Lang's Conjecture implies A see [C-H-M 1]; it uses the fiber power theorem only for fiber dimension equal to one. Statement A implies (in conjunction with Faltings' Theorem applied to each of the Kexceptional curves of genus g) that there is a uniform upper bound for the number of K-rational points of any curve of genus g defined over K. By an intricate (and elegant) argument, Pacelli has shown (in [P]) that Lang's Conjecture implies such a uniformity, where the bound depends only upon the degree d of K (see also [A-V]). Explicitly, Lang's Conjecture implies:
B. (Pacelli): For integers d > 1 and g > 1 there is a number N(d, g) with the following property. Given any number field K of degree < d, and curve C of genus g defined over K, the number of K-rational points on C is < N(d, g). To prove this result, Pacelli studies the loci (Fn, in her terminology) given by the Zariski closure of the set of what she calls (indefinitely) "prolongable" points in the symmetric d-the power of the n-th fiber power of families VT -4 T of curves of genus g > 1. In an appropriate context she shows that if n is appropriately large, the irreducible components of maximal dimension in Fn are of general type. Since, by construction, these Fn have Zariski dense sets of points rational over a fixed number field, Lang's conjecture implies that Fn is empty, giving the uniformity that is sought.
Open problems regarding rational points
247
Remark and further questions. With Pacelli's statement B in mind, let us return to the question we had raised when we first formulated Lang's Conjecture in section 1. Namely, given a variety of general type V with Lang
locus Z, how does the cardinality of the set of K-rational points of V - Z vary with K? More specifically, is there an upper bound for this cardinality that depends only on the degree of K. Statement B is an affirmation of this, in the case where V is a variety of general type of dimension 1. Returning to our original question we might wonder whether the correlation- theoretic techniques we have been discussing, suitably strengthened, may be used to show that Lang's Conjecture "improves itself' for varieties of general type of all dimensions, in the sense that it implies that the set of K-rational points of V - Z depends only on the degree of K ( where Z is the Lang locus of V) for V varying through a "bounded" family of varieties of general type over K. In this connection, see [A-V] Theorems 1.5-1.7. If C is a curve defined over the field of complex numbers, by the gonality of C let us mean the minimal degree S of any nonconstant function on C, i.e., it is the minimal degree of any nonconstant mapping from the Riemann surface C to P1. We refer the reader to [A 3] and to [Y], [L-Y] for discussion and results regarding gonality of modular curves. Recall that if V is a variety over K, a point of degree < d on VIK is an L-rational point of V where L/K is some field extension of degree < d.
Using the correlation theorem that Abramovich proved (i.e., for all fiber dimensions), Abramovich showed that Lang's Conjecture implies:
C. (Abramovich): For integers d, e > 1 and g > 1 there is a number N(d, e, g) with the following property. Given any number field K of degree < d, and curve C defined over K, of genus g and of gonality > 2e, the set of points of C of degree < e over K is less than or equal to N(d, e, g).
6. Counting K-rational isogenies If E is an elliptic curve over K, by a K-isogeny one means a surjective homomorphism E -4 E' where E' is an elliptic curve over K and the homomorphism is defined over K. Let N(K, E) denote the cardinality of the set of j-invariants of all the elliptic curves E' which are K-isogenous to E. It is known, for example, by a result of Kenku that for any elliptic curve E over Q, N(Q, E) < 8. It is also easy to see that the (maximum) value N(Q, E) = 8 is reached infinitely often: we have N(Q, E) = 8, for example, for any elliptic curve E over Q which has no complex multiplications (over C) and which admits a Q-rational isogeny whose kernel is cyclic of order 12; there are infinitely many such E's since Xo(12) is of genus 0.
B. Mazur
248
Proposition 1. Statement A of section 5 (and therefore, also, Lang's Conjecture) implies that there is a "universal" natural number N such that for any number field K, the set
_
the set of j-invariants of elliptic curves
`7(K)' - (E over K for which N(K, E) > N
)
is finite.
Remarks and further questions. 1. Can one take N = 20 in Proposition 1? Without appeal to Lang's Conjecture, finiteness of ,7(K) is not yet known. Is there an upper bound for the cardinality of J(K) which depends only upon the degree of K? 2. A related question, however, replacing K-rational isogenies by Krational torsion points is now known, by the recent work of Merel [Me]; namely, there is an integer M which has the property that for all num-
ber fields K the set of K-isomorphism classes of elliptic curves defined over
K with Mordell-Weil torsion (over K) of order > M is finite. One can, in fact, take M = 18. Lang's Conjecture implies the further fact that the cardinality of this finite set of K-isomorphism classes of elliptic curves with large Mordell-Weil torsion admits an upper bound depending only on the degree of K. Explicitly,
Proposition 2.
Let d > 1 be an integer. Lang's Conjecture implies that there is a finite bound M(d) such that for any number field K of degree d the number of K-isomorphism classes of elliptic curves having Mordell-Well
torsion (over K) of order > 18 is < M(d). Proof. The number M = 18 is chosen to guarantee that the modular curves that parametrize elliptic curves with rational torsion of order > M are all of genus > 1. Merel (cf [Me]) has shown that the order of K-rational torsion in the Mordell-Weil group of any elliptic curve over K is bounded by a quantity U(d) which depends only on the degree d. It follows that the "exceptions", i.e., the set of K-isomorphism classes of elliptic curves with Mordell-Weil torsion
(over K) of order > M is of cardinality no greater than the cardinality of the set of K-rational points on the disjoint union of the following set S(d) of modular curves: For integers m1i m2, denote by X (Ml, m2) the modular curve which classifies pairs (F, a) where F is an elliptic curve, and a is an injective homomorphism of the product of two cyclic groups Z/mi Z x Z/m2 Z into F. Now let S(d) denote the disjoint union of these curves X (MI, m2) where (ml, m2) runs through all pairs of integers with 1 < m1 m2 and such that M < m1 m2 < U(d). Clearly, the cardinality of S(d) admits a finite upper bound depending only on d, and the genus of any curve which is a member of S(d) is > 1 and admits a finite upper bound depending only on d. It follows
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(from Pacelli's Theorem quoted above) , using Lang's Conjecture, that the total number of K-rational points on the disjoint union of the curves in S(d) admits a finite upper bound depending only on d. Proposition 1 will be proved in section 10 below.
7. The modular curves X (E; p) Let p be a prime number. Our proof of Proposition 1 rests on the properties of certain models (over number fields K) of the modular curves X (p)lc for p > 7. Recall that X (p) is the Riemann surface which is the compactification of
Y(p): = (upper half-plane)/F(p)
where r(p) C PSL2(Z) is the kernel of the projection PSL2(Z) to PSL2(Fp)
and the action of r(p) is via the natural action of PSL2(Z) on the upper half-plane. The group PSL2(Fr) then acts naturally on X(p). The quotient space of this action is the projective line P' which we view as parametrized by the elliptic modular function j: (upper half-plane) -+ P1
and we denote by the same letter j the projection
j:X(p) -*P1. The genus gp of X (p), for p > 7, satisfies the formula
84 (gp - 1) _ IPSL2(Fp)I . (7 -
p) =
2(p2
- 1) (7p - 42)
as can be computed from Prop. 1.40 of [Sh] and is therefore > 1.
The full automorphism group of X(p) for p > 7 is PSL2(Fp) and the following paraphrase of this will be particularly useful to us: Proposition. If p > 7, any automorphism of X(p) preserves j.
The appendix below contains an elegant argument due to Serre which proves this Proposition (as Serre remarked, this result is very likely in the literature, but it seems difficult to find an adequate reference for it). For p > 7, and any elliptic curve E over a number field K there is an affine curve Y(E; p) defined over K and its smooth complete model X (E; p) whose underlying Riemann surfaces are isomorphic to the disjoint union of p - 1 copies of Y(p) and of X (p) respectively. For any scheme U over K the U- valued points of Y(E; p) are naturally identified with isomorphism classes of pairs (FLU, a) where Flu is an elliptic curve over U (i.e., an abelian scheme over U of dimension 1) and where
a: E[p]lu -- F[p]
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is an isomorphism of finite flat group schemes over U. Here E[p]lu is the pullback to U of the kernel of multiplication by p, viewed as endomorphism of E and F[p] is the kernel of multiplication by p, viewed as endomorphism of Flu. The above property characterizes the curves Y(E; p) C X (E; p) over K.
Let us take a moment to discuss the fact that the curve X (E; p) is not connected. To any point x E X (E; p) over a K-scheme U, we may associate an element of FP as follows: the Weil pairing identifies the "wedge-squares" over Fr of E[p]lu and of F[p]lu with the group scheme µp over U. Thus A2(a) induces an automorphism A2(a): pp -+ pp,
and since the automorphism group of up is canonically FP, A2 (a) identifies with a well-defined element (call it 8(x)) in F;. For each element u E Fp, there is a (geometrically irreducible) component X (E; p) u of X (E; p) characterized by the property that if x E X (E; p)u, then 8(x) = u. The components X (E; p) u are each defined over K and are isomorphic over C to X (p). The multiplicative group FP acts on X (E; p) by the following rule. If A E F; then A sends the pair (FLU, a) representing a U-valued point in X (E; p) to (Flu, A a). Clearly, then, A sends the component X (E; p) u to X (E; p)a2.u. It follows that among the p - 1 components X (E; p) u (as u ranges through FP*) there are at most two distinct K-isomorphism classes of geometrically irreducible curves (of genus gp) represented, e.g., X (E; p)1 and X (E; p),,, for w a choice of quadratic nonresidue modulo p. The curves X (E; p) 1 and X (E; p).
are twists of one another over K and become isomorphic over the splitting field K(E;p) of the K-Galois module E[p]. Moreover, the curve X (E; p)1 has at least one K-rational point (namely the point represented by the pair (E[p]K, a) where a is the identity mapping.
8. Digression: When are the different components of X (E; p) isomorphic over K? The results of this sections is an account of some conversations with Brian Conrad and Richard Taylor and I am grateful for their permission to include it in this article. Since we have isomorphisms over K between X (E; p)v and X (E; for any u, V E F;, to answer the question in the title of this section it suffices, for EIK an elliptic curve, and p an odd prime, to fix a quadratic non-residue mod p, w E FP and to give "necessary and sufficient" criteria for X(E;p)1 and X (E; p),,, to be isomorphic curves over K. First note that for every prime .£ # p we have a natural "Hecke operator", Tj: X (E; p)v - - -* X (E; p)ve,
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i.e., it is the correspondence of degree £+ 1 going from the curve X (E; p)v over K to the curve X (E; p)ve and characterized by the following property. Let x
be a K-valued point of X (E; p)v. Suppose that x corresponds to the isomorphism class of pairs (FR, a) where FK is an elliptic curve over K and where a: E[p]K -> F[p] is an isomorphism of determinant v E F. For every cyclic subgroup C C F of order t, consider the K-valued point of X (E; p)ve corresponding to the isomorphism class (F/C, ac) where ac: E[p]K -+ (F/C)[p] is the composition of a with the isogeny F -* F/C. Since the determinant of ac is equal to vf, the point xC lies on X (E; p)v,. We have the formula TE(x) = E xC. C
Proposition 1. (the genus zero cases) Suppose that p = 3, or 5. The curves X (E; p)v are isomorphic to P1 over K for all v E F;. Proof: Since p = 3, or 5 the curves X (E; p)v are of genus 0, and since X (E; p)1
has a K-rational point it follows that X(E;p)1 is isomorphic to P1 over K. Since 2 is a quadratic non-residue mod 3 and mod 5, to prove the proposition it suffices to show that X (E; p)v, is isomorphic to P1 for w equal to 2 mod p (p = 3, 5). But the Hecke correspondence T2 is a K-rational correspondence of degree 3 from X (E; p)1 to X (E; p)2. The image of any K-rational point
on X (E; p)1 is a(n effective) K-rational divisor of degree 3 on the genus zero curve X (E; p)2. Since X (E; p)1 has a K-rational point, it follows that X (E; p)2 has a K-rational divisor of odd degree, and is therefore isomorphic to P1 as was to be proved. Here is the answer to the title question in this section.
Proposition 2.
Let E be an elliptic curve over K and p > 7 a prime
number. Denote by
p: Gal(K/K) -* Aut(E[p]) ?' GL2(Fp) the natural continuous Galois representation on p-torsion points of E. Then the curves X (E; p)1 and X (E; p)v, are isomorphic over K if and only if either: a. the image of p is contained in a Cartan subgroup of GL2(Fp);
b. p - -1 mod 4 and the image of p is contained in the normalizer of a (split) Cartan subgroup;
c. p - 1 mod 4 and the image of p is contained in the normalizer of a nonsplit Cartan subgroup.
Proof: First consider the set Isomer(X(E;p)1iX(E;p)v,) of isomorphisms X (E; p)1 -> X (E; p)v, over C, which may be viewed as a torsor over the
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group Autc(X(E; p)1). Identifying X (E; p)1 (taken over C) with the Riemann surface X (p) and applying the Proposition of section 7 above, we see that Isomc (X(E;p)1iX(E;p)w) is naturally a torsor over PSL2(Fp). It follows that any isomorphism t: X (E; p) 1 -f X (E; p)w may be given as follows. There is an automorphism r: E[p] -* E[p] of determinant w such that
t(F, a) = (F, a r). Moreover such an automorphism r is unique up to multiplication by ±1. The correspondence t H {±r} gives us a natural bijection of the sets: Isomo(X(E; p)1i X (E; p)w) - Aut(E[p]),,,/{±1}, where Aut(E[p])w = GL2(Fp)w denotes the SL2(Fp)-coset of automorphisms of determinant w. One easily computes the necessary and sufficient condition for such an isomorphism in Isomer (X (E; p)1i X (E; p)w) to be defined over K. Namely, it is necessary and sufficient that conjugation by elements of p stabilize the set {±r}. Consequently, there exist an isomorphism X (E; p) 1 X(E;p)w over K if and only if there exists an element r E GL2(Fp) whose determinant is a nonquadratic residue mod p, and such that the image of p stabilizes the two-element set {±r}. Such an element r, whose determinant w is a nonquadratic residue mod p, is semi-simple. At this point it suffices to list cases. First suppose that r is diagonalizable.
Then after multiplication by a scalar we may take {±r} to be the set of diagonal (2 x 2) matrices with diagonal elements {±[l, w]}; the group of elements stabilizing this set (under conjugation) is either the group of diagonal
matrices (in which case we are in case a ) or the normalizer of the group of diagonal matrices if w = -1. But w = -1 is allowed as a possibility only if -1 is a nonresidue, i.e. if we are in case b. of the Proposition. Next, suppose that r is not diagonalizable, in which case the Fp-algebra E generated by r in the matrix algebra M2(Fp) is a maximal commutative subfield, and the
hypothesis that the image of p stabilizes the two-element set {±r} C E (under the action of conjugation) implies that the image of p is contained in the normalizer of the corresponding (nonsplit) Cartan subgroup. Again, we have the possibility that the image of p is contained in the (nonsplit) Cartan subgroup itself (in which case we are in case a ) or there is an element g
in the image of p such that grg-1 = -r. Since conjugation by g is a field automorphism of E it follows that u: = r2 E Fp, and since E is a field, we know that u E Fp is a quadratic non-residue. Since w = det r = -u2 is also a quadratic nonresidue, we have that -1 is a quadratic residue, i.e., we are in case c. This establishes the Proposition.
Idle Question. Are there occasions when X(E;p)i is not isomorphic to X (E; p)w over K, and yet the jacobians of these two curves are isomorphic
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over K? Note that we have a good number of homomorphisms defined over
K between these jacobians: choose any prime number 2 = w mod p and consider the homomorphism induced from the Hecke correspondence TL.
9. The isomorphism class of the curve X (E; p) over K Fix the prime p > 7, and the elliptic curve E over K. Note that twisting E by a quadratic character X over K does not change the isomorphism class of X (E; p) over K; i.e., there is a canonical isomorphism
X(E;p)=X(E®X;p) given by the rule
[F, a]'-* [F®X,a®X]. Also, if [F, a] represents a K-rational point of X (E; p), and if the determinant of a is u E FP then for each w E FP there is a canonical isomorphism over K X (F; p)w = X (E; p)u.
Lemma.
.
Let u E F. Let E be an elliptic curve over K such that there is
some K-rational point of X (E; p)u represented by a pair [F, a] where F is an elliptic curve without complex multiplication such that j (F) # 0, 1728). Let E' be an elliptic curve and u' an element in F* such that (i.e.,
X (E;p)u ' X (E';p)u,
Then there is a quadratic character X over K and a Gal(K/K)-equivariant isomorphism 0: E[p] -4 E'[p] 0 X such that the pair [E'(9) X, a'] represents a K-rational point of X (E; p)u.u,-1. X (E'; p)u, be the K-isomorphism, and let [F', a'] be the image of [F, a] under h. By the previous discussion, we have that j (F) = j (F') and since these j-invariants are distinct from 0 and 1728 we Proof. Let h: X (E; p)u -
have that F is a twist of F by a (quadratic) character over K. Let X be this character. Since a: E[p] -* F[p) and a': E'[p] -+ F'[p] = F ® x[p] are Gal(K/K)-equivariant isomorphisms of determinants u and u' respectively, if we denote by a' again the induced isomorphism E'(9 X[p] -* F[p], we see
that
/3=a'-1 a does what we want.
Corollary. Let p > 7, u E F*, and E an elliptic curve over K such that there is some K-rational point of X (E; p)u represented by a pair [F, a] where F is an elliptic curve without complex multiplication. Then the set of j-invariants
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254
of elliptic curves E' over K for which there is an element in u' E FP such that X (E; p)u = X (E'; p)ui is finite.
Proof. By the previous lemma, any E' having the above property must be a quadratic twist of some elliptic curve representing some K-rational point on X (E; p). Since p > 7, the curve X (E; p) is a finite union of curves each of genus > 1 and therefore has only a finite number of K-rational points by Faltings' Theorem.
10. Proof of Proposition 1 Let p be a prime number > 7. Any K-isogeny 0: E -* E' of degree prime to p gives rise to a K-rational point [0] = (E', 0: E[p] -> E'[p]) E Y(E; p).
More precisely, if the degree of the K-isogeny ¢: E -+ E' is 6 (assumed prime to p) then the point [0] lies in the component Y(E; p)u C X (E; p)u where u is the reduction of S modulo p. Let N(K, E, p) denote the number of j-invariants of elliptic curves which are K-isogenous to E via K-isogenies of degree prime to p. Since the set
of these j-invariants is contained in the image (in the j-line) of the set Y(E; p) (K) (under the natural projection (E', 0) H j(E')) we have: N(K, E, p) < card{Y(E; p)(K)} < card{X(E; p) (K)}. To prove Proposition 1 let us assume Statement A of section 5. Let N(g) be the number given to us in Statement A for the genus g > 1. Put
M(g): = (p - 1) max(3. IPSL2(Fp)l, N(gp))
Choose any two distinct prime numbers pl and P2 both > 7 ( pi = 7 and p2 = 11 will do). We will prove Proposition 1 with N = M(gpl) - M(gp2) If we are given any K- isogeny ¢: E -* E' of degree d, writing d = with do relatively prime to pi we may factor the isogeny 0 as in the commutative diagram of K-isogenies below, where the arrows are labelled by their degrees,
and in particular, the horizontal isogenies are of degree prime to pl and the vertical ones are of degree prime to p2:
E d4 E' Pa
Yd
1pi
El do E'
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Since the isogeny E -+ E' is determined by the pair of isogenies E -* El and
E -* Ei, we get that N(K, E) < N(K, E, p1) N(K, E, p2). It follows then that if N(K, E) > M(gp,) M(gp2) we must have either N(K, E, pl) > M(gp, ) or N(K, E, P2) > M(gp2), i.e., we have that one of the curves X (E, p)u (for p = pl or P2, and for some u) has the property that
card{X(E,p)u(K)} > max(3. IPSL2(Fr)I,N(gp)) It follows from this inequality that a. The curve X (E, p)u is a "K-exceptional curve of genus gp", and
b. The set Y(E, p)u(K) contains a point whose associated elliptic curve has j-invariant different from 0 and 1728. Statement A together with the corollary above guarantee that there are at most a finite number of j-invariants of elliptic curves satisfying a) and b), which concludes the proof of Proposition 1.
Appendix: The automorphism group of X (p). (Copied from a letter of J.-P. Serre, June 26, 1996)
Let p be a prime number. Let X (p) denote the quotient of the extended upper half plane H* under the action of the congruence subgroup r(p) = ker(PSL2(Z) -3 PSL2(Fp)). Let G = PSL2(Fp) which acts faithfully on the Riemann surface X (p). Denote by A the group of all automorphisms of X (p) and put m = [A : G]. The genus gp of X (p), for p > 7, satisfies the formula
84. (gp - 1) = IPSL2(Fp)I (7 - 42/p),
and therefore, by the well known Hurwitz bound for the order of the automorphism group of a Riemann surface of given genus, we have m < 7. Lemma. The subgroup G is normal in A. Proof.- The natural action of G on A/G via left multiplication is trivial. This can be seen by first noting that any element s E G of order p acts trivially on
A/G (since p > m). It then follows that the action of (all of) G on A/G is trivial, since G is generated by its elements of order p. Therefore G is normal in A. If e = 2, 3, or p, call Xe the finite subset of points of X = X (p) which have ramification index e under the mapping X -> X/G = P1. Any element t E A
normalizes G and therefore stabilizes the subsets Xe for e = 2, 3, or p and also induces an automorphism t' of P1. This automorphism fixes the three image points of the sets X. (e = 2, 3, and p) and is therefore the identity on P1. It follows that A acts trivially on X/G and consequently A = G.
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PART II. Speculations about the topology of rational points: a further up-date 1. The topological closure of the set of rational points in the real locus A few years ago I formulated some conjectures about the topology of rational points [Ma 1,2,3] the original motivation for them being to examine the ques-
tion of whether or not "Z is Diophantinely definable in Q". That question had been raised by mathematical logicians because an affirmative answer to it would imply (using Matjasevic's work) the non-existence of any algorithmic solution to to the problem of determining if a polynomial in many variables with rational coefficients has a rational solution. Here is the list of those conjectures:
XConjecture 1.
Let V be a smooth variety over Q such that V(Q) is Zariski-dense. Then the topological closure of V (Q) in V (R) is equal to a union of connected components in V(R).
Conjecture 2. Let V be any variety over Q. Then the topological closure of V (Q) in V (R) is homeomorphic to a finite simplical complex.
Conjecture 3. Let V be any variety over Q. Then the topological closure of V (Q) in V (R) possesses at most a finite number of connected components.
Conjecture 4. Z is not Diophantinely definable in Q. Each of the conjectures in the above list implies the subsequent ones.
The reason for the "X" in front of Conjecture 1 above is that recently, a counter-example to it was constructed by Colliot-Thelene, Skorobogatov, and Swinnerton-Dyer [C-T, S, S-D]. In the light of their counter-example, they "repair" XConjecture 1 by making the following modification of it:
Conjecture A (of C-T, S and S-D): Let V be a smooth integral variety over Q and U a connected component of the real locus V (R) such that v (Q) f1 U is Zariski-dense in V. Then V (Q) fl U is topologically dense in U.
Remark. This is Conjecture 5 in the article [C-T, S, and S-D]; the authors also formulate a strengthened version of it (Conjecture 4 of [C-T, S, and S-D]) as follows:
Conjecture B (of C-T, S, and S-D): Let V be a smooth integral variety over Q and U a connected component of the real locus V(R). If W is the topological closure of V (Q) fl U in U, then there is a Zariski-closed set Y C V
defined over Q such that W is a (finite) union of connected components of Y(R).
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Either of these Conjectures of C-T, S and S-D imply Conjectures 2,3 and 4 above; and they both avoid the (counter)-examples that C-T, S and S-D construct. The formulation of the "repaired conjectures" by C-T, S and SD focusses on a phenomenon illustrated by their counter-examples; namely, that there can be a good deal of "possible variation" in the Zariski-closures of V (Q) fl U as U ranges through the connected components of the real locus of V. How much variation is there in the Zariski-closures of V(Q) fl U? To focus
on this, I found it useful to formulate Question I below, an affirmative or negative solution of which would be interesting. First, an affirmative answer
to Question I still implies Conjecture 4. Second, Question I concerns the topology of K-rational points of a variety V defined over K, where K is any number field (not just Q) and concerns the placement of these K-rational points in the topological space of S-adic points of V where S is any finite set of places (not just a real archimedean prime) of K.
2. S-adic topological closure Let V be any variety defined over a number field K. Let S be a finite set of places of K, and consider
Ks=[JK VES
viewed as locally compact topological ring. Let V(Ks) denote the topolog-
ical space of Ks-rational points of K. For every point p E V(Ks) define W (p) C V to be the subvariety defined over K which is the intersection of the Zariski-closures of the subsets V (K) fl U where U ranges through all open
neighborhoods of p in V(Ks). By the noetherian property of V, W(p) is, in fact, the Zariski-closure of V(K) fl U for some open neighborhood of p which is "small enough". Question I. As p ranges through the points of V (Ks) are there only a finite number of distinct subvarieties W (p)?
Proposition. An affirmative answer to Question I implies Conjecture 4. Proof. If Conjecture 4 is false, then there is a variety V over Q and a (Q-) rational function f on V such that the image of V (Q) under f is the subset
Z C Q. For each N E Z choose a rational point PN E V (Q) such that f (PN) = N and note that W (pN) is a nonempty subvariety contained in the fiber off over N E Q; the existence of such an f thus implies that there are an infinite number of distinct W (p)'s and therefore provides an instance where Question I has a negative answer. More generally, an affirmative answer to Question I implies that there is no rational function f such that f(V(Q)) is an infinite subset of Z C Q.
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I will recall the construction of [C-T, S, S-D] below. Their construction exhibits certain varieties V over Q possessing a number of distinct Zariskiclosures of v(Q) n u for distinct connected components U C V(R).
3. Isotrivial families that are trivialized by finite etale extensions of the base The basic idea of [C-T, S, S-D] is to examine isotrivial families VT of varieties
over a parameter space T which is a smooth projective variety, the family becoming trivial over a finite etale cover of T. The variety V of the previous paragraph is the "total space" VT. More specifically, T will be taken to be an elliptic curve E and the finite etale cover of T which trivializes the family will be given by an isogeny E' -4 E. We now prepare for the construction.
Let E' and E be elliptic curves over K, and let ir: E' -+ E be an isogeny of degree two defined over a number field K. The kernel of the isogeny 7r is a cyclic group of order two which we denote by C = {1, c} where c is a point of
order two in E'(K). We shall ignore the group structures of E' and E for a moment and think of E' as C-torsor over the curve E. As such it is classified by an element, denote it ry, in the etale cohomology group H' (E, C). For any K-algebra R and any R-valued point e E E(R) the isogeny it determines a C-torsor Te = 7r-1(e) whose associated cohomology class rye in the etale cohomology group H'(Spec(R), C) is simply the pullback of -y via the section
Alternatively, we might form the exact sequence of sheaves of abelian groups for the etale topology, e.
O-+ C-+ E'-4 E-4 O, and form the corresponding long exact sequence of cohomology over Spec(R),
0 -* H°(Spec(R), C) -+ E'(R) -> E(R) -> H'(Spec(R), C), and think of rye as the image of e under the coboundary mapping
6: E(R) - Hl (Spec(R), C). From this we see that the C-torsor Te classified by 'ye depends only on the coset of E(R) modulo the image of E'(R) and, in fact, the set of isomorphism classes
6(R) of C-torsors "Te obtained from this process as e ranges through the elements of E(R) is in natural one-one correspondence with E(R)/7rE'(R).
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4. Comparing global to local First we consider the global context, and take R = K, a number field. We have
£(K) = E(K)/irE'(K) and since K is a number field, £(K), which is a quotient of E(K)/2 E(K), is a finite abelian group of exponent 2. Now let S be a finite set of places (possibly including archimedean places) of K. We will be interested in the locally compact topological space (group, in fact) E(Ks) = II VES
Put R = Ks so that £(R) is again a finite group of exponent 2. As e ranges through the compact group E(R) = E(Ks), the isomorphism class of the C-torsor Te is "locally constant" and is constant "precisely" on the cosets of E(Ks) modulo the open subgroup of finite index 7r{E'(Ks)} C E(Ks). It is easy to see that one can find finite sets of places S for which the restriction of the natural mapping
£(K) -+£(K5) is injective. When S is such a finite set of places, let us simply say that S is
large enough.
5. Twisting varieties by it: Let Y/K be a variety, equipped with a K-rational involution t. We view the group C as acting on Y (in a K-rational way) by having the generator of C act on Y as the involution t. The particular case of this that is used in [C-T, S, S-D] is when Y is an elliptic curve over K and the involution t is given by y
-y. We will twist the "constant" family over E,
YxE -E,
(1)
by the isogeny it, to get an isotrivial family which we will denote
YRE- E
(2)
Explicitly, Y x E is obtained by passing to the quotient of the product Y x E' by the diagonal action of C; let
u:YxE'-+YxE
(3)
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260
denote this quotient morphism. Then u is an etale morphism of degree two; we may view Y x E' as a C-torsor over Yx- E . Given an R-valued point e of E(R) the fiber Ye of the twisted family (2) over the point e is the variety over R obtained by twisting the pullback of Y over R via that cohomology class in H'(Spec(R), Aut(Y/R)) which is the image of rye under the natural homomorphism
H'(Spec(R), C) -* H1(Spec(R), Aut(Y/R)).
In the case where R = K is our number field, the K-rational points of Ye admit the following somewhat more explicit description. Let e' E E'(K) be a choice of one of the two inverse images of e E E(K) C E(K), the other choice being e' + c. Then Y, (k), the set of K-rational points of Ye, may be naturally identified with the set of couples {(y, e'), (t (y), e' + c)}
(4)
of points in Y(K) x E'(K), this identification being equivariant with respect to Galois action. The above discussion gives rise to the following straightforward Proposition 1. (Topological local constancy of fibers) If S is large enough (in the sense of section 2) then for points e E E(K) the K-isomorphism class of the variety Ye (i.e., the fiber of (2) over the point e) is determined by the coset modulo the open subgroup 7rE'(Ks) C E(Ks) of the image of e under the inclusion E(K) C E(Ks).
To get a more precise picture of the placement of K-rational points in Y x E note that for every e E E(K) we have a degree-two covering ue: Ye x
E' -* Yx- E defined over K, which on K-rational points sends (ye7e') E Ye x E'(K) to the image of either of the points (y, e'+ e') or (t(y), Z+ e+ c) under the mapping u: Y x E' -+ Y x E, where ye is represented by the couple
{(y, e'), (t(y), e' + c)} as in (4) above. We have a cartesian diagram of Krational morphisms,
YexE'
Yx- E
lr'I
E
Tea
I E
where ir' is the composition of projection to the second factor and the isogeny 7r; the unmarked vertical morphism is the natural projection; and Te is trans-
lation by e, i.e., Te(y) = y + e. We see that the fiber of Y<E -4 E over any point e+7r(e') in the coset e+irE'(K) is isomorphic over K to Ye (and, more precisely, is the image Ye x {e'} under the mapping ue). of coset representatives of E(K) modulo Now choose a system 7rE'(K). The above discussion shows
Open problems regarding rational points
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Proposition 2. (Topological local constancy of the set of K-rational points) If S is large enough (in the sense of section 2) the set (YxE)(K) of Krational points of the family YxE breaks up into the disjoint union of v subsets, the i-th subset (i = 1, ... , v) being equal to the image under the morphism ue: of the set Yey(K) x E'(K), of K-rational points of Yes x E'. Under the natural projection YxE -* E, this "i-th" subset maps into the coset of irE'(K) containing ei in E(K). We now examine the topological closure of (Y X_ E) (K) in the topological
space (Y x E) (Ks) when S is a finite set of places of K, which is "large enough" (which means that each i = 1, ... , v also determines a distinct coset of E(Ks) modulo 7rE'(Ks); we refer to this coset as the "i-th coset" of E(Ks) modulo 7rE'(Ks). Of course, it is not necessarily the case that all cosets of E(Ks) modulo 7rE'(Ks) are included in this enumeration. Let Ui C (YxE)(Ks) be the inverse image of the i-th coset of E(Ks) modulo 7rE'(Ks) under the natural mapping
(Y<E)(Ks) -4 E(Ks), so that each Ui is an open compact subset of the compact topological space (Y x E) (Ks). Let Wi C Yey denote the Zariski closure of the subset of Krational points Ye;(K) in Ye,. By Proposition 2, the subset of K-rational points of our family Y x E whose image in (Y x E) (Ks) is contained in Ui is equal to ue,(Wi(K) x E'(K)) C Ui.
In particular, for each of the Ui, the Zariski-closure of the set of K-rational
points of YxE which are contained in Ui is equal to Zi:= uey(Wi x E'). The fun, then, is to find examples where the set of K-rational points of Yey, and hence also the Zariski closures Wi, differ substantially - differ even in dimension - for distinct i's. In diagram 1 overleaf we are schematically
envisioning a case where v = 2, W, consists of four points, and W2 = Y. The cross-hatching in Ul and U2 stand for Zl and Z2 respectively. Let us now prepare more specifically to describe the kind of (counter-)examples given in
[C-T, S, S-D]. Their construction hinges on taking K = Q, S = the real prime, and finding 2-isogenies E' -+ E of elliptic curves over Q with the property that (*) 1. The real locus E(R) has two components; the image of E'(R) in E(R)
is E°(R) the connected component of E(R) containing the identity element.
(*) 2. The Mordell-Weil group E(Q) is infinite and is dense in E(R).
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262
U2
U1
YxE(Ks) _
t
I E(Ks)
= c
el + 7rE'(Ks)
e2 + irE' (Ks) 1
Diagram 1 (*) 3. The natural mapping
E(Q)/imageE'(Q) -4 E(R)/imageE'(R) is an isomorphism of groups of order two.
In such a situation, one can apply Proposition 1 to get that the family Y x E -* E has only two isomorphism classes of Q-rational fibers, the ones lying over points in E°(R) being Q-isomorphic to Y, and the ones lying over the other connected component of E(R), call it E1(R), being Q-isomorphic to a single specific twist Ye of Y. Now take Y to be an elliptic curve over
Q and take the involution t to be y H -y, but consider only such elliptic curves Y with the added property that Y(Q) is infinite and Ye(Q) is finite. Applying Proposition 2 one sees that the intersection of (Y x E) (Q) with the union of connected components of (Y x E) (R) lying over E° (R) is Zariski dense in Y x E while the Zariski closure of the intersection of (Y x E) (Q) with the union of connected components of (Y x E) (R) lying over E' (R) is (non-empty, and is) a finite union of (elliptic) curves.
6. " C.-T., S. and Sw.-D." examples Here is a specific choice of example, following the recipe of C.-T., S. and Sw.D. which produces a surface "out of' elliptic curves of low conductor which have the desired property. For the base curve of the surface take the elliptic curve E displayed below, which admits a 2-isogeny E' --+ E over Q.
E': y2 + xy = x3 + 4x + 1 E: y2 + xy = x3 - X.
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The curves E, E' are the two elliptic curves of conductor 65 and the pair of them satisfy (*)1,2,3. This is the example of smallest conductor (of a pair of elliptic curves) satisfying (*) occurring in Cremona's tables [Cr]. One computes that E(R) is disconnected and E'(R) is connected. Con-
dition 1 of (*) then follows. The point of order two in E(Q) is given by x = 0, y = 0, and lies on the connected component of E(R) not containing the identity. The Mordell-Weil group E(Q) is generated by this point of order two, together with the point (x, y) = (1, -1) which is of infinite order. This gives Condition 2 of (*).
Letting E'/Z and E/Z denote the associated Neron models over Spec Z, we have an exact sequence of group schemes over Z
0-4µ2Z-4E'Z-* E/z -*0 where /µ2/z is the finite flat group scheme over Spec Z which is the kernel of multiplication by 2 in G./z. The exactness of this sequence may be checked fairly directly using the information given in Cremona's listing for conductor 65. Passing to the diagram of long exact sequence of flat cohomology groups, one gets the commutative diagram
E'(Q)
) E(Q)
> H1(Spec Z, /-p2) ij
E'(R)
) E(R)
) H1(SpecR,p2)
)
0
where the horizontal rows are exact. One computes that the vertical mapping labelled j is an isomorphism of groups of order two. This gives Condition 3 of (*). The unique nontrivial element in the group Hl (SpecZ, µ2) "classifies" the p2-torsor given by the extension SpecZ[ ] over SpecZ. Denote by x the quadratic Dirichlet character (of discriminant 4) attached to this extension.
7. An explicit example To complete the construction of an example, we need only take our "fiber" Y to be an elliptic curve over Q which has the property that the Mordell-Weil
rank of Y is 0 while the Mordell-Weil rank of Y 0 x, i.e., of Y twisted by x, is positive. Of course, we take our t to be the involution y i -+ -y. There is no difficulty finding such an elliptic curve: Y = Xo(17) 0 x, for example, has the property that its Mordell-Weil group over Q is of positive rank, while Y 0 x = Xo (17) has finite Mordell-Weil group (it is cyclic of order 4). We therefore focus on the surface V over Q given by V = Y X E -* E (in the notation of section 5.) where E is the elliptic curve of conductor 65 whose
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B. Mazur
equations is y2 + xy = x3 - x, and where Y = X0(17) ® X. The real locus of V consists of two components, Ul (which projects onto the component of E(R) containing the identity element) and U2 (which projects onto the other topological component of E(R)). Both Ul and U2 are compact two-manifolds of genus one. The Q-rational points of V meeting Ul are topologically dense in Ul and therefore the Zariski-closure of v (Q) fl Ul in V is equal to V.
In contrast, the Zariski-closure of v(Q) fl U2 in V is the image of X0(17)(Q) x E'(R) in V. In other words the Zariski-closure of V(Q) fl U2 is the the union of three elliptic curves! (one of which is isomorphic to E' (projecting by a map of degree 2 to the base) and the other two are isomorphic to E). Our example, then, is of the sort envisioned in Diagram 3 (with K = Q, and KS = R, and each of the cosets el +7rE'(K) and e2 +7rE'(K) are topologically homeomorphic to the circle.
References [A 1] Abramovich, D.: Uniformity des points rationnels des courbes algybriques sur les extensions quadratiques et cubiques, C.R. Acad. Sci. Paris 321 (1995) 755-758 [A 2] Abramovich, D.: A high fibered power of a family of varieties of general type dominates a variety of general type. Invent. math. 128 (1997) 481-494
[A 3] Abramovich, D.: A linear lower bound on the gonality of modular curves. International Mathematics Research Notices, 20 (1996) 1005-1013
[A-H] Abramovich, D., Harris, J.: Abelian varieties and curves in Wd(C). Comp. Math. 78 (1991) 227-238 [A-V] Abramovich, D., Voloch, J.F.: Lang's conjecture, fibered powers and uniformity, New York J. Math II (1996) 20-34 [C-H-M 1] Caporaso, L., Harris, J., Mazur, B.: Uniformity of rational points. Journal of the A.M.S. 10 (1997) 1-35; a more expository version available by anonymous ftp:
ftp://ftp.math.harvard.edu/pub/uniformityofrationalpoints.tex [C-H-M 21 Caporaso, L., Harris, J., Mazur, B.: How many rational points can a curve have? Proceedings of the Texel Conference, Progress in Math. 129 Birkhauser, Boston (1995) 13-31 [C-T, S, S-D] Colliot-Thylene, J.-L., Skorobogatov, A.N., Swinnerton-Dyer, P.: Double fibres and double covers: paucity of rational points. To appear.
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[Cr] Cremona, J.: Algorithms for modular elliptic curves. Cambridge Univ. Press (1992)
[D-F] Debarre, 0., Fahlaoui, R.: Abelian varieties in Wd (C) and points of bounded degree on algebraic curves. Compositio Math. 88 (1993) 235-249
[dJ] de Jong, J.: Smoothness, semistability, and alterations. Publ.Math. I.H.E.S. 83 (1996) 51-93
[H-S] Harris, J., Silverman, J.: Bi-elliptic curves and symmetric products. Proc. Amer. Math. Soc. 112 (1991)347-356 [H] Hassett, B.: Correlation for surfaces of general type. Duke Math. J. 85 (1996) 95-107
[L] Lang, S.: Hyperbolic diophantine analysis. Bull. A.M.S. 14 (1986) 159205
[L-Y] Li, P., Yau, S.-T.: A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Inventiones math. 69 (1982) 269-291
[Ma 1] Mazur, B.: The topology of rational points. Journ. Exp. Math. 1 (1992) 35-45
[Ma 2] Mazur, B.: Questions of decidability and undecidability in number theory. J. Symbolic Logic 59 (1994) no. 2, 353-371 [Ma 3] Mazur, B.: Speculation about the topology of rational points: an update. Asterisque 228 (1995) 165-181
[Me] Merel, L.: Bornes pour la torsion des courbes elliptiques sur les corps des nombres. Inventiones Math. 124 437-449
[P] Pacelli, P.: Uniform boundedness for rational points. Duke Math. J. 88 (1997) 77-102
[Sh] Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions. Princeton Univ. Press, 1971 [Y] Yau, S.-T.: An application of eigenvalue estimate to algebraic curves defined by congruence subgroups. Math. Research letters 3 (1996) 167-172 DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY, ONE OXFORD STREET,
CAMBRIDGE, MA 02138, USA
[email protected] Models of Shimura varieties in mixed characteristics BEN MOONEN
Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Shimura varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 . . . . . . . . Canonical models of Shimura varieties . . . . . . 282 Integral canonical models . . . . . . . . . . . . . . . . . . . 293 Deformation theory of p-divisible groups with Tate classes . . . . . 310 Vasiu's strategy for proving the existence of integral canonical models317 Characterizing subvarieties of Hodge type; conjectures of Coleman and Oort . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 333 .
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Introduction At the 1996 Durham symposium, a series of four lectures was given on Shimura varieties in mixed characteristics. The main goal of these lectures was to discuss some recent developments, and to familiarize the audience with some of the techniques involved. The present notes were written with the same goal in mind. It should be mentioned right away that we intend to discuss only a small number of topics. The bulk of the paper is devoted to models of Shimura varieties over discrete valuation rings of mixed characteristics. Part of the discussion only deals with primes of residue characteristic p such that the group G in question is unramified at p, so that good reduction is expected. Even at such primes, however, many technical problems present themselves, to begin with the "right" definitions. Reprinted from `Galois Representations in Arithmetic Algebraic Geometry', edited by A. J. Scholl & R. L. Taylor. ©Cambridge University Press 1998
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There is a rather large class of Shimura data-those called of pre-abelian type-for which the corresponding Shimura variety can be related, if maybe somewhat indirectly, to a moduli space of abelian varieties. At present, this seems the only available tool for constructing "good" integral models. Thus, if we restrict our attention to Shimura varieties of pre-abelian type, the construction of integral canonical models (defined in §3) divides itself into two parts: Formal aspects. If, for instance, we have two Shimura data which are "closely related", then this should have consequences for the existence of integral canonical models. Loosely speaking, we would like to show that if one of the two associated Shimura varieties has an integral canonical model, then so does the other. Most of such "formal" results are discussed in §3. Constructing models for Shimura varieties of Hodge type. By definition, these are the Shimura varieties that can be embedded into a Siegel modular variety. As we will see, the existence of an integral canonical model is essentially a problem about smoothness, which therefore can be studied using deformation theory. We are thus led to certain deformation problems for pdivisible groups. These can be dealt with using techniques of Faltings, which are the subject of §4. This is not to say that we can now easily prove the existence of integral canonical models. Faltings's results only apply under some assumptions, and in the situation where we want to use them, it is not at all clear that these are satisfied. To solve this, Adrian Vasiu has presented an ingenious, but technically complicated strategy. We will discuss this in §5. Unfortunately, it seems that Vasiu's program has not yet been brought to a successful end. We hope that our presentation of the material can help to clarify what technical points remain to be settled. I have chosen to include quite a bit of "basic material" on Shimura varieties, which takes up sections 1 and 2. Most of this is a review of Deligne's papers [Del] and [De3]. I also included some examples and some references to fundamental work that was done later, such as the generalization of the theory to mixed Shimura varieties. The main strategy of [De3] is explained in some detail, since it will reappear in our study of integral models. The only new result in the first two sections concerns the existence of canonical models for Shimura varieties which are not of abelian type. It was pointed out to me by J. Wildeshaus that the argument as it is found in the literature is not complete. We will discuss this in section 2, and we present an argument to complete the proof.
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In §3 we take up the study of integral models of Shimura varieties. The first major problem here is to set up good definitions. We follow the pattern laid out by Milne in [Mi3], defining an integral canonical model as a smooth model which satisfies a certain Neron extension property. The main difficulty is to decide what class of "test schemes" to use. We explain why the class used by Milne in loc. cit. leads to unwanted results, and we propose to use
a smaller class of test schemes which (at least for p > 2 and ramification e < p - 1) avoids this. Our definition differs from the one used by Vasiu in [Va2].
In the rest of §3 we prove a number of "formal" results about integral canonical models, and, inspired by Deligne's approach in [De3], we develop the notion of a connected Shimura variety in the p-adic setting. The main result of this section is Cor. 3.23. It says, roughly, that in order to prove the existence
of integral canonical models for all Shimura varieties of pre-abelian type at primes of characteristic p > 2 where the group in question is unramified, it suffices to show that certain models obtained starting from an embedding into a Siegel modular variety, are formally smooth. As we will explain, there are finitely many primes that may cause additional problems if the group has simple factors of type At. We give full proofs of most statements. Although the reader may find some details too cumbersome, we think that they are quite essential, and that only by going through all arguments we are able to detect some unexpected problems. Some of our results were also claimed in [Va2], but most proofs given here were obtained independently (see also remark 3.24). In §4 we study deformation theory of p-divisible groups with given Tate classes. The main results are based on a series of remarks in Faltings's paper [Fa3], of which we provide detailed proofs. In §5 we attempt to follow Vasiu's paper [Va2]. Our main goal here is to explain Vasiu's strategy, and to explain which technical problems remain to be solved. This section consists of two parts. Up until Thm. 5.8.3, we prove most statements in detail. This leads to a result about the existence of integral canonical models under a certain additional hypothesis (5.6.1). After that we indicate a number of statements that should allow to remove this hypothesis. It is in this part of Vasiu's work that, to our understanding, further work needs to be done before the main result (see 5.9.6) can be accepted as a solid theorem.1 'After completing our manuscript we received new versions of Vasiu's work (A. Vasiu, Integral canonical models of Shimura varieties of Preabelian type, third version, July 15,
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The last section contains a hodgepodge of questions and results, due to various people. We will try to give references in the main text. The main topic here is no longer the existence of integral canonical models per se. Instead, we discuss some results about the local structure of (examples of) such models, in relation to conjectures of Coleman and Oort.
There are some interesting related topics for which we did not find place in this article. Among the casualties are the recent work [RZ] of Rapoport and Zink, examples of bad reduction (see, e.g., [R2]), the Newton polygon stratification of A9 in characteristic p (for an overview, see [Ool], [Oo2]) and the study of isocrystals with additional structure as in [Kol], [Ko3], [RR].
Acknowledgements. In preparing this paper I benefited a lot from discussions with Y. Andre, D. Blasius, C. Deninger, B. Edixhoven, 0. Gabber, J. de Jong, G. Kings, E. Landvogt, F. Oort, A. Vasiu, A. Werner and J. Wildeshaus. I thank them all cordially. Also I wish to thank the referee for several useful comments.
Notations. Superscripts and subscripts: 0 denotes connected components for the Zariski topology, + connected components for other (usually analytic) topologies. A superscript - (as in G(Q)+ for example) denotes the closure of a subset of a topological space. If G is an algebraic group then ad (adjoint group), ab (maximal abelian quotient), der (derived group) have the usual meaning, G(]R)+ denotes the pre image of Gad(R)+ under the adjoint map, and in case G is defined over Q we write G(Q)+ for the intersection of G(Q) and G(IR)+ inside G(IR). For fields, ab denotes the maximal abelian extension.
A superscript P usually denotes a structure "away from p"; a subscript P something "at p". If (X, A) is a g-dimensional principally polarized abelian scheme over a basis S then A gives rise to a Weil pairing eA: X [n] x X [n] -+ An,s. Write On : (Z/nZ)29 x (Z/nZ)2g -* (Z/nZ) for the standard symplectic form. By a Jacobi level n structure on (X, A) we mean an isomorphism rl : X [n] -
(Z/nZ)s such that there exists an isomorphism a: (Z/nZ)s ' Pn,S with a o On o (r) x rl) = eA. We write A9,l,n for the (coarse) moduli scheme over 1997, UC at Berkeley, and Ibid., December 1997, UC at Berkeley.) We have not yet had the opportunity to study this work in detail, and we therefore cannot say whether it can take away all doubts we have about the arguments in [Va2]. We strongly recommend the interested reader to consult Vasiu's original papers.
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Spec(Z[1/n]) of principally polarized, g-dimensional abelian varieties with a Jacobi level n structure. If n > 3 then it is a fine moduli scheme. Let S := Resc/RGm,C. We write p: Gm,c -* Sc for the cocharacter which on complex points is given by C* E) z H (z, 1) E C* X C* = (C OR C)*. The natural inclusion w : Gm,a -3 S is called the weight cocharacter. We write Af for the ring of finite adeles of Q and AL for the ring of (full) adeles of a number field L. We refer to [Pi], 0.6, for an explanation of when a subgroup K C G(Af) (where G is an algebraic group over Q) is called neat, and for some basic properties concerning this notion. Abbreviations: H.S. for Hodge structure, V.H.S. for variation of Hodge structure, d.v.r. for discrete valuation ring, p.p.a.v. for principally polarized abelian variety, i.c.m. for integral canonical model (see 3.3), a.t.s. for admissible test scheme (see 3.5), e.e.p. for extended extension property (see 3.20).
§1 Shimura varieties 1.1 Recall ([De2]) that a pure Hodge structure of weight n with underlying Q-vector space V is given by a homomorphism of algebraic groups h : S -+ GL(V)R such that the weight cocharacter h o w: Gm -4 GL(V)R maps z to
idv. The Tate twist Q(1) corresponds to the norm character Nm: S -* Gm,R. An element v E V is called a Hodge class (in the strict sense) if v is z
purely of type (0, 0) in the Hodge decomposition VV _ ®Vp>9. In other words: the Hodge classes are the rational classes v E V which, as elements of VR, are
invariant under the action of S given by h. The Mumford-Tate group MT(V) of V is defined as the smallest algebraic subgroup of GL(V) x Gm which is defined over Q and such that h x Nm: S -+ GL(V)R x Gm,R factors through MT(V)a. In Tannakian language MT(V) is the automorphism group of the forgetful fibre functor (V, Q(1))® -+ VecQ, where (V, Q(1))® C HdgQ is the Tannakian subcategory generated by V and Q(1). Concretely, this means that for every tensor space V (ri, r2; s) := V®T' ® (V*)®r2 0 Q(S),
the Hodge classes in V(ri, r2; s) are precisely the invariants under the natural action of MT(V). In more classical language one would define a Hodge class to be a rational class v E V which is purely of type (n/2, n/2) in the Hodge decomposition.
Clearly there are in general more Hodge classes in this sense than in the "strict" sense, but the difference is only a matter of weights. If we define the
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272
Hodge group Hg(V) (sometimes called the special Mumford-Tate group) to be the kernel of the second projection map MT(V) --» Gm, then the Hodge classes (in the more general sense) of a tensor space Van ® (V*)®T2 are precisely the invariants of Hg(V). All in all, the Hodge group contains essentially the same
information as the Mumford-Tate group, except that it does not keep track of the weight. The main principle that we want to stress here is the following: if h : S GL(V), defines a Hodge structure on the Q-vector space V, and if we are given tensors t1i ... , tk in spaces of the form V (rl, r2; s), then there is an algebraic group G C GL(V) (depending on the classes ti) such that
tl, ... , tk are Hodge classes
h factors through GR
.
1.2 To illustrate the usage of Mumford-Tate groups, let us discuss some examples pertaining to Hodge classes on abelian varieties. There are at least two reasons why abelian varieties are special: (i) Riemann's theorem tells us that there is an equivalence of categories {complex abelian varieties}
{polarizable Z-H.S. of type (0, 1) + (1, 0)}
,
sending X to H'(X, Z). (This should really be done covariantly, using H1; as we shall later always work with cohomology we phrase everything in terms of H1.) This result has some important variants, in that polarized abelian varieties are in equivalence with polarized Z-H.S. of type (0, 1) + (1, 0), abelian varieties up to isogeny correspond to polarizable Q-H.S. of type (0, 1) + (1, 0), and if S is a smooth variety over C then abelian schemes over S correspond to polarizable Z-V.H.S. of type (0, 1) + (1, 0) over S. (See [De2], section 4.4.) Furthermore, all cohomology of X and of its powers X', can be expressed directly in terms of H'(X, Z): we have natural isomorphisms of Hodge structures k
Hk(Xm, Z) = A (®"` H1(X, Z)) .
(ii) Let V := H'(X,Q), and write Hg(X) := Hg(V). Choose a polarization of X. The corresponding Riemann form co is a Hodge class in Hom(V®2, Q(-1)) = V(0, 2; -1), hence it is invariant under Hg(X). This means that Hg(X) is contained in the symplectic group Sp(V, (p). Next we remark that, because of the above equivalence of categories,
End(X) ®z Q =: End°(X) = {Hodge classes in End(V) } = End(V)Hg(x)
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We conclude that Hg(X) is contained in the centralizer of End°(X) inside Sp(V, cp), and that the commutant of Hg(X) in End(V) equals End°(X). These observations become even more useful if we remark that for abelian varieties of a given dimension, the Albert classification (see [Mu2], section 21)
gives a finite list of possible types for the endomorphism algebra End°(X). When combined with other properties of the Hodge group, knowing End°(X) is in some cases sufficient to determine Hg(X) and its action on V. This then enables us-at least in principle-to determine the Hodge ring of all powers of X. In general, however, the endomorphism algebra does not determine the Hodge group.
Example 1. Main references: [Ri], [Haz], [Ku], [Se2], [Ch]. Let X be a simple abelian variety of dimension 1 or of prime dimension. Then the Hodge
group is equal to the centralizer of End°(X) in Sp(V, cp). (This does not depend on the choice of the polarization.) The Hodge ring of every power of X is generated by divisor classes; in particular, the Hodge conjecture is true for all powers of X. Example 2. Main references: [We], [MZ2]. Suppose k is an imaginary quadratic field, acting on X by endomorphisms. If a and T are the two complex embeddings of k, then H°(X, Q1) = V"° is a module over k ®Q C _ Ci°i x C(T), hence it decomposes as Vl,o = Vl,o(u)®V1,o(T) Suppose that the
dimensions n° = dim V1'°(o) and nT = dim V1'°(r) are equal. This implies that dim(X) is even, say dim(X) = 2n. The 1-dimensional k-vector space 2n
Wk :=/\V k
can be identified with a subspace of A V = H2n(X, Q). Moreover, the condition that n° = nT implies that Wk C Hen (X, Q) consists of Hodge classes. This construction was first studied by Weil in [We]; we call Wk the space of Weil classes with respect to k. Weil showed that for a generic abelian variety X with an action of k (subject to the condition n° = nT), the nonzero classes in Wk are exceptional, i.e., they do not lie in the Q-subalgebra D' (X) C (DH 2i (X, Q) generated by the divisor classes. The construction of Weil classes works in much greater generality. They play a role in Deligne's proof of "Hodge = absolute Hodge" for abelian varieties. In [MZ2] the space WF of Weil classes w.r.t. the action of an arbitrary
field F " End° (X) is studied. We find here criteria, purely in terms of F, End° (X) and the action of F on the tangent space V ',O, of when WF contains Hodge classes, and of when these Hodge classes are exceptional.
