Galois Groups and Fundamental Groups (Mathematical Sciences Research Institute Publications)

This book explores recent research underlining the remarkable connections between the algebraic and arithmetic world of ...

Author: Leila Schneps

53 downloads 858 Views 5MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

This book explores recent research underlining the remarkable connections between the algebraic and arithmetic world of Galois theory and the topological and geometric world of fundamental groups. Arising from an MSRI program held in the fall of 1999, it contains ten articles, all of which aim to present new results in a context of expository introductions to theories that are ramifications and extensions of classical Galois theory. B. H. Matzat and M. van der Put introduce differential Galois theory and solve the differential inverse Galois problem over global fields in positive characteristic; D. Harbater gives a comparative exposition of formal and rigid patching starting from the familiar complex case. S. Mochizuki discusses aspects of Grothendieck’s famous anabelian geometry, while the articles by R. Guralnick, A. Tamagawa, and F. Pop and M. Sa¨ıdi investigate the structure of the fundamental groups of curves over different kinds of characteristic p fields. M. Imbert and L. Schneps study the structure of the Hurwitz spaces and moduli spaces of curves, which are of great importance to Galois theory because of the Galois action on their fundamental groups. The first interesting such group is SL2 (Z), a family of special subgroups of which is studied by F. Bogomolov and Y. Tschinkel. Finally, R. Hain and M. Matsumoto present their result proving part of a conjecture by Deligne on the structure of the Lie algebra associated to the Galois action on the fundamental group of the thrice-punctured projective plane.

Mathematical Sciences Research Institute Publications

41

Galois Groups and Fundamental Groups

Mathematical Sciences Research Institute Publications 1 2 3 4 5 6 7 8 9 10–11 12–13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

Freed/Uhlenbeck: Instantons and Four-Manifolds, second edition Chern (ed.): Seminar on Nonlinear Partial Differential Equations Lepowsky/Mandelstam/Singer (eds.): Vertex Operators in Mathematics and Physics Kac (ed.): Infinite Dimensional Groups with Applications Blackadar: K-Theory for Operator Algebras, second edition Moore (ed.): Group Representations, Ergodic Theory, Operator Algebras, and Mathematical Physics Chorin/Majda (eds.): Wave Motion: Theory, Modelling, and Computation Gersten (ed.): Essays in Group Theory Moore/Schochet: Global Analysis on Foliated Spaces Drasin/Earle/Gehring/Kra/Marden (eds.): Holomorphic Functions and Moduli Ni/Peletier/Serrin (eds.): Nonlinear Diffusion Equations and Their Equilibrium States Goodman/de la Harpe/Jones: Coxeter Graphs and Towers of Algebras Hochster/Huneke/Sally (eds.): Commutative Algebra Ihara/Ribet/Serre (eds.): Galois Groups over Concus/Finn/Hoffman (eds.): Geometric Analysis and Computer Graphics Bryant/Chern/Gardner/Goldschmidt/Griffiths: Exterior Differential Systems Alperin (ed.): Arboreal Group Theory Dazord/Weinstein (eds.): Symplectic Geometry, Groupoids, and Integrable Systems Moschovakis (ed.): Logic from Computer Science Ratiu (ed.): The Geometry of Hamiltonian Systems Baumslag/Miller (eds.): Algorithms and Classification in Combinatorial Group Theory Montgomery/Small (eds.): Noncommutative Rings Akbulut/King: Topology of Real Algebraic Sets Judah/Just/Woodin (eds.): Set Theory of the Continuum Carlsson/Cohen/Hsiang/Jones (eds.): Algebraic Topology and Its Applications Clemens/Koll´ ar (eds.): Current Topics in Complex Algebraic Geometry Nowakowski (ed.): Games of No Chance Grove/Petersen (eds.): Comparison Geometry Levy (ed.): Flavors of Geometry Cecil/Chern (eds.): Tight and Taut Submanifolds Axler/McCarthy/Sarason (eds.): Holomorphic Spaces Ball/Milman (eds.): Convex Geometric Analysis Levy (ed.): The Eightfold Way Gavosto/Krantz/McCallum (eds.): Contemporary Issues in Mathematics Education Schneider/Siu (eds.): Several Complex Variables Billera/Bj¨ orner/Green/Simion/Stanley (eds.): New Perspectives in Geometric Combinatorics Haskell/Pillay/Steinhorn (eds.): Model Theory, Algebra, and Geometry Bleher/Its (eds.): Random Matrix Models and Their Applications Schneps (ed.): Galois Groups and Fundamental Groups Nowakowski (ed.): More Games of No Chance Montgomery/Schneider (eds.): New Directions in Hopf Algebras Buhler/Stevenhagen (eds.): Algorithmic Number Theory Jensen/Ledet/Yui: Generic Polynomials: Constructive Aspects of the Inverse Galois Problem

Q

Volumes 1–4 and 6–27 are published by Springer-Verlag

Galois Groups and Fundamental Groups

Edited by

Leila Schneps Institut de Mathématiques de Jussieu

Series Editor Silvio Levy Mathematical Sciences Research Institute 17 Gauss Way Berkeley, CA 94720 United States

Leila Schneps Institut de Mathématiques de Jussieu 175, rue du Chevaleret 75013 Paris France

MSRI Editorial Committee Michael Singer (chair) Alexandre Chorin Silvio Levy Jill Mesirov Robert Osserman Peter Sarnak

The Mathematical Sciences Research Institute wishes to acknowledge support by the National Science Foundation. This material is based upon work supported by NSF Grant 9810361.

published by the press syndicate of the university of cambridge The Pitt Building, Trumpington Street, Cambridge, United Kingdom cambridge university press The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarc´ on 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org c Mathematical Sciences Research Institute 2003 ° Printed in the United States of America A catalogue record for this book is available from the British Library. Library of Congress Cataloging in Publication data available ISBN 0 521 80831 6 hardback

Galois Groups and Fundamental Groups MSRI Publications Volume 41, 2003

Contents Introduction

ix

Monodromy Groups of Coverings of Curves

1

Robert Guralnick

On the Tame Fundamental Groups of Curves over Algebraically Closed Fields of Characteristic > 0

47

Akio Tamagawa

On the Specialization Homomorphism of Fundamental Groups of Curves in Positive Characteristic

107

Florian Pop and Mohamed Saïdi

Topics Surrounding the Anabelian Geometry of Hyperbolic Curves

119

Shinichi Mochizuki

Monodromy of Elliptic Surfaces

167

Fedor Bogomolov and Yuri Tschinkel

Tannakian Fundamental Groups Associated to Galois Groups

183

Richard Hain and Makoto Matsumoto

Special Loci in Moduli Spaces of Curves

217

Leila Schneps

Cellulation of Compactied Hurwitz Spaces

277

Michel Imbert

Patching and Galois Theory

313

David Harbater

Constructive Dierential Galois Theory B. Heinrich Matzat and Marius van der Put

vii

425


Introduction This volume is the outcome of the MSRI special semester on Galois Groups and Fundamental Groups, held in the fall of 1999. Respecting the famous Greek requirements of unity of place, time, and action, the semester was an unforgettable, four-month-long occasion for all mathematicians interested in and responsible for the developments of the connections between Galois theory and the theory of fundamental groups of curves, varieties, schemes and stacks to interact, via a multitude of conferences, lectures and conversations. Classical Galois theory has developed a number of extensions and ramications into more specic theories, which combine it with other areas of mathematics or restate its main problems in dierent situations. Three of the most important of these extensions are geometric Galois theory, dierential Galois theory, and Lie Galois theory, all of which have undergone very rapid development in recent years. Each of these theories can be developed in characteristic zero, over the eld C of complex numbers, over number elds or p-adic elds, or in characteristic p > 0; various versions of the classical and inverse Galois problems can be posed in each situation. The purpose of this introduction is to give a brief overview of these three themes, which form the framework for all the articles contained in this book. The main focus of study of geometric Galois theory is the theory of curves and the many objects associated to them: curves with marked points, their elds of moduli and their fundamental groups, covers of curves with their ramication information and their elds of moduli, and the nite quotients of the fundamental group which are the Galois groups of the covers, as well as the moduli spaces and Hurwitz spaces which parametrize all these objects. To consider a curve X topologically is tantamount to considering it over the eld of complex numbers C . As an abstract group, the topological fundamental group of the curve depends only on the genus g and the number n of marked points chosen on the curve; it can be identied with the group of homotopy classes of loops on the curve based at a xed (not marked) base point, and is presented by standard generators a1 , . . . , ag , b1 , . . . , bg , c1 , . . . , cn subject to the unique relation

[a1 , b1 ] · · · [ag , bg ]c1 · · · cn = 1. ix

(∗)

x

INTRODUCTION

Note that when n ≥ 1, this group is actually free. The Galois covers of the curve correspond to the nite quotients of this group, which are exactly the nite groups generated by generators ai , bi and cj satisfying (∗), so they are perfectly understood. The algebraic fundamental group of the curve is the Galois group of the compositum of all function elds of nite étale covers of the curve over the function eld of the curve itself; in fact, it is exactly the pronite limit of the nite quotients of the topological fundamental group. This simple situation leads or generalizes very naturally into new regions that contain all kinds of very dicult problems. We sketch some of them:

1. Fundamental groups in characteristic p. When a curve X is dened over a eld in characteristic p, all relations between its fundamental group and any topological notion of `loops' must be forgotten. Over an algebraically closed eld, one denes the algebraic fundamental group directly, exactly as above; it is the pronite limit of the Galois groups (monodromy groups) of nite étale covers of the curve. However, in this situation, for (g, n) dierent from (0, 0) and (1, 0), it is extremely dicult to determine the structure of the fundamental group, or even the weaker question of which nite groups can occur as its quotients. Indeed, this is one of the fundamental problems of geometric Galois theory in characteristic p. In the ane case (n ≥ 1), the complete solution to the weaker problem was conjectured by Abhyankar; this conjecture was proved over the ane line by M. Raynaud, and the proof extended to all curves by D. Harbater. However, the situation remains completely mysterious in the case of complete curves (n = 0, g > 1). Things are better if one considers only the quotients of order prime to p; then a result due to Grothendieck states that the groups of order prime to p which can occur are exactly the nite quotients of order prime to p of the topological fundamental group of type (g, n) dened in (∗), and that in fact the primeto-p quotients of the fundamental groups over C and in characteristic p are isomorphic. In characteristic p, this group is a quotient of the tame fundamental group, which is the largest quotient of the fundamental group having inertia subgroups of order prime to p; this group (which is equal to the fundamental group when n = 0) is easier to work with than the full group for various purposes. But the structure of the tame fundamental group and the set of its nite quotients are absolutely unknown, except in the non-hyperbolic cases (g, n) = (0, 0), (0, 1), (0, 2) and (1, 0). The articles by R. Guralnick, A. Tamagawa, and F. Pop and M. Saïdi all work in the situation of curves dened over an algebraically closed eld of characteristic p. Guralnick works on the problem of determining which groups can occur as Galois groups (or their composition factors) of nite separable covers f : Y → X , where Y is of xed genus g , and seeks groups which can specifically be excluded. Tamagawa shows that given the tame fundamental group, it is possible to recover the type (g, n) of the curve (if (g, n) 6= (0, 0) or (0, 1)).

INTRODUCTION

xi

Results in this and other papers by Tamagawa even tend to imply that in some cases, the tame fundamental group may determine the isomorphism class of the curve completely. This shows how dierent the characteristic p case is from the characteristic 0 case, in which as we saw, curves of many dierent types may have isomorphic fundamental groups (for instance, the fundamental groups of curves of type (2, 2) and (1, 4) are both free of rank 5). Finally, Pop and Saïdi address similar questions, proving, under certain hypotheses on the Jacobians, that at most a nite number of curves can have isomorphic fundamental groups.

2. Anabelian theory. We saw above that the isomorphism class of the topo-

logical or the algebraic fundamental group is very far from determining even the most basic information about a curve in characteristic 0, such as its type (g, n), whereas in characteristic p it determines much more if not all of the information about the specic curve. However, one can also consider the algebraic fundamental group equipped with its canonical outer Galois action, which should provide more information. Indeed, any variety (scheme, stack) dened over an algebraically closed eld can actually be considered as dened over a subeld K , given by the coecients of the equations of a dening model, say, and which is nitely generated over the prime eld and not algebraically closed. Then there is an exact sequence

1 → π1 (X ⊗ K) → π1 (X) → Gal(K/K) → 1,

(∗∗)

where π1 (X ⊗ K) denotes the algebraic fundamental group. The anabelian problem, which was posed by Grothendieck in his famous letter to G. Faltings, asks which varieties are entirely determined by the group π1 (X ⊗K) together with the action Gal(K/K) → Out(π1 (X ⊗ K)). Grothendieck called varieties which are thus determined anabelian varieties, and explicitly stated that hyperbolic curves should be anabelian. This is related to the hitherto unproven section conjecture for a hyperbolic curve X , which states that the sections

Gal(K/K) → π1 (X) of (∗∗) are in bijection with the rational points of X if X is complete, and this set together with the tangential base points if X is not complete. S. Mochizuki proved that hyperbolic curves dened over sub-p-adic elds, that is, elds which are subelds of elds nitely generated over the p-adics, are indeed anabelian. In his article in this volume, he discusses various results related to this theorem, including a partial generalization to characteristic p and a discussion of the section conjecture over the eld of real numbers.

3. Galois action on fundamental groups. In his Esquisse d'un Programme,

completing the letter to Faltings, Grothendieck suggested that not only hyperbolic curves, but also the moduli spaces Mg,n of curves of type (g, n) should be examples of anabelian varieties, and that explicitly investigating the Galois action on their fundamental groups should provide information of an entirely new

xii

INTRODUCTION

type about the elements of Gal(Q/Q); this is known as Grothendieck-Teichmüller theory. The rst non-trivial moduli space Mg,n is the case (g, n) = (0, 4); we have M0,4 = P 1 − {0, 1, ∞}. Following the direction initiated by Grothendieck, the Galois action on the fundamental group of this space (the pronite free group Fb2 on two generators) has been studied in a theory known as dessins d'enfants. The study has focused on coding the conjugacy classes of nite index subgroups of π b1 (P 1 − {0, 1, ∞}), corresponding to the nite étale covers of P 1 − {0, 1, ∞}, as combinatorial objects (the dessins d'enfants), and using the combinatorics to look for invariants identifying the Galois orbits of these covers. The only other moduli space of dimension 1 is M1,1 , the moduli space of elliptic curves (genus one curves with one distinguished point). This space is the quotient of the Poincaré upper half-plane by the proper and discontinuous (but not free) action of SL2 (Z). Finite-index subgroups of SL2 (Z) correspond to covers of M1,1 . As in the case of M0,4 , many specic families of these subgroups have been studied in detail, most familiarly the modular subgroups Γ(N ). Using graphs in the spirit of the theory of dessins d'enfants, F. Bogomolov and Y. Tschinkel characterize another family of very special nite-index subgroups of SL2 (Z), namely those corresponding to elliptic brations. Before passing from these two curves to what can be said in the case of general moduli spaces Mg,n , let us make a brief foray out of the geometric situation into the domain of Lie Galois theory, a subject that originates in the geometric situation but has been linearized by focusing on graded Lie algebras associated to the pronite fundamental groups rather than the groups themselves. A great deal of work has been done in this subject, mainly by Y. Ihara and his school, but we restrict ourselves here to discussing one conjecture which is a paradigm for the manner in which the problems in the domain arise in geometry, but raise their own interesting arithmetic questions. Since as above, we have π1 (P 1 −{0, 1, ∞}) ' Fb2 , the exact sequence (∗∗) gives a canonical homomorphism GQ → Out(Fb2 ). (∗∗∗) As an initial step, the passage from the geometric situation to the Lie situation involves replacing the pronite completions of fundamental groups by their pro-` completions, that is, the completions with respect to all nite quotients which (`) are `-groups for a xed prime `. Denote the pro-` completion of F2 by F2 . This completion is a quotient of the pronite completion by a characteristic subgroup, (`) so that (∗∗∗) yields a homomorphism GQ → Out(F2 ). Following Ihara, dene a ltration on GQ by setting (`)

I m GQ = Ker{GQ → Out(F2 /Lm+1 )} (`)

where Lm denotes the m-th term of the lower central series of F2 , and set

Grm GQ = I m GQ /I m+1 GQ .

INTRODUCTION

xiii

The following conjecture on the structure of the graded Lie algebra associated to the ltration I m GQ was stated (in fuller detail) by Ihara, who attributed it to Deligne. Conjecture.

The Lie algebra

·

M

¸ Grm GQ ⊗ Q `

m>0

is freely generated by generators s3 , s5 , s7 , . . ., where sm ∈ Grm GQ is the so-called Soulé element . Part of this conjecture, namely the fact that the Lie algebra is actually generated by the si , was proved by Hain and Matsumoto. In their contribution to this volume, they discuss the conjecture and show how to t it into a motivic framework. Now let's return to the situation of geometry and consider the (conjecturally anabelian) moduli spaces Mg,n . The geometry of these spaces has been described by explicitly cutting them into simply connected regions called cells, enumerated by objects known as fatgraphs, which are in fact equivalent to dessins d'enfant. The Hurwitz spaces are similar to the moduli spaces, but they parametrize equivalence classes of ramied coverings of Riemann surfaces, where two such coverings f : Y → X and f 0 : Y 0 → X 0 are equivalent if a diagram

Y

f

φ

² Y0

/X ψ

f0

² / X0

commutes for two biholomorphisms φ and ψ . A cellulation of the compactication of these spaces, analogous to that of the moduli spaces, is dened and studied in the article by M. Imbert. The Hurwitz spaces are closely related to special loci in the moduli spaces, namely the loci of points in the moduli spaces corresponding to marked Riemann surfaces admitting a particular group of automorphisms. The article by L. Schneps studies these loci, showing that under certain conditions (which are always fullled in genus zero), the special loci are themselves moduli spaces of smaller type. The morphisms mapping these smaller moduli spaces to the special loci of the larger ones are respected by the canonical Galois action on the fundamental groups of the moduli spaces, so that the addition of these morphisms to those previously studied in Grothendieck-Teichmüller theory adds new combinatorial information on the elements of Gal(Q/Q).

4. Inverse Galois theory. One of the most fundamental problems of Galois

theory is that of determining which nite groups can occur as Galois groups over a given eld K . This problem has been studied in many dierent situations and

xiv

INTRODUCTION

by many dierent methods; by various direct methods, by explicit computation and solution of obstructions to embedding problems, and by geometry. The geometry comes in when studying a eld of the type K(t) for an indeterminate t, and the main tools are curves, since the desired groups are exactly the Galois groups of covers of the projective line over K . When K = C , the inverse Galois problem is solved by Riemann's existence theorem; every nite group occurs. Although no completely general analog to Riemann's existence theorem exists over arbitrary elds, many partial analogs have been developed in dierent situations. The key notion is that of patching; covers are constructed locally over disks, and the pieces are patched together (agree) on the overlaps. Like the theories described above, the inverse Galois problem exists in characteristic 0 and p > 0, necessitating the use of dierent techniques. Patching techniques have been developed using formal schemes (formal patching) and non-archimedean disks (rigid patching) which yield partial or complete solutions to the inverse Galois problem over many dierent elds (large or algebraically closed elds of any characteristic, fraction elds of complete local rings other than elds, complete elds, henselian elds, and so on). In fact, one can obtain stronger results, such as specic information on the structure of the fundamental group for ane curves; in certain cases, it can even be shown that the fundamental group is free, or that it has the property that every split embedding problem has a proper solution. The contribution by D. Harbater contains a comprehensive account of all these methods and results. The domain of dierential Galois theory is an adaptation of the classical inverse Galois problem; instead of considering nite groups as Galois groups of Galois extensions of arbitrary elds, one considers linear algebraic groups as Galois groups of so-called Picard-Vessiot extensions of D-elds, which are elds F equipped with a derivation ∂ : F → F . Over algebraically closed elds of characteristic 0, this problem has been completely solved by the combined results of Ramis, Mitschi, Singer, van der Put and nally Hartmann; any linear algebraic group over such a eld is the dierential Galois group of a PicardVessiot extension. In their contribution to this volume, Matzat and van der Put develop a non-obvious analog of these results in the characteristic p situation; they introduce iterated dierential elds and give a complete formulation and solution to the inverse Galois problem over them.

Leila Schneps Paris, November 2002


Monodromy Groups of Coverings of Curves ROBERT GURALNICK Abstract. We consider nite separable coverings of curves f : X → Y over a eld of characteristic p ≥ 0. We are interested in describing the possible monodromy groups of this cover if the genus of X is xed. There has been much progress on this problem over the past decade in characteristic zero. Recently Frohardt and Magaard completed the nal step in resolving the GuralnickThompson conjecture showing that only nitely many nonabelian simple groups other than alternating groups occur as composition factors for a xed genus. There is an ongoing project to get a complete list of the monodromy groups of indecomposable rational functions with only tame ramication. In this article, we focus on positive characteristic. There are more possible groups but we show that many simple groups do not occur as composition factors for a xed genus. We also give a reduction theorem reducing the problem to the case of almost simple groups. We also obtain some results on bounding the size of automorphism groups of curves in positive characteristic and discuss the relationship with the rst problem. We note that prior to these results there was not a single example of a nite simple group which could be ruled out as a composition factor of the monodromy group of a rational function in any positive characteristic.

Contents

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Introduction The RiemannHurwitz Formula The Tate Module Upper Bounds for Genus Regular Normal Subgroups Minimal Genus for Composition Factors Composition Factors of Genus g Covers Estimates on Inertia Groups Automorphism Groups of Curves The Generalized Fitting Subgroup AschbacherO'NanScott Theorem Aschbacher's Subgroup Theorem Abelian Supplements References

2 6 8 12 14 16 19 21 24 27 29 31 41 43

Mathematics Subject Classication: 14H30, 14H37, 20B15, 20B25. Keywords: curves, coverings of curves, permutation group, automorphisms of curves, genus. The author gratefully acknowledges support from MSRI and NSF grant DMS 9970305.

1

2

ROBERT GURALNICK

1. Introduction Let k be a perfect eld of characteristic p ≥ 0. Suppose that X and Y are smooth projective curves over k and f : X → Y is a nonconstant separable rational map. The (arithmetic) monodromy group A of this cover is dened to be the Galois group of the Galois closure of the extension of function elds k(X)/k(Y ). Let Z denote the curve corresponding to the Galois closure. Let H be the subgroup of A corresponding to X , i.e. X = Z/H . It is possible that k 0 , the constant eld of Z , properly contains k . Let G be the normal subgroup of A consisting of those automorphisms which are the identity on k 0 . We call G the geometric monodromy group of the cover. Then A/G is isomorphic to the Galois group of k 0 /k . The general theme that we wish to stress is that many arithmetic and geometric properties of the cover f can be recast in properties of A and G and their permutation representation on the cosets of H . This program has proved very successful in attacking several problems in particular, exceptional polynomials [16], [42], [34], covers with a totally ramied point [35], exceptional rational functions [31] and the genus question. This approach has three parts. The rst is the translation of the arithmetic or geometric problem to a group theoretic one. The second is the solution of the group theoretic problem. Finally, the third problem is to determine which group theoretic solutions correspond to an actual geometric solution. All three parts may be dicult and interesting. In particular, the classication of nite simple groups and results about primitive permutation groups have been used to solve several outstanding problems (for example, see the above mentioned references). The main focus of this article is to study in more detail the problem of describing the covers if we bound the genus of X . For this problem, we may assume that k is algebraically closed and in particular A = G. There has been great progress when p = 0 or more generally if the cover is tame. See [22]. We will develop approaches here that are valid even in the presence of wild ramication. If the cover is Galois, then there are classical results bounding the order of Aut(X). By a classical result of Hurwitz, if the cover is tame and g(X) > 1, then |G| ≤ 84(g − 1). If p > 0, Stichtenoth [64] showed that |G| < 16g 4 with one explicit family of exceptions see also [56], [57]. The other extreme case is when the cover is indecomposable (or equivalently, the eld extension k(X)/k(Y ) is minimal). Since every cover is a composition of indecomposable covers, this is a critical case. There is in fact a very close connection between the Galois and non-Galois cases. In particular, if G has no genus zero representations, then it cannot act on a curve of small genus (relative to the size of G). This is already apparent in [64]. Let S be a (nonabelian) simple group. We say that S is a genus g group (in characteristic p), if S is a composition factor of the monodromy group of a cover f : X → Y with X of genus at most g . Since there exist covers from X → P1 of

MONODROMY GROUPS OF COVERINGS OF CURVES

3

degree n with monodromy group Sn (or An ) for g = 0 , we will concentrate on Chevalley groups. Thus, we let Ep (g) denote the set of genus g groups (in characteristic p) other than alternating groups. Similarly, let Eta p (g) denote the set of simple groups (other than alternating groups) which are composition factors of monodromy groups of tamely ramied covers X → Y with X of genus at most g . By [40], this problem reduces to the case where f is indecomposable. It is also easy to see that the critical case is when Y has genus 0. If p = 0, there is a recent result answering a question posed in [40] (the nal paper proving this result was done by Frohardt and Magaard [22]; other papers involved in the proof include [21], [32], [40], [58], [6], [49] and [51]) since the proof really only uses the assumption that the cover is tame, the result can be stated as follows: Theorem 1.1.

Eta p (g) is nite for each g .

Indeed, much more precise information is known and hopefully a complete determination of the monodromy groups of the tamely ramied indecomposable covers of genus zero (and in particular, indecomposable rational functions) will be available in the near future. In particular, there will be several innite families and a nite list of other examples. There will be a similar result for any xed genus g . We mention two results which involve special cases of this analysis. The rst is a special case in [31]:

Let f (x) ∈ Q(x) be an indecomposable rational function . Suppose that f is bijective modulo p for innitely many primes p. Aside from nitely many possibilities , the genus of the Galois closure of Q(x)/Q(f ) is at most 1.

Theorem 1.2.

A much more precise version of the theorem is in [31], where an essentially complete list of possibilities is given. After one solves the group theory problem, it is left to determine which possibilities actually arise. This involves a careful analysis of elliptic curves and results about torsion points and isogenies of elliptic curves over Q. The second result is a consequence of [32], [30] and [39].

Let g ≥ 4 and p = 0. Let X be a generic curve of genus g . If f : X → P1 is an indecomposable cover of degree n, then the monodromy group of f is either Sn with n > (g + 1)/2 or An with n > 2g . Theorem 1.3.

This was a problem originally studied by Zariski who proved that if g > 6 and f : X → P1 with X generic of genus g , then the monodromy group of f is not solvable (this is a special case of the result above using the observation of Zariski that any such cover is a composition of an indecomposable cover and covers from P1 to P1 ). A more precise statement of the theorem above is to say that the set of Riemann surfaces of genus g ≥ 4 which have indecomposable covers of degree n to P1 with monodromy group other than An or Sn is contained

4

ROBERT GURALNICK

in a proper closed subvariety of the moduli space of genus g curves. It is well known that Sn does occur as the monodromy group of the generic curve (for n > (g + 1)/2). It has been recently shown [17] that An actually does occur for n > 2g , thus giving a fairly complete picture of the situation when g > 3. If g < 4, there are more group theoretic possibilities. In unpublished work, Fried and Guralnick have considered some possibilities for g = 2. The recent work of Frey, Magaard and Völklein show that there are other examples when g = 3 (all the group theoretic possibilities for g = 3 are known by the results cited above). Until now, it was not known that a single simple group in any positive characteristic could be shown not to be a genus 0 group. In this article, we show that there are innitely many such groups. In particular, we show that: Theorem 1.4. If p does not divide the order of |S|, then S ∈ Ep (g) implies that S ∈ Eta p (g + 2) ⊆ E0 (g + 2). In particular , for any odd prime p and any g , there are innitely many simple groups not in Ep (g).

We also show that there are innitely many simple groups whose order is divisible by p which are not in contained in Ep (g) for a xed p and g . Let µp (S) be the smallest g such that S ∈ Ep (g). Let Chev(r) denote the family of simple groups which are Chevalley groups in characteristic r. Let Chevb (r) denote the groups in Chev(r) which have rank at most b. Indeed, we prove the following result.

Let X be a xed type of Chevalley group . Fix a nonnegative integer g . There are only nitely many pairs (p, q) with p a prime and q a prime power not divisible by p such that X(q) ¡S∈ Ep (g). More¢ precisely , µp (X(q)) → ∞ as q → ∞ for (p, q) = 1 and Ep (g) ∩ r6=p Chevb (r) is nite for each g . Theorem 1.5.

The proof shows that typically µp (X(q)) grows like a polynomial of degree close to b in q (as long as p does not divide q ). Abhyankar ([1], [2], [3], [4], [5]) has shown that many nite groups of Lie type (particularly the classical groups) are genus 0 groups in the natural characteristic and so the exclusion p 6= r is necessary. This led the author to make the following conjecture several years ago the positive characteristic analog of the GuralnickThompson Let Chev(r) denote the set of nite simple groups which are nite groups of Lie type over a eld of characteristic r. ¡S ¢ Conjecture 1.6. Ep (g) ∩ r6=p Chev(r) is nite . Given the classication of nite simple groups, this conjecture says that there are only nitely many simple groups in Ep (g) other than Chevalley groups in characteristic p. The previous theorem goes a long way towards proving the conjecture. Namely, the conjecture is true if we consider Chevalley groups of bounded dimension. The next step would be to prove the same result for xed q and then nally to prove


5

that the genus increases as the rank of the Chevalley group increases irrespective of eld size (all under the assumption that we are considering Chevalley groups in characteristic dierent from the characteristic of the eld). It is not clear what the right answer for exceptional groups in the natural characteristic is. It will also be quite dicult to handle the case of small elds this was already evident in the case for tame covers. Some of the techniques developed here should be useful. We prove two main results and then apply them to obtain the previous theorem. The rst is to show that one can check these questions by reducing to a few minimal congurations and in particular, if p does not divide | Aut(S)|, it reduces to the tame case. The results we obtain here give a much easier reduction for the genus problem even in characteristic zero (but do give less precise information). The analog of the reduction theorem in the tamely ramied case seems out of reach when wild ramication is present. The second is to show that there is a close connection between the genus of the Galois closure of the cover and the genus of X . In particular, let γp (S) be the minimal genus h > 1 of a curve Z (in characteristic p) so that S is a subgroup of Aut(Z). We show that if γp (S)/|S| is large compared to xed point ratios of elements in primitive permutation representations of S (and related groups), then S cannot be a genus g group for g small. This is used in conjunction with the following theorem.

Let X be a type of Chevalley group . Let p and r be distinct primes . There exists a constant c = c(X) such that if X(ra ) acts on a curve of genus g > 1, then g ≥ c|X(ra )|. Theorem 1.7.

Using patching constructions, one can show that the constant c(X) → 0 as the rank of X goes to innity and also that the characteristic assumptions are necessary. Of course if g ≤ 1, we know the automorphism groups. For tame covers, a more specic version of the previous result is the Hurwitz bound on the size of Aut(X). We will explore other bounds on automorphism groups of curves in future work. The paper is organized as follows. In section 2, we discuss the Riemann Hurwitz formula and show the connection between xed points in permutation representations and the genus. In section 3, we indicate the connection between the `-torsion in the Jacobian (and more generally the Tate module) and the genus and use some elementary representation theory to obtain some inequalities on the genus. In section 4, we give upper bounds for µ(S) and also show how to reduce to the case that the cover is indecomposable and the map is to P1 . In section 5, we deal with the case of regular normal subgroups and show how one can reduce to a smaller case (at the expense of possibly slightly increasing the minimal genus).

6

ROBERT GURALNICK

In section 6, using the previous sections, we prove our main reduction result and prove Theorem 1.4. In section 8, we obtain estimates for the RiemannHurwitz formula when we have certain conditions on the inertia groups. In section 9, we indicate the relationship between the genus of the Galois closure and µ(S) and prove our main result about Chevalley groups. In sections 10, 11 and 12, we introduce some group theoretic notation and machinery. In particular, we prove a simple version of the AschbacherO'Nan Scott theorem that we use in the paper. There is a nice proof of this in the literature (see [48]), but our proof is quite short and elementary and gives the result precisely in the form we require. We also include a proof of a version of Aschbacher's theorem on subgroups of classical groups. This has been of fundamental importance in studying primitive permutation groups. In the nal section, we turn to a dierent topic. It does show how group theory plays an important role in studying covers of curves. It gives a simpler example of a group G such that G/Q is an abelian p0 -group on two generators where Q is a quasi-p group (i.e. is generated by its Sylow p-subgroups) but G 6= QA where A is abelian. In the case that G/Q is cyclic, clearly cyclic supplements always exist and this easy fact is used in the proof of the Abhyankar conjecture for curves. This example in conjunction with work of Harbater and Van der Put [44] shows that the strongest form of a conjecture of Abhyankar about covers unramied outside a normal crossing in the ane plane is not true. A much more general but more complicated construction is given by the author in the appendix of [44]. Some of the results stated above do depend on the classication of nite simple groups and we do use that theorem in a few places in this paper. However, for the most part, the results about Chevalley groups do not depend on the classication. In particular, one can give a proof of the minimal genus result for Chevalley groups without reference to the classication. The author wishes to thank MSRI for its generous hospitality. Much of this work was done while the author was a Research Professor at MSRI during the Fall 1999 program on Galois groups and fundamental groups. He would also like to thank the referee for a careful reading of the manuscript.

2. The RiemannHurwitz Formula Let G be a nite group and Ω a G-set of size n. If I ≤ G, dene ind(I) = ind(I, Ω) = n − orb(I), where orb(I) is the number of orbits of I on Ω. Let f (x) = f (x, Ω) denote the size of the set of xed points of x ∈ G. It follows by a result of Burnside (or by Frobenius reciprocity) that


orb(I) = |I|−1

X

7

f (x).

x∈I

If I is a descending sequence of normal subgroups of I = I0 ≥ I1 ≥ I2 ≥ . . . ≥ Im , dene

ρ(I, Ω) =

X

[I : Ii ]−1 ind(Ii ).

i=0

We will often abuse notation and write ρ(I) for ρ(I, Ω). This notation will be used in the case that I is the inertia group of a point on a curve and I is the sequence of higher ramication groups. We can now express the RiemannHurwitz formula in group theoretic notation. Let k be an algebraically closed eld of characteristic p ≥ 0 and X, Y smooth projective curves over k with the genus of X , g(X) = g and g(Y ) = h. Let f : X → Y be a separable nonconstant rational map of degree n. Let Z denote the curve corresponding to the Galois closure and G the monodromy group of the cover. Let B ⊂ Y denote the (nite) set of branch points of the cover. If y ∈ B , let I = I(y) denote the inertia group of some point in Z over y and let Ii (y) denote the ith higher ramication group. See [61] for details about higher ramication groups. The RiemannHurwitz formula can now be stated: Theorem 2.1.

2(g − 1) = 2n(h − 1) +

X

ρ(I(y)).

y∈B

In particular, we record: Corollary 2.2.

If h > 1, then n ≤ (g − 1)/(h − 1) ≤ (g − 1).

Thus, for a xed g , given n and h > 1, there are only nitely many possibilities for G. If h = 1, we have a similar result. This is stated in [32] for characteristic 0; however the proof is identical using the RiemannHurwitz formula. For the second corollary, just note that ind(I) ≥ n/2 for any nontrivial I in the regular representation.

If the cover f : X → Y is indecomposable of degree n and h = 1, then one of the following holds :

Corollary 2.3.

(i) n is prime , G is cyclic of order n, g = 1 and the cover is unramied . (ii) G ∼ = An or Sn . √ (iii) 2(g − 1) ≥ n − 1. Corollary 2.4.

following holds :

If the cover f : X → Y is Galois and h = 1, then one of the

8

ROBERT GURALNICK

(i) G is abelian and the cover is unramied . (ii) 2(g − 1) ≥ n/2. So the critical case is when h = 0. Also, note that the each of the groups I(y) has a normal Sylow p-subgroup, I1 (y), and that I(y)/I1 (y) is a cyclic p0 -group. Let N denote the normal subgroup of G generated by the subgroups I1 (y), y ∈ B . Then G/N is a p0 -group and is the monodromy group of the cover Z → Z/N . Moreover G/N generated by the images of the I(y) and choosing appropriate generators for the (cyclic) images of the I(y), the product of these generators is 1 i.e. we have the description of the monodromy group G/N as in characteristic zero.

3. The Tate Module Let Z be a smooth projective curve of genus g over an algebraically closed eld k of characteristic p ≥ 0. Let J = J(Z) be the Jacobian of Z . So J is the set of formal nite sums of points of Z with weight zero modulo those elements which correspond to divisors of functions on Z . If m is a positive integer, let J[m] denote the m-torsion points on J . If ` is a prime, let T` (Z) denote the inverse limit of J[à ]. So this is a Z` -module and of course, Aut(Z) acts on this module as well. Let T`0 (Z) = T` (Z) ⊗ Q` . The following result is classical.

Let H be a nite subgroup of Aut(Z). Let V = T`0 for any ` 6= p. Then 2g(Z/H) = dim CV (H). If p 6= ` and ` does not divide the order of H , then 2g(Z/H) = dim CJ[`] (H).

Lemma 3.1.

The Jacobian is a g -dimensional abelian variety. Thus, for any m not divisible by p, J[m] has order m2g . Let d = |H| and f : Z → Z/H the covering map of degree d. Let f∗ denote the induced map from J(Z) → J(Z/H). If P y ∈ Z/H , let f ∗ (y) = ny z , where the sum runs over x with f (z) = y and ny is the order of the inertia group of any such z (note this is independent of z ). Then f∗ induces a map from J(Z/H) → Y . In particular, note that the image of f ∗ is contained in J(Z)H . It follows immediately from the denitions that f ∗ f∗ (D) = dD for element D ∈ J(Z)H and similarly that f∗ f ∗ (D) = dD for element D ∈ J(Z/H). In particular, this shows that T`0 (Z)H ∼ = T`0 (Z/H) for all ` 6= p. If ` does not divide the order of H , then the xed points on the Tate module have the same dimension as the xed on points on the `-torsion subgroup of the Jacobian. ¤ Proof.

We could replace V by the `-torsion subgroup for some ` not dividing |H|. We remark that it is well known that the Tate module is really independent of `. Also if ` does not divide the order of H and the genus is at least 2, then H acts faithfully on the `-torsion subgroup.


9

The case ` = p is also interesting but in fact in that case V can be 0 (and in general 0 ≤ dim V ≤ g(Z)). If p = 0, we could also use the module of holomorphic dierentials on Z and remove the 2 in the formula. We point out an interesting consequence. If H is a subgroup of G, let 1G H denote the permutation module for G over C.

Let Z be a curve over k with G a nite group of automorphisms of Z . Suppose that H and K are subgroups of G such that 1G H is isomorphic to . Then g(Z/H) ≤ g(Z/K) . a submodule of 1G K Corollary 3.2.

Let V denote the Tate module for some suciently large prime ` other than the characteristic of the curve. By Frobenius reciprocity, dim CV (H) = G G dim HomG (1G H , V ) and dim CV (K) = dim HomG (1K , V ). Since 1H is a direct G G G summand of 1K , dim HomG (1H , V ) ≤ dim HomG (1K , V ), whence the result. ¤

Proof.

Here are some well known situations where the previous result applies. (i) G = Sn or An . Let H be the stabilizer of a subset of size j and K the stabilizer of a set of size j 0 with 1 ≤ j ≤ j 0 ≤ n/2. (ii) P SL(n, q) ≤ G ≤ P ΓL(n, q). Let H be the stabilizer of a subspace of dimension j and K the stabilizer of the a subspace of dimension j 0 with 1 ≤ j ≤ j 0 ≤ n/2. (iii) G is a classical group and H is the stabilizer of a totally singular 1-space. Then we can take K to be the stabilizer of any totally singular space of less than maximal rank or usually the stabilizer of a nonsingular space as well. See [19] for a precise statement. We now prove some easy representation theoretic facts that will be useful in estimating genera. Lemma 3.3. Let G be a nite group with a normal subgroup E . Let H be a maximal subgroup of G which does not contain any normal subgroup of G contained in E . Assume that E = X1 × . . . × Xt with the Xi = X gi being the set of G-conjugates of X := X1 . Set Y = X2 × . . . × Xt . Let N = NG (X) = NG (Y ). If V is a nite dimensional CG-module , then dim CV (H) ≥ dim CV (NH (X)Y )− dim CV (NG (X)).

Since both sides of the inequality we are proving are additive over direct sums and since V is a completely reducible CG-module, it suces to prove the result for V irreducible. If V is trivial, there is nothing to prove. Suppose that E does not act faithfully on V . Let K denote the kernel of E on V . Since H is maximal and does not contain K , G = HK and NG (X) = NH (X)K and similarly for Y . In this case 0 = CV (G) = CV (HK) = CV (H) and CV (NG (X)) = CV (NH (X)) whence we have equality. So we may assume that E acts faithfully on V . Note that since NG (X) ≥ E , CV (NG (X)) = 0. Proof.

10

ROBERT GURALNICK

We may assume that CV (Y ) = W is nonzero (or the result obviously holds). P P Let Yi = Y gi . Note that i CV (Yi ) is a direct sum (for if vi = 0 with 0 6= v1 T and vi ∈ CV (Yi ), then v1 ∈ CV (Y )∩ i>1 CV (Yi ) = CV (Y )∩CV (X) = CV (E) = 0). Now NG (X) leaves W invariant (since NG (X) normalizes Y ). As we have seen above the distinct images of W under G form a direct sum Also the stabilizer of W is NG (X) (for if gW = W and g is not in NG (X), then hY, Y g i = E would imply that W = CV (E) = 0). It follows that V ∼ = WNHH (X) = WNGG (X) and so V ∼ as H -modules (since as noted G = HNG (X)). So by Frobenius reciprocity, CV (H) ∼ ¤ = CW (NH (X)) = CV (NH (X)Y ). The following variant of the previous result will also be useful. Lemma 3.4. Let G be a nite group with a normal subgroup E . Let H be a maximal subgroup of G which does not contain any normal subgroup of G contained in E . Assume that E = X1 ×. . .×Xt with the Xi = X gi being the set of Q G-conjugates of X := X1 . Let ∆ = {1, . . . , t}. Let δ ⊂ ∆ and set Xδ = i∈δ Xi . Let Yδ = Xδ0 where δ 0 is the complement of δ . Let Nδ = NG (Xδ ). Let V be an irreducible CG-module containing an E -submodule W of the form W1 ⊗ . . . ⊗ Wt with Wi an irreducible Xi module with Wj trivial if and only if j ∈ δ 0 . Then dim CV (H) ≥ dim CV (NH (Xδ )Yδ ) − dim CV (NG (Xδ )).

Note that NH (Xδ )Yδ ≤ NG (Xδ ) and so each term on the righthand side of our desired inequality is non-negative. First suppose that E does not act faithfully on V . Let K denote the kernel of E on V . Since K is normal in G, G = KH . Then CV (H) = CV (HK) = CV (G). If G acts trivially, then the lefthand side is 1 and the righthand side is 0. Otherwise, the lefthand side is 0. Since G = HK , NG (Xδ ) = KNH (Xδ ), whence the righthand side is also 0. So we may assume that E acts faithfully on V . If W = V , Yδ has no xed points on V for δ any proper subset of ∆ (for Yδ contains some Xj and V restricted to Xj is a direct sum of copies of Vj ). Thus, the righthand side of the equation is 0. P Let U := Uδ = CV (Yδ ). So W ⊆ U . By irreducibility, V = Uγ where γ is the orbit of δ . Note that this sum is in fact direct, since the terms are direct sums of irreducible E -modules which are not isomorphic (as they have dierent kernels). Moreover, the stabilizer in G of Uδ is precisely NG (Xδ ) (because of the permutation action on the Xi ). Thus, V is isomorphic to the induced module, G H UN . Since G = NG (Xδ )H , this implies that VH ∼ and so by = UN G (Xδ ) H (Xδ ) Frobenius reciprocity, dim CV (H) = dim CU (NH (Xδ )). Since U = CV (Yδ ), it follows tht CV (NH (Xδ ))(Yδ ) = CU (NH (Xδ )), whence the result. ¤ Proof.

We next deal with diagonal subgroups (see section 11 for terminology). The result is actually more general than we state the condition that X is simple is not necessary.


11

Lemma 3.5. Let G be a nite group with a minimal normal subgroup E = X1 × . . . × Xt with the Xi the set of G-conjugates of the nonabelian simple group X = X1 and t > 1. Let H be a maximal subgroup of G not containing E such that H ∩ E is a diagonal subgroup of E . Let ∆ = {1, . . . , t}. If δ ⊂ Q ∆, set Xδ = i∈δ Xi . Let Yδ = Xδ0 where δ 0 is the complement of δ . Let Nδ = NG (Xδ ). If V is a nite dimensional CG-module , then dim CV (H) ≥ dim CV (NH (X12 )Y12 ) − dim CV (NG (X12 )).

It suces to assume that V is irreducible. Note that the righthand side is always non-negative (since NG (X12 ) ≥ NH (X12 )Y12 ). If V is a trivial G-module, there is nothing to prove. If E acts trivially on V , then CV (H) = CV (HE) = 0. On the other hand, since G = HE , we have NG (X12 ) = ENH (X12 ) and so CV (NH (X12 )Y12 ) = CV (NG (X12 )). So assume that E acts nontrivially on V . In particular, this implies that CV (NG (X12 )) = 0. If Y12 has no xed points on V , then clearly the result holds (since the right hand side of the inequality is 0). Suppose rst that CV (Y1 ) is nonzero. Then as in the previous result, V is the direct sum of the CV (Yi ) = [Xi , V ]. Since H ∩ E is a diagonal subgroup, this implies that (H ∩ E)Yi ≥ E and so H ∩ E has no xed points on [Xi , V ] and so none on V . Since H ∩ E ≤ NH (X12 ), the right hand side is 0, whence the result. Finally, assume that W := CV (Y12 ) 6= 0, but CV (Y1 ) = 0. This implies that every irreducible E -submodule of V is of the form U1 ⊗ . . . ⊗ Ut with Ui an irreducible Xi -module with Ui nontrivial for precisely 2 terms. In particular, it follows that W is a sum of E -homogeneous components. Let Wij = CV (Yij ). Since Wij is also a sum of E -homogeneous components and there are no common irreducibles among the distinct Wij , it follows that V = ⊕Wij and the nontrivial Wij must be a single G-orbit. Clearly, W is invariant under NG (X12 ) and indeed, we see that this is the full stabilizer of W . Since G = HE = HNG (X12 ), H acts transitively on the Wij as well. Thus, V ∼ = WNHH (X12 ) and so by Frobenius reciprocity, CV (H) = CW (NH (X12 )) = CV (NH (X12 )Y12 ). ¤ Proof.

The next lemma gives a bound in certain additional cases.

Let A be a nite group and G a normal subgroup . Let M = NA (M ) be a maximal subgroup of A such that M ∩G is properly contained in the maximal subgroup J of G. Assume moreover that the intersection of any proper subset of M -conjugates of J properly contains M ∩ G. Let V be an irreducible CA-module . Then either G acts trivially on V or dim CV (J) ≤ dim CV (M ). Lemma 3.6.

Since M does not contain G, it follows that A = GM . Let W = CV (J). P Choose a transversal 1 = x1 , . . . xt in M for A/G. We claim that x W is a P i direct sum. If not, then¡there exists a nonzero vector v ∈ W with v ∈ I>1 xi W . ¢ T T xi xi Thus, v ∈ CV (J) ∩ CV . By assumption, i>1 J is not contained in i>1 J Proof.

12

ROBERT GURALNICK

T ® P J and so G = J, i>1 J xi . Hence v ∈ CV (G) = 0. Thus, the map w 7→ xi w is an injection from W to CV (M ). ¤ Note that if M ∩ G is not maximal in G and J is a maximal self-normalizing subgroup of G containing M ∩G and A/G has order 2, the hypotheses are always satised.

4. Upper Bounds for Genus The rst result is classical. Let k be an algebraically closed eld of characteristic p ≥ 0. All covers refer to curves over k . Lemma 4.1.

Sn .

There exist covers f : P1 → P1 of degree n with monodromy group

Let f (x) = x2 h(x), where h is a polynomial of degree n−2 with distinct nonzero roots. Choose h in addition so that f is separable and indecomposable (these are both open conditions on the coecients of h). Then the monodromy group is a primitive group of degree n containing a transposition (consider the inertia group over 0). It is elementary to prove that a primitive permutation group of degree n containing a transposition is Sn . ¤

Proof.

Let S be a nonabelian simple group and let n be the minimal index of a maximal subgroup of S . Then there exists a cover f : X → P1 of degree n with X of genus n + 1. In particular , µ(S) ≤ n + 1. Lemma 4.2.

By [63], there exists an unramied S -cover of a genus 2 curve Y . Thus, there exists a degree n cover X of Y with monodromy group S . By the Riemann Hurwitz formula, X has genus n + 1. ¤ Proof.

With some eort, one should be able show that µ(S) < n/2 at least for p 6= 2. One would need a slight generalization of a the generation result from [29] given below something like given 1 6= x ∈ S , there exists y ∈ S with S = hx, yi such that y and xy have order prime to p. If p 6= 2, this would give µ(S) ≤ (n − 1)/2 and a slightly weaker bound for p = 2. We give the proof of a slightly better result in characteristic zero only. One can do a bit better for most simple groups because they can be generated by elements of order 2 and 3, we can require in this case that (in characteristic 0) X has genus at most n/6 + 1 (and asymptotically for many families that is the best that can be done).

Let G be an almost simple group with socle S acting faithfully on a set of cardinality m. If x is a nontrivial element of G, then there exists a Riemann surface X and f : X → P1 of degree at most m with X of genus g ≤ ind(x)/2 and monodromy group G0 with S ≤ G0 ≤ G. In particular , if n is the minimal degree of a permutation representation of S , then there exists a Riemann surface f : X → P1 of degree m with X of genus g ≤ n/4. Lemma 4.3.


13

By [29], there exists an element y ∈ G so that G0 := hx, yi ≥ S . By passing to a G0 -orbit if necessary we may assume that G = G0 . By Riemann's existence theorem, there exists a 3 branch point cover f : X → P1 with inertia groups generated by x, y and z := (xy)−1 . Since ind(y), ind(z) < m, it follows that g(X) ≤ ind(x)/2. Apply this result to the case that G = S and x is an involution to obtain the last statement. ¤

Proof.

[40] If f : X → Y is a branched covering and f = f1 ◦ f2 , then any composition factor of the monodromy group of f is a composition factor of the monodromy group of fi for i = 1 or 2. Lemma 4.4.

We will need the following result about minimal permutation representations of a groups with a given composition factor.

Let S be a nonabelian simple group . Let n be the smallest cardinality of a faithful G-set for any group G with S as a composition factor of G. Then F ∗ (G) = S and n is the index of the largest proper subgroup of S . Lemma 4.5.

Let Ω be the given G-set of size n. Note rst that G is transitive (for otherwise S is a composition factor in G/K or K where K is the subgroup acting trivially on some G-orbit and either group acts faithfully on a smaller G-set). We claim also that G is primitive on Ω. Otherwise, S is a composition factor of the stabilizer of a block or a composition factor of G acting on the blocks. In either case, we would have a smaller action with S as a composition factor. Let H be a point stabilizer. If G has a normal abelian subgroup or 2 minimal normal subgroups, then H has a smaller faithful orbit and has S as a composition factor. So let L be a simple component of G and let Li , 1 ≤ i ≤ t be the Gconjugates of L. Thus, G has a unique minimal normal subgroup. Suppose that S is not a component of G. Then S embeds in G/K where K is the subgroup normalizing each Li . Thus, t ≥ n, but (cf. section 11), n ≥ 5t , a contradiction. So S is a component of G and G has a unique minimal normal subgroup. Let m be the index of the largest proper subgroup of S . By section 11, one of the following holds: n ≥ mt or n ≥ |S| > m. Thus, t = 1 (as claimed). ¤

Proof.

It is convenient to dene md(S) to be the smallest index of a proper subgroup of S . We remark that md(S) is known for all S cf [45].

Let f : X → Y be a branched covering of degree n with S a nonabelian composition factor of the monodromy group G of f . Assume that Y has genus at least 1. Lemma 4.6.

(i) If Y has genus at least 2, then g(X) − 1 ≥ n ≥ md(S); (ii) If S is not an alternating group , then g(X) − 1 > n/12 ≥ md(S)/12.

14

ROBERT GURALNICK

By Lemma 4.4, we may assume that f is indecomposable. Let h = g(Y ). If h > 1, then the RiemannHurwitz formula yields g − 1 ≥ n(h − 1) ≥ n. By the previous lemma, n ≥ md(S). Suppose that h = 1. Since S is nonabelian, the cover must be ramied. Then the RiemannHurwitz formula yields that g − 1 ≥ (1/2)ρ(J) where J is a nontrivial inertia group. Clearly, ρ(J) ≥ ind(J) and so by [49], ind(J) ≥ n/6 whence the second statement. ¤ Proof.

5. Regular Normal Subgroups In this section, we show that the ane case can be reduced to the other cases. We rst prove two general results.

If G is a nite primitive permutation group with point stabilizer H and N is a regular normal subgroup , then H ∩ H g contains no nontrivial normal subgroup of H for g any nontrivial element of G \ N . Lemma 5.1.

Let K be a nontrivial normal subgroup of H in H ∩H g . Since G = HN , we may assume that g ∈ N . Then H ∩ H g = CH (g) and so NG (K) ≥ hH, gi = G (by the maximality of H ). This contradicts the fact that H contains no nontrivial normal subgroup of G. ¤ Proof.

Lemma 5.2. Let H be a group of automorphisms of a nite group N which is transitive on the set of all nontrivial elements of N . If H is not solvable , then H contains a normal cyclic subgroup C with F ∗ (H/C) simple .

N is characteristically simple and must have all elements of the same order, whence N is an elementary abelian p-group for some prime p. The result follows easily from Aschbacher's theorem on subgroups of classical groups (see the appendix). ¤ Proof.

We now x some notation. Fix a prime p. All curves will be smooth projective curves over an algebraically closed eld of characteristic p. Let S be a nite nonabelian simple group. Fix a non-negative integer d. Let λ(S, d) denote the smallest positive n such that there exists an indecomposable separable branched cover f : X → Y of degree n with monodromy group G such that S is a composition factor of G and X has genus at most d (if the characteristic is not clear, we write λp (S, d)). Similarly, let λ0 (S, d) denote the smallest positive n0 such that there exists an indecomposable separable branched cover of degree f : X → Y of degree n with monodromy group G such that S is a component of G and X has genus at most d. Let λ00 (S, d) denote the smallest positive n such that there exists an indecomposable separable branched cover f : X → Y of degree n with monodromy group G such that F (G) = 1, X has genus at most d and S is a composition factor of G.


15

In particular, to say that any of these quantities is nite is to say that such covers exist.

Let f : X → Y be an indecomposable separable nonconstant map of degree n = λ(S, d) with X of genus g ≤ d. Assume that S is a nonabelian composition factor of the monodromy group G of f . Assume that G contains a regular normal abelian subgroup N . Then Y has genus zero and one of the following holds : Theorem 5.3.

(i) g = 0 and λ0 (S, 2) < n; (ii) g = 0 and λ00 (S, 1) < n; (iii) g = d, G acts transitively on the nontrivial elements of N via conjugation and λ0 (S, d + 1) < λ(S, d). Proof. Let Ω denote the G-set of degree n corresponding to the cover. Let H be the stabilizer of a point ω and let Ωi be the nontrivial H -orbits on Ω. Let Hi be the stabilizer of a point in Ωi . Identifying H with G/N , we may identify Ωi with the G-set G/N Hi . Let Z denote the Galois closure of X/Y and consider the curves Xi := Z/N Hi and let gi denote the genus of Xi . If x ∈ G, then write x = yz with y ∈ H and z ∈ N . Let fix(x, Ω) denote the cardinality of the set of xed points of x on Ω. We note that fix(x, Ω) ≤ P 1 + fix(y, Ωi ). For if x is conjugate to y , this is clear while if x is not conjugate to y , then fix(x, Ω) = 0. Then ind(J, Ωi ) = ind(JN/N, Ωi ). The previous paragraph shows that for P any subgroup J of G, ind(J, Ω) ≤ 1 + i ind(J, Ωi ). Let h denote the genus of Y . Now applying the RiemannHurwitz formula to the curves X and Xi , we obtain:

2(g − 1) = 2n(h − 1) +

X

ρ(J, Ω) ≥ 2n(h − 1) +

J

X

ρ(J, Ωi ).

J,i

Here the sum is over the inertia groups (and higher ramication groups) J of the cover X → Y . Now X 2(gi − 1) = 2ni (h − 1) + ρ(J, Ωi ), and so since n = 1 +

P

J

ni , (g − 1) ≥ (h − 1) +

X

(gi − 1).

Note that the monodromy group of the cover of Xi → Y is G/N ∼ = H (by Lemma 5.1). By minimality, it follows that gi > d ≥ g for each i and so h = 0. This implies that either g = 0 and each gi = 1 or H has only one nontrivial orbit on Ω in which case g = g1 − 1 ≥ d and so g = d.

16

ROBERT GURALNICK

Suppose that the second case occurs and g 6= 0. Now apply Lemma 5.2 to conclude that (3) holds. Now suppose the rst case holds. So g = 0. Now start over and choose an indecomposable cover φ : W → P1 of degree m with W of genus 1 and m minimal with S as a composition factor. Note that m < n. So this gives a primitive permutation group. If this group has a normal elementary abelian group, then we repeat the argument and obtain a cover as in the second case (with the genus at most 2) and so λ0 (S, 2) < n. If not, then we conclude that λ00 (S, 1) < n. ¤

Let f : X → Y be an indecomposable separable nonconstant map of degree n = λ (S, d) with X of genus g ≤ d. Assume that S is a nonabelian composition factor of the monodromy group G of f . Then G has a unique minimal normal subgroup or d = 0 and λ00 (S, 1) < λ00 (S, 0). Lemma 5.4.

00

Assume that G has more than one minimal normal subgroup. Let Ω be the G-set of size n associated with the cover. Let H be a point stabilizer. Let N be a minimal normal subgroup of G. Since there are 2 minimal normal subgroups of G, N is a regular normal nonabelian subgroup. So G = HN is a semidirect product. Dene the curves Xi as in the previous proof. Arguing precisely as above, we have the same possibilities (and note that S is a composition factor of G/N and G/N has no normal abelian subgroups). In this case, H has more than one orbit on the nontrivial elements of N , eliminating that possibility. So it follows that g = 0 and gi = 1 for each i. ¤ Proof.

We note that for most S , the situations in the lemma cannot occur. For example, if S = Am , m ≥ 5, then there is a degree m cover from P1 to P1 and so λ(S, d) = λ0 (S, d) = λ00 (S, d) = m.

6. Minimal Genus for Composition Factors Let S be a nite nonabelian simple group. As in the previous section, all curves considered are over an algebraically closed eld of characteristic p. Let µ0p (S) = µ0 (S) denote the minimal genus g of a curve X so that there exists a cover f : X → Y with f indecomposable and S is a component of the monodromy group of f . Let µ00 (S) denote the minimal genus g of a cover f : X → Y with f indecomposable such that the monodromy group has no normal abelian subgroup and S is a composition factor. Clearly, we have: Lemma 6.1.