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Example 3. Main references: [Mul], [MZ1], [Ta]. Let X be an abelian fourfold with End°(X) = Q. Then either Hg(X) is the full symplectic group Sp8 Q, in which case the Hodge ring of every power of X is generated by divisor classes, or Hg(X) is isogenous to a Q-form of SL2 x SL2 x SL2, where
the representation V is the tensor product of the standard 2-dimensional representations of the three factors. Both possibilities occur. In the latter case, the Hodge ring of X is generated by divisor classes, but for X2 this is no longer true: H4(X2, Q) contains exceptional classes. These are not of the same kind as in example 2, i.e., they are not Weil classes with respect to the action of a field on X2. In case X is defined over a number field, we have "the same" two possibilities for the image of the Galois group acting on the Tate module. In particular, knowing End°(X) here is not sufficient to prove the MumfordTate conjecture. Known by many as "the Mumford example", this is actually the lowest dimensional case where the Mumford-Tate conjecture for abelian varieties remains, at present, completely open. Mumford's example can be generalized to abelian varieties of dimension 4k, see [Ta]. 1.3 Guided by the considerations in 1.1, we can make sense of the problem to study Hodge structures with "a given collection of Hodge classes". How one translates this in purely group-theoretical terms is explained with great clarity in [De3], especially section 1.1. Here we summarize the most important points. Fix an algebraic group GR over R, and consider the space Hom(S, GR) of homomorphisms of algebraic groups h: S -4 G. Its connected components are the G(R)+-conjugacy classes. Given one such component X+, and fixing a representation pR : GR -* GL(VR), we obtain a family of R-Hodge structures on VR, parametrized by X+. From an algebro-geometric point of view, the natural conditions to impose on this family are: (a) the weight decomposition VR = elEZVV does not depend on h E X+,
(b) there is a complex structure on X+ such that the family of Hodge structures on each VR is a polarizable ll -V.H.S. over X+.
Now an important fact is that (a) and (b) can be expressed directly in terms of Gr and X+, and that, at least for faithful representations, they do not depend on pig. If (a) and (b) are satisfied for some (equivalently: every) faithful representation pig, then the complex structure in (b) is unique and X+ is a hermitian symmetric domain. (For all this, see [De3], 1.13-17.) By adding a Q-structure on GR, one is led to the following definition.
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1.4 Definition. A Shimura datum is a pair (G, X) consisting of a connected reductive group G over Q, and a G(R)-conjugacy class X C Hom(S, GR), such that for all (equivalently: for some) h E X, (i) the Hodge structure on Lie(G) defined by Ad o h is of type (-1, 1) + (0, 0) + (1, -1), (ii) the involution Inn(h(i)) is a Cartan involution of G,', (iii) the adjoint group G' does not have factors defined over Q onto which h has a trivial projection. In this definition we have followed [De3], section 2.1. Pink has suggested (cf. [Pi]) to allow not only G(1R)-conjugacy classes X C Hom(S, GR) but also finite coverings of such. We will not use this generalization in this paper. There are some other conditions that sometimes play a role. For instance, condition (i) implies that the weight cocharacter how: Gm,c -+ GC (for which
we sometimes simply write w) does not depend on h E X, and one could require that it is defined over Q. It turns out, however, that the theory works well without this assumption, and that there are rather natural examples where it is not satisfied. 1.5 Let (G, X) be a Shimura datum, and let K be a compact open subgroup of G(Af). We set
ShK(G,X)c = G(Q)\X x G(Af)/K, where G(Q) acts diagonally on X x (G(Af)/K). If X+ C X is a connected component, and if 9 i , .. , gm are representatives in G(Af) for the finite set G(Q)+\G(Af)/K, then we can rewrite ShK(G,X)c as a disjoint sum ShK(G,X)e =
L_L
r2\X+,
where Fa is the image of G(Q)+ fl giKgi 1 inside G '(Q)+, which is an arith-
metic subgroup. By the results of Baily and Borel in [BB], the quotients I'z\X+ have a natural structure of a quasi-projective algebraic variety. For compact open subgroups K1 C K2, the natural map
Sh(K1iK2): ShK,(G,X)c -* ShK2(G,X)c is algebraic. We thus obtain a projective system of (generally non-connected) algebraic varieties ShK(G, X)c, indexed by the compact open subgroups K C
G(Af). This system, or its limit Sh(G, X)c = ]_M ShK (G, X)c, K
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(which exists as a scheme, since the transition maps are finite) is called the Shimura variety defined by the datum (G, X).
1.6 We will briefly recall some basic definitions and results. For further discussion of these topics, see [Del], [De3], [Mi2].
1.6.1 The group G(Af) acts continuously on Sh(G, X),c from the right. The continuity here means that the action of an element g E G(Af) is obtained as the limit of a system of isomorphism g: ShK(G, X )c -24 Sh9-lK9 (G, X )C, see [De3], 2.1.4 and 2.7, or [Mi2], II.2 and 11.10. On "finite levels", the G(Af)action gives rise to Hecke correspondences: for compact open subgroups K1,
K2 C G(Af) and g E G(Af ), set K' = K1 fl gK2g-1; then the Hecke correspondence T from ShK, (G, X )c to ShK2 (G, X )c is given by
T:
ShK,(G,X)o
shi
ShK'(G, X)c -9
ShK2(G,X)C.
1.6.2 A morphism of Shimura data f : (G1, X1) -+ (G2i X2) is given by a homomorphism of algebraic groups f : G1 -a G2 defined over Q which induces a map from X1 to X2. Such a morphism induces a morphism of schemes Sh(f) : Sh(G1i X1)c -> Sh(G2, X2)c
If f : G1 -* G2 is a closed immersion then so is Sh(f ). (See [Del], 1.14-15.)
1.6.3 Let (G, X) be a Shimura datum. Associated to h E X, we have a cocharacter h o p: Gm,c -4 Gc, whose G(C)-conjugacy class is independent of h E X. The reflex field E(G, X) C C is defined as the field of definition of this conjugacy class. It is a finite extension of Q. If f : (G1, X1) -* (G2, X2) is a morphism of Shimura data, then E(G2, X2) C E(G1, X1) C C.
1.6.4 A point h E X is called a special point if there is a torus T C G, defined over Q, such that h : S -+ GR factors through TR. In this case (T, {h}) is a Shimura datum, and the inclusion T " G gives a morphism (T, {h}) -+ (G, X). A point x E ShK(G, X),c is called a special point if it is of the form x = [h, gK] with h special. (This does not depend on the choice of the representative (h, gK) for x.) Here we follow [De3]; the definition in [Del], 3.15, is more restrictive. 1.6.5 Consider a triplet (Gad, G', X+) consisting of an adjoint group Gad over Q, a covering G' of Gad, and a Gad(R)+-conjugacy class X+ C Hom(S, GR )
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such that the conditions (i), (ii) and (iii) in 1.4 are satisfied. Let r(G') be the linear topology on G' (Q) for which the images in G' (Q) of the congruence subgroups in G' (Q) form a fundamental system of neighbourhoods of the identity. The connected Shimura variety Sh°(G', G', X+)c is defined as the projective system Sho (Gaa G1, X+)C
=
r
r\x+
,
where r runs through the arithmetic subgroups of GWI(Q) which are open in r(G'). It comes equipped with an action of the completion Gad(Q)+' of G '(Q)+ for the topology T(G). Given a Shimura datum (G, X) and a connected component X+ C X, we obtain a triplet (G d, Gder X+) as above. The associated connected Shimura variety Sh°(G ', Gder, X+)c is the connected component of Sh(G, X)c containing the image of X+ x {e} C X x G(Af). In particular, we see that this component only depends on Gad, Gder and X+ C X. In the sequel, when working with connected Shimura varieties, we will usually omit G' from the notation. For lack of better terminology, we will refer to a pair (G', X+) as above as "a pair defining a connected Shimura variety". 1.6.6 Let G be a reductive group over a number field L. Write p: G -> Gder for the universal covering (in the sense of algebraic groups) of its derived group. By [Del], Prop. 2.2 and [De3], Cor. 2.0.8, G(L) pd(AL) is a closed subgroup of G(AL) with abelian quotient ir(G) := G(AL)/G(L) pG(AL). (Note: AL is the ring of full adeles of L.) Consequently, the set of connected components 7roir(G) is also an abelian group.
Now let (G, X) be a Shimura datum. If K C G(Af) is a compact open subgroup then ShK (G, X )c is a scheme of finite type over C. For K getting smaller, its number of connected components generally increases. Deligne proves in [De3], 2.1.3 that lro(ShK(G,X)c)
G(Af)/G(Q)+ . K
Woir(G)/K,
where Toir(G) := iro7r(G)/7roG(l[8)+. Passing to the limit one finds that the G(Af)-action on Sh(G,X)c makes iro(Sh(G,X)c) a principal homogeneous space under 7oir(G) = G(Af)/G(Q)+.
1.6.7 Given a Shimura datum (G,X), we can define some other data as follows. Write Xad C Hom(S, Ga) for the G'(R)-conjugacy class containing the image of X under the map Hom(S, GIR) -4 Hom(S, G'). The map
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X -* Xad is not necessarily an isomorphism, but every connected component
of X maps isomorphically to its image. The pair (Gad, Xad) is a Shimura datum, called the adjoint Shimura datum. Similarly, the image Xab of X in Hom(S, Gab) is a Gab(R)-conjugacy class (necessarily a single point), and we have a Shimura datum (Gab Xab) Another construction that is sometimes useful is the following. Suppose the group G is of the form G = ResF/Q(H), where F is a totally real number field and H is an absolutely simple algebraic group over F. (Such is the
case, for example, if G is a simple adjoint group.) Now take an extension F C F' of totally real number fields, and set G2 = ResF'/Q(HF,). There is a unique G2(IR)-conjugacy class X2 C Hom(S, G2,a) such that the natural homomorphism G -* G2 gives a closed immersion of Shimura data (G, X) " (G2, X2).
1.7 One might ask "how many" Shimura varieties there are. A possible approach is to begin by classifying the Shimura varieties of adjoint type. These are products of Shimura varieties Sh(G, X)c, where G is a Q-simple adjoint group. The group Ga is an inner form of a compact group, of one of the types A, B, C, Da, DH, E6 or E7, and given Ga, the possibilities for X are classified in terms of special nodes in the Dynkin diagram. We refer to [De3], sections 1.2 and 2.3 for more details. Given (Gad, Xad), we can list all possibilities for Gdu. As we have seen, the pair (Gd,T, X+) consisting of Gder and a connected component X+ C X, determines the connected components of the Shimura variety. In particular, the "toric part" (Gab, Xab) does not contribute to the geometry of Sh(G, X)c, in the sense that it has no effect on Sh°o, but only on 7ro (Sh(G, X)c). Finally, let us remark that "toric" Shimura data are in bijective correspondence to pairs (Y,,a) consisting of a free Z-module Y of finite rank with a continuous action of Gal(O/Q) (the cocharacter group of the torus), together with an element p E Y. 1.8 The definition of a Shimura variety is set up in such a way that that if : GR -4 GL(Va) is a representation, then we obtain a (direct sum of)
polarizable R VHS V(C)R over X with underlying local system X x Va. If Va = VOQR for a Q-vector space V, and if the weight eow: Gm,R -4 GL(V)a is defined over Q, then V(e)a comes from a polarizable Q-VHS V(e). Under some conditions on G/Ker(e), this VHS descends, for K sufficiently small, to a Q-VHS on ShK(G, X). (It suffices if the center of G/Ker(e) is the almost
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direct product of a Q-split torus and a torus T for which T(R) is compact.) One expects (see [De5], [Mi4]) that these variations of Hodge structure are the Betti realizations of families of motives, and that Shimura varieties, at least those for which the weight is defined over Q, have an interpretation (depending on the choice of a representation ) as moduli spaces for mo-
tives with certain additional structures. What is missing, at present, is a sufficiently good theory of motives. In certain cases, however, the dictionary between abelian varieties and certain Hodge structures (see 1.2 above) leads to a modular interpretation of Sh(G, X). Let us briefly review some facts and terminology.
1.8.1 Siegel modular varieties. Let cp denote the standard symplectic form on q9, and set G = CSp2g. The homomorphisms h: S -* GR which determine a Q-H.S. of type (-1, 0) + (0, -1) on Q29 such that ± is a polarization, form a single G(IR)-conjugacy class S59 . It can be identified with the Siegel double-2-space. The pair (CSp2g, Sjy) is a Shimura datum with reflex field Q. The associated Shimura variety is often referred to as the Siegel modular variety.
For K C G(Af) a compact open subgroup, ShK(CSp29, )9 is a moduli space for g-dimensional complex p.p.a.v. with a level K-structure (as defined, for instance, in [Ko2], §5). Here a couple of remarks should be added. The interpretation of ShK(CSp29, Sj9 )c in terms of abelian varieties up to isomorphism depends on the choice of a lattice A C Q29. This choice also determines the "type" of the polarization; if we want to work with principally polarized
abelian varieties then we must choose A such that VIA has discriminant 1 (e.g., A = Z29). For further details see [Del], §4. In the sequel, we identify Sh(CSp2g, S)9 )c and
,,, Ag,l,n ® C.
1.8.2 Shimura varieties of PEL and of Hodge type. By definition, a Shimura
datum (G, X) (as well as the associated Shimura variety) is said to be of Hodge type, if there exists a closed immersion of Shimura data j : (G, X) " (CSp2g, Sjy) for some g. If this holds, the Shimura variety Sh(G, X)c ' Sh(CSp2g, S59 )c has an interpretation in terms of abelian varieties with certain "given Hodge classes". The precise formulation of such a modular interpretation is usually rather complicated. This is already the case for Shimura varieties of PEL type (see [Del], 4.914, [Ko2]). Loosely speaking, these are the Shimura varieties parametrizing
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abelian varieties with a given algebra acting by endomorphisms.2 Recall (1.2) that endomorphisms of an abelian variety are particular examples of Hodge classes. On finite levels one thus looks at abelian varieties with a Polarization, certain given Endomorphisms, and a Level structure.
Shimura varieties of PEL type are more special in that they represent a moduli problem that can be formulated over an arbitrary basis. For more general Shimura varieties of Hodge type we can only do this if we assume the Hodge conjecture. In 2.10, we will introduce two more classes of Shimura varieties: those of abelian and of pre-abelian type. Among these classes we have the following inclusions
(
Sh. var. of PEL type
C
Sh. var. of Hodge type
)
C
(
Sh. var. of abelian type) C
Sh. var. of )
C
(
pre-ab. type J
C
( general ) Sh. var.
All inclusions are strict; for the first one see 1.2. (A priory, the Shimura variety corresponding to a "Mumford example" could have a different realization for which it is of PEL type. By looking at the group involved over R, one easily shows that this does not happen.) 1.9 Compactifications; mixed Shimura varieties. This is a whole subject in itself, and we cannot say much about it here. We will briefly indicate some important statements, referring to the literature for details. The first compactification to mention is the Baily-Borel (or minimal) compactification, for which we write ShK(G, X)*. (References: [BB], see also [Br], §4 for a summary.) It was constructed by Baily and Borel in the setting of locally symmetric varieties. If r\X+ is a component of ShK(G, X)c, say with K neat so that F\X+ is non-singular, then its Baily-Borel compactification is given as a quotient r\X*. Here X* is the Satake compactification of X+; as a set it is the union of X+ and its (proper) rational boundary components, which themselves are again hermitian symmetric domains. It is shown in [BB] that [ \X* has a natural structure of a normal projective variety. The 'For the reader who has not worked with Shimura varieties before, it may be instructive
to read Shimura's paper [Sh]. Here certain Shimura varieties of PEL type are written down "by hand". Both for understanding Shimura's paper and for understanding the abstract Deligne-formalism we are presenting here, it is a good exercise to translate the considerations of [Sh] to the "(G, X) language".
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stratification of X* by its boundary components F induces a stratification of r\X* by locally symmetric varieties FF\F. As the referee pointed out to us, it is worth noticing that the Baily-Borel compactification ShK(G, X)* is not simply defined as the disjoint union of the F\X*. Instead, one starts with a Satake compactification of the whole of X at once, and one defines ShK (G, X)* as a suitable quotient. Thus, for example, if (G, X) = (GL2, fj'), one does not adjoin the points of F'(Q) to 5+ and Sj- separately but one works with IF' (Q) U S5+ U U. We refer to [Pi], Chap. 6 for further details. The Baily-Borel compactification is canonical. In particular, it is easy to show (see [Br], p. 90) that it descends to a compactification of the canonical model ShK(G, X) (to be discussed in the next section). In general, ShK(G,X)* is singular along the boundary. Next we have the toroidal compactifications3 studied in the monograph
These are no longer canonical, as they depend on the choice of a certain cone decomposition. We will reflect this in our notation, writ[Aea].
ing ShK(G, X; S),c for the toroidal compactification corresponding to a Kadmissible partial cone decomposition S as in [Pi], Chap. 6. From the construction, we obtain a natural stratification of the boundary. For suitable choices of S (and K neat), one obtains a projective non-singular scheme ShK(G, X; S)c such that the boundary is a normal crossings divisor-in this case one speaks of a smooth toroidal compactification. Although both the Baily-Borel and the toroidal compactifications were initially studied in the setting of locally symmetric varieties, it was realized that they should be tied up with the theory of degenerating Hodge structures (e.g., see [Aea], p. iv). For certain Shimura varieties this was done by Brylinski
in [Br], using 1-motives. Subsequently, Pink developed a general theory of mixed Shimura varieties and studied compactifications in this setting. Similar ideas, but in a less complete form, were presented by Milne in [Mi2]. It seems that several important ideas can actually be traced back to Deligne. The main results of [Pi] include the following statements (some of which had been known before for pure Shimura varieties or some special mixed Shimura varieties). We refer to loc. cit. for definitions, more precise statements and of course for the proofs. (i) Let ShK(G, X)c be a pure Shimura variety. It has a canonical model 3Here we indulge in the customary abuse of terminology to call these compactifications,
even though ShK (G, X ; S) is compact only if the cone decomposition S satisfies some conditions.
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Models of Shimura varieties
ShK(G, X) over the reflex field E(G, X) (see §2 below). The Baily-Borel compactification descends to a compactification ShK (G, X) * of this canonical model. The boundary has a stratification by finite quotients of (canonical models of) certain pure Shimura varieties; each such stratum is a finite union of the natural boundary components in the Baily-Borel compactification. Which pure Shimura varieties occur in this way can be described directly in terms of the Shimura datum (G, X). (ii) Next consider K-admissible cone decompositions S for (P, X). If S satisfies certain conditions (such S always exists if K is neat) then the following assertions hold. The toroidal compactification ShK (P, X; S)c descends to a compactification ShK (P, X; S) of the canonical model. It is a smooth projective scheme, and the boundary is a normal crossings divisor. The boundary has a stratification by finite quotients of (canonical models of) certain
other mixed Shimura varieties; each such stratum is a finite union of the strata of ShK(P, X; S) as a toroidal compactification. The natural morphism 7r: ShK(P, X; S) -4 ShK(P, X)* is compatible with the stratifications. If ShK (P', X') is a mixed Shimura variety of which a finite quotient occurs as a boundary stratum C C ShK(P, X; S), then the restriction of it to C is induced by the canonical morphism of ShK (P', X') to the associated pure Shimura variety. Furthermore, Pink proves several results about the functoriality of the structures in (i) and (ii). To conclude this section, let us remark that in some cases (modular curves: Deligne and Rapoport, [DR]; Hilbert modular surfaces: Rapoport, [R1]; Siegel modular varieties: Chai and Faltings, [FC]) we even have smooth compactifications of Shimura varieties over (an open part of) Spec(Z) or the ring of integers of a number field. As Chai and Faltings remark in the introduction to [FC], many of their ideas also apply to Shimura varieties of PEL type; they conclude: "... and as our ideas usually either carry over directly, or we are lead to hard new problems which require new methods, we leave these generalizations to the reader."
§2 Canonical models of Shimura varieties. 2.1 Before turning to more recent developments, we will discuss some aspects of the theory of canonical models of Shimura varieties (over number fields). Our motivation for doing so is twofold. (i) For "most" Shimura varieties, the existence of a canonical model was
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shown by Deligne in his paper [De3]. As we will see, the same strategy of proof is useful in the context of integral canonical models. (ii) The existence of canonical models in general, i.e., including the cases where the group G has factors of exceptional type, was claimed in [Mil] (see also [La], [Bo2], [MS], [Mi2]) as a consequence of the Langlands conjecture on
conjugation of Shimura varieties. It was pointed out to us by J. Wildeshaus that the argument given there is not complete. Below we will explain this in more detail, and we correct the proof.
Recall (1.6.6) that for a reductive group G over a global field K of characteristic 0 we have set 7r(G) = G(AK)/G(K) pG(AK). We have the 2.2
following constructions, for which we refer to [De3], section 2.4. (a) Given a finite field extension K C L, there is a norm homomorphism
NmLIK: 7r(GL) -#7r(G). (b) If T is a torus over K and M is a G(K)-conjugacy class of homomorphisms Tg -* GK which is defined over K, then there is associated to M a homomorphism qM : 7r (T) -+ 7r (G) .
If (G, X) is a Shimura datum with reflex field E = E(G, X) C C, we use this to define a reciprocity homomorphism r(G,X): Gal(O/E) -+ 7rro7r(G) = G(Af)/G(Q)+
as follows. Global class field theory provides us with an isomorphism (2.2.1)
Gal(O/E)a'b
7ro7r(Gm,E)
Applying (b) to the conjugacy class M = {hot: Gm,c Gc I h E X}, which (by definition) is defined over E, we obtain a map qM: 7r(Gm,E) -4 7r(GE). From (a) we get NmE/Q: 7r(GE) -+ 7r(G). Combining these maps we can now define the reciprocity map as (2.2.2)
r(G,X): Gal(Q/E) -* Gal(Q/E)ab
(z
7ro7r(Gm,E)
lra N"m
7roir(G) -* iro7r(G).
For a Shimura datum (T, {h}) where T is a torus, the reciprocity map can be described more explicitly: if v is a place of E dividing p then r(T,{h}) : Gal(O/E)
To7r(T)
= T(A1)/T(Q)
sends a geometric Frobenius element 4% E Gal(O/E) at v to the class of the E T (Q) " T (Af ), where 7r is a uniformizer at v. element NmE/Q
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If (T, {h}) is a Shimura datum with T a torus, then for every compact open subgroup K C T(Af) the Shimura variety ShK(T, {h})e consists of finitely many points. To define a model of it over the reflex field E = E(T, {h}) it therefore suffices to specify an action of Gal(Q/E). Write 2.3
ShK(T, {h}) for the model over E determined by the rule that o E Gal(Q/E) acts on ShK (T, {h})o by sending [h, tK] to [h, r(T,{h}) (v) tK]. It is clear that the transition maps Sh(K',K) descend to E, and we define the canonical model of Sh(T, {h})e to be Sh(T, {h}) = ]{im ShK(T, {h})
.
K
2.4 Definition. Let (G, X) be a Shimura datum. (i) A model of Sh(G, X )c over a field F C C is a scheme S over F together
with a continuous action of G(Af) from the right and a G(Af)-equivariant isomorphism S OF C -+ Sh (G, X )C. (ii) Let F C C be a field containing E(G, X). A weakly canonical model of Sh(G, X) over F is a model S over F such that for every closed immersion of Shimura data i : (T, {h}) y (G, X) with T a torus, the induced morphism Sh(T, {h})c " Sh(G, X)e Sc descends to a morphism Sh(T, {h}) ®E
EF " S OF EF, where E = E(T, {h}), and where Sh(T, {h}) is the model defined in 2.3. (iii) A canonical model of Sh(G, X) is a weakly canonical model over the
reflex field E(G,X). It should be noticed that if S is a model of Sh(G, X) over the field F C C,
then we have an action of G(Af) x Gal(F/F) on SF (i.e., two commuting actions of G(Af) and Gal(F/F).) 2.5 Let f : (G1, X1) - (G2, X2) be a morphism of Shimura data, and suppose there exist canonical models Sh(G1i X1) and Sh(G2i X2). Then, as shown in [Dell, section 5, the morphism Sh(f) descends uniquely to a morphism Sh(G1, Xl) -4 Sh(G2,X2) ®E(G2,x2) E(G1, X1), which we will also denote Sh(f). In particular, it follows that a canonical model, if it exists, is unique up to isomorphism. (The isomorphism is also unique, since the isomorphism
Sh(G, X) ®E C - Sh(G, X)e is part of the data.) 2.6 If Sh(G,X) is a canonical model of a Shimura variety, then the Galois group Gal(i/E) acts on the set of connected components of Sh(G, X)c, which, as recalled in 1.6.6, is a principal homogeneous space under Toir(G).
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Deligne proves in [De3], section 2.6, that the homomorphism Gal(Q/E) -+ To7r(G) describing the Galois action on ir° (Sh(G, X)(c) is equal to the homomorphism r(G,x) defined above. (Strictly speaking, this is only true up to a sign: in [De3] the Galois action on 7r°(Shc) is described to be r(G,x); Milne pointed out in [Mi3], Remark 1.10, that the reciprocity law is given by r(G,x), not its inverse.)
2.7 An important technique for proving the existence of canonical models is the reduction to a problem about connected Shimura varieties. To explain this, let us assume that Sh(G, X) is a canonical model of the Shimura variety associated to the datum (G, X), and let us inventory the available structures. As in all of this section, we are mainly repeating things from Deligne's paper [De3].
The group G(Af) acts continuously on Shc = Sh(G,X)c from the right.
If Z denotes the center of G then Z(Q)- C G(Af) acts trivially. Write G'd(Q)1 := GH(Q) n Im(G(R) -* G"d(R)). The action of Gad on G by inner automorphisms induces (by functoriality) a left action of Gad(Q)1 on Sh(G, X)c. For g E G(Q), the action of g through Gad (Q)1 coincides with the one of g-1 considered as an element of G(Af). In total we therefore obtain a continuous left action of the group
r := (G(Af)/Z(Q) )
G(Q)/Z(Q)
Gad(Q)1 = (G(Af)/Z(Q) )
G(Q)+/Z(Q)
Gad(Q)+
(converting the operation of G(Af) to a left action). The group r operates transitively on ir°(Shc). For any connected component of Shc, the stabilizer of this component is the subgroup
(G(Q)+/Z(Q) )
Gad(Q)+
Gad(Q)+n (rel.
T(Gder))
G(Q)+/z(Q)
where the completion GI(Q)+' is taken relative to the topology T(Gder) The profinite set 7r°(Shc) is a principal homogeneous space under the abelian group G(Af)/G(Q)+ = T°ir(G) . (Cf. 1.6.5 and 1.6.6.)
From now on we fix a connected component X+ C X, and we write Sh°° = Sh°(Gder,X+)c for the corresponding connected Shimura variety, to be identified with a connected component of Sh(G, X )C. We have an action of the Galois group Gal(Q/E) on Shr. As mentioned in 2.6, it acts on 7r°(Shc)
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through the reciprocity homomorphism r(G,X). The subgroup EE(Gder X+) C F x Gal(O/E) which fixes the connected component Shoe is an extension
0 -f Gad(Q)+n
EE(Gder X+) -+ Gal( /E) -> 0.
With these notations, we have the following important remarks. (i) The extension EE(Gder, X+) depends only on the pair (Gder, X+); in particular this justifies the notation. (See [De3], section 2.5.) (ii) Galois descent (see also 2.15 below) tells us that it is equivalent to
give a model of Sh(G, X)e over E or to give a scheme S over 0 with a continuous action of F x Gal(O/E) and a F-equivariant isomorphism S 00
C -* Sh(G,X)e. (iii) Write e E vo(She) for the class of the connected component Shoe. To
give a Q-scheme S as in (ii), which, in particular, comes equipped with a F-equivariant isomorphism 7r°(S) = io(She), is equivalent to giving its connected component Se corresponding to e together with a continuous action of EE(Gder, X+). The idea here is that we can recover S from Se by "induction" from EE(Gder, X+) to F x Gal(Q/E). (See [De3], section 2.7.)
2.8 Definition. (i) Let (G', X+) be a pair defining a connected Shimura variety with reflex field E, let F C Q be a finite extension of E, and write EF(G', X+) for the extension of Gal(Q/F) by Gad(Q)+n (completion for the topology ir(G')) described in [De3], Def. 2.5.7. Then a weakly canonical model of the connected Shimura variety Sh°(G', X+)e over F consists of a scheme S over Q together with a continuous left action of the group EF(G', X+) and an
isomorphism i : S 00 C - Sh° (G', X+)e such that the following conditions are satisfied. (a) The action of EF(G', X+) on S is semi-linear, i.e., compatible with the
canonical action on Spec(O) through the quotient Gal(Q/F). (b) The isomorphism i is equivariant w.r.t. the action of Gad(Q)+' C l EF(G, X+) (which by (i) acts linearly on S). (c) Given a special point h E X+, factoring through a subtorus h: S -4 denote the field of definition of the cocharHe C G' defined over Q, let acter h o p. Delinge defines in loc. cit., 2.5.10, an extension 0 -+ H(Q) -* Gal(O/EE(h)) -+ 0, for which there is a natural homomorphism EF(h) EF(G', X+). Then we require that the point in Sh°(G', X+)e deEF(h) fined by h is defined over 0 and is fixed by EF(h). (ii) A canonical model of the connected Shimura variety Sh°(G', X+)e is a weakly canonical model over the reflex field E.
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Although our formulation of condition (c) in (i) is a little awkward, it should be clear that this definition is just an attempt to formalize the above remarks. In fact, these remarks lead to the following result (= [De3], Prop. 2.7.13).
2.9 Proposition. Let (G, X) be a Shimura datum, and choose a connected component X+ of X. If Sh(G, X) is a weakly canonical model of Sh(G, X)c over F D E(G, X), then the connected component Sh° (G, X )U determined by the choice of X+ is a weakly canonical model of Sh°(Gder, X+)c over F. Conversely, if there exists a weakly canonical model of Sh°(Gder, X+)c over F, then it is obtained in this way from a weakly canonical model of Sh(G, X)r.
2.10 The main result of [De3] is the existence of canonical models for a large class of Shimura varieties (see below). Since the strategy of proof also works for other statements about Shimura varieties, let us present it in an abstract form (following [Mi2], 11.9). So, suppose we want to prove a statement P(G, X) about Shimura varieties. (a) Prove P(CSp29, ,f59) using the interpretation of Sh(CSp29, $9 )c as a moduli space. (b) For a closed immersion i: (G1, X1) y (G2, X2), prove the implication
P(G2, X2) = P(G1i X1). (c) Find a statement P°(G', X+) for pairs (G', X+) defining a connected Shimura variety, such that, for any connected component X+ C X, we have P°(Gder, X+). P(G, X) (d) Given pairs (Gti, Xi ), i = 1, ... , m, prove that Vi PO (GI, Xi+) fIz Xi ). P°(H (e) For an isogeny G' -+ G", prove that P°(G', X+) ==#. PO (G", X
Roughly speaking, the class of Shimura varieties of abelian type is the largest class for which (a)-(e) suffice to prove statement P. (As we will see below, this is not completely true: we may have to modify the strategy a bit, and even then it is not clear whether we obtain property P for all Shimura varieties of abelian type.) More precisely, a Shimura datum (G, X) is said to be of abelian type if there exists a Shimura datum (G2, X2) of Hodge type
and an isogeny Gder - Gder which induces an isomorphism (G', X2) (G', X'). Deligne has analysed which simple Shimura data belong to this class. He showed that if (G, X) is of abelian type with G simple over Q, then the following two conditions hold:
(i) The adjoint datum (Gad, X') is of type A, B or C, or of type Da, or
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of type DH (cf. [De3], section 1.2), and
(ii) For a datum (Gad, Xad) of type A, B, C or DR, let GO denote the universal covering of Gad; for (G', Xad) of type D', let GO be the double covering of Gad which is an inner form of (a product of copies of) SO(2t), cf.
ibid., 2.3.8, and notice that the case D? is defined to exclude the case DR. Then Gdei is a quotient of G. Conversely, if (G', X+) is a pair defining a connected Shimura variety such
that (i) and (ii) hold, then there exists a Shimura datum (G, X) of abelian type with Gde` = G', X+ C X. Finally, we define (G, X) to be of pre-abelian type if condition (i) holds. We see that, as far as connected Shimura varieties is concerned, this class is only slightly larger than that of data of abelian type. 2.11 Let us check steps (a)-(e) above for the statement
P(G, X) :
there exists a canonical model for Sh(G, X )r-.
(a) The scheme A9,1,,, 0 Q is a canonical model for Sh(CSp29, fj9 )C Given the definitions as set up above, this boils down to a theorem of Shimura and Taniyama-see [Del], section 4. (Needless to say, the theorem of Shimura and Taniyama historically came first. The definition of a canonical model was modelled after a number of examples, including the Siegel modular variety.)
(b) This is shown in ibid., section 5. We should note here that, using a modular interpretation, one can prove P(G, X) more directly for Shimura varieties of Hodge type. This was indicated in the introduction of [De3], and carried out in detail in [Br]. For steps (c)-(e), let us work with the statement P°(G', X+) :
there exists a canonical model for Sh°(G', X+)c.
The (d) and (e) follow easily from the definitions (cf. [De3], 2.7.11) As for (c), we see that our strategy is not completely right: to prove P°(G', X+), we want to take a Shimura datum (G2, X2) of Hodge type (for which we know
(G', X+), and then we can P(G2, X2) by (a) and (b)) with (G de`, X2) apply Prop. 2.9. The problem here is that this only gives the existence of a weakly canonical model of Sh°(G', X+)c over E(G2, X2), which in general is a proper field extension of E(Gd, X+). (Notice that (G2, X2) is required to be of Hodge type-without this condition there would be no problem.) Thus we see that our "naive" strategy has to be corrected. This is done in two steps.
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First one assumes that G is Q-simple, and one considers the maximal covering GO -> Gad (as in 2.10) which occurs as the semi-simple part in a Shimura datum of Hodge type. As explained, the Shimura data (G2, X2) of Hodge type with GZe`' GO in general have E(Gad, X+) C E(G2i X2). Deligne shows, however, that by "gluing in" a suitable toric part, the field extension E(G2, X2) can be made in almost every "direction"; for a precise statement see [De3], Prop. 2.3.10. Finally one shows that this is enough to guarantee the existence of a canonical model of Sh°(G',X+)ei one proves (ibid., Cor. 2.7.19): if for every finite extension F C 0 of E = E(G', X+), there exists another finite extension E C_ F' C 0 which is linearly disjoint from F, and such that Sh°(G', X+)e has a weakly canonical model over F', then it has a canonical model. Putting everything together, one obtains the following result. 2.12 Theorem. (Deligne, [De3]) Let (G, X) be a Shimura datum, and let (Gad, Xad) = (G1i X1) x x (G,,,,, be the decomposition of its adjoint datum into simple factors. Suppose that, using the notations of 2.10, Gder is a quotient of GO x x GO. Then there exists a canonical model of Sh(G, X).
Notice that it is not clear whether this statement covers all data (G, X) of abelian type.
To extend this result to arbitrary Shimura data, additional arguments are needed. Since eventually we want to apply a Galois descent argument, it would be useful if we could first descend Sh(G, X)e to a scheme over Faltings has shown that this can be done using a rigidity argument.
2.13 Theorem. (Faltings, [Fal]) Let G be a semi-simple algebraic group over Q, C G(R) a maximal compact subgroup, and F C G(Q) a neat arithmetic subgroup. If X = G(R)/K,,. is a hermitian symmetric domain, then the locally symmetric variety F\X (with its unique structure of an algebraic variety) is canonically defined over 0. The special points on F\X are defined over Q. If 171, F2 C G(Q) are neat arithmetic subgroups, ry E G(Q) an element with ryF1ry-1 C_ F2, then the natural morphism ry: F1\X -+ F2\X is also defined over Q. Next we have to recall Langlands's conjecture on the conjugation of Shimura varieties (now a theorem, due to work of Borovoi, Deligne, Milne, and MilneShih, among others). We will not go into details here; the interested reader can consult [Bol], [Bo2], [Mil], [MS].
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2.14 Theorem. (Borovoi, Deligne, Milne, Shih, ... ) Given a Shimura datum (G,X), a special point x E X, and a T E Aut(C), one can define a Shimura datum (T,xG, T,xX ), a special point Tx E T,xX, and an isomorphism G(Af) - T,xG(Af), denoted g T,xg, satisfying the following conditions (writing T(g) for the action of an element g E G(Af) on Sh(G, X)c) (i) There is a unique isomorphism cpT,x: TSh(G,X)c T(T,xg) with 7'T,x(T [x, 1]) = ["x, 1] and with cpT,x o "T(g) = o cor,x for all Sh(TxG,T,xX)c
g E G(Af) (ii) If X' E X is another special point then there is an isomorphism Sh(T,xG, T,xX)e
Sh(Tx'G, T°x'X)c such that cp(T; x, x') o W',x = cpT,x' and such that go (,r; x, x')oT(T,xg) = y(T,x'g)oco(T; x, x') for all g E G(Af ). cp(T; x, x') :
As explained in [La], section 6 (see also [Mi2], section 11.5), using the theorem one obtains a "pseudo" descent datum from C to E = E(G, X) on Sh(G,X)c. By this we mean a collection of isomorphisms U,: T Sh (G, X )c - Sh(G, X)c}TEAut(c/E)
satisfying the cocycle condition f°T = f° o ° fT. At several places in the literature (e.g., [La], section 6, [Mi2], p. 340, [MS], §7) it is asserted that "by descent theory" this gives a model of Sh (G, X )c over E. (Due to the properties of the fT, this model would then be a canonical model.) We think that this argument is not complete-let us explain why. 2.15 To descend a scheme Xc from C to a number field E C C, it does in general not suffice to give a collection of isomorphisms { fT : T Xc Xc}TEAut(c/E) with ffT = ff o 'f, (or, what is the same, a homomorphism of groups a: Aut(C/E) -+ Aut(Xc) sending T to a r-linear automorphism of Xc). For instance, using the fact that Q is an injective object in the category of abelian groups, we easily see that there exist non-trivial group homomorphisms c: Aut(C/E) -- Q. Taking Xc = 4, on which we let T E Aut(C/E) act as the T-linear translation over c(-r), we get an example of a non-effective
"pseudo" descent datum. The same remarks apply if we replace C by i. (Thus, for example, [Mi4J, Lemma 3.23 is not correct as it stands.) In this context it seems useful to remark the following. Given a Q-scheme XCF, one might expect that a descent datum on XQ- relative to O/E can be expressed as a collection of isomorphisms 1701: T XX - XX}TEGaI(O/E) for which, apart from the cocycle condition c°,, = cpO o °cpT, a certain "continuity condition" holds. To see why a continuity condition should enter, one must realize that a scheme such as Spec(O(&E0) is not a disjoint union of copies of
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Spec(Q) indexed by Gal(Q/E) (which would not be a quasi-compact scheme), but rather a projective limit Spec(O ®E 0) = 1F Spec (0)Gal(F/E), where F runs through the finite Galois extensions of E in Q. (In other words: this is Gal(Q/E) as a pro-finite group scheme.) It seems though that it is not so easy to formulate the desired continuity condition directly. Even if one succeeds
in doing this, however, it should be remarked that descent data relative to Q/E are not necessarily effective (cf. [SGA1], Exp. VIII). Since we are really only interested in effective descent data relative to Q/E, we take a slightly different approach. Let us call a (semi-linear) action a: Gal(i/E) --* Aut(X') of Gal(O/E) on a u-scheme X' continuous if it is continuous as an action of a locally compact, totally disconnected group (see [De3], section 2.7). Since the Galois group is actually compact, the following statement is then a tautology.
2.15.1 The functor X quasi-projective schemes X over E
X' = X ®E Q gives an equivalence of categories eq.
i
quasi-projective schemes X' over Q with a continuous semi-linear action of Gal(Q/E)
We thus see that, in order to prove the existence of canonical models in the general case, we need to show that Theorem 2.14 provides us with a continuous Galois action on Sh(G,X)O. For this we will use the following lemma.
2.16 Lemma. Let (G, X) be a Shimura datum, K C G(Af) a compact open
subgroup, and let S = I\X+ be a connected component of ShK(G,X)G. Then we can choose finitely many special points xl,... , x,,, E S° such that S has no non-trivial automorphisms fixing the xi. Proof. Let j : S " S* denote the Baily-Borel compactification. Every automorphism of S extends to an automorphism of S*. There exists an ample line bundle G on S* such that a*L =' G for every a E Aut(S). In fact, if G has no simple factors of dimension 3 then we can take G := j*Sts, where d = dim(S). In the general case one has to impose growth conditions at infinity: using the terminology of [BB] we can take for G the subsheaf of j*SZS (now taken in the analytic sense) of automorphic forms which are integral at infinity. (So L is
the bundle 0(1) corresponding to the projective embedding of S* as in loc. cit., §10. Mumford showed in [Mu3] that if S is a smooth toroidal compactification and 7r: S -* S* is the canonical birational morphism, then lr*L is the sheaf Q9' (log 9k).)
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Let P be the "doubled" Hilbert polynomial of G, given by P(x) = Pc(2x). Recall from [FGA], Expose 221, p. 20, that the scheme Hom(S*, S*)P given by
Hom(S*, S*)P(T) = {g: S* x c T -+ S* x c T I x((G ®OT g*.C)®") = P(n)}
is of finite type. With the obvious notations, it follows that Aut(S*)P is a scheme of finite type, being a locally closed subscheme of Hom(S*, S*)P. The lemma now follows from the following two trivial remarks:
(i) if a E Aut(S*) fixes all special points of S then a = id, (ii) if x1i ... , x" are special points of S° then Aut(S*; x1,
... , x")' := {a E Aut(S*)P I a(xi) = xi for all i = 1,...
,
n}
is a closed subgroup scheme of Aut(S*)P.
2.17 We now complete the argument showing that Sh(G, X)C has a canonical model. Obviously, the first step is to use Theorem 2.13, so that we obtain a model Sh(G, X)o over Q. We claim that the "pseudo" descent datum { fT : T Sh(G, X)c
-* Sh(G, X)c}TEAut(C/E) considered in 2.14 induces
a semi-linear action of Gal(i/E) on Sh(G, X)U, which is functorial. We can show this using the special points: if Sh(T, {h})c y Sh(G, X)c is a 0dimensional sub-Shimura variety, then the canonical model Sh(T, {h}) over E' = E(T, {h}) gives rise to a collection of isomorphisms { f° : °Sh(T, {h})c Sh(T, {h})C}QEAut(C/E'), and for o E Aut(C/E'), the two maps fo and fo are equal on °Sh(T, {h})C. Using the fact that the special points on Sh(G, X)C
are defined over 0 for the 0-structure Sh(G, X)U, one now checks that the f° induce a system {cpT :
T Sh(G, X)o
Sh(G, X)O}TEGaI(O/E)
with cpoT = cp° o °cp.. What we shall use is that the action on the special points agrees with the one obtained from the canonical models Sh(T, {h}). Now for the continuity of the Galois action on Sh(G,X)U. First let us remark that it suffices to prove that the semi-linear Galois action on each of the ShK (G, X) is continuous, since the transition morphisms then automatically descend. Here we may even restrict to "levels" ShK where K is neat. Furthermore, it suffices to show that there is an open subgroup of Gal(O/E) which acts continuously. In fact, if we assume this then ShK (G, X) descends to a finite Galois extension F of E. On the model ShK(G, X)F thus obtained
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we still have a Galois descent datum relative to F/E, and since this is now a finite Galois extension, the descent datum is effective. Since ShK (G, X )jCF is a Q-scheme of finite type, there exists a finite extension E' of E and a model SEA of ShK (G, X) over E'. This model gives rise to
semi-linear action of Gal(i /E') on ShK (G, X )U, which we can describe as a collection of automorphisms to,: T ShK(G, X)Q-
ShK(G, X)0}1EGaI(0/E)
Observe that cp- o (,T)-1 is a Q-linear automorphism of ShK(G, X)U, and that IT E Gal(U/E) I co = bT} is a subgroup of Gal(Q/E). At this point we apply Lemma 2.16. It gives us special points x1, ... , x, E ShK(G,X)U such that there are no automorphisms of ShK(G,X)U fixing all xi. For each xi, choose a closed immersion ji : (Ti, {hi}) " (G, X) and an element gi E G(Af) such that xi lies in gi Sh(Ti,{hi})U C Sh(G,X)c. Let Ki := ja 1(K) C Ti(Af). There exists a finite extension E" of E', containing the reflex fields E(xi), such that the xi are all E"-rational on the chosen model SEA and such that furthermore all points of ShK;(Ti, {hi}) are rational over E" (for every i = 1,... , n). It now follows from what was said above that the two Galois actions on ShK (G, X )U, given by the cpr and the 0T, respectively, are the same when restricted to Gal(Q/E"). This finishes the proof of the following theorem.
2.18 Theorem. Let (G, X) be a Shimura datum. Then there exists a canonical model Sh(G, X) of the associated Shimura variety. 2.19 Remark. In [Pi], the notion of a canonical model is generalized to the mixed case, and the existence of such canonical models is proven for arbitrary mixed Shimura varieties. Pink's proof essentially reduces the problem to statement 2.18; once we have 2.18, the mixed case does not require any further corrections.
2.20 Remark. There is also a theory of a canonical models for automorphic vector bundles on Shimura varieties. The interested reader is referred to [Ha] and [Mi2].
§3 Integral canonical models 3.1 Let (G, X) be a Shimura datum with reflex field E = E(G, X ), and let v be a prime of E dividing p > 0. We want to study models of the Shimura
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variety Sh(G, X) over the local ring of E at v. In our personal view, the theory of such models is still in its infancy. How to set up the definitions, what properties to expect, etc., are dictated by the examples where the Shimura variety represents a moduli problem that can be formulated in mixed characteristics (notably Shimura varieties of PEL type). Even in the case where G is unramified over Qp, this leaves open some subtle questions. Some of the rules of the game become clear already from looking at Siegel modular varieties. We have seen that the canonical model in this case can be identified with the projective limit 1 Ag,l,n®® Q. Fixing a prime number p, we see that, for constructing a model over Ziri, we run into problems at the levels A9,1,,, with p I n. By contrast, if we only consider levels with p f n, then we have a natural candidate model, viz. ] prn A9,1,n ® Zipi, which has all good properties we can expect. Returning to the general case, this suggests the following set-up. Let (G, X), E and v be as above. We fix a compact open subgroup Kp C G(Qp), and we consider ShKp(G,X) = 11_mShKKXKP(G,X), Kp
where Kp runs through the compact open subgroups of G(AA). It is this scheme ShKp (G, X), the quotient of Sh (G, X) for the action of Kp, of which we shall study models. Notice that we can expect to find a smooth model (to be made precise in a moment) only for special choices of Kp.
3.2 Definition. Let (G, X) be a Shimura datum, E = E(G, X), v a finite prime of E dividing p, and let Kp be a compact open subgroup of G(Qp). Let 0 be a discrete valuation ring which is faithfully flat over 0(v). Write F for the quotient field of 0. (i) An integral model of ShKp (G, X) over 0 is a faithfully flat 0-scheme M with a continuous action of G(Af) and a G(AA)-equivariant isomorphism
M®F=ShKp(G,X) ®E F. (ii) An integral model M of ShKp (G, X) over 0 is said to be smooth (respectively normal) if there exists a compact open subgroup C C G(Af), such that for every pair of compact open subgroups KP C K2 C G(AA) contained in C, the canonical map ,M/KP -4 M/K2 is an etale morphism between smooth (resp. normal) schemes of finite type over 0. It should be clear that an integral model M, if it exists, is by no means unique. For example, given one such model, we could delete a G(AA)-orbit properly contained in the special fibre, or we could blow up in such an orbit,
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to obtain a different integral model. To arrive at the notion of an integral canonical model, we will impose the condition that M satisfy an "extension property", similar to the Neron mapping property in the theory of Neron models (cf. [BLR], section 1.2). This idea was first presented by Milne in [Mi3]. As we shall see, one of the main difficulties in this approach is to find a good class of "test schemes" for which the extension property should hold. Given a base ring 0, we will work with a class of 0-schemes that we call "admissible test schemes over 0", abbreviated "a.t.s.". We postpone the precise definition of the class that we will work with until 3.5.
3.3 Definition. Let (G,X), E, v, K,,, 0 and F be as in 3.2. (i) An integral model M of ShKK (G, X) over 0 is said to have the extension property if for every admissible test scheme S over 0, every morphism SF -4
MF over F extends uniquely to an 0-morphism S -+ M. (ii) An integral canonical model of ShK,, (G, X) at the prime v is a separated smooth integral model over which has the extension property. A local integral canonical model is a separated smooth integral model over 01:= O having the extension property. 3.4 Comments. A definition in this form was first given by Milne in [Mi3]. As admissible test schemes over 0 he used all regular 0-schemes S for which SF is dense in S. Later it was seen that this is not the right class to work with (cf. [Mi4], footnote on p. 513); the reason for this is the following. One wants to set up the theory in such a way that 1 {", A9,1,,, 0 Z(p) is an integral canonical model for the Siegel modular variety. Using Milne's definition, this boils down to [FC], Cor. V.6.8, which, however, is false as it stands. Recall that this concerns the following question: suppose given a regular scheme
S with maximal points of characteristic 0, a closed subscheme Z -4 S of codimension at least 2, and an abelian scheme over the complement U = S\Z.
Does this abelian scheme extend to an abelian scheme over S? In loc. cit.
it is claimed that the answer is "yes"-this is not correct in general. A counterexample, due to Raynaud-Ogus-Gabber, is discussed in [dJO], section 6.
Let us try to explain the gist of the example, referring to loc. cit. for
details. As base scheme we take S = Spec(R), where R = W(Pp)Qx, yJ/((xy)P-1p) . There exists a primitive pth root of unity (p in R. Let s E S be the
closed point, and set U1 = D(x), U2 = D(y), U = S \ {s} = U1 U U2, U12 = D(xy) = U1 f1 U2. We obtain a finite locally free group scheme Gu of
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rank p2 over U by gluing
Gl = (pp X Z/pZ)Ul
and G2 = (pp X Z/pZ)UZ
via the isomorphism G1IU12
GZ1U,Z
given by the matrix
01
1J,
where Q: pp - Z/pZ (over U12) maps (p E F(U12i pp) to 1 E r(U12i Z/pZ). One easily sees from the construction that we have an exact sequence
0-+(Z/pZ)U-24 GU-3pp,U-4 0, and that this extension is not trivial. The group scheme GU extends uniquely to a finite locally free group scheme G over S. Also, the homomorphism yu extends uniquely to a homomorphism
y: (Z/pZ)s -+ G, which, however, is not a closed immersion. (The whole point!) To get the desired example, one only has to embed G into an abelian scheme X over S (using the theorem [BM3], Thm. 3.1.1 by Raynaud), and take YU := Xu/(Z/pZ)u, where (Z/pZ)u is viewed as a subgroup scheme of XU via yu and the chosen embedding G y X. To understand what is going on, the following remarks may be of help. One can show that the fibre Gs is isomorphic to ap x ap. There is a blowing
up 7r: S -+ S with center in s such that (Z/pZ)u' Gu extends to a closed flat subgroup scheme N " G. Over S, the abelian scheme YU extends to the abelian scheme YS := XS/N. When restricted to the exceptional fibre E, we have YSJE = (XS x E)/NE, where NE " (ap x ap)E is a non-constant subgroup scheme isomorphic to ap. Therefore, we cannot blow down Yg to an abelian scheme over S. In order to guarantee that 1 p{-, A9,1,,,,®7G(p) is an i.c.m., we want our a.t.s. to satisfy the following condition. (Here 0 is a d.v.r. with field of fractions F and S is an 0-scheme.) (3.4.1)
for every closed subscheme Z y S, disjoint from SF and of codimension at least 2 in S, every abelian scheme over the complement U = S \ Z extends to an abelian scheme over S.
On the other hand, we want that an integral canonical model, if it exists, is unique up to isomorphism. Thus we want it to be an a.t.s. over itself. The notion that we will work with in this paper is the following.
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3.5 Definition. Let 0 be a discrete valuation ring. We call an O-scheme S an admissible test scheme (a.t.s.) over 0 if every point of S has an open neigbourhood of the form Spec(A), such that there exist 0 C_ 0' C_ AO C A, where -O C_ O' is a faithfully flat and unramified extension of d.v.r. with O'/(ir) separable over 0/(7r), -AO is a smooth O'-algebra, and where -Spec(A) -* Spec(Ao) is a pro-etale covering. We write ATSo for the class of a.t.s. over O.
We want to stress that this should be seen as a working definition, see also the remarks in 3.9 below. Clearly, a smooth model of a Shimura variety over 0 belongs to ATSo. In particular, we have unicity of integral canonical models:
3.5.1 Proposition. Let (G, X) be a Shimura datum, v a prime of its reflex field E dividing the rational prime p, and let Kp be a compact open subgroup ofG(Qp). If there exists an integral canonical model of ShKp(G, X) over then it is unique up to isomorphism. Furthermore, we have the following properties. (3.5.2) If S E ATSo then S is a regular scheme, formally smooth over O. (To prove that the local rings of S are noetherian, we can follow the arguments of [Mi3], Prop. 2.4.)
(3.5.3) If 0 C_ 0' is an unramified faithfully flat extension of d.v.r., then S E ATSo = S E ATSo, and S E ATSo (S ®o 0') E ATSo. Next we investigate whether (3.4.1) holds. For this we use the following two lemmas.
3.6 Lemma. (Faltings) Let 0 be a d.v.r. of mixed characteristics (0, p) with p > 2. Suppose that the ramification index e satisfies e < p - 1. Then every regular formally smooth O-scheme S satisfies condition (3.4.1). Proof (sketch).
As mentioned above, some statements in [FC], section V.6,
are not correct. The mistake can be found on p. 182: the map p- dim(G) . traceG[pn+']/G[nn1 is not a splitting of the map OG[,n] C OGLpn+1], as claimed.
Most arguments in the rest of the section are correct however, and with some
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modifications we can use them to prove the lemma. Let us provisionally write
RFSo for the class of regular, formally smooth 0-schemes. For S E RFSn, we have the following version of [FC], Thm. V.6.4'.
3.6.1 Let S be a regular, formally smooth 0-scheme (0 as above, with e < p - 1), and let U " S be the complement of a closed subscheme Z " S of codimension at least 2. Then every p-divisible group Gu over U extends uniquely to a p-divisible group G over S.
The only step in the proof of [FC], Thm. V.6.4' that we have to correct is the one showing the existence of an extension G in case dim(S) = 2 (loc. cit.,
top of p. 183). So we may assume S = Spec(R) U = S \ {s}, where R is a 2-dimensional regular local ring, and where s is the closed point of S. The cu,n := C9,,[p"]]extend uniquely to an inductive system of finite flat group schemes {Gn;in: G,, -4 Gn+i}. (See [FC], Lemma V.6.2.) We have to prove
that the sequences (3.6.2)
0-+cJ
in
F,n
GGn+l-ICJ',--4 0
are exact. That i,, is a closed immersion needs to be checked only on the closed fibre. The formal smoothness of R over 0 guarantees that there exists an unramified faithfully flat extension of d.v.r. 0 C 0' such that S has a section over 0' with s contained in the image. Pulling back to 0', it then follows from [Ral], Cor. 3.3.6, that i,, is a closed immersion. Finally, this implies that Gn+l/Gn is a finite flat extension of Gu,I, and because of the unicity of such an extension it follows that (3.6.2) is exact.
It remains to be checked that, using 3.6.1 to replace [FC] Thm. V.6.4', all steps in the proof of ibid., Thm. 6.7 go through for S E RFSn. One has to note that in carrying out the various reduction steps, we stay within the class RFSo. At some points one furthermore needs arguments similar to the above ones, i.e., taking sections over an extension 0' and using [Ral], Cor. 3.3.6. We leave it to the reader to verify the details.