µ(S) ≤ µ00 (S) ≤ µ0 (S).

We rephrase Theorem 5.3 in this notation. Lemma 6.2.

Assume that µ(S) < µ00 (S). Then one of the following holds :


17

(i) µ(S) = 0 and µ0 (S) ≤ 2; (ii) µ(S) = 0 and µ00 (S) = 1; (iii) µ(S) > 0, and µ0 (S) = µ(S) + 1. Proposition 6.3.

µ00 (S) = µ0 (S) or µ0 (S) ≤ 2.

Let f : X → Y be a separable branched covering of degree n = λ00 (S, g) with X of genus g = µ00 (S), S a composition factor of the monodromy group G of the cover. We may assume that G has no normal abelian subgroup and that µ0 (S) > 2. Let Z denote the curve corresponding to the Galois closure. Let H be the subgroup with X = Z/H . If S is a component of G, µ00 (S) = µ0 (S). So assume that this is not the case. It follows that S is a composition factor of G/F ∗ (G). Since G has no normal abelian subgroup, we can write E = F ∗ (G) as a direct product of conjugates of a subgroup J . Let J 0 be the direct product of all the other distinct conjugates of J . Let V denote a the complexication of a Tate module for Z . By Lemmas 3.3 and 3.4, dim CV (H) ≥ dim CV (NH (J)J 0 ) − dim CV (NG (J)). Note that S is a composition factor of the monodromy group of the cover Z/NG (J) → Y . It follows that Z/NG (J) has genus g 0 which is at least µ(S). First suppose that g 0 > 1. Then the genus of Z/(NH (J)J 0 ) is at least 5(g 0 − 1) + 1 (by the RiemannHurwitz formula and the fact that the degree of Z/(NH (J)J 0 ) → Z/NG (J) is at least 5). Hence 2g = dim CV (H) ≥ 8(g 0 − 1). Thus, g = µ00 (S) ≥ 4(g 0 − 1) ≥ 4(µ(S) − 1). Thus µ00 (S) > µ(S) and so the previous lemma applies. If µ(S) = 0, then µ00 (S) ≤ 2, contradicting the fact that g ≥ 4(g 0 − 1) ≥ 4. Otherwise, µ00 (S) ≤ µ(S) + 1 and so µ(S) + 1 ≥ 4(µ(S) − 1), whence µ(S) = 1 and g = µ00 (S) ≤ 2, contradicting the fact that g ≥ 4. Next consider the case that g 0 ≤ 1. If g 0 = 1, since Z/(NH (X)Y ) → Z/NG (X) is not an abelian, Z/(NH (J)J 0 ) has genus g 00 ≥ 2. Again, by Lemmas 3.3 and 3.4, g ≥ g 00 − 1 ≥ 1. It follows that µ(S) < µ00 (S) and λ(S, µ(S)) < λ00 (S, µ00 (S)) (because there is a smaller degree cover which yields a cover with composition factor S and genus no larger than g ). Thus, the minimal degree cover achieving µ(S) must have an abelian normal subgroup. It follows by the proof of Theorem 5.3 that λ00 (S, µ(S) + 1) < λ(S, µ(S)) or µ0 (S) ≤ 2. The rst condition does not hold by the inequality at the start of the paragraph and the second does not hold by assumption. This completes the proof. ¤

Proof.

Corollary 6.4.

Either µ(S) ≤ µ0 (S) ≤ 2 or µ0 (S) ≤ µ(S) + 1.

Assume µ0 (S) > 2. Then µ0 (S) = µ00 (S). By Lemma 6.2 µ00 (S) ≤ µ(S) + 1. ¤ Proof.

The previous result allows us to concentrate on computing µ0 (S) i.e. a lower bound for µ0 (S) is very close to the lower bound for µ(S). The next result essentially reduces this to the almost simple case.

18

ROBERT GURALNICK

Let f : X → Y be an indecomposable degree n cover with monodromy group G with S a component of G. Assume that X has genus g = µ0 (S). Moreover , assume that n is minimal with respect to these conditions . Then one of the following holds : Theorem 6.5.

(i) F ∗ (G) = S ; or (ii) F ∗ (G) = S × S and H ∩ F ∗ (G) is a diagonal subgroup of F ∗ (G); or (iii) µ0 (S) ≥ md(S)/12 + 1. Let E = F ∗ (G). Let Z be the curve corresponding to the Galois closure of the cover. Let H be the subgroup of G of index n with X = Z/H . Suppose that we can write E = A1 × . . . × At , where the Ai are all the conjugates of A = A1 ∼ = S m , H ∩ E is the direct product of the subgroups H ∩ Ai and NH (A)C(A) does not contain A. By Lemmas 3.3 and 3.4, g ≥ g(Z/(NH (A)C(A)) − g(Z/NG (A)). If h := g(X/NG (A)) ≥ 2, then by the RiemannHurwitz formula, g(Z/(NH (A)C(A)) − 1 ≥ md(S)(h − 1) and so g ≥ (md(S) − 1)(h − 1) ≥ (md(S) − 1). If h = 1, then the same argument (together with Lemma 2.3 and the fact that we may assume that S is not an alternating group), implies that g ≥ md(S)/12 + 1. So we may assume that h = 0 and so g ≥ g(Z/(NH (A)C(A)). Note that the monodromy group of the cover Z/(NH (A)C(A)) → Z/NG (A) is NG (A)/CG (A) and so has S has a composition factor (since C(A)NH (A) does not contain A). Thus, by minimality, A = E . So we cannot decompose E in such a manner. By the AschbacherO'NanScott Theorem, this implies that either F ∗ (G) = S or E is the unique minimal normal subgroup of G and H ∩ E is a full diagonal subgroup of E . In the latter case, we apply Lemma 3.5. Arguing exactly as above, by minimality, we see that F ∗ (G) = S × S . ¤ Proof.

Putting together the previous results, we obtain:

Let S be a nonabelian simple group . If µ0 (S) < (md(S))/12 + 1, then there exists an indecomposable cover f : X → P1 with X of genus g and with S a composition factor of the monodromy group J of f such that g ≤ µ(S) + 1 or g = 2 and one of the following holds : Theorem 6.6.

(i) F ∗ (J) = S ; or (ii) F ∗ (J) = S × S and H ∩ F ∗ (J) is a full diagonal subgroup of F ∗ (J). If p is a prime that does not divide the order of Aut(S), the previous theorem essentially asserts that the minimal genus of any group involving S does not have order divisible by p, whence we can apply the results about tame covers and so we obtain the following result. Note this says nothing about characteristic 2.

Let S be a nonabelian simple group . If p does not divide the order of Aut(S), then one of the following holds :

Theorem 6.7.


19

(i) µp (S) ≥ md(S)/12 + 1; or (ii) µ0 (S) ≤ 2; or (iii) µp (S) = µta (S). We can extend this result to the case that p does not divide the order of S (rather than Aut(S)) by observing if that holds, then the Sylow p-subgroup P of Aut(S) is cyclic and Aut(S) = P G0 with G0 being a p0 -group. An easy analysis of this situation together with the results of this section yield Theorem 1.4. Since for any odd prime there are innitely many simple groups not divisible by p (for p > 3, consider L2 (r) with r prime and p not dividing r(r2 − 1); for p = 3, consider the Suzuki groups), we have the following corollary.

If p is an odd prime and g is xed , then there are innitely many simple groups with µp (S) > g . Corollary 6.8.

We will obtain much more precise results in the next few sections including results that hold for p = 2.

7. Composition Factors of Genus g Covers We will use Theorem 6.6 to show that there are many groups which are not composition factors of genus g in characteristic p. If G is a nite group, dene γ(G) := min{g(X) − 1|G ≤ Aut(X)}. Of course, γ(G) depends on the characteristic (although if p does not divide the order of G, then by Grothendieck, the description of G-covers is independent of characteristic). We need the following result. There should be a more conceptual proof but we use the classication of nite simple groups. If S is a nite simple group, let fpr(S) be minimum of fix(x, Ω)/|Ω| as where Ω is a faithful G-set for some group G with F ∗ (G) = S and 1 6= x ∈ G.

Let S be a nite simple nonabelian group . If x ∈ Aut(S), the number of elements y in the coset xS with y 2 = 1 is at most |S| fpr(S). Lemma 7.1.

If S is alternating, this is clear (since fpr(S) is very close to 1). The result follows by inspection for the sporadic groups. So assume that S is a Chevalley group. The number of involutions in S (or xS ) is at most |S|/d where d is the smallest degree of a nontrivial complex representation of S . These degrees are known (see [66] and [52]). If S is a classical group, then fpr(S) is approximately 1/q whence we are not even close. If S is an exceptional group, all conjugacy classes of involutions are known and we can get an exact formula for the number of involutions and again the result holds easily. ¤

Proof.

Theorem 7.2.

following holds :

Let S be a nite nonabelian simple group . Then one of the

20

ROBERT GURALNICK

(i) µ(S) ≥ 1 + md(S)/12 ; or (ii) µ(S) ≥ −2 md(G) fpr(G) + md(G)(γ(G))/|G|)(1 − fpr(G)) for some group G with F ∗ (G) = S . We may assume that µ(S) < (md(S) − 1)/6. Then by Theorem 6.6, there exists an indecomposable cover f : X → P1 with X either of genus 2 or genus at most µ(S) + 1 such that the monodromy group G of the cover satises (1) or (2) of Theorem 6.6. Let n denote the degree of f . Let Z denote the Galois closure of the cover and let H be such that Z/H = X . Let h denote the genus of Z . Consider case (1) of 6.6 rst. P Then F ∗ (G) = S and 2 + (h − 1)/|G| = J ρ(J, G)/|G| and 2 + (g − 1)/n = P J ρ(J, Ω)/n. Here the sum is over the inertia groups corresponding to branch points of the cover. P Now ind(J, Ω)/n = 1 − |J|−1 − |J|−1 g∈J # f (g, Ω))/n) ≥ (1 − |J|−1 )(1 − fpr(G)). It follows that ρ(J, Ω)/n ≥ (ρ(J, G)/|G|)(1 − fpr(G)). Thus, (1 − fpr(G))[2 + (h − 1)/|G|] ≤ 2 + (g − 1)/n, or

Proof.

g − 1 ≥ −2n fpr(G) + n(γ(G))/|G|)(1 − fpr(S)). Since n ≥ md(G) and g ≤ µ(S) + 1 or µ(S) = 0 and g = 2, the result follows in this case. Consider case (2). So n = |S|. In this case f (g, Ω)/n ≤ fpr(S) (by the previous lemma and [6]). By Lemma 5.4, we may assume that G does not normalize either component. Precisely as above, it follows that

(1 − fpr(S))(h − 1)/|G| ≤ −2 fpr(S) + (g − 1)/n. Consider the curve Z/S where S is one of the components of G. Then R := NG (S)/S acts on this curve, whence it has genus at least γ(R). Note that R is almost simple with socle S . Thus, 2(h − 1) ≥ 2|S|γ(R) and so

(1 − fpr(S))nγ(R)/|G| = (1 − fpr(S))γ(R)/|R| ≤ −2 fpr(S) + (g − 1)/n. As above, this implies (ii) holds.

¤

We restate this: Corollary 7.3. Let S be a nonabelian nite simple group with n = md(S). Then (µ(S) − 1) ≥ n/12 or µ(S)/n ≥ −2 fpr(G) + (γ(G))/|G|)(1 − fpr(G)) for some group G with F ∗ (G) = S .

In particular, if fpr(G) is small and γ(G) is large, then µ(S) is large. We shall apply this to Chevalley groups of characteristic dierent from p. Here fpr(G) is roughly 1/q and γ(G) is a constant times |G| log(q) (where the constant depends only on the type of G).


21

8. Estimates on Inertia Groups Let k be an algebraically closed eld of characteristic p. Let f : X → Y be a Galois cover with Galois group G. Let I be the inertia group of a point of X and let Ii be the higher ramication groups. We write I = I1 D with D cyclic. Let C = CD (I1 ) and set r = |D : C| and s = |C|. In this section, we obtain estimates for ρ(I)/|G|. The permutation representation is the regular representation for G since we have a Galois cover. Restricting this representation to I gives |G : I| copies of the regular representation. Thus, ρ(I)/|G| is independent of G and can be computed by considering any such cover with the same I (and higher ramication groups). In particular, we want to reduce to the case that I = G. This can be done in several ways. We could just replace G by I and consider the cover X → X/I (and so the point with inertia group I is totally ramied). Alternatively, by Katz Gabber [46], there exists a Galois cover ψ : L → P1 ramied at precisely 2 points with inertia groups I (and the same higher ramication groups) and D. Let g(L) denote the genus of L. Thus, 2(g(L) − 1)/|I| + 2 = ρ(I)/|I| + (|D| − 1)/|D| or ρ(I)/|I| = 1 + 2(g(L) − 1)/|I| − 1/|D|. For the remainder of this section, we assume that G = I ; i.e. there is a totally ramied point. Lemma 8.1.

If Ij 6= Ij+1 , then j ≡ 0 (mod s).

We may assume j > 0. Choose an element x ∈ Ij with x not in Ij+1 and pass to the abelian subgroup C × hxi. Now apply HasseArf [61]. ¤ Proof.

Lemma 8.2.

ρ(I)/|I| ≥ 1 + 1/r − (s + 1)/|I|.

By the previous result, I1 = . . . = Is . Thus, ρ(I)/|I| ≥ 1 − 1/|I| + (1/rs)(1 − 1/|Ii |), whence the result. ¤ i=1

Proof.

Ps

Assume now that I1 is abelian. We change our numbering scheme to keep track of distinct terms among the higher ramication subgroups and count multiplicities. Let I1 = J1 > J2 > . . . > Jm = 1 with the Jj being the distinct higher ramication groups. Let ri denote the number of higher ramication groups equal to Ji . It follows from HasseArf [61] that if Ij 6= Ij+1 , then |CI1 | divides Pj i=1 |Ii | and in particular, ri+1 is a multiple of |CI1 : Ii+1 |. Thus (recall our permutation representation is the regular representation) X ρ(I)/|I| = 1 − 1/|I| + |C|/|D| λi (1 − 1/|Ji |), i=1

where the λi are positive integers. In particular, we see that ρ(I)/|I| > 1 + |C|/|D| − 1/|I| − |C|/|D||I1 | ≥ 1 + |C|/|D| − 1/|D||I1 | − |C|/|D||I1 | ≥ 1 + |C|/2|D|, unless possibly |I1 | = 2 or |I1 | = 3 and C = 1.

22

ROBERT GURALNICK

If |I1 | = 2, then C = D. If D = 1, then ρ(I)/|I| is a positive integer. If D 6= 1, then |I| ≥ 6 and so ρ(I)/|G| ≥ 4/3. If |I1 | = 3 and C = 1, then either D = 1 = I2 and ρ(I)/|I| = 4/3 or ρ(I)/|I| ≥ 1 + 1/|D| = 1 + |C|/|D|. Summarizing we have the following: Lemma 8.3.

If I1 is a nontrivial abelian group , then one of the following holds :

(i) ρ(I)/|I| ≥ 1 + |C|/2|D|; (ii) I = I1 has order 2, I2 = 1 and ρ(I)/|I| = 1; (iii) |I1 | = 2 and ρ(I)/|I| ≥ 4/3; or (iv) I = I1 has order 3, I2 = 1 and ρ(I)/|I| = 4/3. P Keep the assumption that I1 is abelian. Set λ = λi . Then

1 − 1/|I| + (|C|/|D|)(1 − 1/p)λ ≤ ρ(I)/|I| < 1 − 1/|I| + |C|/|D|λ. Lemma 8.4.

Fix r = |D/C|. Assume that I1 is abelian and p > 3.

(i) If λ ≥ 2|D|/|C|, then ρ(I)/|I| ≥ 12/5; (ii) If p is suciently large , then ρ(I)/|I| > 1 + 1/3r; (iii) Let d > 1 be a positive integer with ρ(I)/|I| > 1 + 1/d. If |I| ≥ 8r2 and p is suciently large , then ρ(I)/|I| > 1 + 1/d + 1/(9r2 ); and (iv) If ρ(I)/|I| > 2, |I| ≥ 8r2 and p is suciently large , then ρ(I)/|I| > 2 + 1/(9r2 ). The rst statement follows immediately from the inequality above. So we may assume that λ ≤ 2r. Note that D/C is a cyclic group acting faithfully on I1 . Thus, |I1 | > r and so r − 1/|I| ≥ 1/r − 1/r(r + 1) ≥ 1/2r. Thus, for p suciently large, ρ(I)/|I| > 1 + 1/3r. Suppose that ρ(I)/|I| > 1 + 1/d for d > 1 an integer. If d > 4r, then ρ(I)/|I| + 1 − 1/d > 2 + 1/(12r). So assume that d ≤ 4r and ρ(I)/|I| + 1 − 1/d > 2. Thus, 1 − 1/|I| + λ/r > 1 + 1/d, and λ/r − 1/d ≥ 1/rd ≥ 1/(4r2 ). Hence λ/r − 1/d − 1/|I| ≥ 1/(8r2 ) and so for p suciently large, ρ(I)/|I| > 1 + 1/d + 1/(9r2 ). The same argument yields the last statement. ¤ Proof.

If p is small or to get better bounds, one needs to analyze the above case more closely. However, we will not need this in this article. We need to handle the remaining primes. We rst show that ρ(I)/|I| cannot be too close to 1 − 1/d if |I1 | is small. If |I| itself is small, this is clear since we have bounded the denominator and we can also bound d easily. First note that an easy consequence of HasseArf is the following:

Each ri is a multiple of |C|. Indeed , ri is a multiple of |Ci |, where Ci = CD (Ji /Ji+1 ).

Lemma 8.5.

By passing to the subgroup Ci Ji , we may assume that I = C1 I1 and prove the result for r1 . We can then pass to I/I2 and assume that I2 = 1, whence I is abelian and now HasseArf applies. ¤

Proof.


23

Fix r = |D/C|. Assume that I1 contains an abelian subgroup I 0 of index at most m. Suppose that there are at least t ≥ 5rm distinct terms among the Ij0 = Ij ∩ I 0 . Then ρ(I)/|I| > 5/2.

Lemma 8.6.

Let J1 , . . . , Jt denote the smallest subgroups among the higher ramication groups with xed intersection with I 0 . Let mi denote the number of terms among the higher ramication groups which intersect I 0 in Ji . Then the RiemannHurwitz formula yields: Proof.

ρ(I)/|I| ≥ 1 − 1/|I| + (1/|I|)

X

mi (|Ji | − 1).

Now mi is a multiple of |C| and also by HasseArf mi is a multiple of |I 0 : Ij0 |. Thus,

ρ(I)/|I| ≥ 1 − 1/|I| + (1/rm)

X

(1 − 1/|Ji |) ≥ 1 − 1/|I| + t/(2rm),

whence the result. Lemma 8.7.

¤

Fix r = |D/C|. Assume that |I1 | < N .

(i) Let d > 1 be a positive integer with ρ(I)/|I| > 1 + 1/d. Then ρ(I)/|I| > 1 + 1/d + 1/(32r3 N 2 ); and (ii) If ρ(I)/|I| > 2, then ρ(I)/|I| > 2 + 1/(2rN ); (iii) If ρ(I)/|I| > 1, then ρ(I)/|I| > 1 + 1/(2rN ). Proof.

that

By the previous lemma and the RiemannHurwitz formula, it follows

ρ(I) = |I| − 1 + ai

X

(|Ji | − 1),

with the ai being positive integers that are multiples of s. Thus,

ρ(I)/|I| = 1 − 1/|I| + b/r|I1 |, for some positive integer b. We assume that s > 1. If s = 1, then |I| = r|I1 | and ρ(I)/|I| = 1 + b/r|I1 | and the argument we give below will also be valid. If ρ(I)/|I| > 2, it follows that b > r|I1 | and so ρ(I)/|I| > 2 + 1/r|I1 | − 1/|I| ≥ 2 + 1/(2r|I1 |). We get precisely the same estimate if ρ(I)/|I| > 1. If d ≥ 4rN and ρ(I)/|I| > 1 + 1/d > 1, then ρ(I)/|I| > 1 + 1/(2rN ) ≥ 1/d + 1/(4rN ). So assume that d < 4rN and ρ(I)/|I| > 1 + 1/d. Thus, b/r|I1 | > 1/d + 1/|I|. It follows that b/r|I1 | − 1/|I| ≥ 1/d|I| and so ρ(I)/|I| ≥ 1 + 1/d + 1/d|I|. If s < 2d, this implies that ρ(I) > 1 + 1/d + 1/(32r3 N 2 ). Also, ρ(I) ≥ 1 + 1/d + 1/rd|I1 | − 1/|I|. If s ≥ 2d, this implies ρ(I) ≥ 1 + 1/d + 1/2rd|I1 | > 1 + 1/(8r2 N 2 ). ¤

24

ROBERT GURALNICK

9. Automorphism Groups of Curves Let k be an algebraically closed eld of characteristic p ≥ 0. Let f : Z → Y be a G-Galois cover. Assume that Z has genus h > 1. It is classical that Aut(Z) is nite. We sketch a slightly dierent proof of this than the standard ones. This proof came out of discussions with M. Zieve at MSRI. This topic will be more fully explored in further work by the author and Zieve. First consider the case that Z is dened over a nite eld. We rst prove a weaker result that is valid for any genus curve. Theorem 9.1. Let Z be a curve over k , the algebraic closure of a nite eld . Then Aut(Z) is locally nite .

Let G be a nitely generated subgroup of Aut(Z). Then G is dened over some nite subeld k0 of k . Let f be any nonconstant function on Z . We may enlarge k0 and assume that f is dened over k0 and all the poles and zeroes of f are k0 -rational points. Let H be the subgroup of G which xes all k0 rational points. Since there are only nitely many such points, H has nite index in G and so it suces to prove that H is nite. Since h ∈ H implies that h xes the zeroes and poles of f , it follows that f h = a(h)f where a : H → k0∗ is a homomorphism. Thus, f n is xed by H for some n (for example, n = |k0∗ |). Thus, the xed eld F of H (acting on k(Z)) has transcendence degree 1 over k , whence k(Z)/F is a nite extension and so H is nite. ¤ Proof.

There exists a positive valued integral monotonic function c(g) such that if Z has genus at least 2, then |Aut(Z)| ≤ c(g). In particular , Aut(Z) is nite .

Theorem 9.2.

Proof. Let G be a subgroup of Aut(Z). It suces to show the bound holds when G is nitely generated (for if every nitely generated subgroup has order less than c(g), so does the whole group). So assume that G is nitely generated. First we dene c(g) and show that if G is nite, then |G| ≤ c(g). We can now apply the RiemannHurwitz formula and some relatively easy computations as in [64] or the previous section to obtain some bound (one easily gets cg 5 ; Stichtenoth obtains cg 4 with some extra eort) to see that c(g) can be taken to be a polynomial in g . Alternatively, let W := W` denote the set of `-torsion points on the Jacobian of Z . As we have observed, if ` does not divide the order of G, then 2g(Z/G) = dim W G (this is really the character formula version of the RiemannHurwitz formula see [61]). In particular, if H is the kernel of the action of G on T , we see that g(Z/H) = g(Z). If Z has genus greater than 1, then the Riemann Hurwitz formula shows that Z has no nontrivial separable maps to a curve of genus g(Z), whence H = 1. Thus, G acts faithfully on W for any suciently large prime `. In particular, |G| divides the greatest common divisor of the


25

orders of GL(2g, `) for all suciently large `. Note that this does not depend on how large we need to let ` be (this was observed by Minkowski in his proof of the bound on the orders of nite subgroups of GL(n, Q) or more generally nite subgroups of GL(n, C) with all traces rational). We can take c(g) to be this greatest common divisor. In conjunction with this previous theorem, this proves the result for k the algebraic closure of a nite eld. Consider the general case. Suppose that G has innite order. We may write the function eld k(Z) = k(u, v). Choose a nitely generated subring R of k such that both Z and G are dened over R and that G acts on S = R(u, v). We note that if M is a maximal ideal of R, then R/M is nite (by the Nullstullensatz). Moreover, taking a plane model for Z over R, we see that any reduction will have genus at most some xed g 0 (in fact, one knows that the reduction will have genus at most g ). Enlarging R if necessary (by inverting a nite number of elements), we can also assume that there are c > c(g 0 ) distinct elements of G that remain distinct on (R/M )(u, v) for any maximal ideal M of R and that the genus of (R/M )(u, v) is at least g (all we need is at least 2). Now we have G acting on F (u, v) with F a nite eld. Moreover, the image H of G in this action has order greater than c(g 0 ) since the xi are still distinct. This is a contradiction, whence G is nite. Then |G| ≤ c(g) and the result follows. ¤ The standard RiemannRoch argument shows that no nontrivial element of G xes more than 2g + 2 elements. So if we are over a nite eld, we can enlarge the eld to guarantee the existence of at least 2g + 3 rational points and we see that G acts faithfully on these points and so is nite. In this section, we consider subgroups of Aut(Z) that are isomorphic to Chevalley groups in a characteristic dierent from p and show that the genus of Z must be at least linear in the order of the group. The constant will depend only the type of Chevalley group and not the eld. We will use the results of the previous section. So x a type of Chevalley group L. Let q be a prime power. Let J(q) denote a group with F ∗ (J(q)) = L(q) and J(q) contained in the group of inner-diagonal automorphisms of L(q) (for what we need we could just consider L(q)). Let W denote the Weyl group of L. We need the following facts about J(q). Lemma 9.3.

Let G = J(q) and let U be a p-subgroup of G with p not dividing q .

(i) If p does not divide the order of the Weyl group of L, then the Sylow psubgroup of G is abelian and if U has exponent pd , then |U | ≤ ptd where t is the rank of the corresponding algebraic group . (ii) If the exponent of U is at most pd , then there exist constants e, t (depending only on L) such that |U | ≤ eptd . (iii) There exists a constant δ (depending only on L) such that any p0 -element of NG (U )/CG (U ) has order at most δ .

26

ROBERT GURALNICK

It suces to prove this result for p-subgroups of the corresponding algebraic group G. If p does not divide the order of the Weyl group, then every p-subgroup of G is contained in maximal torus (see [24]). Thus, (i) and (ii) follow in this case. Similarly, it is known that W controls fusion in subgroups of tori and so N (U )/C(U ) embeds in W and (iii) holds as well for these primes. Now consider a prime dividing the order of the Weyl group. There is no harm in embedding the simple algebraic G into GL(n, F ) with F algebraically closed (over a nite eld) and n depending only on L. Then every p-subgroup is conjugate to a subgroup of the monomial group M (since every absolutely irreducible representation of a p-group is induced from a 1-dimensional representation of a subgroup). So we may assume that U ≤ M . If U has exponent pd , then U ∩ T (with T the torus) has order at most pdn and since |U | divides |U ∩ T |n!, (ii) follows. Finally, we prove (iii). Again, it suces to prove this for subgroups of GL(n, F ) = GL(V ) and for primes dividing the order of the Weyl group. In particular, there are only nitely many primes (depending upon L) to consider. By a result of Thompson (see [23], 3.11), we may assume that [U, U ] ≤ Z(U ), U/Z(U ) is elementary abelian and the element is faithful on U/Z(U ). By elementary representation theory, we see that U/Z(U ) has rank at most n, whence the order of U/Z(U ) is bounded as a function of n. Thus (iii) holds. ¤ Proof.

We will now prove:

There exists a constant c = c(L) such that if J(q) is a subgroup of Aut(Z) for some curve Z of genus g > 1 dened over the algebraically closed eld k of characteristic p with p not dividing q , then g ≥ c|J(q)|. Theorem 9.4.

Let G = J(q). Consider Z/G. Let h be the genus of Z/G. If h > 0, the RiemannHurwitz formula together with the fact that ind(I) ≥ |G|/2 for any nontrivial inertia group I gives g − 1 ≥ |G|/4. So we may assume that h = 0. Let I be an inertia group with I1 6= 1. Write I = DI1 with D a cyclic p0 -group and let C = CD (I1 ). By the previous results, we know that |D/C| < r for some constant r = r(L). There are three possibilities for p. First suppose that p is small (depending upon L) but |I1 | is large (large enough so that it contains an abelian subgroup of exponent at least pt with t satisfying the hypotheses of 8.6. Then ρ(I)/|G| ≥ 5/2, whence g − 1 ≥ 5|G|/4 and the result holds. So assume either p is large (and in particular does not divide the order of the Weyl group, whence I1 is abelian) or that |I1 | is bounded. If there is another wildly ramied point or at least 3 ramied branch points, then 2(g − 1)/|G| ≥ −1 + ρ(I)/|I|. It follows by Lemma 8.4 for p large and by Lemma 8.7 if |I1 | is small, that ρ(I)/|I| > 1 + δ for some positive δ (depending only upon r which in turn is bounded in terms of L and the bound on |I1 |).

Proof.


27

Suppose that there is only one ramied point. Then 2(g − 1)/|G| = −2 + ρ(I)/|I|. In particular, ρ(I)/|I| > 2 and Lemma 8.4 for p large and Lemma 8.7 if |I1 | is small, that ρ(I)/|I| > 2 + δ for some positive δ (again depending only upon r). So we may assume that there is precisely one more ramied point in the cover and it is tamely ramied with inertia group cyclic of order d. Thus, 2(g−1)/|G| = −1 − 1/d + ρ(I)/|I|. In particular, ρ(I)/|I| > 1 + 1/d. Again, Lemma 8.4 for p large and Lemma 8.7 if |I1 | is small imply that ρ(I)/|I| > 1 + 1/d + δ for some positive δ depending only upon r This completes the proof. ¤

Let L be a xed type of Chevalley group . There exists a positive constant c = c(L) such that µp (L(q)) ≥ c md(L(q))/ log q for q suciently large with q prime to p. Corollary 9.5.

Proof. Let f : X → Y be a cover with monodromy group G involving L(q) with X of genus µp (L(q)). Let Z be the curve corresponding to the Galois closure. By Theorem 7.2, we may assume that F ∗ (G) = L(q). Note that |G| ≤ 6|J(q)| log(q), where J(q) is the subgroup of G consisting of inner-diagonal automorphisms. By the previous result, this implies that g(Z) ≥ d(L)|G|/ log(q) for some constant d(L). By [49] (excluding L2 (q)), fpr(G) ≤ 4/3q . There is an analogous result for L2 (q). Now apply Corollary 7.3. ¤

10. The Generalized Fitting Subgroup Let G be a nite group. A subgroup H of G is subnormal in G if there is a chain of subgroups H = G0 < G1 . . . < Gm = G with Gi normal in Gi+1 . A group G is called quasisimple if G/Z(G) is simple and G is equal to its own commutator subgroup. A component of G is a quasisimple subnormal subgroup. It is not dicult to show that any two distinct components of G commute (see the next two lemmas). Let F (G) denote the Fitting subgroup of G (the maximal normal nilpotent subgroup). Let E(G) be the subgroup of G generated by the components of G. The generalized Fitting subgroup of G is dened to be E(G)F (G) and is denoted by F ∗ (G). We rst need an elementary result about commutators. This follows from the three subgroup lemma (see [23]).

Suppose that H is a perfect subgroup of G, N is a subgroup of G and [H, N ] is centralized by H . Then H commutes with N .

Lemma 10.1.

Proof. Since H is perfect, [H, N ] = [[H, H], N ]. The three subgroup lemma asserts that [[H, H], N ] ≤ [[H, N ], H] = 1 as claimed. ¤

The next result is standard although the proof is not. Lemma 10.2.

Let H be a component of G.

28

ROBERT GURALNICK

(i) H commutes with every normal subgroup of G not containing H . (ii) If H and K are distinct components , then H and K commute . (iii) H commutes with F (G). Let M be a normal subgroup of G minimal with respect to containing H . So M is generated by the conjugates of H . If M = G, then H is normal in G. If M is proper in G, then by induction, H commutes with all other components of M (any component of M is a component of G). So in either case, M is a central product of the components it contains and HCM . Also, we see that F (M ) = Z(M ). Let N be a normal subgroup of G not containing H . Then [M, N ] ≤ M ∩ N CM . So either [M, N ] ≤ Z(M ) or M ≤ N , whence either H ≤ N or H commutes with N by the previous result. Since H is not contained in F (G), it follows that H commutes with F (G). All that remains to show is that H commutes with any other distinct component K . Let N be a minimal normal subgroup containing K . So N is a central product of all its components, whence if H ≤ N , the result holds. If not, then we have seen that [H, N ] = 1 and so also [H, K] = 1. ¤ Proof.

We next give a dierent characterization of E(G). Lemma 10.3. Let D = Z(F (G)). Let X/D be the product of all the minimal normal subgroups of CG (F (G))/D. Then E(G) = [X, X]. Proof. Note that X/D has no normal abelian subgroups, for if Y /D is abelian, then since D is central in Y , Y is nilpotent and so Y ≤ F (G)∩X = Z(F (G)) = D. Thus, X/D is a direct product of nonabelian simple groups. In particular, X = [X, X]D with D ≤ Z(X) and so [X, X] is perfect and modulo its center is a direct product of nonabelian simple groups S1 × . . . × St . Let Qi be the preimage of Si in X . Then as above, we see that [Qi , Qi ] is perfect and simple modulo its center i.e. a component. We also see that [X, X] is the product of the [Qi , Qi ], whence [X, X] ≤ E(G). By the previous lemma, every component centralizes F (G) and modulo D is simple. Moreover, as we have already seen, its normal closure is a direct product of simple groups, whence X contains every component of G. Thus, E(G) = [X, X]. ¤

The important property of this subgroup is the following (cf [8]) result which follows immediately. Theorem 10.4. ∗

CG (F (G)).

CG (F ∗ (G)) = Z(F (G)) = Z(F ∗ (G)). In particular , F ∗ (G) ≥

Note that D := Z(F ∗ (G)) = Z(F (G)) (since every component commutes with F (G)). Let C = CG (F ∗ (G)). Suppose that C properly contains D. Consider a minimal normal subgroup of CG (F (G)/D contained in C/D. By the Proof.


29

previous result, this minimal normal subgroup is contained in DE(G) and so is contained in the center of DE(G), whence is contained in D, a contradiction. ¤ In particular, this shows that there are only nitely many groups with a given generalized Fitting subgroup (for G/Z(F (G)) embeds in Aut(F ∗ (G)) and so we have a bound on |G|).

11. AschbacherO'NanScott Theorem In this section, we give a proof a version of the structure theorem for primitive permutation groups which we have used extensively. See [9] for a more detailed version. Recall that a group G is said to act primitively on a set Ω of cardinality greater than 1 if G preserves no nontrivial equivalence relations on Ω. In particular, this implies that G is transitive on Ω (consider the equivalence relation of being in the same G-orbit). With this added assumption, it is equivalent to the condition that a point stabilizer is maximal. We include a proof of this well known elementary fact.

Let G act transitively on Ω. Then G is primitive if and only if the stabilizer of a point ω is a maximal subgroup of G.

Lemma 11.1.

Let H be the stabilizer of the point ω . Then Ω can be identied with G/H , the set of left cosets of H . If H is not maximal, consider the natural map π : G/H → G/M for M a maximal subgroup containing H (here π(gH) = gM ). The bers of π dene a G-invariant equivalence relation on G/H . Suppose that H is maximal. Let Γ be the equivalence class of ω in a nontrivial G-invariant equivalence relation. Since H xes ω , H preserves Γ. The same is true for every point of Γ. Since G does not preserve Γ and H is maximal this implies that H preserves each point of Γ. Since G is transitive, this implies that NG (H) is transitive on Γ. Since H is maximal, this implies that H is normal in G and so is trivial. Thus, G has prime order, a contradiction. ¤ Proof.

Let G be a nite group acting primitively on a set Ω of cardinality n. Let H be the stabilizer of a point . Let A be the product of the minimal normal subgroups of G. Then A = F ∗ (G) and one of the following holds : Theorem 11.2.

(i) A is an elementary abelian p-group , G = AH (semidirect ) and H acts irreducibly on A via conjugation and n = pa = |A|. (ii) A = A1 × A2 with A1 ∼ = A2 a direct product of t ≥ 1 isomorphic nonabelian simple groups , H ∩ A = {(a, φ(a)|a ∈ A1 } for some isomorphism φ : A1 → A2 . Moreover , A1 and A2 are the two minimal normal subgroups of G and n = |A1 |. (iii) A is the unique minimal normal subgroup of G, A = L1 × . . . × Lt is the direct product of t copies of isomorphic nonabelian simple groups and one of the following :

30

ROBERT GURALNICK

(i) 1 6= H ∩ A = H ∩ L1 × . . . × H ∩ Lt and n = mt with m = |L1 : H ∩ L1 | and the H ∩ Li all H -conjugate . Moreover , NH (L1 )CG (L1 )/CG (L1 ) is maximal in NG (L1 )/CG (L1 ). (ii) There exists a partition {∆1 , . . . ∆s } of {1, . . . , t} into s < t subsets of size t/s and A ∩ H = K1 × . . . × Ks where Ki ∼ = L1 is a full diagonal of the direct product of the Ai := Lj , j ∈ ∆i . In this case , n = |L1 |t−s . (iii) A ∩ H = 1, t > 1 and n = |L1 |t . Note that H contains no nontrivial normal subgroups (since a normal subgroup xing 1 point xes all points). Let B be any normal subgroup of G. Then G = BH (since H is maximal). Thus, B is transitive, Now CH (B) is normal in H and normalized by B , whence is normal in G and trivial. Now let B be a minimal normal subgroup. Suppose that B is abelian. Then B ≤ CG (B), so CG (B) = B , and so B is the unique minimal normal subgroup. Thus we are in case (1). So we may assume that there are no minimal normal abelian subgroups. So F ∗ (G) = E(G). Let A1 = L1 × . . . × Lt be a minimal normal subgroup with Li conjugate nonabelian simple groups (and components of G). Suppose that there is another minimal normal subgroup A2 . Then A1 and A2 commute. Then G = HAi and H ∩ Ai centralizes Aj for j 6= i. As noted above, CH (Ai ) = 1, whence H ∩ Ai = 1 for i = 1, 2. On the other hand, Ai ≤ G = HAj and so the projections of H ∩ A1 A2 into Aj are both onto and injective. Thus, H ∩ A1 A2 = {(a, φ(a)|a ∈ A1 } for some isomorphism φ : A1 → A2 as required. Thus, we are in case (2). So we may assume that A is the unique minimal normal subgroup of G and A = L1 × . . . × Lt with the Li conjugate nonabelian simple groups (and components). If H ∩ A = 1, there is nothing more to say (except to show that t > 1 this requires the classication of nite simple groups in the form of the Schreier conjecture that outer automorphism groups are solvable see [9] for details). Suppose that H1 = H ∩ L1 6= 1. Since G = AH and A normalizes L1 , it follows that H permutes the Li transitively, whence Hj = H ∩ Lj is conjugate to H1 via H . The maximality of H implies that H is the normalizer of H ∩ A and all that remains to be shown in this case is that NH (L1 )CG (L1 ) is maximal in NG (L1 ). We note the following H1 is the maximal NH (L1 ) invariant subgroup of L1 (otherwise H normalizes the direct product of the conjugates of this NH (L1 ) invariant overgroup of H1 contradicting the maximality of H ). Suppose that K is a maximal subgroup of NG (L1 ) containing NH (L1 )CG (L1 ). Let K1 = K ∩ L1 . Since ANH (L1 ) = NG (L1 ), it follows that |NG (L1 ) : NH (L1 )CG (L1 )| = |L1 : H1 | and similarly |NG (L1 ) : K| = |L1 : K1 |. Clearly, K1 contains H1 and is normalized by NH (L1 ) whence K1 = H1 and K = NH (L1 )CG (L1 ) as required. In the remaining case, H ∩ A 6= 1 = H ∩ Li . Let πi denote the projection from onto Li . Then H normalizes the direct product of these projections and by the Proof.


31

maximality of H , if these are proper H contains them. Thus, each projection is onto. Let Ki be the kernel of πi . If Ki = 1, then H ∩ A ∼ = Li and is a full diagonal subgroup (i.e. of the form {(x, φ2 (x), . . . , φt (x))|x ∈ L1 } where φi an isomorphism from L1 to Li ). If K1 6= 1, let ∆1 be those i such that πi (K1 ) = 1. All other projections of K1 are surjective (because K1 is normal in H ∩A and the projections are surjective). By induction, there is a partition ∆1 , . . . , ∆s such that K1 is a direct product of full diagonal subgroups of Ai , i = 2, . . . , s. Since K1 is normal in A ∩ H and is self normalizing in A2 × . . . × As , it follows that the projection τ of A ∩ H into A2 × . . . × As is the same as that of K1 . Thus, A ∩ H = ker(τ ) × K1 . Since (A ∩ H)/K1 ∼ = L1 , this implies that ker(τ ) is a full diagonal subgroup of A1 . The only remaining point is to show that the ∆i all have the same cardinality. This is clear since H normalizes A ∩ H and acts transitively on the Lj . ¤ Much more can be said particularly in case (3) of the theorem. See [9].

12. Aschbacher's Subgroup Theorem In this section, we prove a version of Aschbacher's Theorem about subgroups of classical groups over nite elds. Roughly, the theorem is that if G is a classical group on a vector space over a nite eld F , then any subgroup either is (modulo its intersection with the center) almost simple (i.e. has a unique minimal normal subgroup which is a nonabelian simple group) or preserves some natural geometric structure on the space. By a natural geometric structure on the space, we include such things as a tensor product decomposition, a subspace, a direct sum decomposition or a eld extension structure. We will make this more precise below. In Aschbacher's statement, there are 8 families of structures considered. In fact, there are dierent ways of organizing the possible structures that one wants to consider (or equivalently, the possible subgroups the stabilizers of these structures). Aschbacher [7] proved a slightly more general theorem in that he considered subgroups of automorphism groups of classical groups. See also [50] for an approach using Lang's theorem. This section is based on notes for a course given at USC in 1998. This theorem has become a standard and important tool in the analysis of subgroups of classical groups. See [33] for one example of how it is used. When one is using the theorem, it can be important to consider as ne a stratication of the possible geometric structures as possible. Indeed, one category that Aschbacher did not consider was the case of tensor decompositions over extension elds. However, the proof of the theorem can be organized in dierent ways. In particular, one does not need to consider all the structures considered by Aschbacher.

32

ROBERT GURALNICK

The theorem is quite a bit simpler to prove in the case that all irreducible submodules for normal subgroups of G are absolutely irreducible (eg, if one works over the algebraic closure). If that fails, then either G preserves a eld extension structure on the space or preserves a direct sum decomposition. Thus, one can give a proof as in the case of an algebraically closed eld except for adding one additional class. One can then study this extra class separately. If there is no form involved, then essentially no extra work is required. We now give a proof of the theorem. In the following subsections, we analyze and classify the groups preserving a eld extension structure (in the case that there is no form preserved, there is essentially nothing more to add). In the last subsection, we give some elementary results about groups preserving forms and other representation theory facts which are used in the proof. Recall that a group G is almost simple if and only if it has a unique minimal normal subgroup which is a nonabelian simple group S (this is equivalent to S ≤ G ≤ Aut(S)). Let F be a eld of characteristic p ≥ 0 which is either nite of algebraically closed (there are variations of this result over more general elds). Let G be a nite subgroup of GL(V ) where V is a vector space of dimension d over F (the statement is valid for algebraic groups as well in the case F is algebraically closed with an identical proof where many of the details become quite a bit easier). Suppose that q is a quadratic form, a unitary form or an alternating form on V . We assume that either q = 0 or that q is nondegenerate (i.e. except for the case of quadratic forms in characteristic 2, the radical of the form is 0). Let X(V, q) denote the subgroup of GL(V ) which preserves q up to scalar multiplication. The nondegeneracy condition implies that X(V, q) acts irreducibly on V . So X(V, q) is one of GL(V ), GO(V, q), GSp(V ) or GU(V ) in the case q = 0, q is a quadratic form, alternating form or unitary form respectively. We let I(V, q) denote the isometry group of q (i.e. the subgroup preserving the form). So I(V, q) is one of GL(V ), O(V, q), Sp(V ) or U (V ). Note that except for the case of quadratic forms, there is only one class of nondegenerate forms. In the case of quadratic forms, there are 2 classes if F is nite and 1 if F is algebraically closed. Moreover, if dim V is odd, the two classes of quadratic forms give rise to the same isometry groups. We say that a group H acts homogeneously on V (over F ) if V is a direct sum of isomorphic simple F H -modules. The homogeneous component corresponding to a simple F H -module W is the sum of all simple submodules isomorphic to W .

Let G be a subgroup of X(V, q) with V a vector space of dimension n over a eld F which is either nite or algebraically closed . Let p denote the characteristic of F . Then one of the following holds : Theorem 12.1.

(R1) G stabilizes a totally singular subspace ; (R2) G stabilizes a nondegenerate subspace ;


33

(R3) if the characteristic is 2 and q is a quadratic form , G stabilizes a 1dimensional nonsingular subspace ; (D1) G leaves invariant a decomposition V = ⊕2i=1 Vi with each Vi totally singular ; (D2) G leaves invariant a decomposition V = ⊕ti=1 Vi with each Vi nondegenerate ; (T1) G leaves invariant a tensor decomposition on V ; i .e . G embeds in

X(W1 , q1 ) ⊗ X(W2 , q2 )

where n = w1 w2 with dim Wi = wi and q = q1 ⊗ q2 or p = 2, each qi is a nondegenerate alternating form and X(W1 , q1 ) ⊗ X(W2 , q2 ) ≤ O(V, q) for the unique (up to scalars ) quadratic form vanishing on all simple tensors ; (T2) G leaves invariant a tensor structure on V ; i .e . G embeds in X(W, q 0 ) o Sr where n = wr , q = q 0 ⊗ · · · ⊗ q 0 (r times ) and dim W = w or p = 2, q 0 is alternating and G preserves a quadratic form on V ; (E) F ∗ (G) = Z(G)E where E is extraspecial of order s1+2a with s prime , n = sa and E acts absolutely irreducibly (and so G is contained in the normalizer of EZ(G)); if s is odd , G preserves no alternating or quadratic form and if s = 2, G will preserve a form . (EXT) G preserves an extension eld structure ; or (S) G/Z(G) is almost simple . Suppose that G acts reducibly. So let U be a proper invariant subspace of minimal dimension. Then rad(U ) is also invariant under G thus, either U ⊆ Rad(U ) or U is nondegenerate (the radical is taken with respect to the corresponding form if it exists in the case q is a quadratic form and F has characteristic 2, we compute the radical with respect to the corresponding alternating form). If U is nondegenerate, then (R2) holds. If U is contained in Rad(U ), then U is totally singular unless possibly p = 2. In the latter case, the set of vectors with q(u) = 0 forms a G-invariant hyperplane of U . By minimality, it follows that either U is totally singular or U is 1dimensional. Thus (R1) or (R3) hold. So we assume that G acts irreducibly. Let N be any normal noncentral subgroup of G. Suppose that N does not act homogeneously on V . Let V1 be a homogeneous component of V for N . If V1 is nondegenerate (or there is no form), then so is every component and so (D1) holds. Otherwise, V1 is totally singular (since ⊕Rad(Vi ) is G-invariant). The irreducibility of G implies that there is a unique component Vi0 so that Vi is not perpendicular to Vi0 . Thus, G permutes the nondegenerate subspaces Vi ⊕ Vi0 . If this is a proper subspace, then (D1) holds. If not, then (D2) holds. So we may assume that every normal subgroup of G acts homogeneously. Let W be an irreducible constituent of N . If W is not absolutely irreducible, then Proof.

34

ROBERT GURALNICK

the center of EndN (V ) is a proper extension eld E/F whence G preserves an extension eld structure on V (this uses the fact that the Brauer group of F is trivial). We will consider this situation in more detail below. So we may assume that every normal noncentral subgroup acts homogeneously and each irreducible constituent is absolutely irreducible. In particular, this means that we are assuming that G is absolutely irreducible. Let G0 be the normal subgroup of G which actually preserves the form (rather than up to a scalar multiple). Since G/G0 Z(G) is cyclic, G0 cannot consist of scalars (unless n = 1). Let N be a normal noncentral subgroup and suppose that N does not act irreducibly. Moreover, if a form is involved, we assume that N ≤ G0 . Let W be an irreducible constituent of V for N . Let G1 be the normalizer in GL(W ) of N . Then a straightforward computation (using the fact that the centralizer of N on W consists of scalars) shows the normalizer of N in GL(V ) is G1 × GLn/d (U ) acting on W ⊗ U with U of dimension n/d. In particular, G embeds in this group. If there is no form involved we are done (i.e. G preserves a tensor decomposition and we note that there is a unique conjugacy class of such subgroups in GL depending only on the dimensions d and n/d). We now consider how the form behaves with respect to this tensor decomposition. Case 1. q is a quadratic form. Then V is self dual as an F N -module and hence so W is self-dual for N . Since N acts absolutely irreducibly on W , there is a unique (up to scalar multiple) bilinear form B on W which is N -invariant (if p 6= 2, this form is symmetric; if p = 2, the form is alternating). An easy dimension computation shows that all the N -invariant bilinear forms on V are of the form (W, B) ⊗ (U, B 0 ). Suppose that p 6= 2. If B is alternating, then q = B ⊗ B 0 where B 0 is an alternating form on B 0 . As we noted above, G ≤ G1 × GL(U ) acting on V = W ⊗ U where G1 normalizes N on W . Thus, G1 ≤ GSp(W ). It follows that G ≤ X(V, q) ∩ G1 × GL(U ) = GSp(W ) ⊗ GSp(U ) < GO(q). Moreover, there is a unique conjugacy class of such subgroups. If B is symmetric, then similarly, we see that G ≤ GO(B) ⊗ GO(B 0 ) < GO(q) with B 0 symmetric as well. There may be several conjugacy classes depending upon the dimension of W and the class of B . If p = 2, we need to proceed in a slightly dierent manner and the answer is actually easier. Let ∆ be the associated alternating bilinear form associated to q . As above, we see that ∆ = B ⊗ B 0 where B 0 is a bilinear form on U and G ≤ GSp(W ) × GSp(U ). Let T denote the image of GSp(W ) ⊗ GSp(U ) in GL(V ). In fact, T is contained in the orthogonal group and not just the symplectic group. Indeed, this last group preserves a unique (up to scalar mul-


35

tiplication) quadratic form. Since T is transitive on all nonzero vectors of the form w ⊗ u, the quadratic form would have to be constant on such vectors. It is straightforward to compute that there exists a unique quadratic form which vanishes on all such vectors and has the corresponding associated alternating form B ⊗ B 0 . The uniqueness shows that the form is T -invariant. Since G is absolutely irreducible, it preserves a unique quadratic form, whence q is the form described above. Case 2. q is alternating. As above, we see that N leaves an essentially unique form B on W . Thus, G1 does as well. Arguing precisely, as above, we see that G ≤ X(W, B)⊗X(U, B 0 ) ≤ X(V, q). If p 6= 2, then we see that B is symmetric and B 0 alternating or vice versa. If p = 2, then we may take both B and B 0 alternating and we see that in fact G preserves a quadratic form (indeed, in Aschbacher's theorem, this is one of the geometric structures allowed a subform). Case 3. q is unitary. In this case F is a nite eld of cardinality m2 . Let F0 be the subeld of F of cardinality m. Since N acts absolutely irreducibly on W and homogeneously on V , it follows that N preserves a unique (up to F0 multiple) unitary form h on W . Arguing as above, we see that this implies that q = h ⊗ h0 and G ≤ X(W, h) ⊗ X(U, h0 ) ≤ X(V, q). So now we may assume that every noncentral normal subgroup acts absolutely irreducibly. Let N be a minimal such subgroup. Thus, CG (N ) = Z(G). It follows that N/(N ∩ Z(G)) is characteristically simple (i.e. has no nontrivial characteristic subgroups). Thus M := N/(N ∩ Z(G) is either an elementary abelian s-group for some prime s or it is a direct product L1 × . . . × Lt where Li ∼ = L is a nonabelian simple group. Suppose that M is an elementary abelian s-group. Since N 0 , the derived group of N , is contained in the center of N , it follows that N 0 is either trivial or has order s. In the rst case, N is abelian and noncentral. Since it acts absolutely irreducibly, it follows that n = 1. So we may assume that N 0 has order s. Since N acts absolutely irreducibly, Z(N ) ≤ Z(G). It follows easily that if s is odd, then N is an extraspecial s-group of order 1+2a s and n = sa . If s is even, then N is of symplectic type (i.e. either N is extraspecial or Z(N ) has order 4 and N = Z(N )E with E extraspecial). Thus, (E) holds. So we may assume that M is a product of t isomorphic nonabelian simple groups. It follows that N is a central product of components Q1 , . . . , Qt . By

36

ROBERT GURALNICK

minimality, each of the Qi are conjugate in G. Also, we may assume that every minimal normal noncentral subgroup has this form. Since CG (N ) = Z(G), it follows that N is unique. So if t = 1, we see that (S) holds. So assume that t > 1. Since N acts absolutely irreducibly on V , it follows that V = W1 ⊗ . . . ⊗ Wt ˆ is a ˆ1 × . . . × Q ˆ t ≤ GL(W1 ) × . . . × GL(Wt ) where Q î ∼ and N embeds in Q =Q covering group of Qi . Since the Qi are conjugate, we may assume that Wi is an ˆ -module and the Wi are isomorphic as Q ˆ -modules. In absolutely irreducible Q particular, they have the same dimension. This is easily seen to be true over the algebraic closure. However, since the character of G is dened over F , the same ˆ acting on Wi . is true for Q It follows that the normalizer of N in GL(V ) is precisely (R1 × . . . × Rt )Symt ˆ i in GL(Wi ) and Symt acts on W1 ⊗ . . . ⊗ Wt by where Ri is the normalizer of Q permuting the coordinates. In particular, G is contained in this product and so G preserves a tensor structure on V . If G preserves a form on V , then so does N and since V restricted to Qi is ˆ i preserves a form on Wi (and the homogeneous (as N = Qi CN (Qi )) and so Q type is the same for each i). So if q is unitary, it follows that G ≤ X(W, h) o St ≤ X(V, q) with h unitary. If V is self dual for N , then it follows that G ≤ X(W, f ) ≤ X(V, q). If p 6= 2, then for t even, necessarily q is symmetric. If t is odd, then q and f are either both symmetric or are both alternating. If p = 2, then we see that G does preserve a quadratic form (necessarily unique) and so we may always take f to be alternating. This completes the proof. ¤ Note that one can state the previous theorem in a dierent manner. Namely, we have produced natural families of subgroups so that any nite subgroup of X(V, q) is either contained in one of those subgroups or is almost simple (modulo the center). One can of course add to this family some almost simple groups and so when analyzing the subgroup structure of X(V, q) (or related groups), one can use this result. See [45] for an analysis of which of these subgroups are maximal. The theorem above gives a very specic list of possibilities and one can analyze the conjugacy classes of such subgroups quite easily and produce some natural invariants. In particular, we note that two irreducible subgroups of X(V, q) are conjugate in X(V, q) if and only if the representations are equivalent (up to an outer automorphism) if and only if the characters are the same (up to an outer automorphism). The one family that we did not analyze so carefully in the proof above is the case where G preserves an extension eld structure on V . If F is algebraically closed, this cannot occur. If q = 0, then it is straightforward to see that the group preserving a eld extension structure corresponding to a eld extension is precisely GLn/d (E).Gal(E/F ) where d = [E : F ] i.e. the subgroup of E -


37

semilinear transformations on V . Note that the only invariant for the conjugacy class is d (and d must divide n). In the next section, we will analyze the case where q 6= 0 and G preserves an extension eld structure on V .

12.1. Field Extension Structures We now make more precise the family of

overgroups occurring in the extension eld case. We will break the proof up into the various cases depending upon the type of q . So we assume that F is a nite eld of order m = pa and that V is a vector space of dimension n over F . As usual, let q denote a form (zero, quadratic, alternating or unitary) on V and we assume that G ≤ X(V, q). We may also assume that G is irreducible on V and that G preserves a eld extension structure on V . More precisely, there is an F -subalgebra E ⊂ EndF (V ) so that G preserves E (and we have an homomorphism from G into Gal(E/F )). Note that if the isomorphism class of E is xed, then E is uniquely determined up to conjugation in GL(V ) (because E has a unique representation of xed dimension). Since the norm map is surjective for nite elds, in fact this conjugation can always be realized in SL(V ). So in the case where q = 0, G preserves an E -structure on V if and only if G is a subgroup of AutE (V ).Gal(E/F ). So we assume that (V, q) is nondegenerate. We also make the blanket assumption throughout this section that every normal subgroup of G acts homogeneously on V (or G will satisfy (D1) or (D2)). Let E/F be an extension of nite elds of degree d > 1. Suppose that B is a nondegenerate bilinear form on the vector space U over E . Let V = U considered as a vector space over F . Then q = tr ◦ B is a nondegenerate bilinear form on V and X(U, B) ≤ X(V, tr ◦ B). We rst discuss the case where q is either alternating or a quadratic form.

Assume that q is a nondegenerate alternating or quadratic form on V where V is a vector space of dimension n over the nite eld F . Let G ≤ X(V, q) be an irreducible subgroup . Assume that every normal subgroup of G acts homogeneously on V . Assume that G normalizes some proper eld extension E/F where E is a subalgebra of EndF (V ). Assume moreover that G preserves no additive decomposition of V . Then G ≤ X(U, q 0 ) where U = V considered as a vector space over some nontrivial eld extension E/F and q 0 is a form on U . Proposition 12.2.

Let Z denote the subgroup of nonzero scalars in GL(V ). Let H = G ∩ I(X, q). Note that either GZ = HZ or GZ/HZ has order 2. We claim that H acts irreducibly. If not, then V = V1 ⊕ V2 with G permuting each of the H invariant subspaces Vi . Since H is homogeneous on V , V1 ∼ = V2 as H -modules. Since V is self dual as an H -module, we may choose V1 and V2 nonsingular. Then G preserves the decomposition V1 ⊕ V2 , a contradiction. Let E = EndH (V ). So E/F is a proper eld extension. Proof.