3.7 Lemma. Let (G, X) be a Shimura datum, and let v be a prime of E(G, X) dividing p. Assume that GQ, is unramified (see 3.11 below). Then v is an unramified prime (in the extension E(G, X) D Q). Proof.
See [Mi4], Cor. 4.7.
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3.8 Corollary. Notations as in 3.2. If p > 2 then every S E ATSO(,) satisfies (3.4.1). In particular, if p > 2 then 1 m t,ti A9,1,,, ®Zipi is an integral canonical model of ShK,, (CSp29,Q,S59 ), where Kp = CSp29(7GP).
Proof. We can follow Milne's proof of [Mi3], Thm. 2.10, except that we have to modify the last part of the proof in the obvious way. Notice that the group CSp29 is unramified everywhere, so that Lemmas 3.6 and 3.7 apply. O
3.9 Remarks. (i) We do not know whether the corollary is also true for p = 2. (Note that in the example in 3.4, the base scheme S is not an a.t.s. over W or W[(].) This is one of the reasons why we do not pretend that Def. 3.5 is in its final form. (ii) Our definitions differ from those used in [Va2]. Vasiu's definition of
an integral canonical model is of the above form, but the class ATSo he works with is the class of all regular schemes S over Spec(O), for which the generic fibre SF is Zariski dense and such that condition (3.4.1) holds. As we have seen above, this contains the class we are working with if p > 2 and
e(O) < p - 1. It seems to us that Vasiu's definition is more difficult to work with. For example, it is not clear to us whether his notion of an a.t.s. is a local one, and whether it satisfies S E ATSo = (S ®o O') E ATSo,. (This is important
for some of the constructions.) On the other hand, if we want that the extension property is preserved under extension of scalars from O to to work with a class ATSo which is not "too small". Here we should draw a comparison with the theory of Neron models: we note that the proof of [BLR], Thm. 7.2.1 (ii) makes essential use of Weil's
theorem, ibid. Thm. 4.4.1, for which we see no analogue in the context of Shimura varieties. This may help to explain why we set up the Def. 3.5 the way we did.
3.10 Proposition. Let (G, X), E = E(G, X), v and Kp be as in 3.2. (i) There exists an integral canonical model of ShKP (G, X) at v if and only if there exists a local integral canonical model. (ii) Suppose that p > 2 and that the prime v is (absolutely) unramified.
Write B for the fraction field of W(lFp), and choose an embedding O ' W (1Fp), where O = O is the completed local ring of OE at v. Suppose there exists a smooth integral model M for ShKP (G, X) ® B over W (Fp) having the extension property. Then there exists an integral canonical model of ShKP (G, X) over O().
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Proof. (i) In the "only if" direction this readily follows from (3.5.3). For the converse, suppose that M0 is a local integral canonical model of ShKP(G,X)
over 0,,. We have MO = 1 m.MKP, where KP runs through the compact open subgroups of G(AA ). Write S' = S= and if = Spec(E) Also write S" = S' X,s S', r7" = 77' x., if, and
write pi (i = 1, 2) for the ith projection S" -> S' (resp. 77" -+ r7'). On the generic fibre MO ® E we have an effective descent datum relative to if 77. If we consider pi (MO (D 77" as a r7'-scheme via P2: eta" -+ if, then this descent datum is equivalent to giving a morphism pi(MI (D E over if. (Here we ignore the cocycle condition for a moment.) Since pi.M0, considered as a S'-scheme via P2: S" -4 S', is an a.t.s. over S', and since M0 was assumed to have the extension property, the descent datum on MO 0 E
extends to one on MO relative to S' -* S. (It is clear that the extended descent datum again satisfies the cocycle condition, MO being separated.) By the arguments of [BLR], pp. 161-162, the extended descent datum is effective. (We can work with each of the MKp separately, and since MO is
a smooth model, we may furthermore restrict our attention to those MKp which are smooth over 0,,.) Thus we obtain a smooth model M over It remains to be shown that this model again has the extension property. This follows easily from property (3.5.3) and the fact that descent data for morphisms are effective ([BLR], Prop. D.4(b) in section 6.2). (ii) The descent from .M to a local i.c.m..MO is done following the same argument. By (i) this suffices. 0
3.11 From now on, we will concentrate on the case where KP C G(Q ) is a hyperspecial subgroup. This means that there exists a reductive group scheme 9zp over Z (uniquely determined by KP) with generic fibre GQ, such that KP = (Zr,). Hyperspecial subgroups of G(Q) exist if and only if G is unramified, i.e., quasi-split over QP and split over an unramified extension. For more on hyperspecial subgroups we refer to [Ti], [Va2]. One can show ([Va2], Lemma 3.13) that the group 9z, is obtained by pull-back from a group scheme 9 over Z (p). This suggests that we define an 11
integral Shimura datum to be a pair (9,X), where 9 is a reductive group scheme over Z(p), and where, writing G = 9Q, the pair (G, X) is a Shimura datum in the sense of 1.44. To (9,X) we associate the Shimura variety 'We hasten to add that one has to be careful about morphisms: if we have two pairs (91,X1) and (92,X2) plus a morphism f : (G1, X1) -* (G2i X2) such that f (KP,1) 9 KP,2 then it is not true in general that f extends to a morphism f : 91 -+ 92; cf. [BT], 1.7 and
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Sh(g,X) := ShK,(G,X), where of course Kp := g(Zp). Suppose G , is unramified. Whether there exists an integral canonical model of ShKP (G, X) does not depend on the choice of the hyperspecial subgroup Kp C G(Qp). This is a consequence of the fact that the hyperspecial subgroups of G(Qp) are conjugate under Ge"(Q,), see [Va2], 3.2.7.
3.12 Examples. (i) Let (T, {h}) be a Shimura datum with T a torus. The group TQ, is unramified precisely if the character group X * (T) is unramified at p as a Gal(O/Q)-module. If this is the case then TQ, extends uniquely to a torus T over Zp, and Kp := T(Zp) is the unique hyperspecial subgroup of T (Qp) . Let Kp C T(AA) be a compact open subgroup. It follows from the description given in 2.2 and 2.3 that ShKpXKP(T, {h}) = Spec(L1 x . . . x L,)
for certain number fields Li D E which are unramified above p. Now set MK,,xKP = Spec(01 x ... x O,), where Oi is the normalization of in Li. Then 1m_ KP MKPXKP is an integral canonical model of ShKP(T, {h}) over O(v) .
(ii) If (G, X) defines a Shimura variety of PEL type, then we can use the modular interpretation of Sh(G, X) to study integral canonical models. As mentioned before, the precise formulation of a moduli problem requires a lot of data, and we refer to [Ko2] for details. We remark that the Shimura varieties
that we are interested in, in general only form an open subscheme of the moduli space studied in loc. cit., section 5. The arguments given there (see also [LR], §6) show that, for primes p satisfying suitable conditions which imply the existence of a hyperspecial subgroup Kp C G(Q.), the Shimura variety ShK,(G, X) has an i.c.m. over
for all primes v of E(G, X) above
p 3.13 Remark. If there exists an i.c.m. M for Sh(G, X ), then one expects that each "finite level" MKP is a quasi-projective This is certainly the case for the examples in 3.8 and 3.12. Moreover, one easily checks that the quasi-projectivity is preserved under all constructions presented in this section.
3.14 Our next goal is to show that if GQ, is unramified, then we can adapt [De3], 2.1.5-8 (which we summarized in 1.6.5) to the present context. The connected component of ShK,(G, X)U containing the image of X+ x {e} is the
projective limit 1'm r\X+, where r = Im([Gder(Q)+fl(KpxKP)] -+ G'(Q)+) for some compact open subgroup Kp C Gder(AA) (Here we use [De3], 2.0.13.) [Va2], 3.1.2.
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Formalizing this, we are led to consider pairs (G', X+) consisting of a semisimple group g' over Z(p) and a CJ'ad(R)+-conjugacy class of homomorphisms
h: S -4 G' (writing G' =
Gad
= GQ := (G')Q) such that conditions (i), (ii) and (iii) in 1.4 are satisfied. For such a pair we define the topology T(g') on Gad(7Gipi) as the linear topology having as a fundamental system of neighbourhoods of 1 the images of the {p, oo}-congruence subgroups Q'(Z(p)) nKp, where Kp is a compact open subgroup of We then write Sh°(c', X+)c := ] r \x+, where r runs through the {p, oo}-arithmetic subgroups of gad(Z'(p)) which are open in r(g'). gad(Z(p))+^ (completion On Sh°(g', X+)c we have a continuous action of rel. r(g')), and by 2.13, these data are all canonically defined over 0. (Even over a much smaller field, as we shall see next.) For an integral Shimura datum (9,X) and a connected component X+ C_ X, the corresponding connected component of Sh(g, X)U is a scheme with continuous gad(Z(p))+^-action, isomorphic to Sho(cder X+)U. Note that Sh°(g',X+) is an integral scheme (use [EGA], IV, Cor. 8.7.3).
3.15 Lemma. Let (G,X) be a Shimura datum, E = E(G,X), v a prime of E dividing p. Assume that GQ, is unramified, and let Kp C G(Qp) be a hyperspecial subgroup. Then the connected components of ShKK (G, X) are defined over an abelian extension k of E which is unramified above p. Proof. First we prove this under the additional assumption that Gder is simply connected. The G(C)-conjugacy class of homomorphisms Gm,e
Gc (for x E X) gives rise to a well-defined cocharacter gab : Gm,e -> G" C, which has field of definition E(Gab Xab) C E. Writing TE = ResE/QGm,E, we get a homomorphism p
NmnogE: TE - Gab,
inducing a map p(A/Q): AE/E* -> Gab(A)/Gab(Q) = 7r (Gab). The assumption that Gder is simply connected implies (see [Del], 2.7) that Toir(G) is a quotient of 7r(Gab). Moreover, the action of Gal(U/E)ab on 7ro(Sh(G, X)) factors through 7rop(A/Q) : lrolr(TE) -3 7ro7r(Gab) By class field theory it therefore suffices to show that the image under p(Qp) of Cp := O;, C TE(Qp) in Gab (%) is contained in Kpb := Im (Kp C G (Qp) -+ Gab (Q) ) The fact that GQ, is unramified implies ([Mi4], Cor. 4.7) that TE is unramified over Qp, so it extends to a torus TE over Zp. Clearly, Cp = TE(7Gp). Write g for the extension of GQ to a reductive group scheme over Zp with Kp = c(Zp). The map p extends to a homomorphism TE _, gab over 7Gp,
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hence we are done if we show that Gab(Zp) = Kpb, i.e., G(Zp) maps surjectively to Gb(Z ). Again using that Gder is simply connected we have H1(Qp, Gder) = {1}, hence G(Qp) - Gab(Qp). For s E Gab(Zp) we thus can lift the corresponding s,7 E Gab(Qp) to s,7 E G(%). Taking the Zariski closure of the image of s,, inside G then gives the desired Zp valued point s of G mapping to s. The general case is reduced to the previous one. An easy generalization of [MS], Application 3.4 shows that there exists a morphism of Shimura data f : (G1i X1) -3 (G, X) such that f der ; Gder -+ Gder is the universal covering of Gder, such that E(G1, X1) = E(G, X), and such that there is a hyperspecial subgroup Kp C G1(Qp) with f (Kp) C_ Kp. This suffices to prove the lemma, since the components of Sh1p (G1i X1) map surjectively to components of ShKP (G, X) and since all components have the same field of definition (being permuted transitively under the G(Af )-action).
3.16 Consider a pair (G', X+) as in 3.14. Write G' = GQ, and write k for the maximal subfield of E(Gad, Xad)ab which is unramified above p. The lemma implies that the connected Shimura variety Sh°(G', X+) has a welldefined "canonical" model over E. Indeed, we can choose an integral Shimura datum (G, X) with G' = Gder, X+ C X and E(G, X) = E(Gad, Xad), and take Sho(Gder X+)E (which makes sense, grace to the lemma) as the desired model.
That this does not depend on the chosen pair (G, X) follows from the facts in 2.7.
3.17 Definition. Write Sh°(G', X+)E for the model over E just defined, and let w be a prime of f above p. We adapt Def. 3.2 to connected Shimura varieties, replacing E by E and G(Af) by G(7Gipi)+^. Then an integral canonical model (resp. local i.c.m.) for Sh°(G', X+)E at w is a separated smooth integral model over O(,,) (resp. Ov,) which has the extension property. Of course, the point of this definition is that a Shimura variety can be recovered from the (or rather: some) corresponding connected Shimura variety by an "induction" procedure. This will enable us to follow the same strategy as in 2.10. We consider the properties P(G, X; v) :
there exists an i.c.m. for Sh(G, X) over
(for (G, X) an integral Shiumura datum, v a prime of E = E(G, X) above p), and
P°(G', X+; w) :
there exists a local i.c.m. for Sh°(G', X+)E over O(w)
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(for (G', X+), E and w as above). Using the induction technique of [De3], Lemma 2.7.3 and Prop. 3.10, we can prove the following statement. We leave the details of the proof to the reader.
3.18 Proposition. Notations as above, with g' = gder, X+ C X. Suppose that v and w restrict to the same prime of E fl E. Then P(9, X; v) PO(gder X+; W)
From now on we restrict our attention to the case p > 2. Recall that it is implicit in our notations that we are working at a prime where the group is unramified, since G and G' are supposed to be reductive group schemes over 7L(p). Write P(G, X) for "P(9, X; v) holds for all primes v of E above p", and similarly for P°(9', X+). We have shown that statements (a) and (c) in 2.10 hold. Furthermore, statement (d) is almost trivially true. By contrast, it is not at all obvious how to prove (b). The only thing we get more or less for free is a good normal model.
3.19 Proposition. Let is (GI, Xl) " (G2i X2) be a closed immersion of Shimura data such that there exist hyperspecial subgroups Kj,p C G;(Qp) with i(K1,p) C K2,p. Suppose there exists an i.c.m. M for ShK,,y(G2i X2) over QE,,( ). If w is a prime of El = E(G1, X1) above v then there exists a normal integral model Al of ShK,,, (Gl, X1) over CAE,,(,,,) which has the exten-
sion property (see Def. 3.3). Proof. Let 9j (j = 1, 2) denote the extension of G; to a reductive group scheme over Z(p) with c, (Z,,) = Kj,p. Write K for the set of pairs (K', K2 P)
of compact open subgroups K, C G,(AA) such that i(Kp) C K2, partially ordered by (Kp, K2) -< (Lp, L2) ifKp D Li and K2 D L. Given (Kp, K2) E K, we have a morphism i(Kp, K2): ShKi (GJ1, XI) -+ ShK2 (c2, X2)
MKs ® QE1,(w)
Write N(Kp, K2) for the (scheme-theoretical) image of i(K', K2), and let N(Kp, K2) be its normalization. For fixed Kp we set
N(K1,K2), Nxi=
NK = K2
Ki
N(K',Kz), MK =
MK2, K2
where the limits run over all K2 such that (Kp, K2) E K. Also we set
N:=1 NKf. Kr
Kf
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First we show that, for Kp D L' sufficiently small, the canonical morphism NLi -4 MKP is etale. For this, we take compact open subgroups C C G; (Af ) with i(Cp) C C2 P, and such that for all KjP D LPj contained in CP (j = 1, 2), the transition morphisms ShLi (g1i Xl) -- ShKi (91, Xl) and MLz _+ MK2 are etale morphisms of smooth schemes over E1 and OE2i(v) respectively. One
checks that for all such Kp D LP, the morphism t: MLi - MKi is again etale, of degree [Kp : LP]. It follows that NLi -* NKi is a pull-back oft, hence etale. Now NKi has finitely many irreducible components, being a schemetheoretical image of ShKP(91iX1), and the normalization of NKi is just NKi. Using this remark, it follows that NLi -> NKi is etale, so that N is a normal model of Sh(91, X1) over OE1,(w). That N has the extension property is seen as follows. We consider an S E ATS0 (with 0 = OE1i(w)) and a morphism aE1 : SE1 -> ME1 on the generic fibre. The fact that OE1,(w) is an unramified extension of OE2i(v) implies, using (3.5.3), that M 0 OE1,(w) has the extension property over OE1,(w), hence aE extends to a morphism
a: S-*N,
) M0OEI,(w)
Now fix (Kp, K2) E IC, and set
S = S(Kp, K2) := S
x N(Kr,Ki )
M(Kp) K2) -° 3 S.
Then S is integral over S, since p is a pull-back of the normalization map M(Kp, KZ) -* N(Kp, K2 P). On the generic fibre, p is an isomorphism. Since S is a normal scheme (being an a.t.s.), it follows that p is an isomorphism, hence a lifts to a: S M.
3.20 Remark. Suppose M is an integral model of a Shimura variety over a d.v.r. 0. We will say that M has the extended extension property (e.e.p.),
if it satisfies the condition that for every S = Spec(01) with 0 C 01 a faithfully flat extension of d.v.r., setting F = Frac(01), every morphism aF: Spec(F) -* MF over 0 extends to an 0-morphism a: S -> M. It follows from the Neron-Ogg-Shafarevich criterion that the model of the Siegel modular variety as in 3.8 enjoys the e.e.p. Also it is clear that in the
situation of 3.19, we have the implication "M has the e.e.p. = M has the e.e.p.", if N is the model constructed in the proof. 3.21 The last step in our strategy is statement (e). So, we consider a pair (G', X+) defining a connected Shimura variety and an isogeny ir : G' -» G".
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We assume that G' (hence also G") is unramified over Q,, so that 7r extends to an isogeny 7r: G' -+ G" of semi-simple groups over 7G(p).
Let us also assume that there exists an i.c.m..M of Sh°(G',X+)E over 0(,,,), where w is a prime of the field k (as in 3.16) above p. We want to show that there exists an i.c.m. N of Sh°(G", X+)E over 0(w). 3.21.1
Set
A := Ker[G'd(Z(p))+' rel. 7-(G')
)
gad(z(p))-4-A
rel. r(G")J .
This is a finite group which acts freely on Sh°(G', X+)E. The canonical morphism Sh(7r) : Sh°(G', X+)-z; -> Sh°(G", X+)E is a quotient morphism for this action. (Cf. [De3], 2.7.11 (b).) Since M has the extension property, the action of A on .ME extends uniquely to an action on M. The natural candidate for an i.c.m. of Sh°(G", X+)E is the quotient Af := .M/0.
3.21.2 Problem. Consider a faithfully flat extension of d.v.r. 7L(p) C 0. Let A be a finite (abstract) group acting on a faithfully flat 0-scheme M which is locally noetherian and formally smooth over 0. Assume the action of A on the generic fibre of M is free. Under what further conditions does it follow that the action of A on all of M is free?
3.21.3 It follows from a result of Edixhoven ([Edl], Prop. 3.4) that, under the previous assumptions, the action of A on all of M is free if p does not divide the order of A. On the other hand, if p does divide JAI, then extra assumptions are needed. Example 1: take 0 = 7Lp[(p], M = Spec(0Qxj) with the automorphism of order p given by x H Sp x - ((p - 1). In this case, the action of Z/pZ on the generic fibre is free (note that x - 1 is a unit in OQxI), but the action on the special fibre is trivial. In order to avoid examples of this kind, we can add the assumption that
p > 2 and e(0/Z(p)) < p - 1. (In the situation where we want to use it, this holds anyway.) That this is not a sufficient condition is shown by the following example that was communicated to us by Edixhoven. Example 2: write A for the 7L module Zp ® Z,[(p], and consider the automorphism of order p given by (x, y) H (x, (p y). This induces a Zp linear automorphism of order p on 1F(A) = P P-1. On the generic fibre there are (geometrically) p fixed points. On the special fibre there is an 1Fp-rational line of fixed points. By removing the closure of the fixed points in the generic
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fibre we obtain a Zr scheme M with a 7G/pZ-action as in 3.21.2, such that the action is not free on the special fibre.
3.21.4 Proposition. In the situation of 3.21, suppose that (i) the action of A on M is free, and (ii) M has the extended extension property (see 3.20). Then A(:= M/O is an i.c.m. of Sh°(W", X+)E over 0(,,,). Proof. Condition (i) implies that N is a smooth model, so it remains to be shown that it has the extension property. Consider an a.t.s. S over O(w) plus
a morphism aE : SE -+ NE. Let TE := (SE X ME) NE
ME
and write T for the integral closure of S in the fraction ring of TE. We have a canonical morphism p: T -* S. If U C_ S is an open subscheme such that extends to au: U -+ N, then p 1(U) = U x,v M, so that p 1(U) U at I UE is etale. We now first consider the special case where S = Spec(A) for some d.v.r. A which is faithfully flat over O(,,,). It then follows from the e.e.p. of M that Of extends to a morphism,3: T -* M which is equivariant for the action of A. On quotients this gives the desired extension a of aE. Back to the general case, it follows from the special case, the remarks preceding it and the Zariski-Nagata purity theorem of [SGA1] Exp. X, 3.1, that p: T -* S is etale, so that T E ATSo(,,,). This again gives an extension O 0 of ,(3E and, on quotients, an extension a as desired.
3.21.5 For a reductive group G over Q, define 6G as the degree of the covering d -+ Gad. (In other words: So is the "connectedness index" of the root system of GjU.) By definition, be depends only on G. We claim that, in the situation of 3.21 and 3.21.1, the order of A is invertible in 7G[1/6], where 6 = 5G' = 6G". To prove this, we need some facts and notations. We write
pl: G -* G' and p2: G
G" for the canonical maps from the universal covering. Writing r1 := p1G(Af) n c'(7Z(P)), r2 := p2G(Af) n 9"(Z(P)), we have
rel. T(g') = p1G(Af) 11* 9'(Z(P))+
(Z(P))+n
y
rel. T(G"). (Cf. [De3], (2.1.6.2).) -* 9"). We claim there is an exact sequence
and similarly for Gad(Z(P))+"
Write K := Ker(p2: (3.21.6)
K(Af)
A
r2/p2c(Z(P))
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Here the map t sends an element g E G with P2(9) = eguu to the element Pi (9) *r, ega, which obviously lies in A. The map u sends an element x *r1 y E
A C p1G(Af) *r1 c'(Z(r))+ to 7r(x) mod p2c(Z(p)); notice that x *r, y E A means that (7r(x), y) = (y-1, ad(-y)) for some y E r2. If x *r1 y E Ker(u) then we can take y = p2(g) for some g E c(Zipi), in which case x *r, y = (x P1 (9)) *r, egad E Im(t). This proves the exactness of (3.21.6). It follows from the definitions that every element of K(A4) has a finite order dividing 6. On the other hand, r2/p2G(Z(p)) is a subgroup of HfpPf(7G(p), IC), in which again all elements are killed by 6. This proves our claim that I A I E 7L[1/b]*.
For simple groups G, the number 6G is given by 6(Ae) = 2 + 1, S(Be) = 2,
6(Ce) = 2, 8(D4) = 4, 6(E6) = 3, 6(E7) = 2. (The other three simple types have 6 = 1 but do not occur as part of a Shimura datum.) In particular, we see that Job is invertible in 7L[1/6] if G does not contain factors of type A. After the technical problems encountered in our discussion of steps (b) and (e), the good news is that we can prove the converse of (e).
3.22 Proposition. Consider the situation as in the first paragraph of 3.21, and assume that Sh°(G", X+)E has an i.c.m. N over O(,,,). Then the nor-
malization M of N in the fraction field of Sh°(G',X+)E is an i.c.m. of Sh°(9',X+)E over 0(,,,). Proof.
First we remark that the action of the group A on Mk extends to an
action on M and that M/0 - N. (We have a map M/0 -* N which is an isomorphism on generic fibres; now use that Al is normal.) We claim that the action of A on M is free. On the generic fibre we know this. The important
point now is that the purity theorem applies, so that possible fixed points must occur in codimension 1. So, suppose A has fixed points. Without loss of generality we may assume that A is cyclic of order p (cf. 3.21.3). Restricting to a suitable open part Spec(A) C M, we then obtain a nontrivial automorphism of order p of the O(,,,)-module A which (using purity and
the fact that the action is free on the generic fibre) is the identity modulo p. But now we have the following fact from algebra, probably well-known and in any case not difficult to prove: if R is a principal ideal domain, p > 2 a prime number with (p) # R, M a flat R-module, and a an R-module automorphism
of M with ap = idM and (a mod p: M/pM -- M/pM) = idMypM, then a = idM. Applying this fact we obtain a contradiction, and it follows that M is a smooth model.
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For the extension property, consider an S E ATSo(,,,) and a morphism
aE: SE -+ M. The projection ,6E: SE -+ NE of aE to N extends to a morphism 3: S -* Al. Set T := S xN M, then T -+ S is a finite etale Galois covering with group A. The section TE +- SE on the generic fibres (corresponding to aE) therefore extends to a section on all of S (recall that S and T are flat over 0(,,,) and normal), which means that aE extends to a morphism a. Combining all the results in this section, we arrive at the following conclusion.
3.23 Corollary. Fix a prime number p > 2. Let (H, Y) be a Shimura datum of pre-abelian type with p f 8H, and let v be a prime of E(H, Y) above p. Sup-
pose that for each simple factor (G', X') of the adjoint datum (Had, Y'), there exist:
(i) a Shimura datum (G, X) covering (G', X'), (ii) a closed immersion i : (G, X) " (CSp29, S59') ,
(iii) a prime w of E(G, X) such that v and w restrict to the same prime ofE(Gad,X'd), (iv) a hyperspecial subgroup Kp C G(Qp) with i(Kp) C CSp29(ZP), such that the normal model Al of ShKK (G, X) constructed in 3.19 is a formally smooth O(,,,)-scheme. Then for every hyperspecial subgroup Lp C H(Q ) there exists an integral canonical model of ShLp (H, Y) over O(v).
3.24 Remark. In this section, we have tried to follow the strategy of [De3] very closely, adapting results to the p-adic context whenever possible. We wish to point out that our presentation of the above material is very different from the treatment in Vasiu's paper [Va2]. In particular, our definitions are different (see 3.9), and models of connected Shimura varieties (which play a central role in our discussion) do not appear in [Va2]. Vasiu claims 3.23 (using his definitions) without the condition that p f SH. We were not able to understand his proof of this (in which one step is postponed to a future publication). It seems to us that at several points the arguments are incomplete, and that Vasiu's proof furthermore contains some arguments which are not correct as they stand. 3.25 Remark. We should mention Morita's paper [Mor]. (See also Carayol's paper [Ca].) Of particular interest, in connection with the material discussed in this section, are the following two aspects.
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(i) Morita proves that certain Shimura varieties (of dimension 1) have good
reduction by relating them to other Shimura varieties which are of Hodge type. (The example is classical-see §6, "Modeles etranges" in Deligne's Bourbaki paper [De3].) In Morita's method of proof we recognize several results that have reappeared in this section in an abstract and somewhat more general form. (ii) The Shimura varieties in question (MO' = MM.XK, (G', X') in the notations of [Ca]) are shown to have good reduction at certain primes p of the reflex field. This includes cases where the group G', in question is ramified. Thus we see that good reduction is possible also if the group KK (the "level at p") is not hyperspecial.
§4 Deformation theory of p-divisible groups with Tate classes In the next section, we will try to approach the smoothness problem appearing in 3.23 using deformation theory. The necessary technical results are due to
Faltings and are the suject of the present section. Here we work out some details of a series of remarks in Faltings's paper [Fa3]. 4.1 To begin with, let us recall a result from crystalline Dieudonne theory. For an exposition of this theory, we refer to the work of Berthelot-Messing and Berthelot-Breen-Messing ([BM1], [BBM], [BM3]); some further results can be found in [dJ]. Let k be a perfect field of characteristic p > 2, let W = W (k) be its ring of infinite Witt vectors, and write or for the Frobenius automorphism of W. We will be working with rings of the form A = W it1 i ... , tj. For such a ring, set AO = k[tl,... , tnl, m = mA = (p, tl, ... , tn), J= JA = (t1, ... , tn), let eA: A - W be the zero section, and define a Frobenius lifting OA by 'A = Q on W, OA (ti) = tP. With these notations we have the following fact: the category of p-divisible
groups over Spf(A) is equivalent to that of p-divisible groups over Spec(A) (see [dJ], Lemma 2.4.4), and these categories are equivalent to the category of 4-tuples (M, Fill, V, F), where -M is a free A-module of finite rank,
-Fill C M is a direct summand, -V: M --> M ®SZA/u, is an integrable, topologically quasi-nilpotent connection,
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-F: M -4 M is a OA-linear horizontal endomorphism, such that, writing M = M +p-1Fil', (4.1.1)
F induces an isomorphism F : 0AM -+ M, and
(4.1.2) Fi11 ®A Ao = Ker(F ® FrobA0 : M ®A Ao --3 M ®A Ao).
(Here, as often in the sequel, we write qA- for -®A,OAA.) Notice that (4.1.1) implies that there is a 0A1-linear endomorphism V : M -+ M such that (4.1.3)
This equivalence is an immediate corollary to [Fa2], Thm. 7.1. One also obtains it by combining the following results: -the description of a Dieudonne crystal on Spf(Ao) in terms of a 4-tuple (M, V, F, V), see [BBM], [BM3], [dJ], -the Grothendieck-Messing deformation theory of p-divisible groups, see [Me],
-the results of de Jong, saying that over formal lFF-schemes satisfying certain smoothness conditions, the crystalline Dieudonne functor for p-divisible groups is an equivalence of categories, see [dJ]. If (M, Fil1, V, F) corresponds to a p-divisible group f over A then
rkA(M) = height(W)
rkA(Fi11) = dim(1l).
,
4.2 The 4-tuples (M, Fill, V, F) form a category similar to the category .M.Fv,1l (A) as in [Fa2], except that we are working here with padically complete, torsion-free modules, rather than with p-torsion modules. More generally, let us write MF[a,b](A) for the category of 4-tuples (M, Fil', V, F), where M and V are as in 4.1, where F is a OA-linear endomorphism M ® A[1/p] -3 M 0 A[1/p], and where Fil' is a descending filtration of M such that Fi12+1 is a direct summand of Filt
and such that, writing M =
,
FilaM = M,
Filb+1 M = 0,
b-a p 'Fi12M,
F induces an isomorphism F : SM A
M.
The arguments of [Fa2], Thm. 2.3, show that for p > 2 and 0 < b - a < p -1, the category M F[a,b] (A) is independent, up to canonical isomorphism, of the chosen Frobenius lifting OA. Every morphism in MF[a,b](A) (the definition
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of which, we hope, is clear) is strictly compatible with the filtrations (cf. [Wi], Prop. 1.4.1(i), in which the subscript "If" should be replaced by "tf"). For a' < a and b' > b, we have a natural inclusion MFv,b](A) C MF[°,,ya(A). We will write MFG ](A) for the union of these categories, i.e., M E MF[,](A) means that M E MF[°,b] (A) for some a and b.
The Tate object A(-n) E MF° n](A) is given by the A-module A with V = d, Fill' = A{n} D Fil"'+1 = (0) and F(a) = p"` - OA (a).
4.3 Before we turn to the deformation theory of p-divisible groups, we need to discuss some properties of 4-tuples (M, Fill, V, F) as in 4.1. 4.3.1 The connection V induces a connection V on 0AM (not on M itself): if m E M and V(m) _ m,, ®w,,, then V (m ®1) = E(ma ®1)® dOA(w«) One checks that this gives a well-defined integrable connection 0. The horizontality of F can be expressed by saying that 0 is the pull-back of V via
F: 0M A -n4 M. 4.3.2 Given (M, Fill, F) satisfying (4.1.1) and (4.1.2), there is at most one connection V for which F is horizontal. Indeed, the difference of two such connections V and V' is a linear form S E End(M) ®S2A/L1, satisfying
Ad(F)(8) = S. Here Ad(F)(8) = (F ® id) o a o F-1, where b = V - V'. One checks that if 8 E JtEnd(M)®S2 tiw, then Ad(F)(8) E so that Ad(F)(8) = 6 implies 6 = 0. Similar arguments show that any connection 6 for which F is horizontal, is integrable and topologically quasinilpotent.
4.3.3 Suppose A = W [tl, ... t,j and B = W [ul, ... u,",] are two rings of the kind considered above. Let f : A -4 B be a W-homomorphism. If 9l is a pdivisible group over A corresponding to the 4-tuple D = (M, Fi1,1N, V M, FM), then the pull-back f *W corresponds to a 4-tuple f *IID = (N, FiIN, VN, FN) described as follows:
Fil = f *Fil , VN = f*VM. (ii) To describe FN, we have to take into account that f may not be (i) N = f *M := M ®A,f B,
compatible with the two chosen Frobenius liftings cbA and OB. First we use the connection VM to construct an isomorphism
C=C(cBof,f oOA): OBf*M -24 f*OAM,
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which, using multi-index notations
0(a)Z = 0(8tl)1' ..... V(atn)=n
,
zi = zip . ... . zn
,
etc.,
is given (for m E M) by
c(m ®1) _ where zz = (cB0 f (tz)- f ogA(t2))/p. Then one defines FN as the composition c
FN:
O N=OBf*M f*¢AM
f*M=N.
4.4 Theorem. (Faltings) Let A = W Qt1, ... tnj and consider a p-divisible group 9{ over A with filtered Dieudonne crystal IID(f) = (M, Fill, VM, FM). Write H = eA9-l, which has Dieudonne module D(H) = (M, Fi1M, FM) _ e* (M, Film, FM). Assume that 9i is a versal deformation of H in the sense that the Kodaira-Spencer map is : W 0tl +
+ W atn -+ Homw (Fill , M/Fill)
is surjective.
Next consider a ring B = WQul.... and a 3-tuple IE' = (Al, Fil , FN) satisfying (4.1.1) and (4.1.2), and such that 9 ®B,eB W =' ]D(H). Then there exists a W-homomorphism f : A -* B such that IE' is isomorphic to the pull-back of (M, Films, FM). In particular, IE' can be completed to a filtered Dieudonne crystal lE by setting Vg = f * VM, and therefore corresponds to a deformation of H. Proof. For every W-homomorphism fl : A -+ B there is an isomorphism of filtered B-modules gl : fl* (M, Film) - (Al, FilN), which is unique up to an element of Aut(Al, Fil ). The map gl induces an isomorphism §j: f; M -
N. By induction on n > 1 we may assume that the two Frobenii
N and (with cl = c(OB o fl, fl o OA) as in 4.3.3) 0a9i
F'iv:
Al
OB.'4
,.+.fiOaM Ii Fm>fiM ^91 +N ci
0*fiM
are congruent modulo J. (For n = 1 this is so by our assumptions.) Because B is JB-adically complete, it suffices to show that we can modify fl and gl such that the new F;v is congruent to FAr modulo JB+1
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Consider an f2: A -* B which is congruent to fl modulo JB. Notice that fl o cbA = f2 0 ¢A and cbB o fl = ¢B o f2 modulo JB+1, so we have canonical isomorphisms
0' fiM 0 B/JB-+1 - ' f 2 M® B/JB-+1 and
fi SAM ® B/JB+1 = f2 *0* M ® B/JB+1
Next we choose an isomorphism h: f2 (.M, Film) - fl* (.M, Fill) which reduces to the canonical isomorphism modulo JB, and we set 92 = gi o h. The first important remark is that, given the above identifications, the two maps c2: cBfzM (i= 1,2) fiO*M are equal modulo JB+1 One can check this using the description of the maps ci given in 4.3.3 and using that fi(tj) E JB. Write v for the automorphism of N such that FN = v o F. The induction hypothesis gives us that v = id,N mod J" End(./V). It follows from the previous remarks that we are done if we can choose f2 and h such that the diagram (4.4.1)
fr FM
yogi
fi O M® B/Jn+1 ^L+ fi M® B/Jn+1
A((& BI jn+l
4
11
f2* Fm
92
f2*M ® B/Jn+1
f2*0*A.M ® B/Jn+1
N ®B/Jn+l
Note that the diagram is commutative modulo JB and that, given f2, we can still change h (and consequently g2) by an element of commutes.
Aut(f2M, f2 Films). The composition gi 1 o v-1 o g2 induces a W-linear map
fi Film ®B/J ^ f2 Film 0 B/J -+ fl* (.M/Fill) ®Jn/Jn+1 , which is independent of the choice of h. Similarly, fi FM o (f2 FM)-1 induces a W-linear map ,q: fl*Fil
® B/J -+ fl* (M/Film) ® Jn/Jn+1
The assumption that fl is a versal deformation of H now implies that we can choose f2 such that 77 = . This means precisely that we can modify g2 by something in Aut(f2* M, f2 Fil ) such that the diagram (4.4.1) commutes.
This proves the induction step.
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4.5 Let H be a p-divisible group over W, with special fibre Ho. Write n = dim(Ho) dim(HH ), and let A = WQt1,... , The formal deformation functor of Hp is pro-represented by A (see [II)), where we may choose the coordinates such that H corresponds to the zero section eA. Write 3l for the universal p-divisible group over A, and let D(W) = (M, Fillet, VM, FM) be its filtered Dieudonne crystal. We will use the previous result to give a more explicit description of D(9-1).
Let (M, Fill, FM) = eA D(1i) be the filtered Dieudonne module of H. Choose a complement M' for Fill C M. Inside the reductive group GL(M) over W, consider the parabolic subgroup of elements g with gM' = M', and let U be its unipotent radical. Notice that U is (non-canonically) isomorphic to Ga,w. Let U = Spf(B) be the formal completion of U along the identity, and choose coordinates B =' W [ul, ... , u7,] such that eB gives the identity section. Over B we define a filtered Dieudonne crystal E = (N, Fi1N, Vg, F,v) as follows. We set
N=M®w B, Fi1N=Fill Ow B, where g : M - N is the "universal" automorphism, i.e., the automorphism given by the canonical B-valued point of U C GL(M). At this point we apply the theorem. This gives us a connection VV and a W-homomorphism f : A -* B such that ]E f *IlD(Il). We claim that the map f is an isomorphism. Since A and B are formally smooth W-algebras of the same dimension, it suffices for this to show that E is a versal deformation of (M, Fil1M, FM). Now we have an isomorphism (4.5.1)
N ®B,4B B/q5(Jn) = ((N ®B,eB W) ®w,, W) ®w B/ct(JB) can
^_' (M ®w,o W) ®w B/W(JB)
On the left hand term we have the connection V; on the right we take 10 d. It follows easily from the definition of V that (4.5.1) is horizontal modulo Jr'. Composing with the isomorphisms Fv and FM then gives a horizontal isomorphism M ®w g: MOB B/Jn-1
B/Jr-1
which, as is clear from the constructions, is just the reduction modulo JP-1 of the automorphism g. Since p > 2, it then follows from the choice of U that E is a versal deformation.
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4.6 Our next goal is to redo some of the above constructions for p-divisible groups with given Tate classes. We keep the notations of 4.5. For r1, r2 E Z>o and s E Z, set
Mh, r2; s)
M®r1
® (M*)®r2 ®
W(S),
with its induced structure of an object of MF[,](W). We will refer to any direct sum of such objects as a tensor space T = T(M) obtained from M. We assume given a polarization V): M ®w M -* W(-1), i.e., a morphism in MF[0,2](W) which on modules is given by a perfect symplectic form. We let CSp(M, 0) act on the Tate twist W(-1) through the multiplier character. Then we consider a closed reductive subgroup G C CSp(M, V5) such that (4.6.1)
there exists a tensor space T and an element t E T such that L = W t is a subobject of T in MF[,](W) isomorphic to W(0), and such that G C CSp(M, 0) is the stabilizer of the line L.
4.7 Remark. In [Fa3], Faltings gives an argument which shows that, for (4.6.1) to hold, it suffices if the Lie algebra g C End(M) is a subobject in MF[_1,_1](W). Since, by assumption, G is a smooth group, an easy argument
then shows that (4.6.1) is equivalent to the condition that g is stable under the Frobenius on End(M). 4.8 We can now construct a "universal" deformation of H such that the Tate class t remains a Tate class. The procedure is essentially the same as in 4.5. First, however, we have to find the right unipotent subgroup Ug C G. For this we use the canonical decomposition M = M° ® Fill defined by Wintenberger in [Wi]. The corresponding cocharacter µQpm,w
GL(M);
µ(z) _
id
on M°
Z-1 id
on Fill M
factors through G. In 4.5 we now take M' = M°, and we set Ug = Uf1G. Then Ug is a smooth unipotent subgroup of G , whose Lie algebra is a complement of Fil°g C g. (Here we use that G is reductive.) Taking formal completions of Ug U along the origin corresponds, on rings, to a surjection
B =WQu1i... ,u,,,j -H C =W[vl,... ,vq], where q = dimw(g/Fil°g). We set
P=M®w C, File=Fi1Mow C,
Oc),
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where h: P - P is the universal element of Ug. As in 4.5, applying Theorem 4.4 gives a connection Vp and a homomorphism fg : A -+ C such that ]Eg := (P, Fil ,, V -p, Fp)
is the Dieudonne crystal of a deformation Ng = f49-l of H over Spf(C) = Ug.
From the fact that Fp is horizontal w.r.t. Vp, one can derive that Vp is of the form Vp = d +)3 with ,Q E gc ®SZc/w C End(M) ®S2C/yj,. It follows that if we extend the space T to an object T E MFV[°,b](C) by applying to
P = M ® C the same linear algebra construction as was used to obtain T from M, then the line L C T extends to a subobject G C T in (C). To finish, let us prove that, conversely, every deformation of H over a ring
D = WQxl,... , x,.] such that the tensor t deforms as a Tate class (i.e., the line L C T extends to an inclusion C C T in MF[a,b] (D) ), can be obtained by pull-
back from flg. The map End(M) -i T/L obtained by sending a E End(M) to the evaluation at t of the induced T(a) E End(T) is a morphism in MF[,)(W), hence strictly compatible with the filtrations. It follows that (4.8.1)
if T(a) maps L into Fil°T, then a E Fil°End(M) + 9.
To prove the claim we can now follow the same reasoning as in 4.4, making use of (4.8.1). Alternatively, it follows from what we did in 4.5 that our deformation of H over D is obtained by pulling back the universal deformation '}lB over B via a homomorphism it : B -+ D. It then suffices to show that
7r,,: B/JB -* D/JD factors via C/Jc for every n. For this we can argue by induction, and because of the way we have chosen Ug and U, the induction step easily follows from (4.8.1). This proves:
4.9 Proposition. Notations and assumptions as above. We have a formally smooth deformation space Ug = Spf(C) -+ U of relative dimension equal to dimw(g/Fil°g) which parametrizes the deformations of H such that the horizontal continuation oft remains a Tate class.
§5 Vasiu's strategy for proving the existence of integral canonical models After our excursion to deformation theory, we return to the problem of the existence of integral canonical models. Our aim in this section is to explain the principal ideas in Vasiu's paper [Va2] (which is a revised version of part
318
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of [Val]). We add that, to our understanding, some technical points are not treated correctly in loc. cit, to the effect that the main conclusions remain conjectural. 5.1 Consider a closed immersion of Shimura data i : (G, X) y (CSp29, .fj9 ).
Let v be a prime of E = E(G, X) above p > 2, and assume that there is a hyperspecial subgroup Kp C G(Qp) with i(Kp) C Cp := CSp2g(ZP). (In particular, Gq, is unramified.) Write A := l pt,, A9,1,,,, ® Z(p) which, as we have seen, is an i.c.m. of Shcp (CSp29, Sig) over Z(p), and let N -* N C be the normal integral model constructed in the proof of 3.19 (so A® JV is the normalization of N). Choose embeddings Q C E C C, and write A -* N C A for the base-change of N, N and A to W := W (ac(v)). Let xo E JV be a closed point mapping to xo E N C A.
We would like to show that J is formally smooth at xo. If this holds (for all xo) then N is an i.c.m. of ShKK (G, X) over O(v). To achieve this, we would like to use Prop. 4.9. This is a reasonable idea: over our Shimura variety we have certain Hodge classes, which, by a results of Blasius and Wintenberger, give crystalline Tate classes (in the sense used in §4). The corresponding formal deformation space of p-divisible groups with these Tate classes is formally smooth and has a dimension equal to that of V. Arguing along these lines one could hope to prove that J is formally smooth at xo. We see at least two obstacles in this argument: (i) in §4 we started from a p-divisible group over a ring of Witt-vectors, and (ii) we need a reductive group G C GL(M) (the generic fibre of which should essentially be our group G). To handle these problems, we will first try to prove the formal smoothness of N under an additional hypothesis (5.6.1). In rough outline, the argument runs as follows. We start with a lifting of xo to a V-valued point of N, where V is a purely ramified extension of W. If (X, A) is the corresponding p.p.a.v.
over V then the associated filtered Frobenius crystal can be described as a module M over some filtered ring Re. We have a closely related ring Re which is a projective limit of nilpotent PD-thickenings of V/pV and on which
we have sections io: Spec(W) " Spec(Re) and i,: Spec(V) " Spec(Re). We will construct a deformation (X,.\) over Re which corresponds to an Revalued point of N and such that i*(X,)s) _ (X, A). Then io(X, will give a lifting of xo to a W-valued point of N, which takes care of problem (i). To construct (X, A), we will use the Grothendieck-Messing deformation
theory; the essential problem is to find the right Hodge filtration on the module M := M ® Re. (We remark that the filtration on M cannot be
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used directly for this purpose: M is filtered free over the filtered ring Re, whereas the desired Hodge filtration should be a direct summand.) One of the key steps in the argument is to show that the Zariski closure of a certain reductive group G1,Re[1/p] `-> GL(M[1/p]) inside GL(M) is a reductive group
scheme-this will also take care of problem (ii). To achieve this, we have to keep track of Hodge classes on X in various cohomological realizations. At a crucial point we use a result of Faltings which permits to compare etale and crystalline classes with integral coefficients. Once we have shown that the closure of G1,&[l/nl is reductive, an argument
about reductive group schemes leads to the definition of the desired Hodge filtration on M. After checking that it has the right properties, this brings us in a situation where the deformation theory of § 4 can be applied. The formal smoothness of JA at xo is then a relatively simple consequence of Prop. 4.9. Sections 5.2 and 5.5 contain the necessary definitions and a brief description of the crystalline theory with values in Re modules. In 5.6 the argument
that we just sketched is carried out, resulting in Thm. 5.8.3. What then remains to be shown is that there exist "enough" Shimura data for which (5.6.1) is satisfied. Vasiu's strategy to solve this problem is discussed briefly from 5.9 on.
5.2 Let 0 be a d.v.r. with uniformizer 7r and field of fractions F. Let W be a finite dimensional F-vector space with a non-degenerate symplectic form 0. Write F(-n) for the vector space F on which CSp(W, ') acts through the nth power of the multiplier character, and consider tensor spaces Wh, r2; s) := W®nl ® (W*)®T2 0 F(s). The fact that z// E W(0, 2; -1) is non-degenerate implies that there exists a class Vi* E W(2, 0; 1) such that (V), V*) = 1 E F = W (0, 0; 0).
Consider a faithfully flat 0-algebra R and a free R-module M with a given identification M OR R[1/7r) = W OF R[1/7r]. An element t E W(rl, r2; s) is said to be M-integral if, with the obvious notations, t 01 lies in the subspace M(rl, r2; s) of W(ri, r2; s) OF R[1/7r]. For example, 0 and 0* are both Mintegral precisely if 0 induces a perfect form )M : M x M --* R.
If t E W (rl, r2; s) then we shall say that t is of type (rl, r2; s) and has degree r1 + r2. In the sequel we shall often use a notation T (W) for direct sums of spaces W (rl, r2; s), and we call such a space a "tensor space obtained from W".
5.3 Definition. Let G C CSp(W, 0) be a reductive subgroup, and consider a collection {ta}aEj of G-invariants in spaces Ta(W). We say that {ta}aEJ
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is a well-positioned family of tensors for the group G over the d.v.r. 0 if, for every R and M as above, we have /,, V)* and {to}
are M-integral
:
the Zariski closure of GR[1/,] inside CSp(M, I'M) is a reductive group scheme over R
If in addition there exists an 0-lattice M C W such that 0, b* and all to are M-integral, then we say that {to}oEJ is a very well-positioned family of tensors. 5.4 Remarks. (i) One should not think of a well-positioned family of tensors as some special family of tensors which cut out the group G (i.e., such that G is the subgroup of CSp(W, 0) leaving invariant all to), since G may be strictly contained in the group cut out by the to. We only use the well-positioned families of tensors to guarantee that certain models of G are again reductive groups.
(ii) For general reductive G C CSp(W, v'), the main difficulty with this notion is to prove the existence of (very) well-positioned families of tensors. We will come back to this point in 5.9 below. 5.5 Consider a purely ramified extension of d.v.r. W = W(]FP) C V. Write
Frac(W)=KoCK=Frac(V),fixKCK-4 C,andlete=e(V/W)=[K: Ko]. Suppose we have a p.p.a.v. (X, A) over V. We write HB z
Hi H1t,zP
Hs(X(C), Z) , HB Hs z ® Q 1 = HdR(XK/K), HdR,C := HdR(Xe/C) = HdR,K ®K C and H' Het (X K1 ZP) H16t := = H4t,z® ®QP
Let TB = T(HB) be a tensor space as in 5.2, obtained from H. We adopt the notational convention that TdR, T6t etc. stand for "the same" tensor space built from the corresponding first cohomology group HdR, H1t etc. In each case, T? naturally comes equipped with additional structures (Hodge struc-
ture/filtration/Galois action/... ), where we interpret F(n) as a Tate twist. (Cf. [De4], Sect. 1 and Sect. 5.5.8 below.) In each theory, the polarization A gives rise to a symplectic form on H;, which, if there is no risk of confusion, we denote by 0 without further indices.
5.5.1 Choose a uniformizer ir of V, and write g = Te + ae_1Te-1 +
+ ao for its minimum polynomial over K0, which is an Eisenstein polynomial. The
PD-hull (compatible with the standard PD-structure on (p)) of W[T] -
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WQTI/(g) = V is the ring Se obtained from WQTI by adjoining all Ten/n!. Let I_:= (p, g) = (p, Te) C S. We define Re as the p-adic completion of Sc and Re as the completion of Se w.r.t. the filtration by the ideals IM. (Thus Re is the nilpotent PD-hull of WQTI -» V/pV.) Notice that these rings only depend on the ramification index e, which justifies the notation. We can identify Re (resp. Re) with the subring of KoITI consisting of all formal power series > a,,, T' such that all In/e j ! an are integral (resp. the coefficients In/ej! a,,, are integral and p-adically convergent to zero for n -* oo). On Re we have
-a filtration by the ideals Filn(Re) :_ (g)!n!, -a or-linear Frobenius endomorphism 0 = OR. given by T H Tp, -a continuous action of Gal(op/Ko), commuting with 0 and respecting the filtration. 5.5.2 Next we briefly recall the definition of the ring Acrys as in [Fo]. (In [Fa3]
and [Va2] the notation B+(V) is used.) Write Oc for the p-adic completion of the integral closure of 7Gp in K = Qp, and let C (= Cp) be its fraction field. Let
Ro, :=
(Oc/pOc E- Oc/POc _ ... - Oc/POc F- ... )
where the transition maps are given by x H xp. It is a perfect ring of characteristic p. Choose a sequence of elements ir() E Oc with ir(') = it (the chosen uniformizer of V) and (7r(m+1))p = 7r(m), and set 7r = (7r(') mod p, 7r(2) mod
p, ...) E Roc. There is a surjective homomorphism 0: W(Ro,) -» Oc whose kernel is the principal ideal generated by := g([a]), where [?r] is the Teichmiiller representative of ?r and where g is the polynomial as in 5.5.1 (see [Fa3], sect. 4). Define Acry$ as the p-adic completion of the PD-hull of W(Ro,) -» Oc, compatible with the canonical PD-structure on (p). Then Acry8 is a W-algebra which comes equipped with -a filtration by the ideals Fil'2(AeTy,,) -a a-linear Frobenius endomorphism 0 = Oflcrys, -a continuous action of Gal(Qp/Ko) commuting with 0 and respecting the filtration. There is a Zr linear homomorphism 7Gp(1) " Fill(A,ys). We let 0 (called tin [Fo]) denote the image of a generator of Zp(1); we have 0(,3) = p 0.
5.5.3 We have ring homomorphisms
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where the sections Re + W and Re -+ W are given by T H 0, and where Re -* V is given by T
ir. Also we have a homomorphism t : Re
+ AcrYs
given by T a [1] (hence g H ). This map is strictly compatible with the filtrations and induces ([Fa3], sect. 4) an isomorphism gr'(Re) ®w 0C gr'(A,,.Y3). Also, t is compatible with the Frobenii, but in general not with the Galois-actions.
5.5.4 Let H be a p-divisible group over V. Its Dieudonne crystal can be described ([BBM], Thm. 1.2.7) as a free Re module M = M(H) of finite rank with an integrable, topologically nilpotent connection V on M as a W[T]module. On M we have -a filtration by Re submodules Fil C M, -a 0&-linear horizontal endomorphism F, such that
-there exists a basis m(°), ... , m.0 )'M(1) l, ...
,
m,(1)
such that FiljM =
FilaRe
-the connection satisfies Griffiths transversality, -F is divisible by pj on F&M, and we can choose the basis {mEP} as above in such a way that the elements F(m;j))/pi form a new Re basis of M. (Modules M with these additional structures are the objects of a category MFjo,1](V), analogous to the categories considered in §4; see [Fa3], §3.)
The Dieudonne module of the special fibre H° := H ®v 1Fp is a free Wmodule M° with a o-linear Frobenius endomorphism F0. We have a canonical isomorphism of Frobenius crystals (5.5.5)
M°=M®Re/T Re.
(Recall that Re/T Re - W.) On the other hand, the reduction (H mod p) on Spec(V/pV) is isogenous, via some power of Frobenius, to H° ®FP V/pV,
so that (5.5.6)
M ®R Re[1/p] `v Mo ®w Re[1/p]
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as Frobenius crystals. Although the homomorphism c from 5.5.3 is in general not compatible with the Galois actions, there is a canonical action of Ga1(ip/K) on M ®p ,t Acrys. (Here one uses that M is a crystal over Re7 see [Fa3], §4.) We also define Het,zp(H)
T'p(H)*,
which is a free Zp module with Gal(Qp/K)-action.
5.5.7 Theorem. (Faltings, [Fa3]) There is a functorial injection p: M(H) ®Re Aerys -+
®zP Aerys
,
which after extension of scalars to Brys := A,rys[1/Q], and using the isomorphism M ®Re Re[1/p] = Mo ®w Re[1lp] gives back Faltings's comparison isomorphism p: MO OW Bcrys * Het(H) ® , Bcrys of [Fa2]. The map p is compatible with the Frobenii, filtrations and Galois actions on both sides. Its cokernel is annihilated by ,Q E Acrys.
5.5.8 We shall try to be precise about Tate twists and polarization forms. We have
-ZB(1) := 2iri Z C C, with H.S. purely of type (-1, -1), -KdR(1) := K with filtration Fil-1 = K D Fil° = (0) (similarly for other fields than K), -7Gp(1) :_ mppn(K) as Gal(K/K)-module (similarly for other fields). Fixing i E C with i2 = -1 we have generators 27ri for 7GB(1) and 1 for KdR(1). Also, the choice of i determines a generator (exp(2lri/p'))-EN for Zp(1) over
C. Via the chosen embedding K'- C this gives a generator S for Zp(1) over K. We have comparison isomorphisms (over C) ZB(1) ®z Z,
Zp(1)
and ZB(1) ®z G
QR(1)
(see [De4], Sect. 1) mapping generators to generators.
If (X, A) is a p.p.a.v. over C then A gives rise to a perfect symplectic form OB: H1 x HB -+ ZB(-1). For de Rham and etale cohomology we have an analogous statement, and the various forms V)? are compatible via the comparison isomorphisms. Using the chosen generators for Tate objects, we can view the forms V)? as "ordinary" bilinear forms with values in the coefficient ring corresponding to ? E {B, dR, et}. This is consistent with our usage of the notation F(n) in 5.2.