38

ROBERT GURALNICK

Let U denote V considered as an EH -module. Note that V 0 := V ⊗F E0 ∼ = ⊕U σ where the sum is taken over σ ∈ Gal(E/F ). Since E is commutative, then centralizer of H in V 0 which is just E ⊗F E is also commutative. It follows that V 0 is multiplicity free, whence U σ and U are nonisomorphic for all nontrivial σ . 2 Since V 0 is self dual, it follows that U τ ∼ = U and so = U ∗ for some τ . Thus, U τ ∼ τ 2 = 1. Suppose rst that τ = 1, i.e. U is self dual. Then there exists a nondegenerate bilinear form B on U which is H -invariant. Moreover, B is unique up to scalar multiplication by E0 . Let I = I(U, B). Note that this is independent of the choice of B . Since G acts naturally on the set of such forms, it normalizes I . Note that I(U, B) preserves the form trE/F ◦ B . Note also that for p 6= 2, this form is the same type as q i.e. B is alternating if and only if q is (because H and I(U, B) have the same centralizer it follows that all forms stabilized by I(U, B) are also stabilized by H and also are of the same type). Thus, I(U, B) ≤ I(X, q) and G is contained in the normalizer of I(U, B) in X(V, q). If p = 2 and q is alternating, precisely the same argument suces. All that remains to show is that if q is a quadratic form, then H preserves a quadratic form on U . Let B be an H -invariant alternating form on U . Then as above we may assume that C = tr ◦ B where C is the alternating form on V associated to q . Since the set of H -invariant forms on V has cardinality |E| and the map B 7→ trE/F ◦ B is injective, we see that we may assume that C := trE/F ◦ B is either q or if p = 2 and q is a quadratic form, C is the associated alternating form. Moreover, in the latter case, by considering Sp(B) ∩ O(V, q), we see that H ≤ I(U, f ) where f is a quadratic form on U whose associated alternating form is B . By a dimension argument, we may assume that q = trE/F ◦ f . Thus, we see that G is contained in the normalizer of I(U, h) where h is a form of the same type as q (i.e. quadratic or alternating). Since q = trE/F ◦ q , it follows that I(U, h) ≤ I(X, q) and the result follows. Next assume that τ 6= 1. It follows that H preserves a unitary form h on U (note the assumption implies that [E : F ] is even). Let E0 denote the xed eld of τ . Since H acts absolutely irreducibly on U , it is contained in precisely one unitary group on U , whence G normalizes this unitary group. ¤ A minor variant on the previous argument shows that:

Assume that h is a nondegenerate hermitian form on V where V is a vector space of dimension n over the nite eld F . Let G ≤ X(V, h) be an irreducible subgroup . Assume that every normal subgroup of G acts homogeneously on V . Assume that G normalizes some proper eld extension of F where contained in EndF (V ). Assume moreover that G preserves no additive decomposition of V . Then G ≤ X(U, h0 ) where U = V considered as a vector space over some nontrivial eld extension E/F and h0 is a hermitian form on U . Proposition 12.3.


39

12.2. Some Elementary Representation Theory Let H be a normal subgroup of nite index in G and assume that V is a homogeneous F H -module and an irreducible F G-module . Let W be an irreducible constituent of V for H . (a) If W is absolutely irreducible , then G embeds in GL(W ) ⊗ GL(U ) acting on W ⊗U ∼ =V. (b) If G/H is abelian and V is absolutely irreducibly as a G-module , then V is an irreducible F H -module . (c) If G/H is abelian of exponent m and all mth roots of 1 are in F , then V irreducible as an F G-module implies that V is irreducible as an F H -module .

Lemma 12.4.

(a) is well known and is straightforward by computing the normalizer of H in GL(V ). We now prove (b). By Frobenius reciprocity, V embeds in the induced module G WH . This has the same composition factors as W ⊗ F [G/H] (for example, we can compute the Brauer character). Since G/H is abelian, over the algebraic G closure, we can nd a chain of G-submodules of WH so that all quotients are isomorphic to W as H -modules. Since V is absolutely irreducible, this implies that dim V ≤ dim W , whence V = W as required. If G/H has exponent m and F contains all mth roots of 1, then the same G argument shows that each F G-composition factor of WH has dimension at most dim W , whence V = W . Thus, (c) holds. ¤ Proof.

Lemma 12.5.

Let G be a nite group .

(i) If V is an irreducible F G-module , then G preserves a nondegenerate symmetric or alternating form on V if and only if V is self dual . Moreover , any two forms are in the same C -orbit where C is the group of units in the centralizer of G in End(V ). (ii) If V is an irreducible F G-module and p = 2, then G preserves a nondegenerate alternating form if and only if V is self dual . If G preserves a nondegenerate quadratic form on V , then there is a single C -orbit of such quadratic forms . (iii) If V is a homogeneous self dual module and W is an irreducible submodule of dimension m and dim EndF G (W ) = d, then the dimension of the G-invariants on the space of bilinear forms is d(n/m)2 .

G leaves invariant a nonzero bilinear form on V if and only if there are nonzero xed points on V ⊗ V . Any nonzero invariant form must be nondegenerate (since Rad(V ) would be invariant). Such a form gives an F G-isomorphism between V and V ∗ . So we may assume that V is self dual. In that case V ⊗V ∼ = V ⊗V ∗ ∼ = End(V ). The nonzero G invariants on End(V ) are precisely C , whence there is a single C -orbit of invariants. Proof.

40

ROBERT GURALNICK

If p 6= 2, then V ⊗ V is the direct sum of alternating forms and symmetric forms so if G has a xed point, there must be one that is either symmetric or alternating. Since C preserves both spaces (as GL(V ) does), it follows that all invariant forms are either symmetric or alternating. If p = 2, then the composition factors on V ⊗ V (as a GL(V )-module) are ∧2 (V ), V 0 and ∧2 (V ) where V 0 is a twist of V (by the Frobenius automorphism). In any case, V 0 is an irreducible G-module, whence if G has xed points, then G must have a xed point on ∧2 (V ). Thus, G always preserves an alternating form. Arguing as above, we see that the nontrivial G xed points is a single C -orbit. If G preserves a quadratic form q , we claim that the only quadratic forms which are G-invariant are Cq . Let Bq denote the associated alternating form (so Bq (v, v 0 ) = q(v + v 0 ) + q(v) + q(v 0 )). We know that the set of G-invariant alternating forms is a single C -orbit. Suppose q 0 is G-invariant. Then replacing q 0 by an element in Cq 0 , we may replace q 0 by something in its orbit so that Bq = Bq0 . An elementary computation shows that the set of elements with q(v) = q 0 (v) is a proper linear G-invariant subspace. Since G is irreducible, it must be 0, whence q = q 0 and the result follows. If V is homogeneous, then we compute the invariants as above. ¤

Let F be the nite eld of order q 2 . Let V be an n-dimensional vector space over F such that V restricted to G is a homogeneous module with irreducible constituent W . Then G xes a unitary form on W if and only if it does so on V if and only if χ(g q ) = χ(g −1 ) for all g ∈ G where χ is the Brauer character associated to W . If V is irreducible , these conditions are also equivalent to χ(g q ) = χ(g −1 ) for all g ∈ G where χ is the character associated to W . Moreover , if G preserves a unitary form on W , and W is absolutely irreducible , then the space of unitary forms on V which are G-invariant is a vector space over Fq of dimension m2 where m = dim V / dim W is the multiplicity of W . Lemma 12.6.

Let ρ denote the eld automorphism x → xq on F . By denition of the unitary group, we know that ρ(g) is similar to g −1 , viewing G as a subgroup of the unitary group on V . Since V is homogeneous as a G-module, this same condition holds in GL(W ). Thus, to complete the rst part of the proof, we need only show that if the character of W satises the hypothesis, then W supports a G-invariant form. The easiest proof is to use Lang's Theorem. Let φ denote the given representation of G into GL(V ). We need to nd S ∈ GLn (F¯ ) so that Sφ(g)S −1 is in the unitary group. This is equivalent to the equation Proof.

(Sφ(g)S −1 )−T = ρ((Sφ(g)S −1 )). By hypothesis, there exists U ∈ GLn (F ) with U φ = φ0 U . By Lang's Theorem, U = Sρ(S −T ) for some S ∈ GLn (F¯ ). This implies that S satises the equation above. ¤


41

13. Abelian Supplements Let p be a prime. Let p(G) denote the normal subgroup of the nite group generated by all its Sylow p-subgroups. A nite group G is said to be a quasi p-group if G = p(G) or equivalently if G has no nontrivial p0 -quotients. This notion has become quite important in studying fundamental groups of varieties in characteristic p. See [43] and [60] for the solution of the Abhyankar conjecture about fundamental groups of ane curves. Certain two dimensional varieties are considered in [44]. For these varieties, Abhyankar observed that for any nite image of the fundamental group we have

1 → p(G) → G → A → 1, where A is abelian generated by 2 elements. Abhyankar also conjectured that these conditions were sucient. Harbater and Van der Put [44] showed that in fact it must also be the case that G = p(G)B for some abelian subgroup B of G. In the appendix of [44], we developed a theory about this situation and gave examples. Combining the examples with the results of [44] shows that the Abhyankar conjecture does not hold. In this section, we give a simpler form of the example. We also give some examples of a similar phenomenon when A is cyclic of bounded order. Recall that if H ≤ G, then B is called a supplement to H in G if G = HB and a complement to B if in addition H ∩ B = 1. Of course, if G/p(G) is cyclic, we can always write G = p(G)B where B is a cyclic p0 -group. If p(G) is a p-group (i.e. there is a unique Sylow p-subgroup of G), then the short exact sequence above splits and so abelian supplements will always exist. The apparent hope was that quasi p-groups have cohomological properties similar to p-groups. However, that is not the case as the examples in the appendix of [44] show. If G/p(G) is abelian of rank larger than 2, it is quite easy to write down a plethora of examples where there is no abelian supplement to p(G). It is a bit harder to nd such examples with the quotient that is abelian of rank 2. A generalization of the following result appears in the thesis of the author. See also [25].

Let r be an odd prime . Let R := Rd be the free group on 2d generators subject to xr = 1 = [[x, y], z] for all x, y, z ∈ R. Then there exist elements in [R, R] which are a product of d commutators but no fewer . Moreover , Z(R) = [R, R] is elementary abelian of order pd(2d−1) and R/Z(R) is elementary abelian of order p2d .

Lemma 13.1.

It is straightforward to verify that |R| = rd(2d+1) and that Z(R) = [R, R] and R/Z(R) are elementary abelian. If we choose generators xi , 1 ≤ i ≤ 2d for R and set yij = [xi , xj ] for i < j , then any element in w ∈ [R, R] can be Q c expressed uniquely as i<j yijij where cij may be viewed as an element of Fr . Proof.

42

ROBERT GURALNICK

This gives a bijection between [R, R] and the set of 2d × 2d skew symmetric matrices over Fr (by sending w to the unique skew symmetric matrix whose i, j entry to be cij for i < j ). On the other hand, a commutator will correspond to a rank two skew symmetric matrix. So if w corresponds to a nonsingular skew symmetric matrix, w is a product of d commutators but no fewer. This also shows that [R, R] has the order mentioned above. ¤ We rst give the example when p = 2 because it is so simple.

Let p = 2 and r be any odd prime . Let H be the semidirect product of R2 and a cyclic group generated by an involution τ , where τ inverts each generator xi . Note that τ centralizes Z := Z(R2 ). Pick a subgroup Y = hyi of Z of order r which contains noncommutators . Let E be the extraspecial group of order r3 and exponent r. Let G be the central product of H and E identifying Z(E) and Y (precisely , G = (H × E)/h(y, w)i where w generates Z(E)). Then Proposition 13.2.

(i) p(G) = H ; (ii) G/p(G) is elementary abelian of order r2 ; and (iii) there is no abelian supplement to H in G. Since R = [τ, R], it follows that the normal closure of τ is H and so the rst assertion holds. Clearly G/H ∼ = E/W is elementary abelian of order r2 , whence the second statement holds. Suppose that B is an abelian supplement to H in G. There is no harm in assuming that B is an r-group (pass to the Sylow r-subgroup of B ) and is generated by two elements u, v (pass the subgroup generated by a pair of elements which generate modulo H ). Then u = ah1 and v = bh2 where a, b generate E/W and hi ∈ R (clearly, we can take hi ∈ H but since u and v are r-elements, so are the hi ). Then 1 = [u, v] = [ah1 , bh2 ] = [a, b][h1 , h2 ]. This implies that y j = [h1 , h2 ] for some nontrivial j (since [a, b] is a nontrivial power of w which we identify with that same power of y ). However, y j is not a commutator in R. This contradiction completes the proof. ¤ Proof.

We now show how to modify the construction for an arbitrary p. Let r be a prime congruent to 1 modulo p. Let c, d ∈ Fr∗ of order p with cd = 1. Then there is an automorphism τ of order p of R2 which sends xi to xci if i is even and xdi if i is odd. Moreover, τ centralizes y = y14 y23 . Note that y ∈ Z(R) and y is not a commutator in R. Let H be the semidirect product R and the group generated by τ . Let E be as above and dene G to be the central product of H and E identify y with a nontrivial central element of E . The proof of the previous result shows that: (i) p(G) = H ; (ii) G/p(G) is elementary abelian of order r2 ; and (iii) there is no abelian supplement to H in G.

Proposition 13.3.


43

More recently, Harbater has become interested in another fundamental group problem. In this case, G is a group with G/p(G) cyclic of order dividing a xed m (with m prime to p). The question is whether there is a cyclic supplement of order dividing m. If G/p(G) cyclic of order exactly m, then any cyclic supplement of order dividing m would have to be a complement of order m. If m is innite, then of course one can also nd such a supplement (just choose any cyclic subgroup which generates G/p(G)). It is not hard to show that for any xed m there are examples with no cyclic supplement of order dividing m. The rst example that comes to mind is G = M10 and m = 2. Let p be 3 or 5. Then p(G) = A6 and G/p(G) has order 2. However, p(G) contains all involutions of G and so there is no supplement of order 2. For convenience, let us take m an odd prime (dierent from p). An obvious modication of the construction gives examples for any m prime to p. Let S be an extraspecial m-group of order m1+2d such that S admits an automorphism τ of order p with CS (τ ) = Z(S). The existence of such an automorphism amounts to nding an element of order p in Sp(V ) that has no trivial eigenvalues. Let H be the semidirect product of S and τ . Let G be the central product of H and J = hwi with J cyclic of order m2 (where we identify the center of S with the subgroup of J of order m). Any cyclic supplement of order m is generated by an element of the form wh for some h ∈ S . Since m is odd, it is straightforward to compute that (wh)m = wm for all h ∈ S , whence any cyclic supplement has order a multiple of m2 . Clearly p(G) = H and G/H is cyclic of order m.

References [1] S. Abhyankar, Nice equations for nice groups. Israel J. Math. 88 (1994), 123. [2] S. Abhyankar, Symplectic groups and permutation polynomials, Part I, preprint. [3] S. Abhyankar, Symplectic groups and permutation polynomials, II, Finite Fields Appl. 8 (2002), 233255. [4] S. Abhyankar, Orthogonal groups and permutation polynomials, preprint. [5] S. Abhyankar and N. Inglis, Galois groups of some vectorial polynomials, Trans. Amer. Math. Soc. 353 (2001), no. 7, 29412869. [6] M. Aschbacher, On conjectures of Guralnick and Thompson, J. Algebra 135 (1990), 277343. [7] M. Aschbacher, On the maximal subgroups of the nite classical groups. Invent. Math. 76 (1984), 469514. [8] M. Aschbacher, Finite Group Theory, Cambridge University Press, Cambridge, 1986. [9] M. Aschbacher and L. Scott, Maximal subgroups of nite groups, J. Algebra 92 (1985), 4480. [10] S. Cohen, Permutation Polynomials in Shum, Kar-Ping (ed.) et al. Algebras and combinatorics, Papers from the international congress, ICAC'97, Hong Kong, August 1997, Singapore, Springer, 133146 (1999).

44

ROBERT GURALNICK

[11] S. Cohen, and R. Matthews, A class of exceptional polynomials, Trans. Amer. Math. Soc. 345 (1994), no. 2, 897909. [12] N. Elkies, Linearized algebra and nite groups of Lie type. I. Linear and symplectic groups. Applications of curves over nite elds (Seattle, WA, 1997), 77107, Contemp. Math., 245, Amer. Math. Soc., Providence, RI, 1999. [13] W. Feit, On symmetric balanced incomplete block designs with doubly transitive automorphism groups, J. Combinatorial Theory Ser. A 14 (1973), 221247. [14] M. D. Fried, Galois groups and complex multiplication, Trans. Amer. Math. Soc. 235 (1978), 141162. [15] M. D. Fried, On a Theorem of MacCluer, Acta Arith. XXV (1974), 122127. [16] M. Fried, R. Guralnick, and J. Saxl, Schur covers and Carlitz's conjecture, Israel J. Math. 82 (1993), 157225. [17] G. Frey, K. Magaard and H. Voelklein, The monodromy group of a function on a general curve, preprint. [18] M. Fried and H. Völklein, Unramied abelian extensions of Galois covers, Proceedings of the Summer Research Institute on Theta Functions (Gunning and Ehrenpreis, editors), Proceedings of Symposia in Pure Mathematics 49 (1989), 675693. [19] D. Frohardt, R. Guralnick and K. Magaard, Incidence matrices, permutation characters, and the minimal genus of a permutation group. J. Combin. Theory Ser. A 98 (2002), 87105. [20] D. Frohardt and K. Magaard, Monodromy composition factors among exceptional groups of Lie type in Group Theory, Proceedings of the Biennial Ohio State-Denison Conference, 134143(eds. Sehgal and Solomon), World Scientic, Singapore, 1993. [21] D. Frohardt and K. Magaard, Grassmanian xed point ratios, Geometriae Dedicata 82 (2000), 21104. [22] D. Frohardt and K. Magaard, Composition factors of monodromy groups, Ann. of Math. 154 (2001), 327345. [23] D. Gorenstein, Finite Groups, Harper and Row, New York, 1968. [24] D. Gorenstein, R. Solomon and R. Lyons, The classication of nite simple groups. Number 3, Part 1. Chapter A, Almost simple K -groups, Amer. Math. Soc., Providence, RI, 1998. [25] R. Guralnick, On a result of Schur, J. Algebra 59 (1979), 302310. [26] R. Guralnick, Subgroups of prime power index in a simple group. J. Algebra 81 (1983), 304311. [27] R. Guralnick, Monodromy groups of rational functions which are Frobenius groups, preprint (1998). [28] R. Guralnick, The genus of a permutation group, in Groups, Combinatorics and Geometry, edited by M. Liebeck and J. Saxl, LMS Lecture Note Series 165, Cambridge University Press, London, 1992. [29] R. Guralnick and W. Kantor, Probabilistic generation of nite simple groups. Special issue in honor of Helmut Wielandt. J. Algebra 234 (2000), 743792. [30] R. Guralnick and P. Müller, Exceptional polynomials of ane type. J. Algebra 194 (1997), 429454.


45

[31] R. Guralnick, P. Müller, and J. Saxl, The rational function analogue of a question of Schur and exceptionality of permutation representations, Mem. Amer. Math. Soc., to appear. [32] R. Guralnick and M. Neubauer, Monodromy groups of branched coverings: The generic case, in Recent developments in the inverse Galois problem (Seattle, WA 1993) 325352, Comtemp. Math.,186, Amer. Math. Soc., Providence, RI, 1995. [33] R. Guralnick, T. Pentilla, C. Praeger, and J. Saxl, Linear groups with orders having certain large prime divisors. Proc. London Math. Soc. (3) 78 (1999), 167214. [34] R. Guralnick, J. Rosenberg, and M. Zieve, A new class of exceptional polynomials in characteristic 2, preprint (2000). [35] R. Guralnick and J. Saxl, Monodromy groups of polynomials. Groups of Lie type and their geometries (Como, 1993), 125150, London Math. Soc. Lecture Note Ser., 207, Cambridge Univ. Press, Cambridge, 1995. [36] R. Guralnick and J. Saxl, Exceptional polynomials over arbitary elds, to appear. [37] R. Guralnick, J. Saxl and M. Zieve, in preparation. [38] R. Guralnick and K. Stevenson, Prescribing ramication in Arithmetic fundamental groups and noncommutative algebra, Proceedings of Symposia in Pure Mathematics, 70 (2002) editors M. Fried and Y. Ihara, 1999 von Neumann Conference on Arithmetic Fundamental Groups and Noncommutative Algebra, August 1627, 1999 MSRI. [39] R. Guralnick and J. Shareshian, Genus of symmetric and alternating groups actions I., preprint (2002). [40] R. Guralnick and J. Thompson, Finite Groups of Genus Zero, J. Algebra 131 (1990) 303341. [41] R. Guralnick and D. Wan, Bounds for xed point free elements in a transitive group and applications to curves over nite elds, Israel J. Math. 101 (1997), 255287. [42] R. Guralnick and M. Zieve, Polynomials with monodromy P SL(2, q), preprint (2000). [43] D. Harbater, Abhyankar's conjecture on Galois groups over curves, Inventiones Math., 117 (1994), 125. [44] D. Harbater and M. van der Put with an appendix by R. Guralnick, Valued elds and covers in characteristic p, in Valuation Theory and its Applications, Fields Institute Communications, vol. 32, edited by F.-V. Kuhlmann, S. Kuhlmann and M. Marshall, 2002, 175204. [45] P. Kleidman and M. Liebeck, The subgroup structure of the nite classical groups. London Mathematical Society Lecture Note Series, 129. Cambridge University Press, Cambridge, 1990. [46] N. Katz, Local-to-global extensions of representations of fundamental groups, Ann. Inst. Fourier (Grenoble) 36 (1986), 69106. [47] H. Lenstra, H. W., Jr. and M. Zieve, A family of exceptional polynomials in characteristic three, Finite elds and applications (Glasgow, 1995), 209218, London Math. Soc. Lecture Note Ser., 233, Cambridge Univ. Press, Cambridge, 1996. [48] M. Liebeck, C. Praeger and J. Saxl, On the O'Nan-Scott theorem for nite primitive permutation groups, J. Austral. Math. Soc. Ser. A 44 (1988), 389396.

46

ROBERT GURALNICK

[49] M. Liebeck and J. Saxl, Minimal degrees of primitive permutation groups, with an application to monodromy groups of covers of Riemann surfaces, Proc. London Math. Soc. (3) 63 (1991), 266314. [50] M. W. Liebeck and G. Seitz, On the subgroup structure of classical groups, Invent. Math. 134 (1998), 427453. [51] M. Liebeck and A. Shalev, Simple groups, permutation groups and probability, J. Amer. Math. Soc. 12 (1999), 497520. [52] F. Lübeck, F., Smallest degrees of representations of exceptional groups of Lie type, Comm. Algebra 29 (2001), 21472169. [53] K. Magaard, Monodromy and Sporadic Groups, Comm. Algebra 21 (1993), 4271 4297. [54] G. Malle, Explicit realization of the Dickson groups G2 (q) as Galois groups in their dening characteristic, Pacic J. Math., to appear. [55] P. Müller, Primitive monodromy groups of polynomials, Recent developments in the inverse Galois problem (Seattle, WA, 1993), 385401, Contemp. Math., 186, Amer. Math. Soc., Providence, RI, 1995. [56] S. Nakajima, p-ranks and automorphism groups of algebraic curves, Trans. Amer. Math. Soc. 303 (1987), 595607. [57] S. Nakajima, On abelian automorphism groups of algebraic curves, J. London Math. Soc. (2) 36 (1987), 2332. [58] M. Neubauer, On monodromy groups of xed genus, J. Algebra 153 (1992), 215 261. [59] M. Neubauer, On primitive monodromy groups of genus zero and one. I. Comm. Algebra 21 (1993), 711746. [60] M. Raynaud, Revêtements de la droite ane en caractéristique p > 0 et conjecture d'Abhyankar, Invent. Math. 116 (1994), 425462. [61] J.-P. Serre, Local elds. Translated from the French by Marvin Jay Greenberg. Graduate Texts in Mathematics, 67. Springer-Verlag, New York-Berlin, 1979. [62] Shih, T., A note on groups of genus zero, Comm. in Alg. 19 (1991), 28132826. [63] K. Stevenson, Galois groups of unramied covers of projective curves in characteristic p, J. Algebra 182 (1996), 770804. [64] H. Stichtenoth, Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik. I. Eine Abschätzung der Ordnung der Automorphismengruppe, Arch. Math. (Basel) 24 (1973), 527544. [65] H. Stichtenoth, Algebraic function elds and codes. Universitext. Springer-Verlag, Berlin, 1993. [66] P. H. Tiep and A. Zalesskii, Minimal characters of the nite classical groups, Comm. Algebra 24 (1996), 20932167. Robert Guralnick Department of Mathematics University of Southern California Los Angeles, CA 90089-1113 United States

[email protected]


On the Tame Fundamental Groups of Curves over Algebraically Closed Fields of Characteristic > 0 AKIO TAMAGAWA

Abstract. We prove that the isomorphism class of the tame fundamental group of a smooth, connected curve over an algebraically closed eld k of characteristic p > 0 determines the genus g and the number n of punctures of the curve, unless (g, n) = (0, 0), (0, 1). Moreover, assuming g = 0, n > 1, and that k is the algebraic closure of the prime eld Fp , we prove that the isomorphism class of the tame fundamental group even completely determines the isomorphism class of the curve as a scheme (though not necessarily as a k-scheme). As a key tool to prove these results, we generalize Raynaud's theory of theta divisors.

Introduction Let k be an algebraically closed eld of characteristic p > 0, and U a smooth, connected curve over k . (A curve is a separated scheme of dimension 1.) We denote by X the smooth compactication of U and put S = X − U . We dene non-negative integers g and n to be the genus of X and the cardinality of the point set S , respectively. In [T2], we proved that the isomorphism class of the (pronite) fundamental group π1 (U ) of U determines the pair (g, n), and that, when g = 0 and k is the algebraic closure Fp of the prime eld Fp , the isomorphism class of π1 (U ) even completely determines the isomorphism class of the curve as a scheme. The aim of the present paper is to generalize these results to the case that π1 (U ) is replaced by its quotient π1t (U ), the tame fundamental group of U (see [SGA1], Exp. XIII and [GM]), as the author announced in [T2], Note 0.3. Thus the main results of the present paper are the following. (See (4.1).) The isomorphism class of the pronite group π1t (U ) determines the pair (g, n), unless (g, n) = (0, 0), (0, 1). Theorem (0.1).

47

48

AKIO TAMAGAWA

Theorem (0.2). (See (5.9).) Assume g = 0, n > 1, and either k = Fp or n ≤ 4. Then the isomorphism class of the pronite group π1t (U ) completely determines the isomorphism class of the scheme U . More precisely , for two such curves Ui /k (i = 1, 2), π1t (U1 ) ' π1t (U2 ) if and only if U1 ' U2 as schemes .

Since it is rather easy to see that the quotient π1t (U ) of π1 (U ) can be recovered group-theoretically from π1 (U ) ([T2], Corollary 1.5), the results of the present paper are stronger than those of [T2]. (i) When g and n are small (more precisely, when 2g + n ≤ 4), (0.1) has been settled by Bouw. See [B] for this and other related results.

Remark (0.3).

(ii) In [T1], a result similar to (0.2) was proved for U ane, smooth, geometrically connected curve (of arbitrary genus) over a nite eld F. In this case, the (arithmetic) tame fundamental group π1t (U ) is an extension of the absolute Galois group Gal(F/F) by the geometric tame fundamental group π1t (U ⊗F F), and we exploited the (outer) Galois action on the geometric tame fundamental group. (0.2) above shows that (for g = 0) the geometric tame fundamental group, without the Galois action, is enough to recover the moduli of the curve. For a pronite group Π, let ΠA denote the set of isomorphism classes of all nite quotients of Π. It is known that the subset ΠA of the set of isomorphism classes of all nite groups completely determines the isomorphism class of the pronite group Π, if Π is nitely generated ([FJ], Proposition 15.4). t We shall write πA (U ) instead of π1t (U )A . Then, since π1t (U ) is nitely genert ated, the information carried by πA (U ) is equivalent to the information carried by t (the isomorphism class of) π1 (U ). Therefore, we can restate the above theorems t t in terms of πA (U ). Moreover, as for (0.1), we can say how πA (U ) determines the pair (g, n) explicitly, by looking carefully at the proofs in the present paper. For this, see [T3]. Remark (0.4).

In [T2], the result corresponding to (0.1) followed from a quick argument combining the Hurwitz formula and the DeuringShafarevich formula, which involves wild ramication. However, in our case, we cannot resort to wild ramication, and we need another strategy. In order to explain our strategy to prove (0.1), rst we shall assume n = 0, or, equivalently, U = X . (Under this assumption, it is elementary to prove (0.1), though. See (4.3)(i).) Note that then we have π1t (U ) = π1 (X). In this case, all the ingredients of our strategy are given by Raynaud's theory of theta divisors ([R1]). (i) The p-rank (or HasseWitt invariant) γX of X is dened to be the dimension of the Fp -vector space Hom(π1 (X), Fp ). More generally, for each surjective homomorphism ρ : π1 (X) ³ G, where G is a nite cyclic group of order N prime to p, Ker(ρ) may be identied with π1 (Y ), where Y → X is the nite étale G-covering corresponding to ρ. Then, G acts on Hom(π1 (Y ), Fp ), and

TAME FUNDAMENTAL GROUPS OF CURVES

49

Hom(π1 (Y ), Fp ) ⊗ k admits a canonical decomposition as a direct sum, corresponding to the decomposition of the group algebra k[G] as the direct product of N copies of k , each of which corresponds to a character G → k × . Now, the dimension of each direct summand of Hom(π1 (Y ), Fp ) ⊗ k is the so-called generalized HasseWitt invariant (see [Ka], [Na], and [B]). (ii) On the other hand, it is well-known that the set of connected nite étale G-coverings of X is in one-to-one correspondence with the set of isomorphism classes of line bundles on X of order N , or, equivalently, the set of points of order N of the Jacobian variety J of X . More precisely, this one-to-one correspondence ∼ is given by xing an isomorphism G→µN (k). For each line bundle L of order N , let f be the order of p mod N in (Z/N Z)× . Then, taking the composite of the pf f f f ∼ th power map L → L⊗p and the isomorphism L⊗p = L ⊗ L⊗(p −1) →L, we get a map L → L, which induces a pf -linear map ϕ[L] : H 1 (X, L) → H 1 (X, L). Now, ∼ the generalized HasseWitt invariant γ[L] with respect to L and G→µN (k) ⊂ k × T r coincides with the dimension of the k -vector space r≥1 Im((ϕL ) ). (iii) Raynaud ([R1]) dened a certain divisor ΘB of J (more naturally, of the Frobenius twist J1 of J ) in a canonical way depending only on X , such that [L] ∈ J belongs to ΘB if and only if the p-linear map H 1 (X, L) → H 1 (X, L⊗p ) induced by the p-th power map L → L⊗p is an isomorphism. In particular, if [L] is a torsion element of order N prime to p as above, ϕ[L] is an isomorphism i (or, equivalently, γ[L] = dimk (H 1 (X, L))), if and only if [L⊗p ] ∈ / ΘB for all i = 0, 1, . . . , f − 1. (iv) More precisely, Raynaud dened a vector bundle B with rank p − 1, degree (p − 1)(g − 1), and EulerPoincaré characteristic 0 to be the cokernel of (the linearization of) the p-th power map OX → OX . Then, he dened ΘB as the theta divisor of B . (That is to say, [L] ∈ / ΘB if and only if H 0 (X, B ⊗ L) = H 1 (X, B⊗L) = 0.) It is easy to see that ΘB is a closed subscheme of codimension ≤ 1 of J , and the main diculty consists in proving that ΘB does not coincide with J . To prove this, Raynaud resorted to a ring-theoretic argument involving the Koszul complex over the (regular) local ring at the origin of J . (v) By using intersection theory, Raynaud proved #(ΘB ∩ J[N ]) = O(N 2g−2 ). From this, we obtain #{[L] ∈ J[N ] | ∃i, s.t. [L⊗i ] ∈ ΘB } = O(N 2g−1 ). So, as a conclusion, we can roughly say that, for `most' prime-to-p-cyclic (nite étale) coverings of X , the generalized HasseWitt invariants are as large as possible. In other words, γ[L] = g − 1 holds for `most' L (unless g = 0). Since the generalized HasseWitt invariants are encoded in π1 (X) by denition, this gives a grouptheoretic characterization of the invariant g − 1. For Raynaud's theory of theta divisors, see also [R2] (a generalization) and [Mad] (an exposition). In this paper, we generalize these arguments to the (possibly) ramied case n > 0. Here, a cyclic (nite étale) covering of U of degree N prime to p corre-

50

AKIO TAMAGAWA

sponds to a pair of a line bundle L and an eective divisor D (whose support is contained in S ) satisfying certain conditions. (In particular, L⊗N ' OX (−D) is required.) Then, as in (ii) above, we can describe the corresponding generalized HasseWitt invariant γ([L],D) in terms of a pf -linear map ϕ([L],D) : H 1 (X, L) → H 1 (X, L). Note that, unlike in the case n = 0 (and L 6' OX ), the dimension of H 1 (X, L) depends on L, since deg(L) varies (among 0, −1, . . . , −(n − 1)). Then, f under certain assumptions on D, we can dene a vector bundle BD depending on f and D, which yields a closed subscheme of J (more naturally, of the f -th Frobenius twist Jf of J ). Now, the main result (2.5) says that this closed subscheme is a divisor if deg(D) = pf −1. As a corollary, we have that, if N = pf −1, for `most' pairs ([L], D) with deg(L) = −1, the generalized HasseWitt invariant γ([L],D) is as large as possible, i.e., coincides with dimk (H 1 (X, L)) = g (if n > 1). On the other hand, by a combinatorial argument, we prove that, if n > 1 and N = pf − 1, for `most' pairs ([L], D), there exists an i = 0, 1, . . . , f − 1 such that deg(Lpi ) = −1. (Here, Lj is a certain modication of L⊗j , so that the cyclic LN −1 (ramied) covering of X corresponding to ([L], D) is the spectrum of j=0 Lj .) Combining these, we can conclude that for `most' prime-to-p-cyclic coverings of U (with degree in the form pf − 1), the generalized HasseWitt invariants coincide with g . This gives a group-theoretic characterization of g . Since it is easy to see that the EulerPoincaré characteristic 2 − 2g − n can be recovered group-theoretically from π1t (U ), this completes the proof of (0.1). Just as in [T2], we can then prove that (for U hyperbolic) the set of inertia subgroups of π1t (U ) can be recovered group-theoretically from π1t (U ), by using (0.1) (see (5.2)). Now, what is missing to prove (0.2) along the lines of [T2] is only to recover the àdditive structures' of the inertia subgroups (see Section 5, (B)). In [T2], this was done by studying wild ramication again. In our case, another usage of our generalization of Raynaud's theory settles the problem. This completes the proof of (0.2). We shall explain briey the content of each section of the present paper, and show in which section each part of the above arguments is contained. The order of the sections does not necessarily follow the order of the above arguments, because it is natural to present the main theorems (which assure the existence of f the theta divisor associated with BD ) as early as possible, and then to present the main results concerning π1 (X) as corollaries of the main theorems. In Section 1, we give a generalization of Raynaud's ring-theoretic argument in (iv) above. The main result is (1.12). In fact, Raynaud's original argument is sucient for the proofs of the main results of the present paper. However, we include this generalization since it is done by just replacing the Koszul complex in Raynaud's proof with the so-called EagonNorthcott complex, and since it is likely that this generalization will be applied to other related problems concerning coverings and fundamental groups.


51

In Section 2, after quickly reviewing Raynaud's theory concerning the theta f divisor ΘB , we dene the vector bundle BD , and prove (by using (1.12)) the main result (2.5) which assures the existence of the theta divisor associated f under certain assumptions. (2.5), together with a slight generalization with BD (2.6), plays a crucial role in the group-theoretic characterization of the genus. Moreover, with another variant (2.13), we investigate the case n ≤ 3 in more detail (2.21). This plays a central role in the group-theoretic characterization of the additive structures of inertia groups. In Section 3, we give a review of generalized HasseWitt invariants, a description (3.5) of prime-to-p-cyclic coverings of X that are unramied on U in terms of line bundles and divisors on X , and a reinterpretation of generalized Hasse Witt invariants via this description. Then, after presenting some inputs from intersection theory (3.10), we prove the main numerical results (3.12) and (3.16) concerning the generalized HasseWitt invariants of prime-to-p-cyclic coverings of U , by using the results of Section 2. Note that so far the eective divisor D is xed. Now, combining these results with a combinatorial result (3.18) (which enables D to vary), we nally establish the summarizing result (3.20) to the efdef fect that most generalized HasseWitt invariants coincides with g 0 , where g 0 = g def (resp. g 0 = g − 1) for n > 1 (resp. n ≤ 1). In Section 4, we apply the results of Section 3 and give a group-theoretic characterization of g 0 in an eective way (4.10) and in an ineective but impressive way (4.11). The latter can be stated as follows. Here, for each pronite group Π and a natural number m, we denote by Π(m) the kernel of Π ³ Πab ⊗ Z/mZ. Theorem (0.5).

(See (4.8), (4.11), and (4.12).) We have

lim γpavf −1 = g 0 ,

f →∞

unless (g, n) = (0, 0), (0, 1), where av γN =

dimFp (π1t (U )(N )ab ⊗ Fp ) . #(π1t (U )ab ⊗ Z/N Z)

Moreover, after settling a few more minor technical problems (for example, the problem that the results above do not give a characterization of g but only give a characterization of g 0 ), we obtain the group-theoretic characterization (4.1) of the pair (g, n). Finally, in Section 5, we give group-theoretic characterizations of the inertia subgroups (5.2) and the àdditive structures' of inertia subgroups (5.3), and present anabelian-geometric results (5.8) and (5.9) for g = 0. In the Appendix, we give a proof of a partial generalization (4.17) of the limit formula (4.11). Here, the theory of uniform distribution (especially, Stegbuchner's higher-dimensional version of LeVeque's inequality) plays a key role.

52

AKIO TAMAGAWA

Acknowledgement. When the author presented the results of [T2] in Kyoto

(May, 1996) and in Oberwolfach (June, 1997), Shinichi Mochizuki and Michel Raynaud, respectively, asked the author about the possibility of generalizing the results of [T2] to the case that the fundamental group is replaced by the tame fundamental group. Although the author had already taken an interest in such possibility, their questions stimulated and encouraged the author very much. The author would like to thank them very much. Also, when the author was trying to prove a more general limit formula (4.16) of the genus, Makoto Nagata suggested the possibility of applying the theory of uniform distribution, which is essential in the Appendix of the present paper. The author would like to thank him very much.

1. A Generalization of the Ring-Theoretic Part of Raynaud's Theory In this section, we shall give a generalization of the ring-theoretic part of Raynaud's theory ([R1], 4.2). The statement of our main result is more general than [R1], Lemme 4.2.3, but the proof is rather similar to Raynaud's proof, if we replace the Koszul complex by the so-called EagonNorthcott complex. Now, let Y be a connected, noetherian scheme and f : X → Y a proper morphism whose bers are of dimension ≤ 1. Let F be a coherent OX -module at over Y . For each y ∈ Y and i = 0, 1, dene hi (y) = hi (F, y) to be the dimension of the k(y)-vector space H i (Xy , F ⊗ k(y)), where Xy denotes the scheme-theoretic ber X ⊗ k(y) of f at y and F ⊗ k(y) denotes the OXy - module obtained as the pull-back of F to Xy . By the local constancy of the Euler Poincaré characteristic ([Mu], § 5, Corollary on p.50), def

χF = h0 (y) − h1 (y) is independent of y . (i) For each i ∈ Z≥0 , we denote by Zi = Zi (F) the closed subscheme of Y dened by the i-th Fitting ideal Fitti (R1 f∗ (F)) of OY . def def (ii) We put W (F) = Z(−χF )+ (F), where x+ = max(x, 0). Definition.

For the denition and the properties of Fitting ideals, we refer to [E], Chapter 20, where only Fitting ideals of modules over rings are treated. However, since the formation of Fitting ideals commutes with localization ([E], Corollary 20.5), we can dene and treat Fitting ideals of coherent sheaves on schemes without any extra eorts. (See [SGA7I], Exp. VI, § 5.) Remark (1.1).

Lemma (1.2).

For each y ∈ Y , we have y∈ / Zi (F) ⇐⇒ h1 (y) ≤ i.

In particular ,

y∈ / W (F) ⇐⇒ min(h0 (y), h1 (y)) = 0.


53

Proof. The rst assertion follows from [E], Proposition 20.6 and [Mu], § 5, Corollary 3 on p. 53. The second assertion follows from the rst, together with the identity

h1 (y) − (−χF )+ = min(h0 (y), h1 (y)).

(1.3)

¤

In the special case that h0 (y) = 1, we have:

Let y be a point of Y , and assume that h0 (y) = 1 and that h (y) ≥ 1. Then , in a certain open neighborhood of y , W (F) is the maximal closed subscheme W on which R1 f∗ (F)|W is locally free of rank h1 (y). Lemma (1.4). 1

By the assumption, we have (−χF )+ = h1 (y) − 1, hence W (F) = Zh1 (y)−1 (F). Put U = Y − Zh1 (y) (F), which is an open neighborhood of y by (1.2). Then, by [E], Proposition 20.8, W (F)∩U is the maximal closed subscheme WU of U on which R1 f∗ (F)|WU is locally free of rank h1 (y), as desired. ¤

Proof.

Next, we shall describe W (F) by using the theory of perfect complexes. For a homomorphism φ : A → B in an abelian category, we denote by (A → B) or simply by (A → B) the complex Definition. φ

φ

··· → 0 → 0 → A → B → 0 → 0 → ···, where A (resp. B ) is placed in degree 0 (resp. 1).

Let y be a point of Y . Then , in some open neighborhood of y , the object Rf∗ (F) (in the derived category of OY -modules ) can be represented by a

Lemma (1.5).

h0 (y) φ

h1 (y)

complex in the form (OY → OY ) with φ ⊗ k(y) = 0. Moreover , for each homomorphism F → F 0 between coherent OX -modules F and F 0 at over Y , the corresponding morphism Rf∗ (F) → Rf∗ (F 0 ) can be represented by a homomorphism of complexes in the form h0 (F ,y) φ

(OY

that is :

h1 (F ,y)

→ OY

0 h0 (F 0 ,y) φ

) → (OY

.. . ↓

h0 (F ,y)

OY φ↓ h1 (F ,y) OY ↓ .. .

h1 (F 0 ,y)

→ OY

),

.. . ↓

→ →

h0 (F 0 ,y)

OY φ0 ↓ h1 (F 0 ,y) OY ↓ .. .

.

Zariski locally on Y , Rf∗ (F) can be represented by a perfect complex in the form ((OY )n0 → (OY )n1 ). (See [Mu], § 5, the second Theorem on p.46. Proof.

54

AKIO TAMAGAWA

We can take a complex in this form since hi (y) = 0 for y ∈ Y and i > 1. See also [SGA6], Exp. IIII.) In particular, we have the exact sequence

0 → R0 f∗ (F) → (OY )n0 → (OY )n1 → R1 f∗ (F) → 0.

(1.6)

On the other hand, consider a minimal free resolution of the OY,y -module R1 f∗ (F)y :

· · · → (OY,y )m0 → (OY,y )m1 → R1 f∗ (F)y → 0. By [E], Theorem 20.2, we can see that the exact sequence

(OY,y )n0 → (OY,y )n1 → R1 f∗ (F)y → 0 obtained by localizing (1.6) is isomorphic to the direct sum of (part of) the minimal resolution

(OY,y )m0 → (OY,y )m1 → R1 f∗ (F)y → 0

(1.7)

proj.

and the complex ((OY,y )a+b → (OY,y )a ) for some a, b ≥ 0. Now, taking the direct sum of (1.7) and the complex ((OY,y )b → 0), we obtain a new complex 0 ((OY,y )m0 → (OY,y )m1 ), where m00 = m0 + b, which is homotopically equivalent to ((OY,y )n0 → (OY,y )n1 ), by denition. Since we are dealing with only nite number of modules and homomorphisms, we can extend this homotopy 0 equivalence to one between ((OY )n0 → (OY )n1 ) and ((OY )m0 → (OY )m1 ), if we replace Y by a suitable open neighborhood of y . By the denition of the minimality, the homomorphism (OY,y )m1 → R1 f∗ (F)y becomes an isomorphism after being tensored with k(y). Then, by [Mu], § 5, Corollary 3 on p.53, we obtain m1 = h1 (y). Thus Rf∗ (F) is represented by the 0 1 complex ((OY )m0 → (OY )h (y) ). Finally, by tensoring this complex with k(y) again, we obtain m00 = h0 (y) and φ ⊗ k(y) = 0, as desired. The second assertion follows from the rst assertion and a standard fact in the theory of derived categories (see, e.g., [E], Exercise A3.54). ¤

In some neighborhood of y , W (F) is dened by the ideal generated by the maximal minors of an h0 (y) × h1 (y) matrix representing φ in (1.5). Corollary (1.8).

Proof.

(1.3).

This follows from the denition of Fitting ideal, together with identity ¤

[Mac]). (See [E], Exercise 10.9.) Let R be a noetherian ring , and let F and G be free R-modules of nite rank . Let φ be an Rhomomorphism F → G, and choose a matrix A with coecients in R that represents φ. Let I be the ideal of R generated by the maximal minors of A. Then , for each minimal prime ideal p containing I , we have ht(p) ≤ |rk(F ) − rk(G)| + 1. ¤ Theorem (Macaulay


55

Definition. In Macaulay's theorem, if, moreover, the equality ht(p) = |rk(F ) − rk(G)| + 1 holds for every minimal prime ideal p containing I , we say that φ is determinantal. (See, e.g., [E], 18.5.) Equivalently, φ is determinantal if and only if either I = R or ht(I) = |rk(F ) − rk(G)| + 1.

The codimension of each irreducible component of W (F) does not exceed |χF | + 1. Corollary (1.9).

Proof.

This follows from (1.8) and Macaulay's theorem above.

¤

We say F is determinantal, if the codimension of every irreducible component of W (F) coincides with |χF | + 1. (Equivalently, F is determinantal if and only if either W (F) = ∅ or codim(W (F)) = |χF | + 1.)

Definition.

Before presenting the main result of this section, we shall establish the following key lemma, which is purely in commutative ring theory. In the special case that R is regular, rk(F ) = rk(F 0 ) = dim(R) and rk(G) = rk(G0 ) = 1, this can be seen in [R1], Lemme 4.2.3. Lemma (1.10).

Let R be a CohenMacaulay local ring . Let F φ↓ G

f

→ g

→

F0 φ0 ↓ G0

(1.11)

be a commutative diagram of R-modules , where F, G, F 0 , G0 are free R-modules of nite rank . Assume that : (a) (b) (c) (d)

rk(F ) − rk(G) ≥ 0 and φ is determinantal ; g is surjective ; either rk(F ) − rk(G) = 0 or φ0 is not surjective ; and rk(F ) − rk(G) ≥ rk(F 0 ) − rk(G0 ).

Then : (i) rk(F ) − rk(G) = rk(F 0 ) − rk(G0 ). (ii) φ0 is determinantal . def (iii) The ber product F1 = G ×G0 F 0 is a free R-module of rank rk(F ), and the determinant of the natural homomorphism F → F1 is not zero . Proof. By replacing R with its completion, we may assume that R is complete. In particular, we may assume that R admits a canonical module ω (see [BH], Corollary 3.3.8). From now on, we write χ and χ0 instead of rk(F ) − rk(G) and rk(F 0 ) − rk(G0 ), respectively. Moreover, we denote by I and I 0 the ideals of R generated by the maximal minors of φ and φ0 , respectively. First, we treat the easier case that φ0 is surjective. Then we must have χ0 ≥ 0. On the other hand, by (c) and (d), we have χ0 ≤ χ = 0. Thus χ0 = χ = 0, which implies (i). Since φ0 is surjective with χ0 = 0, φ0 must be an isomorphism. In

56

AKIO TAMAGAWA

particular, we have I 0 = R, hence (ii) holds. Next, the natural map F1 → G is an isomorphism, so we have rk(F1 ) = rk(G) = rk(F ). Moreover, the natural map F → F1 can be identied with φ. Now, since φ is determinantal, the determinant of φ is non-zero. This complete the proof in the case that φ0 is surjective. def def Next, assume that φ0 is not surjective. Put M = Coker(φ) and M 0 = Coker(φ0 ). Since χ ≥ 0 by the rst half of (a), we have I = Fitt0 (M ) by denition, and I annihilates M by [E], Proposition 20.7a. By (b), the natural map M → M 0 is also surjective, hence I annihilates M 0 . In particular, I 6= R as M 0 6= 0, so, by (a), we have ht(I) = χ + 1 ≥ 1. Now, since M 0 is annihilated by I with ht(I) ≥ 1, we obtain χ0 ≥ 0. (To see this, for example, tensor the (right) exact sequence F 0 → G0 → M 0 → 0 with the residue eld at any minimal prime ideal of R.) Thus we have I 0 = Fitt0 (M 0 ). Note that g is surjective by (b) and G0 is free. Accordingly, g is split surjective, def or, equivalently, if we put K = Ker(g), g is a composite of an isomorphism ∼ G→K × G0 that restricts to the identity on K and the projection K × G0 → G0 . From this, we can easily see that the ber product F1 = G ×G0 F 0 is free and ts naturally into the following commutative diagram whose columns are all exact:

F φ↓ G ↓ M ↓ 0

f1

→ = ³

F1 φ1 ↓ G ↓ M0 ↓ 0

F0 φ ↓ ³ G0 ↓ = M0 ↓ 0.

³

0

Moreover, we have rk(F1 ) − rk(G) = χ0 ≥ 0. Thus, calculating Fitt0 (M 0 ) by using φ1 , we see Fitt0 (M ) ⊂ Fitt0 (M 0 ), or, equivalently, I ⊂ I 0 . Moreover, since I 0 annihilates M 0 6= 0 by [E], Proposition 20.7a, we have I 0 6= R. Thus, we have

χ + 1 = ht(I) ≤ ht(I 0 ) ≤ χ0 + 1, where the last inequality follows from Macaulay's theorem. Combining this with (d), we obtain ht(I 0 ) = χ0 + 1 and χ0 = χ. The former implies (ii), and the latter implies both (i) and the rst half of (iii). Note that φ1 is also determinantal. To see the second half of (iii), we shall compare the EagonNorthcott complexes associated with φ and φ1 . (For EagonNorthcott complexes, see [E], A2.6.) So, consider the following commutative diagram whose rst (resp. second) row is the EagonNorthcott complex canonically associated with φ (resp. φ1 ):

Vr Vs Vs 0 → Dχ ⊗ F → · · · → D0 ⊗ F = F ↓ ↓ Vr Vs Vs 0 → Dχ ⊗ F1 → · · · → D0 ⊗ F1 = F1

Vs

φ

Vs

φ1

k Vs

→ Vs

→

→ →

Vs Vs

⊗R/I → 0 ↓ ⊗R/I 0 → 0,


57

Vs Vs where r = rk(F ) = rk(F1 ), s = rk(G), Di = (Si G)∗ , and = G (' R). Vs Since φ and φ1 are determinantal, the two rows are exact and both ⊗R/I and Vs ⊗R/I 0 are CohenMacaulay R-modules, by [E], Corollary A2.13. Vr Vr Now, suppose that the determinant map F → F1 is zero. Then, the Vr Vr rst vertical arrow Dχ ⊗ F → Dχ ⊗ F1 is also zero. So, calculating (−, ω) by using the EagonNorthcott complexes, we see that the map Extχ+1 R Vs χ+1 Vs 0 Extχ+1 ( ⊗R/I , ω) → Ext ( ⊗R/I, ω) associated with the natural surjecRV R Vs s tion ⊗R/I → ⊗R/I 0 must be also zero. However, since ht(I 0 ) = ht(I) = χ + 1, the duality theory (see [BH], Theorem 3.3.10, (a)⇒(c)) tells us that this Vs Vs implies that the original surjection ⊗R/I → ⊗R/I 0 is zero. This is absurd, Vs 0 since ' R and I 6= R. This completes the proof. ¤ The following is the main result of this section. Theorem (1.12). Let Y be a CohenMacaulay , noetherian , integral scheme . Let f : X → Y be a proper morphism whose bers are of dimension ≤ 1. Let Fi (i = 1, 2, 3) be coherent OX -modules at over Y , and 0 → F1 → F2 → F3 → 0 an exact sequence of OX -modules . Assume that :

(a) F2 is determinantal (in the sense of the Denition following (1.9)); (b) one of the following three conditions holds : χF2 < 0, W (F1 ) 6= ∅; χF2 = 0; χF2 > 0, W (F3 ) 6= ∅; and (c) χF1 · χF3 ≥ 0.

Then : (i) χF1 · χF3 = 0. (ii) F1 and F3 are determinantal . First, we shall treat the case that χF2 ≥ 0. By (1.5), in some neighborhood of each y ∈ Y , the objects Rf∗ (F2 ), Rf∗ (F3 ), and the morphism h0 (F ,y) h1 (F ,y) Rf∗ (F2 ) → Rf∗ (F3 ) can be represented by complexes (OY 2 → OY 2 ), Proof.

h0 (F3 ,y)

(OY

h1 (F3 ,y)

→ OY

), and a commutative diagram h0 (F2 ,y)

OY

OY

→

h1 (F ,y) OY 3

↓

h1 (F ,y) OY 2

h0 (F3 ,y)

→

(1.13)

↓

,

respectively. Put R = OY,y and k = k(y). Localizing (1.13) at y , we obtain a commutative diagram (1.11) of free R-modules of nite rank, where rk(F ) = h0 (F2 , y), rk(G) = h1 (F2 , y), rk(F 0 ) = h0 (F3 , y), and rk(G0 ) = h1 (F3 , y). In particular, we have rk(F ) − rk(G) = χF2 ≥ 0 and rk(F 0 ) − rk(G0 ) = χF3 . We shall check conditions (a)(d) of (1.10) by using our assumptions (a)(c). (For conditions (c) and (d) of (1.10), we need an extra assumption on y . See below.) Condition (a) of (1.10) follows directly from our assumption that χF2 ≥ 0 and our assumption (a). Next, since the ber Xy is of dimension ≤ 1, we see that the map g ⊗ k : G ⊗ k = H 1 (Xy , F2 ⊗ k(y)) → H 1 (Xy , F3 ⊗ k(y)) = G0 ⊗ k

58

AKIO TAMAGAWA

is surjective, hence g is surjective by Nakayama's lemma. Thus condition (b) of (1.10) holds. If χF2 = 0, condition (c) of (1.10) clearly holds. Moreover, in this case, our assumption (c) says −(χF3 )2 ≥ 0, or, equivalently, χF3 = 0. Thus condition (d) of (1.10) holds. If χF2 > 0, we shall put an extra assumption that y ∈ W (F3 ). Then, by (1.2), we have h1 (F3 , y) > 0. Since φ0 ⊗ k = 0, this implies that φ0 ⊗ k is not surjective, hence φ0 is not surjective. Thus condition (c) of (1.10) holds. Moreover, suppose that condition (d) of (1.10) does not hold. Then, we have χF3 > χF2 ≥ 0, and χF1 = χF2 − χF3 < 0. Thus χF1 · χF3 < 0, which contradicts our assumption (c). Now, we may apply (1.10). Conclusion (i) of (1.10) implies χF1 = 0, hence (i) of (1.12) (and χF3 ≥ 0). Conclusion (ii) of (1.10) implies that each irreducible component of W (F3 ) passing through y has codimension |χF3 | + 1 = χF3 + 1. So, (considering all points y ∈ W (F3 )) we obtain that F3 is determinantal. Moreover, conclusion (iii) of (1.10) implies that the map F → G ×G0 F 0 is injective, since R is an integral domain by the assumption that Y is integral. Accordingly, the map

R0 f∗ (F2 )y = Ker(φ) → Ker(φ0 ) = R0 f∗ (F3 )y is injective, or, equivalently, R0 f∗ (F1 )y = 0. Thus, in particular, we obtain h0 (F1 , η) = 0, where η is the generic point of the integral scheme Y . (Here, we have used our assumption (b) for the rst time to choose a point y .) Thus, η ∈ / W (F1 ). Since Y is integral and |χF1 | + 1 = 1, this implies that F1 is determinantal. Next, we shall treat the case that χF2 < 0. In this case, by (1.5), we can take (Zariski locally) a commutative diagram h0 (F1 ,y)

OY

h0 (F2 ,y)

→ OY

↓

h1 (F ,y) OY 1

↓

→

h1 (F ,y) OY 2

(1.14)

representing the morphism Rf∗ (F1 ) → Rf∗ (F2 ). Then, localizing (1.14) at y ∈ Y and taking the dual (= HomR (−, R), where R = OY,y ) of the diagram, we obtain a commutative diagram of free R-modules such as

G0 φ0 ↑ F0

←

G φ↑ ← F

,

where rk(G0 ) = h0 (F1 , y), rk(F 0 ) = h1 (F1 , y), rk(G) = h0 (F2 , y), and rk(F ) = h1 (F2 , y). If we regard this diagram as (1.11), the proof in the case χF2 < 0 can be done just in parallel with that of χF2 > 0. This completes the proof. ¤


59

2. Generalizations of Raynaud's Theorem Let k be an algebraically closed eld and X a proper, smooth, connected curve of genus g over k . For a vector bundle E on X (regarded as a locally free OX -module of nite rank), let rk(E), deg(E) and hi (E) (i = 0, 1) denote the rank of E , the degree of def rk(E)

E (which is dened to be the degree of the line bundle det(E) = ∧ E ), and the dimension (as a k -vector space) of the i-th cohomology group H i (X, E). The RiemannRoch theorem implies the following formula for the EulerPoincaré def characteristic χ(E) = h0 (E) − h1 (E) of E : χ(E) = deg(E) − (g − 1) rk(E).

(2.1)

In [R1], Raynaud investigated the following property of a vector bundle E on X . (?)). We say that E satises (?) if there exists a line bundle L of degree 0 on X such that min(h0 (E ⊗ L), h1 (E ⊗ L)) = 0.

Definition (Condition

First, we shall see the relation between condition (?) and the contents of Section 1. So, Let J be the Jacobian variety of X , and let L be a universal line bundle on X × J . Let prX and prJ denote the projections X × J → X and X × J → J , respectively. Regarding prJ : X × J → J and (prX )∗ (E) ⊗ L as f : X → Y and F in Section 1, respectively, we can apply denitions and results to our situation. Definition.

We denote by ΘE the closed subscheme W ((prX )∗ (E) ⊗ L) of J .

We have the following rst properties of ΘE . Proposition (2.2).

Let the notations be as above .

(i) The denition of ΘE is independent of the choice of L. (ii) Let L be a line bundle of degree 0 on X , and let [L] denote the point of J corresponding to L. Then , [L] ∈ / ΘE if and only if

min(h0 (E ⊗ L), h1 (E ⊗ L)) = 0. (iii) We have the following implications :

ΘE = ∅ or codim(ΘE ) = |χ(E)| + 1 ⇐⇒ (prX )∗ (E) ⊗ L is determinantal w Ä ΘE 6= J

⇐⇒

E satises (?).

Moreover , if χ(E) = 0, the above four conditions are all equivalent . (i) It is known that the dierence of two choices of L comes from a line bundle on J . So, Zariski locally on J , the dierence is resolved. Since the denition of Fitting ideal is of local nature (see (1.1)), this shows the desired well-denedness of ΘE .

Proof.