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For crystalline cohomology (or Dieudonne modules) with values in Re, the Tate twist is given by
-Re(-1) := Re with filtration File (Re(-1)) = Filj-1Re and Frobenius
F=p.0
.
(This should be thought of as an object of a category MF(V), see [Fa3]. We have Re(-1) ^_' M(Gm).) The generator ( E Zp(1) over K determines a generator (* of Zp(-1) and an element 0 E AC1ys as in 5.5.2 (see [Fo], 1.5.4). On Tate twists, the comparison map of Thm. 5.5.7 is (after a suitable normalization) the map 8: Re(1) ®& Acrys + Zp(1) ®zp Ac,ys with 1®1 H S ® ,6. Here we see a factor ,Q entering. In other words: if (X, A) is a p.p.a.v. over V with associated p-divisible group H = X [p'], then A gives polarization forms 'bet : Hdlt,zp x Hit
zp -> Zp(-1) and
bcrys : M(H) x M(H) --+ Re(-1) ,
so that under the map p from 5.5.7 we have 8 0 (Wcrys (D 1) _ ('bet ®a) o (p X p).
5.6 Vasiu's strategy-first part. We return to the situation considered in 5.1. We shall first try to prove the formal smoothness of AT under the following assumption. Here we recall that we write CSp2g for the Chevalley group scheme CSp(7G2g, 0), where V is the standard symplectic form on 7629.
(5.6.1)
There is a collection of tensors {ta E r,,; 0)}aEJ1 of degrees 2ra < 2(p-2) such that this collection is very well-positioned over the d.v.r. Z(p) for the group G (considered as a subgroup of CSp2g Q via the given closed embedding i).
Also, we shall consider a larger collection {ta}aEJ (with Jj C J) of tensors which, together with the tensor V), cut out the group G. (The to with a E again of types (re, ra, 0) but not neccesarily Z )-integral, nor of degree < 2(p - 2).)
5.6.2 Consider triplets (Xc, AC, 6p) consisting of a g-dimensional p.p.a.v. over C with a compatible system of Jacobi level n structures for all n with p { n (which we represent by the single symbol 0"). The modular interpretation ShCn(CSp29,Q, b9)(C)
{(Xc, Ac, Bp)}/
is given as follows. If (h, y) E b9 x CSp2g (Af ), then we can view y as an isomorphism y : qg ®Q Af -L-4 7G2g ®t Af . For XC we take the abelian variety determined by the lattice A := Q29 fl y-1(7G2g) and the Hodge structure h.
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There is a unique q E Q" such that q V) is the Riemann form of a principal polarization AC on XX, and the system of level structures is given by the Z29. isomorphism y : A ® (fJe#p 7Lt) One checks that this gives a Ut p well-defined bijection as claimed. By construction of the model N, there exists a purely ramified extension W C V as in 5.5 such that the closed point xo E A-7 lifts to a V-valued point
x: Spec(V) -* V. Considered as a point of A it corresponds to a p.p.a.v. with a system of level structures (X, A, 9P) over V. We shall use the notations and assumptions of 5.5; in particular we obtain a triplet (Xc, Ac, OP) over C via base-change over the chosen embedding V C K C. The corresponding point of Shoe (CSp2g,Q, $9) (C) can be represented by a pair (h, e) E .)9 x CSp2g (Af ). In particular, we get an identification HB (X (C), Z) O Z (P) = Z'9 The fact that x factors through N now implies that the tensors to as in (5.6.1)
correspond to Hodge classes ta,B on Xo which for a E 9i C J are integral w.r.t. the Z(p)-lattice HB,z ® Z(p) C H. Notice that V) gives an isomorphism (HB,z)*
HB,z(1), so that it is no restriction to assume that all to live in
spaces (Z29) (ra, ra; 0).
By [De4], Prop. 2.9, the de Rham realizations ta,dR (a E J) are defined over a finite extension of K. Possibly after replacing V by a finite extension, we may therefore assume that the ta,dR are defined over K (i.e., they are elements of tensor spaces of the form HaR K(re, ra; 0)). In particular, we obtain a subgroup G1,K C CSp(HdlR,K, 0) such that G1,K OK C = G ®Q C. Write ta,dt for the (p-adic) etale realization of ta,B, which is an element of some tensor space Ta,gt. For a E 91, the class ta,et is H't,z,, integral. Since the ta,B are Hodge classes on Xo, the ta,dR and ta,et correspond to each other via the p-adic comparison isomorphism. More precisely, we have the following result, which was obtained independently by Blasius (see [B!]) and Wintenberger. A simplified proof was given by Ogus in [Og].
5.6.3 Theorem. (Blasius, Wintenberger) Let 0 be a complete d.v.r. of characteristic (0, p) with perfect residue field. Set F = Frac(0), and let
XF be an abelian variety over F with good reduction over 0. Let tdR E HdR(rl, r2; s) and E H1t(rl, r2; s) be the de Rham component and the padic etale component respectively of an absolute Hodge class on X. Under the comparison isomorphism
y: HdR(XF/F) OF BdR -224 Het(XF,Qp) 0 we
have y(tdR 0 1) = t6t 0 /3
BdR
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We remark that in [BI] and [Og], this results is only proven under the additional assumption that XF is obtained via base-change from an abelian variety over a number field. It can be shown that this condition, which appears in the proof of a version of Deligne's "Principle B", is superfluous. In [Va2], Vasiu shows this by using a trick of Lieberman. One can also remark that the "Principle B" is needed only in the situation where we have a family of abelian varieties X -+ S over a variety S over 0, such that XF occurs as the fibre over an F-valued point of S. (The variety S constructed in [De4], Sect. 6 is a component of a Shimura variety.) In this situation, the arguments given in [Bl], Sect. 3 and [Og], Prop. 4.3 suffice. We also remark that the factor ,6-S appears because we choose KdR(1) BdR - Qp (1) 0 BdR to be the map 1 ® 1 H S ® 0-1. (In [Bl] a different normalization is used.) 5.6.4 Set H := X [p'], where X is as in 5.6.2. Notice that Hit z(H) ^-' Hit,zp(XK). There are well-defined F°-invariants ra,crys E M°[1/p](re,ra;0) such that p(T,,,crys ®1) = ta,et ®1. Using (5.5.6), we then obtain horizontal Finvariant classes ta,crys E M[1/p](ra,ra; 0). Also we have polarization forms ocrys and 0et as already mentioned in 5.5.8.
We claim that, writing T,,crys for the tensor spaces obtained from M = M(H), the ta,crys lie in Fil°(Ta,crys[1/p]). To see this, we use that ta,crys is a lifting of ta,dR in the following sense. By (5.5.5) and the isomorphism M° ®w K - HeR,K from [BO], we have M OR K - H'R,K. Combining this with the isomorphism Re(-1) ORe K -* KdR(-1) by 10 1 H 1, we obtain maps M(H) (rl, r2; s) (DRQ K - HAR,K(r1, r2; s). The functoriality of the map p in Theorem 5.5.7 implies that ta,crys ®1 H ta,dR and Ocrys ®1 H V)dR. That
ta,crys E Fil°T,,,,,ys now follows from the fact that Fil1M(H) is the inverse image of Fil1HdlR,K under the map M(H) H M(H) OR. K - HeR,K. In this way we see that V)crys and the ta,crys are crystalline Tate classes, in the sense that they are horizontal, in Fil° and invariant under Frobenius. We will be able to exploit assumption (5.6.1) by using the following supplement to Thm. 5.5.7.
5.6.5 Theorem. (Faltings, [Fa3]) Suppose that r < (p - 2), and consider Tate classes tcrys E M(H)(r, r; 0) ®R. Re[1/p]
and tet E Hit,z,,(H)(r, r; 0) ®z,
with p(tcrys 0 1) = (tet (D 1). Then tcrys is M(H)-integral if and only if tet is Hit zp(H)-integral.
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It follows from this result that the ta,crys with a E J1 are M(H)-integral classes. (Notice that we assumed these classes to have degree < 2(p - 2), as required in Faltings's theorem.) 5.6.6 We are now ready to prove one of the key steps in the argument. Since at this point we were not able to follow [Va2], we present our own explanation of what is going on. The tensors Ta,crys cut out a subgroup G1 of CSp(Mo[1/p],7,bMo). By
Thm. 5.6.3 we have G1 ®Ko K = G1,K (where the latter is the group cut out by the ta,dR that was introduced in 5.6.2 above), so that G1 is reductive. What we would like to show now is that the Zariski closure 71,Re of Gl,xe[1/p] inside CSp(M(H), 'cry,,) is a reductive group scheme over Re. Here we use the identification (5.5.6) to identify G1,R,[l/p] as a subgroup of CSp(M(H), 1LJcrys). Obviously, we will try to achieve our goal by using (5.6.1). If we look
at Def. 5.3 then we see that we already know (grace to Thm. 5.6.5) that oc*rys and the ta,crys with a E 91 are M(H)-integral, and therefore it only remains to show that there exists an isomorphism Q'9 ® Re[1/p] M(H)[1/p] such that V) and the to (a E .I) are sent to 0,,y,, and the t,,;crys ,W,/'cry.,,
respectively. The first step is that we have an isomorphism Q29 (9 Qp -C4 Het sending V) and the to to their etale realizations &et and ta,6t
Notice that we now only have to consider rings with p inverted. Since the ta,crys were obtained from the Ta crys using (5.5.6), it suffices (and will
actually be easier) to show that there exists an isomorphism v :
H1t
Mo[1/pj such that 06t H V)Mo and ta,$t H Ta,crys (This isomorphism is of course not required to have any "meaning".) Ko
By what was explained before, we can compare H't and Moll/p] after extension of scalars to the ring Bcrys, in such a way that the tensors ta,6t and the Ta,crys correspond. We have to be a little more careful about the and 'bet correspond to each other only up to a factor 3 (see 5.5.8). The ring to work over therefore is Bcys[/. ], since the factor 0 allows us to modify the isomorphism Mo[1/p] 0 Bcrys H1t 0 Bcrys in such a way that the forms 0crys and 06t do correspond. This does not affect the tensors ta, since these are of type (r,,, r,,; 0). In any case, we see that there exists a field extension Ko C SZ such that the desired comparison isomorphism exists after extension of scalars to Q. Writing ZT = (Hlt; 0, {ta,et}aE.1) and polarization forms, since
Ybcrys
1XT' = (Mo[1/p]; 1I Mo, {TT,crys}aEJ) the torsor 1som(93, 9Y) is therefore non-
empty. Since the automorphism group of the system '27 is precisely Gam, the
obstruction for finding v then is a class in H1(Gal(Q /Ko), G(Qp)). Now
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the fact that KO is a field of dimension < 1, together with [Se2], Thm. 1 in Chap. III, §2.2, proves that the obstruction vanishes, whence the existence of an isomorphism v as desired. By applying (5.6.1) this gives the following statement. 5.6.7 Proposition. The Zariski closure G1,Re of Gl,R&[llp] inside the scheme CSp(M(H), bcry9) is a reductive group scheme over Re.
It will now rapidly become clear why 5.6.7 is important. For this, we set j W:= M ®Re Re
,
and My := M ®Re V = HjR(X/V) ,
using the maps Re C Re -4 V from 5.5.3. Also we set G1,Re := 91 X Re Re ,
91,v := Gl X& V,
which are reductive groups over k and V respectively. We write Fill (Mv) _ Fill(M) ®Re V for the Hodge filtration on Mv.
5.7 Lemma. (i) There exists a complement for Fill(Mv) C Mv such that the cocharacter p: Gm,v -4 GL(Mv) given by
µ(z) =
id on Mi, z-1 id on Fill(Mv)
factors through Gl,v (ii) The cocharacter l.c: Gm,v - Gl,v lifts to a cocharacter µ: Gm,Re Gl,iie
We will admit this lemma, referring to [Va2], sect. 5.3 for a proof. We remark that the reductiveness of Gl,v and G1,Re is used in an essential way.
The cocharacter µ yields a direct sum decomposition M = M' ®M" with M" ®Re V = Fill (Mv). Since Re is a projective limit of nilpotent PDthickenings of V/pV, we can apply the Grothendieck-Messing deformation theory of abelian varieties (see [Me], in particular Chap. V). This gives us a formal p.p.a.v. (3f, A) over Spf(Re) (with the I-PD-adic topology on Re), the de Rham cohomology of which is given by HaR(3e/Re) = M with Hodge filtration M" and Gauf3-Manin connection induced from the connection on M(H) as a crystal. The fact that we have a polarization on X implies that (3E, A) algebraizes to a p.p.a.v. (X, A) over Spec(Re). We have (X, A) ®Re V = (X, A). Since we only consider level n structures with p f n, the system of level structures Bp extends to a system WP on(X, A). We claim that the morphism
Spec(Re) -> A corresponding to (X, A, Bp) factors through N C A. To prove this, we will use the following lemma.
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5.8 Lemma. Notations as above. Let R := CQzj with its (z)-adic topology, and let y : Spf(R) -+ A 0 C be a morphism corresponding to a (formal) p.p.a.v. (Y,µ,i') over Spf(R). Let io: Spec(C) -4 Spf(R) be the unique Cvalued point (given by z N 0), and assume that yo := y o i° is a point of Sh(G, X )c --+ A ® C. As in 5.6.2, we obtain de Rham classes ta,dR,o E T,
dR,o
for a E J, where the subscript "o" refers to the fact that these are classes on the special fibre Yo. Assume that the formal horizontal continuations of the classes ta,dR,o over Spf(R) remain inside Fil°T,,,dR. Then y factors through Sh(G, X). Proof (sketch).
There exists a p.p.a.v. (Y, µ) over an algebraic curve S such
that the formal completion at some non-singular point so E S gives back (Y, E.c). (In this sketch of the argument we will forget about the level structures.) Over some open disc U y San around so, we can choose a symplectic basis of H1R(Yu/U). By virtue of the Hodge filtration, this gives rise to a map
q : U - S)', where ,fjy (the compact dual of )) is the domain parametrizing g-dimensional subspaces Fill C 0.9 which are totally isotropic for the standard symplectic form. The point q(so) lies on a subvariety k C S5y (where X is the compact dual of the hermitian symmetric domains X+ C_ X as in the given Shimura datum) parametrizing those flags for which the horizontal continuations ta,dR of the t«,dR,o remain in the filtration step Fil°. By consideration of the Taylor series development of the map q at so one shows that q maps U into X, and this implies the assertion.
5.8.1 Proposition. The morphism x: Spec(Re) - A factors through N C A.
Proof (sketch). It suffices to show that the generic point of Spec(Re) maps to Sh (G, X). Consider the homomorphism j : Re * CQzI with T H z + a (7r) (using the chosen embedding Ko C K 1 6 ) C). Note that if we set (Y, p, if) :_ j- (5f-, a, 9P), then (Yo, µo, no) = a*(X, A, 0). The de Rham classes j*ta,dR on
Y are formally horizontal, since a/az (j*t.,dR) = j*(a/aT tQ,dR) = 0. The claim now follows by applying the lemma.
5.8.2 The rest of the argument is easy. Pulling back (X, )., OP) via the morphism Spec(W) y Spec(Re) we obtain a lifting of the closed point xo that we started off with, to a W-valued point of N. If H1 is the corresponding p-divisible group over W then, by specialization, we have a collection {t«,crys,1}aEJ of crystalline Tate classes on H1. Writing M1 := M(H1) = M®ReW and 9:= G1,Re xk,W " CSp(M1i 01), the group 9 is reductive and
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is precisely the group fixing all tensors ta,crys,1 This brings us in a situation Nn where we can apply the results of §4. Writing JVn, and ,AA for the formal completions at xo and x0, and using the notations of 4.5-4.9, the same reasoning as in 5.8.1 above shows that the composition Spf(C) = UgU = ,AA
factors through NA. Now C and 0N,yo are local W-algebras of the same dimension, hence UQ y N' is dominant onto a component of NA and lifts to a morphism Ug- y r. Then OV,io --» C is a surjective homomorphism between local domains of the same dimension, hence an isomorphism. This concludes the proof of the following result.
5.8.3 Theorem. In the situation of 5.1, assume that (5.6.1) holds. Then the model N is an integral canonical model of ShK, (G, X) over 0(V). 5.9 Vasiu's strategy-second part. We continue our discussion of the paper [Va2]. What remains to be done to complete Vasiu's program is to show that, in the situation of Corollary 3.23, there exists a covering (G, X) with is (G, X) ---4 (CSp29, $y) for which the assumption (5.6.1) holds. This is a highly non-trivial problem, and it is not clear to us if one can expect to solve this with the definition of a well-positioned family of tensors as in 5.3. The presentation of this material as it is presently available is too sketchy to convince us of the correctness of all arguments5; we will indicate by a marginal symbol Q statements of which we have not seen a complete proof. In the rest of this section we shall only indicate the main line of Vasiu's arguments, without much further explanation.
5.9.1 Let W be a finite dimensional vector space over a field F of characteristic zero, and consider a semi-simple subgroup G C GL(W). On Lie algebras we have gl(W) = g ® g1, where g' is the orthogonal of g := Lie(G) w.r.t. the form (Al, A2) H Tr(A1A2) on g((W). Write 7r9 for the projector onto g; we view Ire as an element of W(2, 2; 0). Next we consider the Killing form ae : g x g -+ F. Since G is semi-simple, the form 0B is nondegenerate, so that there exists a form 0,*: g* x g* -+ F with (a9, ,60*) = 1. Using the direct sum decomposition gl(W) = g ® g1 and the induced isomorphism g((W)* = g* ® (g1)*, we can view aB and /3e as elements of W(2, 2; 0).
Clearly the tensors Ire, 0g and ,69 are G-invariant. Even better: if G is the derived subgroup of a reductive group H C GL(W) then 7rg, ae and /3 are 5As remarked before, we strongly encourage the reader to read Vasiu's original papers, some versions of which appeared after we completed this manuscript.
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also H-invariant. Finally we define an integer s(g, W). For this we fix an algebraic closure F of F and we choose a Cartan subalgebra t C gF. For a root a E R(gF, t), let sa C op denote the Lie subalgebra (isomorphic to sl2) generated by ga
and g'a. We write da := max{dim(Y) I Y C WF is an irreducible sa-submodule} ,
and we define s(g,W) := max{da I a E R(g,t)}. If E is the set of weights occurring in the g-module W then da = 1 + E E"}- So, if I
a = al, a2i ... , a,. is a basis of R(g, t) and if W is irreducible with highest weight zv = nl zol + + n,. t where wi is the fundamental dominant weight corresponding to ai, then da = dal = 1 + nl. 5.9.2 Claim. Let W be a finite dimensional Q-vector space with a nondegenerate symplectic form V. If G C CSp(W, 0) is a semi-simple subgroup and if p > s(g, W) then {7r9, ,Q9, OB } is a well-positioned family of tensors for the group G over the d.v.r. Z(p).
If one tries to prove a statement like this then a priori one would have to consider an arbitrary faithfully flat Z(p)-algebra R and a free R-module M with M OR R[1/p] ^= W ®Q R[1/p]. Since we are dealing with a finite collection of tensors, however, one easily reduces to the case that R is of finite type over Z(p). Also we may replace R by a faithfully flat covering, since taking a Zariski closure of something quasi-compact commutes with flat base-change. This allows one to reduce to the case that R is a complete local noetherian ring.
It should be noted that in general the Zariski closure of GR[Ipl inside GL(M) is not a subgroup scheme, even if R is a regular local ring. We refer to the work [BT] of Bruhat and Tits, especially loc. cit., 3.2.15, for further theory and a very instructive example.
5.9.3 Corollary. Assume 5.9.2 to hold. Consider a closed immersion of Shimura data i : (G, X) -* (CSp29,Q, fj9) . Let p be a prime number, with p > 5. Assume that the Zariski closure of G inside CSp29,Z(p) is reductive and that the tensors lr9der, ,39der and NBder are Zip)-integral. Then condition (5.6.1) is satisfied. In particular: for every prime v of E = E(G, X) above p and every hyperspecial subgroup Kp C G(Q ), there exists an i.c.m. of ShKK(G, X) Over OE,(v)
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Up to one technical detail, we can derive this corollary from the previous claim by the following argument. By the results of [De3], Sect. 1.3, all highest
weights in the representation i : G " GL(Q29) are miniscule in the sense of [Bou], Chap. VIII, §7, n° 3. It follows from this that s(gdeT,Q9) = 2. Since p > 5, the set of tensors {7rgder, 00der, 09der} is a well-positioned set of G-invariant tensors (of degree 4) for the group Gder over Z(p). Next we consider the Zariski closure 9 of G inside CSp29,Z(p). By assump-
tion, it is reductive. Let Z := Z(g)° be the connected center of g, which is a torus over 7L(p) with generic fibre Z := Z(G)°. Also write C C End(Z29) for the subalgebra of endomorphisms which commute with the action of Q. We claim that the elements of C form a well-positioned collection of G-invariant tensors (of degree 2) for the group Z over Z(p). We will not prove this; the essential idea is to reduce to the situation where Z is a split torus. For details we refer to [Va2]. Now we take T :=I 7rgd., I I39der, ,Q8*der} U C as our collection of G-invariant
tensors. Notice that the condition 2r, < 2(p - 2) in (5.6.1) is satisfied, since we are only using tensors of degrees 2 and 4 and since p > 5. To conclude the proof of the corollary, one considers a faithfully flat Z(p)-algebra R and a free R-module M with an identification M OR R[1/p] Q29 (&Q R[1/p] such that 0 and V)*, as well as all tensors in our collection T are M-integral. Then we know that V induces a perfect form WM on M, that the Zariski closure 91 of Gder ®Q R[1/p] inside CSp(M, OM) is semi-simple, and that the Zariski closure Zl of Z®QR[1/p] inside CSp(M, 'M) is a torus. We are therefore left with the following question. (In [Va2] it is used implicitly that the answer is affirmative.)
5.9.4 Problem. Let R be a faithfully flat Z(p)-algebra and let M be a free R-module of finite rank. If GR[1/p] C_ GL(M[1/p]) is a reductive subgroup scheme such that the Zariski closures yl and Zl of respectively its derived subgroup and its connected center are reductive subgroup schemes of GL(M), does it follow that the Zariski closure of GR[1/p] inside GL(M) is flat over R and therefore again a reductive subgroup scheme?
Perhaps the answer to this question is known to experts in this field, in which case we would be interested to hear about it. If we assume that the answer is affirmative then Cor. 5.9.3 follows by the arguments given above.
5.9.5 Claim. Let (Gad, Xd) be an adjoint Shimura datum of abelian type, and let p > 5 be a prime number such that G' is unramified. Then there
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exists a Shimura datum (G, X) covering (G', X') and a closed immersion i:
(G, X) -+ (CSp29,Q, S59) such that condition (5.6.1) holds for G.
In [Va2] this statement is claimed as a consequence of a whole chain of constructions, reducing the problem to Cor. 5.9.3.
0
5.9.6 Corollary. (Assuming 5.9.2-5.9.5) Let (G, X) be a Shimura datum of pre-abelian type. Let p > 5 be a prime number such that (notations of 3.21.5) p { SG and such that GQP is unramified. Let Kp C G(Q) be a hyperspecial subgroup and let v be a prime of E(G, X) above p. Then there exists an integral canonical model M of ShKK(G, X) over As a scheme, M is the projective limit of smooth quasi-projective with etale coverings as transition maps.
§6 Characterizing subvarieties of Hodge type; conjectures of Coleman and Oort 6.1 We now turn to a couple of problems of a somewhat different flavour. Consider a Shimura variety ShK (G, X). We have seen in § 1 that, depending on the choice of a representation of G, we can view it, loosely speaking, as a "moduli space" for Hodge structures with some given Hodge classes. In this interpretation, the "Shimura subvarieties" would be components of the loci where the Hodge structures have certain additional classes. The type of question that we are interested in here is: "can we give a direct description of these Shimura subvarieties?", and "given an arbitrary subvariety of ShK (G, X), can we say something about "how often" it intersects a Shimura subvariety?". More specific questions will be formulated below. First, however, let us make the notion of a Shimura subvariety more precise.
6.2 Definition. Let (G, X) be a Shimura datum. An irreducible algebraic subvariety S C_ ShK (G, X )c is called a subvariety of Hodge type if there exist an algebraic subgroup H C_ G (defined over Q), an element i E G(Af) and a connected component YH of the locus
YH := {h E X I h: S -4 Ga factors through HR}
such that S(C) is the image of YH x 77K in ShK(G,X)(C) = G(Q)\X x G(Af)/K. If E(G, X) C F C C, then an algebraic subvariety S C ShK(G, X)F is called a subvariety of Hodge type if all components of Sc are of Hodge type. (If S is irreducible then it suffices to check this for one component of Sc.)
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For example: a point x of ShK(G, X), considered as a 0-dimensional subvariety, is of Hodge type if and only if x is a special point. If ShK (G, X) -+ A9,1,n is a Shimura subvariety of Hodge type then these conditions on x are equiva-
lent to saying that x corresponds to an abelian variety of CM-type (in which case we say that x is a CM-point.) If f : (G1, X1) y (G2, X2) is a closed immersion of Shimura varieties and if we have compact open subgroups Ki C Gi(A1) (i = 1, 2) with f (K1) C_ K2, then the connected components of the image of Sh(f) : ShK, (G1, X1) -* ShK, (G2, X2) are called subvarieties of Shimura type. The subvarieties of Hodge type are precisely the irreducible components of Hecke translates of subvarieties of Shimura type. For further details see [Mol], Chap. I or [Mo2], section 1. Now for some of the concrete problems that we are interested in.
6.3 Conjecture. (Coleman, c£ [Co]) For g > 4, there are only finitely many smooth projective genus g curves C over C (taken up to isomorphism) for which Jac(C) is of CM-type.
6.4 Conjecture. (Oort, cf. [Oo3]) Let Z " A9,1,,, ® C be an irreducible algebraic subvariety such that the CM-points on Z are dense for the Zariski topology. Then Z is a subvariety of Hodge type. 6.5 Let us first make some remarks on the status of these conjectures. Coleman's conjecture, as we phrased it here, is false for g = 4 and g = 6: there exist families of curves C -+ S of genus 4 and 6, such that the image of S in A9,1 corresponding to the family of Jacobians Jac(C/S) -+ S is (an open part of) a subvariety of Hodge type of dimension > 0. The known examples of this type are given by explicit polynomial equations. For example, let S be the affine line with coordinate A, and let CN be the smooth curve over S with affine
equation yN = x(x - 1) (x - A). If 3 f N then C is a family of curves of genus N - 1 with an automorphism (N of order N given by (x, y) H (x, e27i/N . Y). For N = 5 (resp. N = 7) we obtain a family of Jacobians JN -+ S with complex multiplication by Q[(5] (resp. Q[(7]), and one computes that the complex embedding given by (N H ek-27ri/N has multiplicity 2,1,1,0 for k = 1, 2, 3, 4 (resp. multiplicity 2, 2, 1, 1, 0, 0 for k = 1, ... , 6) on the tangent space. Now the Shimura variety of PEL type parametrizing abelian 4-folds (resp. 6-folds) with complex multiplication by an order of Q[S5] (resp. Q[(7]) and the given multiplicities on the tangent space is 1-dimensional, so the image of S in A4,1 (resp. A6,1) is an open part of such a subvariety of PEL type. It follows that
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there are infinitely many values of A such that Jac(CA) is of CM-type. For further details, and another example of this kind, we refer to [dJN]. For genera g = 5 and g > 7, Coleman's conjecture remains, to our knowledge, completely open. It is plausible that the known counter examples are exceptional, and that examples of such kind only exist for certain "low" genera.
Let us point out here that in the above example, we do not find a
subvariety of Hodge type if 3 { N and N > 8; this follows from [dJN], Prop. 5.7 and the results of Noot in [No2] (see 6.15 below). 6.6 Oort's conjecture was studied by the author in [Mol]. The results here are based on a characterization of subvarieties of Hodge type in terms of certain "linearity properties". We will discuss this in more detail below. One of the results in loc. cit., is a proof of Oort's conjecture under an additional assumption. This is a general result, which provides further evidence for the conjecture. Unfortunately, the extra assumption is difficult to verify in practice. In another direction, one can try to prove the conjecture in concrete cases. The first non-trivial case is to consider subvarieties of a product of two modular curves. After some reduction steps first proved by Chai, Andre and Edixhoven (see [Ed2]) both found a proof for the conjecture in this case under an additional hypothesis. Both their methods and the hypotheses involved were rather different. Recently, Andre found an unconditional proof, so that we now have the following result (see [An2]).
6.6.1 Theorem. (Andre) Let S1 and S2 be modular curves over C, and let C C S1 x S2 be an irreducible algebraic curve containing infinitely many points (x1, x2) such that both x1 E Sl and X2 E S2 are CM-points (in other words, C contains a Zariski dense set of CM-points). Then C is a subvariety of Hodge type, i.e., either C = Si x {x2}, where x2 is a CM-point of S2, or C = {x1 } x S2, where x1 is a CM-point of S1, or C is a component of a Hecke correspondence. 6.7 One of the motivations for Oort to formulate his conjecture is its analogy with the Manin-Mumford conjecture, now a theorem of Raynaud (see [Ra2]). We recall the statement:
6.7.1 Theorem. (Raynaud) Let X be a complex abelian variety, and let Z " X be an algebraic subvariety which contains a Zariski dense collection of torsion points. Then Z is the translate of an abelian subvariety over a
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torsion point. The analogy is obtained by using the following dictionary:
Oort's conjecture Shimura variety CM-point (or special point) subvariety of Hodge type
"Manin-Mumford" = Raynaud's thm. abelian variety torsion point translate of an abelian subvariety over a torsion point
To push the analogy even further, let us mention that one can formulate a conjecture which contains both Oort's conjecture and "Manin-Mumford" as special cases. The idea here is to look at mixed Shimura varieties. Since we have not discussed these in detail, let us mention the following fact: if S y A9,1,,, is a subvariety of Hodge type, and if X -+ S is the universal abelian scheme over it, then X can be described as a (component of a) mixed Shimura variety. (See [Pi] and [Mi2] for further examples and details.) The special points on X are the torsion points on fibres X, of CM-type. However, the axioms of mixed Shimura varieties are too restrictive for our purposes, since, for example, an abelian variety X which is not of CM-type, cannot be described as a mixed Shimura variety. By loosening the axioms somewhat, we are led to what might be called "mixed Kuga varieties" and to the following conjecture, proposed by Y. Andre in [Anl]. (Andre adds the remark that this is only a tentative statement, which may have to be adjusted.)
6.7.2 Conjecture. Let G be an algebraic group over Q, let K... be a maximal compact subgroup of G(]R), and let r be an arithmetic subgroup of G(Q). Suppose that K,,,:, is defined over Q, that G(R)/K,,,, has a G(R)-invariant complex structure and that the complex analytic space P\G(R)/K,,. is algebraizable. Let us call an irreducible algebraic subvariety s y r\G(R)/K,,. a special subvariety if there exists an algebraic subgroup H C G defined over Q E r\G(R)/K,,. I h E H(R)}. and an element go E G(O) such that S = Then S is a special subvariety if and only if it contains a Zariski dense collection of special points. 6.8 Assume Oort's conjecture to be true. Then Coleman's conjecture becomes the question of whether there are positive-dimensional subvarieties of Hodge type S --4 A9,1 ® C of which an open part is contained in the open Torelli locus 7 (:= the image of the Torelli morphism M9 0 C -* A9,1(& C).
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This seems a difficult question, also if we replace the open Torelli locus by its closure. Hain's paper [H] contains interesting new results about this.' To state Hain's results, let us first consider an algebraic group G over Q which gives rise to a hermitian symmetric domain X (i.e., an algebraic group of hermitian type), and consider a locally symmetric (or arithmetic) variety
S = r\X, where P is an arithmetic subgroup of G(Q). If G is Q-simple then we call S a simple arithmetic variety. We say that S is bad if it contains a locally symmetric divisor (examples: G = SO(n, 2) or G = SU(n, 1), as well as the case dim(S) = 1); otherwise call S good. This is a really a property of G, i.e., it does not depend on r and the resulting S. In the next statement we only consider the simple case; this is not a serious restriction since every arithmetic variety has a finite cover which is a product of simple ones.
6.8.1 Theorem. (Hain) Let S be a simple arithmetic variety which is good in the above sense.
(i) Suppose p: C -> S is a family of stable curves over S such that the Picard group Pic°(C3) of every fibre is an abelian variety (i.e., every fibre is
a "good" curve: its dual graph is a tree), such that the generic fibre Cn is smooth, and such that the period map S -4 A9,1 is a finite map of locally symmetric varieties. Then S is a quotient of the open complex n-ball for some
n. (So GR = PSU(n, 1) x (compact factors).) (ii) Suppose q : Y -* S is a family of abelian varieties, such that every fibre
Y9 is the Jacobian of a good curve, and such that the period map S -- A9,1 is a finite map of locally symmetric varieties. Write Sred (resp. ShYP) for the locus of points such that YS is the Jacobian of a reducible (resp. hyperelliptic) curve, and let S* be the complement of Sred which we assume to be nonempty. Then either S is the quotient of the complex n-ball for some n, or g > 3, each component of Sred has codimension > 2 and S* fl ShYP is a non-empty smooth divisor in S*.
(We point out that a family Y -+ S as in (ii) is not necessarily of the form Jac(C/S) -+ S for a family C -* S as in (i), due to the fact that the Torelli morphism is ramified along the hyperelliptic locus. If, in (i), all fibres are smooth then the condition that S is good can be ommitted.) 6.9 The next issue that we want to discuss is the characterization of subvarieties of Hodge type by their property of being "formally linear". Here we 6 We thank R. Hain for sending us a preliminary version of this paper.
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owe the reader some explanation. Let us first do the theory over C, which works for arbitrary Shimura varieties. Consider a Shimura variety ShK = ShK(G, X)c over C, and let S " ShK be a subvariety of Hodge type. Then S is a totally geodesic subvariety: if u: X+ -+ _Sh°{ is the uniformization of the component _Sh°{ C ShK containing
S, and if S C_ X+ is a component of u-1(S), then S is a totally geodesic submanifold of the hermitian symmetric domain X+. This property does not characterize subvarieties of Hodge type; for a trivial example: any point x E S forms a totally geodesic algebraic subvariety, but {x} C S is a subvariety of Hodge type if and only if x is a special point. Essentially, however, we are dealing with the well-known distinction between "Kuga subvarieties" and subvarieties of Hodge type. In a somewhat less general setting, this distinction was clarified by Mumford in [Mul]. The same idea works in general, and we have the following characterization (see [Mol], Thm. 11.3.1, or [Mo2]).
6.9.1 Theorem. Let S " ShK(G,X)c be an irreducible algebraic subvariety. Then S is a subvariety of Hodge type if and only if (i) S is totally geodesic, and (ii) S contains at least one special point.
Let us mention that one can also give a description of totally geodesic subvarieties in general (i.e., not necessarily containing a special point). It turns out that they are intimately connected with non-rigidity phenomena. For example, let ShK (G, X )c " ShK' (G', X')c be a closed immersion of Shimura varieties, and suppose that the adjoint group Gad decomposes (over Q) as a product, say Gad = Gl x G2. Correspondingly, there is a decomposition Xad = Xl x X2 of X as a product of (finite unions of) hermitian symmetric domains. Fix a component Xl C_ Xl, a point x2 E X2, and a class 77K E G(Af)/K, and let S,1K(Xi , x2) denote the image of Xi x {x2} in ShK(G, X) under the map X 3 x H [xxr7K]. One can show that SnK(Xi , x2)
is a totally geodesic algebraic subvariety of ShK' (G', X')c, and that, conversely, all totally geodesic algebraic subvarieties of ShK' (G', X')c are of this form. After passing to a suitable level (i.e., replacing K by a suitable subgroup of finite index) we can arrange that the component ShK of ShK(G, X)c containing S,,K(X; , x2) is a product variety ShK = S1 x S2, with S,7K(Xi , x2) _
Si x {s2} for some point s2 E S2. Now assume that G2 is not trivial, so that dim(S2) > 0. We see that Sl x {s2} is non-rigid: global deformations are obtained by moving the point s2 E S2. If (G, X) is of Hodge type, say with ShK' (G', X')c = A9,1,,,, 0 C in the above, then we obtain a non-rigid
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abelian scheme over S1. (Notice, however, that the non-rigidity may be of a trivial nature, in the sense that all non-rigid factors of the abelian scheme
in question are isotrivial.) The gist of the results in [Mol], §II.4 (see also [Mo2]) is that all non-rigid abelian schemes, and all their deformations, can be described via the above procedure. We refer to loc. cit. for further details.
6.9.2 We can jazz-up the above characterization of subvarieties of Hodge type. This will lead to a formulation very analogous to the results in mixed characteristics, to be discussed next. The first important remark is that total geodesicness needs to be tested only at one point. More precisely: if Z " ShK(G, X)c is an irreducible algebraic subvariety, and if x E Z is a non-singular point of ShK(G,X)c, then Z is totally geodesic (globally) if and only if it is totally geodesic at the point x. This is true because ShK(G,X)c has constant curvature. Next we define a Serre-Tate group structure on the formal completion 67
x
of ShK(G, X)c at an arbitrary point x. Here we assume that K is neat, so that ShK (G, X) is non-singular. The procedure is the following.
The point x lies in the image Sh° of a uniformization map u: X+ -> ShK (G, X), which, by our assumption on K, is a topological covering. Choose
x E X+ with u(x) = x. We have a Borel embedding
X+ "X =Gad (C)1Pj(C), where Pi C Gad is the parabolic subgroup stabilizing the point x. Using the Hodge decomposition of gc with respect to Ad o ht, we obtain a parabolic subgroup P. C G' opposite to P . Write U. for the unipotent radical of P, , which is isomorphic to Ga for d = dim(X). The natural map U ,,-(C) - X gives an isomorphism of U,, (C) onto its image U C X which is the complement of a divisor D C X. On formal completions we obtain an isomorphism := Ui /11}
u11.i} = X/m
ShOW =: 61J,,,
and in this way C7CI inherits the structure of a formal vector group. This we One checks that it is independent call the Serre-Tate group structure on of the choice of x above x. If Z is a subvariety as above, then by taking the formal completion at x, and we call Z formally linear at x we obtain a formal subscheme 3x " if 3x is a formal vector subgroup of C70.. Using this terminology we have the following result. (See [Mo2], §5.)
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6.9.3 Theorem. Let Z y ShK(G,X)c be an irreducible algebraic subvariety. If Z is totally geodesic then it is formally linear at all its points. Conversely, if Z is formally linear at some point x E Z, then it is totally geodesic. In particular, Z is a subvariety of Hodge type if and only if (i) Z is formally linear at some point x E Z, and (ii) Z contains at least one special point.
6.10 In mixed characteristics, our notion of formal linearity is based on Serre-Tate deformation theory of ordinary abelian varieties. Almost everything we need is treated in Katz' paper [Ka]; additional references are [DI] and [Me]. Without proofs, we record some statements that are most relevant for our discussion. Let k be a perfect field of characteristic p > 0, and let X0 be an ordinary abelian variety over k. Set W = W (k), and write Cw for the category of artinian local W-algebras R with W/(p) = k -24 R/mR. The formal deformation functor DefoXo : Cw -* Sets is given by Defoxo(R) = {(X, cp) I X an abelian scheme over R; cp: X ®k - Xo}/
By the general Serre-Tate theorem, this functor is isomorphic to the formal deformation functor of the p-divisible group X0[p°°]. Since X0 was assumed to be ordinary, the latter is a direct sum Xo[p°°] = Gt, ® Get of a toroidal and an etale part. For R E Cw, these two summands both have a unique lifting, say Gµ and Get respectively, to a p-divisible group over R. We therefore have DefoXo (R) = {a E ExtR(Get, G,y) I alspec(k) is trivial},
and in particular we see that DefoXo has a natural structure of a group functor. Fix an algebraic closure k of k, write W := W (k), and write TPXo for the
"physical" Tate module of X0. The formal deformation functor of Xo 0 k can be given "canonical coordinates": if (X, cp) E DefoXo®k(R) for some R E Cw(k), then one associates to X a Zr bilinear form q(X/R; -, -) : TpXo x TPXo
Gm (R) = 1 + MR
and it can be shown that this yields an isomorphism of functors DefoXo®k
Homz,(TPXo 0 TPXo, dm).
If we identify the double dual Xtt and X, then we have a symmetry formula q(X/R; a, at) = q(Xt/R; at, a). Furthermore, if fo : Xo,k -* Yo k is a
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homomorphism of ordinary abelian varieties over k, then fo lifts to a homomorphism f : X -> Y over R E Cyy if and only if q(X/R; a, ft(/3)) _ q(Y/R; f (a),,6) for every a E TPXo, /3 E TpYo. Let Ao : X0 --* Xo be a principal polarization. Using the induced isomorphism TPXo - TPXo we have Defoxo,k ^= Hom(TPXo 2, Q'm), and by the previous remarks the formal deformation functor Defo(XO,k,),o) of the pair (Xo,k, )o)
is isomorphic to the closed subfunctor Hom(Sym2(TPXo), Gm).
6.11 Let i be a perfect field of characteristic p with p { n, and let x E (A9,1,n (g r ),ra be a closed ordinary moduli point with residue field k. Write (Xo, Ao, Bo) for the corresponding p.p.a.v. plus level structure over Spec(k). The formal completion 2( := (A9,1,n ® W(N))1{X} is a formal torus over Spf (W (k)); since we consider level n structures with p { n, it represents the formal deformation functor Defo(X0,),o). By the above, % has the structure of a formal torus over W (k), called the Serre-Tate group structure. Choose a basis {al, ... , a9} for TPXo, and set qii = q(-; ai, Ao(aj)). We
have %&W Spf(A), where A = W[qij - 1]/(qij - qji) with its m-adic topology, m = (p, qi7 - 1). If X -+ 2( := 2(x®W is the universal formal deformation, then there is an explicit description of the Hodge F-crystal H = HaR(3E/2(): to the chosen basis jai, ... , a9} one associates an A-basis {al, ... , a9, bl, ... , b9} of H such that (i) the Hodge filtration is given by Fil° = H D Fill = A bl +
+ A b9 D
Fi12 = (0),
(ii) the Gauf3-Manin connection is given by O(ai) = 0, O(bi) = Ei ai ® dlog(gij),
(iii) the Frobenius ('H is the PA-linear map determined by 4DH(ai) = ai,
We will need to work with 2t, is a slightly more general setting. For this,
consider a number field F, a finite prime v of F above p, and write A9 = be a closed ordinary moduli point with A := W(k) ®w{.()) 0,,, A := W 0,,. The formal completion 2t := (A9)/{y} now is a formal torus over A. It is simply the pull-back via Spf(A) -* Spf(W) of the formal torus considered A9,,,, 0 O(v) - Let X E (A9 0
ic(v))ora
residue field k. Set 0 =
above.
6.12 The lifting of Xo corresponding to the identity element 1 E 24 (W (k)) is called the canonical lifting, and will be denoted Xoa1. The liftings over
342
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W(k) [(pn ] corresponding to the torsion points of 21,, are called the quasicanonical liftings. Suppose that k is a finite field, so that Xo is an abelian variety of CM-type. The canonical lifting Xoa° is the unique lifting of Xo such that all endomor-
phisms of Xo lift to Xoa". The quasi-canonical liftings of Xo are precisely the liftings of Xo which are of CM-type; they are mutually all isogenous. For proofs see [dJN], section 3, [Me], Appendix, [Moll, §III.1.
6.13 Definition. Suppose, with the above notations, that Z " A9,l,n 0 F is an algebraic subvariety. Let Z " A. denote its Zariski closure inside A9.
y
Suppose that the closed ordinary moduli point x is a point of (Dic(v))°rd. Then we say that Z is formally linear (resp. formally quasi(A9 linear) at x if its formal completion 3 := y 2t is a formal subtorus (resp. if all its (formal) irreducible components are the translate of a formal subtorus over a torsion point). (Z(Dk(V))ord
6.14 Example. Suppose Z is a component of a subvariety of PEL type, parametrizing p.p.a.v. with an action of a given order R in a semi-simple Q-algebra. In particular, we have to : R ' End(Xo). Consider the formal subscheme of 21., parametrizing liftings X of Xo such that to lifts to t : R y End(X). It follows from the facts in 6.10 that this is a union of translates of formal subtori of 2t over torsion points. (The reader is encouraged to verify this.) It follows that Z is formally quasi-linear at x. Moreover, if Z is absolutely irreducible and the order R is maximal at p then Z is formally linear at x.
The relation between formal linearity and subvarieties of Hodge type is expressed by the following two results, which were obtained by Noot in [Nol] (see also [No2]) and the author in [Moll (see also [Mo3]), respectively.
6.15 Theorem. (Noot) Let F be a number field, and let S y A9,1,,, 0 F be a subvariety of Hodge type. Let v be a prime of F above p, and write S for the Zariski closure of S inside A9,l,n ® O(v). Let x be a closed point in the ordinary locus (S (9 ic(v))°rd. Then S is formally quasi-linear at x. For v outside a finite set of primes of OF, the formal completion C7., of S at x is a union of formal subtori of %x.
6.16 Theorem. Let Z " A9,l,n ® F be an irreducible algebraic subvariety over a number field F. Suppose there is a prime v of OF such that the model Z
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of Z (as above) has formally quasi-linear components at some closed ordinary point x E (z (9 ,c(v))°Td Then Z is of Hodge type. We refer to [Moll and [Mo2] for some applications of 6.16 to Oort's conjecture. Given Z as in 6.4 (which then is defined over a number field), one tries
to prove that Z is formally linear at some ordinary point in characteristic p. In general, we do not know how to do this; the main difficulty is that we have little control over the CM-points on Z. With certain additional assumptions, which we will not specify here, one can, however, prove such a statement. See in particular [Mo2], §5. Notice that 6.16 is a "local" version of Oort's conjecture: an algebraizable irreducible formal subscheme of 2t comes from a subvariety of Hodge type if and only if it contains a dense collection of CM-points (= torsion points). (The adjective "algebraizable" is essential.) We think of this local version and of Raynaud's "Manin-Mumford" theorem as "abelian" cases. Morally, the global case of Oort's conjecture is more difficult because it involves nonabelian group structures.
6.17 To finish, let us take one more look at Coleman's conjecture. A naive attempt to disprove it runs as follows: consider the ordinary locus of M g,Fp, and try to lift the corresponding curves to characteristic zero such that the Jacobian remains of CM-type. This does not work so easily, due to the wellknown fact that the canonical lifting of a Jacobian in general no longer is a Jacobian. In [DO], Dwork and Ogus give an "abstract" proof of this. ("Abstract" as opposed to the explicit examples demonstrating this fact given by Oort and Sekiguchi in [OS].) They call an ordinary (smooth projective) curve C over a perfect field k of char. p a pre-W,, canonical curve if, setting
X0 = Jac(C), the canonical lifting X0` mod p"+1 over W,,(k) is a Jacobian. They then show that the locus Ew, of pre-Wl-canonical curves (preW2-canonical in their notations) forms a constructible part of .M°ra which is g>Fn nowhere dense if g > 4.
For our "naive attempt" this still leaves hope, though. As Dwork and Ogus write, "It would be interesting to study the "deeper" subschemes Ew,, for higher n ... ". Coleman's conjecture suggests that Ew. should be a finite set of points. Oort's conjecture together with 6.16 lead to another suggestion. Namely, if we write -r: Mg -4 Ag,1 for the Torelli morphism, and if x E Ew_ then "locally around T(x)", the locus r(Ewy.) should be the largest subvariety which is contained in the Torelli locus T(Mg,FP) and which is formally linear (purely in characteristic p). It seems that one can prove this by "iterating" the
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method of [DO]. Unfortunately, our control of the higher-order deformation theory is as yet insufficient to use this to show that Ewe, is 0-dimensional.
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WESTFALISCHE WILHELMS-UNIVERSITAT MUNSTER, MATHEMATISCHES INSTI-
TUT, EINSTEINSTRASSE 62, 48149 MUNSTER, GERMANY
[email protected]. de
Euler systems and modular elliptic curves KARL RUBIN
INTRODUCTION
This paper consists of two parts. In the first we present a general theory of Euler systems. The main results (see §§3 and 4) show that an Euler system for a p-adic representation T gives a bound on the Selmer group, associated These theorems, which generalize work to the dual module Hom(T, of Kolyvagin [Ko], have been obtained independently by Kato [Kal], PerrinRiou [PR2], and the author [Ru3]. We will not prove these theorems here, or even attempt to state them in the greatest possible generality. In the second part of the paper we show how to apply the results of Part I and an Euler system recently constructed by Kato [Ka2] (see the article of Scholl [Scho] in this volume) to obtain Kato's theorem in the direction of the Birch and Swinnerton-Dyer conjecture for modular elliptic curves (Theorem 8.1).
Part 1. Generalities 1. SELMER GROUPS ATTACHED TO p-ADIC REPRESENTATIONS
A p-adic representation of GQ = Gal(Q/Q) is a free Z, ,-module T of finite rank with a continuous, Zp linear action of GQ. We will assume in addition throughout this paper that T is unramified outside of a finite set of primes. Given a p-adic representation T, we also define V = T ®zp QP,
W = V/T = T ®Qp/Zp.
(Note that T determines V and W, and W determines T and V, but in general there may be non-isomorphic Zp lattices T giving rise to the same vector space V.) The following are basic examples of p-adic representations to keep in mind.
Example. If p : GQ -+ Zp is a continuous character we can take T to be a free, rank-one Zp-module with GQ acting via p (clearly every one-dimensional Reprinted from `Galois Representations in Algebraic Arithmetic Geometry', edited by A. J. Scholl & R. L. Taylor. ©Cambridge University Press 1998
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representation arises in this way). For example, when p is the cyclotomic character we get T = Zp(1) = lim upn.
n
Example. If A is an abelian variety we can take the p-adic Tate module of A T = Tp(A) = limApn. n
This is the situation we will concentrate on in this paper, when A is an elliptic curve.
Example. If T is a p-adic representation, then we define the dual representation T* = Hom(T, Zp(1)) and we denote the corresponding vector space and divisible group by V* = T* 0 Qp = Hom(V, Qp(1)), W* = V*/T* = Hom(T, µp.). If E is an elliptic curve then the Weil pairing gives an isomorphism Tp(E)* Tp(E).
Let Q,,, C Q(µp.) denote the cyclotomic Zp extension of Q. For every n let Q. C Q(µpn+l) be the extension of Q of degree pn in Q,,, and let Qn p denote the completion of Q, at the unique prime above p. Fix a p-adic representation T as above. We wish to define a Selmer group S(Qn, W) C H' (Qn, W) for every n. If v is a place of Qn not dividing p, let I denote an inertia group of v in GQn, let Qn'v denote the maximal unramified extension of Q, ,,,, and define
HS(Qn,v,V) = Hnr(Qn,v, V) = ker(H1(Qn,v, V) -+H1(Qnv,V))
= H1(Qnrv/Qn,v, VI)
For the unique prime of Qn above p, we will ignore all questions about what is the "correct" definition, and we just fix some choice of subspace HS(Qn,p, V) C H'(Qn,p, V). (For example, one could choose HS(Qn p, V) _ H1(Qn,p,V) or HS(Qn,p,V) = 0.) For every place v of Qn we now define HS (Qn,v, W) C H1(Qn,v, W) and HS (Qn,v, T) C H' (Qn,v, T) to be the image and inverse image, respectively, of HS(Qn,v, V) under the maps on cohomology induced by the exact sequence
0-*T-+ V -W-30. Finally, we define
S(Qn, W) = ker (H'(Q W) - ® H'(Qn,v,W)/HS(Qn,v,W) vofQ. Of course, this definition depends on the choice we made for HS(Qn p, V).
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2. EULER SYSTEMS
Fix a p-adic representation T of GQ as in §1, and fix a positive integer N divisible by p and by all primes where T is ramified. Define
R = R(N) = {squarefree integers r : (r, N) = 1} For every prime q which is unramified in T, let Frq denote a Frobenius of q in GQ and define the characteristic polynomial Pq(x) = det(1 - FrgxIT) E ZP[x].
Since q is unramified in T, Pq is independent of the choice of Frobenius element Frq.
Definition. An Euler system c for T is a collection of cohomology classes CQn(pr) E H'(Qn(/-Lr), T)
for every r E R and every n > 0, such that if m > n, q is prime, and rq E R, then CorQn({1rq)/Qn({pr)CQn(b&,.q)
= Pq(Y 1Fq 1)CQn(µr),
CorQm(µr)/Qn(p7)CQm(µr) = CQ, (/pr).
Note that this definition depends on N (since R does), but not in an important way so we will suppress it from the notation. Remarks. Kolyvagin's original method (see [Ko] or [Rul]) required the Euler
system to satisfy an additional "congruence" condition. The fact that our Euler system "extends in the Q,,. direction" (i.e., consists of classes defined over the fields Qn(tr) for every n, and not just over Q(tr)) eliminates the need for the congruence condition. There is some freedom in the exact form of the distribution relation in the definition of an Euler system. It is easy to modify an Euler system satisfying one distribution relation to obtain a new Euler system satisfying a slightly different one. 3. RESULTS OVER Q
We now come to the fundamental applications of an Euler system. For the proofs of Theorems 3.1, 3.2, and 4.1 see [Kal], [PR2], or [Ru3]. In fact, once the setting is properly generalized the proofs are similar to the original method of Kolyvagin [Ko]; see also [Rul] and [Ru2]. For this section and the next fix a p-adic representation T as in §1. Fix also a choice of subspaces HS (Qn,p, V) and HS (Qn,p, V *) for every n, so that
we have Selmer groups as defined in §1. We assume only that these choices satisfy the following conditions:
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HS (Qn,p, V) and HS (Qn,p, V*) are orthogonal complements under the cup product pairing
H'(Qn,p,V) x H'(Qn,p,V*) -+ H2(Qn,p,Qp(1)) = Qp,
ifm>nthen CorQ.,p/Q,..pH8(Qm,p, V) C Hs(Qn,p, V) I ResQm,pHS(Qn,p, V) C HH(Qm,p,V)
and similarly for V*. We will write His (Qn,p)T) = H1 (Qn,,,, T) /HS (Qn,p, T) (and similarly with T replaced by V or W), and we write locpa' for the localization map locna'n : H'(Qn,T) _+ HIS(Qn,p,T).
We will make use of two different sets of hypotheses on the Galois representation T. Hypotheses Hyp(Q,,., T) are stronger than Hyp(Q,,., V), and will allow us to prove a stronger conclusion. Hypotheses Hyp(Qc,, T). (i) There is a r E GQ such that T acts trivially on µpa,, T/(T - 1)T is free of rank one over Zp. (ii) T/pT is an irreducible
(i) There is a r E GQc such that r acts trivially on µp.,
Hypotheses Hyp(Qo,,, V).
dimQp(V/(T - 1)V) = 1. (ii) V is an irreducible Qp[GQ.]-module.
Theorem 3.1.
Suppose c is an Euler system for T, and V satisfies
HYp(Qc,V) If cQ V H'(Q,T)tors and [HIs(Qp,T) : 1ocPan'(H1(Q,T))] is finite, then S(Q, W*) is finite. In particular if locca"(cQ) 0 0 and rankzpH/'s(Qp, T) = 1, then S(Q, W*) is finite. Define S2 = Q(W)Q(µp.), where Q(W) denotes the minimal extension of Q such that GQ(w) acts trivially on W.