60

AKIO TAMAGAWA

(ii) The ber (prJ )−1 ([L]) is naturally identied with X , and the restriction of (prX )∗ (E) ⊗ L to this ber is nothing but E ⊗ L. Now, (ii) is just the second half of (1.2). (iii) The rst ⇐⇒ is just the denition, if we note that χ(prX )∗ (E)⊗L dened in Section 1 coincides with χ(E). The second ⇒ is trivial, and the third ⇐⇒ follows from (ii). Finally, if χ(E) = 0, Macaulay's theorem (see Section 1) says that either ΘE = ∅ or codim(ΘE ) ≤ 1. So, in this case, the converse of the second ⇒ also holds. ¤ From now on, we assume that k is of characteristic p > 0. For an Fp -scheme S , we shall denote by FS the absolute Frobenius endomorphism S → S . We dene X1 to be the pull-back of X by FSpec(k) , and denote by FX/k the relative Frobenius morphism X → X1 over k . We put B = ((FX/k )∗ (OX ))/OX1 , which is a vector bundle on X1 with rk(B) = p − 1 and χ(B) = 0. In [R1], Raynaud proved, among other things, the following: Theorem.

([R1], Théorème 4.1.1.) The vector bundle B on X1 satises (?). ¤

As an application of this theorem, Raynaud proved, roughly speaking, that the p-ranks of the Jacobian varieties of `most' (prime-to-p-)cyclic étale coverings of X are as large as can be expected. In order to generalize such a result to ramied coverings, we need to modify the vector bundle B , so that it involves a divisor whose support is in the ramication locus. So, the aim of this section is to generalize Raynaud's theorem along these lines, and, in the next section, the application to cyclic ramied coverings will be given. Let q = pf be a power of p (f ≥ 1). We dene Xf to be the pull-back of X f by (FSpec(k) )f , and dene FX/k : X → Xf to be the composite of the f relative def

f = FXf −1 /k ◦ · · · ◦ FX1 /k ◦ FX/k . Frobenius morphisms: FX/k P Let D = P ∈X nP P be an eective divisor on X (i.e., nP ≥ 0 for all P ). We shall write ordP (D) instead of nP , which is a non-negative integer. Then, by P denition, deg(D) = P ∈X ordP (D).

Definition.

We put f f BD = ((FX/k )∗ (OX (D)))/OXf .

Lemma (2.3).

condition

f (i) BD is a vector bundle on Xf if and only if the torsion-freeness

ordP (D) < q for each P ∈ X

(TF)

holds . (ii) Assume that (TF) holds . Then we have f f f rk(BD ) = q − 1, deg(BD ) = deg(D) + (g − 1)(q − 1), χ(BD ) = deg(D).


61

More generally , for a line bundle L on Xf , we have f f rk(BD ⊗ L) = q − 1, deg(BD ⊗ L) = deg(D) + (g − 1 + deg(L))(q − 1),

and f χ(BD ⊗ L) = deg(D) + deg(L)(q − 1).

(2.4)

f (i) Since BD is a coherent sheaf on the (smooth) curve Xf , it is a vector f bundle if and only if the stalk (BD )Pf is a torsion-free OXf ,Pf -module for each f Pf ∈ Xf . By denition, we have (BD )Pf = (mX,P )− ordP (D) /OXf ,Pf , where P is the unique point of X above Pf and mX,P denotes the maximal ideal of the local f ring OX,P . So, (BD )Pf is torsion-free if and only if (mX,P )− ordP (D) ∩ k(Xf ) = OXf ,Pf , which turns out to be equivalent to ordP (D) < q . This completes the proof.

Proof.

(ii) We have f f rk(BD ) = rk((FX/k )∗ (OX (D))) − rk(OXf ) = q − 1

and f f χ(BD ) = χ((FX/k )∗ (OX (D))) − χ(OXf ) = (deg(D) + 1 − g) − (1 − g) = deg(D). f From these, deg(BD ) can be calculated by using (2.1). Moreover, for a line bundle L on Xf , we have f f rk(BD ⊗ L) = rk(BD )=q−1

and f f f deg(BD ⊗ L) = deg(BD ) + rk(BD ) deg(L) = deg(D) + (g − 1 + deg(L))(q − 1). f From these, χ(BD ⊗ L) can be calculated by using (2.1).

¤

Now, the following is one of the main results of this section.

Assume deg(D) = q − 1, and let L−1 be a line bundle of degree f −1 on Xf . Then BD ⊗ L−1 is a vector bundle on Xf with χ = 0, and satises (?). Theorem (2.5).

Before proving (2.5), we shall give a slight generalization (which will be used later), assuming (2.5): Corollary (2.6).

s(q − 1) and that

Let s be a non-negative integer . We assume that deg(D) = #{P ∈ X | ordP (D) = q − 1} ≥ s − 1.

f Let L−s be a line bundle of degree −s on Xf . Then BD ⊗ L−s is a vector bundle on Xf with χ = 0, and satises (?).

62

AKIO TAMAGAWA

Let D0 be an eective divisor on X , and Q a point of X which is not contained in the support of D0 . We put D1 = D0 + (q − 1)Q, and consider the following commutative diagram with two rows exact: Proof.

f f 0 → OXf (−Qf ) → (FX/k )∗ (OX (D1 )) ⊗ OXf (−Qf ) → BD ⊗ OXf (−Qf ) → 0 1 k T .. f (FX/k )∗ (OX (D0 − Q)) . ∨ T

0→

OXf

→

f (FX/k )∗ (OX (D0 ))

→

f BD 0

→ 0,

f where Qf denotes FX/k (Q) (∈ Xf ). From this, we can see ∼

f f BD ⊗ OXf (−Qf )→BD . 1 0

Using this isomorphism repeatedly, our assumption

#{P ∈ X | ordP (D) = q − 1} ≥ s − 1 enables us to reduce the problem to the case s ≤ 1. The case s = 1 is just (2.5). The case s = 0 can be reduced to the case s = 1 again by using this isomorphism. (Choose any Q ∈ X .) ¤ Note that (2.6) includes Raynaud's original theorem as the case that f = 1 and s = 0. f (2.5). Since deg(D) = q − 1 < q , (TF) clearly holds, hence BD f is a vector bundle by (2.3)(i). Moreover, we have χ(BD ⊗ L−1 ) = deg(D) + deg(L−1 )(q − 1) = 0 by (2.4). f ⊗ L−1 satises (?) by using (1.12). To do We would like to prove that BD this, let Jf be the Jacobian variety of Xf , and let Lf be a universal line bundle on Xf × Jf . Let prXf and prJf denote the projections Xf × Jf → Xf and Xf × Jf → Jf , respectively. The main diculty is that, unlike Raynaud's original case, we cannot apply (1.12) directly to the exact sequence on Xf ×Jf obtained by taking (prXf )∗ (−)⊗ Lf of the exact sequence

Proof of

f f 0 → L−1 → (FX/k )∗ (OX (D)) ⊗ L−1 → BD ⊗ L−1 → 0

on Xf , because W ((prXf )∗ (L−1 ) ⊗ Lf ) = ∅ and condition (b) of (1.12) is not satised (unless g = 0). (Note that h0 (L−1 ⊗ L) = 0 for all line bundle L of degree 0 on Xf .) This leads us to the following procedure. Since the validity of (2.5) is independent of the choice of the line bundle L−1 of degree −1, we may and shall assume L−1 = OXf (−Qf ), where we x any


63

f Q ∈ X and put Qf = FX/k (Q). By denition, we have the exact sequence f f 0 → OXf (−Qf ) → (FX/k )∗ (OX (D)) ⊗ OXf (−Qf ) → BD ⊗ OXf (−Qf ) → 0. k f (FX/k )∗ (OX (D − qQ)) f f f Let ED,Q be the sum of OXf and (FX/k )∗ (OX (D − qQ)) in (FX/k )∗ (OX (D)). This coincides with the amalgamated sum with respect to OXf (−Qf ), since f f OXf ∩ (FX/k )∗ (OX (D − qQ)) = OXf (−Qf ). Thus the vector bundle ED,Q ts into the following commutative diagram with two rows exact: f f 0 → OXf (−Qf ) → (FX/k )∗ (OX (D − qQ)) → BD ⊗ OXf (−Qf ) → 0 T T (2.7) k f f 0 → OXf → ED,Q → BD ⊗ OXf (−Qf ) → 0.

If ΘB f ⊗OX D

f

(−Qf )

we assume ΘB f ⊗OX D

f

= ∅, the assertion of (2.5) clearly holds. So, from now on, (−Qf )

6= ∅. Then, in order to prove that ΘB f ⊗OX D

f

(−Qf )

6=

Jf , we shall apply (1.12) to the exact sequence on Xf × Jf obtained by taking (prXf )∗ (−) ⊗ Lf of the second row of (2.7). First, we shall check condition (b) of (1.12). Note that χ(pr

Xf

f )∗ (ED,Q )⊗Lf

f f ⊗ OXf (−Qf )) = 1 − g. ) = χ(OXf ) + χ(BD = χ(ED,Q

If g = 0, condition (b) is equivalent to ΘB f ⊗OX D

f

(−Qf )

6= ∅, which we have just

assumed. If g = 1, condition (b) automatically holds. If g > 1, condition (b) is equivalent to ΘOXf 6= ∅, which follows from ΘOXf 3 0. (Note that h0 (OXf ) = 1 and h1 (OXf ) = g .) Moreover, since

χ(pr

Xf

f )∗ (BD ⊗OXf (−Qf ))⊗Lf

f = χ(BD ⊗ OXf (−Qf )) = 0,

condition (c) of (1.12) holds. Finally, we shall check that condition (a) holds, or, equivalently, that either ΘE f = ∅ or codim(ΘE f ) = |1−g|+1. Namely, we have to prove that ΘE f D,Q

D,Q

D,Q

is nite over k (resp. empty) if g > 0 (resp. g = 0). This is the most dicult part of our proof. (In Raynaud's original case, this part was immediate. See (2.11)(ii).) Lemma (2.8).

We have def

f h0max = max{h0 (ED,Q ⊗ L) | [L] ∈ Jf } = 1

and def

f h1max = max{h1 (ED,Q ⊗ L) | [L] ∈ Jf } = g.

64

AKIO TAMAGAWA f Since χ(ED,Q ⊗ L) = 1 − g , it is sucient to prove the rst equality. By (2.7), we get an exact sequence

Proof.

f f 0 → (FX/k )∗ (OX (D − qQ)) → ED,Q → k(Qf ) → 0.

So, we have f f h0 (ED,Q ⊗ L) ≤ h0 ((FX/k )∗ (OX (D − qQ)) ⊗ L) + 1 f = h0 (OX (D − qQ) ⊗ (FX/k )∗ (L)) + 1 = 1, f the last equality following from the fact that deg(OX (D − qQ) ⊗ (FX/k )∗ (L)) = f −1 < 0. Therefore, we have h0max ≤ 1. On the other hand, since ED,Q contains OXf , we have f 1 ≤ h0 (ED,Q ) ≤ h0max .

This completes the proof.

¤ def

We shall return to the proof of (2.5). Put W = ΘE f

D,Q

(⊂ Jf ) for simplicity. If

g = 0, we have W = ∅ by (1.2) and (2.8), as desired. So, we shall assume g > 0 and prove that W is nite over k . Note that, again by (1.2) and (2.8), we have f ⊗ L) = 1} W = {[L] ∈ Jf | h0 (ED,Q f = {[L] ∈ Jf | h1 (ED,Q ⊗ L) = g},

set-theoretically. We dene the divisor D0 on X to be the `prime-to-Q part' of D, namely, X def D0 = ordP (D)P, P ∈X,P 6=Q f and let d0 denote the degree of D0 . Then, by using the denition of ED,Q , we see that the following exact sequence exists: f f f 0 → (FX/k )∗ (OX (D − qQ)) → ED,Q ⊕ (FX/k )∗ (OX (D0 − Q)) f → (FX/k )∗ (OX (D0 )) → 0.

For each [L] ∈ W , we tensor this sequence with L and take the global sections. Then we will obtain f H 0 (Xf , ED,Q ⊗ L) ⊕ |o k

f H 0 (Xf , (FX/k )∗ (OX (D0 − Q)) ⊗ L) k f H 0 (X, OX (D0 − Q) ⊗ (FX/k )∗ (L))

,→

f H 0 (Xf , (FX/k )∗ (OX (D0 )) ⊗ L) k f H 0 (X, OX (D0 ) ⊗ (FX/k )∗ (L)).


65

That is to say, each [L] ∈ W denes a point of f f | OX (D0 ) ⊗ (FX/k )∗ (L) | − | OX (D0 − Q) ⊗ (FX/k )∗ (L) | .

Here, for a line bundle M on X , | M | denotes the (schematized) projective space (H 0 (X, M ) − {0})/k × . In other words, consider the following diagram: Vf

W ⊂ Jf → J

+(D 0 −Q) ∼

→

0

+Q ∼

→

J (d ) ↑

0

+Q

X (d ) S

J (d −1) ↑ X (d −1)

0

(2.9)

0

,→

0

0

X (d ) − X (d −1) where V f denotes the composite of the f Verschiebungs, i.e., V f : [L] 7→ f [(FX/k )∗ (L)], J (r) denotes the degree r part of the Picard variety of X (hence, in particular, J (0) = J ), and X (r) denotes the r-th symmetric power of X . (We put X (0) = Spec(k) and X (−1) = ∅.) In this setting, the above observation tells 0 0 0 us that there exists a natural set-theoretic map W → X (d ) − X (d −1) over J (d ) . 0 0 Now, assume that this map W → X (d ) − X (d −1) can be regarded as a 0 0 morphism (as J (d ) -schemes). Then, since W → J (d ) is a nite morphism, so is 0 0 0 0 0 W → X (d ) − X (d −1) . Now, since X (d ) − X (d −1) = (X − {Q})(d ) is ane, so is W . On the other hand, W is proper over k as a closed subscheme of Jf . Thus W must be nite over k . 0 0 So, it suces to prove that the above set-theoretic map W → X (d ) − X (d −1) is a morphism. To do this, we name the morphisms involved as follows, for the sake of simplicity:

X

ξ

←

X ×W &η

↓π

¤

↓ πW

ª

W,

(2.10)

%η0 Xf

ξ0

←

Xf × W

f where π = FX/k , πW is the base change of π , and ξ, ξ 0 , η and η 0 are projections. We shall apply the functor (from the category of OXf -modules to that of OX×W -modules)

η ∗ η∗0 (ξ 0∗ (−) ⊗ (Lf )W ) f to the natural map ED,Q ,→ π∗ (OX (D0 )). By the at base change theorem and the projection formula, we have

ξ 0∗ (π∗ (OX (D0 ))) ⊗ (Lf )W = (πW )∗ (ξ ∗ (OX (D0 ))) ⊗ (Lf )W = (πW )∗ (ξ ∗ (OX (D0 )) ⊗ (πW )∗ (Lf )W ).

66

AKIO TAMAGAWA

Thus we obtain f η ∗ η∗0 (ξ 0∗ (ED,Q ) ⊗ (Lf )W ) → η ∗ η∗ (ξ ∗ (OX (D0 )) ⊗ (πW )∗ (Lf )W ),

and taking the composite with the natural map η ∗ η∗ (−) → (−), we obtain f η ∗ η∗0 (ξ 0∗ (ED,Q ) ⊗ (Lf )W ) → ξ ∗ (OX (D0 )) ⊗ (πW )∗ (Lf )W . f By (1.2) and (2.8), we have Zg (ξ 0∗ (ED,Q ) ⊗ Lf ) = ∅. So, by (1.4) and its f 1 0 0∗ proof, R η∗ (ξ (ED,Q ) ⊗ (Lf )W ) is a locally free OW -module of rank g . (Note f f that we have R1 η∗0 (ξ 0∗ (ED,Q ) ⊗ (Lf )W ) = R1 η∗0 (ξ 0∗ (ED,Q ) ⊗ (Lf ))|W .) By this f 0∗ 0 and (1.5), we see that Rη∗ (ξ (ED,Q )⊗(Lf )W ) can be represented Zariski locally 0

g f by the complex (OW → OW ). In particular, η∗0 (ξ 0∗ (ED,Q ) ⊗ (Lf )W ) is a locally f ) ⊗ (Lf )W ) ⊗ free OW -module of rank 1, and, for each z ∈ W , η∗0 (ξ 0∗ (ED,Q ∼ f 0 k(z)→H ((Xf )k(z) , ED,Q ⊗ Lz ), where Lz is the line bundle on (Xf )k(z) corresponding to z ∈ W ⊂ Jf . (We denote the base change from k to k(z) by means of a subscript k(z).) Hence the pull-back of the OX×W -module f ) ⊗ (Lf )W ) to Xk(z) can be identied with η ∗ η∗0 (ξ 0∗ (ED,Q f H 0 ((Xf )k(z) , ED,Q ⊗ Lz ) ⊗k(z) OXk(z) .

On the other hand, the pull-back of ξ ∗ (OX (D0 )) ⊗ (πW )∗ (Lf )W to Xk(z) is OX (D0 ) ⊗ (πk(z) )∗ (Lz ), and we can check that the resulting map f H 0 ((Xf )k(z) , ED,Q ⊗ Lz ) ⊗k(z) OXk(z) → OX (D0 ) ⊗ (πk(z) )∗ (Lz )

is the composite of  0  f 0 0  H ((Xf )k(z) , ED,Q ⊗Lz ) → H ((Xf )k(z) , π∗ (OX (D ))⊗Lz )  ⊗ O k   k(z) Xk(z) H 0 (Xk(z) , OX (D0 ) ⊗ (πk(z) )∗ (Lz )) and the natural map

H 0 (Xk(z) , OX (D0 ) ⊗ (πk(z) )∗ (Lz )) ⊗k(z) OXk(z) → OX (D0 ) ⊗ (πk(z) )∗ (Lz ). Here, as in the previous argument, the map f H 0 ((Xf )k(z) , ED,Q ⊗ Lz ) → H 0 (Xk(z) , OX (D0 ) ⊗ (πk(z) )∗ (Lz )) f ⊗ Lz ) is a one-dimensional k(z)-vector is injective. Since H 0 ((Xf )k(z) , ED,Q space, we conclude that f (η ∗ η∗0 (ξ 0∗ (ED,Q ) ⊗ (Lf )W )) ⊗ k(z) → (ξ ∗ (OX (D0 )) ⊗ (πW )∗ (Lf )W ) ⊗ k(z)

is injective. Now, by [EGA4], Proposition (11.3.7), the map f η ∗ η∗0 (ξ 0∗ (ED,Q ) ⊗ (Lf )W ) → ξ ∗ (OX (D0 )) ⊗ (πW )∗ (Lf )W

is injective and its cokernel is at over W . Hence it equips ξ ∗ (OX (D0 )) ⊗ (πW )∗ (Lf )W with a structure of relative eective Cartier divisor on X × W/W .

TAME FUNDAMENTAL GROUPS OF CURVES 0

67

(d0 )

Thus we are given the morphism W → X (d ) over Jf , whose underlying map 0 0 coincides with the set-theoretic map W → X (d ) − X (d −1) in the previous argument. (See [Mi], § 3, especially Proposition 3.13 there.) This completes the proof of the niteness of W . f Now, we can apply (1.12) and conclude that (prXf )∗ (BD ⊗ OXf (−Qf )) ⊗ Lf ¡ f ⊗ is determinantal, or, equivalently, that ΘB f ⊗OX (−Qf ) = W ((prXf )∗ (BD D f ¢ OXf (−Qf )) ⊗ Lf ) 6= Jf . This nally completes the proof of (2.5). ¤ Remark (2.11).

f f (i) When D = (q − 1)Q, ED,Q coincides with (FX/k )∗ (OX ).

f In general, ED,Q is not isomorphic to the direct image of a line bundle on X . In f f fact, suppose that ED,Q is isomorphic to (FX/k )∗ (M ) for some line bundle M on X . Then we have f f χ(M ) = χ((FX/k )∗ (M )) = χ(OXf ) + χ(BD ⊗ OXf (−Qf )) = 1 − g, f hence deg(M ) = 0. On the other hand, since OXf ⊂ (FX/k )∗ (M ), M admits a f non-trivial global section. Thus M ' OX . By the denition of ED,Q , we can see that f f det((FX/k )∗ (M )) ' det((FX/k )∗ (OX (D − qQ))) ⊗ OXf (Qf ),

and since M ' OX , we have f f det((FX/k )∗ (OX (D − qQ))) ⊗ det((FX/k )∗ (OX ))−1 ⊗ OXf (Qf ) ' OXf .

Here the left-hand side is known to be isomorphic to

OXf (Df − qQf ) ⊗ OXf (Qf ) ' OXf (Df − (q − 1)Qf ), def P where Df = P ∈X ordP (D)Pf . Thus it follows that the divisor Df − (q − 1)Qf on Xf should be principal, which does not hold in general. (ii) When D = (q − 1)Q, the subscheme W of Jf is nothing but Ker(V f ), hence its degree over k is pf . In general, the author does not know much about the nite k -scheme W . For example, he does not know its degree over k . In later sections, we use a slight generalization (2.13) of (2.6). First, we shall prove the following: Lemma (2.12).

Let s be a non-negative integer .

(i) Let f be a natural number and D an eective divisor of degree s(pf − 1) on X satisfying (TF) with respect to q = pf . Let f1 be a natural number . Then f the vector bundle BD ⊗ L−s on Xf satises condition (?) of page 59 for some (or , equivalently , all ) line bundle L−s of degree −s on Xf , if and only if the f vector bundle BD ⊗ L1,−s on Xf1 +f satises condition (?) for some (or , f1 equivalently , all ) line bundle L1,−s of degree −s on Xf1 +f .

68

AKIO TAMAGAWA

(ii) For each i = 0, 1, we let fi be a natural number , Di an eective divisor of def degree s(pfi −1) satisfying (TF) with respect to q = pfi . Then D = pf1 D0 +D1 becomes an eective divisor of degree s(pf − 1) satisfying (TF) with respect to def f ⊗ L−s satises condition (?) for q = pf , where f = f0 + f1 . Moreover , BD some (or all ) line bundle L−s of degree −s on Xf , if and only if , for each fi ⊗ Li,−s satises condition (?) for some (or all ) line bundle Li,−s i = 0, 1, BD i of degree −s on Xfi . ∼

(i) Under the natural (pf1 -linear) isomorphism Xf1 →X of schemes, f corresponds to BD and the line bundles of degree −s correspond to the line bundles of degree −s. Proof.

f BD f1

(ii) First the numerical conditions can be checked as follows:

deg(D) = pf1 deg(D0 ) + deg(D1 ) = pf1 s(pf0 − 1) + s(pf1 − 1) = s(pf − 1), ordP (D) = pf1 ordP (D0 ) + ordP (D1 ) ≤ pf1 (pf0 − 1) + (pf1 − 1) = pf − 1. For simplicity, we shall denote by π , π1 and π0,f1 the relative Frobenius morf f1 f0 phisms FX/k : X → Xf , FX/k : X → Xf1 and FX : Xf1 → Xf , respectively, f1 /k so that π = π0,f1 ◦ π1 . We have natural homomorphisms

OXf → (π0,f1 )∗ (OXf1 ((D0 )f1 )) → π∗ (OX (D)) of OXf -modules, and obtain the following exact sequence: f0 f f1 0 → B(D → BD → (π0,f1 )∗ (BD ⊗ OXf1 ((D0 )f1 )) → 0. 1 0 )f 1

From this, the rst assertion follows. Moreover, tensoring this exact sequence with L−s , we obtain f0 f ⊗ L−s 0 → B(D ⊗ L−s → BD 0 )f 1

f1 → (π0,f1 )∗ (BD ⊗ OXf1 ((D0 )f1 ) ⊗ (π0,f1 )∗ (L−s )) → 0. 1

Now, since

deg(OXf1 ((D0 )f1 ) ⊗ (π0,f1 )∗ (L−s )) = s(pf0 − 1) + pf0 (−s) = −s, the second assertion follows from the associated long exact sequence (and (i)). ¤

Let s be a non-negative integer , D an eective divisor of degree s(pf − 1) on X . Assume the following condition :

Corollary (2.13).

(2.14). There exist natural numbers fi and eective divisors Di P P of degree s(pfi − 1) (i = 0, 1, . . . , k ), such that f = ki=0 fi , D = ki=0 pf>i Di , def P where f>i = kj=i+1 fj (f>k = 0), and that , for each i = 0, 1, . . . k , Condition

#{P ∈ X | ordP (Di ) = pfi − 1} ≥ s − 1. f Then , for a line bundle L−s of degree −s on Xf , BD ⊗ L−s is a vector bundle on Xf with χ = 0, and satises (?).

TAME FUNDAMENTAL GROUPS OF CURVES Proof.

Use (2.6) for each Di and apply (2.12) repeatedly.

69

¤

What can we expect for a more general eective divisor D? For the time being, we are interested in vector bundles with χ = 0. So, considering (2.4), we shall f ⊗L−s assume that deg(D) = s(q−1) for some natural number s, and consider BD for a line bundle L−s of degree −s on Xf . Moreover, we have to assume the torsion-freeness condition (TF): ordP (D) < q for each P ∈ X , which does not hold automatically this time. Under these assumptions, can we expect that f BD ⊗ L−s satises (?)? In general, the answer is no. In fact, as Raynaud remarked in [R1], § 0, condition (?) for a vector bundle E with χ(E) = 0 implies that E is semi-stable, in the sense that deg(F )/rk(F ) ≤ deg(E)/rk(E) for all vector subbundles F of f f f E . So, if BD ⊗ L−s satises (?), then BD ⊗ L−s is semi-stable, hence so is BD . Definition.

Let D be an eective divisor on X .

(i) For each natural number n, we put X def [ordP (D)/n]P, [D/n] = P ∈X

which is an eective divisor on X . (ii) For each natural number i, we put X def Di = ordP (D)Pi , P ∈X i where Pi denotes FX/k (P ) ∈ Xi . This is an eective divisor on Xi .

(iii) For n = 0, 1, . . . , pf − 1, let n =

f −1 X

nj pj be the p-adic expansion with

j=0

nj = 0, . . . , p − 1. Identifying {0, 1, . . . , f − 1} with Z/f Z naturally, we put f −1 X def ni+j pj . Now, assume that D satises (TF) with respect to q = pf . n(i) = j=0

Then, we put

def

D(i) =

X

ordP (D)(i) P,

P ∈X

which is an eective divisor on X .

Assume that deg(D) = s(q − 1) for some natural number s and f that D satises (TF) with respect to q = pf . Then , if BD is semi-stable , we have Lemma (2.15).

deg(D(i) ) ≥ deg(D) for each i = 0, 1, . . . , f − 1. (`NSS' means `necessary condition for semi-stability'.) Proof.

f The vector bundle BD on Xf admits the vector subbundles def

f −i f −i B[D/p = ((FX )∗ (OXi ([D/pi ]i )))/OXf i] i /k i

(NSS)

70

AKIO TAMAGAWA

for i = 0, 1, . . . , f − 1. (Note that (Xi )f −i = Xf .) We have f −i deg(B[D/p i] ) i f −i rk(B[D/p i] ) i

=

deg([D/pi ]i ) + (g − 1)(pf −i − 1) pf −i − 1

=

deg([D/pi ]) + g − 1, pf −i − 1

so we must have

deg(D) deg([D/pi ]) ≤ f for each i = 0, 1, . . . , f − 1, f −i p −1 p −1

(2.16)

f since BD is assumed to be semi-stable. Now, it is elementary to check that (2.16) is equivalent to (NSS). ¤

We have deg(D(i) ) ≡ pf −i deg(D) ≡ 0 (mod pf − 1). So, if deg(D) = p − 1, (NSS) automatically holds. (Of course, by (2.5) and [R1], § 0, f we know that BD is then semi-stable.) Remark (2.17). f

Now, we are tempted to ask the following: Question (2.18). Let s be a natural number . Let D be an eective divisor of degree s(q − 1) on X satisfying (TF) and (NSS), and let L−s be a line bundle of f degree −s on Xf . Then , BD ⊗ L−s is a vector bundle on Xf with χ = 0. Does it satisfy (?)?

However, in general this fails, as the following example shows. Example (2.19).

We assume p 6= 2 and let X = P1 . We put f = 1 and let

p−1 {(0) + (1) + (λ) + (∞)}, where λ ∈ k − {0, 1}, so that s = 2. Then 2 1 BD ⊗ L−2 satises (?)(if and) only if the elliptic curve y 2 = x(x − 1)(x − λ) is ordinary. (We omit the proof, which uses some contents of the next section.)

D=

Considering Bouw's work ([B]), we might hope that the following is armative. Question (2.20).

Is (2.18) true for U generic (in the moduli space )?

Finally, the following proposition shows to what extent our results can be applied, in the case where #(Supp(D)) is small. This analysis is a key to recover àdditive structures' of inertia subgroups of tame fundamental groups in Section 5 (B). Here, for each natural number N , we denote by IN the set {0, 1, . . . , N −1}.

Let s be a non-negative integer and D an eective divisor of degree s(pf − 1) satisfying (TF) with respect to q = pf . We assume that D can be written as D = n1 P1 + n2 P2 + n3 P3 , where P1 , P2 , P3 are three distinct points of X . Then :

Proposition (2.21).

(i) nh ∈ Ipf holds for each h = 1, 2, 3, and 0 ≤ s ≤ 3 holds . (ii) If s 6= 2, D satises (2.14).


71

(iii) Assume s = 2. Then , (NSS) is equivalent to

n1,j + n2,j + n3,j = 2(p − 1) for each j = 0, 1, . . . , f − 1, P −1 where nh = fj=0 nh,j pj is the p-adic expansion with nh,j ∈ Ip (h = 1, 2, 3). (iv) Assume s = 2 and (NSS). def

(iva) If n1 ∈ pIf = {pb | b ∈ If }, then either n2 = pf − 1 or n3 = pf − 1 holds , and D satises (2.14). def

(ivb) If n1 ∈ Ip−1 pIf = {apb | a ∈ Ip−1 , b ∈ If }, then D satises (2.14) if and only if either n2 = pf − 1 or n3 = pf − 1. (ivc) For each

½ If

n1 ∈ /p

If

∪ Ip−1 p

=

Ip−1 pIf , Ip pIf ,

if p 6= 2, if p = 2,

there exist n2 , n3 ∈ Ipf −1 such that D = n1 P1 +n2 P2 +n3 P3 satises (2.14). (i) The rst assertion just says that D is eective and satises (TF). The second assertion follows from the rst, since s(q−1) = deg(D) = n1 +n2 +n3 .

Proof.

(ii) If s ≤ 1, (2.14) requires nothing. If s = 3, then we must have n1 = n2 = n3 = q − 1, which implies (2.14). (Take k = 0, f = f0 , and D = D0 .) (iii) (NSS) is equivalent to saying that (j)

(j)

(j)

n1 + n2 + n3 ≥ n1 + n2 + n3 = 2(pf − 1) holds for j = 0, 1, . . . , f −1. Here, by denition, the left-hand side is congruent to the right-hand side modulo pf − 1, hence it is a multiple of pf − 1. On the other hand, it is less than or equal to 3(pf − 1). Moreover, if it is equal to 3(pf − 1), (j) (j) (j) each of n1 , n2 , n3 must be pf − 1, which implies that each of n1 , n2 , n3 is pf − 1. This contradicts the assumption n1 + n2 + n3 = 2(pf − 1). Thus (NSS) turns out to be equivalent to saying that (j)

(j)

(j)

n1 + n2 + n3 (= n1 + n2 + n3 ) = 2(pf − 1) def

holds for j = 0, 1, . . . , f − 1. Now, put ν = n1 + n2 + n3 = 2(pf − 1) and def νj = n1,j + n2,j + n3,j . Since we have (j+1)

nh

(j)

=

nh − nh,j 1 (j) pf − 1 + nh,j pf −1 = nh + nh,j , p p p

we see that

ν=

1 pf − 1 ν+ νj , i.e., νj = 2(p − 1) p p

is a necessary condition for (NSS). It is clear that this condition is also sucient for (NSS).

72

AKIO TAMAGAWA

(iva) If n1 = pb (b ∈ If ), we must have ½ 2(p − 1), if j 6= b, n2,j + n3,j = 2(p − 1) − 1, if j = b, by (iii), or, equivalently,

½ (n2,j , n3,j ) =

(p − 1, p − 1), (p − 1, p − 2) or (p − 2, p − 1),

if j 6= b, if j = b.

From this (iva) follows. (ivb) If n1 = apb (a ∈ Ip−1 , b ∈ If ), we have n1,j = 0 (resp. a) for j 6= b (resp. j = b). Accordingly, by (iii), we must have n2,j = n3,j = p − 1 for j 6= b and n2,b + n3,b = 2(p − 1) − a. First, if either n2 or n3 coincides with pf − 1, then it is clear that D satises (2.14) for k = 0. Conversely, suppose that there exist k , fi and Di as in (2.14). Since deg(Di ) = 2(pfi − 1) and ordP (Di ) = pfi − 1 for some P = P1 , P2 , P3 , we have ordP (Di ) ≤ pfi − 1 for all P = P1 , P2 , P3 . From this, (considering the p-adic expansion of nh = ordPh (D)) we conclude Pfi −1 nh,f>i +j pj for each h = 1, 2, 3. Thus, that ordPh (Di ) should coincide with j=0 for some h = 1, 2, 3 (depending on i), we must have nh,f>i +j = p − 1 for j = 0, . . . , fi − 1. Now, taking the unique i such that f>i ≤ b < f>i−1 , we see that nh,b = p − 1 holds for some h. Since n1,b = a < p − 1, we must have either n2,b = p−1 or n3,b = p−1, which implies n2 = pf −1 or n3 = pf −1, respectively. This completes the proof of (ivb). (ivc) Assume n1 ∈ / pIf ∪ Ip−1 pIf . In particular, n1 6= 0, hence there exists b = 0, 1, . . . , f − 1 with n1,b > 0. Now, we put   (p − 1, p − 1 − n1,j ), if j 6= b, def (n2,j , n3,j ) = (p − 1 − n1,b , p − 1), if j = b and n1,b < p − 1,  (p − 2, 1), if j = b and n1,b = p − 1. (Note that (NSS) holds by (iii).) Since p − 1 ∈ {n1,j , n2,j , n3,j } for each j = 0, 1, . . . , f − 1, we see that D satises (2.14). Finally, since n2,b < p − 1 by denition, we have n2 < pf − 1. On the other hand, suppose n3 = pf − 1. Then we must have p − 1 − n1,j = p − 1 for j 6= b, and 1 = p − 1 if n1,b = p − 1. Namely, we have n1 = n1,b pb and either p = 2 or n1,b < p − 1. This contradicts the assumption n1 ∈ / pIf ∪ Ip−1 pIf . This completes the proof of (ivc). ¤

3. The p-Ranks of p0 -Cyclic Ramied Coverings As in Section 2, let k be an algebraically closed eld of characteristic p > 0 and X a proper, smooth, connected curve of genus g over k . Let S be a nite (possibly empty) set of closed points of X and denote by n the cardinality of S . We put U = X − S . In this section, we investigate the p-ranks of the Jacobian varieties of p0 -cyclic coverings of X , étale over U and possibly ramied over S .


73

Cyclic coverings and generalized HasseWitt invariants. Let N be a natural number prime to p. We consider the elements of the étale cohomology 1 group Hét (U, µN ), where µN = µN (k) is the group of N -th roots of unity. In terms of fundamental groups, 1 Hét (U, µN ) = Hom(π1 (U ), µN ) = Hom(π1t (U ), µN ), 1 and, in terms of torsors, Hét (U, µN ) can be identied with the set of isomorphism classes of (étale) µN -torsors of U . We shall consider the p-ranks for such µN torsors, or µN -coverings. 1 Let V be a µN -torsor of U and [V ] the corresponding element of Hét (U, µN ). Let Y be the normalization of X in V , to which the µN -action on V extends uniquely. We dene the p-rank (or the HasseWitt invariant) γ[V ] to be the 1 (Y, Fp ). dimension of the Fp -vector space Hét To obtain ner invariants, we consider the following canonical decomposition of the group algebra k[µN ]: Y ∼ k[µN ] → k,

∪ µN ∪p ζ

i∈Z/N Z

∪p 7→

(3.1)

(ζ i )i∈Z/N Z .

Corresponding to this decomposition, each k[µN ]-module M admits a canonL i ical decomposition M = i∈Z/N Z Mi , where ζ ∈ µN acts on Mi as the ζ multiplication. We shall denote by γi (M ) the dimension of the k -vector space Mi . Moreover, for an Fp [µN ]-module M , we shall write γi (M ) instead of γi (M ⊗Fp k). In the latter case, γpa i (M ) = γi (M ) holds for each integer a. (Observe that the p-th power map of k maps (M ⊗ k)i isomorphically onto (M ⊗ k)pi .) 1 Now, since µN naturally acts on the Fp -vector space Hét (Y, Fp ), we can dene as follows: Definition.

def

1 γ[V ],i = γi (Hét (Y, Fp )).

These invariants essentially coincide with the so-called generalized HasseWitt invariants (see [Ka], [Na], and [B]). Of course, we have X γ[V ],i . γ[V ] = i∈Z/N Z

We shall present another description of these invariants, which is also wellknown. Let ψ denote the structure morphism Y → X . Corresponding to the decomposition (3.1), we obtain a decomposition of the sheaf ψ∗ (OY ) on X : M ψ∗ (OY ) = Li . (3.2) i∈Z/N Z

74

AKIO TAMAGAWA

Let f be the order of p mod N in the multiplicative group (Z/N Z)× . The p-th power map of OY sends Li into Lpi , hence the pf -th power map of OY sends Li into itself, which induces a pf -linear map

ϕ[V ],i : H 1 (X, Li ) → H 1 (X, Li ) 0 on the Zariski cohomology group H 1 (X, Li ). We denote by γ[V ],i the dimension T r of the k -vector space r≥1 Im((ϕ[V ],i ) ). Then, ArtinSchreier theory, together 0 with the well-known properties of pf -linear maps, implies γ[V ],i = γ[V ],i

Cyclic coverings and line bundles. Next, in order to apply the results of

Section 2, we shall give a description of µN -torsors of U in terms of line bundles and divisors on X , which is essentially widely known (possibly in slightly dierent forms). We denote by Pic(X) the Picard group of X and by Z[S] the group of divisors whose supports are contained in S , which can be identied with the free Z-module with basis S . We denote by Z/N Z[S] the free Z/N Z-module with basis S , hence Z/N Z[S] = Z[S]/N Z[S]. Let (Z/N Z)∼ denote the set {0, 1, . . . , N − 1}, and (Z/N Z)∼ [S] the subset of Z[S] consisting of the elements whose `coecients' are contained in (Z/N Z)∼ . Consider the following (short) complex of abelian groups: βN

α

(3.3)

N Z[S] → Pic(X) ⊕ Z[S] → Pic(X),

where αN (D) = ([OX (−D)], N D) and βN (([L], D)) = [L⊗N ⊗ OX (D)]. We dene the abelian group PN = PN (X, S) to be the homology group Ker(βN )/ Im(αN ) of the complex (3.3). Definition.

We can easily see that the following exact sequence exists: a

b

c

(3.4)

N N N 0 → Pic(X)[N ] → PN → Z/N Z[S] → Z/N Z,

where [N ] means the N -torsion subgroup, and

aN ([L]) = ([L], 0) mod Im(αN ), bN (([L], D) mod Im(αN )) = D mod N, cN (D mod N ) = deg(D) mod N. (2)

From this, PN turns out to be isomorphic to (Z/N Z)⊕2g+n−1+b , where ½ 1, if n = 0, def b(2) = 0, if n > 0, is the second Betti number of U . We shall dene two maps 1 1 iN : PN → Hét (U, µN ), jN : Hét (U, µN ) → PN .


75

To do this, we need some more notations. First, we denote by Z/N Z[S]0 the kernel of cN in (3.4) and by (Z/N Z)∼ [S]0 the subset of (Z/N Z)∼ [S] corresponding ∼ to Z/N Z[S]0 under the natural bijection (Z/N Z)∼ [S]→Z/N Z[S]. We dene PÑ ∼ 0 to be the inverse image of (Z/N Z) [S] under the projection Ker(βN ) → Z[S] (see (3.3)). Then we can easily see that the modulo-Im(αN ) map PÑ → PN is a bijection. We denote by ˜bN the projection PÑ → (Z/N Z)∼ [S]0 . Now, rst, take ([L], D) in PÑ . We have L⊗N ⊗OX (D) ' OX , or, equivalently, ⊗N L ' OX (−D). These isomorphisms are unique up to multiplication by an ∼ element of k × . We x such an isomorphism L⊗N →OX (−D), which induces an ∼ isomorphism (L|U )⊗N →OX (−D)|U = OU . Then, by using this isomorphism, L we can equip the locally free OU -module i∈(Z/N Z)∼ (L|U )⊗i with a structure of étale OU -algebra, as usual. This OU -algebra admits a µN -action: ζ ∈ µN acts on (L|U )⊗i as the ζ i -multiplication. The nite U -scheme corresponding to this OU -algebra, together with this µN -action, denes an étale µN -torsor of U . It is easy to check (by using the surjectivity of the N -th power map k × → k × ) that the isomorphism class of the µN -torsor we have just constructed is independent ∼ of the choice of the isomorphism L⊗N →OX (−D). This gives the denition of 1 a map ˜ıN : PÑ → Hét (U, µN ). Composing this with the canonical bijection ∼ ˜ 1 PN ←PN , we obtain iN : PN → Hét (U, µN ). Next, take a µN -torsor V /U , and let ψ : Y → X be the normalization of V → U , as above. Then, as we have seen, the locally free OX -module ψ∗ (OY ) L can be canonically decomposed as a direct sum i∈Z/N Z Li . Using the fact that V is a µN -torsor of U , we can see that each Li is a line bundle on X and that L0 = OX . Since µN acts on ψ∗ (OY ) as an OX -algebra, the multiplication of Li and Li0 is contained in Li+i0 . In particular, we are given an OX -linear map L⊗N → L0 = OX . Since V is a µN -torsor of U , the restriction (L1 |U )⊗N → 1 OU is an isomorphism. Therefore the map L⊗N → OX is injective, and it 1 ∼ factors as L⊗N →O (−D) ⊂ O for some (uniquely determined) eective divisor X X 1 D ∈ Z[S]. We claim that the eective divisor D belongs to (Z/N Z)∼ [S]. This comes from the fact that Y is the normalization of X . In fact, the N -th power of a (local) section of L1 ([D/N ]) belongs to OX (N [D/N ] − D) ⊂ OX , hence should belong to ψ∗ (OY ). (See Section 2 for the denition of the divisor [D/N ].) Considering the µN -action, we have L1 ([D/N ]) ⊂ L1 , which implies [D/N ] = 0, or, equivalently, D ∈ (Z/N Z)∼ [S]. Now, since ([L1 ], D) falls in the kernel of βN by denition, ([L1 ], D) is an element of PÑ . This gives the denition of a map ∼ 1 Ñ : Hét (U, µN ) → PÑ . Composing this with the canonical bijection PÑ →PN , 1 we obtain jN : Hét (U, µN ) → PN . Proposition (3.5). The canonical maps iN , jN are group isomorphisms , which are inverse to each other .

By the denition of multiplication of two torsors, we see that iN is a group homomorphism. (See, for example, [Mi1], III, Remark 4.8(b).) Proof.

76

AKIO TAMAGAWA

In the above construction concerning jN , the canonical map L⊗i 1 → Li (i ∈ ∼ (Z/N Z)∼ ) becomes an isomorphism after restricting to U : (L1 |U )⊗i →Li |U , since V is a µN -torsor of U . Using this fact, we can check that iN ◦ jN = id. Finally, for ([L], D) ∈ PÑ , let V be the corresponding µN -torsor of U , and ψ : Y → X its normalization. Then, since the N -th power of each (local) section of L belongs to OX (−D) ⊂ OX , ψ∗ (OY ) should contain L, by the denition of normalization. Moreover, observing the µN -action, we can conclude that L should be contained in L1 ⊂ ψ∗ (OY ). Now, the N -th power map induces a commutative diagram ∼ L⊗N → OX (−D) ∩ ∩ L⊗N → OX . 1 Since D ∈ (Z/N Z)∼ [S], this implies that L = L1 , hence that Ñ ◦ ˜ıN = id, or, equivalently jN ◦ iN = id. From this jN is also a group isomorphism. (To prove 1 jN ◦ iN = id, we may also resort to the fact #(PN ) = #(Hét (U, µN )), which 2g+n−1+b(2) equals N . ¤

Generalized HasseWitt invariants via line bundles. Now, we can de-

scribe the generalized HasseWitt invariants in terms of PN , as follows. For ∼ an element ([L], D) of PÑ , x an isomorphism L⊗N →OX (−D) (unique up to f k × -multiplication). Taking the composite of the pf -th power map L → L⊗p and ³ pf − 1 ´ f f ∼ L⊗p = L ⊗ L⊗(p −1) →L ⊗ OX − D ,→ L, N we get a map L → L, which induces a pf -linear map

ϕ([L],D) : H 1 (X, L) → H 1 (X, L). We denote by γ([L],D) the dimension of the k -vector space Then, by the various denitions, we see

T r≥1

Im((ϕ([L],D) )r ).

γ([L],D) = γ[V ],1 , where [V ] = ˜ıN (([L], D)). Remark (3.7).

By the RiemannRoch theorem, we have

dimk (H 1 (X, L)) = g − 1 − deg(L) + dimk (H 0 (X, L)) 1 =g−1+ deg(D) + dimk (H 0 (X, L)) N · ¸ n(N − 1) ≤g−1+ + dimk (H 0 (X, L)) N h ni =g+n−1+ − + dimk (H 0 (X, L)). N

(3.6)


77

From this, we obtain the following rough estimate: ½ g, if ([L], D) = ([OX ], 0), γ([L],D) ≤ (2) g + n − 2 + b , otherwise. More generally, we can describe γ[V ],i in terms of line bundles and divisors. First, we shall determine the Z-action on PÑ induced by the natural Z-action on the abelian group PN . In PN , i times ([L], D) is ([L⊗i ], iD) (mod Im(αN )) for each i ∈ Z. Since the element of (Z/N Z)∼ [S] that is equivalent to iD modulo N is iD − N [iD/N ], we can see that the i-action on PÑ is given by

([L], D) 7→ (L⊗i ([iD/N ]), iD − N [iD/N ]). We shall denote L⊗i ([iD/N ]) and iD − N [iD/N ] by L(i) and D(i), respectively. Now, let V be the µN -torsor of U corresponding to ([L], D). Then we have the following generalization of (3.6): Claim (3.8).

γ([L(i)],D(i)) = γ[V ],i .

In fact, consider the decomposition (3.2). By the denition of iN , Y is the normalization of X in the nite X -scheme corresponding to the OX -algebra L ⊗i ⊗i ,→ Li , for each i∈(Z/N Z)∼ L , hence we have the canonical injection L i ∈ (Z/N Z)∼ . We have more: L(i) = L⊗i ([iD/N ]) ,→ Li , since the N -th power of a (local) section of L⊗i ([iD/N ]) is contained in OX (N [iD/N ] − iD) = OX (−D(i)) ⊂ OX . (Note that Y is normal.) In fact, we have L(i) = Li . Otherwise, Li would be strictly bigger than L(i), hence we could nd a (local) section of Li whose N -th power would not belong to OX . This is absurd. Thus, we can identify ϕ([L(i)],D(i)) with ϕ[V ],i , which implies our claim.

Digression: Torsion points on divisors of abelian varieties. As in [R1],

we need some inputs from intersection theory to deduce our main (numerical) results concerning p-ranks of cyclic coverings from the results of Section 2. Lemma (3.9). Let A be an abelian variety of dimension d > 0 over an algebraically closed eld k , and D an eective divisor on A. For each natural number N not divisible by char(k), we put

cfin (D, N ) = min{(D . C) |C : irreducible , reduced curve in A,

such that C 3 0 and D ∩ NA−1 (C) is nite .} and cirr (D, N ) = min{(D . C) |C : irreducible , reduced curve in A,

such that C 3 0 and NA−1 (C) is irreducible .} Then : (i) For each irreducible , smooth curve C in A such that π1 (C) surjects onto π1 (A), we have cirr (D, N ) ≤ (D . C).

78

AKIO TAMAGAWA

(ii) For each very ample divisor H on A, we have cfin (D, N ) ≤ (D . H d−1 ), where H r denotes the r-th self-intersection product H . . . H}. | . .{z r times

(iii) We have #(Supp(D) ∩ A[N ]) ≤ cfin (D, N )N 2d−2 . (iv) If cfin (D, N ) < N 2 , then we have cfin (D, N ) ≤ cirr (D, N ). Proof.

(See [R1], Lemme 4.3.5 and the proof of [R1], Théorème 4.3.1.)

(i) The condition π1 (C) ³ π1 (A) and the number (D . C) do not change if C is translated by an element of A. So, we may assume that C passes through 0. Now, since π1 (C) ³ π1 (A), NA−1 (C) must be irreducible. Thus the inequality holds. (ii) Since H is very ample, a general member C1 of H d−1 that passes through 0 has nite intersection with NA (D), hence D ∩ NA−1 (C1 ) is also nite. Let C be an irreducible component of C1 that passes through 0. Then, regarding C as a reduced scheme, we obtain

cfin (D, N ) ≤ (D . C) ≤ (D . C1 ) = (D . H d−1 ). (iii) Take an irreducible, reduced curve C in A with (D . C) = cfin (D, N ) such that C 3 0 and that NA−1 (C) ∩ D is nite. Then we have

#(Supp(D) ∩ A[N ]) ≤ (D . NA−1 (C)) = N 2d−2 (D . C) = cfin (D, N )N 2d−2 . Here, The inequality follows from the fact that C 3 0 and that NA−1 (C) ∩ D is nite, and the rst equality follows from intersection theory as in the proof of [R1], Lemme 4.3.5. (iv) Take an irreducible, reduced curve C in A with (D . C) = cirr (D, N ) such that C 3 0 and that NA−1 (C) is irreducible. By (iii) and the assumption cfin (D, N ) < N 2 , we have #(Supp(D) ∩ A[N ]) < N 2d , hence A[N ] 6⊂ D, and, a fortiori, NA−1 (C) 6⊂ D. Since NA−1 (C) is irreducible by assumption, this implies that D ∩ NA−1 (C) is nite. Thus we have cfin (D, N ) ≤ (D . C) = cirr (D, N ). ¤

Let X be a proper , smooth , connected curve of genus g over k and J the Jacobian variety of X . Let E be a vector bundle on X with χ(E) = 0, and assume that E satises (?) of page 59. Then , ΘE is a divisor on J , and : Corollary (3.10).

(i) If g > 0, we have cirr (ΘE , N ) ≤ g rk(E). (ii) If g > 0, we have cfin (ΘE , N ) ≤ 3g−1 g! rk(E). (iii) We have #(Supp(ΘE ) ∩ J[N ]) ≤ 3g−1 g! rk(E)N 2g−2 . (iv) If 3g−1 g! rk(E) < N 2 , we have #(Supp(ΘE ) ∩ J[N ]) ≤ g rk(E)N 2g−2 . First, we note that, by [R1], Proposition 1.8.1 (2), ΘE is algebraically equivalent to rk(E)Θ, where Θ is the classical theta divisor (the image of X (g−1) in J ). Proof.


79

(i) This is obtained by applying (3.9)(i) to C = X , which is embedded into A = J by means of an Albanese morphism. In fact, then π1 (X) ³ π1 (J) is well-known, and we have (ΘE . X) = rk(E)(Θ . X) = rk(E)g. (ii) We rst note that 3Θ is very ample by [Mu], Section 17, Theorem, since Θ is ample. Now, (ii) is obtained by applying (3.9)(ii) to H = 3Θ. In fact, then we have (ΘE . (3Θ)g−1 ) = rk(E)3g−1 (Θg ) = rk(E)3g−1 g!. (iii) We may assume g > 0, since ΘE = ∅ for g = 0. Then, (iii) follows from (ii) and (3.9)(iii). (iv) We may assume g > 0, as in (iii). Then, (iv) follows from (i), (ii) and (3.9)(iii)(iv). ¤ In [R1], Lemme 4.3.5, it was necessary to assume that D ∩ −1 lA (C) is nite. (Counterexample: C : a one-dimensional abelian subvariety of A, D: the inverse image in A of a divisor of A/C that contains the whole (A/C)[l].) Accordingly, in Théorème 4.3.1, loc. cit., the condition l + 1 ≥ (p − 1)g had to be modied. (For example, l + 1 ≥ (p − 1)3g−1 g! is sucient.) Similarly, 2g −l2g−1 in [T1], Lemma (1.9), the condition lm > l l2g −1 (p − 1)g had to be modied 2g 2g−1 l −l m g−1 as l > l2g −1 (p − 1)3 g!, and, in its proof, we should have assumed that m −1 D ∩ (lA ) (C) is nite. Remark (3.11).

Main numerical consequences. Until the end of this section, with the ex-

ception of (3.17), we restrict ourselves to the case that N = q − 1, where q is a (positive) power of p. (Note that, in this case, we have pf = q .) Then, we get some numerical consequences of the results of Section 2, as follows. We note that, for each element D of (Z/N Z)∼ [S]0 , deg(D) = s(D)N for some integer s(D) with 0 ≤ s(D) ≤ n − 1 + b(2) , and that the cardinality of ˜b−1 N (D) is N 2g . Theorem (3.12).

Put

½ def

C(g) =

0, 3g−1 g!,

if g = 0, if g > 0.

(3.13)

Then , for each D ∈ (Z/N Z)∼ [S]0 with s(D) ≤ 1, the following statements hold . (i) We have

#{[L] ∈ Pic(X) | ([L], D) ∈ PÑ and ϕ([L],D) is bijective} ≥ N 2g − C(g)N 2g−1 . (ii) We have

#{[L] ∈ Pic(X) | ([L], D) ∈ PÑ and γ([L],D) ≥ g − 1 + s(D)} ≥ N 2g − C(g)N 2g−1

80

AKIO TAMAGAWA

and #{[L] ∈ Pic(X) | ([L], D) ∈ PÑ and γ([L],D) = g − 1 + s(D)} ½ 2g N − C(g)N 2g−1 − 1, ≥ N 2g − C(g)N 2g−1 ,

if s(D) = 0, if s(D) = 1.

For simplicity, we shall write s instead of s(D). Since the degree of L ∈ ˜b−1 N (D) is −s, we see that ½ g, if s = 0 and L ' OX , (3.14) γ([L],D) ≤ dimk (H 1 (X, L)) = g − 1 + s, otherwise, Proof.

as in (3.7). In particular, the statements clearly hold for g = 0. From now on, we shall assume g > 0. (i) First, recall the following commutative diagram (see Section 2):

X

=

X

f FX/k ↓

ª

f ↓ FX

→

∼

X

¤

↓

Xf ↓ Spec(k)

(FSpec(k) )f ∼

→

Spec(k). ∼

We shall denote by ι the q -linear isomorphism Xf →X in this diagram. Note that f the pullback by FX of a line bundle L on X is canonically isomorphic to L⊗q . f ∗ In fact, we can easily check that the OX -linear map (FX ) (L) → L⊗q induced ⊗q by the q -th power map L → L is an isomorphism. Take any [L0−s ] ∈ ˜b−1 N (D). Then we have

(L0−s )⊗N ' OX (−D) (hence deg(L0−s ) = −s) and

˜b−1 (D) = {[L0 ⊗ L0 ] | [L0 ] ∈ Pic(X)[N ]}. −s N def ∗

We put L−s = ι

(L0−s ).

(3.15)

Then we have deg(L−s ) = −s and

f (FX/k )∗ (L−s ) = (L0−s )⊗q ' OX (−D) ⊗ L0−s . def

f Now, by (2.6), the vector bundle E = BD ⊗ L−s on Xf satises condition (?) of page 59. So, applying (3.10), we get

#{[L] ∈ Pic(Xf )[N ] | h0 (E ⊗ L) = h1 (E ⊗ L) = 0} ≥ N 2g − C(g)N 2g−1 . f By the denition of BD , the condition

h0 (E ⊗ L) = h1 (E ⊗ L) = 0


81

implies ∼

f H 1 (Xf , (FX/k )∗ (OX (D)) ⊗ L−s ⊗ L) k f 1 H (X, OX (D) ⊗ (FX/k )∗ (L−s ⊗ L)).

H 1 (Xf , L−s ⊗ L) →

In terms of L0 = ι∗ (L) (hence L = ι∗ (L0 )), this is equivalent to: ∼

H 1 (X, L0−s ⊗ L0 ) → H 1 (X, OX (D) ⊗ (L0−s ⊗ L0 )⊗q ). Considering (3.15), these imply the inequality in (i). (ii) Immediate from (i) and (3.14). (The term −1 in the case s = 0 comes from the trivial line bundle.) ¤ We have the following slight generalization of (3.12). See Section 2 for the denition of the divisor D(i) .

Let N = pf − 1 and D ∈ (Z/N Z)∼ [S]0 . Assume that there exists i ∈ {0, 1, . . . , f − 1} such that s(D(i) ) = 1. Then we have Corollary (3.16).

#{[L] ∈ Pic(X) | ([L], D) ∈ PÑ and γ([L],D) = g} ≥ N 2g − C(g)N 2g−1 . Proof.

Notations being as in (3.8), we can deduce

γ([L(pi )],D(pi )) = γ[V ],pi = γ[V ],1 = γ([L],D) , where the rst and the third equalities follows from (3.8) and the second from the remark just before the denition of γ[V ],i . Since N = pf − 1, we have the coincidence D(i) = D(pi ). Thus, we obtain (3.16) by applying (3.12) to D(i) = D(pi ). (Note that the pi -action on PÑ is bijective.) ¤ Remark (3.17).

In this remark, we do not assume N = pf − 1.

(i) For general N , the same argument as in the proof of (3.12) shows that (3.12) holds if we replace N 2g −C(g)N 2g−1 by N 2g −C(g)(pf −1)N 2g−2 . (Apply (2.6) to pf −1 (i) by D(pi ), and N 2g −C(g)N 2g−1 N D .) Similarly, (3.16) holds if we replace D 2g f 2g−2 by N − C(g)(p − 1)N . However, the resulting inequalities say nothing, unless N 2 > C(g)(pf − 1). For g p > 0, the last condition forces N to be a rather big divisor of pf − 1 (N > pf − 1). (ii) Following [R1], we can improve the inequalities for s(D) = 0 in (3.12) (for N general). This can be achieved by considering the p-linear maps L1 → Lp , Lp → Lp2 , . . . , Lpf −1 → Lpf = L1 step by step, instead of considering the whole pf linear map L1 → L1 at a time. Then, the right-hand sides of both the inequality of (3.12)(i) and the rst inequality of (3.12)(ii) become N 2g −C(g)(p−1)f N 2g−2 . This time, the results say something nontrivial, if N 2 > C(g)(p − 1)f . The last condition is satised for N > C(g)(p − 1). In particular, they say something nontrivial for almost all N .

82

AKIO TAMAGAWA

(iii) What is the counterpart of (ii) above in the case s(D) = 1 (or s(D(pi )) = 1)? To state it, put

{a1 , . . . , am } = {a = 0, 1, . . . , f − 1 | s(D(pa )) = 1}, where we assume (0 ≤)a1 < a2 < · · · < am (≤ f − 1). We dene the natural numbers fi by ½ ai+1 − ai if 1 ≤ i < m, def fi = a1 + f − am if i = m, Pm so that i=1 fi = f . Now, by considering the pfi -linear maps Lpai → Lpai+1 step by step as in (ii), we obtain the improvement (for N general)

µ

≥N

2g

¶

m X − C(g) (pfi − 1) N 2g−2 i=1

in the statements of (3.12)(i)(ii) and (3.16). (For (3.12), we assume a1 = 0.) The inequalities say something nontrivial, if N 2 is greater than ¡Pimproved ¢ m fi C(g) i=1 (p − 1) . However, the right-hand side of the last condition depends not only on p, g and f but also on the coecients of D ∈ Z[S]. Assuming N = pf − 1 again, we shall give a rough estimate of the number of D to which (3.12) or (3.16) can be applied. (See Appendix for a related result for general N .) Note that ½ 1 if n ≤ 1, ∼ 0 n−1+b(2) #((Z/N Z) [S] ) = N = N n−1 if n > 1. Proposition (3.18).

(i) If n ≤ 1, the value s for the unique element of

(Z/N Z) [S] is 0. (ii) If n > 1, there exist M > 0 and 0 ≤ α < 1 depending only on p and n, such that © ª # D ∈ (Z/N Z)∼ [S]0 | s(D(i) ) = 1 for some i = 0, 1, . . . f − 1 ∼

0

≥ N n−1 (1 − αf ) − 1

for all f ≥ M . More precisely , let k be any positive integer ≥ logp (n−1) and ε any positive real number < 1. Then we can take µ µ k ¶¶(1−ε)/k k 1 p . M = , α = 1 − k(n−1) ε n−1 p Proof.

(i) Clear. ((Z/N Z)∼ [S]0 consists of the trivial divisor.) def

(ii) We choose any Q ∈ S and put S 0 = S − {Q}, whose cardinality is n0 = n − 1. The projection (Z/N Z)∼ [S]0 → (Z/N Z)∼ [S 0 ], D 7→ D0 is bijective. We have » ¼ 1 1 s(D) = deg(D) = deg(D0 ) , N N


83

where dxe denotes the smallest integer ≥ x. Therefore, we have

s(D) ≤ 1 ⇐⇒ deg(D0 ) ≤ N. When n0 = 1, deg(D0 ) ≤ N for all D0 . Then the statements clearly hold (α = 0). (Note that the term −1 comes from the trivial divisor.) So, from now on, we shall assume n0 > 1. Let k be a positive integer ≥ logp (n0 ) and ε any positive real number < 1. def We assume f ≥ M = kε . Let D0 be any element of (Z/N Z)∼ [S 0 ], and, for each P ∈ S 0 , consider the p-adic expansion 0

ordP (D ) =

f −1 X

nP,j pj ,

j=0

with nP,j ∈ {0, 1, . . . , p − 1}. If we have X X k−1

nP,f −k+j pj ≤ pk − n0 ,

P ∈S 0 j=0

then we obtain

 X

deg(D0 ) =



f −k−1 X

P ∈S 0 0



j=0

f −k

≤ n (p

f −1 X

nP,j pj  +

nP,j pj

j=f −k f −k

− 1) + p

k

(p − n0 )

= pf − n0 ≤ pf − 1 = N. In the same way, if we have

X k−1 X

nP,f −hk+j pj ≤ pk − n0

P ∈S 0 j=0

for some h = 1, 2, . . . ,

h i f k , then we obtain deg((D0 )(−(h−1)k) ) ≤ N.

In other words, if we suppose that deg((D0 )(i) ) > N for all i = 0, 1, . . . , f − 1, we must have X k−1 X nP,f −hk+j pj > pk − n0 (3.19) P ∈S 0 j=0

£ ¤ for all h = 1, . . . , fk . Now, since #{E ∈ (Z/pk Z)∼ [S 0 ] | deg(E) ≤ pk − n0 } =

¶ µ k¶ µ k p (p − n0 ) + n0 = n0 n0

84

AKIO TAMAGAWA

(repeated combination), we obtain

#{D ∈ (Z/N Z)∼ [S]0 | s(D(i) ) > 1 for all i = 0, 1, . . . , f − 1} µ µ k ¶¶[ fk ] f 0 0 0 p kn0 ≤ p − p(f −k[ k ])n − (pf n − (pf − 1)n ). 0 n 0

0

Here, the term (pf n − (pf − 1)n ) is the cardinality of the set

(Z/pf Z)∼ [S 0 ] − (Z/N Z)∼ [S 0 ]. (Note that each element of (Z/pf Z)∼ [S 0 ] − (Z/N Z)∼ [S 0 ] automatically satises (3.19), since we have assumed n0 > 1.) By applying the identity

(B 0 − A0 )(B 0 − B) B 0 0 0 A − {A − (B − B)} = B0 B0 ³ ´ ¡ k ¢ [ fk ] (f −k[ f ])n0 0 0 0 k to B = (pf − 1)n , B 0 = pf n and A0 = pkn − pn0 p , we obtain #{D ∈ (Z/N Z)∼ [S]0 | s(D(i) ) > 1 for all i = 0, 1, . . . f − 1} µ k ¶¶[ fk ] 0 µ f 0 (pf − 1)n p kn0 ≤ p − p(f −k[ k ])n pf n0 n0 µ µ ¶¶[ fk ] 1 pk n0 =N 1 − kn0 p n0 µ µ k ¶¶(1−ε)f /k 0 1 p ≤ N n 1 − kn0 , p n0 where the last inequality follows from our assumption f ≥ k/ε: · ¸ f (1 − ε)f f ≥ −1≥ . k k k Now, the statements of (ii) follow immediately.

¤

Finally, we shall summarize (3.12), (3.16) and (3.18) in terms of µN -torsors, via (3.5) and (3.6). Recall that we are assuming N = pf − 1. Theorem (3.20).

Let C(g) be as in (3.13).

(i) If n ≤ 1, we have 1 #{[V ] ∈ Hét (U, µN ) | γ[V ],1 = g − 1} ≥ N 2g − C(g)N 2g−1 − 1

(ii) If n > 1, we have 1 #{[V ] ∈ Hét (U, µN ) | γ[V ],1 = g} ≥ (N 2g − C(g)N 2g−1 ){N n−1 (1 − αf ) − 1}

for f ≥ M , where M and α are as in (3.18)(ii).

¤

Roughly speaking, (3.20) says that the generalized HasseWitt invariants for `most' (pf − 1)-cyclic coverings are g (resp. g − 1) if n > 1 (resp. n ≤ 1).


85

4. A Group-Theoretic Characterization of Genera In this section, we shall prove that the genus of a curve over an algebraically closed eld of characteristic > 0 can be recovered group-theoretically from the tame fundamental group of the curve. More precisely, we shall prove the following:

For each i = 1, 2, let pi be a prime number , ki an algebraically closed eld of characteristic pi , Xi a proper , smooth , connected curve of genus gi over ki , Si a nite (possibly empty ) set of closed points of Xi with cardinality ni , and Ui = Xi − Si . If π1t (U1 ) ' π1t (U2 ) (as topological groups ), then we have : Theorem (4.1).

(i) p1 = p2 , unless gi = 0, ni ≤ 1 for i = 1, 2; (ii) g1 = g2 ; and (iii) n1 = n2 , unless {(g1 , n1 ), (g2 , n2 )} = {(0, 0), (0, 1)}. To specify the notion of being recovered group-theoretically, we need to introduce two curves (see [T2], § 1, Denition). However, the following proof involves only one curve, and what we shall do is to extract its various invariants from its tame fundamental group by purely group-theoretic procedure. Remark (4.2).

Now, let p, k, X, g, S, n, U be as in Section 2 and Section 3. Recall that the i-th Betti number b(i) of U , dened as the Zl -rank of the l-adic étale cohomology i group Hét (U, Zl ) (l: a prime number 6= p), is given in terms of (g, n) as: ½ 1 if n = 0, (0) (1) (2) (2) b = 1, b = 2g + n − 1 + b , b = 0 if n > 0. First, we shall settle some minor things. Lemma (4.3).

π1t (U ).

(i) The invariant b(1) can be recovered group-theoretically from

(ii) We have

b(1) = 0 ⇐⇒ (g, n) = (0, 0), (0, 1) b(1) = 1 ⇐⇒ (g, n) = (0, 2)

) ⇒ g = 0.

(iii) Except for the case b(1) = 0, the invariant p can be recovered from π1t (U ) group-theoretically . Proof.

(ii) is trivial. As is well-known, π1t (U )ab is isomorphic to Y (1) Zbl × Zγp , l6=p

where γ is the p-rank of (the Jacobian variety of) X (see [T1], Corollary (1.2)). From this, (i) follows. Since (0 ≤)γ ≤ g ≤ 2g ≤ 2g + n − 1 + b(2) = b(1) ,

86

AKIO TAMAGAWA

γ = b(1) holds (if and) only if g = n−1+b(2) = 0 holds, or, equivalently, b(1) = 0. In other words, except for the case b(1) = 0, γ < b(1) holds, so we can extract the invariant p from the above description of π1t (U )ab (see [T2], Proposition (1.2)). Thus, (iii) follows. ¤ By this lemma, we obtain (4.1)(i). Moreover, by (i) and (ii) of this lemma, we may assume that b(1) > 1 when we prove (4.1)(ii). In particular, we may use the invariant p freely. The essence of (4.1)(ii) is in (3.20), which says, roughly speaking, that the generalized HasseWitt invariants for `most' (pf −1)-cyclic coverings are g (resp. g − 1) if n > 1 (resp. n ≤ 1). However, a few problems remain. The rst problem is that, strictly speaking, µN = µN (k) is not a group-theoretic object. Namely, even if we are given an isomorphism π1t (U1 ) ' π1t (U2 ), we are not given any natural isomorphism µN (k1 ) ' µN (k2 ), a priori, hence we do not have any natural isomorphism 1 1 Hét (U1 , µN ) ' Hét (U2 , µN ). 1 Moreover, in order to dene γ[V ],1 for each [V ] ∈ Hét (X, µN ), we have used not only the group µN but also the natural embedding µN ,→ k and the eld structure of k , which are also not group-theoretic objects. In fact, by means of (3.8), any xed isomorphism µN (k1 ) ' µN (k2 ) will turn out to work for our purpose. However, to avoid confusion and to make things clear, in this section, we will use the set of open normal subgroups H of π1t (U ) such that π1t (U )/H is a cyclic group of order dividing N , instead of using 1 Hét (U, µN ), and rewrite (3.20) in purely group-theoretic terms. The second problem is that, if n ≤ 1, the generalized HasseWitt invariant for a general (pf − 1)-covering is g − 1, and, if n > 1, it is g , and that we do not know, a priori, in which case we are. We overcome this second problem by considering not only the base curve U but also suitable (tame) coverings of U .

Now, we shall start with the rst problem. Assume that a cyclic group G of order prime to p and an Fp [G]-module M are given. As in Section 3, as soon as we are given a character χ : G → k × for some def eld k of characteristic p, we can dene γχ (M ) = dimk ((M ⊗ k)(χ)), where def

(M ⊗ k)(χ) = {x ∈ M ⊗ k | σ · x = χ(σ)x for all σ ∈ G}. However, the case is that only G and M are given. In this situation, only certain sums of γχ (M ) can be well-dened, as follows. Definition.

We dene the primitive part of M by X def M prim = M/( M hσi ), σ6=1


87

def

where M hσi = {x ∈ M | σ · x = x} for each σ ∈ G. We put def

γ prim (M ) = dimFp (M prim ). (i) Let k be a eld of characteristic p containing all #(G)-th roots of unity. Then we can check: X γχ (M ). γ prim (M ) =

Remark (4.4).

χ:G,→k×

(ii) Assume that M is nite-dimensional as an Fp -vector space. We can naturally def regard the dual vector space M ∗ = HomFp (M, Fp ) as a G-module by (σ ·φ)(x) = φ(σ −1 · x), where σ ∈ G, φ ∈ M ∗ , and x ∈ M . Then we can check:

γ prim (M ∗ ) = γ prim (M ). (One way to check this: γχ (M ∗ ) = γχ−1 (M ).) We return to the tame fundamental group π1t (U ). Let H be an open normal subgroup of π1t (U ) such that π1t (U )/H is cyclic of order prime to p. The conjugation induces an action of π1t (U )/H on the Fp -vector space H ab /p, whose dimension we denote by γH . Definition.

def

prim = γ prim (H ab /p). γH

For each natural number N prime to p, we dene HN to be the set of open normal subgroups H of π1t (U ) such that π1t (U )/H is cyclic of order dividing N . def

1 P

γ prim . N b(1) H∈HN H In this denition, av means average. In fact, we have a reinterpretation of av γN : Definition.

av γN =

Lemma (4.5).

We have av γN = def

=

Proof.

Average 1 (U,µ ) [V ]∈Hét N

γ[V ],1

1

X

γ[V ],1 . 1 (U, µ )) #(Hét N [V ]∈H 1 (U,µ ) N ét

By (4.4)(ii), prim γH = γ prim (H ab /p) = γ prim ((H ab /p)∗ ) 1 = γ prim (Hom(π1t (UH ), Fp )) = γ prim (Hét (XH , Fp )),

where UH is the tame covering of U corresponding to H ⊂ π1t (U ) and XH is the normalization of X in UH , and then by (4.4)(i), X prim 1 γH γχ (Hét (XH , Fp )). = χ:π1t (U )/H,→k×

88

AKIO TAMAGAWA

On the other hand, we have the following bijection: ∼

{(H, χ) | H ∈ HN , χ : π1t (U )/H ,→ k × }→ Hom(π1t (U ), µN ),

(4.6)

χ

where (H, χ) goes to (π1t (U ) ³ π1t (U )/H ,→ µN ). Now, let [V ] be the element 1 of Hét (U, µN ) = Hom(π1t (U ), µN ) corresponding to (H, χ). Then we claim 1 γ[V ],1 = γχ (Hét (XH , Fp )).

(4.7)

In fact, let Y be the normalization of X in V . Then, since UH is the Im(χ)χ ∼

torsor of U corresponding to (π1t (U ) ³ π1t (U )/H → Im(χ)), we see that Y coincides with µN ×Im(χ) XH , the quotient of µN × XH by the Im(χ)-action 1 (ζ, x)ζ0 = (ζ0−1 ζ, xζ0 ). From this, we can deduce that Hét (Y, Fp ) is the induced 1 Fp [µN ]-module of the Fp [Im(χ)]-module Hét (XH , Fp ). Now our claim (4.7) follows immediately. (1) 1 (U, µN ) ' (Z/N Z)b Now, bijection (4.6), identity (4.7) and the fact Hét complete the proof. ¤ av Here is another simple reinterpretation of γN . For a pronite group Π and a natural number m, we shall denote by Π(m) the kernel of Π ³ Πab /(Πab )m , or, equivalently, Π(m) is the topological closure of the subgroup [Π, Π]Πm of Π. Moreover, we shall denote by U (m) the tame covering of U corresponding to the subgroup π1t (U )(m) of π1t (U ), so that π1t (U (m)) = π1t (U )(m), and X(m) the normalization of X in U (m). (Note that this last notation is somewhat confusing: X(m) does not coincide with the (étale) covering of X corresponding to π1 (X)(m) in general.) Then we have

Remark (4.8).

av γN =

dimFp (π1t (U )(N )/(π1t (U )(N ))(p)) . (π1t (U ) : π1t (U )(N )) (1)

In fact, the denominator of the right-hand side is N b , while the numerator is 1 dimFp (Hét (X(N ), Fp )). Since π1t (U )/π1t (U )(N ) = π1t (U )ab /N is abelian of order prime to p, we see that the following canonical decomposition exists: M t t 1 1 Hét (X(N ), Fp ) = (Hét (X(N ), Fp )H/π1 (U )(N ) )(π1 (U )/H)-prim , H∈HN

where (π1t (U )/H)-prim means the primitive part as a (π1t (U )/H)-module. Since t 1 1 (X(N ), Fp )H/π1 (U )(N ) = Hét (XH , Fp ) (see the proof of (4.5) for the denition Hét of XH ), we obtain the desired equality

1 dimFp (Hét (X(N ), Fp )) =

X

prim γH

H∈HN

(see the beginning of the proof of (4.5)). The following is a variant of (3.20). Recall that we are assuming b(1) > 1. Theorem (4.9).

Assume N = pf − 1. Let C(g) be as in (3.13).


89

(i) If n ≤ 1, we have

g−1−

C(b(1) /2)(b(1) /2 − 1) 1 av ≤ γN ≤ g − 1 + b(1) . N N

(ii) If n > 1, let k be any positive integer ≥ logp (b(1) ), ε any positive real number < 1, and put µ µ k ¶¶(1−ε)/k k 1 p M = , α = 1 − kb(1) (1) . ε b p

Then , we have ½µ µ· (1) ¸¶ · (1) ¸ ¶ · (1) ¸ ¾ b b 1 b av g− C +1 + α f ≤ γN ≤ g + (b(1) − 1)αf 2 2 N 2 for all f ≥ M . Proof.

(i) We rst note that b(1) = 2g holds in this case. By (3.7)(and (3.6)),

we have

½ γ[V ],1 ≤

g, if [V ] = 0, g − 1, otherwise,

1 for each [V ] ∈ Hét (U, µN ). From this, X (1) (1) γ[V ],1 ≤ (N b − 1)(g − 1) + g = N b (g − 1) + 1, 1 (U,µ ) [V ]∈Hét N

hence

1 . N b(1) On the other hand, by (3.12)(ii) and (3.18)(i), we have X (1) (1) γ[V ],1 ≥ (g − 1)(N b − C(g)N b −1 ), av γN ≤g−1+

1 (U,µ ) [V ]∈Hét N

hence

C(g)(g − 1) . N Since g = b(1) /2, this completes the proof. P (ii) Dividing the sum [V ]∈H 1 (U,µN ) γ[V ],1 into the three parts: (0) D = 0; (1) av γN ≥g−1−

ét

s(D(i) ) = 1 for some i = 0, 1, . . . , f − 1; (2) otherwise, we obtain X γ[V ],1 1 (U,µ ) [V ]∈Hét N

≤ ((g − 1)N 2g + 1) + gN 2g (N n−1 (1 − αf ) − 1) + (g + n − 2)N 2g N n−1 αf (1)

− (N 2g − 1) + (n − 2)N b αf

(1)

+ (b(1) − 1)N b αf

= gN b

≤ gN b

(1)

(1)

90

AKIO TAMAGAWA

for f ≥ M , by (3.7) and (3.18)(ii). (Here, note that

k ≥ logp (b(1) ) ≥ logp (n − 1) and

µ α= 1−

1

µ

pk b(1)

pkb(1)

¶¶(1−ε)/k

µ ≤

1−

µ

1 pk(n−1)

¶¶(1−ε)/k

pk n−1

,

since n − 1 ≤ b(1) .) Therefore av γN ≤ g + (b(1) − 1)αf .

Similarly, considering the cases (0) and (1), we have X γ[V ],1 1 (U,µ ) [V ]∈Hét N

≥ (g − 1)(N 2g − C(g)N 2g−1 ) + g(N 2g − C(g)N 2g−1 )(N n−1 (1 − αf ) − 1) = gN b

(1)

− C(g)gN b

(1)

−1

− gN b αf + C(g)gN b

(1)

(1)

≥ gN b

(1)

− C(g)gN b

(1)

−1

− gN b αf − N 2g

≥ gN b

(1)

≥ gN b

(1)

− (C(g)g + 1)N b −1 − gN b αf µ µ· (1) ¸¶ · (1) ¸ ¶ · (1) ¸ (1) (1) b b b − C + 1 N b −1 − N b αf 2 2 2

−1 f

α − N 2g + C(g)N 2g−1

(1)

(1)

(1)

£ ¤ for f ≥ M , by (3.16), etc. (For the last inequality, note that g ≤ b(1) /2 and that C(g) is monotone increasing.) Therefore, we have ½µ µ· (1) ¸¶ · (1) ¸ ¶ · (1) ¸ ¾ b b 1 b av C γN ≥ g − +1 + αf 2 2 N 2 for all f ≥ M . Definition.

¤

We dene

½ def

g0 =

g − 1, g,

if n ≤ 1, if n > 1.

The following (together with (4.3)) gives a group-theoretic characterization of the invariant g 0 . (We are assuming b(1) > 1.) Let M and α be as in (4.9), and C(g) as in (3.13). Then ½µ µ· (1) ¸¶ · (1) ¸ ¶ · (1) ¸ ¾ b b 1 b 0 av g− C +1 + αf ≤ γN 2 2 N 2 µ ¶ 1 0 (1) f ≤ g +max , (b −1)α N b(1)

Corollary (4.10).

for all f ≥ M .


91

In particular , if f is suciently large (e .g ., if µ µ µ· (1) ¸¶ ¶¶ log(3b(1) ) b (1) f ≥ max M, , log b + 5 C p log(α−1 ) 2 holds ), then g 0 can be characterized as the unique integer in the interval · µ ¶ 1 av (1) f − max , γN , (b − 1)α N b(1) ½µ µ· (1) ¸¶ · (1) ¸ ¶ · (1) ¸ ¾¸ b b 1 b av C γN + +1 + αf . 2 2 N 2 By (4.9), g 0 falls in the interval for f ≥ M . µ µ µ· (1) ¸¶ ¶¶ log(3b(1) ) b (1) Assume f ≥ max M, , log C b + 5 . Then the p log(α−1 ) 2 length λ of the interval satises: µ ½µ µ· (1) ¸¶ · (1) ¸ ¶ · (1) ¸ ¾ 1 b b 1 b + C + 1 + αf , λ = max 2 2 N 2 N b(1) ½µ µ· (1) ¸¶ · (1) ¸ ¶ · (1) ¸ ¾¶ b b 1 b (1) f (b − 1)α + C +1 + αf 2 2 N 2 ½µ µ· (1) ¸¶ · (1) ¸ ¶ · (1) ¸ ¾ b b 1 b 1 C +1 + αf < b(1) + (b(1) − 1)αf + 2 2 N 2 N µ µ· (1) ¸¶ · (1) ¸ ¶ b b 1 3b(1) f < C +2 + α 2 2 N 2 Proof.

≤

1 2

+

1 2

= 1.

This implies the desired uniqueness.

¤

At the cost of sacricing eectivity, we also obtain the following more impressive characterization of g 0 . Corollary (4.11).

lim γpavf −1 = g 0 .

f →∞

¤

It is easy to see that (4.11) is also valid for b(1) = 1. In order to include the case that b(1) = 0, the formula should be modied as Remark (4.12).

lim γpavf −1 = (g 0 )+ ,

f →∞ def

where x+ = max(x, 0). Now, we shall treat the second problem: the group-theoretic characterization above is not for g but for g 0 . First, we introduce the following temporary invariant, which can be recovered group-theoretically from π1t (U ) (assuming b(1) > 0): def

ntemp = b(1) − 2g 0 + 1.

92

AKIO TAMAGAWA

Next, let m be a natural number prime to p, and U (m) the tame covering of U as in (4.8). We dene n(m) (resp. ntemp (m)) to be the invariant n (resp. ntemp ) for the curve U (m). Note that ntemp (m) can also be recovered group-theoretically from π1t (U ). Lemma (4.13).

Assume b(1) > 0.

(i) We have

½ n

temp

=

3, n,

if n ≤ 1, if n > 1.

(ii) If m > 1, we have

  3, (1) ntemp (m) = mb ,  b(1) −1 m n, Proof.

if n = 0, if n = 1, if n > 1.

(i) Immediate from the denitions.

(ii) By (i), we have

½ n

temp

(m) =

3, n(m),

if n(m) ≤ 1, if n(m) > 1.

On the other hand, we have

½ n(m) =

(1)

mb n, if n ≤ 1, b(1) −1 m n, if n > 1.

These give the desired equality.

¤

Finally, we can prove the following: Theorem (4.14).

p. Then

n=

Assume b(1) > 0. Fix any natural number m 6= 1, 3 prime to

 temp n ,    0,  1,   3,

if if if if

ntemp ntemp ntemp ntemp

6= 3, = 3 and ntemp (m) = 3, (1) = 3 and ntemp (m) = mb , (1) = 3 and ntemp (m) = 3mb −1 .

In particular , the invariant n can be recovered group-theoretically from π1t (U ) (except for the case b(1) = 0). The rst statement follows from (4.13). Since ntemp , ntemp (m) and b(1) can be recovered group-theoretically from π1t (U ), the second statement follows. (More precisely, we have to consider two cases separately. If b(1) = 1, then we (1) have n = ntemp = 2. Otherwise, i.e. if b(1) > 1, then the numbers 3, mb and (1) 3mb −1 are distinct from one another.) ¤

Proof.


93

(4.1). As we have already seen, (4.3) implies (4.1)(i) and that we may assume b(1) > 0 when we prove (4.1)(ii)(iii). Now, (4.14) implies (4.1)(iii). Since b(2) is determined by n, (4.14) and (4.3), together with the equality b(1) = 2g + n − 1 + b(2) , End of proof of

implies (4.1)(ii). This completes the proof of (4.1). Remark (4.15).

¤

We might hope for the more general limit formula av lim γN = g0 .

N →∞ p-N

(4.16)

For the present, we can only prove òne half' of (4.16): av lim sup γN ≤ g0 , N →∞ p-N

(4.17)

by using a higher-dimensional version of LeVeque's inequality, due to Stegbuchner ([S]), in the theory of uniform distribution modulo 1. See Appendix for this. It may be interesting to ask if (3.17)(iii) gives an approach to the other half of (4.16). As in [T2], Remark (1.11), not only π1t (U ) but also a suitable quotient is enough to determine g (and n). For example, π1t (U )/D(D(D(π1t (U )))) is enough, where, for a pronite group G, D(G) denotes the (topological) commutator subgroup of G, or, equivalently, the kernel of G ³ Gab . Remark (4.18).

Finally, as a direct consequence of (4.1), we have:

The quotient π1 (X) of π1t (U ) can be recovered grouptheoretically from π1t (U ). Corollary (4.19).

As in [T2], Corollary 1.10, this follows from (4.1)(ii) and the Hurwitz formula. ¤ Proof.

5. Applications (A) A group-theoretic characterization of inertia groups. Since we have

established (4.1), we can prove, as in [T2], that the set of inertia subgroups of π1t (U ) can be recovered group-theoretically from π1t (U ) for U hyperbolic. (We say that the curve U is hyperbolic, if the EulerPoincaré characteristic b(0) − b(1) + b(2) = 2 − 2g − n of U is negative.) ˜ t to be the maximal Let K be the function eld k(U ) = k(X), and dene K sep Galois extension of K in a xed separable closure K , unramied over U and at ˜ t /K). most tamely ramied over S . We may and shall identify π1t (U ) with Gal(K t t t ˜ to be the normalization of X in K ˜ and S˜ to be the inverse image We dene X t t ˜ ˜ ˜ of S in X . For each P ∈ S , we denote by IP˜ the inertia subgroup of π1t (U )

94

AKIO TAMAGAWA

associated to P˜ , i.e. the stabilizer of P˜ . We have IP˜ 6= {1} if and only if (n > 0 and) (g, n) 6= (0, 1) (see [T1], Lemma (2.2)). Lemma (5.1).

Assume that U is hyperbolic .

˜ be two points of S˜t distinct from each other . Then the intersec(i) Let P˜ and Q tion of IP˜ and IQ˜ is trivial in π1t (U ). In particular , for any σ ∈ π1t (U ) − IP˜ , the intersection of IP˜ and σIP˜ σ −1 is trivial . (ii) The map S˜t → Sub(π1t (U )), P˜ 7→ IP˜ is injective , where , for a pronite group G, Sub(G) denotes the set of closed subgroups of G. Moreover , for each P˜ ∈ S˜t , the normalizer of IP˜ in π1t (U ) is IP˜ itself . Proof.

(i) [T2], Lemma (2.1). (ii) [T2], Corollary (2.2).

¤

Let I t be the set of inertia subgroups in π1t (U ), namely the image of the map S˜t → Sub(π1t (U )), P˜ 7→ IP˜ .

If U is hyperbolic , then the set I t can be recovered grouptheoretically from π1t (U ). More precisely , let the notations and the assumptions be as in (4.1), and assume further that 2 − 2gi − ni < 0 for some i = 1, 2. Then , if an isomorphism π1t (U1 ) ' π1t (U2 ) (as topological groups ) is given , the induced bijection Sub(π1t (U1 )) ' Sub(π1t (U2 )) induces a bijection I1t ' I2t , where Iit denotes the set I t for the curve Ui , for each i = 1, 2.

Theorem (5.2).

Proof.

See [T2], Proposition (2.4) and Remark (2.6). (Use (4.1)(iii).)

¤

(B) A group-theoretic characterization of àdditive structures' of inertia groups. Let P˜ ∈ S˜t . As is well-known, IP˜ can be canonically identied with the Tate module

b 0 (1) def Z = lim µm (k) ← − p-m

of the multiplicative group k × , where µm (k) is the group of m-th roots of unity in k . So, IP˜ ⊗Z (Q/Z)0 , where (Q/Z)0 denotes the prime-to-p part of Q/Z, can be canonically identied with [ def (Q/Z)0 (1) = µm (k) = F × , def

where F denotes the algebraic closure of the prime eld Fp in k . Thus, FP˜ = ` (IP˜ ⊗Z (Q/Z)0 ) {∗} (where {∗} means a one-point set) can be identied with F , hence carries a structure of eld, whose multiplicative group is IP˜ ⊗Z (Q/Z)0 and whose zero element is ∗. Now, we have the following proposition. Unlike in (A) above, the proof here is quite dierent from [T2], Proposition 2.8, even after we have established (4.1) and (5.2). (See [T2], Remark 2.10(ii).) Proposition (5.3).

FP˜ = IP˜ ⊗Z (Q/Z)0

`

Assume that U is hyperbolic . Then the eld structure of {∗} can be recovered group-theoretically from π1t (U ).


95

We may assume n > 0. First, we shall reduce the problem to the case n ≥ 3. If g = 0, this follows automatically from the hyperbolicity condition. For g > 0, take any natural number m prime to p such that m2g n ≥ 3. Then, replacing π1t (U ) by the kernel of π1t (U ) → π1 (X)ab /m (whose index in π1t (U ) is m2g ), we have n ≥ 3. (Note that IP˜ is contained in the kernel.) Next, by (5.1)(i), the set I t divided by the conjugacy action of π1t (U ) consists of n orbits. Choosing any 3 orbits among these n orbits such that one of them is the conjugacy class of the given IP˜ , and dividing π1t (U ) by the subgroup (topologically) generated by all members I of I t whose conjugacy class is among the other n − 3 orbits, we may reduce the problem to the case n = 3. (Observe that these reduction steps are purely group-theoretic, by (4.1), (4.19) and (5.2).) From now on, we assume n = 3 and we shall use (2.21). For each natural number f , let Fpf ,P˜ denote the unique subeld of FP˜ with cardinality pf . Since Fpf ,P˜ × = IP˜ /(pf − 1), the subeld Fpf ,P˜ can be recovered group-theoretically as a (multiplicative) submonoid. Fix any eld Fpf with cardinality pf (unrelatedly to Fpf ,P˜ ). Then the set Hom(Fpf ,P˜ × , Fpf × ) = Hom(groups) (Fpf ,P˜ × , Fpf × ) is group-theoretic; recovering the eld structure of Fpf ,P˜ is equivalent to recovering Hom(Fpf ,P˜ , Fpf ) = Hom(fields) (Fpf ,P˜ , Fpf ) as a subset of Hom(Fpf ,P˜ × , Fpf × ). Moreover, it is sucient to recover this subset for f in a conal subset of Z>0 with respect to division. To do this, we shall consider the two maps

Proof.

Resf : Hom(π1t (U )ab /(pf − 1), Fpf × ) → Hom(Fpf ,P˜ × , Fpf × ) and

Γf : Hom(π1t (U )ab /(pf − 1), Fpf × ) → Z≥0 . The rst map Resf is the restriction with respect to the canonical inclusion Fpf ,P˜ × = IP˜ /(pf −1) ,→ π1t (U )ab /(pf −1). The second map Γf is dened to send def 1 (XH , Fp )), where H = Ker(χ). (For χ ∈ Hom(π1t (U )ab /(pf − 1), Fpf × ) to γχ (Hét the denitions of γχ and XH , see Section 4, especially (4.5) and its preceding paragraphs. Strictly speaking, in Section 4, we use the notation γχ only for a character χ of a cyclic group. However, the same denition goes well for characters of general nite groups, or we can replace π1t (U )ab /(pf − 1) by Im(χ). See the proof of (4.5).) Now we can state the following claim, which completes the proof of (5.3). Let m0 be the product of all prime numbers ≤ p−2. (For p = 2, 3, m0 = 1.) Let f0 be the order of p mod m0 in the multiplicative group (Z/m0 Z)× . For each f > logp (C(g) + 1) divisible by f0 , we have Claim (5.4).

Hom(Fpf ,P˜ , Fpf ) = Surj(Fpf ,P˜ × , Fpf × ) − Resf (Γ−1 f ({g + 1})) ¡ ¢ × × ⊂ Hom(Fpf ,P˜ , Fpf ) .

96

AKIO TAMAGAWA

To prove this claim, we x any embedding Fpf → k as elds. Then we have 1 Hom(π1t (U )ab /(pf − 1), Fpf × ) = Hét (U, µN ), def where N = pf − 1. By (3.5), this can be identied with PN (or PÑ ), in the notation of Section 3. On the other hand, FP˜ can be canonically identied with the algebraic closure of Fp in k . This identication, together with the xed embedding Fpf → k , species one identication Fpf ,P˜ = Fpf . By using this, we obtain Hom(Fpf ,P˜ × , Fpf × ) = Z/N Z.

By using various denitions, we see that the map PN → Z/N Z coming from Resf is nothing but the composite of bN : PN → Z/N Z[S] (see (3.4)) and Z/N Z[S] → Z/N Z, D mod N 7→ ordP (D). Thus we can reformulate (5.4) as follows: For each n ∈ (Z/N Z)∼ ,

n ∈ {pb | b = 0, 1, . . . , f − 1} ⇐⇒ (n, N ) = 1 and @([L], D) ∈ PÑ s.t. ordP (D) = n and γ([L],D) = g + 1. (5.5) First, by (3.7), we see that γ([L],D) ≤ g+1 always holds and that γ([L],D) = g+1 holds if and only only if γ([L],D) = dimk (H 1 (X, L)) (or, equivalently, ϕ([L],D) is f bijective) and deg(D) = 2N . Moreover, if ϕ([L],D) is bijective, then BD should satisfy condition (?) of page 59, and, in particular, it should be a semi-stable vector bundle. By this observation, the `⇒' part of (5.5) follows from (2.21)(iva). More specically, let us denote by P1 = P, P2 , P3 the three points of S . Then (2.21)(iv a) implies that either ordP2 (D) = N or ordP3 (D) = N holds, which is impossible as D ∈ (Z/N Z)∼ [S]. To prove the `⇐' part of (5.5), let n be a natural number ∈ (Z/N Z)∼ such that (n, N ) = 1, and suppose that n ∈ / {pb | b = 0, 1, . . . , f − 1}. Then we have to prove that there exists ([L], D) ∈ PÑ such that ordP (D) = n and γ([L],D) = g + 1. Since f is assumed to be divisible by f0 , N is divisible by all prime numbers ≤ p − 2. Therefore, by the assumption (n, N ) = 1, we must have n ∈ / Ip−1 pIf = {apb | a = 0, 1, . . . , p − 2, b = 0, 1, . . . , f − 1}. Now, by (2.21)(ivc), there exists D ∈ (Z/N Z)∼ [S] with degree 2N which satises (2.14) and to which (2.13) can be applied. Then, as in the proof of (3.12), we obtain

#{[L] ∈ Pic(X) | ([L], D) ∈ PÑ and γ([L],D) = g + 1} ≥ N 2g − C(g)N 2g−1 > 0, where the last inequality comes from the assumption f > logp (C(g) + 1). This completes the proof. ¤ Remark (5.6). A similar technique as in the proof of (5.4) gives an alternative proof of (4.1). More precisely, x an algebraic closure Fp of the prime eld Fp , and put def

×

1 γ max = max{γχ (Hét (XKer(χ) , Fp )) | χ ∈ Hom(π1t (U ), Fp ), χ 6= 1}.


97

(For the sake of convenience, we shall dene max ∅ = −1.) Then: Claim (5.7).

We have

γ max = g + n − 2 + b(2) . If we assume (5.7), def

g = b(1) − γ max − 1, n0 = n + b(2) = 2γ max − b(1) + 3 can be recovered group-theoretically. Moreover, to recover n (assuming b(1) > 0), we have only to note that ½ 0 n , if n0 (m) > 1, n= 0, if n0 (m) ≤ 1, where m is an arbitrary natural number > 1 prime to p and n0 (m) is the invariant n0 for the curve U (m). (See the paragraph preceding (4.13).) For the proof of (5.7), rst, the inequality γ max ≤ g + n − 2 + b(2) follows from (3.7) (together with (3.6) and (4.7)). For the opposite inequality, let f be a natural number > max(n − 1, logp (C(g) + 1)) and put N = pf − 1. Write the set S of cardinality n as {P−1 , P0 , P1 , . . . , Pn−2 } and dene D ∈ (Z/N Z)∼ [S] by ½ Pn−2 j n−2 X def j=0 p , if i = −1, D = ni Pi , ni = N − pi , if i = 0, 1, . . . , n − 2. i=−1 (D = 0 for n ≤ 1.) Then, we see that D satises (2.14) (with s = n − 1 + b(2) ). Now, as in the proof of (3.12), we deduce that

#{[L] ∈ Pic(X) | ([L], D) ∈ PÑ and γ([L],D) = g + n − 2 + b(2) } ≥ N 2g − C(g)N 2g−1 > 0, where the last inequality comes from the assumption f > logp (C(g) + 1). This completes the proof.

(C) The genus 0 case. By means of the above results, we can prove that the

isomorphism class of the scheme U can be recovered group-theoretically from the tame fundamental group π1t (U ) in the case where g = 0 and k = Fp . More precisely, we have:

Let k be an algebraically closed eld of characteristic > 0 and F the algebraic closure of Fp in k . Let U be a smooth , connected curve over k . For each given smooth , connected curve U0 over F whose smooth compactication is of genus 0 and whose number of punctures is greater than 1, we can detect whether U is isomorphic to U0 ⊗F k as a scheme or not , group-theoretically from π1t (U ). Theorem (5.8).

Corollary (5.9). For each i = 1, 2, let ki be an algebraically closed eld of characteristic > 0 and Ui a smooth , connected curve over ki . Let (gi , ni ) denote (g, n) for Ui . Assume k1 ' k2 . For some i = 1, 2, assume that gi = 0, ni > 1,

98

AKIO TAMAGAWA

and either (a ) Ui is dened over Fi , the algebraic closure of Fp in ki or (b ) ni ≤ 4. Then π1t (U1 ) and π1t (U2 ) are isomorphic as topological groups if and only if U1 and U2 are isomorphic as schemes . (5.8) and (5.9). With (4.1), (5.2), (5.3), etc., the same proofs as those of [T2], Theorem 3.5 and Corollary 3.6 work for π1t (U ). ¤ Proof of

Appendix: Proof of (4.17) First, we recall some notations in the text. Let k be an algebraically closed eld of characteristic p > 0, and let U be a smooth, connected curve over k . We denote by X the smooth compactication of U and put S = X − U . We dene non-negative integers g and n to be the genus of X and the cardinality of the point set S , respectively. We put ½ g − 1, if n ≤ 1, def g0 = g, if n > 1. Moreover, see Section 4 for the denition of the i-th Betti number b(i) of U . av In Section 4, we introduced the invariant γN of U for each natural number N prime to p, as a certain average of generalized HasseWitt invariants of N -cyclic étale coverings of U . Now, the following is the main result of this Appendix.

Assume b(1) > 0 (or , equivalently , (g, n) 6= (0, 0), (0, 1)). Then (4.17) holds , that is , we have

Theorem (A.1).

av lim sup γN ≤ g0 . N →∞ p-N

We devote the rest of this Appendix to proving (A.1). Let N be a natural number prime to p. Recall that for each divisor D in def (Z/N Z)∼ [S]0 , s(D) = deg(D)/N is an integer with 0 ≤ s(D) ≤ n − 1 + b(2) . Moreover, for each integer a, we denote by D(a) the element of (Z/N Z)∼ [S]0 that is equivalent to aD modulo N . Let f be the order of p mod N in the multiplicative group (Z/N Z)× . We put def

MN = #{D ∈ (Z/N Z)∼ [S]0 | s(D(pj )) ≤ 1 for some j = 0, 1, . . . f − 1} def

(2)

and EN = #((Z/N Z)∼ [S]0 ) − MN = N n−1+b term and error term.) Now, we have the following: Lemma (A.2).

− MN . (M and E mean main

(i) If n ≤ 1, we have av γN ≤g−1+

1 . N b(1)

(ii) If n > 1, we have av γN ≤ g + (b(1) − 1)

EN . N n−1

TAME FUNDAMENTAL GROUPS OF CURVES Proof.

99

(i) Just the same as the rst half of the proof of (4.9)(i).

1 (ii) Let [V ] be an element of Hét (U, µN ), and ([L], D) the element of PÑ corresponding to [V ]. (See Section 3, especially (3.5) and paragraphs preceding it.) Then, as in the proof of (3.16), the remark just before the denition of γ[V ],i (at the beginning of Section 3) and (3.8) imply

γ[V ],1 = γ[V ],pj = γ([L(pj )],D(pj )) for each i. So, if s(D(pj )) ≤ 1 for some j = 0, 1, . . . , f − 1, we have γ[V ],1 ≤ g , as in (3.7). From this, we obtain X (1) γ[V ],1 ≤ gN 2g MN + (g + n − 2)N 2g EN = gN b + (n − 2)N 2g EN 1 (U,µ ) [V ]∈Hét N

by (3.4) and (3.7). Thus we have av γN ≤ g + (n − 2)

EN EN ≤ g + (b(1) − 1) n−1 , N n−1 N

as desired.

¤

(A.2)(i) settles the proof of (A.1) for n ≤ 1, while (A.2)(ii) reduces the proof of (A.1) for n > 1 to

EN = o(N n−1 ), that is,

lim

N →∞ p-N

EN = 0. N n−1

(A.3)

Note that (A.3) depends only on the nite set S , so it no longer involves the geometry of U . To prove (A.3), we need some knowledge of the theory of uniform distribution modulo 1, which we shall recall here. (For more details, see [KN].) def

(i) I = [0, 1) = {x ∈ R | 0 ≤ x < 1}. For each x ∈ R, we denote by {x} the fractional part x − [x] ∈ I of x. (ii) Let s be a positive integer. Let a = (a1 , . . . , as ) and b = (b1 , . . . , bs ) be elements of Rs . We say that a < b (resp. a ≤ b) if ai < bi (resp. ai ≤ bi ) for each i = 1, . . . , s. We put

Definition.

def

[a, b) = {x ∈ Rs | a ≤ x < b}. If a ≤ b, the Lebesgue measure λ([a, b)) of [a, b) is given by (b1 − a1 ) . . . (bs − as ). Note that I s = [0, 1), where 0 = (0, . . . , 0), 1 = (1, . . . , 1). For each x = (x1 , . . . , xs ) ∈ Rs , we put def

{x} = ({x1 }, . . . , {xs }), def

kxk = max |xi |, i=1,...,s Y def r(x) = |xi | (r(0) = 1). i=1,...,s xi 6=0

100

AKIO TAMAGAWA

For each x = (x1 , . . . , xs ), y = (y1 , . . . , ys ) ∈ Rs , we put def

hx, yi = x1 y1 + . . . + xs ys . (iii) Let x1 , . . . , xM be a sequence of length M of elements of Rs . For each subset E of I s , put def

A(E; M ; x1 , . . . , xM ) = #{j = 1, . . . , M | {xj } ∈ E}. Moreover, we dene the discrepancy DM of the sequence x1 , . . . , xM by

¯ ¯ ¯ A(J; M ; x1 , . . . , xM ) ¯ def DM = DM (x1 , . . . , xM ) = sup ¯¯ − λ(J)¯¯ , M J where J runs over the subsets of I s in the form [a, b) with a, b ∈ Rs , 0 ≤ a < b ≤ 1. (Observe that 0 ≤ DM ≤ 1.) Now, we can state the following higher-dimensional version of LeVeque's inequality, due to Stegbuchner. [S]). Let x1 , . . . , xM be a sequence of length M of

Theorem (Stegbuchner

elements of Rs . Then

Ã DM (x1 , . . . , xM ) ≤

Cs

X h∈Zs −{0}

¯

¯ !1/(s+2)

2 M 1 ¯¯ 1 X 2πihh,xj i ¯¯ e ¯ r(h)2 ¯ M j=1

,

where Cs is a positive constant depending only on s. More precisely , we may 2 take Cs = s2(s+2) 2s 9s +3s+1 . ¤ For various improvements of the constant Cs , see, e.g., [GT], Theorem 3 and [DT], Theorem 1.28. As in the proof of (3.18)(ii), we choose any Q ∈ S and put S 0 = S − def {Q}, whose cardinality is n0 = n − 1(> 0). The projection (Z/N Z)∼ [S]0 → ∼ 0 0 (Z/N Z) [S ], D 7→ D is bijective. For each a ∈ Z, we have D(a)0 = D0 (a). Moreover, we see that

s(D) =

» ¼ 1 1 deg(D) = deg(D0 ) , N N

where dxe denotes the smallest integer not less than x. Therefore, we have

s(D) ≤ 1 ⇐⇒ deg(D0 ) ≤ N. 0

0

Taking S 0 as a basis, we may identify (Z/N Z)∼ [S 0 ] = ((Z/N Z)∼ )n ⊂ Zn .


101

Then, for each D ∈ (Z/N Z)∼ [S]0 , we can apply Stegbuchner's theorem to s = n0 , M = f , and xj = D(pj−1 )0 /N and obtain µ ¶ D(p0 )0 D(pf −1 )0 def Df (D) = Df ,..., N N

Ã

≤

Cn0

h∈Zn0 −{0}

Ã ≤

X

Cn0

h∈Zn0 −{0}

So

X

0

Df (D)n +2

D∈(Z/N Z)∼ [S]0

¯ !1/(n0 +2)

¯

X

2 f −1 1 ¯¯ 1 X 2πihh, D(pj )0 i ¯¯ N e ¯ r(h)2 ¯ f j=0

¯ f −1 ¯2 !1/(n0 +2) j hh,D 0 i ¯ X ¯ p 1 ¯1 . e2πi N ¯¯ r(h)2 ¯ f j=0

¯ ¯2 ¯ f −1 ¯ j hh,D 0 i ¯ X ¯ p 1 ¯1 2πi ¯ N e ≤ Cn 0 ¯ r(h)2 ¯¯ f j=0 ¯ 0 0 0 ∼ n n D ∈((Z/N Z) ) h∈Z −{0} ¯ ¯2 ¯ fX ¯ X X ¯ 1 −1 2πi pj hh,D0 i ¯ 1 ¯ ¯ . N = Cn 0 e ¯f ¯ 2 r(h) ¯ ¯ 0 0 j=0 h∈Zn −{0} D 0 ∈((Z/N Z)∼ )n X

X

Here, we have ¯2 ¯ ¶µ f −1 ¶ µ f −1 ¯ ¯ fX X X ¯ 1 −1 2πi pj hh,D0 i ¯ pj hh,D 0 i pj hh,D 0 i 1 1 2πi 2πi ¯ = ¯ N N N e e e ¯ ¯f f j=0 f j=0 ¯ ¯ j=0

=

f −1 1 X 2πi (pj −pj0 )hh,D0 i N . e f2 0 j,j =0

Now, since

χh,j,j 0 : D0 mod N 7→ e2πi

0 (pj −pj )hh,D 0 i N

0

is a character of the abelian group (Z/N Z)n , we have ½ X 0, if χh,j,j 0 = 6 1, χh,j,j 0 (D0 ) = n0 0 = 1. N , if χ h,j,j 0 D 0 ∈((Z/N Z)∼ )n

Moreover, we have 0

χh,j,j 0 = 1 ⇐⇒ (pj − pj )h ≡ 0 (mod N ) ⇐⇒ fh | (j − j 0 ). Here, fh is the order of p mod Nh in the multiplicative group (Z/Nh Z)× , where 0 Nh is the order of h mod N ∈ (Z/N Z)n . Thus, in summary, we get X 0 Df (D)n +2 D∈(Z/N Z)∼ [S]0

102

AKIO TAMAGAWA

X

≤ Cn0

h∈Zn0 −{0}

X

= Cn0

h∈Zn0 −{0}

X

= Cn0

h∈Zn0 −{0}

= Cn0 N n

0

f −1 1 1 X r(h)2 f 2 0

X

j,j =0

χh,j,j 0 (D0 )

D 0 ∈((Z/N Z)∼ )n0

0 1 1 #({(j, j 0 ) | j, j 0 = 0, . . . , f − 1, fh | (j − j 0 )})N n 2 2 r(h) f

1 1 f 2 n0 N r(h)2 f 2 fh

X h∈Zn0 −{0}

1 . r(h)2 fh

Taking a positive integer K (which we x later), we divide the last innite sum into the sum of the innite sum with khk > K and the nite sum with khk ≤ K . For the former, we have

X khk>K

X 1 1 ≤ 2 r(h) fh r(h)2 khk>K 0

n X

X

1 r(h)2 i=1 h s.t. |hi |>K     n0 Y X X X 1 1     = 2 2 |h | max(|h |, 1) i j i=1

≤

j6=i

|hi |>K

≤

n0 µ X

Z

∞

2 K

i=1

dx x2

hj ∈Z

¶ 0

(1 + 2ζ(2))n −1 0

= 2n0 (1 + 2ζ(2))n −1

1 . K

For the latter, we need an estimate of fh . Since N | Nh h, we have Nh khk ≥ N , unless h = 0. So, if khk ≤ K and h 6= 0, we have Nh ≥ N/K . On the other hand, since Nh | pfh − 1 < pfh , we have fh ≥ log(Nh )/ log(p). These two inequalities imply

fh ≥

log(N/K) . log(p)

Therefore, we have (assuming N/K > 1)

X 0 0, this implies (A.3). This completes the proof of (A.1).

References [B] I. Bouw, Tame covers of curves: p-ranks and fundamental groups, thesis, Univ. Utrecht, 1998. [BH] W. Bruns and J. Herzog, CohenMacaulay rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, 1993. [DT] M. Drmota and R. F. Tichy, Sequences, discrepancies and applications, Lecture Notes in Mathematics 1651, Springer, Berlin, 1997. [E] D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics 150, Springer, New York, 1994.

104

AKIO TAMAGAWA

[EGA4] A. Grothendieck, Éléments de Géométrie Algébrique IV: Étude locale des schémas et des morphismes de schémas, Publications Mathématiques de l'IHES 20, 24, 28, 32, 19641967. [FJ] M. D. Fried and M. Jarden, Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3. Folge) 11, Springer, Berlin and New York, 1986. [GT] P. J. Grabner and R. F. Tichy, Remark on an inequality of ErdösTurán Koksma, Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. 127 (1990), 1522. [GM] A. Grothendieck and J. P. Murre, The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme, Lecture Notes in Mathematics 208, Springer, Berlin and New York, 1971. [Ka] H. Katsurada, On generalized HasseWitt invariants and unramied Galois extensions of an algebraic function eld, J. Math. Soc. Japan 31 (1979), 101125. [KN] L. Kuipers and H. Niederreiter, Uniform distribution of sequences, Wiley, New York-London-Sydney, 1974. [Na] S. Nakajima, On generalized HasseWitt invariants of an algebraic curve, pp. 69 88 in Galois groups and their representations (Nagoya, 1981), Advanced Studies in Pure Mathematics 2, edited by Y. Ihara, North-Holland, Amsterdam, and Kinokuniya, Tokyo, 1983. [Mac] F. S. Macaulay, The algebraic theory of modular systems, Cambridge University Press, Cambridge, 1916. [Mad] D. A. Madore, Theta divisors and the Frobenius morphism, pp. 279289 in Courbes semi-stables et groupe fondamental en géométrie algébrique (Luminy, 1998) Progr. Math. 187, edited by J.-B. Bost, F. Loeser and M. Raynaud, Birkhäuser, Basel, 2000. [Mi1] J. S. Milne, Étale cohomology, Princeton Mathematical Series 33, Princeton Univ. Press, Princeton, New Jersey, 1980. [Mi2] , Jacobian varieties, pp. 167212 in Arithmetic geometry (Storrs, 1984), edited by G. Cornell and J. H. Silverman, Springer, New York-Berlin, 1986. [Mu] D. Mumford, Abelian varieties, Oxford University Press, London, 1970. [R1] M. Raynaud, Sections des brés vectoriels sur une courbe, Bull. Soc. math. France 110 (1982), 103125. [R2] , Revêtements des courbes en caractéristique p > 0 et ordinarité, Compositio Math. 123 (2000), 7388. [S] H. Stegbuchner, Eine mehrdimensionale Version der Ungleichung von LeVeque, Monatsh. Math. 87 (1979), 167169. [SGA1] A. Grothendieck and Mme. M. Raynaud, Revêtements étales et groupe fondamental, Séminaire de Géometrie Algébrique du Bois Marie 196061 (SGA 1), Lecture Notes in Mathematics 224, Springer, Berlin and New York, 1971. [SGA6] P. Berthelot, A. Grothendieck and L. Illusie, Théorie des intersections et théorème de RiemannRoch, Séminaire de Géométrie Algébrique du Bois Marie 19661967 (SGA 6), Lecture Notes in Mathematics 225, Springer, Berlin and New York, 1971.


105

[SGA7I] A. Grothendieck, M. Raynaud and D. S. Rim, Groupes de monodromie en géométrie algébrique, Séminaire de Géométrie Algébrique du Bois Marie 19671969 (SGA 7I), Lecture Notes in Mathematics 288, Springer, Berlin and New York, 1972. [T1] A. Tamagawa, The Grothendieck conjecture for ane curves, Compositio Math. 109 (1997), 135194. [T2] , On the fundamental groups of curves over algebraically closed elds of characteristic > 0, Internat. Math. Res. Notices (1999), no. 16, 853873. [T3] , Fundamental groups and geometry of curves in positive characteristic, pp. 297333 in Arithmetic fundamental groups and noncommutative algebra (Berkeley, 1999), Proceedings of Symposia in Pure Mathematics 70, edited by M. D. Fried and Y. Ihara, American Mathematical Society, Providence, 2002. [Y] Y. Yoshino, CohenMacaulay modules over CohenMacaulay rings, London Mathematical Society Lecture Note Series 146, Cambridge University Press, Cambridge, 1990. Akio Tamagawa Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502 Japan

[email protected]


On the Specialization Homomorphism of Fundamental Groups of Curves in Positive Characteristic FLORIAN POP AND MOHAMED SAïDI

Introduction Recall that for proper smooth and connected curves of genus g ≥ 2 over an algebraically closed eld of characteristic 0 the structure of the étale fundamental group πg is well known and depends only on the genus g . Namely it is the pronite completion of the topological fundamental group of a compact orientable topological surface of genus g . In contrast to this, the structure of the étale fundamental group of proper smooth and connected curves of genus g ≥ 2 in positive characteristic is unknown, and it depends on the isomorphy type of the curve in discussion. The aim of this paper is to give new evidence for anabelian phenomena for proper curves over algebraically closed elds of characteristic p > 0. Before going into the details of the results we are going to prove, we set some notation and recall well known facts. Let k be an algebraically closed eld of characteristic p > 0. Let X be a projective smooth and connected curve of genus g ≥ 2 over k , and let J be the Jacobian of X . We denote by π1 (X), π1p (X), and 0 π1p (X) the étale fundamental group of X , its pro-p quotient, and its prime to p quotient. Then: (1) The structure of π1p (X) is given by Shafarevich's Theorem; see [Sh]. It is isomorphic to the pro-p free group on r := rX generators, where rX is the p-rank of J . 0 (2) The structure of π1p (X) is well known by Grothendieck's Specialization Theorem; see [SGA-1]. It is the prime to p completion of the topological fundamental group of a compact orientable topological surface of genus g . (3) In contrast to this, the structure of the whole fundamental group π1 (X) is a big mystery! Its structure is not known in any single case. However, by Grothendieck's Specialization Theorem we know that π1 (X) is the quotient of 107

108

FLORIAN POP AND MOHAMED SAïDI

the pronite completion Πg of the topological fundamental group of a compact orientable topological surface of genus g . In particular π1 (X) is topologically nitely generated. Since such groups are completely determined by the set of their nite quotients, another interpretation of (1) is the following: If two curves as above have the same p-rank, then there is a bijection between the set of their Galois étale covers with Galois group a p-group. In the same way, the interpretation of (2) is that for two curves of the same genus there is a bijection between the set of their Galois étale covers having a Galois group of order prime to p. In order to approach the complexity of π1 of proper curves in positive characteristic we introduce the following: Let Mg → Spec Fp be the coarse moduli space of proper and smooth curves of genus g in characteristic p. It is well known that Mg is a quasi-projective and geometrically irreducible variety. Let k be an algebraically closed eld of characteristic p; thus Mg (k) is the set of isomorphism classes of curves of genus g over k . For x ¯ ∈ Mg (k) let Cx¯ → Spec k be a curve classied by x ¯, and let x ∈ Mg such that x ¯ : Spec k → Mg factors through x. We set

π1 (x) := π1 (Cx¯ ),

π1p (x) := π1p (Cx¯ ),

0

0

π1p (x) = π1p (Cx¯ ).

We remark that the structure of π1 (x) as a pronite group depends only on x and not on the concrete geometric point x ¯ ∈ Mg (k) used to dene it. Indeed, let κ be the algebraic closure of the residue eld κ(x) at x in k . Then, if Cx is the curve classied by Spec κ → Mg , then Cx¯ is the base change Cx¯ ' Cx ×κ k of Cx to k . Hence π1 (Cx¯ ) ' π1 (Cx ) by the geometric invariance of the fundamental group for proper varieties; see [SGA-1]. Second, the isomorphy type of Cx as an Fp -scheme does depend only on x, and not the concrete choice of the algebraic closure κ of κ(x). We further remark that by the comments above, if Jx¯ is the Jacobian of Cx¯ , then the p-rank of Jx¯ as well as Jx¯ being a simple abelian variety depends only on x and not on the geometric point x ¯. Indeed, in the notations above, if Jx is the Jacobian of Cx , then Jx¯ ' Jx ×κ k ; and for dierent choices of the algebraic closure of κ(x), the corresponding curves are isomorphic as Fp -schemes. Hence their Jacobians too are isomorphic as Fp -schemes. Coming back to the fundamental group we thus have maps

π1 : Mg → (Prof.groups),

x → π1 (x),

and the induced maps

π1p : Mg → (Prof.groups),

x → π1p (x)

and 0

π1p : Mg → (Prof.groups),

0

x → π1p (x),

FUNDAMENTAL GROUPS OF CURVES IN POSITIVE CHARACTERISTIC

109

where (Prof.groups) are the objects of the category of pronite groups. The last two maps are not very interesting: rst, the isomorphy type of the images of π1p depends only on the p-rank; and second, the isomorphy type is constant on the 0 image of π1p . To nish our preparation we remark that for points x, y ∈ Mg such that x is a specialization of y , by Grothendieck's specialization theorem there exists a surjective continuous homomorphism Sp : π1 (y) → π1 (x). In particular, if η is the generic point of Mg , then Cη is the generic curve of genus g ; and every point x of Mg is a specialization of η . Thus, for every x ∈ Mg , there is a surjective homomorphism Spx : π1 (η) → π1 (x) which is determined up to Galois-conjugacy by the choice of the local ring of x in the algebraic closure of κ(η). For every x ∈ Mg we x such a map once for all; in particular, if x is a specialization of y , there exists a specialization homomorphism Spy,x : π1 (y) → π1 (x) such that Spy,x ◦ Spy = Spx . (In order to obtain Spy,x one has to choose the local ring of x to be contained in the local ring of y .) Finally, let S a.s. ⊂ Mg be the set of closed points corresponding to curves a.s. Cx having an absolutely simple Jacobian Jx . Further, let S≥g−1 ⊂ S a.s. be the a.s. subset of points x ∈ S such that the p-rank of Cx equals g or g−1. Concerning the set S a.s. , Chai and Oort proved the following (see [Se-1] for facts concerning Dirichlet density):

The subset S a.s. is non empty and has a positive Dirichlet density . In particular , S a.s. is Zariski dense . Theorem ([Ch-Oo]).