Theorem 3.2. Suppose c is an Euler system for T, and T satisfies Hyp(Q., T). If p > 2 and loc,a'(cQ) # 0 then IS(Q, W*)I /S(QP+T)/A[loc ' (cQn)] is a torsion A-module. Define G = char(H.1 /s(Qn,
7')/A[locpaz"(cQn)])
Then
(i) X,,. is a torsion A-module, (ii) there is a nonnegative integer t such that char(X,,,,) divides ptG, (iii) if T satisfies Hyp(Q,,, T) then char(X,,.) divides G.
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Part 2. Elliptic curves The `Heegner point Euler system' for modular elliptic curves used by Kolyvagin in [Ko] does not fit into the framework of §2, because the cohomology classes are not defined over abelian extensions of Q. However, Kato [Ka2] has constructed an Euler system for the Tate module of a modular elliptic curve, using Beilinson elements in the K-theory of modular curves. We now describe how, given Kato's Euler system and its essential properties, one can use the general results above to study the arithmetic of elliptic curves. The main result is Theorem 8.1 below. 5. LOCAL COHOMOLOGY GROUPS
Suppose E is an elliptic curve defined over Q, and take T = Tp(E), the p-adic Tate module of E. Then V = Vp(E) = Tp(E) 0 Qp and W = Ep00. The Weil pairing gives isomorphisms V ^_' V*, T ^_' T*, and W ^_' W*.
If B is an abelian group, we will abbreviate B ® Zp = lim B/p"B, the p-adic completion of B, and
B ®Qp = (1imB/p"B) ®z,, Q,, B® Qp/Zp = (1imB/p"B) ®zp Qp/Zp. n
n
For every n define
HH(Q",p) V) = image(E(Q",p) ® Qp `+ H'(Q"p,V)),
image under the natural Kummer map. Since V = V*, this also fixes a choice of Hs(Qp, V*), and this subgroup is its own orthogonal complement as required. Let UI(E/Qn) denote the Tate-Shafarevich group of E over Q,,.
Proposition 5.1. With Hs(Q" p, V) as defined above, S(Q", Ep00) is the classical p-power Selmer group of E over Q", so there is an exact sequence 0 -4 E(Qn) ® Qp/Zp -4 S(Qn, Ep00) -+
ru(E/Qn)p'° -+ 0
Proof. If v { p then the p-part of E(Q",,,) is finite, and one can check easily that H. (Q",,,, Vp(E)) = 0. Therefore for every v, HS(Q",,,, Vp(E)) is the image of E(Q",,,) 0 Qp under the Kummer map. It follows that for every v, Hs(Q",,,, Ep00) is the image of E(Q",,,) ® Qp/Zp under the corresponding Kummer map, and so the definition of S(Q", Ep00) coincides with the classical definition of the Selmer group of E.
For every n let tan(E/Qn,p) denote the tangent space of E/Qn,p at the origin and consider the Lie group exponential map
expE : tan(E/Qn,p) - E(Qn,p) ® Q,. Fix a minimal Weierstrass model of E and let WE denote the corresponding holomorphic differential. Then the cotangent space cotan(E) is Q",pWE, and
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we let wE be the corresponding dual basis of tan(E). We have a commutative diagram
224
tan(E/Q.,P)
E(Qn,P) ® QP
WET
T
)'E(Pn) > E(lfin)
El(Qn,p)
)
where t is the formal group of E, AE is its logarithm map, pn is the maximal ideal of Q,,,p, E1(Qn,p) is the kernel of reduction in E(Qp), and the bottom maps are the formal group exponential followed by the isomorphism of [T] Theorem 4.2. (Note that E(pn)tors = 0 because Qp(Ep)/Qp is totally ramified of degree p - 1, so AE is injective.) Extending AE linearly we will view it as a homomorphism defined on all of E(Qn,p) ® Qp. Since V = V*, Hom(E(Qn,p), Qp) N Hom(HH(Qnp, V), Qp) ti H/lS(Qnp, V).
Thus there is a dual exponential map expE : H1 (Qn,p, V) -> cotan(E/Q,.,,) = Qf,PWE. V) -4 Qn,p for the composition wE o expE Since His (Qn,p, T) injects into His (Qn,p, V ), expE is injective on His (Qn,p, T ). The local pairing allows us to identify We write exp*,E :
Hjs(Qn,P) V) -L--+ Hom(E(Qn,p), QP)
I
(1)
T
H%s(Qn,p)T) -> Hom(E(Qn,p),Zp) Explicitly, z E His (Qn,p, V) corresponds to the map (2)
x H TrQn P/QPAE(x) exp,*i1E(z)
Proposition 5.2. exp,*,E(HH8(Qp,T)) = [E(Qp) : Ei(Qp)+E(Qp)t.]p 1Zp. Proof. By (2), exp,,,E(H%s(QP,T)) = p°ZP
where
AE(E(Qp)) = p °Zp. We have )E(El(Qp)) = pZp and, since rankzPE(Qp) = 1, [AE(E(Qp)) : XE(E1(Qp)] = [E(QP) : E1(Qp) +E(Qp)tors].
Thus the proposition follows.
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6. THE p-ADIC L-FUNCTION
Let ann-s
L(E, s) =
= 11 Qq(q-s)-1
n>1
q
denote the Hasse-Weil L-function of E, where Qq(q-9) is the usual Euler factor
at q. If N E Z+ we will also write
LN(E,s) = >2 an -s = fltq(q-s)-1 (n,N)=1
qfN
for the L-function with the Euler factors dividing N removed. If F is an abelian extension of Q of conductor f and -y E Gal(F/Q), define the partial L-function LN (E, y, F/Q, s) _ >2
ann-s
nHry
where the sum is over n prime to f N which map to y under
(Z/ f Z)" -4 Gal(Q(µ f)/Q) - Gal(F/Q). If x is a character of GQ of conductor fx, and ker(x) = Gal(Q/Fx), let LN(E, x, s)
_
1:
x(n)ann-s
(n,fx N)=1
1]
= I q(q_sx(q))-1 gtfx N
x(y)LN(E, y, Fx/Q, s)
ryEGal(Fx/Q)
When N = 1 we write simply L(E, x, s), and then we have (3)
LN (E, x, s) = JJ £q(q-sx(q))L(E, x, s). qIN
If E is modular then these functions all have analytic continuations to C. Fix a generator [(pn]n of lim µpn. Write Gn = Gal(Qn/Q) = Gal(Qn,p/Qp). If x is a character of Gal(Qc/Q) of conductor pn define the Gauss sum
7(x) =
>2
x(y)(P .
'yEGa1(Q(µpn )/Q)
Fix also an embedding of Qp into C so that we can identify complex and p-adic characters of GQ. The following theorem is proved in [MSD] in the case of good ordinary reduction. See [MTT] for the (even more) general statement.
Theorem 6.1. Suppose E is modular and E has good ordinary reduction or multiplicative reduction at p. Let a E ZP and /3 E pZp be the eigenvalues of p) (resp. Frobenius if E has good ordinary reduction at p, and let (a, /3) (-1, -p)) if E has split (resp. nonsplit) multiplicative reduction. Then there
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is a nonzero integer cE independent of p, and a p-adic L-function GE E cE1A such that for every character x of Gal(Q,,,,/Q) of finite order,
if x = 1 and E has good reduction at p x(LE) _ (1 - a-1)L(E,1)/liE if x = 1 and E is multiplicative at p Ia-nT(x)L(E, x-1,1)/iE if x has conductor pn > 1 1(1 - a-1)2L(E, 1)/SZE
where 11E is the fundamental real period of E.
If N E Z+, define
LE,N = fi £q(q-'Fr ')LE E A. qI N,q$p
Using (3) and Theorem 6.1 one obtains analogous expressions for x(LE,N) in terms of LN(E, x 1, 1) 7. KAro's EULER SYSTEM
The following theorem of Kato is crucial for everything that follows.
Theorem 7.1 (Kato [Kal], see also [Scho]). Suppose that E is modular, and let N be the conductor of E. There are a positive integer rE independent of p, positive integers D $ 1, D' 0 1 (mod p), and an Euler system c = c(D, D') for Tp(E), {cQ (µr) E H1(Qn(pr), Tp(E)) : r squarefree, (r, NpDD') = 1, n > 0}
such that for every such n > 0 and every character x : Gal(Qn/Q) -* C", x('y) exp*,E(locP' 7EGal(Qn/Q)
= rEDD'(D - x-1(D))(D' - x-'(D')) LNp(E, x, 1)/lE. Proof. See the paper of Scholl [Scho] in this volume (especially Theorem 5.2.7)
for the proof of this theorem when p > 2 and E has good reduction at p. 0
Using the Coleman map Cole : H. /S(Qn p, T) -+ A described in the Appendix, we can relate Kato's Euler system to the p-adic L-function.
Corollary 7.2. With hypotheses and notation as in Theorems 7.1 and 6.1, there is an Euler system c for Tp(E) such that (1) expu,E(lo p (CQ)) = rELNP(E, 1)/QE, (11) Col ([lo p'(CQ..)]) = rELE,N.
Proof. Let c be the Euler system of Theorem 7.1 for some D, D' $- 1 (mod p). denote the automorphism (H (D for ( E µp,,, Let oD E and similarly for QD,.
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Since D, D' # 1 (mod p), (D - CD) (D' - QD,) is invertible in A. Let PD,D' E Zp[[GQ]] be any element which restricts to (D - OD)-1(D' in A, and define D-1D'-1
cQ,
It is clear that
UD')-1
4
)=
PD,DlCQ.(µr)
is still an Euler system, and for every n > 0 and
every character /X : Gal(Qn/Q) -+ C"
X(PD,D') = X((D - cD)(D' - aD'))-1 = (D - X(D))-'(D- X(D'))-1, so by Theorem 7.1
X(") exp;E(locPT'(4 )) = rELNP(E, X, 1)/QE 7EGal(Q /Q)
When X is the trivial character this is (i), and as X varies (ii) follows from Proposition A.2 of the Appendix along with the definition (Theorem 6.1) of LE and (3). 8. CONSEQUENCES OF KATO'S EULER SYSTEM
Following Kato, we will apply the results of §§3 and 4 to bound the Selmer group of E.
8.1. The main theorem. Theorem 8.1 (Kato [Ka2]). Suppose E is modular and E does not have complex multiplication.
(i) If L(E, 1) # 0 then E(Q) and M(E) are finite. (ii) If L is a finite abelian extension of Q, X is a character of Gal(L/Q), and L(E, X, 1) # 0 then E(L)x and IH(E/L)X are finite. Remarks. We will prove below a more precise version of Theorem 8.1(i). Kato's construction produces an Euler system forTp(E)®X for every character X of GQ of finite order, with properties analogous to those of Theorem 7.1. This more general construction is needed to prove Theorem 8.1(ii). For simplicity we will not treat this more general setting here, so we will only prove Theorem 8.1(i) below. But the method for (ii) is the same. Theorem 8.1(i) was first proved by Kolyvagin in [Ko], using a system of Heegner points, along with work of Gross and Zagier [GZ], Bump, Friedberg, and Hoffstein [BFH], and Murty and Murty [MM]. The Euler system proof given here, due to Kato, is `self-contained' in the sense that it replaces all of those other analytic results with the calculation of Theorem 7.1.
Corollary 8.2. Suppose E is modular and E does not have complex multiplication. Then E(Q.) is finitely generated.
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Proof. A theorem of Rohrlich [Ro] shows that L(E, x,1) 0 0 for almost all characters x of finite order of Gal(Qoo/Q). Serre's [Se] Theoreme 3 shows that E(Qoo)torg is finite, and the corollary follows without difficulty from Theorem 8.1(ii).
Remark. When E has complex multiplication, the representation Tp(E) does not satisfy Hypothesis Hyp(Q,, V)(i) (see Remark 8.5 below), so we cannot apply the results of §§3 and 4 with Kato's Euler system. However, Theorem 8.1 and Corollary 8.2 are known in that case, as Theorem 8.1 for CM curves can be proved using the Euler system of elliptic units. See [CW], [Ru2] §11, and [RW].
8.2. Verification of the hypotheses. Fix a ZP basis of T and let PE,p : GQ -> Aut(T) - GL2(Zp)
be the p-adic representation of GQ attached to E with this basis.
Proposition 8.3.
(i) If E has no complex multiplication, then Tp(E) satisfies hypotheses Hyp(Qoo,V) and Hl(Q(Ep00)/Q,Ep00) is finite. (ii) If the p-adic representation PE,p is surjective, then Tp(E) satisfies hypotheses Hyp(Qoo, T) and Hl(Q(Ep00)/Q, Ep00) = 0.
Proof. The Weil pairing shows that PE,p(GQ(µp_)) = PE,p(GQ) n SL2(Zp)
If E has no complex multiplication then a theorem of Serre ([Se] Theoreme 3) says that the image of PE,p is open in GL2(Zp). It follows that Vp(E) is an irreducible GQ_ -representation, and if PE,p is surjective then Ep is an irreducible Fp [GQ, ]-representation.
It also follows that we can find r E GQ(µp.) such that PE,p (T) = (0 1 )
with x 0 0, and such a T satisfies Hypothesis Hyp(Qoo, V)(i). If PE,p is surjective we can take x = 1, and then T satisfies Hypothesis Hyp(Q., T)(i). We have
H'(Q(Ep-)/Q, Ep-) = H'(PE,p(GQ), (Qp/Zp)2) and the rest of the proposition follows. Remark 8.4. Serre's theorem (see [Se] Corollaire 1 of Theoreme 3) also shows
that if E has no complex multiplication then PE,p is surjective for all but finitely many p.
Remark 8.5. The conditions on r in hypotheses Hyp(Q,,., V)(i) force PE,p(T)
to be nontrivial and unipotent. Thus if E has complex multiplication then there is no r satisfying Hyp(Qoo, V)(i).
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8.3. Bounding S(Q, Ep-). Theorem 8.6. Suppose E is modular, E does not have complex multiplication, and L(E, 1) 0 0. (i) E(Q) and III(E)p- are finite. (ii) Suppose in addition that p f 2rE and pE,p is surjective. If E has good reduction at p and p { jE(Fp)I (where k is the reduction of E modulo p), then IIH(E)p-I divides
LN(E, 1) QE
where N is the conductor of E.
Proof. Recall that £q(q-9) is the Euler factor of L(E, s) at q, and that by Proposition 5.1, S(Q, Ep.) is the usual p-power Selmer group of E. Since Qq(q-1) is nonzero for every q, Corollary 7.2(i) shows that locca'(cQ) # 0. By Proposition 8.3(i) and Proposition 5.2 we can apply Theorem 3.1 to conclude that S(Q, Ep.) is finite, which gives (i). If E has good reduction at p then pip(p1) = IE(Fp)I and [E(QP) : E1(QP) + E(QP)tors]
divides
IE(FP)I
Therefore if p { rE It (Fp) I then exp,E(HIS (QP,TP(E)))) = p 1ZP
p 1(LN(E, 1)/QE)ZP
by Proposition 5.2 and Corollary 7.2(i) By Proposition 8.3(ii), if p # 2 we can apply Theorem 3.2, and (ii) follows. Remarks. In Corollary 8.9 below, using Iwasawa theory, we will prove that Theorem 8.6(ii) holds for almost all p, even when p divides IE(Fp)I. This is needed to prove Theorem 8.1(i), since IE(Fp)I could be divisible by p for infinitely many p. However, since IE(Fp)I < 2p for all primes p > 5, we
see that if E(Q)to,, # 0 then IE(Fp)I is prime to p for almost all p. Thus Theorem 8.1(i) for such a curve follows directly from Theorem 8.6. The Euler system techniques we are using give an upper bound for the order of the Selmer group, but no lower bound.
8.4. Bounding S(Q,,,,, Ep-). Define S(Qoo, Ep00) = lim S(Q,,, EP-) C H1 (Q., Ep00 ),
and recall that X.. = Hom(S((:I,,,, Ep00), Qp/Zp). Let rE be the positive integer of Theorem 7.1 and let N be the conductor of E. Theorem 8.7. Suppose E is modular, E does not have complex multiplication, and E has good ordinary reduction or nonsplit multiplicative reduction
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at p. Then Xoo is a finitely-generated torsion A-module and there is an integer
t such that char(X,,,,) divides ptLE,NA. If pE,p is surjective and p f rE HqjN q#P $q(q-1) then char (Xoo) divides LEA.
If E has split multiplicative reduction at p, the same results hold with
char(Xoo) replaced by Jchar(Xoo) where ,7 is the augmentation ideal of A.
Proof. Rohrlich [Ro] proved that LE 54 0. Thus the theorem is immediate from Propositions 8.3 and A.2, Corollary 7.2, and Theorem 4.1. Corollary 8.8. Let E be as in Theorem 8.7. There is a nonzero integer ME such that if p is a prime where E has good ordinary reduction and p { ME, then X has no nonzero finite submodules. Proof. This corollary is due to Greenberg [Gr1], [Gr2]; we sketch a proof here.
Let E be a finite set of primes containing p, oo, and all primes where E has bad reduction, and let QE be the maximal extension of Q unramified outside of E. Then there is an exact sequence (4) 0 # S(Q00, E p.) + H'(QE/Qoo, Ep°°) -4 ®qEE ®vlq H, 8(Q.,., Epo)
Suppose q E E, q # p, and v q. If p f JE(Qq)to,Bl then it is not hard to show that E(Qoo,,,) has no p-torsion, and so by [Grl] Proposition 2, I
Hl (Qoo,,,, Ep00) = 0. Thus for sufficiently large p the Pontryagin dual of (4) is lim E(Qn,p) 0 Zp -> Hom(H' (QE/Qoo, E p.), Qp/Zp) -) Xoo -4 0.
Since Qoo/Q is totally ramified at p, lim E(Qn,p) 0 Zp = lim El (Qn,p) = lim E(lfin)
n
n
n
and this is free of rank one over A (see for example [PR1] Theoreme 3.1 or [Schn] Lemma 6, §A.1). It now follows, using the fact that Xoo is a torsion A-module (Theorem 8.7) and [Grl] Propositions 3, 4, and 5 that Hom(H'(QE/Qoo, Ep00), Qp/Zp) has no nonzero finite submodules, and by the Lemma on p. 123 of [Grl] the same is true of Xoo. Corollary 8.9. Suppose E is modular, E does not have complex multiplication, E has good reduction at p, p f 2rEME 11q1N £q(q-1) (where rE is as in Theorem 7.1 and ME is as in Corollary 8.8), and pE,p is surjective. Then IIU(E)po0 divides
L(SE,1)
Proof. First, if E has supersingular reduction at p then IE(Fp)I is prime to p, so the corollary follows from Theorem 8.6(ii).
Thus we may assume that E has good ordinary reduction at p. In this case the corollary is a well-known consequence of Theorem 8.7 and Corollary
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8.8; see for example [PR1] §6 or [Schn] §2 for details. The idea is that if X. has no nonzero finite submodules and char(X,,.) divides LEA, then
IS(Q,, E00
)Gal(Qoo/Q)I
divides
Xo(LE,N),
where Xo denotes the trivial character, and with a as in Theorem 6.1
Xo(LE,N) = (1 - a-1)211Qe(4-1)(L(E,1)/lE) 9IN
On the other hand, one can show that the restriction map Ep_)Gal(Q-/Q)
S(Q, Ep_) -* S(Q.,
is injective with cokernel of order divisible by (1 - a-1)2, and the corollary follows.
Proof of Theorem 8.1(i). Suppose E is modular, E does not have complex multiplication, and L(E, 1) # 0. By Theorem 8.6, E(Q) is finite and IH(E)poo is finite for every p. By Corollary 8.9 (and using Serre's theorem, see Remark 8.4) IH(E)po0 = 0 for almost all p. This proves Theorem 8.1(i). We can also now prove part of Theorem 8.1(ii) in the case where E has good ordinary or multiplicative reduction at p and L C Q. For in that case, by Theorem 8.7, X(char(Hom(S(Q,,,,, Ep00), Qp/Zp))) is a nonzero multiple of L(E, X, 1)/SZE. If L(E, X, 1) # 0 it follows that S(Q,,,, Ep.)X is finite. The kernel of the restriction map S(L, Epoo) -+ S(QOO, Ep00) is contained in the finite group H' Ep- °° ), and so we conclude that both E(L)X and Iu(E/L)po are finite. APPENDIX. EXPLICIT DESCRIPTION OF THE COLEMAN MAP
In this appendix we give an explicit description of the Coleman map from
H./S(Q,,,p,T) to A. This map allows us to relate Kato's Euler system with the p-adic L-function ICE-
Suppose for this appendix that E has good ordinary reduction or multiplicative reduction at p. As in Theorem 6.1, let a E Zp and 0 E pZp be the eigenvalues of Frobenius if E has good ordinary reduction, let a = 1, ,Q = p if E has split multiplicative reduction and let a = -1,,3 = -p if E has nonsplit multiplicative reduction. Recall we fixed before Theorem 6.1 a generator [('p,.]n of l4 m µp . For every
n > 0 define '.r.,y = a -n-
Lemma A.1.
In k=0
(pn+l-k - 1
h
(i) If n > m then TrQ P/Qm,Pxn = xm..
a E Q7L,]l*
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(ii) If X is a character of Gn then tryn
X(
a--,r(X)
if X has conductor ptm > 1 if X = 1.
Y)
7EGn
Proof. Exercise.
Proposition A.2.
(i) For every n > 0 there is a ZP[Gn]-module map Coln : Hj1s(Qnp)T) -+ ZP[Gn]
such that for every z E HI's (Qn,P,T) and every nontrivial character X of Gn of conductor pn`, X(Coln (z)) = a-mr(X)
X-1 (-y) eXp,E (z7) ryEGn
If Xo is the trivial character then
Xo(Coln(z)) = (1 - a-1)(1 -,Q-1)-1 > expWE(z1). 7EGn
(ii) The maps Coln are compatible as n varies, and in the limit they induce a map of A-modules Col". : H, Is(Qp, T) * A.
(iii) The map Cole is injective. If E has split multiplicative reduction at p then the image of Cole is contained in the augmentation ideal of A. Proof. The proof is based on work of Coleman [Co]. For the curves E which we are considering, k is a height-one Lubin-Tate formal group over Zp for the uniformizing parameter Q. It follows that, writing R for the ring of integers of the completion of the maximal unramified extension of Qp, k is isomorphic over R to the multiplicative group G,-n. Fix an isomorphism q : Gm 4 E, 77 E R[[X]]. We define the p-adic period Q p of
E sip = 9'(0) E R". This period is unique up to Zp , and is also characterized by the identity AE(77(X)) = Slp log(1 + X).
(5)
By [dS] §1.3.2 (4), if 0 is the Frobenius automorphism of R/Zp then Opm = a-1S2p.
By [Co] Theorem 24 (applied to the multiplicative group) there is a power series g in the maximal ideal (p,X)R[[X]] of R[[X]] such that 1
log(1+g(X))=cP1 N
1
k_O 6Eµy_iczy
3pk Xk - 1
N
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In particular if we set X = [;pn+1 - 1 and use (5), AE(77(9(Cpn+ - 1))) = S2plog(1+9((pn+l - 1)) = an+lxn
Observe that E(p R)tor = 0 because Qp(Ep)/Qp is totally ramified of degree p - 1. Therefore AE is injective on E(p R) and so
x, E
AE(pnR)Gal(Qn,pR/Qn,p) =.E(Pn) C.E(E(Qn,p))
Define Col, on Hl', (Q.,P) T) by
Col,(z) = E xn Y 7EGn
exp:E (z7)y-i 7EGn
(Trgn,p/Q ,xn eXPWE (z))'Y 7EGn
Clearly this gives a Galois-equivariant map H/S(Qn,p)T) Qp[Gn]. Since x,7 E .E(E(Qn,p)), (1) and (2) show that Coln(z) E Zp[G,]. The equalities of (i) follow from Lemma A.1(ii), and (ii) follows easily.
For (iii), suppose first that E has good ordinary reduction or nonsplit multiplicative reduction at p, so that a # 1. Then the injectivity of Coln (and of Col,,,,) follows from (i), the nonvanishing of the Gauss sums, and the injectivity of expWE.
If E has split multiplicative reduction at p, then a = 1. In this case it follows from (i) that ker(Coln) = HHS(Qn,p,T)Gn, which is free of rank one over Zp, for every n. But one can show using (1) that H,Q/S(Qp,T) has no A-torsion, so Cole must be injective in this case as well. The assertion about the cokernel is clear from (i), since a = 1. REFERENCES
[BFH] Bump, D., Friedberg, S., Hoffstein, J.: Eisenstein series on the metaplectic group and nonvanishing theorems for automorphic L-functions and their derivatives, Annals of Math. 131 (1990) 53-127. [CW] Coates, J., Wiles, A.: On the conjecture of Birch and Swinnerton-Dyer, Inventiones math. 39 (1977) 223-251. [Co] Coleman, R.: Division values in local fields. Invent math. 53 (1979) 91-116 [dS] de Shalit, E.: The Iwasawa theory of elliptic curves with complex multiplication, (Perspec. in Math. 3) Orlando: Academic Press (1987) [Grl] Greenberg, R.: Iwasawa theory for p-adic representations, Adv. Stud. in Pure Math. 17 (1989) 97-137. [Gr2] Greenberg, R.: Iwasawa theory for p-adic representations II, to appear. [GZ] Gross, B., Zagier, D.: Heegner points and derivatives of L-series, Inventiones math. 84 (1986) 225-320 [Kal] Kato, K.: Euler systems, Iwasawa theory, and Selmer groups, to appear [Ka2] : to appear Kolyvagin, V. A.: Euler systems. In: The Grothendieck Festschrift (Vol. II), P. [Ko] Cartier, et al., eds., Prog. in Math 87, Boston: Birkhauser (1990) 435-483.
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[MSD] Mazur, B., Swinnerton-Dyer, H.P.F.: Arithmetic of Weil curves, Inventiones math. 25 (1974) 1-61 [MTT] Mazur, B., Tate, J., Teitelbaum, J.: On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. math. 84 (1986) 1-48. [MM] Murty, K., Murty, R.: Mean values of derivatives of modular L-series, Annals of Math. 133 (1991) 447-475.
[PR1] Perrin-Riou, B.: Theorie d'Iwasawa p-adique locale et globale, Invent. math. 99 [PR2] [Ro]
(1990) 247-292. : Systemes d'Euler p-adiques et theorie d'Iwasawa, to appear. Rohrlich, D.: On L-functions of elliptic curves and cyclotomic towers, Invent. math. 75 (1984) 409-423.
[Rul] Rubin, K.: The main conjecture. Appendix to: Cyclotomic fields I and II, S. Lang, Graduate Texts in Math. 121, New York: Springer-Verlag (1990) 397-419. [Ru2] : The "main conjectures" of Iwasawa theory for imaginary quadratic fields, Invent. math. 103 (1991) 25-68. [Ru3] : Euler systems, to appear. [RW] Rubin, K., Wiles, A.: Mordell-Weil groups of elliptic curves over cyclotomic fields. In: Number Theory related to Fermat's last theorem, Progress in Math.26, Boston: Birkhauser (1982) 237-254. [Schn] Schneider, P.: p-adic height pairings, II, Inventiones math. 79 (1985) 329-374. [Scho] Scholl, A.: An introduction to Kato's Euler systems, this volume [Se] Serre, J-P.: Proprietes Galoisiennes des points d'ordre fini des courbes elliptiques, Invent. math. 15 (1972) 259-331. [T] Tate, J.: Algorithm for determining the type of a singular fiber in an elliptic pencil. In: Modular functions of one variable (IV), Lecture Notes in Math. 476, New York: Springer-Verlag (1975) 33-52. DEPARTMENT OF MATHEMATICS, OHIO STATE UNIVERSITY, COLUMBUS, OH 43210
USA
E-mail address: rubin'math.ohio-state.edu
Basic notions of rigid analytic geometry PETER SCHNEIDER The purpose of my lectures at the conference was to introduce the newcomer to the field of rigid analytic geometry. Precise definitions of the key notions
and precise statements of the basic facts were given. But, of course, the limited time did not allow to include any proofs. Instead the emphasis was placed on motivating and explaining the shape of the theory. The positive response from the audience encouraged me to write up the following notes which reproduce my lectures in an essentially unchanged way. I hope that they can serve as a means to quickly grasp the basics of the field. Of course, anybody who is seriously interested has to go on and has to dig into the proper literature. Rigid or non-archimedean analysis takes place over a field K which is complete with respect to a non-archimedean absolute value 1 1. The most important examples are the fields of p-adic numbers QP where p is some prime number. For technical purposes we fix throughout an algebraic closure K of K and denote by K its completion which again is algebraically closed. The absolute value I I extends uniquely to an absolute value I I of K.
Fix a natural number n E N, and let us consider the "n-dimensional polydisk"
B":={(z1i...,ztt) E K":max Iz1l 0
be an arbitrary formal power series with coefficients in K. The following two properties are quite immediate: * The power series f converges on all of B" if and only if the coefficients tend to zero, i.e., I av1.....v. I -+ 0 if vl +... + v" -+ 00. * If f converges on B" then we have f (B" (L)) C L for any intermediate field K C L C_ K which is finite over K; here 13"(L) denotes the set of those vectors in B" with coordinates in L. The subalgebra
T" :=If E K[[T1, ... , T"]]
:
f converges on B' j
of the algebra K[[T1, ... , T"]] of formal power series over K is called a Tate
algebra. We may and will say that any f E T" induces an "analytic function on B" defined over K". Why is this a good notion? At first glance
it does not seem to be local at all! Certainly we do not want to give up completely the possibility of recognizing the analyticity of a function locally.
In order to prepare the way out of this apparent trap we first collect a number of properties of the algebra T". The two most basic ones are the following:
1) T" is a K-Banach algebra w.r.t. the multiplicative norm IfI := max Iav1,...,vnl
2) The Maximum Modulus Principle holds:
IfI=ma If(z)I in particular: If f (z) = 0 for any z E B" then f = 0. The proof of the Maximum Modulus Principle is actually very easy: By scaling we may assume that If I = 1. We then can reduce f modulo the maximal ideal of K obtaining, because of the convergence criterion, a nonzero polynomial f over the residue class field of K. Since the residue field of K is infinite we find a point z with coordinates in the latter such that 1(i) 54 0. Any lifting z E B" of z then satisfies If (z) I = 1.
Basic notions of rigid analytic geometry
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Next one shows that the Weierstrass theory (preparation theorem, ...) works for T. This eventually leads to many ring theoretic properties of T'":
3) T' is noetherian and factorial. 4) T" is Jacobson, i.e., for any ideal a C_ T' its radical ideal Va- is the intersection of all the maximal ideals containing a. 5) Any ideal in T' is closed. 6) For any maximal ideal m in T'", the residue field T'"/m is a finite extension
of K. This last property is an analogue of Hilbert's Nullstellensatz. It has the interesting consequence that the map
Galois orbits in B'"(K) - Max(T'") z
-+
mx :
= If f( z ) = 0} :
is a bijection. The inverse map is obtained as follows: For a maximal ideal
m let cp denote the composite of the projection T" -* T" /m and some embedding T"/m -* K and put z1 := cp(TZ).
In this way the maximal ideal spectrum Max(T') appears as an algebraically defined model for the space B'". This suggests we should proceed as Grothendieck did in algebraic geometry and define a category of analytic spaces as maximal ideal spectra of certain algebras. (The property 4) is the reason that maximal ideals in contrast to arbitrary prime ideals will suffice.)
Definition: A K-algebra A is called a noid if A = T'°/a for some n E N and some ideal a.
For any affinoid algebra A we have: * A is noetherian and Jacobson (by 3) and 4)).
* A is a K-Banach algebra with respect to any residue norm (by 5)). Moreover:
- The topology on A is independent of the chosen residue norm; - any homomorphism between affinoid K-algebras is automatically continuous. We put Max(A) := set of all maximal ideals of A .
By 6), this set depends functorially on A. Also by 6) we may define the so-called supremum or spectral seminorm on A by If IsuP
sup xEMax(A)
If WI
372
Peter Schneider
where f (x) := f + m,; E A/mi -4 K. It is obviously bounded above by any
residue norm. The general Maximum Modulus Principle says that If(sup=XEMax
If(x)I
If A is reduced then I Isup is a norm which is equivalent to any residue norm.
Using the above description of Max(T") as the Galois orbits in B"(K) we obtain (from the metric topology on K) a "canonical" Hausdorif topology on Max(A). Of course it is totally disconnected so that our initial problem persists. In order to emphasize the geometric intuition we write X := Max(A) from now on. For any functions g, fl, ... , E A without common zero (i.e., generating the unit ideal (g, f$) = A) we introduce the open subset
x(9'):={xEX X: maIf=(x)I A(9) is a homeomorphism onto X(11).
Comments: - A(T1, ... , is the algebra of all power series in the variables with coefficients in A tending towards 0. T1,.. . , - The affinoid algebra A(9) can be characterized by a universal property which is given solely in terms of the rational subdomain X(9 ). 1. - In A(9) we have 19 I = l residue class of T j - The assertion becomes wrong without the assumption that (g, fa) is
the unit ideal (look at A = K(T), m = 1, g = T, and fl = 0). This observation allows to define a presheaf Ox at least on the rational subdomains in X by
dx(X(g)) := A(g)
Basic notions of rigid analytic geometry
373
Main Theorem of Tate: , Yr C X are rational subdomains such that Y = Yl U ... U YY, then Ox satisfies the sheaf property for that covering, i.e.,
If Y, Yi,
0 -* Ox (Y) -* 11 Ox (Y)
[J Ox (Yi n Yj)
is exact.
This means that the notion of an analytic function on X, i.e., an element of A, is local as far as finite coverings by rational subdomains are concerned! This picture can be "enlarged" by a completely formal construction:
* A subset U C X is called admissible open if there are rational subdomains Ui C X for i E I such that i. U = U Ui (in particular U is open in the canonical top.) and iEI
ii. for any map a : Y := Max(B) -* X = Max(A) induced by a homomorphism of affinoid K-algebras A -> B with im(a) C U the covering Y = U a-'(Ui) has a finite subcovering. iEI
* Let V and Vj, for j E J, be admissible open subsets of X such that V = U Vj; this covering of V is called admissible if for any map a : Y -* jEJ X as above with im(a) C V the covering Y = U a-'(Vj) can be refined into a finite covering by rational subdomains. jEJ These notions define a Grothendieck topology on X (which is considerably coarser than the canonical topology). The presheaf Ox extends in a purely formal way (once one knows Tate's theorem) to a sheaf on X with respect to this Grothendieck topology; it is called the structure sheaf of X.
Definition: The triple Sp(A) := (X, Grothendieck topology, Ox) is called an a noid variety over K.
Fact: Any homomorphism of affinoid K-algebras A -* B induces a "morphism of affinoid varieties" Sp(B) -+ Sp(A). In order to illustrate these concepts let us go back to the 1-dimensional disk X = Max(K(T)) = "{z E K : Izi _< 1}" and look at the simplest
example. By construction we have Ox(X) = K(T). Clearly the "unit
Peter Schneider
374
circle" V := {x E X : IT(x)I = 1} = X(T) = "{z E
IzI = 1}" is a
rational subdomain with
Ox(V) = K(T)(T')/(TT' - 1) = K(T,T-1) {EvEZ avTv
:
Iavl -+ 0 if IvI -+ oo}
We now look at the "open unit disk"
U:=X\V= {xEX : IT(x)I < 1} = "{zEK: IzI Xan
Basic notions of rigid analytic geometry
375
together with a natural morphism of locally G-ringed spaces
anx:Xa"-+X which has the universal property that any morphism of locally G-ringed spaces Y -+ X where Y is a rigid K-analytic variety factorizes through Xa'', i.e., we have a commutative triangle
- - -- _
Y
Xan
Kanx
X
By a locally G-ringed space we mean a set equipped with a Grothendieck topology consisting of subsets and a sheaf of K-algebras whose stalks are local rings.
In order to demonstrate the existence of Xa" it suffices, by a gluing argument (made possible by the universal property), to consider the case of an affine K-scheme of finite type X = Spec(A). As a set we put Xan :_ Max(A). To define the analytic structure we fix a representation of A as a quotient
A = K[T1i...,Td]/a of some polynomial algebra and we fix a c E K with Icl > 1. Put
Un := {x E Xa" : max lTj(x)I < jcI"} for n > 0 so that
Xan =
UUn. n>O
We define affinoid K-algebras
An := K(T1i ... ,Td)l(P(cnTj) : P E a)
From the commutative diagram Max(An)
c-^Tj 1. Hence we have redX1(Max(A[f-1])) =
X(1)
.
Note that also the preimage redj1(V (f)) = {x E X : If (x) l < 11 of the Zariski closed zero set V (f) off is admissible open (but i.g. not affinoid).
Basic notions of rigid analytic geometry
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So far I have described the "classical" approach to rigid analytic geometry which was invented by Tate. More details and full proofs for everything which was said can be found in the book [BGR]. Later on Raynaud saw that rigid geometry can be developed entirely within the framework of formal algebraic geometry. Because of the conceptual as well as technical importance of this approach I want to finish by explaining Raynaud's point of view ([R] or [BL]).
For simplicity we assume that is a discrete valuation on K. As before o denotes the ring of integers in K. We fix a prime element 7r in o. Let o{{Ti,... , be the ring of restricted formal power series over o; recall that a formal power series over o is restricted if, for any given m > 1, almost all its coefficients lie in 7rtmo. An o-algebra A of the form is called topologically of finite type. It is a 7rAA = o{ {Ti, ... , adically complete topological ring and gives rise to the affine formal scheme Spf(A) over o which is the set of all open prime ideals of A equipped with the
Zariski topology and a certain structure sheaf constructed by localization and completion. Roughly speaking one has
Spf(A) = "lim" Spec(A/irmA) M
The first basic observation is that, for an o-algebra A topologically of finite type, the tensor product A := AO K is an affinoid K-algebra. The affinoid O
variety Sp(A) is called the general fibre of the formal scheme Spf(A). Can we describe the datum Sp(A) directly in terms of the algebra A ? The set Max(A): Consider any prime ideal p g A such that 1. p is not open in A, and 2. A/p is a finitely generated o-module. We claim that p 0 K is a maximal ideal in A. Condition 1) ensures that
the obvious map o -4 A/p is injective. Condition 2) implies that A/p is an integral domain finite over o. It follows that A/p is a local ring which is finite and flat over o. Hence (A/p) 0 K = A/(p 0 K) is a finite field extension of K. In this way one obtains a bijection
{p C A prime ideal with 1), 2) } => Max(A) p --a p ®K . The rational subdomains (i.e., the Grothendieck topology on Max(A)):
Peter Schneider
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Consider any rational subdomain X(L) X := Max(A). There is no s have , fr E A. That g, no common zero means that the ideal I := ' Ag+Afi+...+.Af,. is open in A. On the other hand, for any open ideal I C A and any element g E I, there is a universal construction called formal blowing-up of a homomorphism A -+ Aj,g of o-algebras topologically of finite type which is universal with respect to making I into a principal ideal generated by g. The morphism of formal schemes Spf (Aj,9) -+ Spf (A) induces in the general fibre the inclusion X (f*) C X. loss of generality in assuming g that g, fl,.
*
The structure sheaf: As is more or less clear from the above description of the rational subdomains of X = Sp(A) the structure sheaf Ox can be reconstructed from the structure sheaves of all the formal blowing-ups of the formal scheme Spf (A).
Theorem of Raynaud: The above construction of the "general fibre" induces an equivalence of categories between the category of all formal flat o-schemes topologically of finite type in which all formal blowing-ups are inverted and
the category of all quasi-compact and quasi-separated rigid K-analytic varieties.
References [BGR] S. Bosch, U. Giintzer, R. Remmert, Non-Archimedean Analysis, Berlin-Heidelberg-New York 1984 [BL]
S. Bosch, W. Lutkebohmert, Formal and rigid geometry I. Rigid spaces, Math. Ann. 295, 291-317 (1993)
[R]
M. Raynaud, Geometrie analytique rigide d'apres Tate, Kiehl,.. . Table ronde d'analyse non archimedienne, Bull. Soc. Math. France Mem. 39/40, 319-327 (1974)
MATHEMATISCHES INSTITUT, WESTFALISCHE WILHELMS-UNIVERSITAT, EINSTEINSTR.
62, D-48149 MONSTER, GERMANY
[email protected]. de http://www.uni-muenster.de/math/u/schneider
An introduction to Kato's Euler systems A. J. SCHOLL to Bryan Birch
Contents 1
Kato-Siegel functions and modular units 1.1 1.2
Review of modular forms and elliptic curves Kato-Siegel functions . . . . . . . . . . . . .
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Reprinted from `Galois Representations in Arithmetic Algebraic Geometry', edited by A. J. Scholl & R. L. Taylor. ©Cambridge University Press 1998
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5 The Euler systems 5.1 5.2
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Introduction In the conference there was a series of talks devoted to Kato's work on the Iwasawa theory of Galois representations attached to modular forms. The present notes are mainly devoted to explaining the key ingredient, which is the Euler system constructed by Kato, first in the K2 groups of modular curves, and then using the Chern class map, in Galois cohomology. This material is based mainly on the talks given by Kato and the author at the symposium, as well as a series of lectures by Kato in Cambridge in 1993. In a companion paper [29] Rubin explains how, given enough information about an Euler system, one can prove very general finiteness theorems for Selmer groups whenever the appropriate L-function is non-zero (see §8 of his paper for precise results for elliptic curves). Partly because of space, and partly because of the author's lack of understanding, the scope of these notes is limited. There are two particular restrictions. First, we only prove the key reciprocity law (Theorem 3.2.3 below), which allows one to compute the image of the Euler system under the dual exponential map, in the case of a prime p of good reduction (actually, for stupid reasons explained at the end of §2.1, we also must assume p is odd). Secondly, we say nothing about the case of Galois representations attached to forms of weight greater than 2. For the most general results, the reader will need to consult the preprint [17] and Kato's future papers. Kato's K2 Euler system has its origins in the work of Beilinson [1] (see also [30] for a beginner's treatment). Beilinson used cup-products of modular units to construct elements of K2 of modular curves. He was able to compute the regulators of these elements by the Rankin-Selberg method and relate them to the L-function of the modular curve at s = 2, in partial confirmation of his general conjectures [1; 27] relating regulators and values of L-functions. Kato discovered that, by using explicit modular units, one obtained normcompatible families of elements of K2. These modular units are the values,
at torsion points, of what are called here Kato-Siegel functions. These are canonical (no indeterminate constant) functions on an elliptic curve (over any base scheme) with prescribed divisors, which are norm-compatible with respect to isogenies. Such functions were, over C, first discovered by Siegel - the associated modular units were studied in depth by Kubert and Lang [19]. Over C generalisations of these functions were found by Robert [28]. It
An introduction to Kato's Euler systems
381
was Kato [16] who first found their elegant algebraic characterisation. In §1 I have given an "arithmetic" modular construction of these functions, which is more complicated than Kato's but at least reveals the key fact behind their
existence - namely, the triviality of the 12th power of the sheaf w on the modular stack. (The Picard group of the modular stack was computed by Mumford [25] many years ago.)
In §2 we turn to K-theory, and give a fairly general construction of the Euler system in K2 of modular curves, and the norm relations. It is relatively formal to pass from this to an Euler system in Galois cohomology of (say)
a modular elliptic curve. The hard part is to show that the cohomology classes one gets are non-trivial if the appropriate L-value is nonzero. In his 1993 Cambridge lectures, Kato explained how this can be regarded as a consequence of a huge generalisation of the explicit reciprocity laws (Artin, Hasse, Iwasawa, Wiles ... ) to local fields with imperfect residue field. This is the subject of the preprint [17]. At the Durham conference he sketched a slightly different proof, using the Fontaine-Hyodo-Faltings approach to p-adic Hodge theory. In §3 we give a stripped-down proof of a weak version of one of the reciprocity laws in [17] in the case of good reduction, using a minimal amount of p-adic Hodge theory. In §4 we explain how Kato uses the Rankin-Selberg integral (very much
as Beilinson did) to compute the projection of the the image of the dual exponential into a Hecke eigenspace. Finally in §5 we tie everything together for a modular elliptic curve.
The appendix to §2 (which is the author's only original contribution to this work) is an attempt to extend Kato's methods to other situations. We construct an Euler system in the higher K-groups of (a suitable open part of) Kuga-Sato varieties. This is a precise version of the construction used in [32] (see also §5 of [9] for a summary) to relate archimedean regulators of modular form motives and L-functions. The p-adic applications of these elements remain to be found.
I have many people to thank for their help in the preparation of this paper. Particular mention is due to Jan Nekovar. He encouraged me to think about norm relations in 1994, although in the end that work was overtaken by events, and all that remains of it is the appendix to §2. It is only because of his insistence that §§3-5 exist at all, and his careful reading of much of the paper eliminated many errors (although he is not to be held responsible for those that remain). I am also grateful to Amnon Besser, Spencer Bloch, Kevin Buzzard, John Coates, Ofer Gabber, Henri Gillet, Erasmus Landvogt and Christophe Soule for useful discussions. Karl Rubin read the original draft of the manuscript and made invaluable suggestions. Above all, it is
a great pleasure to thank Kato, the creator of this beautiful and powerful mathematics, for encouraging me to publish this account of his work and for
A. J. Scholl
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pointing out some blunders in an earlier draft. This paper was begun while the author was visiting the University of Munster in winter 1996 as a guest of Christopher Deninger, and completed during a stay at the Isaac Newton Institute in 1998. It is a pleasure to thank them for their hospitality.
Notation If G is a commutative group (or group scheme) and n E Z then [x n]o : G -+G is the endomorphism "multiplication by n", written simply [x n] if no confusion can occur. We also write nG and G/n for the kernel and cokernel of [xn], respectively. Throughout this paper we use the geometric Frobenius, and normalise the reciprocity laws of class field theory accordingly (see §3.1 below for precise conventions, as well as the remarks following Theorem 5.2.1). The symbol "=" is used to denote equality or canonical isomorphism. We
use the usual notation ":=" to indicate that the right-hand expression is the definition of that on the left (and "=:" for the reflected relation).
1
1.1
Kato-Siegel functions and modular units Review of modular forms and elliptic curves
We review some well-known facts about the moduli of elliptic curves. See for example [7; 8; 18, Chapter 2]. For any elliptic curve f : E -> S, with zero-section e, we have the standard invertible sheaf WE/S .- f*QE/S =
e*Q1
From the second description (as the conormal bundle of the zero-section of
E/S) we have the isomorphism WE/S = e*OE(-e). Because M IS is free along the fibres of f, in fact WE/S = x*Q1IS for any section x E E(S). The formation of W E/S is compatible with basechange - in fancy language, w is a sheaf on the modular stack M of elliptic curves. A (meromorphic) modular form of weight k is a rule which assigns to compatible with basechange. By definition this each E/S a section of
is the same as an element of I'(M, w®"). The discriminant 0(E/S) is a nowhere-vanishing section of WEl/s compatible with basechange, and it defines an invertible modular form A of weight 12. From this it follows in particular
that The set of nowhere-vanishing sections of w®12d is {±Od}, for any integer d.
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Let N > 1 be an integer. The modular stack Mro(N) classifies pairs (E/S, a) where a: E -+ E' is a cyclic isogeny of degree N of elliptic curves over S. (When N is not invertible on S the definition of cyclic can be found in [18, §3.4].) The functor (E, a) H E defines a morphism c:Mro(N) + M. A (meromorphic) modular form on FO(N) of weight k is a section of c*wok over Mro(N). Equivalently, it is a rule which associates to each cyclic Nisogeny a : E -+ E' of elliptic curves over S a section of wE/S, compatible with arbitrary basechange S' -> S. As well as A, one has the modular form O(N) of weight 12, defined by
O(N)(E -24 E') = a*O(E').
It is invertible exactly where a is etale. In particular, it is invertible on S ® Z[1/N].
Suppose N = p is prime. The reduction of Mro(p) mod p has two irreducible components, one of which parameterises pairs (E/S, a) where a is Frobenius, and the other those pairs where a is Verschiebung. On the first component 0(p) vanishes, and on the second it does not.
Let m be the denominator of (p - 1)/12. Then A(P) A` is the mt' power of a modular function up E r(Mro(p), 0), which is invertible away from characteristic p by the previous remarks. It is a classical fact [26] that r(Mr(,(p) ®Q, O*) = (q, up). and therefore
r(Mro(p)' O*) =
{±1}.
Recall the Kodaira-Spencer map (see e.g. [18, 10.13.10]); if E/S is an elliptic curve and S is smooth over T, one has an Os-linear map KS = KSE/S: WE/S -+ StS/T.
If T = Spec Q and E/S is the universal elliptic curve over the modular curve Y(N), N > 3 (the definition is recalled in §2.2 below), then KS is an isomorphism.
If S -* S, E " E is an extension of E to a curve E/S of genus 1 (not necessarily smooth), and the identity section e E E(S) extends to a a section e: S -> E whose image is contained in the smooth part, then WE/3 := e*QE/S is an invertible sheaf on S extending WE/S. If S is smooth over the base scheme T, and S is the complement in S of a divisor S°° with relative normal crossings, the Kodaira-Spencer map extends to a homomorphism KSE/S: WE/S -> SZs/T(logS°°).
(1.1.1)
X(N)I, for N > 3 and E is the regular minimal model of the universal If elliptic curve, then (1.1.1) is an isomorphism.
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1.2
Kato-Siegel functions
If D is a principal divisor on an elliptic curve over (say) a field, there is in general no `canonical' function with divisor D. For certain special divisors, such canonical functions do exist. In their analytic construction they have been used extensively in the theory of elliptic units. Kato observed that they have a completely algebraic characterisation. Here we give a slightly more general, modular, description of such a class of functions. Theorem 1.2.1. Let D be an integer with (6, D) = 1. There is one and only one rule 19D which associates to each elliptic curve E -+ S over an arbitrary base a section VDE") E O*(E - ker[xD]) such that:(i) as a rational function on E, 19DE/s) has divisor D2(e) - ker[xD];
(ii) if S' -+ S is any morphism, and g : E' = E x s S' -+ E is the basechange, 19DE-/S'); then
(iii) if a: E -+ E' is an isogeny of elliptic curves over a connected base S whose degree is prime to D, then a*19DEls)
-
19DE,lS)
(iv) 19 D = 19D and i91 = 1. If D = MC with M, C > 1 then [x M]
1
= z9M2 and t90 o [x M] = 19,oll&
In particular, [xD]*?9D = 1.
(v) if r E C with Im(T) > 0 and E,r/C is the elliptic curve whose points are C/Z + TZ, then -9DE,./c) is the function
(-1) 2 e(u, T)D2e(Du, T)-1 where
e(u,T) = q'r (t2 - t-2) fl(1 - q"t)(1 - qnt-1) n>O
and q = e2niT t = .2riu.
Remarks. (i) We do not require that D be invertible on S. (ii) Locally for the Zariski topology, any elliptic curve may be obtained by basechange from an elliptic curve over a reduced base. It is therefore enough to restrict to reduced base schemes S.
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i9DE/s) uniquely; any (iii) Properties (i) and (iii) alone already determine other function with the same divisor is of the form ut9, for some u E O*(S), and applying (ii) for the isogenies [x2], [x3] would give u4 = u = u9, whence u 1. (iv) In down-to-earth terms, if S = Speck for an algebraically closed field
k then for a separable isogeny a: E -+ E', the property (iii) is just the distribution relation J1 19D(E/k)(x)
= 99(D'Eh/k)(y), for any y E E'(k).
xEE(k)
a(x)=y
(v) Over C this theorem was obtained by Robert (28], who proves rather
more: he shows that for any elliptic curve E/C and any finite subgroup P C E of order prime to 6, there is a certain canonical function with divisor #P(e) - P and properties generalising those of 19D. One can prove his more general result in a manner similar to the proof of 1.2.1; in place of the modular OD2-1 form one should use A(E)#P/l3*A(E/P), where /0: E -4 E/P is the quotient map.
Proof. We begin with the first two conditions. First observe that if S is a spectrum of a field, then the divisor ker[xD] - D2(e) is principal (because D is odd, the sum of ker[xD] - D2(e) in the Jacobian is zero). To give a rule t9D satisfying (i) and (ii) is equivalent to giving, for any elliptic curve E/S, an isomorphism of line bundles on E
OE(ker[xD]) -+ OE(D2e)
(1.2.2)
compatible with basechange. We have just observed that the line bundles are isomorphic when restricted to any fibre of E/S. Since we can assume (by remark (ii) above) that S is reduced, the seesaw theorem tells us that to give an isomorphism (1.2.2) is equivalent to giving an isomorphism of the restriction of the bundles to the zero-section. In other words, the existence of dD is equivalent to finding, for each E/S, a trivialisation of the bundle e*OE(ker[xD]) ® e*OE(D2e)" = e*[xD]*OE(e) ® e*OE(-D2e)
= e*OE(e)®(1-D2) _ ®(D2-1) - wE/S compatible with base-change. Note that (6, D) = 1 implies D2 = 1 (mod 12). compatible There are then exactly two non-vanishing sections of with arbitrary basechange, namely ±(E/S)()2_1)/12. Choose one of them,
and let 4(E/s) be the corresponding function on E - ker[xD]. So the rule
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(0: E/S H (p(EIS)) satisfies properties (i) and (ii). In a moment we will see that exactly one of ± satisfies (iii). (See also remark 1.2.3 below). By the basechange compatibility (ii), we are free to make any faithfully flat basechange in order to check (iii). There exists such a basechange over which a factors as a product of isogenies of prime degree. It is therefore enough to verify (iii) when deg a = p is prime. The quotient gp(E/S, a) =
a*O(E/s)(O(E'/s))_1
E O*(S)
is compatible with basechange. It therefore defines a modular unit gp E I'(Mro(p), 0*), and so gp(E/S, a) E {±1} for every (E/S, a). Moreover the sign depends only on p. To determine the sign, evaluate gp(E/1Fp, FE) for an elliptic curve over 1Fp
and its Frobenius endomorphism. The norm map FE*: r,(E)* -+ r,(E)* is then the identity map, so gp(E, FE) = 1, and therefore if p is odd we have gp = +1. Notice that replacing O(E/s) by -qs(E/s) does not change gp for p odd, but replaces g2 by -g2. Therefore for exactly one choice 19D = ±0 it will be the case that g2 = +1, so exactly one of these choices satisfies (iii). Now for property (iv). Evidently 19-D also satisfies the characteristic properties (i) and (iii), hence 19_D = 19D. Also t91 = 1 for the same reason. The function [x M],19D has divisor M2 (C2 (e) - ker [x C]) and is compatible with base change, so we can write [x M]*19D = e19CM2 for some e = ±1. Now property (iii) gives '519C2
= [xM]* 9D = [xM]*[x2].i9D = =[x2]*[XM]*t9D=[x2]*(e''C2)-641902-l9C2
and so e = 1. The same calculation works for M = D by writing t91 = 1. If D = MC then the functions 19c o [x M] and 19D/19, E2 both have divisor
C2 ker[xM] - ker[xD]
hence their ratio is a unit compatible with basechange. The norm compatibility (iii) then shows that this unit equals 1, as in Remark (iii) above. This proves property (iv). Finally we check (v). Classical formulae (as can for example be found in [39] - the function a is essentially the same as the Jacobi theta function t91) show that
F(u, r) = O(u,r)D2e(Du,T)-1 is a function on E1. with divisor D2(e) - ker[xD], and is SL2(Z)-invariant.1 Hence flu, ,r) is a constant multiple (independent of r) of 19DE'/C . As a 'The SL2(Z) action is:
I
a dJ c
: (U, 7-) y
( +d cr+d)
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formal power series,
F=
(1 -
q(D2-1)I12t-D(D-1)I2jj
n>o
gnt)D2
r, (1 - gnt_1)D2
1 - gntD n>o
1 - qnt-D
is a unit in the ring of Laurent q-series with coefficients in Z[t,1/t(1 - tD)]. So by the q-expansion principle, the constant has to be ±1. To determine the sign, consider any elliptic curve Er defined over R with 2 real connected components. For such a curve one can assume that Re(T) = 0. (To be definite, take E to be the curve
Y2=X3-X for which r = i.) The real components of E1. are the images of the line segments [0, 1] and [T/2,1+T/2] in the complex plane. We compute [x2]*F(u, r) for such a curve. The explicit formula for 8(u, ,r) shows that the first nonvanishing u-derivative of F(u, T) at the origin is real and positive. On the interval [0, 1], flu, ,r) has simple poles at u = k/D (1 < k < D -1) and so by calculus (_1)(D-1)/2F(1/2, T) > 0. On the segment [T/2,1 + r/2], F is real, finite and non-zero, hence the product F(T/2, T)F((1 +,r)/2,,r) is positive. Therefore at the origin,
1, r)F,(u+T',)F(u+ 1 +T'T)
[x2]*F(u,T) = F(u,T)F(u+ 2 2 2 , (_1)(D-1)/2 x (positive real) x iD2-1 and so [x2]*F(u,T) =
2
(-1)(D-1)/2F(u,T).