We now come to the main results of the present article. We remark that for genus g = 2, even stronger results were proven by Raynaud. This is Raynaud's theory of the theta divisor of the sheaf of locally exact dierentials for curves in positive characteristic; see [Ra-1] the main tool that we use in our approach.

For all points s ∈ S a.s. , the specialization homomorphism Sps : π1 (η) → π1 (s) is not an isomorphism . More precisely , every cyclic étale cover of Xη of order prime to p is ordinary , whereas there exist such covers of Cs that are not ordinary . Theorem A.

a.s. If a point y ∈ Mg specializes to some point s ∈ S≥g−1 with s 6= y , then the specialization homomorphism Spy,s : π1 (y) → π1 (s) is not an isomorphism . a.s. In particular , for a given point s ∈ S≥g−1 there exist only nitely many points 0 a.s. 0 s ∈ S≥g−1 such that π1 (s ) ' π1 (s).

Theorem B.

As an application we have the following corollary answering a question raised by David Harbater:

There is no nonempty open subset U ⊂ Mg such that the isomorphy type of the geometric fundamental group π1 (x) is constant on U .

Corollary.

We conclude with a question:

110 Question.


Is it true that the specialization homomorphism Sp : π1 (y) → π1 (x)

to points y 6= x with x closed is never an isomorphism ? If this is the case the same proof as that of Corollary 4.4 below would imply the following niteness result: Given a closed point x ∈ Mg there exists at most nitely many closed points x0 in Mg such that π1 (x0 ) ' π1 (x). One could ask the preceding question more generally, without the condition that the point x be closed. However, the condition that x be closed is essential in the proof of our results.

Acknowledgments. Parts of this paper were written while Saïdi was a post-

doc in the Graduiertenkolleg of the Mathematics Institute, University of Bonn, from September 1998 to June 1999. He would like very much to thank the members of the Mathematics Institute for their hospitality and the very good working conditions. The manuscript was completed during the fall 1999 at the Mathematical Sciences Research Institute in Berkeley. The authors would like to express their gratitude for the support from MSRI and the wonderful working conditions there.

1. Preliminaries and Notations 1.1. The sheaf of locally exact dierentials in characteristic p > 0 and the associated theta divisor. We recall here the denition of the sheaf of lo-

cally exact dierentials associated to an algebraic curve in positive characteristic and its associated theta divisor, mainly following Raynaud (see [Ra-1], 4). Let X be a proper smooth and connected algebraic curve of genus gX := g ≥ 2, over an algebraically closed eld k of characteristic p > 0. Consider the Cartesian diagram X 1 −−−−→ X

  y

  y

F

Spec k −−−−→ Spec k where F denotes the absolute Frobenius morphism. The projection X 1 → X is a scheme isomorphism, in particular X 1 is a smooth and proper curve of genus g . The absolute Frobenius morphism F : X → X induces in a canonical way a morphism π : X → X 1 called the relative Frobenius which is a radicial morphism of k -curves of degree p. The canonical dierential π∗ d : π∗ OX → π∗ Ω1X is a morphism of OX 1 -modules. Its image BX := B := Im(π∗ d) is the sheaf of locally exact dierentials. One has the exact sequence

0 → OX 1 → π∗ OX → B → 0,


111

and B is a vector bundle on X 1 of rank p − 1. Let c : π∗ (Ω1X ) → Ω1X 1 be the Cartier operator; this is a morphism of OX 1 -modules. The kernel ker(c) of c is equal to B , and the following sequence of OX 1 -modules is exact (see [Se], 10):

0 → B → π∗ (Ω1X ) → Ω1X 1 → 0 Let L be a universal Poincaré bundle on X 1 ×k J 1 where J 1 := Pic0 (X 1 ) is the Jacobian of X 1 . This is a line bundle such that its restriction to X 1 × {a} for any a ∈ J 1 (k) is isomorphic to the invertible sheaf La which is the image of a under the natural isomorphism J 1 (k) ' Pic0 (X 1 ). Let h : X 1 × J 1 → X 1 and f : X 1 × J 1 → J 1 be the canonical projections. As Ri f∗ (h∗ B ⊗ L) = 0 for i ≥ 2, the total direct image Rf∗ (h∗ B ⊗ L) of (h∗ B ⊗ L) by f can be realized by a complex u : M0 → M1 of length 1, where M0 and M1 are vector bundles on J 1 , ker u = R0 f∗ (h∗ B ⊗ L), and coker u = R1 f∗ (h∗ B ⊗ L). Moreover as the Euler-Poincaré characteristic χ(h∗ B ⊗ L) = 0, the vector bundles M0 and M1 have the same rank. In [Ra-1], théorème 4.1.1, it has been proved that the determinant det u of u is not identically zero on J 1 , hence one can consider the divisor θ := θX on J 1 , which is the positive Cartier divisor locally generated by det u, it is the theta divisor associated to the vector bundle B (note that the denition of θX is independant on the above chosen complex u). By denition a point a ∈ J 1 (k) lies on the support of θ if and only if H 0 (X 1 , B ⊗ La ) 6= 0.

1.2. p-Rank of cyclic étale covers of degree prime to p. We use the

same notations as in 1.1. The p-rank rX of X is the dimension of the maximal subspace of H 1 (X, OX ) on which the absolute Frobenius F acts bijectively. By duality it is also the dimension of the maximal subspace of H 0 (X, Ω1X ) on which the Cartier operator c is bijective (see [Se-1], 10). The p-rank rX of X is also the rank of the maximal pro-p-quotient π1p (X) of the fundamental group π1 (X) of X , which is a free pro-p-group (see [Sh]). The relative Frobenius morphism π : X → X 1 induces a canonical isomorphism π1 (X) → π1 (X 1 ) between fundamental groups (see [SGA-1]). In particular for any positive integer n which is prime to p one has a one to one correspondence between µn -torsors of X 1 and µn -torsors of X . More precisely the 1 1 canonical homomorphism Het (X 1 , µn ) → Het (X, µn ) induced by π is an isomorphism. Consider a µn -torsor f : Y → X with Y connected. By Kummer theory f ⊗i is given by an invertible sheaf L of order n on X , and Y := Spec(⊕n−1 i=0 L ). Thus 1 1 0 there exists an invertible sheaf L on X of order n, such that if f : Y 1 → X 1 is the associated µn -torsor we have a Cartesian diagram f

Y −−−−→ X

 

 

π0 y

πy f0

Y 1 −−−−→ X 1

112


Let JY (resp. JX ) denote the Jacobian variety of Y (resp. the Jacobian of X ). The morphism f : Y → X induces a homomorphism f ∗ : JX → JY between Jacobians. Let J new := JY /X denote the quotient of JY by the image f ∗ (JX ) of JX , that is the new part of the Jacobian JY of Y with respect to the morphism f .

1.3. Denition. The µn -torsor f : Y → X is said to be new-ordinary if the new

part J new of the Jacobian of Y with respect to the morphism f is an ordinary abelian variety. Since the dimension of the abelian variety J new is h = gY − gX , it follows that J is ordinary if the étale part of the kernel of the multiplication by p in J new has order ph . This is also equivalent to the fact that the absolute Frobenius F acts bijectively on H 1 (J new , OJ new ). One has H 1 (JY , OJY ) ' H 1 (Y, OY ), and ⊗i 1 new H 1 (Y, OY ) = H 1 (X, f ∗ OY ) = H 1 (X, ⊕n−1 , OJ new ) ' i=0 L ). Moreover H (J n−1 1 ⊗i H (X, ⊕i=1 (L) ) and these identications are compatible with the action of Frobenius. Hence the kernel of Frobenius on H 1 (J new , OJ new ) is isomorphic ⊗i to the kernel of Frobenius acting on H 1 (X, ⊕n−1 i=1 L ). On the other hand 0 0 ∗ 0 as f is étale (f ) (BX ) = BY , thus also (f )∗ (BY ) = BX ⊗ (f 0 )∗ (OY 1 ) = 1 ⊗i ⊕n−1 i=0 (BX ⊗ (L ) ). Now by duality, the kernel of the Frobenius acting on 1 ⊗i H 1 (X 1 , ⊕n−1 ) is isomorphic to the kernel of the Cartier operator on H 0 (X 1 , i=1 L n−1 1 1 ⊗i 0 1 1 ⊗i π∗ ΩX ⊗ (⊕i=1 (L ) )), which is ⊕n−1 i=1 H (X , BX ⊗ (L ) ). Thus the above 0 µn -torsor f : Y → X is new-ordinary if and only if H (X 1 , B ⊗ (L1 )⊗i ) = 0 for i ∈ {1, . . . , n − 1}, which is also equivalent to the fact that the subgroup hL1 i generated by L1 in J 1 intersects the support of the theta divisor θX associated to BX at most at the zero point 0J 1 of J 1 . new

2. µn -Torsors of Curves over Finite Fields and Ordinariness In this section we consider curves over the algebraic closure Fp of the prime eld Fp . We establish that after nite étale covers the theta divisor associated to the sheaf B of locally exact dierentials contains innitely many torsion points of order prime to p. This indeed gives information on the fundamental group of these curves.

Let A be an abelian variety of dimension ≥ 2 over Fp , and let Y be a closed sub-variety of A of dimension ≥ 1. Assume either A is a simple abelian variety , or Y (Fp ) contains the zero point 0A of A. Then Y (Fp ) contains an innity of torsion points of pairwise prime order . Proposition 2.1.

First note that the abelian group A(Fp ) = A(Fp )tor is torsion. We will use the following result from [An-In]:

Proof.

Proposition. Let C be a proper smooth and connected curve over Fp of genus g ≥ 1, and let J := Pic0 (C) be its Jacobian . Let φ : C → J be the embedding of C in J associated to a point x0 ∈ C(Fp ). For any integer m denote by m J(Fp ) the m-primary part of the torsion group J(Fp ) (i .e ., m J(Fp ) := ⊕l (l J(Fp )) the


113

sum being taken over all primes ` dividing m), and let λ : J(Fp ) →m J(Fp ) be the canonical projection . Then the map λ ◦ φ : C(Fp ) →m J(Fp ) is surjective . This was proved in [An-In] only in the case where m = l is a prime number, but it is easy to check that the proof there works also in the case of any positive integer m. It follows immediately from the above result that C(Fp ) contains innitely many points which have pairwise prime orders, in particular it contains innitely many points of order prime to p. Indeed, If {x1 , . . . , xn } are nitely many points of C(Fp ), r is the least common multiple of the orders of the points {x1 , . . . , xn }, and if s > 1 is an integer which is relatively prime to r, and x 6= 0 is an s-torsion point on J , then by the above result one can nd a point on C(Fp ) whose s primary part equal x and whose r-primary part equals 0, in particular such a point has an order which is prime to r. ¤ For the proof of 2.1, let y ∈ Y (Fp ) be a closed point in Y and let C be an irreducible sub-scheme of Y of dimension 1 which contains y . We endow C with its reduced structure. Let C˜ be the normalization of C which is a smooth and connected curve of genus ≥ 1, and let J˜ be its Jacobian. One has a commutative diagram: f J˜ −−−−→ A

x 

x 

˜ φ

i ˜

f C˜ −−−−→ C where f˜ is the normalization morphism, φ˜ is the embedding of C˜ in its Jacobian associated to a point y˜ above y , and f is the morphism induced by the universal property of J˜, which is a composition of a homomorphism g and a translation τy by the point y . If y is a point of order prime to p then the image via f of ˜ C) ˜ (which exists and are an innity by the the points of order prime to p on φ( above result) yields innitely many points in C(Fp ) which have pairwise prime orders. Moreover if 0A ∈ Y (Fp ) and one takes y = 0A , then with the same ˜ C) ˜ having pairwise prime notations as above, the images via f of the points of φ( orders yield innitely many points on C having pairwise prime orders. Assume now that A is a simple abelian variety. Then the above homomorphism g is necessarily surjective, in particular there exists x in J˜ such that g(x) = y , and ˜ , where τx denotes the translation by x inside J˜. On the other C = g(τx (C)) hand it is easy to see, using the above result in [An-In] in the same way that was ˜ also contains innitely many points which have pairwise used above, that τx (C) prime orders in J˜ hence the result in this case.

With the same hypothesis as in Proposition 2.1 let Yi be an irreducible component of Y which has dimension ≥ 1, and denote by 0 A(Fp )(p ) the prime to p-part of the torsion group A(Fp ). Then :

Proposition/Definition 2.2.

0

(1) either Yi (Fp ) ∩ A(Fp )(p ) is Zariski dense in Yi , in which case we call Yi an abelian like sub-variety of A, or

114

FLORIAN POP AND MOHAMED SAïDI 0

(2) Yi (Fp ) ∩ A(Fp )(p ) is empty in which case Yi must be a translate of an abelian like sub-variety of A by a point which necessarily has order divisible by p. After eventually a translation we can assume that Yi contains the zero 0 point of A and then we can assume by 2.1 that Yi (Fp ) ∩ A(Fp )(p ) is non empty. 0 Assume that the closure Zi of Yi (Fp ) ∩ A(Fp )(p ) is distinct from Yi . Let x be a 0 point in Yi (Fp ) ∩ A(Fp )(p ) and y ∈ Yi (Fp ), but y is not contained in Zi . Then one can nd a curve C which contains both y and x (see [Mu], lemma on p. 56). It follows then from the same argument used in the proof of 2.1 that C contains innitely many points of order prime to p, hence Yi − Zi contains such a point which contradicts the fact that Zi 6= Yi . ¤ Proof.

Here is an immediate consequence of these propositions:

Let X be a proper smooth and connected curve over Fp . Let θX be the theta divisor associated to the sheaf BX of locally exact dierentials on X (see Section 1.1). Assume : either the Jacobian J of X is a simple abelian variety , or that the curve X is not ordinary which is equivalent to the fact that 0 ∈ θX (Fp ). Then θX (Fp ) contains innitely many torsion points of the Jacobian J 1 of X 1 having pairwise prime orders . In general , if θX (Fp ) contains a torsion point of order prime to p, then θX (Fp ) contains innitely many torsion points of order prime to p. In both cases θX has an irreducible component which is an abelian like sub-variety of J 1 . Proposition 2.3.

In the general case where the conditions of 2.3 are not satised one has the following.

Let X be a proper smooth and connected curve over Fp . Then there exists an étale Galois cover Y → X with Galois group G of order prime to p such that the theta divisor θY associated to the sheaf of locally exact dierentials on Y contains innitely many Fp -torsion points of pairwise prime order . Proposition 2.4.

By a result of Raynaud (see [Ra-2]) there exists an étale Galois cover Y → X with Galois group G of order prime to p such that Y is not ordinary. In particular the theta divisor θY associated to the sheaf of locally exact dierentials on Y contains the zero point of JY1 . Hence the result follows from 2.3. ¤ Proof.

3. On the Theta Divisor θ of Curves with Simple Jacobians Let A be a simple abelian variety of dimension g ≥ 2 over an algebraically closed eld K of characteristic p > 0. Assume that A is not dened over a nite eld , and that the p-rank of A equals g or g − 1. Let D be a closed sub-variety of codimension ≥ 1 of A. Then D(K) contains at most nitely many torsion points of A(K)tor of order prime to p.

Theorem 3.1.

Since A is simple, any K -homomorphism from A to an abelian variety is either trivial or an isogeny. In particular, the Fp -trace of A is either trivial

Proof.


115

or isogenous to A in which case the kernel of such an isogeny is automatically dened over a Fp because of the condition on the p-rank of A (see [Oo], 3.4), hence the Fp -trace of A equals 0 necessarily, since A is not dened over a nite 0 eld by assumption. Let D(p ) be the closure in A of the intersection of D with the prime to p-part of the torsion group J(K)tor . By the results of Hrushovski on the analog of the Mordell-Lang conjecture over function elds in positive 0 characteristic D(p ) is a nite union ∪i (ai + Ai ) of translates of abelian subvarieties Ai of A (see [Hr], Corollary 1.2). As A is simple, dim Ai = 0 and hence 0 D(p ) consists of at most nitely many points. ¤

Let X be a proper smooth and connected curve of genus g ≥ 2 over an algebraically closed eld K of characteristic p > 0. Assume that X is not dened over a nite eld . Let θX be the theta divisor associated to the sheaf of locally exact dierentials BX on X 1 (see Section 1.1). Assume that the Jacobian J of X is a simple abelian variety and that the p-rank of X equals g or g − 1. Then θX (K) contains at most nitely many torsion points of order prime to p. Corollary 3.2.

Since X is not dened over a nite eld this is also the case for its Jacobian J By Torelli's theorem [We]; hence 3.2 follows from 3.1. ¤ Proof.

4. Proof of Theorem A, Theorem B, and Corollary We reformulate the assertions of the theorems as follows: Let x, y be points of Mg with x a specialization of y . Thus the local ring OMg ,x of the point x contains a prime ideal Py corresponding to y , and OMg ,y is the localization of OMg ,x at Py . Let K be an algebraic closure of κ(y). Then there exits a valuation ring R of K dominating the factor ring OMg ,x /Py inside κ(y) ⊂ K , such that the residue eld of R is an algebraic closure κ of κ(x). Thus y = Spec K is the generic point, and x ¯ = Spec κ is the closed point of Spec R. We choose a smooth projective curve f : X → Spec R so that we have a morphism g : Spec R → Mg such that the induced morphisms y¯ → Mg and x ¯ → Mg dene the generic ber Xy¯ → Spec K , respectively the special ber Xx¯ → Spec κ as points in Mg (K), respectively Mg (κ). We can identify π1 (Xκ ) with π1 (x), and π1 (XK ) with π1 (y) respectively, in such a way that the Grothendieck's specialization homomorphism π1 (XK ) → π1 (Xκ ) is exactly the specialization homomorphism Sp : π1 (y) → π1 (x). Now we suppose that the points y and x are of a special nature, as in Theorem A and/or Theorem B. This means in particular, that y might be the generic a.s. . Assuming that Spy,x point η of Mg , and x is a point s in S a.s. or S≥g−1 is an isomorphism, we will get a contradiction by showing that the morphism g : Spec R → Mg is constant.

116


Concerning Theorem A. In the above notations, let x = s and y = η , thus κ is an algebraic closure of the nite eld κ(s), and K is the algebraic closure of κ(η). We denote by Js = Jκ the Jacobian of Xs := Xκ , respectively by Jη = JK the Jacobian of Xη := XK . Further let θs , respectively θη be the theta divisor in (Js )1 associated to the sheaf of locally exact dierentials on Xs , respectively the theta divisor in (Jη¯)1 associated to the sheaf of locally exact dierentials on Xη . It follows from 2.3 that θs contains innitely many torsion points of order prime to p. Let L be an invertible sheaf of order n prime to p on X → S . Let Lη , respectively Ls be the restriction of L to Xη , respectively its restriction to Xs . The assumption that Sp : π1 (Xη ) → π1 (Xs ) is an isomorphism implies in particular: The µn -torsor associated to Lη is new-ordinary (in the sense of 1.3) if and only if the µn -torsor associated to Ls is new ordinary. In other words: The subgroup hL1η i generated by L1η intersects the theta divisor θη at a non zero point if and only if the subgroup hL1s i generated by L1s intersects the theta divisor θs at a non zero point. Hence we deduce from Proposition 2.3, it follows that θη contains innitely many torsion points of Jη of order prime to p. On the other hand, it is well known that all cyclic étale covers Y → Xη of degree n prime to p (and even without this condition) are new-ordinary (see [Na], for instance). This means that the theta divisor θη contains no torsion point of order prime to p. Thus a contradiction in this case. Concerning Theorem B. One proceeds as above, but without using the

assumption that y is the generic point of Mg . In the above notations we then have: Let J → Spec R be the Jacobian of the projective smooth curve X → Spec R. Thus J → Spec R is an abelian scheme over Spec R, and Js = J ×R κ is the special ber of J , and Jy = J ×R K is the generic ber of J . Since Js is a simple abelian variety (by the hypothesis on s), it follows that its generic ber Jy is simple too. Since f is non iso-trivial, it follows that Xy := XK is not dened over a nite eld. Hence Corollary 3.2 implies that the theta-divisor θy of Xy1 is such that θy (λ) contains at most nitely many torsion points of order prime to p. This is a contradiction, so Sp cannot be an isomorphism in this case. a.s. We next prove the second assertion of Theorem B. Let x ∈ S≥g−1 be a closed point of Mg . By contradiction, suppose that there exists innitely many points a.s. x0 ∈ S≥g−1 such that π1 (x) ' π1 (x0 ). Let Sx denote the subset of those points, and S x be the closure of Sx in Mg . Then S x is a closed sub-scheme of Mg of dimension d ≥ 1. Let z be a point of S x which is not a closed point. By hypothesis there exists a point x0 ∈ Sx such that z specializes in x0 , and hence there exists a continuous surjective homomorphism Sp : π1 (z) → π1 (x0 ). In particular one has an inclusion of sets πA (x0 ) ⊂ πA (z). On the other hand it is well known that every nite group G ∈ πA (z) belongs to πA in an open neighborhood of z (see [St]), and as each such a neighborhood contains a point of Sx one deduces in fact that one has an equality πA (x0 ) = πA (z), and the above homomorphism Sp : π1 (z) → π1 (x0 ) is an isomorphism (this follows from the


117

Hopan property for nitely generated pronite groups; see [Fr-Ja], Prop. 15.4). a.s. But this can not be the case by the rst half of Theorem B since x0 ∈ S≥g−1 .

Concerning the Introduction's Corollary. We nally come to the proof of

the Corollary. First, the fact that the subset S a.s. of closed points with absolutely simple Jacobian has positive Dirichlet density implies in particular that S a.s. ∩ U is dense in U for every open (nonempty) subset U of Mg (see [Se-2]). Further, since the Jacobian of the generic curve Cη is ordinary, it follows that every curve Cx with π1 (x) ∼ = π1 (η) is ordinary too. Thus we have: If π1 is constant on U , a.s. then S a.s. ∩ U = S≥g ∩ U is dense in U , in particular innite. This in turn is a contradiction by the second part of Theorem B.

References [An-In] G. W. Anderson and R. Indik, On primes of degree one in function elds, Proc. Amer. Math. Soc. 94 (1985), 3132. [Bo-Lu-Ra] S. Bosch, W. Lütkebohmert and M. Raynaud, Néron Models, Ergebnisse der Mathematik (3. Folge) 21, Springer, Berlin, 1990. [Ch-Oo] C. L. Chai and F. Oort, A note on the existence of absolutely simple jacobians, Jour. Pure Appl. Alg. 155:23 (2001), 115120. [D-M] P. Deligne and D. Mumford, The irreducibility of the space of curves with a given genus, Publ. Math. IHES 36 (1969), 75110. [EGA] A. Grothendieck and J. Dieudonné, Élements de la géométrie algébrique, Chap. 2, Pub. Math. IHES 8 (1961). [Fr-Ja] M. Fried and M. Jarden, Field Arithmetic, Ergebnisse der Mathematik (3. Folge) 11, Springer, Berlin 1986. [Hr] E. Hrushovski, The Mordell-Lang conjecture for function elds, J. Amer. Math. Soc. 9:3 (1996), 667690. [Mu] D. Mumford, Abelian Varieties, Oxford Univ. Press, Oxford, 1970. [Na] S. Nakajima, On generalized HasseWitt invariants of an algebraic curve, Adv. Stud. Pure Math. 12 (1987), 6988. [Oo] F. Oort, The isogeny class of a CM-type abelian variety is dened over a nite extension of the prime eld, J. Pure Appl. Alg. 3 (1973), 399408. [Ra-1] M. Raynaud, Section des brés vectoriel sur une courbe, Bull. Soc. Math. France, 110 (1982), 103125. [Ra-2] M. Raynaud, Revêtements des courbes en caractéristique p > 0 et ordinarité, Comp. Mathematica 123 (2000), 7388. [SGA-1] A. Grothendieck, Revêtements étales et groupe fondamental, Lecture Notes in Math. 224, Springer, Berlin, 1971. [Se-1] J.-P. Serre, Sur la topologie des variétés algébriques en caractériqstique p > 0, pp. 2453 in Symposium Internacional de Topologia Algebraica (Mexico, 1958), Universidad Nacional Autonoma de Mexico and UNESCO, Mexico City, 1958. [Se-2] J.-P. Serre, Zeta and L-functions, pp. 8292 in Arithmetical Algebraic geometry, Harper and Row, New York, 1965.

118


[Sh] I. Shafarevich, On p-extensions, Mat. Sb. Nov. Ser. 20 (62) (1947), 351363; AMS Transl. Series 2, 4 (1965), 5972. [St] K. Stevenson, Quotients of the fundamental group of algebraic curves in positive characteristics, J. Alg. 182 (1996), 770804. [We] A. Weil, Zum Beweis des Torellischen Satzes, in Collected papers, Volume II, (19511964). Florian Pop Mathematisches Institut Universität Bonn Beringstraÿe 6 53115 Bonn Germany

[email protected]

Mohamed Saïdi Department of Mathematical Sciences University of Durham Science Laboratories South Road Durham DH1 3LE United Kingdom

[email protected]


Topics Surrounding the Anabelian Geometry of Hyperbolic Curves SHINICHI MOCHIZUKI

Contents

Introduction 1. The Tate Conjecture as a Sort of Grothendieck Conjecture 1.1. The Tate conjecture for non-CM elliptic curves 1.2. Some pro-p group theory 2. Hyperbolic Curves As Their Own Anabelian Albanese Varieties 2.1. A corollary of the Main Theorem of [Mzk2] 2.2. A partial generalization to nite characteristic 3. Discrete Real Anabelian Geometry 3.1. Real complex manifolds 3.2. Fixed points of antiholomorphic involutions 3.3. Hyperbolic curves and their moduli 3.4. Abelian varieties and their moduli 3.5. Pronite real anabelian geometry 4. Complements to the p-adic Theory 4.1. Good Chern classes 4.2. The group-theoreticity of a certain Chern class 4.3. A generalization of the main result of [Mzk2] References

119 122 122 126 128 128 129 132 132 137 139 140 141 147 147 152 157 163

Introduction We give an exposition of various ideas and results related to the fundamental results of [Tama1-2], [Mzk1-2] concerning Grothendieck's Conjecture of Anabelian Geometry (which we refer to as the Grothendieck Conjecture for short; see [Mzk2], Introduction, for a brief introduction to this conjecture). Many of these ideas existed prior to the publication of [Tama1-2], [Mzk1-2], but were not discussed in these papers because of their rather elementary nature and secondary importance (by comparison to the main results of these papers). Nevertheless, it is the hope of the author that the reader will nd this article useful as a supplement to [Tama1-2], [Mzk1-2]. In particular, we hope that the discussion of this 119

120

SHINICHI MOCHIZUKI

article will serve to clarify the meaning and motivation behind the main result of [Mzk2]. Our main results are the following: (1) In Section 1, we take the reverse point of view to the usual one (i.e., that the Grothendieck Conjecture should be regarded as a sort of (anabelian) Tate Conjecture) and show that in a certain case, the Tate Conjecture may be regarded as a sort of Grothendieck Conjecture (see Theorem 1.1, Corollary 1.2). In particular, Corollary 1.2 is interesting in that it allows one to express the fundamental phenomenon involved in the Tate and Grothendieck Conjectures using elementary language that can, in principle, be understood even by high school students (see the Introduction to Section 1; the Remarks following Corollary 1.2). (2) In Section 2, we show how the main result of [Mzk2] gives rise to a purely algebro-geometric corollary (i.e., one which has nothing to do with Galois groups, arithmetic considerations, etc.) in characteristic 0 (see Corollary 2.1). Moreover, we give a partial generalization of this result to positive characteristic (see Theorem 2.2). (3) In Section 3, we discuss real analogues of anabelian geometry. Not surprisingly, the real case is substantially easier than the case where the base eld is p-adic or a number eld. Thus, we are able to prove much stronger results in the real case than in the p-adic or number eld cases (see Theorem 3.6, Corollaries 3.7, 3.8, 3.10, 3.11, 3.13, 3.14, 3.15). In particular, we are able to prove various real analogues of the so-called Section Conjecture of anabelian geometry (which has not been proven, at the time of writing, for any varieties over p-adic or number elds) see [Groth], p. 289, (2); [NTM], § 1.2, (GC3), for a discussion of the Section Conjecture. Also, we note that the real case is interesting relative to the analogy between the dierential geometry that occurs in the real case and certain aspects of the p-adic case (see [Mzk4], Introduction, § 0.10; the Introduction to Section 3 of this article). It was this analogy that led the author to the proof of the main result of [Mzk2]. (4) In Section 4, we show that a certain isomorphism version (see Theorem 4.12) of the main result of [Mzk2] can be proven over generalized sub-p-adic elds (see Denition 4.11), which form a somewhat larger class of elds than the class of sub-p-adic elds dealt with in [Mzk2]. This result is interesting in that it is reminiscent of the main results of [Tama2], as well as of the rigidity theorem of MostowPrasad for hyperbolic manifolds of real dimension 3 (see the Remarks following the proof of Theorem 4.12). Although we believe the results of Section 4 to be essentially new, we make no claim of essential originality relative to the results of Sections 13, which may be proven using well-known standard techniques. Nevertheless, we believe that it is likely that, even with respect to Sections 13, the point of view of the discussion

ANABELIAN GEOMETRY OF HYPERBOLIC CURVES

121

is likely to be new (and of interest relative to understanding the main result of [Mzk2]). Finally, before beginning our exposition, we pause to review the main result of [Mzk2] (which is the central result to which the ideas of the present article are related). To do this, we must introduce some notation. Let Σ be a nonempty set of prime numbers. If K is a eld, denote its absolute Galois group Gal(K/K) (where K is some algebraic closure of K ) by ΓK . If X is a geometrically connected K -scheme, recall that its algebraic fundamental group π1 (X) (for some choice of base-point) ts into a natural exact sequence 1 → π1 (X ⊗K K) → π(X) → ΓK → 1. Denote by ∆X the maximal pro-Σ quotient of π1 (X ⊗K K) (i.e., the inverse limit of those nite quotients whose orders are products of primes contained in Σ). The pronite group ∆X is often referred to as the (pro-Σ) geometric fundamental group of X . Note that since the kernel of π1 (X ⊗K K) → ∆X is a characteristic subgroup of π1 (X ⊗K K), it follows that it is normal inside π1 (X). Denote the quotient of π1 (X) by this normal subgroup by ΠX . The pronite group ΠX is often referred to as the (pro-Σ) arithmetic fundamental group of X . (When it is Σ necessary to specify the set of primes Σ, we will write ∆Σ X , ΠX .) Thus, we have a natural exact sequence

1 → ∆X → ΠX → ΓK → 1. In [Mzk2] we proved the following result:

Let K be a sub-p-adic eld (i .e ., a eld isomorphic to a subeld of a nitely generated eld extension of Q p ), where p ∈ Σ. Let XK be a smooth variety over K , and YK a hyperbolic curve over K . Let Homdom K (XK , YK ) be the set of dominant K -morphisms from XK to YK . Let Homopen ΓK (ΠX , ΠY ) be the set of open , continuous group homomorphisms ΠX → ΠY over ΓK , considered up to composition with an inner automorphism arising from ∆Y . Then the natural map Theorem A.

open Homdom K (XK , YK ) → HomΓK (ΠX , ΠY )

is bijective . Theorem A as stated above is a formal consequence of Theorem A of [Mzk2]. In [Mzk2], only the cases of Σ = {p}, and Σ equal to the set of all prime numbers are discussed, but it is easy to see that the case of arbitrary Σ containing p may be derived from the case Σ = {p} by precisely the same argument as that used in [Mzk2] (see [Mzk2], the Remark following Theorem 16.5) to derive the case of Σ equal to the set of all prime numbers from the case of Σ = {p}. Remark.

122

SHINICHI MOCHIZUKI

Acknowledgements. The author would like to thank A. Tamagawa and T.

Tsuji for stimulating discussions of the various topics presented in this article. In particular, the author would especially like to express his gratitude A. Tamagawa for discussions concerning removing the hypothesis of monodromy type from Theorem 4.12 in an earlier version of this article. Also, the author greatly appreciates the advice given to him by Y. Ihara (orally) concerning the Remark following Corollary 1.2, and by M. Seppala and C. McMullen (by email) concerning the Teichmüller theory used in Section 3.

1. The Tate Conjecture as a Sort of Grothendieck Conjecture In this section, we attempt to present what might be referred to as the most fundamental prototype result among the family of results (including the Tate and Grothendieck Conjectures) that states that maps between varieties are essentially equivalent to maps between arithmetic fundamental groups. The result given below, especially in the form Corollary 1.2, is interesting in that it allows one to express the fundamental phenomenon involved using elementary language that can, in principle, be understood even by high school students (see the Remark following Corollary 1.2). In particular, it does not require a knowledge of the notion of a Galois group or any another advanced notions, hence provides a convincing example of how advanced mathematics can be applied to prove results which can be stated in simple terms. Also, it may be useful for explaining to mathematicians in other elds (who may not be familiar with Galois groups or other notions used in arithmetic geometry) the essence of the Tate and Grothendieck Conjectures. Another interesting feature of Corollary 1.2 is that it shows how the Tate conjecture may be thought of as being of the same genre as the Grothendieck Conjecture in that it expresses how the isomorphism class of a curve (in this case, an elliptic curve) may be recovered from Galois-theoretic information.

1.1. The Tate conjecture for non-CM elliptic curves. Let K be a number eld (i.e., a nite extension of Q ). If E is an elliptic curve over K , and N is a natural number, write K(E[N ]) for the minimal nite extension eld of K over which all of the N -torsion points are dened. Note that the extension K(E[N ]) will always be Galois. Then we have the following elementary consequence of the Tate Conjecture for abelian varieties over number elds proven in [Falt]:

Let K be a number eld . Let E1 and E2 be elliptic curves over K such that neither E1 nor E2 admits complex multiplication over Q . Then E1 and E2 are isomorphic as elliptic curves over K if and only if K(E1 [N ]) = K(E2 [N ]) for all natural numbers N . Theorem 1.1.


123

Remark. The equality K(E1 [N ]) = K(E2 [N ]) is to be understood in the sense of subelds of some xed algebraic closure of K . The substance of this expression is independent of the choice of algebraic closure precisely because both elds in question are Galois extensions of K .

If E1 ∼ = E2 over K , then it is clear that K(E1 [N ]) = K(E2 [N ]) for all natural numbers N . Thus, assume that K(E1 [N ]) = K(E2 [N ]) for all natural numbers N , and prove that E1 ∼ = E2 over K . In this proof, we use the notation and results of Section 1.2 below. Since we assume that K(E1 [N ]) = K(E2 [N ]), we denote this eld by K[N ]. Also, if p is a prime number, then we write K[p∞ ] for the union of the K[pn ], as n ranges over the positive integers. Finally, for n ≥ 0, we denote the Galois group Gal(K[p∞ ]/K[pn ]) by Γ[pn ]; the center of Γ[pn ] by ZΓ[pn ]; and the quotient Γ[pn ]/ZΓ[pn ] by P Γ[pn ]. Let p be a prime number. Then by the semisimplicity of the Tate module, together with the Tate conjecture (both proven in general in [Falt]; see also [Ser2], IV), the fact that neither E1 nor E2 admits complex multiplication over Q implies that there exists an integer n ≥ 1 such that the Galois representation on the p-power torsion points of E1 (respectively, E2 ) induces an isomorphism β1 : [n] [n] [n] Γ[pn ] ∼ = GL2 (Z p )), where GL2 (Z p ) ⊆ = GL2 (Z p ) (respectively, β2 : Γ[pn ] ∼ GL2 (Z p ) is the subgroup of matrices that are ≡ 1 modulo pn . Since the kernel [n] [n] [n] of GL2 (Z p ) → PGL2 (Z p ) is easily seen to be equal to the center of GL2 (Z p ), it thus follows that β1 , β2 induce isomorphisms Proof.

[n] α1 : P Γ[pn ] ∼ = PGL2 (Z p ),

[n] α2 : P Γ[pn ] ∼ = PGL2 (Z p ).

Thus, in particular, by Lemma 1.3 of Section 1.2 below, we obtain that (after def [n] possibly increasing n) the automorphism α = α1 ◦ α2−1 of PGL2 (Z p ) is dened by conjugation by an element of PGL2 (Z p ). In particular, we obtain that there exists a Z p -linear isomorphism

ψ : Tp (E1 ) ∼ = Tp (E2 ) between the p-adic Tate modules of E1 and E2 with the property that for σ ∈ Γ[pn ], we have ψ(σ(t)) = λσ σ(ψ(t)) (∀t ∈ Tp (E1 )), for some λσ ∈ Z p × which is independent of t. On the other hand, since the determinant of ψ is clearly compatible with the Galois actions on both sides (given by the cyclotomic character), it thus follows (by taking determinants of both sides of the equation ψ(σ(t)) = λσ σ(ψ(t))) that λ2σ = 1. Since the correspondence σ 7→ λσ is clearly a homomorphism (hence a character of order 2), we conclude:

(∗) There exists a nite extension K 0 of K over which the Gal(K/K 0 )modules Tp (E1 ) and Tp (E2 ) become isomorphic . (Here, K 0 is the extension of K[pn ] (of degree ≤ 2) dened by the kernel of σ 7→ λσ . In fact, if p > 2, then this extension is trivial (since Γ[pn ] is a pro-p-group).) Thus, by the Tate Conjecture proven in [Falt], we obtain that HomK 0 (E1 , E2 )⊗Z

124

SHINICHI MOCHIZUKI

Z p contains an element that induces an isomorphism on p-adic Tate modules. On def the other hand, since HK 0 = HomK 0 (E1 , E2 ) (the module of homomorphisms (E1 )K 0 → (E2 )K 0 over K 0 ) is a nitely generated free Z -module of rank ≤ 1 (since E1 , E2 do not have complex multiplication over Q ), we thus obtain that HK 0 is a free Z -module of rank 1. Let ε ∈ HK 0 be a generator of HK 0 . Then ε necessarily corresponds to an isogeny E1 → E2 that induces an isomorphism on p-power torsion points. def Now write HK = HomK (E1 , E2 ). Then the above argument shows that HK is a free Z -module of rank 1 with a generator ε that induces an isomorphism on p-power torsion points for every prime number p. But this implies that ε : (E1 )K → (E2 )K is an isomorphism, i.e., that E1 and E2 become isomorphic over K . Thus, it remains to check that E1 and E2 are, in fact, isomorphic over K . Let p ≥ 5 be a prime number which is suciently large that: (i) K is absolutely unramied at p; (ii) the Galois representations on the p-power torsion points of E1 and E2 induce isomorphisms β1 : Γ[p0 ] ∼ = GL2 (Z p );

β2 : Γ[p0 ] ∼ = GL2 (Z p )

(the existence of such p follows from the modulo l versions (for large l) of the semisimplicity of the Tate module, together with the Tate conjecture in [Mord], VIII, § 5 ; see also [Ser2], IV). Now we would like to consider the extent to which def the automorphism β = β1 ◦ β2−1 of GL2 (Z p ) is dened by conjugation by an element of GL2 (Z p ). Note that by what we did above, we know that the mor[n] phism induced by β on PGL2 (Z p ) (for some large n) is given by conjugation by some element A ∈ GL2 (Z p ). Let γ : GL2 (Z p ) → GL2 (Z p ) be the automorphism of GL2 (Z p ) obtained by composing β with the automorphism given by conju[n] gation by A−1 . Thus, γ induces the identity on PGL2 (Z p ). But this implies (by Lemma 1.4 below) that γ induces the identity on PGL2 (Z p ). In particular, it follows that there exists a homomorphism λ : GL2 (Z p ) → Z p × such that γ(σ) = λ(σ) · σ (∀σ ∈ GL2 (Z p )). Next, recall that since p ≥ 5, the topological group SL2 (Z p ) has no abelian quotients (an easy exercise). Thus, λ factors through the determinant map GL2 (Z p ) → Z p × . Moreover, (see the argument at the beginning of the proof involving arbitrary p) since the composites of β1 , β2 with the determinant map are given by the cyclotomic character, we obtain that λ2 = 1. In particular, we obtain that λ is trivial on the index 2 subgroup of GL2 (Z p ) of elements whose determinant is a square. Put another way, if we write Kp for the quadratic extension of K determined by composing the cyclotomic character Gal(K/K) → Z p × (which is surjective since K is absolutely unramied at p) with the unique surjection Z p × ³ Z/2Z , then over Kp , the Tate modules Tp (E1 ), Tp (E2 ) become isomorphic as Galois modules, which implies that HomKp (E1 , E2 ) 6= 0. But this implies that E1 and E2 become isomorphic over Kp .


125

On the other hand, for distinct primes p, p0 as above, Kp , Kp0 form linearly disjoint quadratic extensions of K (as can be seen by considering the ramication at p, p0 ). Thus, the fact that both Gal(K/Kp ) and Gal(K/Kp0 ) act trivially on HomK (E1 , E2 ) implies that Gal(K/K) acts trivially on HomK (E1 , E2 ), so ¤ E1 ∼ = E2 over K , as desired. Remark.

The above proof beneted from discussions with A. Tamagawa and

T. Tsuji. In the preceding proof (see also the arguments of Section 1.2 below), we use in an essential way the strong rigidity properties of the simple p-adic Lie group PGL2 (Z p ). Such rigidity properties are not shared by abelian Lie groups such as Z p ; this is why it was necessary to assume in Theorem 1.1 that the elliptic curves in question do not admit complex multiplication. Remark.

There is a nite set CM ⊆ Z such that if E1 and E2 are arbitrary elliptic curves over Q whose j -invariants j(E1 ), j(E2 ) do not belong to CM, then E1 and E2 are isomorphic as elliptic curves over Q if and only if Q(E1 [N ]) = Q(E2 [N ]) for all natural numbers N . Corollary 1.2.

In light of Theorem 1.1, it suces to show that there are only nitely many possibilities (all of which are integral see, e.g., [Shi], p. 108, Theorem 4.4) for the j -invariant of an elliptic curve over Q which has complex multiplication over Q . But this follows from the niteness of the number of imaginary quadratic extensions of Q with class number one (see, e.g., [Stk]), together with the theory of [Shi] (see [Shi], p. 123, Theorem 5.7, (i), (ii)). (Note that we also use here the elementary facts that: (i) the class group of any order surjects onto the class group of the maximal order; (ii) in a given imaginary quadratic extension of Q , there are only nitely many orders with trivial class group.) ¤ Proof.

According to an (apparently) unpublished manuscript of J.-P. Serre ([Ser3]) whose existence was made known to the author by Y. Ihara, the set CM of Corollary 1.2, i.e., the list of rational j -invariants of elliptic curves with complex multiplication, is as follows: Remark.

d = 1, f = 1 =⇒ j = j(i) = 26 · 33 d = 1, f = 2 =⇒ j = j(2i) = (2 · 3 · 11)3 √ d = 2, f = 1 =⇒ j = j( −2) = (22 · 5)3 d = 3, f = 1 =⇒ j =

√ j( −1+2 −3 )

√

([Weber], p. 721)

=0

d = 3, f = 2 =⇒ j = j( −3) = 24 · 33 · 53 d = 3, f = 3 =⇒ j =

√ j( −1+32 −3 ) = −3 √ j( −1+2 −7 ) = −33

([Weber], p. 721) 15

·2

· 53

([Weber], p. 462)

· 53

([Weber], p. 460)

d = 7, f = 2 =⇒ j = j( −7) = (3 · 5 · 17)3

([Weber], p. 475)

d = 7, f = 1 =⇒ j = d = 11, f = 1 =⇒ j =

√

√ j( −1+2 −11 )

= −215

([Weber], p. 462)

126

SHINICHI MOCHIZUKI

d = 19, f = 1 =⇒ j = −(25 · 3)3 6

d = 43, f = 1 =⇒ j = −(2 · 3 · 5)

([Weber], p. 462) 3

([Weber], p. 462)

5

d = 67, f = 1 =⇒ j = −(2 · 3 · 5 · 11)3

([Weber], p. 462)

6

d = 163, f = 1 =⇒ j = −(2 · 3 · 5 · 23 · 29)3 ([Weber], p. 462) √ Here Q( −d) is the imaginary quadratic extension of Q containing the order in question, f is the conductor of the order, and the reference given in parentheses is for the values of the invariants f and f1 of [Weber], which are related to the j -invariant as follows: j = (f 24 − 16)3 /f 24 = (f124 + 16)3 /f124 . Thus, if one denes elliptic curves over Q using cubic equations, constructs the group law on elliptic curves by considering the intersection of the cubic with various lines, and interprets the notion of isomorphism of elliptic curves (over Q ) to mean being dened by the same cubic equation, up to coordinate transformations, then Corollary 1.2 may be expressed as follows: Remark.

Except for the case of nitely many exceptional j -invariants , two elliptic curves E1 , E2 over Q are isomorphic if and only if for each natural number N , the coordinates (∈ C ) necessary to dene the N -torsion points of E1 generate the same subeld of C i .e ., collection of complex numbers closed under addition , subtraction , multiplication , and division as the coordinates necessary to dene the N -torsion points of E2 . (Here, of course, the j -invariant is dened as a polynomial in the coecients of the cubic.) In this form, the essential phenomenon at issue in the Tate or Grothendieck Conjectures may be understood even by high school students or mathematicians unfamiliar with Galois theory.

1.2. Some pro-p group theory. Let n ≥ 1 be an integer. In this section

we denote by PGL2 the algebraic group (dened over Z ) obtained by forming the quotient of GL2 by Gm (where Gm ,→ GL2 is the standard embedding by scalars), and by [n] PGL2 (Z p ) ⊆ PGL2 (Z p )

the subgroup of elements which are ≡ 1 modulo pn . Write pgl2 (Z p ) for the quotient of the Lie algebra M2 (Z p ) (of 2 by 2 matrices with Z p coecients) by the scalars Z p ⊆ M2 (Z p ). Thus, pgl2 (Z p ) ⊆ pgl2 (Z p ) ⊗Z p Q p = pgl2 (Q p ). [n] Write pgl2 (Z p ) ⊆ pgl2 (Z p ) for the submodule which is the image of matrices [n] in M2 (Z p ) which are ≡ 0 modulo pn . Thus, for n suciently large, pgl2 (Z p ) [n] maps bijectively onto PGL2 (Z p ) via the exponential map (see [Ser1], Chapter V, § 7). [n]

[n]

Let α : PGL2 (Z p ) → PGL2 (Z p ) be an automorphism of the [n] [m] [m] pronite topological group PGL2 (Z p ) such that α(PGL2 (Z p )) = PGL2 (Z p ) for all m ≥ n. Then there exists an element A ∈ PGL2 (Z p ) such that for some m ≥ n, the restriction α|PGL[m] (Z p ) is given by conjugation by A. Lemma 1.3.

2

ANABELIAN GEOMETRY OF HYPERBOLIC CURVES Proof.

127

Write

A : pgl2 (Q p ) → pgl2 (Q p ) for the morphism on Lie algebras induced by α. By [Ser1], Chapter V, § 7, 9, after possibly replacing n by a larger n, we may assume that α is the homomorphism obtained by exponentiating A. Moreover, by the well-known theory of the Lie algebra pgl2 (Q p ), it follows that A may be obtained by conjugating by some A0 ∈ PGL2 (Q p ). (Indeed, this may be proven by noting that A induces an automorphism of the variety of Borel subalgebras of pgl2 (Q p ). Since this variety is simply P 1Q p , we thus get an automorphism of P 1Q p , hence an element of PGL2 (Q p ), as desired.) On the other hand, it follows immediately from the structure theory of nitely generated Z p -modules that A0 may be written as a product A0 = C1 · A00 · C2 , where C1 , C2 ∈ PGL2 (Z p ), and A00 is dened by a matrix of the form µ ¶ λ1 0 , 0 λ2 where λ1 , λ2 ∈ Q p × . [n] Now observe that the fact that A arises from an automorphism of PGL2 (Z p ) [n] implies that A induces an automorphism of pgl2 (Z p ) (see the discussion at the beginning of this section). Since conjugation by C1 and C2 clearly induces [n] automorphisms of pgl2 (Z p ), it thus follows that conjugation by A00 induces an [n] automorphism of pgl2 (Z p ). Now, by considering, for instance, upper triangular matrices with zeroes along the diagonal, one sees that A00 can only induce an def [n] 0 automorphism of pgl2 (Z p ) if λ1 = λ2 · u, where u ∈ Z p × . Let A = λ−1 1 ·A . Then clearly A ∈ PGL2 (Z p ), and conjugation by A induces A. Thus, by using the exponential map, we obtain that for some m ≥ n, the restriction α|PGL[m] (Z p ) 2 is given by conjugation by A, as desired. ¤ The following lemma was pointed out to the author by A. Tamagawa:

Let α : PGL2 (Z p ) → PGL2 (Z p ) be an automorphism of the pronite topological group PGL2 (Z p ) such that for some integer m ≥ 1, the restriction α|PGL[m] (Z p ) is the identity . Then α itself is the identity . Lemma 1.4.

2

First let us show that α is the identity on the image in PGL2 (Z p ) ¡ ¢ of matrices of the form 10 λ1 , where λ ∈ Z p . For m ≥ 0 an integer, write Um ⊆ PGL2 (Z p ) for the subgroup of images in PGL2 (Z p ) of matrices of the ¡ ¢ form 10 λ1 , where λ ∈ pm · Z p . Since, by hypothesis, α preserves Um for some m, it follows that α preserves the centralizer Z(Um ) of Um in PGL2 (Z p ). On the other hand, one checks easily that Z(Um ) = U0 . Thus, α preserves U0 , i.e., induces an automorphism of the topological group U0 ∼ = Z p which is the identity m on p · Z p . Since Z p is torsion free, it thus follows that α is the identity on U0 ,

Proof.

128

SHINICHI MOCHIZUKI

as desired. Moreover, let us observe that since conjugation commutes with the operation of taking centralizers, one sees immediately that the above argument implies also that α is the identity on all conjugates of U0 in PGL2 (Z p ). Next, observe that α is the identity on the subgroup B ⊆ PGL2 (Z p ) consisting of images of matrices of the form µ ¶ µ1 λ 0 µ2 (where λ ∈ Z p , µ1 , µ2 ∈ Z p × ). Indeed, since B is generated ¡ by ¢ U0 and the subgroup T ⊆ PGL2 (Z p ) of images of matrices of the form µ0 01 , it suces to see that α is the identity on T . But T acts faithfully by conjugation on U0 , and α is the identity on U0 . This implies that α is the identity on T , hence on B . Moreover, as in the previous paragraph, this argument implies that α is the identity on all conjugates of B in PGL2 (Z p ). Since PGL2 (Z p ) is generated by the union of the conjugates of B , it thus follows that α is the identity on PGL2 (Z p ). ¤

2. Hyperbolic Curves As Their Own Anabelian Albanese Varieties In this section, we present an application (Corollary 2.1) of the main theorem of [Mzk2] which is interesting in that it is purely algebro-geometric, i.e., it makes no mention of Galois actions or other arithmetic phenomena.

2.1. A corollary of the Main Theorem of [Mzk2]. We x a nonempty set of prime numbers Σ, and use the notation of the discussion of Theorem A in the Introduction. Now Theorem A has the following immediate consequence: Let K be a eld of characteristic 0. Let C be a hyperbolic curve over K , and let ψ : X → Y be a morphism of (geometrically integral ) smooth varieties over K which induces an isomorphism ∆X ∼ = ∆Y . Write Homdom K (−, C) for the set of dominant K -morphisms from − to C . Then the natural morphism of sets dom Homdom K (Y, C) → HomK (X, C)

Corollary 2.1.

induced by ψ : X → Y is a bijection . By a standard technique involving the use of subelds of K which are nitely generated over Q , we reduce immediately to the case where K is nitely generated over Q . (We recall for the convenience of the reader that the essence of this technique lies in the fact that since we are working with K -schemes of nite type, all schemes and morphisms between schemes are dened by nitely many polynomials with coecients in K , hence may be dened over any subeld of K that contains these coecients of which there are only nitely many!) Next, observe that since the morphism ψ : X → Y induces an isomorphism between the respective geometric fundamental groups, it follows from the exact Proof.


129

sequences reviewed in the Introduction that it induces an isomorphism ΠX ∼ = ΠY . By Theorem A of the Introduction, it thus follows that the morphism of dom sets under consideration i.e., Homdom K (Y, C) → HomK (X, C) is naturally isomorphic to the morphism of sets given by open Homopen ΓK (ΠY , ΠC ) → HomΓK (ΠX , ΠC )

which is bijective.

¤

As stated above, Corollary 2.1 is interesting in that it is a purely algebro-geometric application of Theorem A, i.e., it makes no mention of Galois actions or other arithmetic phenomena. The observation that Corollary 2.1 holds rst arose in discussions between the author and A. Tamagawa. Typical examples of morphisms ψ : X → Y as in Corollary 2.1 are: Remark.

(1) the case where X → Y is a ber bundle in, say, the étale topology, with proper, simply connected bers; (2) the case where Y ⊆ P nk is a closed subvariety of dimension ≥ 3 in some projective space, and X is obtained by intersecting Y with a hyperplane in P nk . In these cases, the fact that the resulting morphism on geometric fundamental groups is an isomorphism follows from the long exact homotopy sequence of a ber bundle in the rst case (see [SGA1], X, Corollary 1.4), and Lefshetztype theorems (see [SGA2], XII, Corollary 3.5) in the second case. Since this consequence of Theorem A (i.e., Corollary 2.1) is purely algebro-geometric, it is natural to ask if one can give a purely algebro-geometric proof of Corollary 2.1. In Section 2.2 below, we give a partial answer to this question.

2.2. A partial generalization to nite characteristic. Let k be an algebraically closed eld. Let C be a proper hyperbolic curve over k . Suppose that we are also given a connected, smooth closed subvariety Y ⊆ P nk of projective space, of dimension ≥ 3, together with a hyperplane H ⊆ P nk such T def that the scheme-theoretic intersection X = H Y is still smooth. Note that X is necessarily connected (see [SGA2], XII, Corollary 3.5) and of dimension ≥ 2. If k is of characteristic p > 0, and S is a k -scheme, then let us write ΦS : S → S for the Frobenius morphism on S (given by raising regular functions on S to the power p). If k is of characteristic 0, then we make the convention that ΦS : S → S denotes the identity morphism. If T is a k -schemes, we dene

HomΦ (T, C) to be the inductive limit of the system

(T, C) → · · · , (T, C) → · · · → Homdom (T, C) → Homdom Homdom k k k

130

SHINICHI MOCHIZUKI

where the arrows are those induced by applying the functor Homdom (−, C) to the k morphism ΦT . Thus, in particular, if k is of characteristic 0, then HomΦ (T, C) = Homdom (T, C). k Now we have the following partial generalization of Corollary 2.1 of Section 2.1 to the case of varieties over a eld of arbitrary characteristic: Theorem 2.2.

Let k , C , X , and Y be as above . Then the natural morphism HomΦ (Y, C) → HomΦ (X, C)

induced by the inclusion X ,→ Y is a bijection . Proof. Denote by AX , AY , and AC the Albanese varieties of X , Y , and C , respectively. We refer to [Lang], Chapter II, § 3, for basic facts concerning Albanese varieties. Thus, the inclusion X ,→ Y induces a morphism AX → AY . I claim that this morphism is a purely inseparable isogeny. Indeed, by various well-known Leftshetz theorem-type results (see, [SGA2], XII, Corollary 3.5), the inclusion X ,→ Y induces an isomorphism π1 (X) ∼ = π1 (Y ); since (by the universal property of the Albanese variety as the minimal abelian variety to which the original variety maps) we have surjections π1 (X) ³ π1 (AX ), π1 (Y ) ³ π1 (AY ), we thus obtain that π1 (AX ) ³ π1 (AY ) is a surjection. Moreover, since X → AX , Y → AY induce isomorphisms on the respectively étale rst cohomology groups with Z l -coecients (where l is prime to the characteristic of k ), we thus obtain that π1 (AX ) ³ π1 (AY ) is a surjection which is an isomorphism on the respective maximal pro-l quotients. Now it follows from the elementary theory of abelian varieties that this implies that AX → AY is a isogeny of degree a power of p. Finally, applying again the fact that π1 (AX ) ³ π1 (AY ) is surjective (i.e., even on maximal pro-p quotients), we conclude (again from the elementary theory of abelian varieties) that this isogeny has trivial étale part, hence is purely inseparable, as desired. Note that since AX → AY is an isogeny, it follows in particular that it is faithfully at. Now let γX : X → C be a dominant k -morphism. Write αX : AX → AC for the induced morphism on Albanese varieties. If γX arises from some γY : Y → C , then this γY is unique. Indeed, γY is determined by its associated αY , and the composite of αY with AX → AY is given by αX (which is uniquely determined by γX ). Thus, the fact that αY is uniquely determined follows from the fact that AX → AY is faithfully at. This completes the proof of the claim, and hence of the injectivity portion of the bijectivity assertion in Theorem 2.2. Now suppose that γX is arbitrary (i.e., does not necessarily arise from some γY ). The surjectivity portion of the bijectivity assertion in Theorem 2.2 amounts to showing that, up to replacing γX by the composite of γX with some power of ΦX , γX necessarily arises from some γY : Y → C . Now although αX : AX → AC itself might not factor through AY , since AX → AY is purely inseparable, it follows that the composite of αX with some power of ΦAX will factor through AY . Thus, if we replace γX by the composite of γX with some power of ΦX , then


131

αX will factor (uniquely) through AY . Denote this morphism by αY : AY → AC . Thus, in order to complete the proof of surjectivity, it suces to show:

The restriction αY |Y of αY to Y (relative to the natural morphism Y → AY ) maps into the subvariety C ⊆ AC . Before continuing, we make some observations: (1) The assertion (∗) for characteristic zero k follows immediately from the assertion (∗) for k of nite characteristic. Indeed, this follows via the usual argument of replacing k rst by a nitely generated Z -algebra, and then reducing modulo various primes. Thus, in the following, we assume the k is of characteristic p > 0. (2) The assertion (∗) will follow if we can show that the restriction αY |Yˆ (where we write Yˆ for the completion of Y along X ) maps into C ⊆ AC . Now we show that (up to possibly composing γX again with a power of Frobenius), γX extends to Yˆ . If I is the sheaf of ideals on Y that denes the closed def subscheme X ⊆ Y , then let us write Yn = V (I n ) ⊆ Y for the n-th innitesimal def neighborhood of X in Y , and J = I|X ∼ = OX (−1). Write T for the pull-back of the tangent bundle of C to X via γX . Since T −1 is generated by global sections, it thus follows that T −1 ⊗J −1 is ample, hence, by Serre duality (see, e.g., [Harts], Chapter III, Theorem 7.6), together with the fact that dim(X) ≥ 2, that there exists a natural number N such that N

N

H 1 (X, T ⊗p ⊗ J ⊗p ) = 0. Note that this implies that for all n ≥ pN , we have: N

H 1 (X, T ⊗p ⊗ J ⊗n ) = 0. (Indeed, it suces to assume that n > pN . Then since J −1 ∼ = OX (1) is very ample, it follows that there exists a section s ∈ Γ(X, OX (1)) whose zero locus def Z = V (s) ⊆ X is smooth of dimension ≥ 1. Thus, s denes an exact sequence N

N

N

0 → T ⊗p ⊗ J ⊗n → T ⊗p ⊗ J ⊗n−1 → T ⊗p ⊗ J ⊗n−1 |Z → 0, whose associated long exact cohomology sequence yields N

N

N

H 0 (Z, T ⊗p ⊗ J ⊗n−1 |Z ) → H 1 (X, T ⊗p ⊗ J ⊗n ) → H 1 (X, T ⊗p ⊗ J ⊗n−1 ) N

N

But H 0 (Z, T ⊗p ⊗ J ⊗n−1 |Z ) = 0 since T ⊗p ⊗ J ⊗n−1 |Z is the inverse of an N ample line bundle on a smooth scheme of dimension ≥ 1, while H 1 (X, T ⊗p ⊗ J ⊗n−1 ) = 0 by the induction hypothesis.) N Next, observe that ΦN Y N : YpN → YpN factors through X (since ΦY N is p

p

induced by raising functions to the pN -th power). Thus, if we compose γX : X → C with ΦN X , we see that this composite extends to a morphism YpN → C . Moreover, since the pull-back to X via this composite of the tangent bundle on

132

SHINICHI MOCHIZUKI N

C is given by T ⊗p , it follows that the obstruction to extending this composite to Yn+1 for n ≥ pN is given by an element of the cohomology group N

H 1 (X, T ⊗p ⊗ J ⊗n ), which (by the above discussion concerning cohomology groups) is zero. Thus, in summary, if we replace the given γX by its composite with ΦN X , the resulting γX extends to a morphism Yˆ → C . This completes the proof of (∗), and hence of the entire proof of Theorem 2.2. ¤ Remark.