0
Remark 1.2.3. As we saw in the proof, t9D corresponds to one of the two nowhere-vanishing modular forms of weight (D2 - 1)/12. Using (v) it is easy to determine which. The form arises by restriction to the zero-section of the composite isomorphism OE(D2e)
OE(ker[xD])
[xD]*OE(e)
X
since a*[xD]* = e*. Therefore the q-expansion is D times the leading coefficient in the expansion of,&DE,/C) in powers of t, which from (v) is easily seen to be n)2D2-2 = (D-1 D2-1 (_1)D-1qD2-1 a 1a 11(l - q) _ _1) 2 Q(T)122
n>O
Remark. Suppose that E/S is an elliptic curve over an integral base S, and that P C E(S) is a finite group of sections. Let
0 = I: mx(x) E Z[P] xEP
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be a divisor with E mx = 0 and E mxx = e. In the case when S is the spectrum of a field, D is principal, but in general this will not be the case. For example, suppose that P = {e, x} for a section x of order 2, disjoint from 02 e. Then Z = 2(x) - 2(e) is principal if and only if WEI S = e*O(Z) is trivial. Consider a Dedekind domain R containing 1/2, and an ideal A C R which has order 4 in Pic R. Let A4 = (a) and let E/R be the elliptic curve given by the affine equation Y2
= x(x2 - a)
over the field of fractions of R. Take an open U C Spec R over which A becomes principal, locally generated by a, say. Then a = a4e for some unit e E O(U)*, and an equation for E over U is (y/a3)2
= (x/a2)((x/a2)2 - E).
Therefore WE/R is locally generated over U by
d(x/a2) = dx y/a3 y i.e. WEIR
1.3
A. So the divisor 2(0,0) - 2(e) is not principal on E/R.
Units and Eisenstein series
Let E be an elliptic curve over an integral base S, let D > 1 be an integer prime to 6, and x E E(S) a section. If x is disjoint from ker[xD], then one obtains a unit t9D(x) = x*19D E O*(S) on the base. In particular, suppose that
x is a torsion section of order N > 1, with (N, D) = 1. Since S is integral, x has order N at the generic point. Under either of the following conditions it is automatic that x fl ker[x D] = 0: N is invertible on S (then x has order N in every fibre); or N is divisible by at least two primes.
In the classical setting one takes S to be a modular curve (over C) and the functions 19D(x) are the Siegel units, studied extensively (see for example [19]).
There are at least two ways to form a logarithmic derivative from the pair (V9D, x). The simplest is to form
dlog(t9D(x)) E I'(S, Sts)
which in the classical setting gives weight 2 Eisenstein series. The other way, which leads to weight 1 Eisenstein series, is to first form the "vertical" logarithmic derivative dlogt, z9D E I'(E - ker[xD], QEis)
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Since wE/s = x'SlEIs (see §1.1) we obtain
DEis(x) = DEis(E/S, x) := x'
E r(S, x*SzE/s) = r(S,WEIS),
a modular form of weight one. Notice that in this construction one can start with any function whose divisor is D2(e) - ker[xD], since it will be of the form g'9D for some g E O*(S), and dlog g = 0. From property 1.2.1(iv) we have ('dD)D12
t9D, o [xD] = (19D,)D2 t9D o [x D']
and therefore D'2 dlog i9D - [xD']* dlog OD = D2 dlog,, t9D, - [x D]` dlog. 19D,.
(1.3.1)
Now [xD]* is multiplication by D on global sections of QEls. Hence (1.3.1) gives
Di2 . DEis(EIS, x) - D'- DEis(EIS, D'x)
= D2 D,Eis(E/S, x) - D DEis(EIS, Dx). It follows that for any D =- 1 (mod N), the section
Eis(E/S, x) := D2
D DEis(EIS, X) E r(S ®Z[D(D -1)]' w)
(1.3.2)
is independent of D. Now if p%2N, there exists D > 1, D =- 1 (mod N) with (D, 6) = 1 and p%D(D - 1), so one can glue the various Eis(E/S, x) for the different D to get a section Eis(E/S, X) E r(S 0 Z[1/2N], w). For any D one then has
DEis(EIS, x) = D2 Eis(E/S, x) - D Eis(E/S, Dx). Suppose E = C/A is an elliptic curve over C, with A = Zw1 + Zw2. Let u be the variable in the complex plane. Using the function a(z, A)D2/a(Dz, A) (Weierstrass a-function) in place of t9D gives
dlog 19D = (D2((u, A) - DS(Du, A)) dz
and if x E E(C) - {e} is the torsion point (alwl + a2w2)/N E N-lA/A, with (N, D) = 1, then
Eis(E/C, x) = E n m;E N tZ
1
(mlw1 + m2w2) Imlwl + m2w21
8
du. 8=0
(1.3.3)
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On the Tate curve Tate(q) over AN = Z[,N]((q't')) there is the canonical differential dt/t, and the level N structure (Z/NZ)2 -+ ker[xN],
(a1, a2) mod N
Sri qa2/N.
If x is the point (N qa2/N then by explicit differentiation of the infinite product in Theorem 1.2.1(v)
Eis(Tate(q)/AN, x) = [B1
\\(
I-
/
sgn(d)(N ) qn/NJ dt/t
I
n>0 \\ dEZ, din e =_a2 mod N
if 0 < a1 < n, 0 < a2 < n. Here B1(X) = X -1/2 is the Bernoulli polynomial.
(In the case a2 = 0 # a1 the constant term is somewhat different.) In particular, Eis(Tate(q)/AN, x) is holomorphic at infinity. One can also compute the logarithmic derivative of the unit VD(x) E AN. The result is most interesting if one works with absolute differentials, that is in the module of (q-adically separated) differentials !AN/Z = AN . d(q'IN) ® AN/IN . d(N
where IN C AN is the annihilator of d(N (and equals the ideal generated by the different of Q(i N)). The key point is that the logarithmic derivative of a typical term in the infinite product for 19D(x) is
dlog (1 - (
a1q(m±a2/N))
_(tat (m±a2/N) (a1 dlog (N + (a2 ± mN) dlog ql/N) 1
4(m±a2/N)
whereas the corresponding term for the vertical logarithmic derivative is -(Na1 q(m±a2 /N)
dlog(1 - W.")
a-SN 9-2/N
= f 1 - (Na1g(mfa2/N) dlogt.
Comparing gives the following striking congruence:
Proposition 1.3.4. If x = (ri qa2/N E Tate(q) (AN), then dlog 19D (x) =
DEis(Tate(q)/AN, x) (al dlog (N + a2 dlog ql/N) dlog t
mod N.
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Norm relations Some elements of K-theory
For a regular, separated and noetherian scheme X, the Quillen K-groups KaX, i > 0, together with the cup-product
U : KX x
K,+3 X
define a graded ring KX, which is a contravariant functor in X- for any morphism f : X' -* X of regular schemes there is a graded ring homomorphism f * : K*X -* K.X. 'If f is proper, then there are also pushforward maps f.: KaX' -> KKX (group homomorphisms) which satisfy the projection formula
f*(f*a u b) = a u f*b. For i = 1 there is a canonical monomorphism
O*(X) -4 K1X.
(2.1.2)
For arbitrary f, the restriction of the pullback map f * to the image of (2.1.2) is pullback on functions; if f is finite and flat, then the pushforward map f* restricts to the norm map on functions. In this section we are concerned with K2. The cup-product in this case is the universal symbol map
O*(X) ®O*(X) - K2X H {u,v} u®v which is alternating and satisfies the Steinberg relation: {u,1 - u} = 0 if u, 1 - u E O* (X) If X = Spec F for a field F, then the symbol map induces an isomorphism -
K2X = K2F
A2F*/(Steinberg relation)
by Matsumoto's theorem. Returning for a moment to the general situation, let Y be a smooth (not necessarily proper) variety over a number field F. Write Y = Y ®F Q,
GF = Gal(Q/F), and let p be prime. Then if Hi+1(Y,Q (n)) has no GF invariants, there is an Abel-Jacobi homomorphism
Ken-i-lY -# H'(GF, Hi(Y,QP)(n)) The condition that H°(GF, Hj+1(Y, Q(n))) = 0 can often be checked just by considering weights; if for example Y is proper, then by considering the action
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of an unramified Frobenius and using Deligne's theorem (Weil conjectures) one sees that it holds if j + 1 # 2n. In the case of interest here, Y is a curve and j = 1, and n = 2. Then the Abel-Jacobi map is even defined integrally: AJ2: K2Y -* H'(GF, H1(Y, ZP) (2)).
(2.1.3)
It is constructed as follows. There is a theory of Chern classes from higher K-theory to etale cohomology: these are functorial homomorphisms, for each
q>0andnEZ: c,,.: KY -+
H2n-4(Y,
Zp(n)) Here the cohomology on the right-hand side is continuous etale cohomology. These maps are not multiplicative, but can be made into a multiplicative map by the Chern character construction.. All we need to know here is that if a, a' E K1Y then c1,(a) U c1,1(a') = -c2,2(a U a')
(2.1.4)
(see for example [33, p.28]). One writes ch = _C2,2' The etale cohomology of Y is related to that of Y by the Hochschild-Serre spectral sequence: Ez'3 = Ht (GF) H2 (Y, ZP) (n))
Ht+' (Y, ZP(n))
Let Y -4 X be the smooth compactification of Y, so that Y = X - Z for a finite Z C X. The ZP module H° (Y, ZP) = H° (X, ZP) is free of rank equal to the number of components of X, and H2(X, ZP) = Ho (X, Z)(-1). The module Hi (X,ZP) is the Tate module of the Jacobian of X, hence is free. There is an exact sequence 0 -+ H'(X, ZP) -+ H1 (F, ZP) -+ H°(Z, Z)(-1)
-4 H2(X, ZP) -* H2(Y, ZP) -+ 0. The map ry is the Gysin homomorphism, mapping the class of a point z E Z
to the class of the component of X to which it belongs. Therefore all the modules Hi (Y, ZP) are free2. Moreover if a is an eigenvalue of a geometric Frobenius acting on Hi (Y, Q) at a prime v %p of good reduction, then a is an algebraic integer satisfying 11
lal =
N(v) 1/2 or N(v)
N(v)
if j = 0, if j = 1, and if j = 2.
2In the case to be considered later, Y is actually affine, in which case one even has H2 (?, ZP) = 0.
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Therefore when n = 2 the first column {EE'"} of the spectral sequence vanishes. The exact sequence of lowest degree terms then becomes: o - H2(GF, H°(Y1 ZP) (2)) -+ H2(Y, ZP(2))
14 H1(GF,H1(Y,Z)(2)) -*H3(GF,H°(V,Z)(2)). Composing the "edge homomorphism" e2 with ch = -C2,2 defines the AbelJacobi homomorphism (2.1.3) - the minus sign is chosen because of (2.1.4). Notice also that the last group H3 (GF, H° (Y, Z) (2)) is zero if p is odd, and killed by 2 in general (see for example [24]). We also need the Chern character into de Rham cohomology. For a Noetherian affine scheme X = Spec R there are homomorphisms for each q > 0 dlog = dlogR : KqR -3 QRIZ
satisfying: (i) dlog(a U b) = dlog a A dlog b;
(ii) If b E R' C K1R then dlogb = b-ldb E S2R/Z;
(iii) On K°R, dlog is the degree map. (iv) If R'/R is a finite flat extension of regular rings, then trR,/R o dlogR, _ dlogR o trJ/R. In (iv), trK IR : KqR' -> KqR is the proper push-forward for Spec R' -* Spec R
(also called the transfer), and
QR,/z -> Q'R/Z is the trace map for
differentials. Since this compatibility does not seem to be documented in the literature we make some remarks about it. What follows was suggested in conversation with Gillet and Soule. To check the compatibility we can work locally on Spec R, and thus assume that R is local. Therefore R' is a free R-module of rank d say. Choosing a basis gives a matrix representation µ: R' y Md(R). We get for every n > 1 GL d(R), which in the limit give an corresponding inclusions GL (R') inclusion GL(R') '--+ GL(R). This induces by functoriality the transfer on
Kq(-) = 7rq(BGL(-)+). One way to define the map dlog is to use Hochschild homology (see for example [22, 1.3.11ff.]). There is a simplicial R-module C,(R) with Cq(R) = R®q+1 (tensor product over Z), whose homology is Hochschild homology HHH(R). There is also a pair of R-linear maps SZR/z 14 HHq(R) -14 S2RIZ, whose composite is multiplication by q!. The map 7rq is given by r°®r1®...®rq"
r°drjn..Arq.
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There is a map Dtr: H*(GL(R),Z) -+ HH*(R), the Dennis trace (see [22, 8.4.3], which maps r E R* C Hl (GL(R), Z) to the homology class of r-1 0 r E C1(R). Assume that q! is invertible in R. Then composing on one side with the Hurewicz map Kq(R) -+ Hq(GL(R), Z), and on the other with (q!)-17rq, defines the map dlog: Kq(R) --* S2RIa, for any q > 0. It is not too hard to check directly (an exercise from [22, Ch.8]) that if a, b E R* then dlog{a, b} = (ab)-lda A db, which is the only part of (i) needed in what follows.
There is a trace map trHH on Hochschild homology: the representation R '---> Md(R) induces by functoriality a map HH*(R) -* HH*(Md(R)), and by Morita invariance [22, 1.2.4] we have an isomorphism HH*(Md(R) -n4 HH*(R). Still under the hypothesis that q! is invertible, the maps eq and (q!)'17rq make S2RIz a direct factor of HHq(R). One can then define the trace map trR,,R as the composite (q!)-17rq o oeq. (This approach to trace maps is due to Lipman [21] - see also Hiibl's thesis [13].) It now is a simple exercise to check the compatibility (iv), the essential point being the transitivity [22, E1.2.2] of the generalised trace. We need all of this only for q < 2. This means that in the reciprocity law 3.2.3 and all its consequences we need to assume that p is odd.
2.2
Level structures
Let E/S be an elliptic curve. Then for every positive integer N which is invertible on S, there exists3 a moduli scheme S(N), which is finite and etale over S, and which represents the functor on S-schemes T ( )( ) level N structures on E x s T S N T = a: ker[xN] IT
More generally, for pairs (M, N) of positive integers invertible on S there is a scheme S(M, N) which represents the functor S(M, N) (T)
monomorphisms of S-group schemes
_I
a: (Z/M x 7G/N)IT -+ EIT
The group GL2(Z/N) acts freely on S(N) on the right, with quotient S. One has S(N, N) = S(N); in general S(M, N) is a quotient of S(N') where N' = lcm(M, N). One usually writes S1(N) for S(N, 1), and we will also write S,'(N) for S(1,N). Of course S1(N) and S'(N) are isomorphic, but they are different as quotients of S(N). We have a lattice of subgroups of GL2(Z/N7G), and a corresponding diagram of quotients of S(N): 'To avoid overloading the notation we do not include the dependence on E in the
notation.
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"S(N)/S1(N)
(N)
Sl IWS1(N)/SO(N)
S
Over S(N) there is a canonical level N structure aN: (7G/N)2 -* ker[xN] c E, and we let YN, yN E E(S(N)) be the images of the generators (1, 0), (0,1). Then yN already belongs to E(S1(N)), and yN to E(S,'(N)). S, (N) is canonically isomorphic to the open subscheme of ker[x N] consisting of points in the kernel whose order is exactly N; and
ker[xN] = H S,(M).
(2.2.1)
MIN
The scheme S0(N) parameterises I'0(N)-structures on E/S; in other words, S0(N)(T) is functorially the set of cyclic subgroup schemes of rank N of E xsT. In the case T = S1(N), the morphism'rsl(N)/so(N): S1(N) -3 S0(N) classifies the cyclic subgroup generated by yN. If MOM' and NON' then there is a canonical level-changing map 7rs(M',N')/S(M,N) : S(M', N') -* S(M, N)
induced by the inclusion Z/M x 7G/N -* Z/M' x Z/N'. One has yM = (M'/M)yM, and likewise for y'. We also recall that all the above moduli schemes can be defined for integers M, N which are not invertible on S, using Drinfeld level structures, see [18, passim]. They are finite and flat over S. Recall finally that for a positive integer N which is the product of two coprime integers > 3, there is a universal elliptic curve with level N structure over the modular curve Y(N)/7G. We shall use the standard notations Y,(N), Y(M, N) without comment.
2.3
Norm relations for r(t)-structure
Now fix an integer D > 1 which is prime to 6. On each basechange E x s T (where T is one of the above moduli schemes) there is the canonical function
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which by Theorem 1.2.1(ii) is simply the pullback of i9DE/s) Since E and D will be fixed in the discussion that follows we shall write all these functions simply as V. Consider the case of prime level B. Write y = ye, y' = yP, and abbreviate S, = S,($) (? = 1 or 0). Fix x E E(S) such that D2x does not meet the zero section of E. Let A : E x s So -+ E be the quotient by the canonical subgroup scheme of rank f, generated by y. Let x E E(So) be the composite 19DET /T ),
S0-E+ ExSS0-E. Write t9 = '0("s-) E F(E - ker[xD], 0*).
Lemma 2.3.1. Ns,ls(i9(x + y))
=19(2x)19(x)-1.
Ns(e)/Sl (&9(x + y')) =19(2x) JJ t9(x + ay)-1.
(Ni) (N2)
aEZ/t
Ns,lso(19(x + y)) =
(N3)
Ns(c)ls, (19(x + y')) =
(N4)
Nsa/s(t9(x))
=19(x)e19(2x).
Proof. (N1) By (2.2.1) there is a Cartesian square Sl I IS -(x+ Y--% E
I
I(Xe)
S
E
Hence
Ns,/5(t9(x + y))t9(x) = Nslus/s((x + y, x)*19) = (ex)*[xe]*19 since the square is Cartesian = (2x)*t9 by 1.2.1(iii) = t9(ex).
(N2) The same argument, applied to the Cartesian square
S(e) II 1I Sl
(x+y',x+ay)
E x S Sl
aEZ/l
I Sl
ex
a E x S S1
(N5)
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(N3) This comes from the Cartesian square: Sl I I So
I So
(x
E xs
l
So
E
The remaining relations (N4) and (N5) are obtained by combining (Nl)-(N3) and using
NSl/So (i9(x + y)) = [J i9(x + ay). aE(Z/l)*
Lemma 2.3.2. The norm relations (Nl)-(N5) hold without the hypothesis that £ is invertible on S. Proof. Choose an auxiliary integer r > 2 prime to £. Then after replacing S by an etale basechange there exists a level r structure Or: (Z/r)2 -* ker[xr] on E, with r invertible on S. Let £u"iv -+ Y(r) be the universal elliptic curve with level r structure over Z[1/r]. Then there is a unique morphism e: S -+ £""iv - ker[x£D] which classifies the triple x): there is a Cartesian square
E J
S£
)
£univ XY(r) (£univ - ker[x£D]) tPr2
£" - ker[xtD]
such that Or is the inverse image of the canonical level r structure on £°"iv and x is the pullback of the diagonal section £u"iv -+ £univ x By the basechange compatibility of X90, it is enough to verify the norm £""iv.
relations in this universal setting; but the inclusion Y(r) ® Z[1/f] y Y(r) induces an injection on 0*. Thus we reduce to the case in which £ is invertible on S.
Now consider two auxiliary integers D, D' with (6t, DD') = 1, and write d = t9oE/S) and t9' _ t9DE/s) Following tradition we write N,/, for the push-
forward maps 7r,/,* on K2, but the group operation in K2 will be written additively (for consistency with the higher K-theory case to be considered below).
Proposition 2.3.3. In K2S the following identity holds: Ns(l)/s{i9(x + y), t9'(x' + y')} = {0(£x), i9'(£x )} + £{O(x),t9'(x)} - Nsols{t9(x),
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Proof. Compute using the projection formula (2.1.1) and the norm relations 2.3.1: Ns(e)/s{19(x + y), 19'(x' + y')}
= NN11s{19(x + y), Ns(e)ls, ?9'(x' + y')} = Nsl/s{19(x + y), 19'(ex')29'(x)-1}
by (N4)
= NNo18{29(1)19(x)-1, i9'(2x)17(x)-1}
by (N3)
= -Nsols{19(1),19'(1')} - (e + 1){19(x), 19'(.2x')} + {19(x), Nso1829'(. )} + {Nso/829(x),19'(ex')}
_ -NNo18{'19(x),19'(1 )} - (e + 1){19(x), ?9'($x')} + {19(x), 19'(x')19'(ex')} + {19(x)E19(ex),19'(ex')}
by (N5)
_ -Nsols{29(x), t7(x )} + e{19(x),19'(x)} + {19(ex),19'(ex')}
Now suppose that S is a modular curve of level prime to e, and E is the universal elliptic curve. Therefore S = YH := Y(N)/H for some subgroup H C GL2(Z/N), E _ £univ YH, and S0(e) = 1'H,e := YO(e, N)/H. It is then possible to rewrite the above norm relation using the Hecke and the diamond operators, whose definitions we briefly recall. The centre (Z/N)* C GL2(7G/n) acts on YH and £aniv on the right, defining the diamond operators (a) E Aut YH, (a)8 E Aut £ for a E (7G/N)*. In modular language, the B-valued points of YH are pairs (X/B, [an]H), where X/B is an elliptic curve and [aN]H is an H-equivalence class of level N structures
(Z/N)2 -* X. Then (a) : (X/B, [aN]H) H (X/B, [aaN]H) is an automorphism of YH. The B-valued points of £univ are triples (X/B, [aN]H, z) with z E X (B), and the automorphism (a)8 of £univ is given by (X/B, [aN]H, z) H (XI B, [aaN]H, z).
Recall also [18, (9.4.1)] that the eN pairing defines a morphism eN : YH -+ Spec Z[,N]det H
(EI S, aN) '---p eN(aN
(11N)
, aN (1/N))
and the restriction of (a)* to Z[,uN]det H is then the map ( -4 det(a) = a2). The is a commutative diagram [5, (3.17)] £univ
£univ () £univ
1'H
(e)
(2.3.4)
YH
(2.3.5)
(d2
(since
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and if x: YH -f Enniv is any N-torsion section, (L)E o Lx = x o (L). Therefore by the basechange property 1.2.1(ii), 19D(Lx) = (Lx)*9D = (Lx)*(L)Et9D = (L)*t9D(x) = (L-1)* 9D(x).
The scheme So is the quotient YH,e := Y0(NL, N)/H, and Y0(NL, N) classifies
triples (X, aN, C) with C C X a subgroup of rank L. One then has the standard diagram [5, (3.16)] W6
p i Euniv XYH YH
Euniv
YH
Euniv v )
e
YH,2
(
YH,2
c
Euniv X yH YH e
Euniv
YH,£ ) YH
Spec Z[µN]H S 4 Spec Z[PN]H in which the first, third and fourth squares in the top row are Cartesian. The Hecke operator TI is by definition the correspondence c*(cw)* on YH, and the correspondence cE*(cwe)* on Enni". All the horizontal arrows are compatible
with the level N structure on En"" and the quotient level N structure on Enni", hence cs o ws o x= x o c o w, and therefore 19(x)
= (A o x)*19 = (A o x)* (ce o v)*9 = (cow)*9(x)
using as always the basechange property 1.2.1(ii). Therefore Te{19(x), 19'(x')} = c*(c o w)*{19(x), 19'(x')} = Nso(e)/s{19(x),
Finally write z = x + ye, z' = x' + y'e, so that Lx = Lz and x = (Lz) o (L-1) Observe that the norm relation is invariant under the action of GL2(Z/L), so that ye, ye can be replaced by any basis for the L-torsion of E. This yields the following reformulation of 2.3.3:
Proposition 2.3.6. If S = YH is a modular curve of level prime to f and z, z' are torsion sections of E/S(L) whose projections onto ker[xL] are linearly independent, then NS(e)/S{19(z),19'(z')} = (1 - TI o (L)* + L(L>*){19(Lz),19'(Lz')}.
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2.4 Norm relations for r(r)-structure We now consider norm relations in the tower {S(em, In)}_
Lemma 2.4.1. If m > 1, n > 0 and x E E(S(em, t")) is any section, then NS(tm,en)/S(tm-1,tn) : t9(x + ytm) -+ t9(ex + yem-1)
(N6)
Proof. If a is invertible on S this follows from the Cartesian square S(tm, en) 1
S(em_l, e")
x±vm
E
tM
ex+Y'--', E
In the general case one reduces to the universal situation exactly as in 2.3.2.
Proposition 2.4.2. If (6e, DD') = 1 and m, n > 1 then for all x, x' E E(S(em-i, In`)),
NS(tm,tn)/S(tm-i,tn-1) : {t9(x + yem), t9'(x'+ yen)}
H {V9(& + yt,n-i), t9'(ex +
Proof. This follows from the lemma since NS(tm,en)/S(tm-l,tn-1){19(x + yen), t9'(x' + yin)} = NS(tm,tn-i)/S{19(x + ytm), NS(tm,1n)1s(em,en_1)19/(x, + yen)}
= NS(em,tn-1)/S{79(x +p ytm), t9'(tx' + y't._i)}
= {t9(ex + ytm-i), 79'(ex, + y'tn-i)J
If E/S is a modular family over S = YH of level prime to e, then E(YH) is finite of order prime to e. Therefore E(YH(t', tn))torsion = (Z/tm X Z/tn) x (prime to e),
so there is a well-defined projection onto (Z/t)2 = ker[xt]. Computing as in 2.3.6 we get:
Proposition 2.4.3. Suppose that S = YH is a modular curve of level prime to a and that z, z' are torsion sections of E over YH(tm, £' ), with m, n > 1. If the projections of {z, z'} into ker[xt] are linearly independent, then NYH(em,en),YH(em-i,en_1){19(z), i9'(z')} _
{9(ez),19'(ez')}.
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401
Norm relations for products of Eisenstein series
We shall repeat the construction of the last paragraph for products of the form
DEis(E/S, x) 'D, Eis(EIS, X') E I'(S, WE/S)
If g : S' -+ S is a finite and flat morphism of smooth T-schemes and E' _ E x5 S' then there are trace maps
tr9 = trs,/S: 9*d5' -+ 0s, 9.Q'
QS/T
as well as a trace map on modular forms, defined to be the composite
tr9 = trs,/S: r(S',WE /s,) = r(S' 9*WE'S) tr
= r(S, g*Os, ®os WE/S) -* r(S, WE/S).
The Kodaira-Spencer map (§1.1) and the trace are not compatible. Proposition 2.5.1. The diagrams below commute: 02
KSE'ls'
KSE,/s,
02 WE '/S'
1 QS'/T
WE'/S'
T9.
Tt
9E2S
Tt 9-*KSEIS
g
QS
02
T
deg(g).g*KSE/s
*Q1 SIT
9
9*W E /S
Proof. The first commutes because of the functoriality of the Kodaira-Spencer map. Then applying tr9 o g* = deg g gives the second.
Lemma 2.5.2. The notation as in Lemma 2.4.1, trs(tm,en)/s(,-n-11n) : DEis(x + ye.,.) H 2-1DEis(tx + yl...-1).
Proof. Since [x2]*,9D = VD, we have tr[xt]: dlog19D H dlog?9D. But on global sections of IZE/S, tr[xf] is multiplication by f. Therefore the diagram
I'(E - ker[xD], Q1)
(x+vtn)i
r(S(t'', p"-1),W) I Itrs(rn,tn-1)/g(tn-1)
tr[xt]
1
r(E - ker[xD], 1')
(tx+y)
r(S(e -1), w)
commutes, which gives the result.
Corollary 2.5.3. The notations being as in 2.4.3, let g be the projection g: 1H(Q, B") -+ yH(2'"-1, e"-1). Then
H t-2DEis(tz) D,Eis(&') tr9: KS(DEis(z) D,Eis(z)) H t2KS(DEis(fz) D,Eis(ez')) tr9: DEis(z) D,Eis(z')
Proof. Follows from the preceding two lemmas, since deg(g) = P4.
(2.5.4) (2.5.5)
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A Appendix: Higher K-theory of modular varieties A.1
Eisenstein symbols
Let f : E -4 S be an elliptic curve, and assume from now on that S is a regular scheme. For any integer k > 0, write E' for the fibre product E x . . . XS E - it is an abelian scheme of dimension k over S. In [2], Beilinson discovered a family of canonical elements of Kk+l(Ek) More precisely, he defined a canonical map Q[Etore]degree=O -+ Kk+1 (Ek) ® Q
which he called the Eisenstein symbol. Here we make a modified construction which gives a norm-compatible system. Let rk be the semidirect product of the symmetric group C5k and µ2, which acts on Ek as follows: C5k acts by permuting the copies of E;
the ith copy of µ2 acts as multiplication by ±1 in the ith factor of the product. There is a character ek : rk --> A2 which is the identity on each factor µ2 and the sign character on the symmetric group. rk has a natural realisation in GLk (Z) as the set of all permutation matrices with entries ±1. Geometrically it is the group of orthogonal symmetries of a cube in n-space. In terms of this representation, Ek is just determinant. For any Z[rk]-module M, write M(Ek) for the Ek-isotypical component of M 0 Z[1/2. k!]. For x E E(S) we shall consider the inclusion Ek -* Ek+1 (u1,
... , uk) H (x - u1, u1 - u2, ... , uk-1 - Uk, uk)'
whose image is the subscheme k+1
S (v1, ... , vk+1)
l vti=x}CEk+I.
JJ1
For any integer D 54 0 such that x is disjoint from ker[xD], we define the following open subschemes of Ek: UD,X = ix1(E - ker[xD])k+2)
UD,z = n 'Y(UD,x)' ,yErk
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Observe that UDx and UDx are stable under translation by ker[xD]k, and there is an etale covering [x D] : UD x -3 U1,Dx'
We prove below the following lemma.
Lemma A.1.1. If z E E(S) is any section disjoint from e, the inclusion Ul Z " (E - {±z})k induces an isomorphism K*(E - {±Z})k(sk)
K*Ui z(ek)
Using this lemma we/ define K-theory elements, whenever (6, D) = 1: l9Dk]
(l)tgD](x) (2)tgDk](x)
(3)19Dk](x)
= prl(9D) U ... U prk+l(t9D)
E Kk+1(E - ker[XD])k+1
= Zx\ D]I
E Kk+1UD,x
=
E Ek('Y)2f*((l)t9J (x)) E Kk+l(UD,x)(Ek)
7# k 'YErk
=
E Kk+l(E - {±Dx})k(Ek)
We call (t)t9Dk](x) Eisenstein symbols. For k = 0 we simply define ix to be the
section x E E(S), and the Eisenstein symbol then becomes a Siegel unit: (i)?9o1(x)
= z9D(x) E (7*(S).
If al, ... , ak+1: E -* E are isogenies of degree prime to D, then by repeated application of 1.2.1(iii) one get the norm-compatibility 19,k]
(A.1.2)
Actually this is only of interest when all the ai are equal.
Proof of Lemma A.1.1. For any T/S U1, (T) _
{(u1.. .,Uk) E E(T)k
for all i, ui # e, ±z; for all i
j, u1 ± U.
0.
The complementary divisor Ek - Ul z is the union of the two divisors Vk = {(ui) I for some i, ui = e} U {(ui) I for some i 0 j, ui ± uj = e} and
Wz = {(ui) I for some i, ui = ±z}
As S is regular, the K-groups in the lemma can be computed in K'-theory.
From the localisation sequence, it is then enough to show that K;(Vk Wz)(Ek) vanishes. This is a special case of the following:
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Lemma A.1.3. Let V' C Vk be any I'k-invariant open subscheme. Then K; (V') (&k) is trivial.
Proof. Define a sequence of reduced closed subschemes Vk = V1]
Vk] D ...
Vk+l] = 0
inductively, by writing V,k+l] for the smallest closed subset of VTR such that Urj - VT+1] is smooth over S. Write VT] = V,, fl V. Then from the definition of Vk it is easy to see that: (i) V*] is a union of closed subsets each given by the vanishing of a certain collection of expressions ui, ui ± u,, which are permuted by rk;
(ii) This gives a decomposition of Vr] - Vr+1] as a disjoint union LI V[,.],,, of open and closed pieces, permuted by Fk, in such a way that for each
µ there is some ry,, E I'k which acts trivially on Vr]µ and for which sk ('Yµ) = -1.
This forces K;(Vr] - Vr+1])(sk) = 0 for each r > 1. In fact, if
c=
cµ E K;(V;.] - Vr+1]) 0 Z[1/2. k!] K; (V.]A) 0 Z[1/2. k!]
then ryµ(c) = sk(ryµ)c = -c, whereas the µ-component of yµ(c) is evidently +c,, by (ii). Now using the long exact sequences K.(V .1) -* K(Vr] - Vr+1]) -> .. inductively (beginning with r = k - 1) we deduce that K;(V')(ek) = 0.
0
A.2 Norm relations in higher K-groups Here we find norm relations for the Eisenstein symbols and for cup-products, analogous to those in sections 2.3 and 2.4. For the I'(2)-structure norm relations, we use the same notation as in 2.3. In addition, write A : E -+ E x s So for the isogeny dual to A, and Ak, Ak for the isogenies on Ek, Ek. Consider the push-forward for the morphisms: Ek [x e] X lrs, /s : Ek x s S1
-
Ak
Ak
x irs1/so : Ek XS S1 -) Ek x 7rso/s : Ek -+ Ek
Fix i E {1, 2, 3} and abbreviate O9[k](x) _ (')19Dk](x). The symbol Dk] will denote the analogue on Ek of t9Dk]
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Lemma A.M. The following relations hold in Kk+1: ([X t] (A k
(EN1)
X 7rsl1S)*19Dk](x+y) = 19Dk](2x) - [X f]*19[k] (x)
-
(EN3)
x 7rSl/So)*19Dk](x + y) = Dk](x) (Ak x 7rS.1S)*'dD](i) =1[xP]*i9Dk](x) + 19Dk1 yX).
(EN5)
Proof. Here (EN1) and (EN3) are to be understood on UD x, and (EN5) on (Ak)-1(UDx). The relations (EN1) and (EN3) are proved just as (N1) and (N3), by considering the Cartesian diagrams: Ek X S S1 II Ek
(`s±y
Ek+1
l[xi
([xt]x71S1/S>[xe]) I
it
Ek
Ek+1
and
Ek x s S1 II Ek x s So
(4+1114)
Ek+l X S SO
(Ak xirsl/so,Ak)J
Ek
and using the norm-compatibility (2.5.4). Applying Ak x 7rso/s to (EN3) gives (EN5). We now consider cup-products of the form 19Dk] U 19D, in Kk+2. Consider the factorisation of multiplication by 1: Ek xs S(2)
idx1rS(t)/S1
Ek x
sl -
Akxxrsl/so
)E
k
akx1rso/s
) Ek
[xlx1rs(t)/S
We compute: (Ak x1rS(e)/So)*[19Dk](x+y)U19D,(x +y)]
= (Ak X 1rS1/SO)*[79Dk](x + y) U (19D,(ex) - 19Dr(x ))]
_
a
VDk'(x))
U (VD- (ex') -19D-(x'))
by (N4)
by (EN3)
We need to compute the image of this cup-product under (Ak x7rso/s)*. Taking
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the terms in turn: X 7r so/S)*: 19Dk)(2) U 19D
*19Dk)(x) U 19D
by (EN5)
(fx') y ([x f]*19Dk)(x)t + 19Dkl (2x)) U 19D, (&x)
by (N5)
H [x f]*19Dk)(x) U (e19D (x) +19D (Qx))
Ak 'k)(x) U 19D, (x') H (t + 1)([xf]*19Dk)(x) U 19D
(2x))
as deg7rso/S = P+ 1
Combining these gives the required generalisation of 2.3.3:
Proposition A.2.2. ([X Q] X 7rs(t)1S)* (19Dk (x + y) U VD' (x + Y)) =
-
t9Dk]
(2x) U 19D
&x')
X 7rSoIS)*(19Dk)(x) U'9D'(x')) +' fQXf1*19Dk)(x) U t9D'(x')).
Having got this far the analogue of 2.4.2 presents no further difficulty:
Proposition A.2.3. If n > 1 and x, x' E E(S(F -1)) then ([xt] x7rs(p)/S(en-'))*(19Dk)(x+yen)U19D'(x'+yt^))
=
3 3.1
19k)(2x + Y,._1) U'OD'(&x +
The dual exponential map Notations
In this section K will denote a finite extension of QP with ring of integers o. We fix an algebraic closure K of K. Write o for the integral closure of o in K, and GK for the Galois group of K over K. We normalise all p-adic valuations such that v(p) = 1. Let K be the completion of K, and write o for its valuation ring. Fix a uniformiser 7rK of o. _ We fix for each n > 0 a primitive pn-th root of unity (P. in K such that SPn+1 = (Pn. Write Kn = and denote by on the valuation ring of Kn. Put an = the relative different of Kn/K. For a topological GK-module M write H'(K, M) for the continuous Galois cohomology groups [38]. The cyclotomic character xcycl: GK -4 z; is defined by 9((P>,) =
(pnYa(9)
for every g E GK and n > 0. Its logarithm is a homomorphism from GK to 7GP, often viewed as an element of H'(K, ZP). We normalise the reciprocity law of local class field theory in such a way
that if L/K is unramified, then the norm residue symbol (7rK, L/K) equals the geometric Frobenius (inverse of the Frobenius substitution x H x9). This implies that for any u E o* we have x,y,,(u, Kab/K) = NK/Q (u).
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407
The dual exponential map for H1 and an explicit reciprocity law
Let V be a continuous finite-dimensional representation of Gx over QP. Suppose that V is de Rham (for generalities about p-adic representations, see for example [12]). Let DR(V) = (BdR ®Q V)tK be the associated filtered K-vector space, with the decreasing filtration DR(V) (induced from the filtration on BdR). Then Kato has defined a dual exponential map [16, §11.1.2]
exp*: H1(K,V) -* DR°(V) which is the composite:
H'(K, V) -* H'(K, B°R ®QP V) = H'(K, Fil°(BdR ®x DR(V))) = DR°(V).
The last isomorphism comes from Tate's computation [37] of the groups Hi(K, K(j)): Y.
(K, R (j)) = 0 unless j = 0 and i = 0 or 1; and K = H°(K, K) u109X`Y`i H1(K,
K)
(3.2.1)
together with the isomorphisms BaR/BaRI K(j). The group H1(K, V) classifies extensions 0 --* V -* V (0) -+ 0 of p-adic Galois representations, and the extension V is de Rham if and only if its class lies in ker(exp*). (This follows from [3], remark before 3.8 and Lemma 3.8.1.) In particular, the kernel of exp* is the Bloch-Kato subgroup
H9(K,V) C H1(K,V). In some cases one can define and study the dual exponential map without
reference to BdR. For example, if V = H'(A,O.(1)) for an abelian variety A/K, it can be defined just using the exponential map for the analytic group A(K). More generally, if the filtration on DR(V) satisfies DR'(V) = 0, then one only needs to use the Hodge-Tate decomposition
KOQP V - ® K(-i) ®x gra DR(V)
(3.2.2)
iEZL
since then by (3.2.1) exp* is the natural map from H1(K,V) to
H'(K, K ®QP V) (3-)®Hl(K, K(-i) (&x grz DR(V )) DR°(V)
=H1(K, K) ®x DR°(V) (3.2.1)
A. J. Scholl
408
In what follows we shall be concerned with the case V = H'(Yk, Q)(1) for a smooth o-scheme Y, which is the complement in a smooth proper o-scheme X of a divisor Z with relatively normal crossings. Write
HdR(Y/o) = H`(X, cz/o(Z)) (the hypercohomology of the de Rham complex of differentials with logarithmic singularities along Z). Then DR(V) is just de Rham cohomology with a shift of filtration:
DR-1(V) = DR(V) = HdR(Y/o) ®o K DR°(V) = H°(X, S0X/o(logZ)) ®0 K = Fill HdR(Y/o) ®o K
DR'(V) = 0 Moreover the Hodge-Tate decomposition has an explicit description, essentially thanks to the work of Fontaine [11] and Coleman [4]. To compute exp* one just needs to know the projection
K®oH°(X,1l 10(logZ))®K(1)®oHl(X,Ox)
7r1: K®Q V (3.2.2)
K ®o H°(X, fZX/o (log Z) )
It is the limit of the maps given by the diagram Hl (YK, ILp^)
( + Hz°u(YR, 0' /P-)
r. H2°ar(Yo, O*
(E
,
I dlog
(mod p^)
H° (X, Sllx/o (log Z)) ®o/p"
Remarks. (i) The isomorphism labelled (1) comes about as follows. Generally, let S be a scheme on which m is invertible, with µm C Os. An element of H'(S, µm) is an isomorphism class of finite etale coverings S' -* S, Galois with group µm. Given such an S'/S, there is an open (Zariski) covering {UJ of S and units f= E O*(U2) such that S' x UE = U1[ m f;]. It is easy to see that { fi} is a well-defined element of H°(YK, Os/m), and moreover that the map thus obtained fits into an exact sequence Hz'. (S, NAM) -> Hdt(S, lam)
((
HZar(S, 0S* /m) -+ HZar(S, A.).
If S is irreducible (as is the case here) then every non-empty Zariski open subset is connected, so µm is flasque for the Zariski topology, and the map (1) is an isomorphism.
An introduction to Kato's Euler systems
409
(ii) The inclusion j : YR -* Yo induces an isomorphism HI (Y-6, 0* lp) -m=+
H°(Yk, 0* /p) denoted (2) in the diagram. To see this, consider the effect of multiplication by p" on the exact sequence
0-0Yo ->j*0
-*Q,, -*0
(the last map is the p-adic valuation along the special fibre, taking values in the constant sheaf Q. This shows that OYo/p" (j*OYK)/p". It is therefore enough to show that (j*O* )/p" j*(O* /p"), because then H°(Yo, O*/p") -* H°(Yo, j*(0
/p" )) = H°(YK, O*/p" )
By passing to the direct limit, we can replace K by a finite extension of K. Now consider more generally an open immersion U y S, where S is a separated noetherian scheme which is integral and regular in codimension 1. Suppose that m is invertible on U, and that µm C Ov. Then Rf j*µm = 0 for i > 0 as µm is Zariski flasque. The exact sequences
0- Am_+ 0*-*(O*)"`-*0 0->(O*)"`-*0*-*0*/m-*0. give a short exact sequence
0 -* (j*Ov)/m -+ j*(Ov/m) -+ m1'j*Oa 4 0. But because S is regular in codimension one, the divisor sequence 1 -+ Os -+ Ks
d'"
ll
ix*z -> 0
codim(x)=1
is exact, and therefore Rl j*OU = 0. (iii) The simplest case (which is, however, not enough for our purposes) is when Y = X is proper, when this recipe reduces to that given by Coleman: an element of HI (XR, Pic XR[p"] is the class [D] of a divisor D on Xk such that p"D = div(g) is principal. One can assume that the divisor of g on X. is precisely the closure DC of D. Put w = dlog g E Ho(Xo, SlXo/o(supp Dc)).
Then because the residues of w at supp D are
0 (mod p"), one has
w (mod p") E H°(X,1lx/o) ®u/p" and this defines ir1([D]) (mod p") = w (mod p"). (Coleman even defines such a map in the case of bad reduction.) Unfortunately we know of no reference for this description of the Hodge-Tate decomposition in the non-proper case.
A. J. Scholl
410
Now assume that X is a smooth and proper curve over o, and that Y is affine. Then Z is a finite etale o-scheme. Recall (see also the following section) that the different an = DKn/K is the annihilator in on of Q,./o. If K/Qp is unramified, then on = o[('p,.] and therefore Q0,/0 is generated by dlog (pn., and moreover an = pn((p - 1)-1On.
Theorem 3.2.3. Suppose that K/Qp is unramified (and that p > 2). There exists an integer c such that for every n > 0 the following diagram commutes up to pc-torsion:
K2(Y®on)®µp 1
ch
H2(Y 0 Kn, µpn)
dlog
t Hochschild-Serre
Hl(Kn, Hl (Y (9 R, µpn))
H°(X 0 on, QX®u,./o(log Z))(-1)
tir1 (mod pn-1) 11
Hl(Kn, 0/pn-1) ®o Fill HHR(Y/o)
Qln/0(-1) ® Fill HHR(Y/o)
dlog(pn®[(pn]-1H1
T U(1/pn) log Xcycl
HdR(Y/0) On/an ®o Fill
4
On/p"-100 Fill HdR(Y/o)
Corollary 3.2.4. (The explicit reciprocity law) The following diagram commutes:
__(K2(Y(9on)®µp lim
l)
HS-ch
llim Hl (Kn, Hl (Y ®K, µpn) )
n
n d1ogI
I
limH°(X ®on, Qx®u,./0(log Z))(-1)
llim Hl (K,,, Hl (Y ®K, µpn) ) n>m I
limIl' ,,(-1) 0 Fill H'R(Y/o)
Hl(Km, Hl(Y 0 k, Zp)(1))
n lexp'
I
lim On/ an ®o Fill HHR(Y/0)
( per
tr
"nlKm ) n>
Km ®o Fill HHR(Y/o)
nn
Remarks. (i) The assumption that K/Qp is unramified is not essential for the proof, and is only included to simplify the statement. In general the situation is completely analogous to 3.3.15 below. The case p = 2 is excluded only because we do not know a reference for the compatibility of the trace maps in this case, cf. §2.1.
An introduction to Kato's Euler systems
411
(ii) The maps "Hochschild-Serre" comes from the Hochschild-Serre spec-
tral sequence with finite coefficients (cf. §2.1); since Y is affine, H2(Y 0 K,1APn) = 0.
(iv) For a discussion of the map dlog, see §2.1. A priori its target is the group H°(Y ®on, SZ2X®On/O)(-1). We just explain why its image is contained
in the submodule of differentials with logarithmic singularities along Z. By making an unramified basechange, one is reduced to the case when Z is a union of sections. Let A be the local ring of X ® on at a closed point of Z, and t a local equation for Z. Then by the localisation sequence, one sees that K2A[t-1] is generated by K2A and symbols {u, t} with u E o,*,, and dlog{u, t} = u-ldu A dlogt. Proof. First we explain precisely what are the transition maps in the various inverse systems in the diagram. In the Galois cohomology groups they are given by corestriction and reduction mod pn. The finite flat morphisms Y 0 on+1 -* Y 0 on induce compatible trace maps (cf. §2.1)
K2(Y 0 on+l) -a K2(Y ®on) and
,LY®On+1/O -> Sty®On/O
which are the maps in the first and second inverse systems in the left-hand side of the diagram. In the system (S2on/O)f1 the transition maps are trace, and in the remaining system (on/cl,l)," the maps are 1 trKn+,/Kn (For the compatibility of these various maps, see 3.3.12 below.) From the discussion above, the diagram below commutes: H1(K,n, Hl(Y (9 K, Q)(1))
Hl (K,,,, K ®o Fill HdR(Y/o))
exp' l
11
Km ®O Fill HdR(Y/o)
H1(K,n, K) ®o Fill HHR(Y/o) U log Xcycl
To deduce the corollary from the theorem it is thus only necessary to take inverse limits and use the commutativity (cf. Proposition 3.3.10 below) of the following diagram logXcycl
K.
H1(Kn, K)
1
pn_m trKn/Km
K.
1cor
log XCHl
(3.2.5)
(K,,,, K).
Remarks. (i) Consider the special case Y = A' - {0} = Spec o[t, t-1]. Let (Un) E 14im on be a universal norm. By applying the corollary to the normcompatible symbols {un, t} E K2(Y ® on) one recovers a form of Iwasawa's
A. J. Scholl
412
cyclotomic explicit reciprocity law, which will be proved more directly in 3.3.15 below. (ii) Theorem 3.2.3 is proved in section 3.4 below. It is much easier than the general cases considered by Kato in [17], first because one is not working with
coefficients in a general formal group, and secondly because the assumption that X/o is smooth makes for considerable simplifications. In the non-smooth case there is an analogous statement which is needed to compute the image of Kato's Euler system when p divides the conductor.
3.3
Fontaine's theory
We shall review here some of the theory of differentials for local fields developed by Fontaine [11], and as a warm-up for the next section, show how it gives a version of Iwasawa's explicit reciprocity law. Recall (see for example [34, §111.6-7]) that if K'/K is a finite extension then its valuations ring o' equals o[x] for some x E o'. This implies that the module of Kahler differentials 520,/0 is a cyclic o'-module, generated by dx, and that its annihilator is the relative different ZK'/K The module Q0/0 equals the direct limit of 520,/0 taken over all finite ex-
tensions of K in K. In particular, it is torsion. Theorem 3.3.1. [11] There is a short exact sequence of o-modules
0-+a(1)-*K(1)4Q210-0 where a = ao10 is the fractional ideal a0/0 = ((P - 1)-lil-1 K1QP o
C K
and where a: R(1) := Z (1)®K -* S2o/0 is the unique 5-linear map satisfying
a([(pm]m ®p n) = dlog(,. =
d( n Spn
foranyn>0. Remark 3.3.2. In particular, for any n > 0 the annihilator of dlog C. E 520/0 is pn a n id.
From 3.3.1 we get the fundamental canonical isomorphism a0/0(1)
TP520/0
which is 5-linear, and maps ((pn)n E Z P(1) C a(1) to (dlog(Pn),,.
(3.3.3)
An introduction to Kato's Euler systems
413
Suppose that K"/K'/K are finite extensions. Then there is an exact sequence of differentials .1 G,, /, ®,, o" -+ J L,,, /, -+ Q
-+0
(the "first exact sequence", [23, 26.H]), which is exact on the left as well by the multiplicativity of the different (or alternatively by the argument in the footnote on page 420). Passing to the direct limit over K" gives a short exact sequence 0
52,,/0 00''d -+ cz /, -+ 52010, -a 0
(3.3.4)
At this point, recall that for any short exact sequence 0 -+ X -# Y -+ Z -* 0 of abelian groups, there is an inverse system of long exact sequences
0-+ P.X -*P.Y-+ P.Z-4X/p"-+ Y/p"-*Z/p"-40.
(3.3.5)
If the inverse systems P,.M (for M = X, Y, Z) satisfy the Mittag-Lefer condition (ML) then the inverse limit sequence
0-aTPX -* TPY -+ TPZ-3 limX/p"-> lim Y/p"-# is also exact (a special case of EGA 0, 13.2.3). Note that (pnM) satisfies (ML) in two particular cases:
the torsion subgroup of M is p-divisible (then pn M tive);
pn-1 M is surjec-
the p-primary torsion subgroup of M has finite exponent (then (pnM) is ML-zero).
Applying these considerations to (3.3.4), since 9-,,1. and Q.,/o is killed by a power of p, we get an exact sequence
are divisible and
0 -* Tp52o/, -> T, TZ 10, -+ 520,/, 00,'6 -a 0.
(3.3.6)
Now pass to continuous Galois cohomology. This gives a long exact sequence since the surjection in (3.3.6) has a continuous set-theoretic section (this is obvious here as S2o'/o®o,o is discrete). We are only interested in the connecting map, and define 8 to be the composite homomorphism: 8 = bK, : 52,,/, '-1 H°(K', 52,,/, (D,, 0)
connecting
H'(K', TPS251,).
The map "reduction mod p"" : TP52310 pnQo/o induces a map on cohomology, which when composed with 6K, gives 6K, (mod p") : 52,,/, -4 H1(K',pJlo/0).
A. J. Scholl
414
Lemma 3.3.7. (i) The following diagram commutes: Kummer
o'*
H1(K', µpn )
dlog t
Q" /0
bK' mod pn
1
dlog pn(
Hl(K',
""0/0)
(ii) For any nonzero x E of 8K' (dx) (mod pn) = x dlog(Kummer(x))
Proof. (i) Simply compute: if u E o'* then fix a sequence (um) in o with uo = U, u +1 = u, The composite dlog o "Kummer" maps u to the class of the cocycle g H dlog(un-1) E pnQo/0.
Now compute the effect of SK, on dlog u: first lift dlog u in the exact sequence
(3.3.6) to the element (dlogu,,n). E Tp(Sto/0,), then act by g - 1 to get the desired cocycle. So the commutativity is trivial. (ii) If x is a unit this is equivalent to (i). For the general case one simply calculates as in (i).
Lemma 3.3.8. Let n _> 1 and assume that µpn C K'. If p # 2, then the diagram
o'/pn(1)
u
log xcyc,
p ) H1(K',
ao/0/p")(1)
1®[Spn]Hdlog(pn l ((
i
I (3.3.3)
LO,/0
6K, mod pn
commutes. For p = 2 it commutes mod
r H1(KI)
`l-
pn Q0/0)
2i-1
Proof. All the maps are o'-linear, so it is enough to compute the image of 1 ® [(pn]. We have x,Y,l(g) - 1 (mod pn) for all g E GK, hence log x1(g) 0 (mod pn) and so if p 2 then pn log xY,:1(g) = pn (X,', (g)
- 1) mod pn.
In the proof of 3.3.7 one can take u,.,, = Spm+n for all m > 0, and then
SK, (dlog (pn) E H1(K', Tp (S2o/0)) is represented by the cocycle g H (dlog(Pm+,.)7Tt E 7p(Q'10)
An introduction to Kato's Euler systems
415
and (p-+n = (m'+;.(9)-' _ (pm"'i9i-1)/pry. Applying the inverse of (3.3.3) maps this to the class of the cocycle 1
g'-f pn (X,Yc,(g) - 1) 0 (Cpm)m
E ao/o(1)
mod pn.
pn log XcYCI (g)
The reader will make the necessary modifications when p = 2.
We now need some elementary facts about cyclotomic extensions of local fields. Our chosen normalisation of the reciprocity law of local class field theory identifies the homomorphisms logXcYcl E H1(K,7Gp) = Hom,ts(Gal(K/K),Z,) and
log oNK/
,
: K* -+ Zr,.
As observed in the proof of the previous lemma, if µpm C K then log Xcycl
0 (mod p).
Lemma 3.3.9. Suppose that µpm C K. Then for any finite extension K'/K the diagram U=1_ log Xcyci
o
H
trK'/K t
1
(K,,
o)
t cor
HI(K,
0 U pm log Xcycl
commutes.
Proof. The statement follows from the projection formula for cup-product in group cohomology, since on H° the corestriction
cor: H°(K', o) = o' -* H°(K, o) = 0 equals trK,/K
Recall that Kn := K((,.). Let £ be the largest integer such that µpz C K. Then if n > m > 2, direct calculation gives trK,JKm
=
[(pmJ (o[(p'J)
=1
pn_ o
if M
Define, for any n > m > 0 1
tom := pn-m trKn/Km : Kn --+ Km.
0.
416
A. J. Scholl
Proposition 3.3.10. If n > m > max(t, 1) the diagram U ynl T log Xcycl
H1(Kn,O)
OK A tn,m
COr
U m_T 1oBXcyc1
H1 (Km, o0)
0[SP,n]
is commutative. If n > t = 0, the diagram U pn-T log Xcycl
o[(Pn]
H1(Kn, o) cor
Ptn,o t U log Xcycl
o
111 (K, o)
is commutative.
Proof. For n = 1, the second diagram commutes by 3.3.9 with K' = K1. By transitivity of trace and corestriction, the lemma will be proved if we verify the commutativity of the first diagram for n = m + 1 > 1. Take the diagram of 3.3.9 for Km+1/Km and factorise: up-m log
XcYcl
O [(Pm+1 ]
trl{,n+1 /Km
Um trKm+1/Km .r1 rO[(pm] if xP
UP
-SACYCI
4 H1(Km+ Om)
UPl-m 109Xcycl
O[(Pm]
The bottom triangle commutes since H1 (Km, om) = Hom,.(Gal(K/Km), om)
is torsion-free. Hence the entire diagram is commutative, and going round the outside gives what we need. Now consider on = oKn/K. From the definition of an 1 as the largest fractional ideal of Kn whose trace is contained in o, it is an easy exercise to check
01 n C p n0 [(pn]. By [37, Propn. 5] the difference vv(t7n) - n is bounded, so for some c independent of n, pcon C 01(p-] C On.
An introduction to Kato's Euler systems
417
Since Ston/o is cyclic with annihilator On, the homomorphism
r
0 [(Pn
(
n xHx dlog Spn
llp
Q Onlo
(3 . 3 . 11)
is well-defined, and its kernel and cokernel are killed by a bounded power of p, by remark 3.3.2.
Proposition 3.3.12. Let n > m > max(t,1). Then the diagram Pn]
xdlog4pn
0[(
On/O
tn,mI
Itr x dlogCpm
01(P-1
n pm/O
commutes.
Proof. It is enough to compute what happens when m = n - 1. Taking 1, (Pn, ... , n 1 as basis for o[(P.] over o[(Pn_1], for 1 < j < p
tr(cpn dlog(Pn) = tr(j-Id(pn) = j-1d(tr(pn) = 0
and for j = 0 tr(dlog(Pn) = dlog(NKn/Kn_1(pn) = dlog("Pn-1.
0
Therefore passing to the inverse limit gives a homomorphism
liM o[(Pn] = liM 0[CPn]/pn - l4 S2,n/0
(3.3.13)
tr
which becomes an isomorphism when tensored with Q. (If K/QP is unramified, then (3.3.13) is itself an isomorphism.) By [38, Proposition 2.2], the canonical map H' (Km, ZP) - lim H1 (Km, Z/pn) is an isomorphism. Invertn
ing both of these arrows yields a diagram Kummer
Q
n
Q ®ltlm Hl (Kn, µn) n
norm
ICpn Hl
dlogI Q(8)
trace
Q ® l4lm H1(Kn, Z/pn)
0n/o
n
dlog Cpn H 1 Ii
tcor
Q ® lam H1 (Km, Z/pn)
Q ®lim 0[S ,,]/pn
t--
n
(tn,m )n
I
K.
H1 (Km, K) Ups 109Xcycl
(3.3.14)
A. J. Scholl
418
where down the right-hand side all the inverse limits are with respect to the corestriction maps and reduction mod pn. We then have the following version of the classical explicit reciprocity law of Artin-Hasse and Iwasawa. Without loss of generality we can assume m chosen so that µp-+, ¢ K.
Theorem 3.3.15. The diagram (3.3.14) is commutative. Proof. At finite level, replacing Z/pn with o/pn, one has the diagram: Kummer
On
Hl (Kn, µpn) dlog
dloog 1
mod pn L0n/O
Hl (Kn, p^Q0/0)
bKR
1-+dlog Sp. T
L-+dlog Spn log Xcycl
1l
,
0[Cpn]/pn
tom !
p Hl(Kn,,6/n)
Spn H1
r
t cor
i
P Hl (Km, U -m log Xcycl
0[(pm]lpn
This is for m > 0; for m = 0 the bottom arrow should read pto/pn -* Hl(K, p to/pn). The top two squares commute by 3.3.7, 3.3.8 respectively. The bottom square commutes up to p-torsion by (3.2.5). All maps are compatible with passing to the inverse limit. As remarked after equation (3.3.11), the left-hand map labelled "1 N dlog (pn" has cokernel and kernel killed by a bounded power of p, and by (3.3.3) the same is true for the one on the right. Therefore passing to the limit and tensoring with Q one obtains the theorem. Remark. One can use 3.3.7(ii) to describe the image of an arbitrary element of Kn under the Kummer map in a similar way. Here is the relation with the usual form of the explicit reciprocity law. Let u = (un)n E 1{ m on* be a universal norm. Its image down the left hand side of the diagram (3.3.14) equals (with an obvious abuse of notation) dlog un
-D(u) := lim
n-*oo pn1-m
E Km.
dlog (P-
Going round the other way, use the expression of the Kummer map in terms of the Hilbert symbol, which we write as a bilinear map [-, -In: Kn X Kn -+ Z/pn given by ( pV
.")(a'K.nb/Kn)-1
=
(Ria].
An introduction to Kato's Euler systems
419
Thus un is mapped to the cocycle in H' (K., Z/pn) which takes the norm residue symbol (a, KK/Kn) to [un, a]n. By the compatibility of the norm residue symbol with norm and corestriction, one gets that the image of the family u in H' (Km, ZP) is represented by the cocycle (i.e. homomorphism) (a, Kamb/Km) H lim [un, a]n E ZP n->oo
Therefore the reciprocity law says that this homomorphism, and the homomorphism
g --> p-"b (u) log x1(g) represent the same cohomology class in H1(Km, K).
Proposition 3.3.16. [35, III.A7 ex. 2] Let cK : HomQP (K, K) -4 K be the unique map such that for all T E HomQP (K, K) and all x E K, tr([xx] o T) = trK/Q (xCK(T)).
Then the diagram
K
HomQP (K, K) o log
j
Homcts(o*, K) local
2
log Xcycl
CFTt1
Homcts(GK, K)
H'(GK, K)
is commutative.
Remark. Because of the normalisation of the reciprocity law of local class field theory used here (see §3.1), this differs from the statement in [35] by a sign.
Now the composite
HomQP(K,Q)
HomQP(K,K)
CK
K
is the inverse of the isomorphism K -3 HomQP (K, %) given by the trace form. Therefore, for every a E o;,`, im [un a]n = p -m trKm/Q
log a)
which is the "limit form" of the classical explicit reciprocity law [20, Ch. 9, Thm. 1.2].
A. J. Scholl
420
3.4
Big local fields
This section reviews the generalisation by Hyodo [14, esp. §4] and Faltings [10, §2] of Fontaine's theory to local fields with imperfect residue field. We consider fields L D Q, such that:
L is complete with respect to a discrete valuation, and its residue field e satisfies [P : Pp] = p' < oo.
(3.4.1)
Fix such a field L, and write A for its ring of integers. If R C A is any subring, define
PAIR := li f2A/R/P
A/R.
Fix also an algebraic closure L of L, and let A be the integral closure of A in L. For any B with A C B C A and any subring R C B set OB/R = lM PA'/A'nR A'
the limit running over all finite extensions A'/A contained in B.
Let K C L be a finite extension of %, with ring of integers o and uniformiser 7rK. Then 7rK is prime in A if and only if A/o is formally smooth (by [23], (28.G) and Theorems 62, 82). Let L'/L be a finite extension with valuation ring A'. Then A' is finite over A (being the normalisation of a complete DVR in a finite extension), and is a relative complete intersection (by EGA IV 19.3.2). Therefore the first exact sequence of differentials is exact on the left as well' 0 -4 A'0AQA/o - f2A'/o
QA'/A -+ 0.
"More generally, if A'/A is a relative complete intersection of integral domains which is generically smooth, then for any R C A the first exact sequence is exact on the left. For an elementary proof, write A' as the quotient B/I of a polynomial algebra B over A by an ideal I generated by a regular sequence. Then one has a split exact sequence
O -+S2A/R ®B -+ SIB/R 4 B/A -1 0
(1)
as well as exact sequences, for ? = A or R,
I/I2-+ 12B/?®B A' -QA'/?->0.
(2)
Applying the tensor product ®BA' to (1), and using (2) and the snake lemma, gives the exact sequence
NA,/A->A'®A PA/R-4 QA'/R-4 A'/A-+ 0. where NA,/A = ker(I/I2 -+ I1B/A ®B A'). Since A'/A is generically smooth the map I/I2 -+ f1B/A is generically an injection, hence NA,/A is torsion. Now I/I2 is projective since I is a regular ideal; therefore NA,/A = 0.
An introduction to Kato's Euler systems
421
As in (3.3.5), we get an exact sequence of inverse systems p'
A'/A + A' ®A QA/./p" _+ QA'/o/P _+ 0A'/Ale -4 0.
Since QA'/A is a finite A'-module, the inverse system and so passing to the inverse limit gives an exact sequence: 0-+ A'OA QA/o_+QA'/o_+QA'/A-+0.
is ML-zero,
(3.4.2)
Proposition 3.4.3. (i) SZA/o is a finite A-module, generated by elements of the form dy, y E A'. (ii) If Tl,... T,. E A are elements whose whose images in t form a p-basis, then {dlogTi} is a basis for the vector space 11A/o ®A L. (iii) If 7rK is prime in A, then QA/0 is free over A.
Proof. By [23] pp. 211-212, A is a finite extension of a complete DVR B in which p is prime. Then A0 = Bo is a complete DVR with uniformiser 7rK' and A/A0 is finite and totally ramified. Let k = o/7rKO. One knows (loc. cit., Thm. 86) that the image of {dTi} is an 2-basis for Qt/k, and therefore (by Nakayama's lemma) QA 0/0 = ® Ao dTi = ®A0 dlog Ti, proving (iii).
To deduce (i) and (ii), it is enough to apply the exact sequence (3.4.2) to
0
A/Ao/o.
Taking the direct limit of (3.4.2) over all finite extensions L'/L, one gets an exact sequence O-4 A®AIA/o-4 QA/o-4 QA/A-4 0
of A-modules. Now apply (3.3.5) again. Since x dy = pzp-lx dz if y = zp, one sees (using 3.4.3(i)) that QA/0 and HA/A are divisible. Therefore, since f2A/o is finitely generated, one can pass to the limit to get an exact sequence of A-modules 0 -* Tp(IIA/o) _+ Tp(dA/A)
A ®AA/o _+ 0.
(3.4.4)
Because QA/0 is a finite A-module, the map 7r has a continuous set-theoretic
section (write SIA/o = P ® N with P free and N torsion; over A (9 P one has a continuous linear section of a by freeness, and A ® N is discrete, so over it one can take any section). One then has Hyodo's generalisation [14, (4-2-2)] of 3.3.1 (see also [10, §2b)]):
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Proposition 3.4.5. Let ao/o be as in 3.3.1 above, and put aA/o = ao/oA C L. Then there is an exact sequence of A-modules and Galois-equivariant maps
0->aA/o(1) -L(1) 4S2A/o 4L'r-+ 0 where a is given by the same formula as in 3.3.1, and where the map 0 is a split surjection, with right inverse Lr
QA/"
(a1/p',. .. , ar/p") '- E ai dlog(Tp n)
(a i E A).
Remark. Hyodo states this only in the case K = QP, but his proof works in general. The key point (which underlies Faltings' approach to p-adic Hodge theory) is that the extension L/L(Ap-, Tip--) is almost unramified (cf. the proof of Proposition 3.4.12 below), which shows that QA/,, is generated as an
A-module by the forms dlog(T,n, dlogT'
Corollary 3.4.6. There is a unique isomorphism aA/o (1)
TP (Q A/o )
( 3 . 4. 7)
which maps ((pn),f4 E ZP(1) to
Remark. Comparing (3.3.3) and (3.4.7) we have in particular p'//A/o
(3.4.8)
= PnSto/o ®o A.
Now consider as before the connecting homomorphism attached to the Galois cohomology of (3.4.4), for a (not necessarily finite) extension L'/L contained in L: 8L'/L: PA/. ®A A' -+ Hl(L',TPQA/o)
(3.4.9)
For L' = L we write 8L for 8L,/L. If L'/L is finite the maps 8L,, 411L are related by a commutative diagram QA/. (gA A' canonical
PA'/.
1/1_
H1(L','
(3.4.10) A/o)
(because the exact sequence (3.4.4) is functorial in A). If L'/L is infinite, for H1(L',TPQ-A/o) as the direct limit of the maps we define 8L': A,/o finite subextensions L C L" C L'; the analogue of (3.4.10) still holds. The following lemma is proved just the same way as 3.3.7.
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Lemma 3.4.11. For any algebraic extension L'/L, the following diagram commutes: A'*
Kummer
H'(L', µpn)
dlog t 5L, mod pn
A,/O
1
dlog
1
H (L , pn' 2A/0)
Proposition 3.4.12. Let L./L be an algebraic extension which contains all p-th power roots of unity, with valuation ring A., and whose residue field extension is separable. Suppose that r = 1, so that [1: lp] = p. Then for j > 2, Hi (L., TpSZAIO) is killed by the maximal ideal m. C A., and the kernel and cokernel of 4_1L: QAlo ®A AQ -+ HI (Loo,
are killed by a power of p.
Proof. Initially there is no need to make any assumption on r. Choose units T,,. .. , T,. E A* whose images in l form a p-basis. Consider the extensions
M = L(TZ'--,...Tp--) and M. = ML.. Let B, B,,. be the valuation rings of M, M. Then the residue field of Mis perfect, so Tate's theory [37] applies; in particular, the groups H'(MM, A) are m,,.-torsion for i > 0. Therefore, using the Hochschild-Serre spectral sequence and the fact that mC = mom, the inflation map
H'(MM/L., B.) = H'(MM/L., H°(M., A)) -* H1 (L., A)
(3.4.13)
is an isomorphism up to m.-torsion. Now by Kummer theory and the hypothesis on the residue fields, Gal(MM/L.) _- Z,(1)' (the isomorphism being determined by the choice of {Ti}). Therefore if r = 1 H'(M,,./L., BOO(1)) = 0 for all j > 1, and
(3.4.14)
HI(Moo/Loo, Boo(1))
(3.4.15)
(Boo)zp(1)
Now by 3.4.6 there exists a (non-canonical!) isomorphism of Gal(L/L.)modules TpSZA/O - A. Combining this and equations (3.4.14) and (3.4.13), one sees that that H' (LQ,, TTSlAlo) is killed by m. for all j > 1. For the second part, we compute the coinvariants in (3.4.15). First observe that the ring A' = A[Tp-n] is finite over A, and that 7rAA' is a maximal ideal
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424
in it. Therefore A' is a discrete valuation ring, hence is the valuation ring of L(Tp "). It follows that any element of B. has the form b=
baT° aEQ /Zv
where T = T1 and ba E A.-, with ba -* 0 as Jalp - oo. Let y E Gal(MM/L.) be the topological generator for which y(Tl/Pr) _ CprT1/p', for each r > 1. If b is divisible by (1 - (P), then b = bo + (1 - y) Y, where
(1 -
b' _
(P,)-1bx/prTx1P'
E B,.
Oj4x/p'EQQ/Z,,
From this one sees that the inclusion A. C B. induces an injection
A.
H'(MM/L.,
(3.4.16)
whose cokernel is killed by (1 - (P). Now there is a diagram
A.
(3
H1(M.1L., B.(1)) tinfl
1- dlog T
QA/. ®A.
b
Hi (L., A(1)) t(3.4.7)
H1 (L., TPSZA/o)
in which the vertical arrows have kernel and cokernel killed by a power of p (by 3.4.3, 3.4.6 and (3.4.13)). It remains to check that it is commutative, which having got this far is an easy exercise. A similar computation can be carried out for all r > 1, using the isomorphism Gal(M./L.) _- Z,(1)' and the Koszul complex. In this way Hyodo computed the cohomology of L(j) over L, generalising Tate's result. His final result (not needed here) is:
Theorem 3.4.17. [14, Theorem 1] There are canonical isomorphisms H9(L, L(j))
1QA/o®Q ifj=q pAj ®Q if j = q - 1 10
otherwise
compatible with cup-product. For j = q - 1 = 0 it is given by cup-product with logXY,l and for q = j = 1 by (3.4.7) and (3.4.9).
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3.5
425
Proof of Theorem 3.2.3
Theorem 3.2.3 is proved by reducing to the setting of the previous section. Recall that X is a smooth and proper curve over the ring of integers o of a finite unramified extension K/Qp. Assume that X is connected and that I'(X, OX) = o (otherwise first replace K by an unramified extension). Let rl E X be the generic point of the special fibre. Write also: A = OX,1, L = field of fractions of A;
Ln = L(µpn); A. = integral closure of A in Ln; The fields L, Ln satisfy the hypothesis (3.4.1), with r = 1. There is an obvious localisation map 0: Spec A -+ Y. Note that since A/o is formally smooth we actually have A. = A 0 on; and by 3.4.3, SZA/o is a free A-module of rank 1.
Now use the fact that the map ¢* :
Fill HdR(Y/o) = H°(X, Qx/o (log Z))
PA/o
is injective and its cokernel is torsion-free (this holds because the fibres of X/o are connected). This means that the diagram in Theorem 3.2.3 can be localised to Spec A without losing information. We shall write down the localised diagram and then explain why it implies 3.2.3.
Proposition 3.5.1. There exists an integer c such that for every n > 0 the following diagram commutes up to pc-torsion:
(K2(An) 0 i
1)0
ch
H2(LnI, µpn)o
dlogI
I Hochschild-Serre
H1(LK, JJJ
H1(Kn,
QAn/o(-1)
µpn)) 11
HI(Kn, (Ao)*/pn)
ion/o(-1) ®u A/o dlogCp..®[C..]'-4i
on/Dn ®o k/o
I dlog U(1/pn) log XcYcl
H1(Kn, QA/o ®0/pn-1)
Remarks. (i) Since A/o is formally smooth, the valuation ring of LK is simply
A. (ii) We have written H2(Ln, µpn)° = ker (res: H2(Ln, µpn) -+ H2(LK, µpn)) (K2(An) (9 µp 1)0 = ker (ch: K2(An) ®µp 1 -+ H2(LK, lupn))
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426
The map marked "Hochschild-Serre" is then the first edge-homomorphism from the Hochschild-Serre spectral sequence. (iii) Concerning the bottom right-hand corner: the natural map is dlog: (Ao)*/pn -4 pAo/o/pn
but as A/o is formally smooth QA-0/0 = (QA/0 ®o) ED (Qulo (&,, A)
and the second summand is divisible. (iv) To deduce Theorem 3.2.3 from the proposition, it is enough, by what
has already been said, to show that there is a map from the diagram in 3.2.3 to the diagram above. Since the composite K2(Y 0 on) -4 K2(An) H2(LK, ,z) factors through H2 (y (& K, µP 2) = 0, one obtains the map K2(Y 0 on) -4 K2(An)°. The only remaining thing to check is that the diagram
H'(Kn,H'(LK,Ilpn))
H' (Kn, H'(Y (9 K, µp.))
11
H1(Kn, (Ao)*/pn)
1r1 (mod pn)
t dlog
H1(Kn, o(1)/pn) 00 Fill HHR(Y/o)
Hl(Kf, S2A/o ®o/p")
commutes, but this follows from the description of 7r1 given in §3.2.
Proof of 3.5.1. We reduce the diagram to the (smaller) diagrams in the following three lemmas. By (3.4.7), pnQA/o is free over A/pn of rank one, and by pn Ado we mean its tensor square as A/pn-module.
Lemma 3.5.2. For any m, n the diagram below commutes:
K2(Am) - H2(Lm'Np2 dlog 2
PAm /o
n2)
1 H2(Lm, p" A/o
in which the unlabelled arrow is induced by dlog: tip. - pnS2A/o. Proof. Since Am is local the symbol A,*n ® A;, -+ K2 (Am) is surjective. Since the Chern character is compatible with cup-product, the compatibility follows by Lemma 3.4.11.
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427
This reduces the computation to Galois cohomology. Write H2(Ln,p'
H2(LK,P'
ker [H2(Ln,pnS2A/o)
Ado)]
Lemma 3.5.3. (i) The composite map n2
Anl
H2(LK, p, SA/o2 )
H2 (L,,,
equals zero.
(ii) The following diagram is commutative: QOn/o QA/o=
0A.n/o
H2(Ln,pnS2 2 )o
^2
5Kn/K®id j
H1(Kn, pn k/o ®0 PA/0) H1(id®5LK/L)
Hochschild-Serre
I
Hl(Kn,p.4/0 ®o H1(LK,pnSA/,))
H'(Kn,H1(LK,pnS2V ))
(iii) The map H1 (id (DcLk/L) has kernel and cokernel killed by a bounded power of p.
Proof. (i) The cup-product A2SLn factorises as S2 on/o
0
A/JK
o
H'(KnipJo/o ) ® H'(LnipofA/)o -4 H2(Lnipn-/ A o)
and so its composition with the restriction to LK is zero (it factors through H1(K, pnQ5/o) 0 Hl (LK, pJlA/a) = 0). (ii) The bottom equality comes from (3.4.8). The commutativity is a general fact. We have groups
r = Gal(L/Ln) D A = Gal(L/LK),
r/O = Gal(LK/Ln) = Gal(K/Kn)
and two exact sequences of r-modules
0-*A -*B -+ C -+ 0 0-*A'--+ B'-+ C'->0 given by (3.3.6) and (3.4.4) respectively. On the first A acts trivially. So we
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428
have the following diagram
H° (r/0, C) ® H° (r,
6®'
C)
> H' (r/0, A) ®H' (r, A') lu
ker[res: H2(r,A(g A') -+ H2(A,A®A')]
50id
I HS
H'(r/o, A) 0 H°(r, c)
H'(r/o, H1 (A, A 0 A'))
uI T
Hlp
H' (r/A,A®H°(A,C'))
H'(F/A,AOH'(A,A'))
and it is a simple, if tedious, exercise to check this commutes. (iii) Follows from Proposition 3.4.12 applied to L. = LK.
Lemma 3.5.4. The following diagram commutes: dlogodlog
H'(LK(',µp2)
H'(LK,pn Ado)
Kummer Ii
II
(Au)`/pn(1)
pn5?y/o ®o H'(LK, posh/o) lid 06LRIL
dlogI
o/pn (1) (D PA'.
x®Spn0w- xdlogSpn®w
pA/o ®o k/o
Proof. This follows from Lemma 3.4.11.
As K/Qp is unramified, we have a = a./o = (Cp -1)-'o by 3.3.1, so ano = pna and o/(0no a/pn. We now can make a big diagram: (K2An ®7G/pn)° -+ H2(Ln, µp 2)°
.
1
H2(Ln,pn A,o)o
22
/a ®o A/o)
QIn/o ® OA/. ---+ H'(Kn, pn dlogspnHl®Spn ®nA/o
On/an(1)
I H'(Kn,H'(LKrpn6Afo)) II
II
1I1
H'(KK, H'(LK, µP" )) E
(t) H'(Kn
H'(Kn, post./a (9o H'(LK, pJ ZA/o))
if
a/p"(1)
®o nA/o)
H'(KK, (A'5)*/Pn)(1)
t
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To save space we have not labelled most of the arrows: they can be found in the corresponding places in the subdiagrams 3.5.2-3.5.4, apart from the arrow labelled (t), which is U(1/p") log Xcycl. The top left square commutes by 3.5.2, and the top right square by functoriality of the Hochschild-Serre spectral sequence. The rectangle in the middle commutes by 3.5.3, and the bottom left square by 3.3.8. The remaining part of the diagram (the righthand hexagon) commutes by 3.5.4 Going round the outside of the diagram in both directions gives two maps (K2A® ® Z/p")° -* H'(KK, a/p"(1)
®o SZA/o)
and it is enough to show that their difference is killed by a bounded power of p. This follows from the commutativity of the diagram, since the kernel of the arrow marked (*) is killed by a bounded power of p, by 3.5.3(iii).
4 The Rankin-Selberg method In this section we calculate the projection of the product of two weight one Eisenstein series onto a cuspidal Hecke eigenspace, using the Rankin-Selberg integral. In order to separate the Euler factors more easily, we work semiadelically, regarding modular forms as functions on (C -118) x GL2(Af). The passage from classical to adelic modular forms is well-known, but we review the correspondence briefly in §4.2 since there is more than one possible normalisation. The same applies to the discussion of Eisenstein series in section §4.3.
4.1
Notations
G denotes the algebraic group GL2, with the standard subgroups
p =
(0 *) '
U
=
(0 1)
J(0 ,
Z=
a/ }
If R is a ring and H is G or any of the above subgroups, write HR for the group of R-valued points of H. If R C 118 then HR denotes {h E HR I det(h) > 0}. The ring of finite adeles of Q is Af = 7G®ZQ. If X =11 XP : A !f /Q>0 -> C* is a character (continuous homomorphism) and M is a multiple of the conductor of X, then Xmod M : (Z/MZ)* -> C* denotes the associated (not necessarily
primitive) Dirichlet character: for a E t*, x(a) = XmodM(a mod M). Of course this means that if (p, M) = 1 then XmodM(p mod M) = XP(p)-i Write finite idelic and p-adic modulus as I - If, I - IP, and archimedean absolute value as I-I.. If there can be no confusion we drop the subscripts.
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Write also Hf, HP in place of HA1, HQ , and define the standard congruence subgroups
GGDKK=GZ,DKO(pv)={hEKKI h= (0
modpV}
DKi(PV)={hEKpIh°(0 *) mod PV}. Haar measure on all the groups encountered is to be normalised in the usual
way: on Q, the additive measure dx gives Z, measure 1, and on g the multiplicative measure d*x gives 7L; measure 1. On Gf7 GP the subgroups Gi, Kp have measure 1. C* by requiring Fix additive characters 0p : Q -+ C* , Of = fl OP: Af oP(x/ph`) = e21rix/p° for every x E Z.
5 denotes the upper-half plane, and 9j} = C - R. The group GR acts on Fj± by linear fractional transformations. Put j (y, T) = det ry (cr +
d)-1
if T E
}, ry = ( d) E GR
so that j(ry,T)(1,-r)ry 1 = (1,-7(r)). Write S(Af) for the space of locally constant functions Af -+ C of compact
support. The group Gf acts on S(Af) by the rule
0 E S(Af), x E Af.
(gO)(x) = 0(g-lx),
(4.1.1)
If S E AA and ¢ E S(Af ), write [S]0 for the function
[S]¢: X H O(6 ' ).
(4.1.2)
So in particular, if 0 is the characteristic function of an open compact subset X C Af, then [6]0 is the characteristic function of 6X.
4.2
Adelic modular forms
In the adelic setting, a holomorphic modular form of weight k is a function F: S5
xGf -iC
which is holomorphic in the first variable, and satisfies: (i) For every ry E GQ, F(y(T), -yg) = j (ry, T)-kF(r, g);
(ii) There exists an open compact subgroup K C Gf such that F(T, gh) _ F(T, g) for all h E K;
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431
(iii) F is holomorphic at the cusps. Any modular form F has a Fourier expansion
F
T E 5, 9 E Gf mEQ
where am(g) = 0 when m < 0 (this is the meaning of condition (iii)). Put A(g) = al(g), the Whittaker function attached to F. Then A is a locally constant function on G f which satisfies A
(('
i)
g) = Vif(-b)A(g)
for all b E Af.
(4.2.1)
One can recover the remaining Fourier coefficients (apart from the constant term) from A(g) by
\\
am(g)=mkA0Igl if0<mEQ. ((0
It is convenient to introduce the normalised Whittaker function (4.2.2)
A*(g) = A(9)Idetgl fk/2
The group Gf acts on adelic modular forms by right translation, and the translates of F by G f generate an admissible representation, call it 7r. From the definition 7r is isomorphic to the representation generated by A(g); the representation generated by A* (g) is isomorphic to the twist it 0 Idet I f k/2. If 7r is irreducible it has a factorisation 7r = ®'-7rp, and the centre of G f acts on
the space of 7r via a character. With the normalisation used here, there is a (finite order) character e: C* with e(-1) = (-1)k such that
7r(0 a)
=e(a)IaIt
for all aEAf.
This means that g) = e(a)A*(g) A* ((a a) nJewform
for all a E Af and g E G f.
on r1(N) then A(g) is factorisable: there are If F comes from a functions AP: Gp -+ C, satisfying AP(1) = 1, [Ap(gp) = A(g) for all g = (gp) E Gf and such that AP
1)
((a0
a) (0
g)
=
E
EQ. (4.2.3)
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432
Suppose AP is Kp-invariant. Then irp is unramified, and its local L-function is given by Av pr
L (?rp , s) r>°
O 1)p
r(k-s)
A* r>°
p(P'
0
1)pr(k/2-s)
1
(1 - app-s)(1 - Q,p-s)
where ap + all = pk/2AP I
1I
and
apap, = Ep(p)pk-1
(this is the normalisation of L-functions which gives the functional equation for s ++ k - 1 - s; it differs from that of Jacquet-Langlands by a shift). Complex conjugation of Fourier coefficients defines an involution of the space of modular forms. In representation theoretic terms, this becomes the isomorphism it
7r 0 C.
(4.2.4)
If A: Af/Q>0 -4 C* is any character of finite order and F is an adelic modular form of weight k, so is
F 0 A: (r, g) -* A(det g)F(T, g).
(4.2.5)
To go from adelic to classical modular forms, let K(n) be the standard level n subgroup of Gi. Then
G,\.fj} x G1/K(n) = Gz\Sjt x G:1/K(n) = Y(n)(C). where the last isomorphism is normalised in such a way that the point (7, h) E
Sj} x GL2(Z/nZ) corresponds to the elliptic curve E,r = C/(Z + -rZ) with level structure
arh: vH(1/n,-z/n) - h - vv (Z/nZ)2 _+(1Z+ 1Z)/(Z+TZ) = ker[xn]E Write z for the coordinate on E and let F be an adelic modular form which is invariant under K(n). It corresponds to the classical modular form over C (E,r, ac,h) H F(T, h)du®k E HO (E,, Wok)
where h E Gz/nz and h E Gi is any lifting of h. The map (2.3.4) eN : Y(N) -a Spec W(AN) is then given on complex points by eN :
(z, g) r+ o f (± det g/N) if g E G2
(the sign depends on the normalisation of the eN-pairing).
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4.3
433
Eisenstein series
Here we establish notations for Eisenstein series in the framework of the previous section. The results quoted can be obtained easily from those found in classical references (probably [31, Chapter VII] is closest to what is found here). Let 0 E S(Af), with the action (4.1.1) of Gf. The series Ek,$ (0) (T, g) = E (go) (m) (ml - m2T)-k I m1 _ m2T
1-2s
O0mEQ2
is absolutely convergent for k+2 Re(s) > 2, with a meromorphic continuation, and satisfies Ek,e(o)(7T,7g) = ('y, T)-kI (7, T)
I-2,Ek,,(O)(7,g)
for all 7 E GQ.
The functions Ek(4) := Ek 0(q) are holomorphic (and therefore modular of
weight k) if k > 3 or k = 1; if k = 2 they are holomorphic provided that fA,0=o.
The map 0 -* Ek , (q5) is G f-equivariant. In particular, if b = d E q, then in the notation of (4.1.2) (4.3.1)
Ek,. ([dj0) = d-k I dI3 sEk,s (o).
One can rewrite the Eisenstein series as a sum over the group. If f : G f -* C is a locally constant function satisfying
f ((a
d) g) =a klal-9f(g) for all a,dEQ*,bEA1
(4.3.2)
then define
Ek,,,f(T,g) = E f(7g)j (7,T)kIj (7,r)I28.
(4.3.3)
ryEPQ \GQ
The relation between the two definitions is that Ek, (¢) = Ek,,, f with
(go) (o) x-kIxl s.
f (g) _ XEQ
Moreover every Ek, f is an Ek,, (cb) for some ¢. In the normalisation used here, the Whittaker function of Ek (0) is
B(g) =
(27rz1)I
l0
E Yzv
f Of(-x/y)(9o) (y) Af
dx
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which can be obtained without too much difficulty from the classical formulae - see e.g. [31, pp.156-7 & 164ff.). One can decompose the Eisenstein series under the action of the centre of Gf. It is more convenient to replace f and B by the normalised functions (cf. (4.2.2) above) f(g)Idetglf(k+zs)/z,
f*(g) =
B(g)Idetglfk/2
B*(g) =
and then to write
B(g) _ I:BX (g)
f * (g) _ : f; (g),
x
X
where the functions ff(g), BX(g) are zero unless X(-1) = (-1)k, in which case 1_1
ff(g) = f X(a)f*
(a0
a-,) g) d*a
A f /Q">o
= 21detgl f(k+zs)lz
fx(x)IxI28(g)()
d*x
Af
and
BX(g) =
f
X(a)B*
a0'1
Af /Q">o
_2((2-ri1)fldetglfk/2
a0 -1) g) d*a
f
X(y)Iylf+zs-lVf(-x/y)(go)
(X)
Af XAf
dx d*y s=0
If ¢ = 11 OP is factorisable, with OP equal to the characteristic function of 7G2P for almost all p, then the expressions above factorise and one has fX (g) = 2 rj fXp (9P),
BX(g) = 2
k ((27ri1)!
11 BXp (gP)
for g = (gp) E Gf
where the functions fXp, BXp are given by local integrals fXp(gP) = IdetgPlp(k+2s)/2f QP*
W d*x
(4.3.4)
An introduction to Kato's Euler systems
BXp (gP) = Idet 9Pl p
k/2
435
(g'O')
f XP(y)
(x)
dxd*y s=O
QxQ
= Idetggl-k/2
f
XP(y)lylk+2,OP(-x)(gPOP) (7) dxdyQ
Qs=0
x
(4.3.5)
In fact the integral in (4.3.5) is a finite sum, because the x-integral is a finite linear combination of integrals of the form
J t+Pvap
V),(-x/y) dx
which vanish if y is sufficiently close to 0. So one can omit s from the formula. Because of (4.3.2) and (4.2.3) the functions fXp and BXp are determined
by their restrictions to the subgroup COP
)K,CG,
and these are given by 00)
XP(m)Iml(k+23)/2fXp(h)
h) =
0)
JJ h111 l =
(4.3.6)
(x)
ImI-k/2+1
QxQ (4.3.7)
= Iml -k/2+1 n
f
(7) dxd*y
J Q xQ
(4.3.8)
4.4
The Rankin-Selberg integral
Let F, G be adelic modular forms of weights k + 1, k respectively, at least one
of which is a cusp form, and let E181 be the Eisenstein series (4.3.3). The product El,s,fGFyk+l+s-2Idet
g)-k-t-sdr
A dT = is a left G+-invariant form on Si x Gf, and the aim of this and the following sections is to compute the inner product SZ
(Et s, fG, F) :=
J
GQ\sxGf
Q dg
A. J. Scholl
436
which is a Rankin-Selberg integral.
Proposition 4.4.1. Let A(g), B(g) be the Whittaker functions of F and G. Then (E1
s
fG, F)
ir(k+1+s-1)
_
(4 .)k+t+/s-1
f ((0 1) h)B(
x AA xGy
\\
?)h)A(( p
\\
/
01
I h I imI -k-1-s-1d*m dh.
Proof. A very similar calculation is done in [30, §5]; here we simply write the equations with little comment:
f
(E1,s,fG,F) =
E
f(7g)j(y,T)1Ii(ry,T)Its
GQ\S3xGf 7EPQ\GQ g)yk+1+s-2ldet gI fk-'-adT
x G(T, g)F(T,
= -2i f f (g)G(r, g)F(T,
g)yk+1+s-2ldet
A dT dg
gl fk-l-sdx dy dg
PQ \fjxGf
= -2i
f
f (g)
PQ\fjxGf
_ -2i
BA ((MO mEQ >o
g) 01)
x e-47rmyyk+l+s-2ldet gI f k-l-sdx dy dg
f
fk-l-sdxdy
dg
ZQUQ\S3xG f
f
2i1'(k + l + s - 1)
(4 .)k+1+s-1
dx dg f (g)B(g)A(g)Idet gl -k-l-, f
ZQNQ\RxG f
To get the final result, use the parameterisation
ir: Af x AA x Gj/{±1} -4 Gf/ZQ 1
(b, m, h) t-a (0
1) (0
0) h
in terms of which integration is given by
f Gf/ZQ
fi(g) dg =
f AfxAfxGy/{f1}
(7r*fi)(b,m,h)lyl-1dbd'mdh
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437
Since f (g)B(g)A(g) is invariant by g H I
above splits as a product
J
dx db
Q\RxAf
1
b
I g, the integral in the last line
\\\
f BA
J
((r
I
h) d*m dh
AxGtl{ti}
and the first factor equals 1 by choice of Haar measure.
0
Now suppose:
F is a cusp form, belonging to an irreducible ir = ®'irp, with central character e, whose Whittaker function A(g) = f A,(gp) is factorisable; G =7Ek(q5'),/is an Eisenstein series and f = f4, for factorisable functions 011Op,`Y'=rj OpES(Af).
Then the integral in the previous proposition can be decomposed under the action of the centre and then factorised, giving:
Proposition 4.4.2. Under the above hypotheses: (El,$(0)Ek(cb'), F) = C E J1 II(Xp, XP') x,x'
ik-lr(k + l + s - 1)
where C = 2k+21+2s-4(k - 1)! Ip(Xp, Xp) =
and
f fXpBXDAp I Qp* xKp
P
I0
\\\\
h I jmj-ld*m dh ///
1111
and ffp, Bx*, are as in (4.3.4), (4.3.5) above. The sum is over all pairs of
characters X, X': Af/%,o -+ C* such that XX' = e and X(-1) = (-1)1. Remark 4.4.3. It will become clear in the computation that follows that the sum over characters is actually a finite sum.
4.5
Local integrals
Write char[y] for the characteristic function of a subset X x Y C A. The next proposition will compute the local integral II(Xp, XP) for almost all primes.
A. J. Scholl
438
Proposition 4.5.1. Suppose that Op = 0, = Op := char [ZPJ
and that AP is Kp-invariant, with AP(1) = 1. Then Ip(x" Xp) =
if XP is unramified otherwise.
0
Remark 4.5.2. Since 7rp has a Kp-invariant vector, s is unramified at p. Therefore since XX' = s, either both or neither of XP, XP are unramified.
Proof. (See [15, §15.9].) If 4p = p = OP then by (4.3.6) for h E Kp, m E Q,p
0)
Xp(m)ImI(d+2s)/2
h)
I
Xp(x)Ixll+2sd*x
Zp-J{0}
- L(Xp, l + 2s)
if XP is unramified otherwise.
0 Moreover by (4.3.7)
BXp
((m 0
0) 1
f
h/J = Iml1-k/2
X%(y) I yl '-lip(-mx/y)dx dy
ZyxZP {0}
where the x-integral equals 1 is m/y E 7Gp and vanishes otherwise. The yintegral then vanishes if Xn1ZP # 1, giving
p B*xp
(( 0
rk/2 E XpPI O<j 0 and Xp is unramified otherwise.
0
(4.5.3)
Thus I'(Xp, XP) = 0 unless both x, x' are unramified at p, which we now assume. Then ffp, BXp are Kp invariant and
I'(Xp, XP') = Eprf*PB*, rEZ
= L(Xp, 1 +
( 0r
2s) E r>o
0) BXp AP
(pr 0
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439
Now from (4.5.3)
Tr E B*x, (pr0 0) 1 r>o
1 (1 - X'P(p)p-k/2T)(1 - pk/2-17 ' )
and therefore by [15, Lemma 15.9.4] one gets
I ( xP , x') = L(XP 14- 2s) Since E = XX' and Tr
L(jrp®x
x', k+l+s-1)L(,rP®xP,l+s) L(EP 1XP P, l+ 2s)
7r ®E-1 (4.2.4) the result follows.
Corollary 4.5.4. Under the hypotheses of Proposition 4.4.2, let S be a finite set of primes such that, for every p S, OP = OP = Op, AP is KP-invariant and AP(1) = 1. Then (El,s(0)Ek(c5'), F) =
Ls(ir0 X'-1,1+ s) 11 IP(XP,xp) PES
X,X'
where the sum is over characters X, X' unramified outside S, with XX' = E
and X(-1) = (-1)l. Here Ls denotes the L-function with Euler factors at all p E S removed. At other primes we use the following choice for OP:
Proposition 4.5.5. Let t E QP with vp(t) = -v < 0. Suppose that 6P = Op 1,t "t := char Lt
Let m
QP', h = (a
m0
1x
(( 0 1h)
db
+ ZPI ZP
) E KP. Then
11 (1
\ 0
1P)-pv(1+2s-I)XP(amt)jMj1/2+s
if cond XP < v and h E K0 (p") otherwise.
Proof. Straightforward calculation from (4.3.4).
At the bad primes for F we are going to choose ¢p in such a way as to make the local factor be simply a constant.
A. J. Scholl
440
The standard way to achieve this is to use a suitable Atkin-Lehner operator to replace the coefficient of q' in the q-expansion by zero whenever p1n. In representation-theoretic language, this means to use the vector in the Kirillov model which is the characteristic function of z;. (See [6, Thm. 2.5.6], and also compare [30, 4.5.4].) For Eisenstein series it is easy to write down a parameter 4'p which does the trick, although possibly this does not give the best constant in 4.6.3 below.
Proposition 4.5.6. Suppose 0P't'
- char
- 1 char [ti + p*
Lt' + 7Gp1
IL
L
zp i
p
where t' E q, vp(t') _ -µ < 0. Then for all h =
pJ
n
(a
c d) E Ko(p,,+1)f1Ko(p2)
X'p (-amt') BXp ((0
f Xy(y)-'OP(y) d*y n' µz; ifcondX' :5 pand mE7Gp
?)h)=
otherwise.
0
Remark. In the special case p = 0, this becomes
p=
O2,1
= char I ZPJ - 1 char p Zp P
L zn
J
and for all h E Ko(p2) and all m E qP
xp ((m B* 0
O) h) = 1
J
is unramified and Imi = 1
1
if XP
0
otherwise.
Proof. First consider what happens when Y'p = char h E K0 (p µ+1) one has hO, = char
t
7Gp .
For every
p
[at' + 7Gpj
ZpJ
which gives
Bx1 ((0 m 10) h) =
Iml1-k/2
f
(at'+Z y) X Zy
%(y)+,p(-mx/y) dx d*y
(4.5.7)
An introduction to Kato's Euler systems
441
and the x-integral vanishes for m V Z,, and equals gyp(-amt'/y) otherwise. Therefore for m E Z, (4.5.7) becomes ml1-k/z I
f Xp(y)Op(-amt'/y)d*y Zp
= Imll-k/2XP(-amt') f XP(y)-10p(y) d*y mt'Z;'
Now if B: Gp -+ C is any Whittaker function (i.e. satisfies (4.2.1) and is locally constant) which is invariant under K1(p"), some v > 1, then the function 1
B=B- 1
10 p11 B
pxmod p
satisfies, for every h E Ko(p") n Ko(pz),
1IhI =charz;(m)BII 0 1I h). Since
'Zp)
(t' +p
P-1 x
TTTT
x Z* = L1
xmodp
[(t, + Zp) x z,*]
0
the result follows.
From Propositions 4.5.5 and 4.5.6 one obtains:
Proposition 4.5.8. Suppose that AP is invariant under K1(p") and that AP(1) = 1. Let t, t' E % with vp(t) = -v, vp(t') = -µ and v > y > 0, v > 2. Then if ¢p = Op,t and Op' = OP>t',
( I, (x" Xp) =
\1
1
pz)
p"(1+zs-z)Xp(t)Xp(-tl) f Xp(y)-10p(y)d*y n °Zp
(0 Remark. Since E = XX' one has cond Xp
if cond XP'
µ
otherwise.
u
cond Xp < v.
Proof. If cond% > p then I,(Xp,X%) = 0. Otherwise, if h E K0 (p') and AP is K1(p")-invariant, one has A;(h) = sp(a)AP(1) = X,Xp(a)Ap(1), so that
A. J. Scholl
442
IP(xp, xP) equals 1
(1 -
pv(t+2s-1) f
D-
x (y)-10P(y) d*y J
P-'-ZP
= vol Ko(pv) (1
xp(at)xp(-at')A;(h) dh
Ko(P') 1
-
pv(c+2s-1)
f xP(y)-1' P(y) d*yxp(t)Xp(-t')Ap(1)
P-µZ;
and vol Ko(pv) = [Kp: Ko(pv)]-1 = (1 + 1/p)-1p-v.
There is just one more case to consider, in order to compute the image of the Euler system in the cyclotomic tower.
Proposition 4.5.9. Suppose that `gy'p=(O1,t)tr=char
m0 x;
((0
h)
-
(1
-
,
1
1
)pu(k_2)ImI(1_k/2)X(dtI)p(_mb/d)
1)
if cond XP < v and vP(m) > -v otherwise.
0
Proof. One has hoP' = char by
[btdt' + '
(4 3 7) .
vv(t')=-v µp > 0, vp > 2, and AP is K1(p"P)-invariant.
(iv) For each p E T, elements tp, tp E Q*p with vp(tp) = vp(tp) = -vp where vp > max(cond AP, 1).
Put
N=flp'p, M=11 p,'P, R=flp"P. pES
pES
pET
Denote by t, t' the finite ideles whose components at primes p E S U T are tp, tp, and which are 1 elsewhere. Pick y E 7G such that
__
tp
Ntp
y
mod p"P
for all p E S
(4.6.1)
(note that the right-hand side belongs to Z) - thus y is well-defined mod M. The integral to compute is (El,s(c5)Ek(cb') ® A-', F) = (El,$(O)Ek(c'), F (9 A)
where F ® A(T, g) = A(det g)F(T, g)
is the twist of F by A, and ¢, 0' are given as follows: For p E S, lop =
For p E T, Op =
4'
and 4p = Op'tP. Y'p'tP and 4p
ForpSUT,¢p=Op=¢P.
=
,
(,/pl'tP)tr
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445
We can then assemble the previous calculations. Put x = eMB and x' = AB-1 for a variable character 0 - thus xx' = eA2, the central character of A* ®a. Then only those 0 satisfying the following conditions contribute to the sum:
9(-1) = (-1)kA(-1); If p V S then Bp is unramified; If p E S then cond OP < µp. So cond 9 M, cond e l N and cond )AIR. This gives:
x
LSUT(lr(99,l+s) II I'(xp,xp) B(-1)=(-1)ka(-1)
PESUT
cond 01M
by 4.5.4, 4.5.8 and 4.5.9, where N1+2s-2
IP (XP, XPpl) =
pES C1
PES
- p21
tP,) f
XP(tP)XP(
XP(y)-1
P(y) d*y
p 'PZP
with Ap(p)-AP-VPep(tp)0
fJ xp(tp)xp(-tp) = H
(-tP/tp)
pES
pES
=
emodM(y)[Jep(tp)9p(p)'PAp(MN)-1,
pES
and
II Ip(xP, xp) _ PET
Rk+l+2s+4
[
xp(tp)xp(tp)L(irp ®6p, l + s)
C1
p2/
pE
and for each p E T, xp(tp)xx(tp) = eP(tp))p(tptp). This gives (El,s(0)Ek(0'), F (9 A) = Nl+2s-2Rk+1+2s-4 fi (1
C
- p2)H 1 -
pESUT
PET
xe(t)[JA pET
x
E
(io'`P f x (y)-1'Ip(y) d*y
0(-1)=(-1)ka(-1) PIM cond Ol M
X OmodM(y)
pl'Pz; -L,q(7r00,1+s)).
(4.6.2)
A. J. Scholl
446
In the third line of this expression, the product over pIM can be rewritten in terms of a classical Gauss sum as W(M)-1
M(x)e2arix1M
Bmod
xE(Z/MZ)`
(Here cp is Euler's totient function.) The sum over characters B in (4.6.2) then becomes (combining odd and even characters) [1 + (-1) cO) (-1)1 condo M xE(Z/M7L)`
m>1 (m,N)=1
X BmodM(xy-1m 1)e2rix/Mamm-l-s
[Of (my/M) + (_1)'\(_1), f(_my/M)] amm-t-s m> 1 (m,N)=1
2
by the character orthogonality relations. For a E Af write Ls(7r, s; a) =
f(ma)amm-s
m>>1
(m,N)=1
for the twisted Dirichlet series.
Theorem 4.6.3. Under the above hypotheses CNl+2s-2Rk+1+2$#GL2(7L/RZ)-1
(E1,,(¢)Ek(O') ® ), F) =
//
X e(t) f Ap(MNtptp) f I 1 pET
pES
\\\
\ a) p
LSUT(7r ®A, l + k + s - 1)
x (LS(7r, l + s; y/M) + (-1)'k.\(-1)LS(7r,1 + s; -y/M)) with C as in Proposition 4.4.2.
5 5.1
The Euler systems Modular curves
We can at last give Kato's construction of an Euler system in the Galois cohomology of the modular curve Y(N) over a family of abelian extensions of Q. We assume throughout that p is a prime not dividing N.
An introduction to Kato's Euler systems
447
Pick auxiliary integers D, D' > 1 which are prime to 6Np, and put RP = {squarefree positive integers prime to NpDD'}
RP={r=ropmI roERP, m>1} We suppose that, for each r E RP, we are given points zr, Z. E £°"'v(Y(Nr)) ^J (7G/Nr)2, such that:
If r and rs E R,, then sz,(') = z,(.') - i.e., one has elements of the inverse limit
(zr)r, (zr)r E lm ker[xNr]
7G2 X
rER.P
fl
7G/.f
t%NPDD'
of torsion points on the universal elliptic curve.
For every r E RP, the points Nzr and Nzr generate ker[xr] (in particular, the orders of zr, z' are multiples of r). If r = pm then the orders of zr, z;. are divisible by a prime other than
p Remarks. (i) The first condition implies that there exists e E 7GP such that for every r = ropm E RP, the Weil pairing of zr and zr is eNr(zr, Zr)
3) the scheme X (N) is regular. Therefore the morphism eN: X (N) --f Spec Z[/N] is a local complete intersection (being a flat morphism of finite type between regular schemes, EGA IV 19.3.2). Therefore the sheaf of relative differentials extends to an invertible sheaf on X(N)/z, namely the relative dualising sheaf (sheaf of regular differentials), and one can then show that (1/N)KSN extends to an isomorphism of invertible sheaves on all of X(N)/z NKSN: w2 -+ SZX(N)/z[I`N](log cusps).
This is not needed in what follows.
A. J. Scholl
450
Because Y(N)°rd is smooth over Z[µN], one has Y(N)ord/Z _ Y(N)ord1Z[,LN) ®,LZ'IAN1IZ
and Q1Z[aN11Z is killed by N and generated by dlog((N).
Proposition 5.1.5. Let z, z' E EUR1"(Y(N)lz) be disjoint from ker[xDD']. In s22Y(N)ord/z the identity dlog{t9D(z),19D,(z')} = 1 KSN(DEis(z) D,Eis(z'))
0dlogeN(z,z')
holds.
Proof. This can be checked on q-expansions. Suppose that on the completion of Y(N) along a cusp we have fixed an isomorphism of Eu"v with the Tate curve Tate(q) over Z[1AN]((g11N)), and that z, z' are the points z = (Na ga21N, z' = (Ngbz1N. Applying the congruence 1.3.4 and the fact that eN(z, z') = bl-aib2 one get the desired result. (We have normalised the eN-pairing as 01 in [18, (2.8.5.3)].) We can now give Kato's description of the image of the Euler system {ir} under the dual exponential map (see §3.2 above)
expp: H1TP,N) Q ®QQ(µr) ®Q Fil' HHR(Y(N)/Q) recalling that Fill HHR(Y(N)/Q) = H°(X (N),1lX(N)IQ(log cusps)).
Define the following differentials on the modular curve in terms of the weight 1 Eisenstein series:
wr = 1 KSNr(DEis(z,,) D,Eis(z')) E H°(X(Nr),1l1(log cusps)). wr = trX(Nr)IX(N)®Q(A,,,) wr E H°(X (N) 0 Q(µr),121(1og cusps))
(5.1.6)
Theorem 5.1.7. For every r E RP, expp* r =
e
e Wr
r
where e E 7Gp is as in (5.1.1).
Proof By 5.1.5 we have in H°(X (Nr)ord QX(Nr)Iz(log cusps)) the identity dlog Ur = wr ®dlog eNr (zr, zr).
Now take r = r°pm and tensor with Z1,. Then by (5.1.1) dlogQr = ro lever 0 dlog( m E H°(X(Nr)ord 0 Z,, 112 (log cusps)).
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451
Taking the trace to X (N) ® Q(µ,,) gives, using the compatibility (§2.1) of trace and transfer dlog v,, = ro le w,, ® dlog 0 and (M, y) = (M, p) = 1, and for which (5.2.4) is non-zero. By what has been just said, there will be a finite
An introduction to Kato's Euler systems
455
collection of such y/M which will cover all possible choices of p. We then take
N= rl q"9,
vq = max(2, ordq(NE), ordq(M) + 1).
gIMNE
Step 3: Fix auxiliary integers D, D' > 1 with (DD', 6pNE) = 1 and D D' - 1 (mod M). Let r = ropm E Rp; thus m > 1 and ro > 0 is squarefree and coprime to pDD'N. In the notation of §4.6 we put R = r, T = {qlr}, S = {qIN} and choose the ideles t, t' E AA to have local components
t =
f1
ifq%Nr (Nr) -1 if gINr
t'q _
'
1
if q%Mr
-r-1ylMl q
if q1M
(Mr)-1
if qjr
Then (4.6.1) holds, and t E (Nr)-17G*, t' E (Mr)-17G*. In §4.6 this data then determines functions 0, ¢' E S(Af). Let 6 E Z* be the finite unit idele
D if qjNr;
S =
11
otherwise
and set, by analogy with (1.3.2),
DO=D20-D[6]0 in the notation of (4.1.2). Likewise define 6' and D'4' in the obvious way. Since (Nr, D) = 1, if cond(A)I r we have (5.2.5)
A(6) = 11 )q(D) = Amodr(D). qjNr
Step 4: We have
char[(t + 7L) x 7L] =
char[(Nr)-1
+ 7G x 7G]. Choose
zr E E°°'°(Y(Nr)) to be the point which in complex coordinates is
Nr
E (N-Z+ Nr9G)/(Z+TZ)
(Z/NrZ)2.
For different r the points Zr are compatible: fzer = zr. We then can use (1.3.3) to write the Eisenstein series in terms of the complex parameterisation as DEis(zr) = El (DO) du.
(5.2.6)
A. J. Scholl
456
Step 5: The function ¢' has local components if q%Nr; if qjr;
char[zq X Zq]
char[Zq x (1/Mr + Z )] q rq
char
- qchar ftq +z;'Zqj
Z+*Zq]
J
q
q
L
if q1N.
J
The last expression can be rewritten as char
[tq
Zq] - [q]char 1q
tq + q
+9
Zq]
q
- q-'char
r
1
rq + q-'Zq] + q-' [q]char I q-'tq + q J Zq Z9
1
2Zgl
((
Now by (4.3.1) there exist a finite set of points zr,.1 E Funiv(Y(Nr)) and constants bj E N-'Z which are independent of r, such that
E bj i
D,Eis(zr,i)
= E, 64) du
Moreover the differences z, i complex coordinates Nz,,j will be the point and Qzl,,,j =
zr,j.
- zr i will be N-torsion, and in
(-Nt' mod 7G)T E (-Z + -Z)/(Zi + TZ) ^' (Z/rZ)2. It follows that eN,(z,, z', 7 ) =
(Mro) P
x (prime-to p root of 1)
and thus that the constant e of (5.1.1) equals (-M-') E z;.