The above proof beneted from discussions with A. Tamagawa.

The other case discussed in the remark at the end of Section 2.1, i.e., the case of a ber bundle with proper, simply connected bers also admits a purely algebro-geometric proof: namely, it follows immediately from the theory of Albanese varieties that there do not exist any nonconstant morphisms from a simply connected smooth proper variety to an abelian variety. Remark.

The role played by the Albanese variety in the proof of Theorem 2.2 given above suggests that the property proven in Corollary 2.1 and Theorem 2.2 might be thought of as asserting that a hyperbolic curve is, so to speak, its own anabelian Albanese variety. This is the reason for the title of Section 2. Remark.

3. Discrete Real Anabelian Geometry The original motivation for the p-adic result of [Mzk2] came from the (differential ) geometry of the upper half-plane uniformization of a hyperbolic curve. This point of view and, especially, the related idea that Kähler geometry at archimedean primes should be regarded as analogous to Frobenius actions at p-adic primes is discussed in detail in [Mzk4], Introduction (especially Section 0.10; see also the Introduction of [Mzk3]). In the present section, we attempt to make this motivation more rigorous by presenting the real analogues of various theorems/conjectures of anabelian geometry. The substantive mathematics here i.e., essentially the geometry of the Siegel upper half-plane and Teichmüller space is not new, but has been well-known to topologists, Teichmüller theorists, and symmetric domain theorists for some time. What is (perhaps) new is the formulation or point of view presented here, namely, that these geometric facts should be regarded as real analogues of Grothendieck's conjectured anabelian geometry.

3.1. Real complex manifolds. We begin with the following purely analytic denition: Let X be a complex manifold and ι an antiholomorphic involution (i.e., automorphism of order 2) of X . A pair such as (X, ι) will be referred to as a real complex manifold. If X has the structure of an abelian variety whose origin is xed by ι, then (X, ι) will be referred to as a real abelian variety. If dimC (X) = 1, then Definition 3.1.


133

(X, ι) will be referred to as a real Riemann surface. A real Riemann surface (X, ι) will be called hyperbolic if the universal covering space of X is isomorphic def (as a Riemann surface) to the upper half-plane H = {z ∈ C | Im(z) > 0}. If XR is a smooth algebraic variety over R , then XR (C) equipped with the antiholomorphic involution dened by complex conjugation denes a real complex manifold (X, ι). Moreover, one checks easily that XR is uniquely determined by (X, ι). Conversely, any real complex manifold (X, ι) such that X is projective arises from a unique algebraic variety XR over R . Indeed, this follows easily from Chow's Theorem (that any projective complex manifold is necessarily algebraic) and the (related) fact that any holomorphic isomorphism between projective algebraic varieties (in this case, the given X and its complex conjugate) is necessarily algebraic. Thus, in summary, one motivating reason for the introduction of Denition 3.1 is that it allows one to describe the notion of a (proper , smooth ) algebraic variety over R entirely in terms of complex manifolds and analytic maps.

Remark.

In the case of one complex dimension, one does not even need to assume projectivity: That is, any real Riemann surface (X, ι) such that X is algebraic arises from a unique algebraic curve XR over R . Indeed, this follows easily by observing that any holomorphic isomorphism between Riemann surfaces associated to complex algebraic curves is necessarily algebraic. (This may be proven by noting that any such isomorphism extends naturally to the onepoint compactications of the Riemann surfaces (which have natural algebraic structures), hence is necessarily algebraizable.) It is not clear to the author whether or not this can be generalized to higher dimensions. Remark.

In the following, we shall consider various groups G with natural augmentations G → Gal(C/R). In this sort of situation, we shall denote the inverse image of the identity element (respectively, the complex conjugation element) in Gal(C/R) by G+ (respectively, G− ). If X is a complex manifold, we shall denote by

Aut(X) → Gal(C/R) the group of automorphisms of X which are either holomorphic or antiholomorphic, equipped with its natural augmentation (which sends holomorphic (respectively, antiholomorphic) automorphisms to the identity (respectively, complex conjugation element) in Gal(C/R )). Thus,

Aut+ (X), Aut− (X) ⊆ Aut(X) denote the subsets of holomorphic and antiholomorphic automorphisms, respectively. In many cases, X will come equipped with a natural Riemannian metric which is preserved by Aut(X). The principal examples of this situation are:

134

SHINICHI MOCHIZUKI

Example 3.2 (the Siegel upper half-plane).

Siegel upper half-plane Hg is the set

Let g ≥ 1 be an integer. The

def

Hg = {Z ∈ Mg (C) | Z = t Z; Im(Z) > 0}, where t denotes the transpose matrix, and > 0 means positive denite. (Thus, H1 is the usual upper half-plane H.) We shall regard Hg as a complex manifold (equipped with the obvious complex structure). Set µ ¶ 0 Ig def Jg = ∈ M2g (R), −Ig 0 where Ig ∈ Mg (R) is the identity matrix. Write def

GSp2g = {M ∈ M2g (R) | M · J · t M = η · J, η ∈ R × } for the group of symplectic similitudes. Thus, we have a natural character

χ : GSp2g → Gal(C/R) that maps an M ∈ GSp2g to the sign of η (where η is as in the above denition − of GSp2g ). In particular, χ denes GSp+ 2g , GSp2g . Then we have a natural homomorphism

φ : GSp2g → Aut(Hg ) given by letting M =

¡A B ¢ C D

∈ GSp2g act on Z ∈ Hg by

Z 7→ (AZ χ(M ) + B)(CZ χ(M ) + D)−1 . Thus, φ is compatible with the augmentations to Gal(C/R). Now it is clear that the kernel of φ is given by the scalars R × ⊆ GSp2g . In fact, φ is surjective. Indeed, this is well-known when +'s are added to both sides (i.e., for holomorphic automorphisms see, e.g., [Maass], § 4, Theorem 2). On the other hand, since φ is compatible with the augmentations to Gal(C/R), the surjectivity of φ thus follows from the 5-Lemma. Thus, in summary, we have a natural isomorphism

GSp2g /R × ∼ = Aut(Hg ) Moreover, the space Hg admits a natural Riemannian metric. Relative to this metric, any two points Z1 , Z2 of Hg can be joined by a unique geodesic (see [Maass], § 3, Theorem). Moreover, this Riemannian metric is preserved by the action of GSp2g on Hg . (Indeed, this follows from [Maass], § 4, Theorem 1, in the holomorphic case. As for the antiholomorphic case, it suces to check that the metric is preserved by a single antiholomorphic map. But this is clear from [Maass], § 4, Theorem 1, for the map Z 7→ −Z¯ .)


135

Let g, r ≥ 0 be integers such that 2g − 2 + r > 0. Denote by Tg,r the Teichmüller space of genus g Riemann surfaces with r marked points. Thus, Tg,r has a natural structure of complex manifold. Moreover, Tg,r is equipped with a natural Kähler metric, called the WeilPetersson metric, whose associated Riemannian metric has the property that any two points t1 , t2 ∈ Tg,r may be joined by a unique geodesic (see [Wolp], § 5.1). Write Modg,r

Example 3.3 (Teichmüller space).

for the full modular group, i.e., the group of homotopy classes of homeomorphisms of a topological surface of type (g, r) onto itself. Note that Modg,r is equipped with an augmentation Modg,r → Gal(C/R) given by considering whether or not the homeomorphism preserves the orientation of the surface. The quotient Tg,r /Mod+ g,r (in the sense of stacks) may be identied with the moduli stack Mg,r of hyperbolic curves of type (g, r) over C , and the WeilPetersson metric descends to Mg,r . Moreover, the Riemannian metric arising from the Weil Petersson metric on Mg,r is preserved by complex conjugation. Indeed, this follows easily, for instance, from the denition of the WeilPetersson metric in terms of integration of the square of the absolute value of a quadratic dierential (on the Riemann surface in question) divided by the (1, 1)-form given by the Poincaré metric (on the Riemann surface in question) see, e.g, [Wolp], § 1.4. If (g, r) is not exceptional (i.e., not equal to the cases (0, 3), (0, 4), (1, 1), (1, 2), or (2, 0)), then it is known (by a theorem of Royden see, e.g., [Gard], § 9.2, Theorem 2) that one has a natural isomorphism

Modg,r ∼ = Aut(Tg,r ), which is compatible with the natural augmentations to Gal(C/R). Now I claim that (at least if (g, r) is nonexceptional, then) Aut(Tg,r ) preserves (the Riemannian metric arising from) the WeilPetersson metric. Indeed, since Tg,r /Mod+ g,r = Mg,r , and the WeilPetersson metric descends to Mg,r , it thus follows that Mod+ g,r preserves the WeilPetersson metric. Thus, the claim follows from the fact (observed above) that (the Riemannian metric arising from) the Weil Petersson metric on Mg,r is preserved by complex conjugation. We now return to our discussion of an arbitrary real complex manifold (X, ι). By analogy with the case when (X, ι) arises from a real algebraic variety (see the Remark following Denition 3.1), we will refer to the xed point locus of ι as the real locus of (X, ι), and use the notation

X(R) for this locus. Observe that X(R) is necessarily a real analytic submanifold of X of real dimension equal to the complex dimension of X . (Indeed, this follows immediately by considering the local structure of ι at a point x ∈ X(R).)

136

SHINICHI MOCHIZUKI

Moreover, at any x ∈ X(R), the involution ι induces a semi-linear (i.e., with respect to complex conjugation) automorphism ιx of order 2 of the complex vector space Tx (X) (i.e., the tangent space to the complex manifold X at x). That is to say, ιx denes a real structure Tx (X)R ⊆ Tx (X)R ⊗R C = Tx (X) on Tx (X). Put another way, this real structure Tx (X)R is simply the tangent space to the real analytic submanifold X(R) ⊆ X . Since ι acts without xed points on X\X(R), it follows that the quotient of X\X(R) by the action of ι denes a real analytic manifold over which X\X(R) forms an unramied double cover. In the following, in order to analyze the action of ι on all of X , we would like to consider the quotient of X by the action of ι in the sense of real analytic stacks. Denote this quotient by X ι . Thus, we have an unramied double cover

X → Xι which extends the cover discussed above over X\X(R). The Galois group of this double cover (which is isomorphic to Z/2Z ) may be identied with the Galois group Gal(C/R). Thus, this double cover induces a short exact sequence of fundamental groups

1 → π1 (X) → π1 (X ι ) → Gal(C/R) → 1 where we omit base-points, since they are inessential to the following discussion. (Here, by π1 we mean the usual (discrete) topological fundamental group in the sense of algebraic topology.) ˜ → X for the universal covering space of X . Thus, X ˜ also Now write X has a natural structure of complex manifold, and ι induces an antiholomorphic ˜ , which is uniquely determined automorphism ˜ι (not necessarily of order 2!) of X ˜ → X . Since X ˜ is up to composition with the covering transformations of X ι also the universal cover of the real analytic stack X , it thus follows that by ˜ → X ι , we get a considering the covering transformations of the covering X natural homomorphism ˜ π1 (X ι ) → Aut(X) which is compatible with the natural projections of both sides to Gal(C/R). Thus, if, for instance, (X, ι) is a hyperbolic real Riemann surface, then by ˜ ∼ Example 3.2, there is a natural isomorphism Aut(X) = PGL2 (R) = GSp2 /R × (well-dened up to conjugation by an element of PGL+ 2 (R)). Thus, we obtain a natural representation

ρX : π1 (X ι ) → PGL2 (R) which is compatible with the natural projections of both sides to Gal(C/R). Definition 3.4.

Let (X, ι) be a hyperbolic real Riemann surface. Then the

representation

ρX : π1 (X ι ) → PGL2 (R)


137

just constructed (which is dened up to composition with conjugation by an element of PGL+ 2 (R)) will be referred to as the canonical representation of (X, ι). Remark. The point of view of Denition 3.4 is discussed in [Mzk3], § 1, Real Curves, although the formulation presented there is somewhat less elegant.

3.2. Fixed points of antiholomorphic involutions. Let T be a (nonempty) complex manifold which is also equipped with a smooth Riemannian metric. Assume also that the Riemannian metric on T satises the following property: (∗) For any two distinct points t1 , t2 ∈ T , there exists a unique geodesic joining t1 and t2 . Then we have the following result, which is fundamental to the theory of the present Section 3: Lemma 3.5. Let T be a (nonempty ) complex manifold equipped with a smooth Riemannian metric satisfying the condition (∗). Let ιT : T → T

be an antiholomorphic involution of T which preserves this Riemannian metric . def Then the xed point set FιT = {t ∈ T | ιT (t) = t} of ιT is a nonempty , connected real analytic submanifold of T of real dimension equal to the complex dimension of T . By the discussion of Section 3.1, it follows that it suces to prove that ιT is nonempty and connected. First, we prove nonemptiness. Let t1 ∈ T def be any point of T , and set t2 = ιT (t1 ). If t1 = t2 , then t1 ∈ FιT , so we are done. If t1 6= t2 , then let γ be the unique geodesic joining t1 , t2 . Then since the subset {t1 , t2 } is preserved by ιT , it follows that γ is also preserved by ιT . Thus, it follows in particular that the midpoint t of γ is preserved by ιT , i.e., that t ∈ FιT , so FιT is nonempty as desired. Connectedness follows similarly: If t1 , t2 ∈ FιT , then the unique geodesic γ joining t1 , t2 is also clearly xed by ιT , i.e., γ ⊆ FιT , so FιT is pathwise connected. ¤ Proof.

The idea for this proof (using the WeilPetersson metric in the case of Teichmüller space) is essentially due to Wolpert ([Wolp]), and was related to the author by C. McMullen. We remark that this idea has been used to give a solution of the Nielsen Realization Problem (see the Introduction of [Wolp]). It is easiest to see what is going on by thinking about what happens in the case 2 2 when T = H (the upper half-plane) equipped with the Poincaré metric dx y+dy . 2 1 Also, we remark that in the case when T = P C , both the hypothesis and the conclusion of Lemma 3.5 are false! (That is, the hypothesis is false because there will always exist conjugate points, and the conclusion is false because it is easy to construct examples of antiholomorphic involutions without xed points.) Remark.

Now assume that (X, ι) is any real complex manifold equipped with a smooth Riemannian metric (i.e., X is equipped with a smooth Riemannian metric preserved by ι) such that the induced Riemannian metric on the universal cover

138

SHINICHI MOCHIZUKI

def ˜ T = X satises (∗). Let Y ⊆ X(R) be a connected component of the real analytic manifold X(R). Then since ι acts trivially on Y , the quotient of Y by the action of ι forms a real analytic stack Y ι whose associated coarse space is Y itself, and which ts into a commutative diagram:

Y ↓ X

→ →

Yι ↓ Xι

Moreover, the mapping Y ι → Y (where we think of Y as the coarse space associated to the stack Y ι ) denes a splitting of the exact sequence

1 → π1 (Y ) → π1 (Y ι ) → Gal(C/R) → 1, hence a homomorphism Gal(C/R) → π1 (Y ι ). If we compose this homomorphism with the natural homomorphism π1 (Y ι ) → π1 (X ι ), then we get a morphism

αY : Gal(C/R) → π1 (X ι ) naturally associated to Y , which is well-dened up to composition with an inner autormorphism of π1 (X). In particular, the image of complex conjugation under αY denes a conjugacy class of involutions ιY of π1 (X ι ). Thus, to summarize, we have associated to each connected component Y ⊆ X(R) of the real locus of (X, ι) a conjugacy class of involutions ιY in π1 (X ι ). Now we have the following immediate consequence of Lemma 3.5:

Let (X, ι) be a real complex manifold equipped with a smooth Riemannian metric (i .e ., X is equipped with a smooth Riemannian metric preserved by ι) such that the ˜ satises (∗). Then the induced Riemannian metric on the universal cover X correspondence Y 7→ ιY denes a bijection Theorem 3.6 (General Discrete Real Section Conjecture).

π0 (X(R)) ∼ = HomGal(C/R) (Gal(C/R), π1 (X ι ))

from the set of connected components of the real locus X(R) to the set of conjugacy classes of sections of π1 (X ι ) → Gal(C/R) (or , equivalently , involutions in π1 (X ι )). Moreover , the centralizer of an involution ιY ∈ π1 (X ι ) is the image of π1 (Y ι ) in π1 (X ι ). Indeed, let ιT ∈ π1 (X ι ) be an involution. Then ιT may be thought def ˜ of as an antiholomorphic involution of T = X . By Lemma 3.5, the xed point locus FιT of ιT is nonempty and connected. Thus, FιT maps into some connected component Y ⊆ X(R). (In fact, the morphism FιT → Y is a covering map.) By functoriality (consider the map of triples (T, ιT , FιT ) → (X, ι, Y )!), it follows that ιY = ιT . Thus, every involution in π1 (X ι ) arises as some ιY . Next, let us show uniqueness. If ιT arises from two distinct Y1 , Y2 ⊆ X(R), then it would follow that the xed point locus FιT contains at least two distinct connected components (corresponding to Y1 , Y2 ), thus contradicting Lemma 3.5. Finally, Proof.


139

if α ∈ π1 (X ι ) commutes with ιY , then α preserves FιY , hence induces an automorphism of FιY over Y ι . But since FιY → Y ι is a covering map, this implies that α is in the image of π1 (Y ι ) in π1 (X ι ). This completes the proof. ¤ Thus, Theorem 3.6 is a sort of analogue of the so-called Section Conjecture of anabelian geometry for the discrete fundamental groups of real complex manifolds see [Groth], p. 289, (2); [NTM], § 1.2, (GC3), for more on the Section Conjecture. Remark.

Theorem 3.6 generalizes immediately to the case where X is a complex analytic stack. In this case, Y ι is to be understood to be the real analytic stack whose stack structure is inherited from that of the real analytic stack X ι . We leave the routine details to the reader.

Remark.

3.3. Hyperbolic curves and their moduli. By the discussion of Examples 3.2 (in the case of H), 3.3, in Section 3.1, together with Theorem 3.6 of Section 3.2, we obtain: Corollary 3.7 (Discrete Real Section Conjecture for hyperbolic

real riemann surfaces). Let (X, ι) be a hyperbolic real Riemann surface . Then the correspondence Y 7→ ιY of Section 3.2 denes a bijection


from the set of connected components of the real locus X(R) to the set of conjugacy classes of sections of π1 (X ι ) → Gal(C/R) (or , equivalently , involutions in π1 (X ι )). Some readers may nd it strange that there is no discussion of tangential sections (at the points at innity of X ) in Corollary 3.7. The reason for this is that in the present real context, where we only consider connected components of the set of real points, every tangential section arising from a real point at innity may be obtained as a limit of a sequence of real points that are not at innity (and, which, moreover, may be chosen to lie in the same connected components of the real locus), hence is automatically included in the connected component containing those real points. Remark.

Corollary 3.8 (Discrete Real Section Conjecture for moduli of

Let g, r ≥ 0 be integers such that 2g − 2 + r > 0. Write (Mg,r , ιM ) for the moduli stack of complex hyperbolic curves of type (g, r), equipped with its natural antiholomorphic involution (arising from the structure of Mg,r as an algebraic stack dened over R ). If (X, ι) arises from a real hyperbolic curve of type (g, r), then the exact sequence hyperbolic curves).

1 → π1 (X) → π1 (X ι ) → Gal(C/R) → 1

denes a homomorphism M α(X,ι) : Gal(C/R) → π1 (Mιg,r ) = Modg,r ⊆ Out(π1 (X))

140

SHINICHI MOCHIZUKI

(where Out(−) denotes the group of outer automorphisms of the group in parentheses ). This correspondence (X, ι) 7→ α(X,ι) denes a bijection M π0 (Mg,r (R)) ∼ )) = HomGal(C/R) (Gal(C/R), π1 (Mιg,r

from the set of connected components of Mg,r (R) to the set of conjugacy classes M M of sections of π1 (Mιg,r ) → Gal(C/R), or , equivalently , involutions in π1 (Mιg,r ). ιM Moreover , the centralizer of an involution ιY ∈ π1 (Mg,r ) is the image of π1 (Y ι ) M in π1 (Mιg,r ). The injectivity portion of the bijection of Corollary 3.8, together with the determination of the centralizer of an involution (the nal sentence in the statement of Corollary 3.8), may be regarded as the discrete real analogue of the so-called Strong Isomorphism Version of the Grothendieck Conjecture. (For the convenience of the reader, we recall that the Strong Isomorphism Version of the Grothendieck Conjecture is the statement of Theorem A in the Introduction, except with K -morphism (respectively, homomorphism) replaced by K -isomorphism (respectively, isomorphism).) Remark.

The author was informed by M. Seppala that results similar to Corollary 3.8 have been obtained in [AG]. Remark.

3.4. Abelian varieties and their moduli. Lemma 3.9. Let (X, ι) be a real complex manifold such that X is an abelian variety over C . Then there exists a translation-invariant Riemannian metric on X which is preserved by ι.

By the Remark following Denition 3.1, (X, ι) arises from a projective algebraic variety XR over R . Write X c for the complex conjugate of the complex manifold X (i.e., X c and X have the same underlying real analytic manifold, but holomorphic functions on X c are antiholomorphic functions on X ). Since X is an abelian variety over C , it follows that X c is also an abelian variety over C . Thus, the holomorphic isomorphism ι : X ∼ = X c is the composite of an isomorphism of abelian varieties (i.e., one which preserves the group structures) with a translation. In particular, it follows that ι preserves the indef variant dierentials V = Γ(X, ΩX ) on X . Thus, ι induces a semi-linear (with respect to complex conjugation) automorphism of V , i.e., ι induces a real structure VR ⊆ VR ×R C = V on V . Then any inner product on the real vector space VR induces an ι-invariant inner product on the underlying real vector space of V which, in turn, induces a translation-invariant Riemannian metric on X which is preserved by ι, as desired. ¤

Proof.

˜ arising from a Riemannian metric as Any Riemannian metric on X ˜ which is isomorphic to in the conclusion of Lemma 3.9 induces a geometry on X Euclidean space, hence enjoys the property that any two points are joined by a unique geodesic. Remark.


141

Now if we apply Theorem 3.6 using Lemma 3.9, Example 3.2, we obtain: Corollary 3.10 (Discrete Real Section Conjecture for real abelian

Let (X, ι) be a real abelian variety . Then the correspondence Y 7→ ιY of Section 3.2 denes a bijection varieties).


from the set of connected components of the real locus X(R) to the set of conjugacy classes of sections of π1 (X ι ) → Gal(C/R) (or , equivalently , involutions in π1 (X ι )). Corollary 3.11 (Discrete Real Section Conjecture for moduli of abelian varieties). Let g ≥ 1 be a positive integer . Write (Ag , ιA ) for the moduli stack of principally polarized abelian varieties of dimension g , equipped with its natural antiholomorphic involution (arising from the structure of Ag as an algebraic stack dened over R ). If (X, ι) is a real abelian variety of dimension g , then the exact sequence

1 → π1 (X) → π1 (X ι ) → Gal(C/R) → 1

denes a homomorphism α(X,ι) : Gal(C/R) → π1 (AιgA ) ∼ = GSp(π1 (X)) (where GSp denotes the automorphisms that preserve , up to a constant multiple , the symplectic form dened by the polarization ). This correspondence (X, ι) 7→ α(X,ι) denes a bijection π0 (Ag (R)) ∼ = HomGal(C/R) (Gal(C/R), π1 (AιgA ))

from the set of connected components of Ag (R) to the set of conjugacy classes of sections of π1 (AιgA ) → Gal(C/R) (or , equivalently , involutions in π1 (AιgA )). Moreover , the centralizer of an involution ιY ∈ π1 (AιgA ) is the image of π1 (Y ι ) in π1 (AιgA ). Proof.

The bijectivity of the natural morphism

π1 (AιgA ) → GSp(π1 (X)) follows from the fact that it is compatible with the projections on both sides to Gal(C/R) (where the projection GSp(π1 (X)) → Z × = Gal(C/R) is given by looking at the constant multiple to which the symplectic form arising from the polarization is mapped), together with the well-known bijectivity of this morphism on the + portions of both sides. ¤

3.5. Pronite real anabelian geometry. So far we have considered the

real analogue of Grothendieck's anabelian geometry given by using the discrete fundamental groups of varieties. Another real analogue of anabelian geometry is that given by using the pronite fundamental groups. Just as in the discrete, the fundamental result was an existence theorem for real points in the presence

142

SHINICHI MOCHIZUKI

of involutions (i.e., Lemma 3.5), in the pronite case, the fundamental existence is given by the following theorem of Cox (see [Frdl], Corollary 11.3): Lemma 3.12. Let X be a connected real algebraic variety . Then X(R) 6= ∅ i i if and only if Het (X, Z/2Z) 6= 0 (where Het denotes étale cohomology ) for innitely many i.

In particular, if the complex manifold X(C) is a K(π, 1) space (i.e., its universal cover is contractible), and, moreover, its fundamental group π1 (X(C)) is good (i.e., the cohomology of π1 (X(C)) with coecients in any nite π1 (X(C))-module is isomorphic (via the natural morphism) to the cohomology of the pronite completion of π1 (X(C)) with coecients in that module), then we obtain: Remark.

i (∗) X(R) 6= ∅ if and only if Het (π1alg (X), Z/2Z) 6= 0 for innitely many integers i.

(Here π1alg (X) denotes the algebraic fundamental group of the scheme X .) Also, if the projection π1alg (X) → Gal(C/R) possesses a splitting, then the fact that i (Gal(C/R), Z/2Z) 6= 0 for innitely many i implies that Het i (π1alg (X), Z/2Z) 6= 0 Het

for innitely many i. Since hyperbolic curves and abelian varieties satisfy the conditions of the preceding remark, we obtain: Corollary 3.13 (Profinite Real Section Conjecture for real hy-

Let X be a hyperbolic curve over R . Then the pronite version of the correspondence Y 7→ ιY of Section 3.2 denes a bijection

perbolic curves).

alg π0 (X(R)) ∼ = HomGal(C/R) (Gal(C/R), π1 (X))

from the set of connected components of the real locus X(R) to the set of conjugacy classes of sections of π1alg (X) → Gal(C/R) (or , equivalently , involutions in π1alg (X)). Surjectivity follows from the above Remark, using the technique of [Tama1]: Namely, given a section α : Gal(C/R) → π1alg (X) of π1alg (X) → Gal(C/R), the family of open subgroups of π1alg (X) containing Im(α) denes a system of coverings {Xi → X} (as i varies over the elements of some index set I ) such that (by the above Remark) each Xi (R) 6= 0. Since each Xi (R) has only nitely many connected components, it thus follows that there exists a compatible system (indexed by I ) of connected components of Xi (R). But this amounts to the assertion that α arises from some connected component of X(R), as desired (see [Tama1], Corollary 2.10). Proof.


143

Injectivity follows from the fact that involutions arising from distinct connected components dene distinct elements of H 1 (π1alg (X), Z/2Z) see [Schd], § 20, Propositions 20.1.8, 20.1.12. ¤ To the author's knowledge, the rst announcement in the literature of a result such as Corollary 3.13 (in the proper case) appears in a manuscript of Huisman ([Huis]). (In fact, [Huis] also treats the one-dimensional case of Corollary 3.14 below.) Unfortunately, however, the author was not able to follow the portion of Huisman's proof ([Huis], Lemma 5.7) that corresponds to the application of Cox's theorem (as in the Remark following Lemma 3.12). Remark.

Corollary 3.14 (Profinite Real Section Conjecture for real abelian

Let X be an abelian variety over R . Then the pronite version of the correspondence Y 7→ ιY of Section 3.2 denes a bijection varieties).

alg π0 (X(R)) ∼ = HomGal(C/R) (Gal(C/R), π1 (X))

from the set of connected components of the real locus X(R) to the set of conjugacy classes of sections of π1alg (X) → Gal(C/R) (or , equivalently , involutions in π1alg (X)). Surjectivity follows as in the proof of Corollary 3.13. Injectivity follows, for instance, from the discrete result (Corollary 3.10), together with the injectivity of the natural morphism

Proof.

H 1 (Gal(C/R), π1 (X(C))) → H 1 (Gal(C/R), π1alg (X ⊗R C)) ˆ , where itself a consequence of the fact that π1alg (X ⊗R C)) = π1 (X(C)) ⊗Z Z ˆ Z is the pronite completion of Z (hence a faithfully at Z -module). ¤ As for the case of moduli, the above argument breaks down in most cases since it is either false that or unknown whether or not the fundamental group of the corresponding moduli stacks is good. More precisely, π1 (Ag ) = Sp(2g, Z) is known not to be good if g ≥ 2 (see Lemma 3.16 below). (If g = 1, then the pronite real section conjecture for Ag is essentially contained in Corollary 3.13 above.) On the other hand, to the author's knowledge, it is not known whether or not π1 (Mg,r ) is good if g > 2. If g ≤ 2, then, up to passing to nite étale coverings, Mg,r may be written as a successive extension of smooth families of hyperbolic curves, hence has a good fundamental group. Thus, we obtain: Corollary 3.15 (Profinite Real Section Conjecture for moduli of hyperbolic curves of genus ≤ 2). Let g, r ≥ 0 be integers such that 2g − 2 + r > 0, g ≤ 2. Write (Mg,r )R for the moduli stack of complex hyperbolic curves of type (g, r) over R . If X is a real hyperbolic curve of type (g, r), then X denes a section α(X,ι) : Gal(C/R) → π1alg ((Mg,r )R ). This correspondence (X, ι) 7→ α(X,ι) denes a bijection alg π0 ((Mg,r )R (R)) ∼ = HomGal(C/R) (Gal(C/R), π1 ((Mg,r )R ))

144

SHINICHI MOCHIZUKI

from the set of connected components of (Mg,r )R (R) to the set of conjugacy classes of sections of π1alg ((Mg,r )R ) → Gal(C/R) (or , equivalently , conjugacy classes of involutions in π1alg ((Mg,r )R )). Moreover , the centralizer of an involution ιY ∈ π1alg ((Mg,r )R ) is the image of the pronite completion of π1 (Y ι ) in π1alg ((Mg,r )R ). Proof. Since (as just remarked) the fundamental groups involved are good, surjectivity follows as in Corollaries 3.13, 3.14. As for injectivity, we reason as follows. Given two real hyperbolic curves X , Y of the same type (g, r) which induce the same section α (up to conjugacy) of π1alg ((Mg,r )R ) → Gal(C/R), we must show that they belong to the same connected component of (Mg,r )R (R). First, observe that since [C : R] = 2, it follows that the marked points of X and Y over C consist of: (i.) points dened over R ; (ii) complex conjugate pairs. Moreover, the combinatorial data of which points are dened over R and which points are conjugate pairs is clearly determined by the section α. Thus, there exists an ordering of connected components of the divisor of marked points over R of X , Y , which is compatible with α. Write

N → (Mg,r )R for the nite étale covering dened by the moduli stack (over R ) of hyperbolic curves equipped with such an ordering. Note, in particular, that the injectivity assertion under consideration for (Mg,r )R follows formally from the corresponding injectivity assertion for N . Moreover, N may be written as a successive extension

N = Nr → Nr−1 → · · · → N1 → N0 of smooth families (i.e., the Nj+1 → Nj ) of either hyperbolic curves (where we include curves which are stacks see the remark in parentheses following the list below) or surfaces (of a special type, to be described below) over the stack N0 , where N0 may be described as follows: (1) If g = 0, then N0 is the moduli stack of 4-pointed curves of genus 0, equipped with an ordering type T , where T is one of the following: a total ordering of the four points; a total ordering of two points, plus a pair of conjugate points; a total ordering of two pairs of conjugate points. In each of these three cases, one sees that N0 is a hyperbolic curve over R , so we conclude the corresponding injectivity assertion for N0 from Corollary 3.13. (2) If g = 1, then N0 is either the moduli stack of 1-pointed curves of genus 1 (which is a hyperbolic curve, so we may conclude the corresponding injectivity assertion for N0 from Corollary 3.13), or N0 is the moduli stack of 2-pointed curves of genus 1, where the two points are unordered. In the latter case, by considering the group of automorphisms of the underlying genus 1 curve which preserve the invariant dierentials, we get a morphism N0 → (M1,1 )R , which is a smooth family whose ber over the elliptic curve E is the stack


145

given by forming the quotient of E\{0E } (where 0E is the origin of E ) by the action of ±1. (Indeed, this ber parametrizes the dierence of the two unordered points.) In particular, (M1,1 )R , as well as these bers over (M1,1 )R are hyperbolic curves, so the corresponding injectivity assertion for N0 follows from Corollary 3.13. (3) If g = 2, then N0 is the moduli stack of 0-pointed curves of genus 2. Moreover, by using the well-known morphism (M2,0 )R → (M0,6 )R (given by considering the ramication points of the canonical double covering of the projective line associated to a curve of genus 2), this case may be reduced to the case g = 0, which has already been dealt with. (We remark here that even though some of the hyperbolic curves appearing above are in fact stacks, by passing to appropriate nite étale coverings which are still dened over R and for which the real points in question lift to real points of the covering, injectivity for such stack-curves follows from injectivity for usual curves as proven in Corollary 3.13.) Finally, the surfaces that may appear as bers in the families Nj+1 → Nj appearing above are of the following type: If C is a hyperbolic curve over R , write ∆C ⊆ C ×R C for the diagonal. Then the surfaces in question are of the form {(C ×R C)\∆C }/S2 (where S2 is the symmetric group on two letters permuting the two factors of C , and we note that this quotient is the same whether taken in the sense of schemes or of stacks). Now by passing (as in the one-dimensional case) to appropriate nite étale coverings of these surfaces which are still dened over R and for which the real points in question lift to real points of the covering, the corresponding injectivity assertion for such surfaces follows from injectivity for surfaces that may be written as a smooth family of hyperbolic curves parametrized by a hyperbolic curve, hence is a consequence of Corollary 3.13. Thus, by dévissage we conclude the desired injectivity for (Mg,r )R . Before proceeding, we observe that the above argument shows that the bijectivity assertion of Corollary 3.15 also holds for any nite étale covering of Mg,r which is dened over R . The nal statement on centralizers may be proven as follows: Given an involution ιY , write MY → (Mg,r )R for the pro-covering dened by the subgroup generated by ιY in π1alg ((Mg,r )R ). Then the statement on centralizers follows from the fact that the conjugates of ιY in π1alg ((Mg,r )R ) are in bijective correspondence with the connected components of the inverse images of Y in MY (where we note that this bijective correspondence follows from the observation of the preceding paragraph). ¤ In many respects the pronite theory is more dicult and less elegant than the discrete theory, where everything follows easily from the very general Lemma 3.5. It is thus the feeling of the author that the discrete theory provides a more natural real analogue of anabelian geometry than the pronite theory. Remark.

146

SHINICHI MOCHIZUKI

Lemma 3.16. Let g ≥ 2, and let H ⊆ Sp(2g, Z) be a subgroup of nite index . Then there exists a subgroup H 0 ⊆ H which is normal and of nite index in Sp(2g, Z) such that the cohomological dimension of the pronite completion of H 0 is > dimR (Ag ) = 2 · dimC (Ag ) = g(g + 1). (This estimate holds even if one restricts to H 0 -modules of order equal to a power of p, for any xed prime number p.) In particular , if g ≥ 2, then Sp(2g, Z) is not good .

First, note that if Sp(2g, Z) is good, then so is any subgroup H of nite index. But there exist H such that if we write AH 0 → Ag for the nite étale covering dened by a nite index subgroup H 0 ⊆ H , then AH 0 is a complex manifold (i.e., not just a stack). The cohomology of H 0 is then given by the cohomology of AH 0 . Moreover, the cohomological dimension of AH 0 is = dimR (AH 0 ) = dimR (Ag ). Thus, if the cohomological dimension of the pronite completion of H 0 is > dimR (Ag ), it follows that the cohomology of H 0 and of its pronite completion (with coecients in a nite module) are not isomorphic in general, i.e., that H 0 is not good. But this implies that Sp(2g, Z) is not good, as desired. Next, assume that we are given H as in the statement of Lemma 3.16, and prove the existence of an H 0 as stated. First, observe that since the congruence subgroup problem has been resolved armatively for Sp(2g, Z) (see [BMS]), it follows that Y ˆ = Sp(2g, Z)∧ = Sp(2g, Z) Sp(2g, Z p ) Proof.

p

(where the ∧ denotes the pronite completion, and the product is taken over all prime numbers p). Thus, it follows that the cohomological dimension of Sp(2g, Z)∧ is ≥ the cohomological dimension of Sp(2g, Z p ) for any prime p. In particular, in order to complete the proof of Lemma 3.16, it suces to show that Sp(2g, Z p ) admits a collection of arbitrarily small normal open subgroups whose p-cohomological dimension is > g(g + 1). But this follows from the theory of [Laz]: Indeed, by [Laz], V, § 2.2.8, it follows that that the p-cohomological dimension of any p-valuable group is equal to the rank r of the group. Here, a p-valuable group (see [Laz], III, § 2.1.2) is a topogical group with a ltration satisfying certain properties. In the present context, the topological group Sp[n] (2g, Z p ) (i.e., symplectic matrices which are ≡ to the identity matrix modulo pn ), equipped with the ltration dened by the Sp[m] (2g, Z p ) for m ≥ n, will satisfy these properties. Moreover, the rank r of a p-valuable group (see [Laz], III, § 2.1.1, § 2.1.3) is the Q p -dimension of the Lie algebra sp(2g, Q p ) of Sp(2g, Z p ). Thus, in this case,

r = dimQ p (sp(2g, Q p )) = dimR (sp(2g, R)) = dimR (Sp(2g, R)) > dimR (Hg ) = dimR (Ag )


147

(where Hg is the Siegel upper half-plane see Example 3.2). Indeed, the inequality here follows from the fact that Sp(2g, R) acts transitively on Hg , with positive dimensional isotropy subgroups. This completes the proof. ¤

4. Complements to the p-adic Theory In this section, we present certain complements to the p-adic theory of [Mzk2] which allow us to prove a certain isomorphism version of Theorem A of [Mzk2] (see Section 0 of the present article) over a somewhat larger class of elds K than was treated in [Mzk2]. This larger class of elds which we refer to as generalized sub-p-adic consists of those elds which may be embedded as sub¯ p ) (the ring elds of a nitely generated extension of the quotient eld of W (F of Witt vectors with coecients in the algebraic closure of F p , for some prime number p).

4.1. Good Chern classes. In this section, we work over a base eld K , which we assume (for simplicity, although it is not absolutely necessary for much of what we shall do) to be of characteristic 0. Let XK be a smooth, geometrically connected variety over K . If p is a prime number, and n ≥ 1 an integer, then we may consider the Kummer sequence on XK , i.e., the exact sequence of sheaves on (XK )et (i.e., the étale site of XK ) given by

0 → (Z/pn Z)(1) → Gm → Gm → 0 (where the (1) is a Tate twist, and the morphism from Gm to Gm is given by raising to the pn -th power.) The connecting morphism induced on étale cohomology by the Kummer sequence then gives us a morphism 1 2 δp,n : Het (XK , Gm ) → Het (XK , (Z/pn Z)(1))

Now suppose that L is a line bundle on XK . Then applying δp,n to L ∈ 1 Het (XK , Gm ) gives us a compatible system of classes 2 (XK , (Z/pn Z)(1)), δp,n (L) ∈ Het 2 ˆ (XK , Z(1)) . and hence (by letting p, n vary) a class c1 (L) ∈ Het 2 ˆ We shall refer to c1 (L) ∈ Het (XK , Z(1)) as the (pronite, étale-theoretic) rst Chern class of L. If N ≥ 1 is an integer, then we shall refer 2 to c1 (L) mod N ∈ Het (XK , (Z/N Z)(1)) as the (étale-theoretic ) rst Chern class of L modulo N .

Definition 4.1.

Next, write

π1 (XK )

148

SHINICHI MOCHIZUKI

for the (algebraic ) fundamental group of XK (where we omit the base-point since it will not be explicitly necessary in our discussion). Also, assume that we are given a quotient π1 (XK ) ³ Q (where Q is pronite, and the surjection is continuous). Then we make the following crucial denition: Let N ≥ 1 be an integer. For i, j ∈ Z , a cohomology class i (XK , (Z/N Z)(j)) will be called good if there exists a (nonempty) nite η ∈ Het i étale covering Y → XK such that η|Y ∈ Het (Y, (Z/N Z)(j)) is zero. Next, suppose that π1 (XK ) ³ Q is a surjection such that the composite of the natural surjection π1 (XK ) ³ ΓK with the cyclotomic character ΓK → (Z/N Z)× factors through Q. Then we shall say that η is Q-good if this covering Y → XK may be chosen to arise from a quotient of π1 (XK ) that factors through π1 (XK ) ³ Q. If L is a line bundle on XK , then we will say that its Chern class is good (respectively, Q-good ) modulo N if the Chern class of L modulo N in 2 Het (XK , (Z/N Z)(1)) is good (respectively, Q-good). Definition 4.2.

Recall that a discrete group Γ is said to be good if the cohomology of Γ with coecients in any nite Γ-module is isomorphic (via the natural morphism) to the cohomology of the pronite completion of Γ with coecients in that module. Then the justication for the terminology of Denition 4.2 is the following:

Suppose that K is a subeld of C (the complex number eld ); that def the topological space X = XC (C) is a K(π, 1) space (i .e ., its universal cover is contractible ); and that the topological fundamental group π1top (X ) is good . Then i it follows that all cohomology classes η ∈ Het (XK , (Z/N Z)(j)) are good . Lemma 4.3.

def

def

Write XC = XK ⊗K C , XK = XK ⊗K K . Since nite étale coverings of XC are always dened over a nite extension of K , and (by well-known elementary properties of étale cohomology) the natural morphism Proof.

i i Het (XK , (Z/N Z)(j)) → Het (XC , (Z/N Z)(j))

is an isomorphism, one sees immediately that it suces to prove Lemma 4.3 when K = C , j = 0. But then top i i Het (XC , Z/N Z) ∼ (XC , Z/N Z) ∼ = Hsing = H i (π1 (X ), Z/N Z)

(where the second isomorphism (between singular and group cohomology) follows from the fact that X is a K(π, 1) space). Thus, the fact that η vanishes upon restriction to a (nonempty) nite étale covering follows from the fact that π1top (X ) is assumed to be good. ¤ Remark. Thus, under the hypotheses of Lemma 4.3, every cohomology class is good. In general, however, we would like to work with varieties XK that do not satisfy the hypotheses of Lemma 4.3, but which nonetheless have the property that the cohomology classes that we are interested in are good.


149

Now let L be a line bundle on XK . Write L → XK for the geometric line bundle associated to L (i.e., the spectrum over XK of the symmetric algebra over OXK of L−1 ). Also, write L× ⊆ L → XK for the complement of the zero section in L. Thus, L× → XK is a Gm -torsor, and we have an exact sequence of (algebraic) fundamental groups

ˆ Z(1) = π1 ((Gm )K ) → π1 (L× ) → π1 (XK ) → 1 (where we omit base-points since they will not be explicitly necessary in our discussion). In general, however, it is not necessarily the case that the rst arrow is injective. (Consider, for instance, the line bundle O(−1) on P 1 (over, say, an algebraically closed eld K of characteristic zero), in which case L× = A2 \{0} has trivial fundamental group.) Lemma 4.4. Suppose that the Chern class of L is good modulo all powers of p. ˆ ˆ Then the restriction to Z p (1) ⊆ Z(1) of the morphism Z(1) → π1 (L× ) is injective , so the above exact sequence denes an extension class ∈ H 2 (π1 (XK ), Z p (1)). If , moreover , the Chern class of L is Q-good modulo all powers of p, then (for all integers n ≥ 1) the reduction modulo pn of this extension class arises from an element ∈ H 2 (Q, (Z/pn Z)(1)). Proof.

Indeed, let

Y → XK be a nite étale covering such that the Chern class of L modulo pn vanishes upon restriction to Y . Going back to the denition of the Chern class using the Kummer exact sequence, one thus sees that there exists a line bundle P on Y such n that P ⊗p ∼ = L|Y . Write P × → Y for the complement of the zero section in the geometric line bundle corresponding to P . Thus, P × → Y is a Gm -torsor with the property that, if we execute the change of structure group Gm → Gm given def by raising to the pn -th power, we obtain the Gm -torsor L× |Y = L× ×XK Y → Y . In particular, we obtain a nite étale covering

P × → L× |Y whose restriction to the geometric bers of L× |Y → Y is (isomorphic to) the covering of Gm given by raising to the pn -th power. Sorting through the denitions of the various fundamental groups involved, one thus sees that the existence of such coverings implies that Z p (1) → π1 (L× ) is injective, and, moreover, that if the covering Y → XK arises from a quotient of Q, then the extension class ∈ H 2 (π1 (XK ), (Z/pn Z)(1)) dened by π1 (L× ) arises from a class ∈ H 2 (Q, (Z/pn Z)(1)), as desired. ¤ In the Q-good portion of Lemma 4.4, it was necessary to use nite coecients Z/pn Z (i.e., rather than Z p ) since it is not clear that the various group extensions of Q by (Z/pn Z)(1) form a compatible system as n varies. Remark.

150

SHINICHI MOCHIZUKI

(When Q = π1 (XK ), one need not worry about this since one already has a natural group extension, namely, that arising from π1 (L× ).) Lemma 4.5.

Write

Suppose that the Chern class of L is good modulo all powers of p. 2 cgp 1 (L) ∈ H (π1 (XK ), Z p (1))

for the extension class of Lemma 4.4. Then the image of cgp 1 (L) under the natural map 2 H 2 (π1 (XK ), Z p (1)) → Het (XK , Z p (1)) is equal to the (p-adic portion of the ) rst Chern class c1 (L) of Denition 4.1. If , moreover , the Chern class of L is Q-good modulo all powers of p, then (for all integers n ≥ 1) there exists a class ∈ H 2 (Q, (Z/pn Z)(1)) whose image in 2 Het (XK , (Z/pn Z)(1)) is equal to c1 (L) mod pn . Proof. If Γ is a pronite group, denote by B(Γ) the classifying site of Γ, i.e., the site dened by considering the category of nite sets with continuous Γ-action (and coverings given by surjections of such sets). Thus, if M is a nite abelian group with continuous Γ-action, then M denes a sheaf of abelian groups on this site whose cohomology may be identied with the usual group cohomology of Γ with coecients in M . Next, note that relative to this notation, there is a tautological morphism

(XK )et → B(π1 (XK )) determined by the well-known equivalence between nite étale coverings of XK and nite sets with continuous π1 (XK )-action. Put another way, this morphism is the étale analogue of the well-known tautological morphism (determined up to homotopy equivalence) in topology from a topological space to the classifying space of its fundamental group. By functoriality, we thus obtain a commutative diagram (L× )et → B(π1 (L× )) ↓ ↓ (XK )et → B(π1 (XK )) (where the horizontal morphisms are the tautological morphisms just discussed, and the vertical morphisms are those induced by functoriality from L× → XK ). Next, observe that both vertical morphisms of the above commutative diagram give rise to LeraySerre spectral sequences on cohomology with coecients in Z p (1). (Here, we note that the fact that the vertical morphism on the right gives rise to such a spectral sequence follows from the injectivity assertion of Lemma 4.4.) In particular, if we consider the dierential on the E2 -term of these spectral sequences we obtain a commutative diagram ˆ Z p = H 0 (π1 (XK ), H 1 (Z(1), Z p (1))) → Z p = H 0 (XK , H 1 ((Gm ) , Z p (1))) et

↓ H 2 (π1 (XK ), Z p (1))

et

K

↓ →

2 Het (XK , Z p (1))


151

(where the vertical morphisms are the dierentials of the spectral sequence, and the horizontal morphisms are induced by the morphisms of sites just discussed). On the other hand, sorting through the denitions, one sees that it is a tautology that the image of 1 ∈ Z p under the vertical morphism on the right (respectively, left) is the Chern class c1 (L) (respectively, cgp 1 (L)). Thus, the assertion that the 2 (L) in H (X , Z (1)) is equal to c1 (L) follows from the commuimage of cgp K p et 1 tativity of the above diagram. The corresponding assertion in the Q-good case follows by replacing π1 (XK ), Z p in the above argument by Q, Z/pn Z , respectively, and applying the Q-good portion of Lemma 4.4. ¤ Lemma 4.6.

Suppose that XK is equipped with a smooth , proper morphism X K → ZK

(where ZK is a smooth , geometrically connected variety over K ) which admits a section σ : ZK → XK . Also , assume that K may be embedded as a subeld of C , and that if z¯ is any geometric point of ZK such that k(¯ z ) may be embedded as a def subeld of C , then the geometric ber Xz¯ = XK ×ZK z¯ satises the hypotheses of Lemma 4.3 (i .e ., its complex valued points form a K(π, 1) space with good topological fundamental group ). Then the Chern class of any line bundle L on XK for which the pull-back σ ∗ (L) is trivial (as a line bundle on ZK ) is good modulo all integers N ≥ 1. First, observe that we may always replace ZK (respectively, XK ) by a nite étale covering of ZK (respectively, XK , possibly at the expense of also replacing ZK by some new nite étale covering of ZK ) without aecting the validity of either the hypotheses or the conclusion of the lemma. Next, x a geometric point z¯ of Z as in the statement of Lemma 4.6, and write Xz¯ for the resulting geometric ber. Then the existence of the section σ implies that we obtain an exact sequence of fundamental groups:

Proof.

1 → π1 (Xz¯) → π1 (XK ) → π1 (ZK ) → 1 Indeed, in general (i.e., in the absence of hypothesis that σ exist) the morphism π1 (Xz¯) → π1 (XK ) need not be injective. That is to say, its kernel is naturally isomorphic to the cokernel of the morphism (induced by XK → ZK via functoriality) between certain étale-theoretic second homotopy groups of XK and ZK (see [Frdl], p. 107, Theorem 11.5). On the other hand, since XK → ZK admits a section, it thus follows (by functoriality) that this morphism between second homotopy groups also admits a section, hence that it is surjective (i.e., its cokernel is trivial). This implies the injectivity of the morphism π1 (Xz¯) → π1 (XK ). Stated in words, the injectivity of this morphism implies that (after possible base-change to a nite étale covering of ZK ) any nite étale covering of Xz¯ may be realized as the restriction to the ber Xz¯ of a nite étale covering of XK . Next, consider the LeraySerre spectral sequence associated to the morphism XK → ZK for étale cohomology with coecients in (Z/pn Z)(1) (for some p, n

152

SHINICHI MOCHIZUKI

as in Denition 4.1). If we consider the E2 -termof this spectral sequence, we 2 see that the cohomology group Het (XK , (Z/pn Z)(1)) gets a natural ltration whose highest subquotient (i.e., the subquotient which is, in fact, a quotient) 0 2 is a submodule of Het (ZK , Het (Xz¯, (Z/pn Z)(1))). On the other hand, by the 2 assumptions placed on Xz¯, it follows that all the classes of Het (Xz¯, (Z/pn Z)(1)) vanish upon restriction to some nite étale covering of Xz¯. Moreover, by the discussion of the preceding paragraph, it follows that this covering may be realized as the restriction to Xz¯ of a nite étale covering of XK . Thus, in conclusion, by replacing XK and ZK by appropriate nite étale coverings, we may assume that the image of the class c1 (L) mod pn in the highest subquotient 2 of Het (XK , (Z/pn Z)(1)) vanishes. 2 The next highest subquotient of the natural ltration on Het (XK , (Z/pn Z)(1)) induced by the LeraySerre spectral sequence may be regarded naturally as a sub1 1 module of Het (ZK , Het (Xz¯, (Z/pn Z)(1))). Thus, by an argument similar to that of the preceding paragraph, we conclude that we may assume that the image of 2 the class c1 (L) mod pn in the next highest subquotient of Het (XK , (Z/pn Z)(1)) also vanishes. The next subquotient (i.e., the subquotient which is, in fact, a submodule) 2 of the natural ltration on Het (XK , (Z/pn Z)(1)) induced by the LeraySerre 2 spectral sequence is given by the submodule of Het (XK , (Z/pn Z)(1)) which is the 2 2 (XK , (Z/pn Z)(1)) (via pull-back relative to (ZK , (Z/pn Z)(1)) in Het image of Het XK → ZK ). Note that the existence of the section σ implies that this pull-back 2 2 morphism Het (ZK , (Z/pn Z)(1)) → Het (XK , (Z/pn Z)(1)) is injective. Thus, we conclude that the class c1 (L) mod pn arises as the pull-back to XK of a 2 class in Het (ZK , (Z/pn Z)(1)). On the other hand, by pulling back via σ (and applying the functoriality of the formation of the Chern class of a line bundle), 2 we thus see that this class in Het (ZK , (Z/pn Z)(1)) is simply c1 (σ ∗ (L)) mod pn , ∗ which is = 0 (since σ (L) is assumed to be trivial). This completes the proof of Lemma 4.6. ¤

4.2. The group-theoreticity of a certain Chern class. In this section, we

let K be a eld of characteristic 0 (until further notice). Also, we x a prime number p and an integer g ≥ 2. Denote by A the moduli stack of principally polarized abelian varieties of dimension g . Write

G→A for the tautological abelian scheme over A, and ε : A → G for its identity section. Also, denote by LG the tautological line bundle on G that denes the principal polarization. Thus, LG is relatively ample over A, and we assume that it is rigidied by some isomorphism ε∗ (LG ) ∼ = OA . Also, we x a geometric point a ¯ ∈ A(K), and denote the fundamental group π1 (Ga¯ ) of the geometric ber Ga¯


153

by π1 (G/A). Thus, we obtain an exact sequence of fundamental groups

1 → π1 (G/A) → π1 (G) → π1 (A) → 1 equipped with a section π1 (ε) : π1 (A) → π1 (G) which allows us to identify π1 (G) with the semi-direct product π1 (G/A)oπ1 (A). Next, observe that since π1 (G/A) ˆ -module of rank 2g , we may write is a free Z (p)

(6=p)

π1 (G/A) = ΠG/A × ΠG/A

for the natural decomposition of π1 (G/A) as a product of a Z p -free module (p) (6=p) i.e., ΠG/A and a module ΠG/A which is free over the product of all Z p0 , where p0 ranges over the prime numbers not equal to p. Moreover, the above exact sequence shows that π1 (A) acts naturally on π1 (G/A) in a way that respects this decomposition. In particular, we may push forward the above exact sequence (p) via the quotient π1 (G/A) ³ ΠG/A to obtain exact sequences (6=p)

(p/A)

1 → ΠG/A → π1 (G) → ΠG (p)

(p/A)

1 → ΠG/A → ΠG (p/A)

where ΠG

→ 1,

→ π1 (A) → 1,

is dened by the rst exact sequence, and the second exact sequence (p/A)

admits a section that allows us to identify ΠG

with the semi-direct product

(p) ΠG/A

o π1 (A). Now consider π1 (A) in greater detail. First, recall that the action of π1 (A) on (p) ΠG/A preserves the symplectic form dened by the tautological principal polarization on the family of abelian varieties G → A up to multiplication by a scalar. Denote by (p) GSp(ΠG/A ) (p)

the group of Z p -linear automorphisms of ΠG/A which preserve this symplectic form up to multiplication by a scalar. Thus, we obtain a natural commutative diagram in which the rows are exact:

1

→

1

→

Sp(π1 (G/A)) → ↓ (p) Sp(ΠG/A ) →

π1 (A) → ↓ (p) GSp(ΠG/A ) →

ΓK ↓ Zp×

→

1

→

1

ˆ -linear automorphisms of π1 (G/A) that preserve Here, Sp denotes the group of Z the symplectic form in question precisely; the homomorphism (p)

GSp(ΠG/A ) → Z p × is the map that assigns to an element of the domain the scalar by which this element acts on the symplectic form in question. (Also, we note that here, we apply two well-known facts: (i) when K = C , the topological fundamental group of A is equal to Sp(2g, Z); (ii) for g ≥ 2, the congruence subgroup problem for Sp(2g, Z) has been resolved armatively (see the proof of Lemma 3.16; [BMS]).)