Step 6: Put Qr j =
(z,), t9D, (zr j) }, and let er,j(E) be the associated Euler system for TP(E). The required Euler system is then cr =
i
bier>i(E) E H'(Q(IAr),TP(E))
We can now compute the dual exponential of cr. Put W,,, for the differential on Y(Nr) constructed from (Zr, The Kodaira-Spencer map takes (dt/t)®2 to dq/q, and therefore du®2 to (27ri)-'d7-. Therefore bi
(2N)_l r El (DO)El(D,q') dT.
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457
We then get mod, (7)
exppcrr
7EGa1(Q(/A.)/Q)=(Z /rZ)*
(-M-lhE
bj J
f (AmodNr o eNr) ' Wr,j A (PE,Nr'E WE Y(Nr)(q
_ hE#GL2(Z/NrZ)
(,
MNr2S2EflE
F)WE.
By Theorem 4.6.3, taking k = 1 = 1 and s = 0, (E1(0)E1(gY) ® A, F) =
C'r-2#GL2(Z/rZ)-1 TT Aq(MNtgt'q)-1 qtr
x LNr(E) Amodr, 1)(LN(E,1; y/M) - A(-1)LN(E,1; -y/M)) for some C' E Q*, depending only on E, M and N. Moreover, using (5.2.5) and the hypothesis that D D' - 1 (mod M), (El(DO) El(D'O') ® A, F) =
C'r2#GL2(Z/rZ)-1
x [JAq(MNtgtq)-1DD'(D - Amodr(D)-1)(D' qlr x LNr(E, Amodr, 1) (LN(E, 1; y/M) - A(-1)LN(E,1; -y/M)). Amodr(D')-1)
Now for qI r we have tgtq = (MNr2)-1, so rjqjr Aq(MNtgtq) = 1 since cond(A)jr. Combining everything one gets the final result:
Theorem 5.2.7. Let E/Q be a modular elliptic curve of conductor NE. Fix a non-zero 1-form WE E '(E/Q), with real and imaginary periods SZE, Q-E' Let p be a prime not dividing NE. Then there is an integer M prime to
p, and for every pair of integers D, D' > 1 with (DD', 6pNE) = 1 and D - D' - 1 (mod M) an Euler system: cr = cr(E, p, D, D') E H1(Q(µr),Tp(E)),
r = ropm, ro squarefree and prime to pMNE, m > 1
such that for each r and each character A: Gal(Q(µr)/Q) ^ (Z/rZ)* -a C* with A(-1) = ±1
E 'YEGa1(Q({+.)/Q)
A('Y) expn c, _ (E, A, 1) L CE DD'(D - A(D)-')(D'- A(D')-1) rMNQ}
WE
E
for some constant CE, depending only on E.
In the special case r = ptm this is (with minor modifications of notation) Theorem 7.1 of [29].
A. J. Scholl
458
References [1] A. A. Beilinson: Higher regulators and values of L -functions J. Soviet Math. 30 (1985), 2036-2070
[2] - : Higher regulators of modular curves. Applications of algebraic Ktheory to algebraic geometry and number theory (Contemporary Mathematics 5 (1986)), 1-34 [3] S. Bloch, K. Kato: Tamagawa numbers of motives. In: The Grothendieck
Festschrift, Vol. I. Progress in Mathematics 86, 333-400 (Birkhauser, 1990)
[4] R. Coleman: Hodge- Tate periods and Abelian integrals. Inventiones math. 78 (1984) 351-379
[5] P. Deligne: Formes modulaires et representations .£-adiques Sem. Bourbaki, expose 355. Lect. notes in mathematics 179, 139-172 (Springer, 1969)
[6] - : Formes modulaires et representations de GL2. In: Modular functions of one variable II. Lect. notes in mathematics 349, 55-105 (Springer, 1973)
[7] - : Courbes elliptiques: formulaire. In: Modular functions of one variable IV. Lect. notes in mathematics 476, 79-88 (Springer, 1975) [8] -, M. Rapoport: Les schemas de modules des courbes elliptiques. Modular functions of one variable II. Lect. notes in mathematics 349, 143-316 (Springer, 1973)
[9] C. Deninger, A. J. Scholl: The Beilinson Conjectures. In: L-functions and Arithmetic (ed. J. H. Coates, M. J. Taylor), Cambridge University Press (1991), 173-209 [10] G. Faltings: Hodge- Tate structures and modular forms. Math.Ann. 278 (1987) 133-149
[11] J-M. Fontaine: Formes differentielles et modules de Tate des varietes abeliennes sur les corps locaux. Inventiones math. 65 (1982), 379-409
[12] - : Representations p-adiques semi-stables. In: Periodes p-adiques. Asterisque 223 (1994) [13] R. Hiibl: Traces of differential forms and Hochschild homology. Lect. notes in mathematics 1368, 1989
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[14] O. Hyodo: On the Hodge-Tate decomposition in the imperfect residue field case. Crelle 365 (1986), 97-113 [15] H. Jacquet Automorphic forms on GL(2) H. Lect. notes in mathematics 278 (Springer, 1972)
[16] K. Kato: Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via BdR. In: Arithmetic Algebraic Geometry. Lect. notes in mathematics 1553, 50-163 (Springer, 1993)
[17] - : Generalized explicit reciprocity laws. Preprint, 1994 [18] N. Katz, B. Mazur: Arithmetic moduli of elliptic curves. Ann. of Math. Studies 108 (1985)
[19] D. Kubert, S. Lang: Units in the modular function field H. Math. Ann. 218 (1975), 175-189 [20] S. Lang: Cyclotomic Fields I and H. 2nd edition, Springer-Verlag, 1990
[21] J. Lipman: Traces and residues of differential forms via Hochschild homology. Contemporary Mathematics 61. American Math. Society, 1987 [22] J-L. Loday: Cyclic homology. Springer-Verlag, 1992
[23] H. Matsumura: Commutative Algebra. 2nd edition, Benjamin/Cummings, 1980
[24] J. S. Milne: Arithmetic duality theorems. Academic Press, 1986 [25] D. Mumford: Picard groups of moduli problems. In: Arithmetic Algebraic Geometry (0. F. G. Schilling, ed.). Harper and Row (1965)
[26] A. Ogg: Rational points on certain elliptic modular curves. Proc. Symp. Pure Math. 24, 221-231 (AMS, 1973)
[27] M. Rapoport, N. Schappacher, P. Schneider (eds.): Beilinson's conjectures on special values of L -Junctions. Academic Press, 1988
[28] G. Robert: Concernant la relation de distribution satisfaite par la fonction cp associee a un reseau complexe. Inventiones math. 100 (1990), 231-257
[29] K. Rubin: These proceedings, 351-366 [30] N. Schappacher, A. J. Scholl: Beilinson's theorem on modular curves. In [27], 273-304
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[31] B. Schoeneberg: Modular forms. Springer, 1974
[32] A. J. Scholl: Higher regulators and L-functions of modular forms. Book in preparation [33] P. Schneider: Introduction to the Beilinson conjectures. In [27], 1-35 [34] J-P. Serre: Corps locaux. Hermann, 1968
[35] - : Elliptic curves and abelian f-adic representations. Benjamin/Cummings, 1968
[36] G. Shimura: On the periods of modular forms. Math. Annalen 229 (1977), 211-221
[37] J. Tate: p-divisible groups. Proceedings of a conference on local fields, Driebergen (T. A. Springer, ed.), 158-183, Springer-Verlag (1967)
[38] - : Relations between K2 and Galois cohomology. Inventiones math. 36 (1976), 257-274
[39] E. Whittaker, G. L. Watson: A course in modern analysis. 4th edition, Cambridge, 1958 DEPARTMENT OF MATHEMATICAL SCIENCES, UNIVERSITY OF DURHAM, SOUTH
ROAD, DURHAM DH1 3LE, ENGLAND
[email protected] La distribution d'Euler-Poincare d'un groupe profini JEAN-PIERRE SERRE a John Tate Lorsqu'on desire calculer des groupes de cohomologie, it y a interet a disposer de resultats generaux simples, du genre "dualite" ou "formule d'EulerPoincare". Un exemple typique (du a Tate, [14], §2) est celui de la cohomologie
du groupe profini G = Gal(K/K), on K est une extension finie de Qp. Un autre exemple (du a Lazard, [7], p.11) est celui oil G est un groupe de Lie p-adique compact sans torsion. Pour calculer des caracteristiques d'Euler-Poincare dans d'autres cas (par exemple celui d'un groupe p-adique compact pouvant avoir de la torsion), it est commode de definir une certain distribution PG,p sur le groupe G considers (p designant un nombre premier fixe). Cette distribution est la distribution d'Euler-Poincare de G. Tout revient ensuite a determiner PG,p, par exemple a montrer que c'est 0 pour certains couples (G, p). C'est la 1'objet du present travail. Les principaux resultats sont resumes au §1 ci-apres.
Table des matieres §1. Enonce des resultats §2. Caracteres de Brauer : rappels §3. La distribution PG §4. Representations Zp lineaires et Qp lineaires §5. Quelques proprietes de PG §6. Exemples galoisiens §7. Groupes de Lie p-adiques §8. Une application Bibliographie
Reprinted from `Galois Representations in Arithmetic Algebraic Geometry', edited by A. J. Scholl & R. L. Taylor. ©Cambridge University Press 1998
462 465
469 474 480 483 485 490 492
462
Jean-Pierre Serre
§1. Enonce des resultats 1.1. Notations La lettre p designe un nombre premier, fixe dans tout ce qui suit. On note G un groupe profini (cf. [12]), et UG 1'ensemble des sous-groupes ouverts normaux de G. Le groupe G est limite projective des groupes finis G/U, pour U E UG. Un element s de G est dit regulier (ou "p-regulier") si ses images dans les G/U (U E UG) sont d'ordre premier a p. Cela revient a dire que 1ordre de s dans G (au sens profini, cf. [12], 1.1.3) est premier a p. L'ensemble des elements reguliers de G est note Greg. C'est une partie compacte de G. On a Greg = lim(G/U)reg
1.2. G-modules et caracteres de Brauer On note CG(p), ou simplement CG, la categorie des G-modules discrets qui sont des Fp-espaces vectoriels de dimension finie. C'est la limite inductive des categories CGIu(p), pour U E U. Si A est un objet de CG (ce que nous ecrirons "A E CG" ), on note SPA: Greg 4 Zp
le caractere de Brauer de A (n03.3). C'est une fonction localement constante sur Greg.
1.3. Cohomologie Si A E CG, les groupes de cohomologie Hi(G, A) sont des Fp espaces vectoriels, nuls pour i < 0. Nous ferons dans tout ce § les deux hypotheses suivantes sur le couple (G, p) :
(1.3.1) On a dim Hi(G, A) < oo pour tout i E Z et tout A E CG. (Par "dim" on entend la dimension sur le corps Fp.)
(1.3.2) On a cdp(G) < oo, autrement dit it existe un entier d tel que Hi(G, A) = 0 pour tout i > d et tout A E CG, cf. [12], 1.3.1. Ces deux hypotheses permettent de definir la caracteristique d'Euler-Poincare
de A: e(G, A) = E(-1)i dim Hi(G, A). C'est un entier, qui depend de facon additive de A.
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1.4. Distribution d'Euler-Poincare Si X est un espace compact totalement discontinu, une distribution Ir sur X, a valeurs dans Qp, est une forme lineaire
f H = f f (x)µ(x) sur l'espace vectoriel des fonctions localement constantes sur X, a valeurs dans Qp (n°3.1).
Theoreme A (n°3.4)-Il existe une distribution uG et une seule sur l'espace GTeg, a valeurs dans Qp, qui a les deux proprietes suivantes :
(1.4.1) Pour tout A E CG, la caracteristique d'Euler-Poincare de A est donnee par : e(G, A)=
(1.4.2) PG est invariante par les automorphismes interieurs s H gsg(g E G), ainsi que par s H sp. La distribution PG sera appelee la distribution d'Euler-Poincare de G. On la note pG,p lorsqu'on veut preciser p. D'apres (1.4.1), la caracteristique
d'Euler-Poincare d'un G-module A s'obtient en "integrant" le produit du caractere de Brauer WA de A par la distribution µG.
1.5. Exemples (1.5.1) Dans le cas, du a Tate, ou G = Gal(K/K), K etant une extension finie de Qp, on a e(G, A) = -d dim A, avec d = [K : Qp], cf. [14]. Comme dim A = WA (1), cela signifie que PG = -d 6l, ou 61 est la distribution de Dirac en 1'element 1 de G, cf. n°6.1. (1.5.2) Si l'ordre du centre de G est divisible par p, on aµG = 0 (cf. n°5.2), autrement dit e(G, A) = 0 pour tout A E CG.
1.6. Determination de µG a partir des HH(U, Qp) Si U E UG, on definit (au moyen de cochaines continues) les groupes de cohomologie Hi(U, Qp). Ce sont des Qp-espaces vectoriels de dimension finie, nuls si i > cdp(G). Le groupe fini G/U opere de facon naturelle sur HH(U, Qp). Soit hU: G/U --+ Qp le caractere de la representation ainsi obtenue. On pose
hp = E(-1)'h2U. Le caractere virtuel hu est nul en dehors de Greg. De plus, la distribution PG est egale a la limite des hu, au sens suivant (0°4.4, th.4.4.3) :
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a:
Theorisme B-Soit f : Greg -* QT, une fonction constante (mod U). On
= (G: U)-'E f (s)hu(s), oil la somme porte sur les elements s de (G/U)reg (Une fonction f sur Greg, ou sur G, est dite "constante (mod U)" si f (s) ne depend que de 1'image de s dans G/U.)
1.7. Le cas analytique Supposons maintenant que G soit un groupe de Lie p-adique compact sans element d'ordre p. D'apres Lazard [7], le groupe G possede les proprietes (1.3.1) et (1.3.2), avec cdp(G) = dim G. La distribution µG peut alors s'expliciter de la maniere suivante : Soit Lie G 1'algebre de Lie de G. Si g E G, notons Ad(g) l'automorphisme de Lie G defini par 1'automorphisme interieur x i-+ gxg-1, et posons :
F(g) = det(1- Ad(g-1)). La fonction F ainsi define sur G est localement constante, et nulle en dehors de GTeg. De plus, elle est egale a hu pour tout U E UG assez petit (n°7.2). Vu le th.B, on en deduit (cf n°7.3) : Theoreme C-La distribution µG est egale an produit de F par la distribution de Haar dg de G (normalisee pour que sa masse soit egale a 1). En d'autres termes, on a la formule : e(G, A) = f cpA(g) det(1 - Ad(g-1))dg pour tout A E CG.
Corollaire-On a µG = 0 si et seulement si le centralisateur de tout element de G est de dimension > 0. C'est le cas lorsque G est un sous-groupe ouvert de G(Qp), ou G est un QP groupe algebrique connexe de dimension > 0, cf n°7.4.
1.8. Une application Soit G un sous-groupe ouvert compact de GLn(Zp), et soit I le G-module discret (Qp/Zp)n. Supposons 1 < n < p - 1. On demontre (cf n°8.3) : Theoreme D-(a) Les groupes de cohomologie H=(G, I) sont des p-groupes
finis, null pour i > n2. (b) Si l'on pose hi(G, I) =1ogp I Hi(G, I)1, on a >(-1)'h=(G, I) = 0. (Autrement dit, le produit alterne des ordres des H=(G, I) est egal a 1.) Lorsque n = 2 et p > 5, ce resultat m'avait ete commande par J. Coates, qui en avait besoin pour des calculs de caracteristiques d'Euler-Poincare de groupes de Selmer, cf. [4].
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§2. Caracteres de Brauer : rappels 2.1. Representants multiplicatifs et trace de Brauer Soit k un corps parfait de caracteristique p, et soit W (k) l'anneau des vecteurs
de Witt de k. On note K le corps des fractions de W (k). Le cas le plus important pour la suite est celui on k = Fp, W (k) = Zp et K = Qp. Si x E k, on note x son representant multiplicatif dans K, autrement dit 1'element (x, 0, 0, ...) de W (k). Lorsque x est une racine de l'unite, on peut caracteriser x comme l'unique racine de l'unite de W (k), de meme ordre que x, et dont 1'image dans k par l'isomorphisme W (k) /pW (k) = k est egale a x. L'automorphisme de Frobenius x H xp de k definit un automorphisme de W (k) (et aussi de K) que nous noterons F. Pour tout x E k, le representant multiplicatif de xp est F(Y) = (x)P. Un element w de W (k) est fixe par F si et seulement si 1'on a w E Z. Soit n un entier > 0, et soit f (T) = T" + ... un polynome unitaire de degre n a coefficients dans k. Ecrivons f sous la forme
.f (T) = [J(T - xi), Oil x1i ... , xn appartiennent a une extension galoisienne finie k' de k. Definissons un polynome 7(T) par 7(T) = li(T - Y_). Les coefficients de 7(T) sont
des elements de W (k') invariants par Gal(k'/k) ; ils appartiennent donc a
W(k). On a f - f (mod p). Soit A un espace vectoriel de dimension finie n sur k, et soit u un element
de End(V). Soit f (T) = det(T - u) le polynome caracteristique de u et soit
7(T) = T" - a1T"-1 +... le polynome correspondant; on a f E W(k)[T]. Le coefficient a1 sera appele la trace de Brauer de u, et note TrBr(u) ; c'est la somme des representants multiplicatifs des valeurs propres de u (dans une extension convenable de k). On a par construction :
Trnr(u) = Ti(u)
(mod p)
et
TrBr(up) = F(TrBr(u))
Remarque. Lorsque k = Fp, on a TrBr(u) E Zp. Si l'on represente u par une matrice (u23) et si U = (U2j) est une matrice a coefficients dans Zp telle que UZj - u$j (mod p) pour tout i, j, on verifie facilement la formule : TrBr(u) = limTr(U'"`)
pour m -> 00,
la limite etant prise pour la topologie naturelle de Zr,. Cela fournit une definition de TrBr(u) "sans sortir de Zp".
2.2. Caracteres de Brauer des groupes finis Supposons G fini, et ecrivons son ordre sous la forme pam, avec (p, m) = 1. Le corps k est dit "assez gros pour G" (cf. [11], Chap.14) s'il contient toutes
Jean-Pierre Serre
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les racines m-iemes de l'unite. Notons CG,k la categorie des k[G]-modules de type fini; lorsque k = Fp, c'est la categorie notee CG au n°1.2. Si A est un objet de CG,k, et si s E G, on note $A 1'automorphisme correspondant du k-espace vectoriel A. On pose CPA(S) = TrBr(sA)
Bien que cette definition ait un sens pour tout s c G, on se borne a s E Greg, i.e. s d'ordre premier a p (les autres elements de G ne fournissent pas d'information supplementaire : si s E Greg est la p'-composante d'un element g de G, on a WA(g) = CPA(s), comme on le verifie facilement). La fonction CPA: Greg + W (k)
est appelee le caractere de Brauer de A. Les proprietes suivantes sont bien connues (cf e.g. [5], Chap.IV ou [11], Chap.18) : (2.2.1) CPA est une fonction centrale (i.e. invariante par automorphismes interieurs) . (2.2.2) On a VA(5") = F(VA(s)) pour tout s E Greg. (2.2.3) CPA ne depend que des quotients de Jordan-Holder de A (i.e. le semi-simplifie de A a meme caractere de Brauer que A). (2.2.4) Si A et A' sont semi-simples et ont meme caractere de Brauer, ils sont isomorphes. (2.2.5) Les caracteres de Brauer des differents k[G]-modules simples sont lineairement independants sur K. Si k est assez gros, Us forment une base de l'espace vectoriel des fonctions centrales sur Greg.
2.3. Le cas oil k = Fp Supposons k = Fp, auquel cas le caractere de Brauer CPA d'un objet A de CG est a valeurs dans Z. D'apres (2.2.1), CPA est une fonction centrale sur Greg, et d'apres (2.2.2), on a : (2.3.1) CPA(sp) = CPA(s) pour tout s E Greg.
Choisissons un ensemble de representants EG des classes d'objets simples de CG.
Proposition 2.3.2-Les caracteres cos, S EEG, forment une Qp-base de l'espace des fonctions centrales sur Greg, a valeurs dans Qp, invariantes par S H sp. Soit F 1'espace des fonctions en question. Les cos appartiennent a F et sont lineairement independants d'apres (2.2.5). Il reste a voir que tout element f de F est combinaison lineaire des CPS. Choisissons un corps fini k contenant Fp, et assez gros pour G (cf n°2.2). D'apres (2.2.5), on peut ecrire f sous la forme
f .=
aT CPT
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467
oil les modules T sont des k[G]-modules, et les coefficients aT appartiennent au
corps des fractions K de W(k). Chacun des modules T definit par restriction des scalaires a Fr un Fp[G]-module T°. Si n = [k : Fr], on a :
= (2.3.3)
(PTO(8)
n-1
E cPT(sp`). :=o
On deduit de la : n. f = E aTCPTo.
(2.3.4)
En decomposant les To en combinaisons lineaires des ccs, cela donne (2.3.5)
f = E bsccs, avec bs E K.
Mais f et les ccs sont des functions a valeurs dans Qp, et les cps sont lineairement independants ; on a donc bs E Qp pour tout S, ce qui acheve la demonstration. Remarque. Soit R(CG) le groupe de Grothendieck de la categorie CG. La prop.2.3.2 revient a dire que l'application A -+ cpA definit un isomorphisme de Qp ® R(CG) sur l'algebre des fonctions centrales sur G1eg, a valeurs dans Qp et invariantes par s H sp.
2.4. Le cas ou k = Fp (suite) Proposition 2.4.1-Soit c: EG -* Qp une application. Il existe une fonction centrale Oc et une seule sur G, a valeurs dans Qp, qui ait les proprietes suivantes : (2.4.2) 61 est nulle en dehors de Greg (2.4.3) 9c est invariante par s H sp. (2.4.4) Pour tout S E EG, on a c(S) = 1 26C(s)cps(s) IGI
(Dans (2.4.4), on considere le produit BCcos comme une fonction sur G nulle en dehors de GTeg) et la somme porte sur tous les elements s de G.) Soit R(s, s') la relation d'equivalence sur GTeg : "il existe i E Z tel que s' et sp' soient conjugues". Soit sl,... , s,. un systeme de representants de Greg (mod R) ; notons C2 la classe d'equivalence de si ; l'ensemble GTeg est reunion disjointe des C. Soit fz la fonction egale a 1/I G I sur CG et a 0 ailleurs. D'apres la prop.2.3.2 on pent ecrire fz sous la forme
fi =
ai,s cps,
avec ai,s E Qp.
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On definit alors 0° en donnant sa valeur en si pour tout i :
0`(si) _ IGI E ai,s c(S).
(2.4.5)
S
On verifie par un simple calcul que la fonction O` ainsi define a les proprietes voulues ; son unicite resulte du fait que les cps forment une base de 1'espace des fonctions sur Greg (mod R). Remarque. On peut ecrire 0, en termes de caracteres de modules projectifs de la maniere suivante : Si S E EG, notons Ps son enveloppe projective (cf. [11], Chap. 14) ; c'est un Fp[G]-module projectif dont le plus grand quotient semi-simple est isomorphe a S ; it est unique, a isomorphisme pres. Ce module est la reduction modulo
p d'un Zp[G]-module projectif Ps, de type fini, dont le caractere (au sens usuel) est note 4bs. On sait ([11], Chap.18) que 4bs est nul en dehors de Greg
et que sa restriction a Greg est le caractere de Brauer de Ps. Notons -bs la fonction s H ts(s-1) ; c'est le caractere du dual de Ps. Soit Ds l'algebre des endomorphismes de S ; c'est un corps, extension finie de Fp ; posons ds = [Ds : Fp]. La fonction 0° de la prop.2.4.1 peut s'ecrire de facon simple comme combinaison lineaire des fis
Proposition 2.4.6-On a : 0e =
oii la somme porte sur les elements S de EG. Cela se voit en remarquant que, pour tout T EEG, on a (cf. [11], Chap. 18)
G
:
' S(8)(PT(8) = dimHomG(S, T) = ds15sT,
Oil 8ST est le symbole de Kronecker (1 si S = T et 0 sinon).
Corollaire 2.4.7-Si c(S)ds1 appartient a Z pour tout S, la fonction 0e est le caractere dun Zp[G]-module projectif "virtuel" (i.e., c'est le caractere d'un element du groupe de Grothendieck P(G) de la categorie des Zp[G]modules projectifs de type fini). C'est clair. Nous aurons besoin dans la suite d'une propriete de "passage au quotient" pour 0' : Soit N un sous-groupe normal de G. L'ensemble EGIN s'identifie a une partie de EG. L'application c: EG -+ Qp definit donc des "fonctions 0c" a la fois pour G et pour GIN; notons ces fonctions 0G et OGIN
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Proposition 2.4.8-Pour tout x E GIN, on a : BGIN(x)
=
1 E 9G(s)
INI
oic la somme porte sur les s E G d'image x dans GIN. En effet, si l'on note 9' la fonction sur GIN definie par le membre de droite, it est clair que 9' a les proprietes reclamees dans la prop.2.4.1, relativement a GIN ; on a done bien 9' = Bc/N
§3. La distribution
µG
3.1. Distributions sur un espace compact totalement discontinu Soit X un espace compact totalement discontinu. On sait (cf e.g. [2], TG.II.32, cor.a la prop.6) que les ouverts fermes de X ("clopen subsets") forment une base de la topologie de X, de sorte que X est limite projective d'ensembles finis. Soit R un anneau commutatif. On note C(X; R) la R-algebre des fonctions
localement constantes sur X, a valeurs dans R. On a :
C(X; R) = R 0 C(X; Z). Une distribution µ sur X, a valeurs dans R, est une R-forme lineaire p: C(X; R) -* R.
Si f est un element de C(X; R), µ(f) est egalement note , ou aussi
f f (x)µ(x) Remarque. Si l'on prefere "mesure" a "integration", on peut voir une distribution comme une fonction U H u(U), definie sur les ouverts fermes de X, a valeurs dans R, et additive :
µ(UuU') =µ(U)+µ(U') si UnU'=0. Le support d'une distribution p se definit a la facon habituelle : c'est la plus
petite partie fermee Y de X telle que p soit nulle sur X - Y (i.e. = 0 pour toute f nulle sur Y). Le support de p se note Supp(p). Si X' est une partie fermee de X, les distributions sur X' peuvent etre iden-
tifiees (par prolongement par 0) aux distributions sur X a support contenu dans X. On fera souvent cette identification par la suite.
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3.2. Exemples de distributions 3.2.1. Si x est un point de X, la distribution de Dirac Jx en x est la forme lineaire f H f (x). 3.2.2. Soit G un groupe profini. Supposons que R soit une Q-algebre. Si f est une fonction localement constante sur G, a valeurs dans R, choisissons U E UG tel que f soit constante mod U, et posons
µ(f) _
(G
1
U)
EU f (x)
II est clair que p(f) ne depend pas du choix de U. On obtient ainsi une distribution sur G. Si Z est une partie ouverte et fermee de G, µ(Z) n'est autre que la mesure de Z relativement a la mesure de Haar de G (normalisee
pour que sa masse totale soit 1). Pour cette raison, nous appellerons p la distribution de Haar de G. 3.2.3. Si µ est une distribution sur X, et F une fonction localement constante, on definit le produit F.µ de F et de p par la formule :
=
.
3.2.4. Si h: X -+ X' est une application continue de X dans un espace compact totalement discontinu X', et sip est une distribution sur X, l'image hp de u par h est definie par la formule
pour tout f' E C(X', R).
3.3. Distribution associee a une fonction additive de modules On revient maintenant aux notations du §1, et l'on note G un groupe profini. Si A est un objet de CG, on note WA son caractere de Brauer, defini par (3.3.1) cpA(s) = TrBr(sA) pour s E Greg. (Pour la definition de TrBr, voir n°2.1.) L'action de G sur A se factorise par un quotient fini G/U, avec U E UG. II en resulte que coA est constant mod U. C'est donc une fonction localement constante sur Greg, a valeurs dans Zp. De plus, coA depend additivement de A : si
0-4A- B-3C-a0
est une suite exacte dans CG, on a con = cpA + cpcNotons EG un ensemble de representants des objets simples de CG, et soit
c: EG -+ Qp une application. Si A E CG a une suite de Jordan-Holder dont les quotients successifs sont S 1 ,-- . , S,,, E EG, on pose c(A) = c(Si) +
+ c(Sm).
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La fonction c ainsi definie sur CG est additive au sens ci-dessus. On va voir qu'on peut 1'exprimer en termes des cPA :
Theoreme 3.3.2-Il existe une distribution a' sur Greg, a valeurs dans Qp, et une seule, telle que : (3.3.3) c(A) = pour tout A E CG.
(3.3.4) je est invariante par faction des automorphismes interieurs, ainsi que par s H sp. (Noter que 1'application s H sp est un homeomorphisme de 1'espace compact Greg sur lui-meme, ce qui donne un sens a (3.3.4).) Lorsque G est fini, ce resultat a deja ete demontre (prop.2.4.1). On va se ramener a ce cas : Soit U E UG. D'apres la prop.2.4.1 appliquee a G/U, it existe une fonction BU sur G/U, et une seule, qui soft a valeurs dans Qp et possede les proprietes suivantes : a) BU est une fonction centrale, invariante par s sp. b) Pour tout A E CGlU, on a c(A)
1
(G U)
SE1: OU(s)cPA(s)
Si f E C(Greg; Qp) est constante (mod U), on definit < f, ,u' > par la formule (3.3.5)
= (G 1 U) E BU(s)f (s),
ou la somme porte sur les elements s de (G/U)Teg. Il resulte de la prop.2.4.8 que ne depend pas du choix de U. La fonction f ->
ainsi definie repond evidemment aux conditions imposees. Son unicite se demontre par un argument analogue : on se ramene au cas on G est fini, deja traite dans la prop.2.4.1.
3.4. Demonstration du theoreme A du n°1.4 Supposons que G satisfasse aux hypotheses du n°1.3, a savoir cdp(G) < oo et dim H'(G, A) < oo pour tout i E Z et tout A E CG. La caracteristique d'Euler-Poincare e(G, A) = E (-1)a dim H' (G, A)
est alors definie pour tout A E CG, et c'est une fonction additive de A, a valeurs dans Z (donc aussi a valeurs dans Qp). Le th.3.3.2, applique a
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c: A H e(G, A), fournit alors une distribution pt' sur Greg ; c'est la distribution d'Euler-Poincare pG cherchee. D'apres (3.3.3), on a : e(G, A) = f cpA(s)PG(s)
(3.4.1)
pour tout A E CG.
Exemples.
Je me borne a deux cas elementaires ; on en verra d'autres plus loin. (3.4.2) Supposons G fini. Son ordre est premier a p (sinon, cdp(G) serait infini). On a alors e(G, A) = dim H°(G, A), ce qui peut aussi s'ecrire e(G, A) =
E co(s), avec m = GI, cf. [11], 18.1.ix. s
m (3.4.1), on voit que pG = m E, 8 oil b8 est la distriEn comparant avec bution de Dirac en s. En d'autres termes, µG est la distribution de Haar de G.
(3.4.3) Supposons que G soit un pro-p-groupe. Le seul element regulier de G est 1'element neutre. Il en resulte que µG est un multiple E.51 de in distribution de Dirac en ce point. Comme WA(1) = dim A, la formule (3.4.1) s'ecrit : e(G, A) = E. dim A.
En prenant A = Fr (avec action triviale de G-d'ailleurs aucune autre n'est possible), on voit que E = e(G, Fp). L'entier E est appele la caracteristique d'Euler-Poincare du pro-p-groupe G, c£ [12], I.4.1.exerc.
3.5. Interpretation de AG en termes de modules projectifs Revenons a in situation du th.3.3.2, dans le cas oii c est la fonction A H e(G, A). Si U E UG, on a associe a U (et c) une certaine fonction centrale eU sur G/U, qui determine UG sur les fonctions f qui sont constantes (mod U) : (3.5.1)
_ (G 1 U) E OU(s)f (s)
Proposition 3.5.2-La fonction BU est le caractere dun Zp[G/U]-module projectif virtuel. D'apres le cor.2.4.7, it suffit de montrer que, pour tout objet simple S de CG/U, le nombre E Z,. Soit U E UG tel que f soit constante (mod U). On a = (G 1 U) E Ou(s)f (s),
et 1'on vient de voir que 9u est le caractere d'un element de P(G/U). On est donc ramene a prouver le resultat suivant :
Lemme 3.6.2-Soient r un groupe fini et x le caractere dun Zp[I']module projectif de type fini. Si f est une fonction centrale sur r, a valeurs
dans Z,, on a r E x(s) f (s) E Z. Il suffit de considerer le cas ou f est la fonction caracteristique d'une classe de conjugaison C de F. Si x est un element de C, et Z(x) son centralisateur, on a
I1 >x(s)f(s) =
ICI
x(x) = IZ1
x(x)
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Or on sait ([5], Chap.IV,cor.2.5) que ce nombre a une valuation p-adique > 0. D'ou le resultat cherche.
Remarque. Une distribution a valeurs dans ZP est parfois appelee une mesure p-adique (cf e.g. [9], n°1.3). On peut l'utiliser pour integrer des fonctions plus generales que les fonctions localement constantes, par exemple des fonctions continues p-adiques. Dans le cas de Cl G, une telle mesure peut s'interpreter comme un element du groupe de Hattori-Stallings T(Zp[[G]]), quotient de Zp[[G]] par 1'adherence du sous-groupe additif engendre par les xy - yx, cf. [9], loc. cit. De ce point de vue, le fait que le support de µA soit contenu dans l'ensemble des classes regulieres (et aussi que µ°G soit invariante par s N sP) est a rapprocher des resultats de Zaleskii [16] et Bass [1] dans le cas discret.
§4. Representations Zp lineaires et Qp lineaires Les G-modules consideres jusqu'ici etaient des groupes abeliens finis de type (p,.. . , p). Nous allons maintenant nous occuper de cas plus generaux, par exemple de QP espaces vectoriels de dimension finie. Cela permettra d'ob-
tenir une caracterisation simple de la distribution µG (n°4.4). C'est cette caracterisation qui sera utilisee par la suite.
4.1. Cohomologie continue : le cas des ZP modules de type fini On suppose que G satisfait a la condition de finitude (1.3.1) dim H' (G, A) < oo pour tout i E Z et tout A E CG.
Soit L un Zp module de type fini, sur lequel G opere continument (pour la
topologie p-adique de L). Pour tout n > 0, L/p"L est un G-module discret, au sens usuel. Les groupes de cohomologie continue HH(G, L) sont definis de Tune des deux facons (equivalentes) suivantes (4.1.1)
H,'(G,
L) = llim Ht(G, L/eL),
la limite projective etant prise pour n -+ oo. (4.1.2) Si CC(G, L) designe le complexe des cochaines continues sur G a valeurs dans L, on definit HH(G, L) comme H'(CC(G, L)). L'equivalence de ces deux definitions se voit en remarquant que CC(G, L) est limite projective des complexes de cochaines C(G, L/p'tL). Comme les groupes de cohomologie de ces complexes sont finis (grace a l'hypothese faite sur G), les homomorphismes naturels
Hi(CC(G,L)) -)1EmH'(C(G,L/p"L))
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sont des isomorphismes (cf. par exemple [6], 13.1.2 et 13.2.3). D'ou le resultat cherche. Par construction, les H'(G, L) sont des pro-p-groupes commutatifs, done des Zp-modules topologiques compacts.
Si 0 -+ L' -+ L -* L" -* 0 est une suite exacte, on a une suite exacte de cohomologie correspondante :
HH+'(G L')-*...; c'est clair si l'on utilise la definition (4.1.2).
Proposition 4.1.3-Les HH(G, L) sont des Zp-modules de type fini. En utilisant la suite exacte ci-dessus, on se ramene au cas oil L est sans torsion. Si l'on pose H' = HH(G, L), on a alors une suite exacte de cohomologie :
H' 24 H' -+ Hi(G, L/pL)
.. .
Comme H'(G, L/pL) est fini, it en resulte que H'/pH' est fini. Comme H' est un pro-p-groupe commutatif, cela entraine que H' est topologiquement de type fini, d'ou la proposition.
4.2. Caracteristique d'Euler-Poincare des ZP modules de type fini On conserve les hypotheses precedentes, et l'on suppose en outre que cdp(G) < oo, de sorte que les caracteristiques d'Euler-Poincare e(G, A) sont definies, ainsi que la distribution µc. Soit L un Zp module de type fini, sur lequel G opere continument. Les HH(G, L) sont des Zp modules de type fini, nuls pour i > cdp(G). Notons rg Hi (G, L) le rang de HH(G, L) comme Zp-module, autrement dit la dimenL). sion du Qp espace vectoriel Qp 0
Proposition 4.2.1-Supposons L sans torsion. On a alors E(-1)'rg HH(G, L) = e(G, L/pL) = E(-1)' dim H'(G, L/pL). Posons, comme au n°4.1, H' = HH(G, L) et utilisons la suite exacte de cohomologie
...-4 H'2* H'->H'(G,LlpL)-+... On obtient ainsi des suites exactes
0 -4 H'/pH' -+ HI (G, L/pL) -4 HP+' -4 0,
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ou Xp design le noyau de la multiplication par p dans le groupe abelien X. On en deduit
e(G, L/pL) = >(-1)i(dim H'/pH' - dim HP). Or, si M est un Zp module de type fini, on a rg M = dim M/pM - dim Mp. La formule ci-dessus peut donc s'ecrire
e(G, L/pL) _
(-1)irg Hi,
ce qui demontre la proposition. Remarque. Les arguments donnes ci-dessus sont standard ; ils s'appliquent chaque fois qu'un foncteur cohomologique possede des proprietes de finitude (par exemple en cohomologie t-adique). Avec les hypotheses ci-dessus, posons (4.2.2)
e(G, L) _ (-1)irg H. (G, L),
de sorte que la prop.4.2.1 revient a dire que e(G, L) = e(G, L/pL). Notons XL le caractere de la representation de G dans L (ou dans Qp (9 L, cela revient au meme). C'est une fonction centrale sur G, continue (mais pas necessairement localement constante), et a valeurs dans Zp.
Proposition 4.2.3-La restriction de XL d Greg coincide avec le caractere de Brauer c°L/pL de L/pL. C'est la une propriete bien connue des caracteres de Brauer. Rappelons la demonstration : Si m = rg L, on peut identifier L a Zp x . . . x Zp (m facteurs), et faction de G sur L est donnee par un homomorphisme continu
p: G + GLm(Zp). Si s est un element de G1eg, p(s) est regulier dans GLm(Zp), donc d'ordre fini r premier a p; en effet, la composante premiere a p de dordre de GLm,(Zp)
est finie. Les valeurs propres de p(s) dans une extension convenable de Qp sont des racines de 1'unite d'ordre divisant r ; ce sont les representants multiplicatifs de leurs reductions mod p. La trace de p(s) est donc egale a la trace de Brauer de sa reduction (mod p), ce qui demontre la formule cherchee. Corollaire 4.2.4-Le caractere XL est localement constant sur Greg, et l'on a : e(G, L) = f XL(s)LG(s) Cela resulte des prop.4.2.1 et 4.2.3, compte tenu du fait que e(G, L/pL) = f c0L/pL(S)AG(S)
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4.3. Le cas des QP espaces vectoriels Soit V un Qp espace vectoriel de dimension finie, sur lequel G opere continument. Comme G est compact, it laisse stable un reseau L de V, autrement dit
un Zp sous-module de type fini de V qui engendre V. Si l'on note QG, V) le complexe des cochaines continues de G a valeurs dans V, on a (4.3.1)
CC(G,V) = UC,(G,p 'L).
Les groupes de cohomologie de ce complexe sont notes Hi(G, V). Il resulte de (4.3.1) que 1'on a : (4.3.2)
He',(G,
V) = Qp 0 HH(G, L).
En particulier, les H,,(G, V) sont des Qp espaces vectoriels de dimension finie, nuls pour i > cdp(G). On pose (4.3.3)
e(G, V) _ (-1)tdimHH(G, V),
ou dim designe la dimension sur Qp. Notons Xv le caractere de la representa-
tion de G dans V. On a Xv = XL et les formules ci-dessus, combinees au cor.4.2.4, donnent :
Proposition 4.3.4-Le caractere Xv est localement constant sur Greg, et l'on a e(G,V) = f Xv(s)i o(s)
4.4. Determination de pG a partir des HH(U, Qp) Soit U E UG. Le groupe U satisfait aux memes hypotheses de finitude que G. Les groupes de cohomologie continue HH(U, Qp) sont donc definis (faction
de U sur Q. etant triviale). Posons : (4.4.1)
Hu =
(U, Qp).
Les HU sont des Qp espaces vectoriels de dimension finie ; par exemple, HU = Qp, et Hp = Hom,,(U, Qp), groupe des homomorphismes continus de G dans Qp. Le groupe G/U opere de facon naturelle sur chaque HU (a cause de faction de G sur U par automorphismes interieurs). On obtient ainsi des representations lineaires de dimension finie de G/U. Notons hU les caracteres de ces representations, et posons (4.4.2)
hu = E(-1)=h'u.
La fonction hu est un caractere virtuel de G/U, a valeurs dans Qp (et meme dans Zp).
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Theoreme 4.4.3-Le caractere hu est egal a la fonction eU introduite au 3, n°3.5.
Cela revient a dire que, si f est une fonction sur G constante (mod U), on a:
(4.4.4)
hu(s)f(s). _ (G 1 U) E1:
Notons tout de suite une consequence de th.4.4.3 :
Corollaire 4.4.5-Le caractere hu est nul en dehors de (G/U)reg. (Bien entendu, ceci ne s'etend pas aux hU ; par exemple, pour i = 0, on a
hU=1.) Demonstration du theoreme 4.4.3.
On a tout d'abord :
Lemme 4.4.5-La formule (4.4.4) est vraie lorsque f = Xv, of V est une representation Qp-lineaire de dimension finie de G/U. Le membre de gauche de (4.4.4) est :
(4.4.6)<Xv,/ a>= e(G,V) = E(-1)idimHH(G,V), cf. 4.3.4. D'autre part, la suite spectrale des extensions de groupes (ou un argument de corestriction) montre que : (4.4.7)
HH(G, V) = H°(G/U,
V)).
Comme U opere trivialement sur V, on a : (4.4.8)
H,' (U, V) = HH(U, Qp) ® V = HU ® V.
Le caractere de la representation de G/U sur cet espace est hU.Xv. La dimension du sous-espace fixe par G/U est donc (lJ) E hU(s)Xv(s). Vu (4.4.7), cela donne (4.4.9)
dimHH(G,V) = (G1U) EhU(s)Xv(s)
En combinant ceci avec (4.4.6), on obtient bjen la formule cherchee
<Xv, Aa> _ (G 1 U) E hu(s)Xv(s)
Posons maintenant 0 = hU - 0U. D'apres ce que l'on vient de voir, on a >V)(s)X(s) = 0 pour tout caractere X d'une representation de G/U sur Qp. Or hu et 8u sont des caracteres de representations virtuelles de G/U sur
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Qp : c'est clair pour hu, et pour Ou, cela resulte de la prop.3.5.2. Le lemme elementaire suivant montre alors que = 0 (ce qui demontre 4.4.3) :
Lemme 4.4.10-Soient r un groupe fini, K un corps de caracteristique zero, et V le caractere dune representation virtuelle de r sur K. Supposons que Ei(s)X(s) = 0 pour tout caractere x de F sur K. Alors V = 0. Soit 1' _ E nsXs la decomposition de 0 en combinaison lineaire de caracteres de K[r]-modules simples. Si l'on prend pour V le dual de la representation simple S, on a E b(s)Xv(s) = E '
(s)Xs(s-1) = IPl.ds.ns,
oil ds = [End(S) : K]. L'hypothese faite sur ' entraine donc ns = 0, d'ou
0=0.
4.5. Interpretation de AG en termes de representations admissibles Soit K un corps de caracteristique 0. Une representation admissible de G sur K est un K-espace vectoriel E sur lequel G agit de fagon K-lineaire, en satisfaisant aux deux conditions suivantes : (a) L'action de G est continue, autrement dit le fixateur d'un point de E est un sous-groupe ouvert de G. (b) Pour tout U E UG, le sous-espace Eu de E fixe par U est de dimension finie.
(Noter que E est reunion des EU, d'apres (a).) A une telle representation de G est associee une distribution-trace µE, a valeurs dans K, caracterisee par la formule (4.5.1)
ILE(f) = TrfE,
pour toute fonction localement constante f sur G, a valeurs dans K, fE designant 1'endomorphisme de E defini par f. (Si x E E, on choisit U fixant x tel que f soit constante (mod U), et l'on definit fE(x) comme la moyenne sur G/U des f(s)sx; l'operateur fE est de rang fini, ce qui donne un sens a (4.5.1).)
Pour tout i > 0, posons E2 = lim HC(U, Qp), on la limite inductive est prise par rapport aux homomorphismes de restriction res: HH(U, Qp) -*
Qp)
(pour U' C U),
qui sont des inclusions. L'espace Ez est une representation admissible de G ; on
a E° = H'(U, Qp) pour tout U. Soit pj la distribution-trace correspondante. L'enonce suivant n'est qu'une reformulation du th.4.4.3 :
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Theoreme 4.5.2-On a µG = E(-1)iµi. Exemples. (i = 0,1) Pour i = 0, on a Eo = Qp, avec action triviale de G. La distribution-trace associee po est la distribution de Haar de G, cf. 3.2.2. Pour i = 1, on a El = Hom,(U, Qp) ; ainsi, E1 est l'espace vectoriel des germes d'homomorphismes continus de G dans Qp, avec faction naturelle de G sur cet espace (provenant des automorphismes interieurs). Generalisation.A la place de la representation triviale de G sur Qp, on peut prendre une representation Qp lineaire continue V de dimension finie,
et definir EE (V) comme la limite inductive des Hi, (U, V). La Somme alternee des distributions-traces des Ei (V) est une distribution sur G, qui est egale a XV.PG, ou Xv est le caractere de V ; cela se verifie par un calcul analogue a celui du n°4.4.
§5. Quelques proprietes de
µG
Dans ce §, on suppose que G possede les proprietes de finitude (1.3.1) et (1.3.2) permettant de definir PG.
5.1. Restriction a un sous-groupe ouvert Proposition 5.1.1-Si G' est un sous-groupe ouvert de G, on a µG' = (G : G' designe la restriction de PG d G'. Il faut prouver que, si f est une fonction localement constante sur Greg, nulle en dehors de GCeg, on a : (5.1.2)
= (G : G').
Choisissons U E UG contenu dans G', et tel que f soit constante (mod U). Si 1'on note hu (resp. h'u) le caractere virtuel de G/U (resp. de G'/U) defini au n°4.4, on a d'apres (4.4.4) 0, on a µG = 0 puisque Greg = {1} et que le centralisateur de 1 est infini; on retrouve un resultat connu, cf. [12], 1.4.1, exerc.(e).
2) Si n = 0, G est fini d'ordre premier a p, on a Lie G = 0, F = 1, et µG est la distribution de Haar de G, cf. 3.4.2. 3) Supposons p # 2, et soit G le groupe diedral p-adique, produit semidirect d'un groupe {1, c} d'ordre 2 par un groupe U isomorphe a Z,, faction
de c sur U etant u H u-1. On a: n = dim Lie G = 1.
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L'action de U sur Lie G est triviale alors que celle de c est x H -x. D'ou F = 0 sur U, et F = 2 sur cU = G - U. On en deduit que PG a pour support G - U (qui est la classe de conjugaison de c), et que sa masse totale est 1. Autrement dit, si A E CG, on a : (7.3.7)
e(G, A) = WA (C) = 2. dim Ae - dim A,
ou A° est le sous-espace de A fixe par c.
7.4. Un cas ou AG = 0 Proposition 7.4.1-Soit G un groupe algebrique connexe sur Qr, de dimension > 0. Soit G un sous-groupe ouvert compact de G(Qr) ne contenant pas d'element d'ordre p. Alors µG = 0. Il faut montrer que la fonction F: G - Z, est egale a 0. Tout d'abord on a F(1) = det(1-1) = 0 puisque dimG > 0. Comme F est localement constante, cela montre que F = 0 dans un voisinage U de l'element neutre. Mais F est la restriction a G de la fonction "morphique" F: g H det(1 - Ad(g-1)), definie sur la variete G. Comme G est lisse et connexe, U est dense dans G pour la topologie de Zariski. Le fait que F soit 0 sur U entraine donc F = 0, d'ou le resultat cherche. Variante. On peut aussi demontrer directement que, pour tout point g de G, la dimension du centralisateur de g est > 0 (puisqu'il en est ainsi pour 1'element generique). Remarques.
1) Un argument analogue montre que PG = 0 si dim G > 0, et si 1'image de
Ad: G - GL(Lie G) est connexe pour la topologie de Zariski de GL(Lie G). 2) Supposons que 7.4.1 s'applique, donc que PG = 0. Par definition de PG, cela signifie que e(G, A) = 0 pour tout A E CG. Plus generalement, soit A un G-module discret qui snit un p-groupe fini, et soit x(G, A) sa caracteristique d'Euler-Poincare au sens multiplicatif, autrement dit le produit alterne des ordres I Hi (G, A) 1. On a alors : (7.4.2)
x(G, A) = 1.
En effet, comme x(G, A) est une fonction multiplicative de A, on peut se ramener par devissage au cas ou pA = 0, de sorte que A appartient a CG, et que (7.4.3)
puisque e(G, A) = 0.
x(G, A) = pe(G'A) = 1,
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§8. Une application Dans les §§ precedents, it ne s'agissait que de G-modules de type fini (sur Fp, Zp ou Qp, suivant les cas). Or le probleme du calcul de la caracteristique d'Euler-Poincare se pose pour tout G-module A dont les groupes de cohomologie sont de type fini (sur l'anneau de base considere), cc qui n'entraine nullement que A soit lui-meme de type fini. C'est d'un cas de cc genre que nous allons nous occuper.
8.1. Les donnees Comme au n°7.4, on part d'un groupe algebrique G sur Qp, et d'un sousgroupe ouvert compact G de G(Qp). On se donne une representation lineaire p: G -+ GLV, on V est un Qp espace vectoriel de dimension finie. On note L un ZP reseau de V stable par G (il en existe, puisque G est compact). Le G-module auquel on va s'interesser
est I = V/L ; si m = dim V, I est isomorphe comme groupe abelien a la somme directe de m copies de Qp/Zp. On se donne egalement un plongement h du groupe multiplicatif Gm dans le centre de G, et l'on suppose que V est homogene de poids # 0 pour faction de Gm : autrement dit, it existe un entier r # 0 tel que p(h(t)).v = t''v pour tout point t de Gm et tout point v de V. (Remarque. Les donnees ci-dessus sont celles que 1'on rencontre (ou que l'on espere rencontrer) dans le theorie des motifs sur un corps de nombres K : G est le groupe de Galois motivique, h est l'homomorphisme de Gm dans G associe au poids, V est la realisation p-adique du motif, L est une Zp-forme
de V, et G est l'image de Gal(K/K) dans G(Q,,).) Dans cc qui suit, on identifie Gm a un sous-groupe de G au moyen de h, et l'on note PG le groupe quotient G/Gm. Soit PG 1'image de G dans PG(Qp). On fait les hypotheses suivantes : (8.1.1) G est connexe, de dimension > 1. (8.1.2) Le groupe PG n'a pas de p-torsion. (8.1.3) Si p = 2, G ne contient pas h(-1). On a alors :
Theoreme 8.1.4-Sous les hypotheses ci-dessus, les groupes de cohomologie H'(G, I) sont des p-groupes finis, nuls si i > dim G, et le produit alterne de leurs ordres :
X(G,I) = [I IH'(G,I)I(-1)i est egal a 1.
(Rappelons que I = V/L.)
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8.2. Demonstration du theoreme 8.1.4 Soit N le noyau de la projection G -+ PG. Comme N est un sous-groupe ouvert compact de G ..(QP) = QP, c'est un sous-groupe ouvert de Zp ; de plus, si p = 2, N ne contient pas -1 d'apres (8.1.3). On peut donc ecrire N sous la forme (8.2.1)
N = C x U,
on C est cyclique d'ordre divisant p - 1, et U est isomorphe a Zp. On va utiliser la suite spectrale relative a 1'extension de groupes :
1- N- G-*PG-*1. Comme cdp(N) = 1, nous n'avons a nous occuper de H'(N, I) que pour i = 0
et i = 1. On a: Lemme 8.2.2-(i) H°(N, I) est un p-groupe fini. (ii) H1 (N, I) = 0.
Comme H(N, I) se plonge dans H'(U, I), it suffit de prouver (i) et (ii) avec N remplace par U. Or, si u est un generateur topologique de U, on a (8.2.3)
H°(U, I) = Ker(u - 1: I -+ I)
(8.2.4)
H1(U, I) = Coker(u - 1: I -+ I).
et
Mais, si 1'on identifie u a un element de Zp, on sait que u agit sur I par Ur, ou r est un entier 0. Puisque u est d'ordre infini, on a u'' - 10 0, et comme I est un groupe divisible, u - 1: I -+ I est surjectif, d'oit (ii). De plus, le noyau de u - 1: I -+ I est fini, ce qui demontre (i). (De faCon plus precise, si v est la valuation p-adique de u'' - 1, H°(U, I) est isomorphe a L/p" L.) Vu (8.2.2), la suite spectrale H(PG, Hi(N, I)) = H1+i(G, I) degenere en un isomorphisme : (8.2.5)
H'(G, I) = Hi(PG, A),
ou A = H°(N, I) est un PG-module fini d'ordre une puissance de p. Or PG est un sous-groupe ouvert compact de PG(Qp), sans p-torsion d'apres (8.1.2). On a donc cdp (PG) = dim PG = dim G - 1, ce qui montre
que les H' (PG, A) sont finis pour tout i, et nuls si i > dim G - 1. Enfin, 1'hypothese (8.1.1) equivaut a dire que PG est connexe de dimension > 0. En appliquant a PG la prop.7.4.1, on voit que IPG = 0, d'ou X(PG, A) = 1 d'apres (7.4.2). Comme X(G, I) = X(PG, A) d'apres (8.2.5), cela acheve la demonstration.
Jean-Pierre Serre
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8.3. Demonstration du theoreme D du n°1.8 Le theoreme D est le cas particulier du th.8.1.4 on G = GL, et V = (Qp)n, L = (Zp)n, I = (QP/Zp)n. Le plongement h: Gm -1 G est le plongement evident (homotheties). On a PG = PGLn. La condition (8.1.1) est satisfaite puisque dim G = n2 > 1. La condition (8.1.3) est satisfaite puisque p > n + 1 > 2. Le fait que (8.1.2) soit satisfaite resulte du lemme elementaire suivant:
Lemme 8.3.1-Si p > n + 1, les grouper GLn(Qp) et PGLn(Qp) ne contiennent pas d'element d'ordre p. Si s E GLn(Qp) est d'ordre p, l'une de ses valeurs propres, z, est une racine primitive p-ieme de l'unite. Mais on sait que z, z2, ... , zp-1 sont conjuguees par Gal(QP/Qp). Donc z, z2, ... , zp-1 sont des valeurs propres de s, ce qui
entraine n > p - 1. Si s E PGLn(Qp) est d'ordre p, soit x un representant de s dans GLn(Qp), et soit d = det(x). Notons t l'homothetie xP. On a
dP = det(xp) = det(t) = tn.
Mais n et p sont premiers entre eux, puisque p > n + 1. II en resulte que t est de la forme t = BP, avec 0 E Q. L'element y = 8-1x est alors un element d'ordre p de GLn(Q,), ce qui est impossible comme on vient de le voir. Cela acheve la demonstration du theoreme D. Remarque. On laisse au lecteur le soin d'etendre le lemme 8.3.1 a d'autres
groupes reductifs que GLn et PGLn. Par exemple, si G est un groupe de type E8 sur Qp, G(Qp) n'a pas de p-torsion si p 2, 3, 5, 7,11,13,19, 31.
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