154

SHINICHI MOCHIZUKI

Finally, we observe that the vertical morphism (on the right) ΓK → Z p × is simply the cyclotomic character of K . (p/A) In particular, by applying the identication of ΠG with the semi-direct (p)

product ΠG/A o π1 (A), we obtain a continuous homomorphism (p/A)

ΠG

(p) def

→ ΠG

(p)

(p)

= ΠG/A o GSp(ΠG/A )

which is surjective whenever the cyclotomic character ΓK → Z p × is surjective. As is well-known (see, e.g., [Ser3], Chapter XIV, § 7), this is the case, for instance, when K = Q, Q p . In the present discussion, we would like to concentrate our attention on the prime p, in the case K = Q . Thus, we have surjections: (p/A)

π1 (G) ³ ΠG

(p)

³ ΠG

Denote (by abuse of notation relative to Denition 4.1) the p-adic component of the rst Chern class of LG by: 2 c1 (LG ) ∈ Het (G, Z p (1))

Lemma 4.7.

Suppose that K = Q . Then relative to the tautological morphisms (p)

Get → B(π1 (G)) → B(ΠG ) 2 (see the proof of Lemma 4.5), the class c1 (LG ) mod pn ∈ Het (G, (Z/pn Z)(1)) (p) 2 n arises from a class ∈ H (ΠG , (Z/p Z)(1)), for all integers n ≥ 1.

Indeed, Lemma 4.7 follows from Lemmas 4.4, 4.5, together with the proof of Lemma 4.6 (where, in Lemma 4.6, we take G → A as our smooth, proper morphism XK → ZK ). In fact, if we apply Lemmas 4.4, 4.5, 4.6, literally as stated, we already obtain that c1 (LG ) arises from a class ∈ H 2 (π1 (G), Z p (1)). (p/A) (6=p) Since the kernel of the surjection π1 (G) ³ ΠG is equal to ΠG/A , which is a pronite group of order prime to p, it thus follows immediately that this class (p/A) arises from a class ∈ H 2 (ΠG , Z p (1)). To see that, in fact, c1 (LG ) mod pn Proof.

(p)

arises from a class ∈ H 2 (ΠG , (Z/pn Z)(1)), it suces to observe that, in the proof of Lemma 4.6, the only coverings of G that were necessary to annihilate c1 (LG ) modulo pn were coverings of G that restricted to arbitrary p-power cov(p) erings of the abelian variety Ga¯ . But it is clear from the denition of ΠG (see the discussion above) that such coverings may be constructed from quotients of (p/A) (p/A) (p) ΠG that factor through the quotient ΠG ³ ΠG . Thus, we conclude by (p)

taking Q to be the quotient π1 (G) ³ ΠG in Lemmas 4.4, 4.5. This completes the proof of Lemma 4.7. ¤ Next, denote by ∆ a copy of the maximal pro-p quotient of the pronite completion of the fundamental group of a (Riemann ) surface of genus g . Since, as


155

is well-known (see, e.g., [Tama1], Proposition 1.11), ∆ is center-free, we have an exact sequence of pronite groups: def

def

1 → ∆ → A∆ = Aut(∆) → O∆ = Out(∆) → 1 If we form the quotient A0∆ of A∆ by the kernel of the quotient ∆ ³ ∆ab (to the maximal abelian quotient of ∆), then we obtain a natural commutative diagram in which the rows are exact:

1

→

1

→

∆ ↓ ∆ab

→ →

A∆ ↓ A0∆

→ →

O∆ ↓ O∆

→

1

→

1

In particular, we obtain a natural action of O∆ on ∆ab which preserves the natural symplectic form on ∆ab i.e., the symplectic form determined by the cup product on group cohomology H 1 (∆, Z p )×H 1 (∆, Z p ) → H 2 (∆, Z p ) ∼ = Z p (where 1 ab we think of H (∆, Z p ) as the Z p -linear dual to ∆ ) up to multiplication by a scalar. Denote by

GSp(∆ab ) the group of Z p -linear automorphisms of ∆ab which preserve this symplectic form up to multiplication by a scalar. Thus, we obtain a natural homomorphism

O∆ → GSp(∆ab ) together with an O∆ -equivariant action of ∆ab on A0∆ (via the inclusion ∆ab ,→ A0∆ ), which determines an isomorphism of pronite groups

(∆ab o O∆ ) ×O∆ A0∆ ∼ = A0∆ ×O∆ A0∆ ; in more geometric language, A0∆ is a ∆ab -torsor over O∆ . In particular, by applying this isomorphism, we obtain a natural projection:

A0∆ ×O∆ A0∆ → (∆ab o O∆ ) → (∆ab o GSp(∆ab )) (p) Moreover, any choice of symplectic isomorphism ∆ab ∼ = ΠG/A determines an isomorphism (p) (∆ab o GSp(∆ab )) ∼ =Π G

which, up to composition with an inner automorphism, is independent of our choice of symplectic isomorphism. In a similar vein, write M for the moduli stack of smooth, proper curves of genus g over K , and

C→M for the tautological curve over M. Also, write J → M for the Jacobian of C → M and (for d ∈ Z ) Jd → M for the J -torsor over M that parametrizes line bundles on C of relative degree d over M. By assigning to a point of the

156

SHINICHI MOCHIZUKI

curve C the line bundle on C dened by that point (regarded as an eective divisor), we obtain a natural morphism:

C → J1 Moreover, the action of J on J1 determines an isomorphism

J ×M J1 ∼ = J1 ×M J1 hence also a projection J1 ×M J1 → J . In particular, by composition, we obtain a morphism C ×M C → J1 ×M J1 → J (over M), which may be thought of, in terms of S -valued points (where S is a scheme), as the morphism that maps a pair of points (x, y) ∈ C(S) ×M(S) C(S) to the line bundle of degree 0 on C determined by the divisor x − y . Finally, if we write M → A for the Torelli morphism, i.e., the classifying morphism of the abelian scheme J → M equipped with its natural principal theta polarization, we thus obtain a natural commutative diagram

J → ↓ M →

G ↓ A

which is, in fact, cartesian, and, moreover, (by composition) induces a commutative diagram C ×M C → G ↓ ↓ M → A Denote by LC×M C the pull-back of LG to C ×M C . Next, we consider fundamental groups. Fix a geometric point m ¯ ∈ M(K), and a set Σ of prime numbers such that p ∈ Σ. Write π1 (C/M) for π1 (Cm ¯ ), and π1 (C/M) ³ ΠC/M for the maximal pro-Σ quotient of π1 (C/M). Since this quotient is characteristic, its kernel is also normal when regarded as a subgroup of π1 (C); denote the quotient of π1 (C) by this kernel by ΠC . Thus, if we write def ΠM = π1 (M), then we obtain an exact sequence:

1 → ΠC/M → ΠC → ΠM → 1 Moreover, the morphism C ×M C → G considered in the preceding paragraph induces a morphism on fundamental groups: (p)

ΠC ×ΠM ΠC → ΠG Finally, consider the diagram

ΠC ×ΠM ΠC ↓ A∆ ×O∆ A∆

→ →

(p)

ΠG ↓ (∆ab o GSp(∆ab ))


157

where the lower horizontal morphism is the morphism constructed above; the vertical morphism on the left arises from the natural action by conjugation of ΠC on ΠC/M (a pronite group whose maximal pro-p quotient is isomorphic to ∆); and the vertical morphism on the right is the isomorphism (well-dened up to composition with an inner automorphism) discussed above It is an immediate formal consequence of our denitions that this diagram commutes up to composition with an inner automorphism. In particular, by pulling back along the morphisms in this commutative diagram the group cohomology classes discussed in Lemma 4.7, it thus follows formally from Lemma 4.7 that: Corollary 4.8.

phisms

Suppose that K = Q . Then relative to the tautological mor-

(C ×M C)et → B(ΠC ×ΠM ΠC ) → B(A∆ ×O∆ A∆ ) 2 (see Lemma 4.7), the class c1 (LC×M C ) mod pn ∈ Het (C ×M C, (Z/pn Z)(1)) 2 n arises from a class ∈ H (A∆ ×O∆ A∆ , (Z/p Z)(1)), for all integers n ≥ 1.

4.3. A generalization of the main result of [Mzk2]. In this section, we

maintain the notation of Section 4.2, except that we again allow K to be an arbitrary eld of characteristic 0 (until further notice). Assume that we are given two hyperbolic curves X1 , X2 of type (g, r) over K . For i = 1, 2, write π1 ((Xi )K ) ³ Π(Xi ) for the maximal pro-Σ quotient K of π1 ((Xi )K ), and π1 (Xi ) ³ ΠXi for the quotient of π1 (Xi ) by the kernel of π1 ((Xi )K ) ³ Π(Xi ) . Thus, for i = 1, 2, we obtain exact sequences K

1 → Π(Xi )K → ΠXi → ΓK → 1. Next, assume that we are given an isomorphism

α : ΠX1 ∼ = ΠX2 which preserves and induces the identity on the quotients ΠXi ³ ΓK . Thus, α induces isomorphisms:

ΠαK : Π(X1 )K ∼ = Π(X2 )K , αH 2 : H 2 (X1 , Z p (1)) ∼ = H 2 (Π et

et

X1 , Z p (1))

2 ∼ (X2 , Z p (1)). = H 2 (ΠX2 , Z p (1)) ∼ = Het

Let X be a proper hyperbolic curve over K equipped with a nontrivial automorphism σ : X ∼ = X over K . Denote the Jacobian of X by JX . Lemma 4.9.

Then the morphism

δσ : X → JX

that maps an S -valued point x ∈ X(S) (where S is a scheme ) to the degree 0 line bundle determined by the divisor x − σ(x) is nonconstant . Assume (without loss of generality) that X(K) is nonempty. Then we may think of JX as the Albanese variety associated to X . Write λ : X → JX for the morphism exhibiting JX as the Albanese variety of X . Then (by the universal property of the Albanese variety) δσ necessarily factors through λ, Proof.

158

SHINICHI MOCHIZUKI

inducing a morphism δσλ : JX → JX which is nonconstant if and only if δσ is nonconstant. On the other hand, δσλ is easily computed to be (up to composition with a translation) equal to the morphism 1 − Jσ , where Jσ is the automorphism induced on JX by σ . Thus, it suces to verify that Jσ is not equal to the identity. But this follows again (formally) from the universal property of the Albanese variety. ¤ Corollary 4.10. Suppose that r = 0. Then for i = 1, 2, there exist ample 2 (X1 , Z p (1)) maps to line bundles Pi on Xi with the property that c1 (P1 ) ∈ Het 2 2 . c1 (P2 ) ∈ Het (X2 , Z p (1)) under αHet

First, observe that by replacing the Xi by nite étale Galois coverings that correspond via α, we may assume (without loss of generality) that Xi admits a K -automorphism σi such that σ1 , σ2 correspond via α. Indeed, once the necessary line bundles are dened over these coverings, one obtains line bundles on the original curves with the desired properties by simply taking the norm of the line bundles on the coverings. Since Xi denes a classifying morphism κi : Spec(K) → M, we let Pi0 be the pull-back (where we note that Xi = C ×M,κi Spec(K)) of the line bundle LC×M C of Corollary 4.8 to Xi ×K Xi . Moreover, by Lemma 4.9, it follows that the pull-back Pi of Pi0 via the morphism (1, σi ) : Xi → Xi ×K Xi given by the product of the identity and the automorphism σ is ample on Xi . Now consider the homomorphism on fundamental groups (well-dened up to composition with an inner automorphism)

Proof.

ΠXi → ΠC induced by κi . Note that the composite

ΠXi → A∆ of this homomorphism with the natural homomorphism ΠC → A∆ of Section 4.2 may be constructed entirely group-theoretically (from the action by conjugation of ΠXi on its normal subgroup Π(Xi ) ). Thus, in particular, it follows that (for K i = 1, 2) these composites are compatible with α. This compatibility implies that the composite of the homomorphism

ΠXi ×ΓK ΠXi → ΠC ×ΠM ΠC (induced by κi ) with the homomorphism ΠC ×ΠM ΠC → A∆ ×O∆ A∆ of Section 4.2 is a homomorphism

ΠXi ×ΓK ΠXi → A∆ ×O∆ A∆ which is compatible with α. Thus, it follows formally from Corollary 4.8 that the Chern classes of the Pi0 correspond via α. Since the pull-back via (1, σi ) may also be dened entirely group-theoretically, we thus conclude that the Chern


159

classes of the Pi0 correspond via α, as desired. This completes the proof of Corollary 4.10. ¤ We are now ready to state and prove the main result of the Section 4. We say that K is generalized sub-p-adic if K may be embed¯ p) ded as a subeld of a nitely generated extension of the quotient eld of W (F (the ring of Witt vectors with coecients in the algebraic closure of F p ).

Definition 4.11.

Sub-p-adic elds are always generalized sub-p-adic. On the other hand, elds such as the maximal algebraic extension of Q which is unramied over p are generalized sub-p-adic, but not sub-p-adic. Remark.

Suppose that K is generalized sub-p-adic, where p ∈ Σ. Then note that even if we do not know a priori that the hyperbolic curves X1 , X2 are of the same type (g, r), the mere existence of an isomorphism

Remark.

α : ΠX1 ∼ = ΠX2 (which preserves and induces the identity on the quotients ΠXi ³ ΓK ) already implies that X1 and X2 are of the same type. Indeed, to see this, we reduce im¯ p ) and then mediately to the case where K is nite over the quotient eld of W (F consider the dimension of the weight 0 portion of the HodgeTate decomposition of the maximal pro-p abelian quotient of Π(Xi ) . This dimension gives back the K genus g ; then r may be recovered from the fact that Π(Xi ) is free on 2g + r − 1 K generators (respectively, not free) if r > 0 (respectively, r = 0). Thus, there is no loss of generality in assuming (as we did in the above discussion) that the Xi are of the same type (g, r). Theorem 4.12 (Isomorphism version of the Grothendieck conjecture

over generalized sub-p-adic fields). Suppose that K is a generalized sub-p-adic eld , where p ∈ Σ. Let X1 , X2 be hyperbolic curves over K .

Write Isom(X1 , X2 ) for the set of K -isomorphisms between X1 and X2 , and IsomΓK (ΠX1 , ΠX2 ) for the set of continuous group isomorphisms ΠX1 ∼ = ΠX2 over ΓK , considered up to composition with an inner automorphism arising from Π(X1 ) or Π(X2 ) . Then the natural map K

K

Isom(X1 , X2 ) → IsomΓK (ΠX1 , ΠX2 )

is bijective . One formal consequence of Theorem 4.12 (see [Mzk2], Theorem C) is the following: Remark.

If K is generalized sub-p-adic , and M denotes the moduli stack of hyperbolic curves of type (g, r) over K , then the natural morphism M(K) → SectΓK (ΓK , ΠM )

is injective .

160

SHINICHI MOCHIZUKI

(Here, ΠM is as dened in Section 4.2 (where we note that the same formal denition can be made even in the case r > 0), and SectΓK (ΓK , ΠM ) is the set of sections of the projection ΠM ³ ΓK considered up to composition with an inner automorphism of ΠM arising from ΠMK .) In a word, once one has Corollary 4.10, the proof of Theorem 4.12 is entirely similar to the proof of Theorem A in [Mzk2], so we will only sketch details. First, we reduce immediately to the case where K is a nite extension of the ¯ p ) (see Lemmas 4.13, 4.14, below; [Mzk2], § 15), and X1 , quotient eld of W (F X2 are proper of genus g (see [Mzk2], proof of Theorem 14.1). Now, the main idea of the proof is to replace the portion of the proof of [Mzk2] given in [Mzk2], § § 16, by Corollary 4.10, by using the following argument. First, write def 2 Hi = Het (Xi , Q p (1)) for i = 1, 2, Proof.

and Gi ⊆ Hi for the geometric part of Hi , i.e., the Q p -subspace generated by rst Chern classes of line bundles on Xi . Also, write Ji for the Jacobian of Xi , and T (Ji ) for its associated p-adic Tate module. Note that α induces an isomorphism αH : H1 ∼ = H2 . Also, observe that since K has cohomological dimension 1 (see Lemma 4.13 below), applying the LeraySerre spectral sequence (for Galois (p) cohomology with coecients in Q p (1)) to the surjection ΠXi ³ ΓK gives rise to an exact sequence:

0 → H 1 (K, T (Ji ) ⊗ Q p ) → Hi → Q p → 0 (where the T (Ji ) on the left should, strictly speaking, be the Cartier dual of T (Ji ), but we identify T (Ji ) with its Cartier dual via the standard principal polarization on the Jacobian Ji ; the Q p on the right arises from the isomorphism (p) H 0 (K, H 2 (Π(Xi ) , Q p (1))) ∼ = Q p dened by the degree map). K Now I claim that αH (G1 ) = G2 . Indeed, by Corollary 4.10, there exists a line bundle Pi of nonzero degree on Xi such that αH (c1 (P1 )) = c1 (P2 ). Thus, to show that αH (G1 ) = G2 , it suces to show that αH preserves rst Chern classes of line bundles of degree 0. Since we are working over Q p , we may always replace K by a nite extension of K without aecting the validity of the claim. In particular, we may assume that the Xi have semi-stable reduction over K . Write Ji for the unique semi-abelian scheme over OK whose generic ber is Ji . Now if Li is a line bundle of degree 0 on Xi , it denes a point [Li ] ∈ Ji (K), hence, by Kummer theory (i.e., considering the obstruction to the p-power divisibility of [Li ] see [Mzk2], § 6, the discussion following Denition 6.1), determines an element κ(Li ) ∈ H 1 (K, T (Ji ) ⊗ Q p ). If we regard this Galois cohomology group as a subspace of Hi via the exact sequence of the preceding paragraph, then this class κ(Li ) coincides with the rst Chern class c1 (Li ) of Li (see [Mzk2], the Remark preceding Denition 6.2). On the other hand, if [L1 ] ∈ J1 (K) arises from a point ∈ J1 (OK ) which is equal to the zero section modulo mK , then κ(L1 )


161

corresponds, via α, to κ(L2 ) for some degree 0 line bundle L2 on X2 dened by a point ∈ J2 (OK ) (which will also be equal to the zero section modulo mK ). Indeed, this follows by applying Tate's theorem (see [Tate], Theorem 4) as in the argument of the proof of [Mzk2], Theorem 7.4, to the p-divisible groups dened by the formal groups associated to J1 , J2 . Moreover, for any point ¯ p = OK /mK is a union of nite elds ∈ J1 (K) it follows from the fact that F that some nonzero multiple of this point arises from a point ∈ J1 (OK ) which is equal to the zero section modulo mK . Thus, since we are working with Q p coecients, we thus conclude that αH maps the Q p -subspace of H1 generated by rst Chern classes of line bundles of degree 0 onto the corresponding subspace of H2 . This completes the proof of the claim. Before proceeding, we note here that the argument of the preceding paragraph is the only place in this proof where we use that the original base eld is a subeld ¯ p ) i.e., as opposed of a nitely generated extension of the quotient eld of W (F to W (k), where we permit k to be an arbitrary perfect eld of characteristic p. ¯ p replaced The arguments to be used in the remainder of the proof are valid for F by an arbitrary such k . Also, we remark here that (not surprisingly) the portion of the argument of [Mzk2] that corresponds to what was done in the preceding paragraph is given in [Mzk2], § § 16, where it was necessary, especially for the arguments of [Mzk2], Lemma 4.1, § 6, to assume that the residue eld be nite (i.e., not an arbitrary perfect eld of characteristic p). Now that we know that α preserves (up to Q p -coecients) rst Chern classes of line bundles over nite extensions of K , the rest of the argument of [Mzk2] goes through without much change. Namely, [Mzk2], Proposition 7.4, follows by the same argument as that given in loc. cit. (except that instead of working over a K which is nite over Q p as in loc. cit., we work over the present K , ¯ p )). We remark that in the present which is nite over the quotient eld of W (F context, it is not necessary to distinguish between F -geometricity and F I geometricity as was done in [Mzk2], since we are working with proper curves to begin with (see the Remark at the end of [Mzk2], § 7). Then [Mzk2], § 8, goes through without change (except that the nite eld k is to be replaced ¯ p ). The convergence arguments of [Mzk2], § 9, 10, are entirely valid when by F k is any perfect eld of characteristic p, so no changes are necessary in these two § 's. [Mzk2], § 11, is unnecessary in the present context since we are working with proper curves to begin with. Finally, the arguments of [Mzk2], § § 1214, go through without essential change (except that they are much easier in the present context since we are working with proper curves to begin with). This completes the proof of the bijectivity assertion of Theorem 4.12 (see [Mzk2], Corollary 14.2). ¤ Thus, Theorem 4.12 states (roughly) that the isomorphism class of ¯ p ) may a hyperbolic curve over a nite extension of the quotient eld of W (F be recovered from the outer action of the Galois group on its geometric fundaRemark.

162

SHINICHI MOCHIZUKI

¯ p /F p ) (as was done mental group. In particular, one need not make use of Gal(F in [Mzk2]). In this sense, Theorem 4.12 is reminiscent of the main results of [Tama2], which state that in certain cases, the isomorphism class of a hyperbolic ¯ p is completely determined by the isomorphism class of its geometric curve over F fundamental group (see the results of [Tama1], which make essential use of the ¯ p /F p )). It would be interesting to see if the relationship between action of Gal(F Theorem 4.12 and [Tama2] could be understood more explicitly. Another interesting aspect of Theorem 4.12 is the following: Note ¯ p ), then its absolute that, if K is a nite extension of the quotient eld of W (F Galois group ΓK has cohomological dimension 1 (see Lemma 4.13 below). On the other hand, if X is a hyperbolic curve over K , then ΠXK has cohomological dimension 2. Thus, the cohomological dimension of ΠX is equal to 3. Since, roughly speaking, Theorem 4.12, states that the structure of X is determined by ΠX , Theorem 4.12 is reminiscent of the rigidity theorem of MostowPrasad for hyperbolic manifolds of real dimension 3 (see the discussion of [Mzk4], Introduction, § 0.10, 2.2.3, 2.2.6). Remark.

The following lemma is, in essence, well-known:

¯ p ). Then Let K be a nite extension of the quotient eld of W (F is center-free and has cohomological dimension equal to 1.

Lemma 4.13.

ΓK

¯ p ), and L0 for the union First, write L for the quotient eld of W (F (inside L) of the quotient elds of the W (k), as k ranges over all nite extensions of F p . Then the ring of integers OL0 of L0 is the union of the W (k), hence stictly ¯ p ) is the p-adic completion of OL0 , it henselian. Moreover, since OL = W (F follows immediately from the general theory of henselian rings that ΓL = ΓL0 . In particular, since K is a nite extension of L, it follows that there exists a nite extension K 0 of L0 such that K = L ⊗L0 K 0 . Moreover, since OK 0 is strictly henselian, with completion equal to OK , we have ΓK = ΓK 0 . Thus, it suces to prove that ΓK 0 is center-free and of cohomological dimension equal to 1. In the remainder of the proof, to simplify notation, we shall simply write K for K 0 . Next, observe that by considering the maximal tamely ramied extension K tm of K , we obtain an exact sequence Y Z l (1) → 1 1 → ΓK tm → ΓK → Proof.

l6=p

(where the product ranges over all prime numbers not equal to p). Moreover, recall that ΓK tm is a pro-p-group which is center-free and of cohomological dimension 1 (see, e.g., [Mzk2], the proof of Lemma 15.6). We thus obtain immediately that ΓK is of cohomological dimension 1. To show that ΓK is center-free, it sufces to show Gal(K tm /K) acts faithfully on Γab K tm ⊗ Q p (where ab denotes the maximal abelian quotient). But this may be done as follows: Let K0 ⊆ K be a


163

nite extension of Q p such that K/K0 is unramied. Thus, it follows that K tm is also the maximal unramied extension of K0 , so Gal(K tm /K) ,→ Gal(K tm /K0 ). Now let L0 /K0 be a nite, Galois, totally tamely ramied extension of K0 . Then L0 ⊆ K tm , so we obtain a surjection: ab wild Γab ⊗ Qp K tm ⊗ Q p ³ (ΓL0 )

(where the superscript wild denotes the wild inertia subgroup). But by the wild class eld theory of nite extensions of Q p , one knows that (Γab ⊗ Q p is L0 ) naturally isomorphic (via the p-adic logarithm see, e.g., [Mzk5], § 2) to L0 , so wild the action of Gal(L0 /K0 ) on (Γab ⊗ Q p is faithful. Since arbitrary nite L0 ) tm quotients of Gal(K /K) may be realized as Gal(L0 /K0 )'s for appropriate choices of K0 , L0 , it thus follows that Gal(K tm /K) acts faithfully on Γab K tm ⊗ Q p , as desired. This completes the proof of Lemma 4.13. ¤ Lemma 4.14.

Let K be generalized sub-p-adic . Then ΓK is center-free .

Let K be an arbitrary generalized sub-p-adic eld. If XL is any hyperbolic curve of type (g, r) over a nite extension L of K , and σ ∈ Z(ΓK ) (i.e., the center of ΓK ), then we have an isomorphism

Proof.

Π(XL )K ∼ = Π(XL )σK (induced by conjugating by σ ) which is compatible with the outer actions of ΓL on both sides. With this observation in hand, it follows that Lemma 4.14 may be derived from Lemma 4.13 by means of Theorem 4.12 using exactly the same argument as that used to derive [Mzk2], Lemma 15.8, from [Mzk2], Lemma 15.6, by means of [Mzk2], Corollary 15.3. ¤

References [AG] N. Alling and N. Greenleaf, Foundations of the theory of Klein surfaces, Lecture Notes in Mathematics 219, Springer, Berlin, 1971. [BMS] H. Bass, J. Milnor, and J.-P. Serre, Solution of the congruence subgroup problem for SLn , (n ≥ 3) and Sp2n , (n ≥ 2), Publ. Math. IHES 33 (1967), 59 137. [Falt] G. Faltings, Endlichkeitssätze für Abelschen Varietäten über Zahlkörpern, Inv. Math. 73 (1983), 349366. [Frdl] E. Friedlander, Étale homotopy of simplicial schemes, Annals of Math. Studies 104, Princeton University Press, 1982. [Gard] F. Gardiner, Teichmüller theory and quadratic dierentials, Wiley, New York, 1987. [Groth] A. Grothendieck, letter to G. Faltings (June 1983) in Lochak, L. Schneps, Geometric Galois actions, 1: Around Grothendieck's Esquisse d'un programme, London Math. Soc. Lect. Note Ser. 242, Cambridge Univ. Press, 1997.

164

SHINICHI MOCHIZUKI

[Harts] R. Hartshorne, Algebraic Geometry, Graduate Texts in Math. 52, Springer, New York, 1977. [Huis] J. Huisman, On the fundamental group of a real algebraic curve, manuscript. [Kerck] S. Kerckho, The Nielsen realization problem, Bull. Amer. Math. Soc. (N.S.) 2 (1980), 452454. [Krav] S. Kravetz, On the geometry of Teichmüller spaces and the structure of their modular groups, Ann. Acad. Sci. Fenn. Ser. A I 278 (1959), 35. [Lang] S. Lang, Abelian varieties, Springer, New York, (1983). [Laz] M. Lazard, Groupes analytiques p-adiques, Publ. Math. IHES 26 (1965), 389 603. [Maass] H. Maass, Siegel's Modular Forms and Dirichlet Series, Lecture Notes in Mathematics 216, Springer, Berlin, 1971. [Mord] L. Szpiro, Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell, Astérisque 127, Soc. Math. France (1985). [Mzk1] S. Mochizuki, The pronite Grothendieck Conjecture for closed hyperbolic curves over number elds, J. Math. Sci., Univ. Tokyo 3 (1996), 571627. [Mzk2] S. Mochizuki, The local pro-p anabelian geometry of curves, Inv. Math. 138 (1999), 319423. [Mzk3] S. Mochizuki, A theory of ordinary p-adic curves, Publ. of RIMS 32 (1996), 9571151. [Mzk4] S. Mochizuki, Foundations of p-adic Teichmüller Theory, AMS/IP Studies in Advanced Mathematics 11, American Mathematical Society/International Press (1999). [Mzk5] S. Mochizuki, A version of the Grothendieck conjecture for p-adic local elds, The International Journal of Math. 8:4 (1997), 499506. [NSW] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of number elds, Grundlehren der math. Wissenschaften 323, Springer (2000). [NTM] H. Nakamura, A. Tamagawa, S. Mochizuki, The Grothendieck conjecture on the fundamental groups of algebraic curves, Sugaku 50 (1998), 113129; English translation in Sugaku Expositions (to appear). [Schd] C. Scheiderer, Real and étale cohomology, Lecture Notes in Mathematics 1588, Springer, Berlin, 1994. [Ser1] J.-P. Serre, Lie algebras and Lie groups, Lecture Notes in Mathematics 1500, Springer, Berlin, 1992. [Ser2] J.-P. Serre (with the collaboration of Willem Kuyk and John Labute), Abelian l-adic Representations and Elliptic Curves, Addison-Wesley, Reading (MA), 1989. [Ser3] J.-P. Serre, Liste des courbes elliptiques à multiplication complexe dont l'invariant modulaire j est rational, manuscript. [Ser4] J.-P. Serre, Local Fields, Graduate Texts in Mathematics 67, Springer, Berlin, 1977. [SGA1] A. Grothendieck et al., Revêtements étales et groupe fondamental, Lecture Notes in Mathematics 224, Springer, Berlin, 1971.


165

[SGA2] A. Grothendieck et al., Cohomologie locale des faisceaux cohérents et théorèmes de Lefshetz locaux et globaux, North-Holland, Amsterdam, 1968. [Shi] G. Shimura, Introduction to the arithmetic theory of automorphic forms, Publ. Math. Soc. of Japan 11, Iwanami Shoten and Princeton University Press (1971). [Stk] H. Stark, A complete determination of the complex quadratic elds of classnumber one, Michigan Math. J. 14 (1967), 127. [Tama1] A. Tamagawa, The Grothendieck conjecture for ane curves, Compositio Math. 109:2 (1997), 135194. [Tama2] A. Tamagawa, On the tame fundamental groups of curves over algebraically closed elds of characteristic > 0, pp. 49107 in Galois groups and fundamental groups, edited by Leila Schneps, Cambridge University Press, New York, 2003. [Tate] J. Tate, p-divisible groups, pp. 158183 in Proceedings of a conference on local elds (Driebergen, 1966), edited by T. A. Springer, Springer, Berlin, 1967. [Weber] H. Weber, Lehrbuch der Algebra, Bd. 3: Elliptische Funktionen und algebraische Zahlen, Vieweg, Brauschweig, 1912; reprinted Chelsea, New York, 1961. [Wolp] S. Wolpert, Geodesic length functions and the Nielsen problem, J. Dierential Geom. 25 (1987), 275296. Shinichi Mochizuki Research Institute for Mathematical Sciences Kyoto University Kyoto 606-8502 Japan

[email protected]

!"$#$% &' ()* + ,-/.021334

57698:6=?69@BAC69DFEHG2GJI'K>L?I2MONQP>=!DR9MTSVU W?XZY\[^]`_V[ba\[^cd[fe![hgjifk^Yml^n^]Vo9pfqrTsô*k^tuXZe

vxwzy{}|~}{$zx'/)2')22 z* ,* 'J)' * 2? 'V') x}2)* z}¡2¢J''£ z* ¥¤!¦TV* '2 ))§'

¨ª©¼½Ñ Ì ¼½º¾ · ¼ÊÂ¼ Ì\Í Ë½Â¼½ÎÇÀ\¼½Ñ£µJÄÁÄ§¿§Ç¶¿ÁÆ:È Â¸¶¿Á¼}º À\ÕVÖ¿ÁÆÖÔÕh¼½ÎÄÁÊÔÖµJÄ§Çå¶¼Ú¸}ºÀÕhµ/ÂÀ¼ Ì µ>º¸¶ÎÂ¸}Ä · ç Îµ/À¶¿§¼½ºÀ/Ï[^ÎÂZ¼½Â¿§Ë½¿§º ¸Ä Ì ¼}¶¿§Ã}¸¶¿§¼½ºQÕ£¸}À!¶¼9Êµ/ÀÆ2Â¿ · µV¶ÖµfÀµ2¶h¼½ÑË½Â¼}ÎÇ ÀÆ2¼½ÂÂµzÀÇ,¼½º Ê¾ ¿§ºËØ¶¼ØÂ¸¶¿§¼½º ¸Ä¼½Âut\ã:µJÄ§ÄÁ¿§Ç¶¿ÁÆQÀÎÂÑ¸½Æ2µ/À µ2ÒÇÄÁ¸}¿§º»Ö¼zÕè¶¼¹Æ2¼ Ì ÇÎ¶µÜ¶ÖµQÊ¿ Ì µJº À¿§¼½º À Ù ¶Ö¿ÁÀÆJÄÁ¸½ÀÀ9ÕV¿¡¶ÖéË½¿¡Ã°µJº ¼½Ñx¶Öµ>ÀÇ ¸½Æ2µ/À9¼½Ñ Ì ¼ÊÎÄ§¿h¼½ÑVÀÎÂÑ¸½ÆJµ/À\¿Áº» Ì ¼½º¼ÊÂ¼ Ì\Í Ë½Â¼½ÎÇ µ2¶Æ½ÏîfÀ^¸>Ê¿ÁÂµ/Æ'¶b¸}ÇÇÄ§¿ÁÆ/¸¶¿§¼½ºÚ¼}Ñ?¶Öµ Ì µ2¶Ö¼ÊÀ^Êµ2Ã°µJÄ§¼½Ç,µzÊÐ¿Áº¹¶Öµ\ÇÂµ/ÀµJº°¶×Ç¸}ÇµJÂV¶Öµ ¸}Î¶Ö¼½ÂÀ¸}º ÊàpbÏ½ê?µ2¶Â¼zÃÜÖ¸/Ã°µ£¼ ¶¸}¿§ºµ/Ê]f¿§µ Ì ¸}ººÐÀÎÂÑ¸}ÆJµ×¿ÁÀ£¸ ¾¶Â¿Á¸}ºË½ÎÄÁ¸¶¿Á¼}ºªÏsf¼zÕxµ2Ã°µJÂ j º¼}¶!¸ÄÁÄ ¾)¶Â¿Á¸}ºË½ÎÄÁ¸¶¿§¼½ºÀ!¸}Â¿ÁÀµh¿Áº9¶Ö¿ÁÀ¥Õx¸ Í ÏháoºÇ ¸}Â¶¿ÁÆJÎÄÁ¸}Â ÕxµhÆ2¼½º À¿ÁÊµJÂ Ì ¼½ÂµhË½µJºµJÂ¸}Ä Ù j Ù ¶Â¿Á¸}ºË½ÎÄ ¸¶¿§¼½º À×¶Ö ¸}( º 'Î À¶ ¸}Â Í Æ2µ/º°¶Â¿ÁÆàÀÎ Ê¿¡Ã¿ÁÀ¿§¼½º Àb¼½ Ñ )_TµJÄ Í +¿ *Ü¶Â¿Á¸}ºË½ÎÄÁ¸¶¿Á¼}º ÀJÏ â [fÑ · · Æ2¼½ÎÂÀµ ¶Öµ>À¶ÎÊ Í ¼}Ñ£À¿ Ì ÇÄÁ¿ÁÆ2¿Á¸}ÄxÊµzÆ2¼ Ì Ç,¼°À¿¡¶¿§¼½ºÀ\¼½Ñh¼½Â¿§µJº$¶µ/Ê Ì ¸}º¿§Ñ¼½ÄÁÊÀ\Ë½¼$µzÀ ¸½Æ-, · ¸¶uÄ§µz¸}ÀÙ ¶u¶¼¹ifÄ§µJÒ¸}º Êµ/Â /Ý .2ß áÕVÖ¼ÐÇÂ¼zÃ°µ/À×¶Ö ¸¶\¿§1 º 03254àÊ¿ Ì µ/ºÀ¿Á¼}º ¶Öµ ¸}Â Í Æ2µJº$¶Â¿ÁÆ · n ÀÎ Ê¿¡Ã¿ÁÀ¿§¼½ºÉ¼½Ñ¸QË½¿¡Ã°µJºæÀ¿ Ì ÇÄ§¿ÁÆJ¿Á¸}Ä!ÊµzÆ2¼ Ì Ç¼°À¿¡¶¿§¼½ºæ¿§ºÊÎÆJµ/À^¸QÀ¿ Ì ÇÄ§¿ÁÆJ¿Á¸}Ä Ì ¸}Ç¹¶¼Q¶Öµ · ¾Ê¿ µJº À¿§¼½º ¸}Ä?ÀÇÖµ/Âµ ÕV¿¡¶ÖÚ¿¡¶À×À¶¸}ºÊ¸}ÂÊæÀ¿ Ì ÇÄÁ¿ÁÆ2¿Á¸}ÄÊµ/Æ2¼ Ì Ç¼½À¿¡¶¿§¼½ºÚ¿§º$¶¼:¶*Õh¼ØÀ¿ Ì ¾ n Ì ÇÄ§¿ Æ2µ/Àâ'ÏcÉ¼}Âµ^Âµ/ÆJµJº°¶Ä Í Ù Æ2¼½ºÀ¶ÂÎ Æ'¶¿Á¼}º Àh¼½Ñ¶Ö¿ÁÀT¶ Í ÇµfÕhµJÂµ×Âµ/Ê¿ ÀÆ2¼zÃ°µJÂµ/Ê ÍQÌ ¸º Í ¸}Î¾ · ¶Ö¼½ÂÀ¿§ºQÆJ¼½ººµ/Æ'¶¿§¼½ºàÕV¿¡Ù ¶Ö>_Tµ/Ä Í 7¿ 6 ÀZ¶ÖµJ¼½Âµ Ì ¸}ºÊ>a^Â¼¶ÖµJºÊ¿§µ/8Æ , 6 À )Y×µ/ÀÀ¿§º ÀT Ê 6 µJºÑ¸}º$¶9À * ÇÂ¼½Ë½Â¸ Ì á Ý ã ß Ý :ß ¸}º Êd¶Öµ>ÂµJÑµJÂµJº Æ2µ/À\¶Öµ/ÂµJ¿§ºâÏÚi ¾)¶Â¿Á¸}ºË½ÎÄÁ¸¶¿Á¼½ºé¼½ÑV¸É]V¿§µ Ì ¸}ºº j ÀÎÂÑ¸½Æ2µ ¿§º ÊÙ Î Æ2µ/ÀV¸ÜË½Â¸}ÇÖ ¼½º ¾T¸}ºÊ ÕVÖ¿ÁÆÖÐ¿ÁÀ£¼ ¶¸}¿§ºµ/Ê Í Âµ Ì ¼zÃ¿§ºË¼½Ñ¶Öµ ¾¶Â¿ ¸ºË½ÎÄ ¸¶¿§¼½ºdáÀµJµ ;Ý .$DPRQSCUTVGWXCD@BEYHJI _T¼½Ë½¼ Ì ¼½ÄÁ¼zÃuÕ£¸}ÀÇ ¸}Â¶¿Á¸}ÄÁÄ Í ÀÎÇÇ¼½Â¶µzÊ ¶Öµhkuq$WxÏpÀÆÖ¿§º,½µ/Ä · Í Õx¸½ÀTÇ ¸Â¶¿Á¸}Ä§Ä Í ÀÎÇÇ¼½Â¶µ/Ê Í ¶Öµ×kuqi9Ï°Ó»µ×¸}Âµ^Ë½Â¸¶µJÑÛÎÄ ¶¼ÅÏÊµ×Å½¼½ºËÜ¸}º Ê¶¼Ü¶ÖµbÂµJÑµJÂµJµuÑÛ¼½Â Ì ¸}º Í Î ÀµJÑÎÄÀÎË½Ë½µ/À¶¿§¼½ºÀ/Ï

D #

K

!Y #B!

!©,¬ ®

*´°´

oºÉ¶Ö¿ À×ÀµzÆ'¶¿Á¼}ºæÕxµË½¿¡Ã°µÜ¸ Â¿§µJÑTÀÎ ÌàÌ ¸}Â Í ½¼ Ñt^¼Ê¸}¿§Â¸6 Àf¶ÖµJ¼½Â Í ¼½Ñ!µ/Ä§Ä§¿§Ç¶¿ ÆÜÈ Â¸¾ · · ¶¿§¼½º ÀJÏÀµ/Æ'¶¿§¼½ºªÏfpxÖµ9µ ç Î¿¡Ã}¸}Ä§µ/ºÆ2µÆJÄÁ¸½ÀÀf¼½Ñ 0 ÎºÊµJÂfË½Ä§¼ ¸}ÄÊ¿§ÑÛ¾ Ù E · ÑÛµJ¼ Ì ¼½ÂÇÖ¿ÁÀ Ì ÀT¿§º ÊÎ Æ2¿§ºËÀ Ì ¼$¼}¶Ö>¿ÁÀ¼ Ì ¼½ÂÇÖ¿ÁÀ Ì Àh¼½ºQµ/¸½ÆÖQÈ µJÂT¿ÁÀxÊµ2¶µJÂ Ì ¿§ºµ/Ê Í ¶Öµ · · ¶¼½Ç,¼½ÄÁ¼}Ë Í ¼½Ñ 0 ¸}ºÊ Í ¶Öµ9Ö¼ Ì ¼ Ì ¼½ÂÇÖ¿ÁÀ Ì áÂµJÇÂµzÀµJº$¶¸¶¿§¼½º,â£¼½Ñ¶Öµ9ÑÎºÊ¸ Ì µ/º°¶¸}Ä · B Ë½Â¼½ÎÇ ¿§º$¶¼Ø¶ÖµàË½Â¼½ÎÇd¼½ÑTÖ¼ Ì ¼}¶¼½Ç Í Æ2Ä ¸½ÀÀµ/Àu¼½ÑT¼½Â¿§µJº$¶¸¶¿§¼½º¾ÇÂµ/ÀµJÂÃ¿§ºË¹¸}Î¾ π (B 0 ) ¶¼ Ì ¼½ÂÇÖ1¿ÁÀ Ì Àu¼½ÑZ¶ÖµÜ¶¼½ÂÎ À 1 áÕVÖ¿ÁÆÖ»¿ÁÀuµ ç Î ¸Ä!¶¼ âÏ>pxÖ$Î À×Õhµ
H I .4¿ÁºÊµ2Ò»¼½Ñx¶Öµ · · j Ë½Ä§¼ ¸}Ä Ì ¼½º¼ÊÂ¼ Ì\Í Ë½Â¼½ÎÇ ˜ ¿§º ÏT[fºÄ Í ¸ÜÈº¿¡¶µuº$Î Ì µJÂ£¼½ÑÇ¼°ÀÀ¿ Ä§µbË½ÄÁ¼ ¸}Ä · · · · Γ SL(2, Z)

"!$#%"&

¤

Ì ¼½º¼ÊÂ¼ Ì\Í Ë½Â¼½ÎÇ À Γ ˜ ¸}ºÊà¼½ºÄ Í Ñµ2ÕFÖ¼ Ì ¼½Ä§¼½Ë½¿ÁÆJ¸}Ä¿§º$Ã}¸}Â¿Á¸}º$¶ÀZÆ/¸}ºà¼ÆJÆ2ÎÂ¿§Ñ,Õhµ£ÈÒ¶Öµ ¿ Ì ¸Ë½µV¼½Ñ ˜ ¿§º Ï?XZÄ§Ä§¿§Ç¶¿ ÆVÀÎÂÑ¸½Æ2µ/À?ÕV¿¡¶ÖÜ¶ÖµfÀ¸ Ì µ ¾*¿§º$Ã}¸}Â¿Á¸}º$¶ Î¶ZÊ¿,µJÂµ/º°¶ j Γ PSL(2, Z) · Ö¼ Ì ¼½Ä§¼½Ë½¿ÁÆJ¸}ÄZ¿§º$Ã}¸}Â¿Á¸}º$¶À9¸}ÂµµJÄ§µ Ì µJº$¶À Z, Z/3 · ¼½Ñ ¶ÖµfÀ¸ Ì µV¼½ÂÊµ/Â¿§º 0 ÕhµV¼ ¶¸}¿§º¼½º>µ2Ã°µJÂ Í ÆJ¼ Ì Ç¼¾ V · ºµJº$¶T¼½Ñ Ï!pxÖµJÂµf¿ÁÀh¸\Æ2¼½ÂÂµzÀÇ,¼½ºÊµJº Æ2µ µ2¶*ÕxµJµJº>ÀÎ ÆÖ>Ê¼½Î Ä§µ^Æ2¼zÃ°µJÂ¿§ºË°ÀT¸}ºÊ>ÀÇ,µ/Æ2¿ ¸}Ä D · · ÆÖ ¸Â¸}Æ2¶µJÂÀ È

·

χ ∈ Hom(π1 (V o \ D), Z/2).

pxÖµVË½Â¼½ÎÇ Ö¸½ÀZ¸ ç Î¼}¶¿§µ/º°¶ZÕVÖ¿ ÆÖÕV¿¡¶Ö χ(E) = 12r · · Ù Ù µJÄ§¼½ºË°À¥¶¼ Ï?pxÖµhÑ¸ Ì ¿§Ä Í ¿ ÀÇ ¸}Â¸ Ì µ2¶Â¿ Jµ/Ê Í ¶ÖµxÆ2¼µ µJÄÁÄ§¿§Ç¶¿ÁÆ · · È Â¸¶¿Á¼}º ¿ À ¿§Â¸¶¿§¼½º¸}Äf¶¼é¸ Ì ¿Áº¿ Ì ¸}Ä^µ/Ä§Ä§¿§Ç¶¿ ÆæÈ Â¸¶¿§¼½º ¸}º Ê ¶Öµ ¾ Ì ¸}Ç Ñ¼½Â ¼}¶Ö j · · · · È Â¸¶¿Á¼}º Àx¿ÁÀx¶Öµ\À¸ Ì µ°Ï · [^ºÐ¶Öµ9¼¶ÖµJÂfÖ ¸º Ê ¶Öµ\XZÎÄ§µJÂ^ÆÖ¸}Â¸½Æ'¶µJÂ¿ÁÀ¶¿ÁÆ Æ/¸º µ9Æ2¼ Ì ÇÎ¶µzÊ¹¸½À^¸QÀÎ Ì χ(E) · ¼½ÑÆ2¼½º$¶Â¿ Î¶¿Á¼}º À£ÑÛÂ¼ Ì Ù À¿§ºË½ÎÄÁ¸}ÂVÈ µ/ÂÀ95 · · χb I0 In II III IV

n 2 3 4

χb I∗0 6 ∗ In n+6 IV∗ 8 III∗ 9 . II∗ 0

s µ/Âµ f ¿ÁÀà¸ÚÀ Ì ¼¼}¶Ö È µJÂ ¿ÁÀà¸ Ì ÎÄ§¶¿§ÇÄ§¿ÁÆ/¸¶¿¡Ã°µ¹È µJÂÜÕV¿¡¶Ö ¾*¿§ÂÂµ/ÊÎÆJ¿ ÄÁµÐÆ2¼ Ì ¾ · Ù In · · I0 n Ç,¼½ºµ/º°¶ÀJÏépxÖµ:¶ Í Çµ/À ¸}º Ê Æ2¼½ÂÂµ/ÀÇ¼½ºÊÔ¶¼Ú¶ÖµÐÆJ¸½ÀµÐ¼½ÑVÇ¼}¶µJº$¶¿Á¸}Ä§Ä Í Ë½¼$¼Ê II, III IV Âµ/ÊÎÆ'¶¿§¼½ºªÏVcÉ¼}Âµ9ÇÂµ/ÆJ¿ÁÀµJÄ Í ¶Öµ9ºµJ¿§Ë½Ö ¼½ÂÖ¼¼ÊÐ¼½ÑÀÎ ÆÖæ¸QÈ µJÂf¿ À^¸Éá)Êµ/À¿§ºË½ÎÄÁ¸}Â¿+/¸¾ · · ¶¿§¼½ºÚ¼½ÑZ¸$â ç Î¼¶¿§µJº$¶b¼½ÑZ¸>Ä§¼Ù ÆJ¸}ÄÈ Â¸¶¿§¼½ºæÕV¿¡¶ÖdÀ Ì ¼$¼}¶ÖÚÈ µJÂÀ Í ¸}ºÚ¸}Î¶¼ Ì ¼½ÂÇÖ¿ÁÀ Ì · · · ¸}º Ê ¿Áº¹¶ÖµÜÆJ¸½Àµ/À ¼½Ñ!Èº¿¡¶µ9¼½ÂÊµJÂzÏfpxÖµÆJ¼½ÂÂµ/ÀÇ,¼½º Ê¿§ºËQ¼½ÂÊµJÂ^¿ÁÀ Ñ¼½ÂV¶Öµ9Æ/¸½Àµ 4 III 3 Ï£pxÖµ\È µJÂÀV¼}Ñ?¶ Í Ç,µ ∗ áÂµ/ÀÇÏ ∗ ∗ ∗ ∗ ¸}Âµ\¼ ¶¸}¿§ºµ/Ê¹ÑÛÂ¼ Ì È µJÂÀ II, IV I , I , II , III , IV ) I · · · ¶ÕV¿Á0À¶¿§ºËdáÆÖ¸}ºn Ë}¿ÁºË
j

M &($(') )/0'3 ;) 3G0/ 7;I .4 1, dn = ords=1−n ζF (s) = r1 + r2 (31)  r2 when n is even, where ζF (s) denotes the Dedekind zeta function of F and r1 , and r2 denote the number of real and complex places of F , respectively. m For all n and m, Hmot (XF,S , Q(n)) is a nite dimensional rational vector space whose dimension is given by ( d + #S when m = n = 1,

Theorem 3.1.

m dim Hmot (XF,S , Q(n)) =

1

dn when m = 1 and n > 1, 0 otherwise . Proof. First suppose that S is empty. Quillen [37] showed that each K -group Km (XF,S ) is a nitely generated abelian group. It follows that each of the groups j Hmot (XF,S , Q(n)) is nite dimensional. The rank of K0 (XF,S ) is 1 and the rank of K1 (XF,S ) is r1 + r2 − 1 by the Dirichlet Unit Theorem. The ranks of the remaining Km (XF,S ) were computed by Borel [10]. It is zero when m is even and > 0, and dn when m = 2n − 1 > 1. It is easy to see that 0 Hmot (XF,S , Q(0)) = K0 (XF,S ) ⊗ Q ∼ = Q.

Borel [11] constructed regulator mappings

K2n−1 (XF,S ) → R dn ,

n > 0,

and showed that each is injective mod torsion. Beilinson [1] showed that Borel's regulator is a non-zero rational multiple of the regulator mapping 1 chn : K2n−1 (XF,S ) → HD (XF,S , R(n)) ∼ = R dn

to Deligne cohomology. The properties of the Chern character and Borel's injectivity together imply that 1 Hmot (XF,S , Q(n)) = K2n−1 (XF,S )(n) = K2n−1 (XF,S ) ⊗ Q

188

RICHARD HAIN AND MAKOTO MATSUMOTO

m and that Hmot (XF,S , Q(n)) vanishes when m > 1, and when m = 0 and n 6= 0. The result when S is non-empty follows from this using the localization sequence [39], and the fact, due to Quillen [38], that the K -groups of nite elds are torsion groups in positive degree. Together these imply that each prime removed adds one to the rank of K1 and does not change the rank of any other K -group. ¤

Denote the Galois group of the maximal algebraic extension of F , unramied outside S , by GF,S . In this paper, a nite dimensional GF,S -module means a nite dimensional Q ` -vector space with continuous GF,S -action. Denote the category of Q mixed Hodge structures by H. Denote the ext functor in the category of nite dimensional GF,S -modules by ExtGF,S , and the ext functor in H by ExtH . The results on regulators of Borel [11] and Soulé [41] can be stated as follows.

The natural transformation from motivic to étale cohomology induces isomorphisms Theorem 3.2.

1 Hmot (XF,S , Q ` (n)) ∼ = Ext1GF,S (Q ` , Q ` (n)) = Hé1t (XF,S , Q ` (n)) ∼

for all n ≥ 1. The natural transformation from motivic to Deligne cohomology induces injections

· 1 Hmot (XF,S , Q(n))

,→

1 HD (XF,S , Q(n))

=

M

¸Gal(C/R) Ext1H (Q, Q(n))

. ¤

ν:F ,→C

Thus each element x of K2n−1 (XF,S ) determines an extension

0 → Q ` (n) → E`,x → Q ` (0) → 0 of `-adic local systems over XF,S and a Gal(C/R)-equivariant extension

0 → Q(n) → EHodge,x → Q(0) → 0 of mixed Hodge structures over XF,S ⊗ C . One can think of these as the étale and Hodge realizations of x ∈ K2n−1 (XF,S ).

4. Mixed Tate Motives As mentioned earlier, one approach to motivic cohomology is to postulate that to each suciently nice scheme X (say, smooth and quasi-projective over a eld, or regular over a ring OF,S of S -integers in a number eld) one can associate a category T (X) of mixed Tate motives over X . This should satisfy the following conjectural properties. (i) T (X) is a (neutral) tannakian category over Q with a ber functor ω : T (X) → VecQ to the category of nite dimensional rational vector spaces.

TANNAKIAN FUNDAMENTAL GROUPS ASSOCIATED TO GALOIS GROUPS 189

(ii) Each object M of T (X) has an increasing ltration called the weight ltration · · · ⊆ Wm−1 M ⊆ Wm M ⊆ Wm+1 M ⊆ · · · , whose intersection is 0 and whose union is M . Morphisms of T (X) should be strictly compatible with the weight ltration that is, the functor M M Wm M/Wm−1 M GrW GrW m M := • : M 7→ m

m

to graded objects in T (X) should be an exact tensor functor. (iii) T (X) contains the Tate motive Q(1) over Spec R where R is the base ring (here either a eld or OF,S ). This can be considered as the dual of the local system R1 f∗ (Q) over Spec R, where f is the structure morphism of the multiplicative group Gm , i.e., f : Gm ⊗ R → Spec R. Put Q(n) := Q(1)⊗n for n ∈ Z . (Negative tensor powers are dened by duality.) (iv) There should be realization functors to various categories such as `-adic étale local systems over X[1/`] := X ⊗R R[1/`] (where R is the base ring and ` does not divide the characteristic of R), variations of mixed Hodge structure over X , etc. These functors should be faithful, exact tensor functors. These functors are related by natural comparison transformations. The Betti, de Rham, `-adic and crystalline realizations of Q(1) should be the Betti, de Rham, `-adic and crystalline versions of H1 (Gm ). W (v) For each object M , GrW 2m+1 M is trivial and Gr2m M is isomorphic to the direct sum of a nite number of copies of Q(−m). The last property characterizes mixed Tate motives among mixed motives. The category T (X), being tannakian, is equivalent to the category of nite dimensional representations of a proalgebraic Q -group π1 (T (X), ω), which represents the tensor automorphism group of the ber functor ω . We denote it simply by π1 (T (X)), if the selection of ω does not matter. There are several approaches to constructing the category T (X), at least when X is the spectrum of a eld or X = XF,S , such as those of Bloch-Kriz [8], Levine [32], and Goncharov[18]. We follow Deligne [14] and Jannsen [29], who dene a motive to be a compatible set of realizations of geometric origin. This is a tannakian category. Deligne does not dene what it means to be of geometric origin, but wants it to be broad enough to include those compatible realizations that occur in the unipotent completion of fundamental groups of varieties in addition to subquotients of cohomology groups. We refer the reader to Section 1 of Deligne's paper [14] for the denition of compatible set of realizations. One example is Q(1), dened as H1 (Gm/Z ), another is the extension Ex of Q(0) by Q(n) coming from x ∈ K2n−1 (XF,S ) described in the previous section. The hope is that m Hmot (X, Q(n)) ∼ = Extm T (X) (Q(0), Q(n))

190


holds when X is the spectrum of a eld or XF,S .3 This covers the cases of interest for us. In general, one expects that motivic cohomology groups of X can be computed as hyper-exts: m Hmot (X, Q(n)) ∼ = H m (X, Ext •T (Q(0), Q(n))).

Deligne's conjecture (Conjecture 5.5) will be a consequence of: If X = XF,S , there is a category of mixed Tate motives T (X) over X with the above mentioned properties. It has the property that there is a natural isomorphism

Postulate 4.1.

m Hmot (X, Q(n)) ∼ = Extm T (X) (Q(0), Q(n)),

which is compatible with Chern maps.

Examples of mixed Tate motives over Spec Z . One of the main points

of [14] is to show that the unipotent completion of the fundamental group of P 1 − {0, 1, ∞} is an example of a mixed Tate motive (actually a pro-mixed Tate motive), smooth over Spec Z . → − As base point, take 01, the tangent vector of P 1 based at 0 that corresponds to ∂/∂t, where t is the natural parameter on P 1 − {0, 1, ∞}. Deligne − → [14] shows that the unipotent completion of π1 (P 1 − {0, 1, ∞}, 01) is a mixed Tate motive over Spec Z by exhibiting compatible Betti, étale, de Rham and crystalline realizations of it. It is smooth over Spec Z essentially because the − → pair (P 1 − {0, 1, ∞}, 01) has everywhere good reduction. There is an interesting relation to classical polylogarithms which was discovered by Deligne (cf. [14], [3], [20]). There is a polylog local system P , which is a motivic local system over P 1Z − {0, 1, ∞} in the point of view of compatible realizations. Its Hodge-de Rham realization is a variation of mixed Hodge structure over the complex points of P 1 − {0, 1, ∞} whose periods are given by log x and the classical polylogarithms: Li1 (x) = − log(1 − x), Li2 (x) (Euler's dilogarithm), Li3 (x), and so on. Here Lin (x) is the multivalued holomorphic function on P 1 (C) − {0, 1, ∞} whose principal branch is given by

Lin (x) =

∞ X xk kn

k=1

in the unit disk.4 → − The ber P− of P over the base point 01 is a mixed Tate motive over Spec Z → 01 and has periods the values of the Riemann zeta function at integers n > 1. In

Q

3 If this is true, then H m (Spec F, (n)) will vanish when n < 0 and m = 0, and when mot n ≤ 0 and m > 0. This vanishing is a conjecture of Beilinson and Soulé. It is known for

number elds, for example. 4 This goes back to various letters of Deligne. Accounts can be found, for example, in [3] and [20].


fact, P− is an extension → 01

0→

M

Q(n) → P− → Q(0) → 0 → 01

n≥1

and thus determines a class

(en )n ∈

M

Ext1H (Q(0), Q(n)).

n≥1

The class en is trivial when n = 1 and is the coset of ζ(n) in

C/(2πi)n Q ∼ = Ext1H (Q(0), Q(n)) when n > 1. Since ζ(2n) is a rational multiple of π 2n , each e2n is trivial. Deligne computes the `-adic realization of P− in [14] and shows that the → 01 polylogarithm motive is a canonical quotient of the enveloping algebra of the Lie → − algebra of the unipotent completion of π1 (P 1 − {0, 1, ∞}, 01). (See also [3] and [20].)

5. The Motivic Lie Algebra of XF,S and Deligne's Conjectures Assume that X is as in Section 4, and that there is a category of mixed Tate motives T (X) with properties (i)(v) in Section 4. Since T (X) is tannakian, it is determined by its tannakian fundamental group π1 (T (X)), which is an extension of Gm by a prounipotent Q -group

1 → UX → π1 (T (X)) → Gm → 1

(51)

as we shall now explain. The category of pure Tate motives is the tannakian subcategory of T (X) generated by Q(1). By the faithfulness of realization functors, it is equivalent to the category of nite dimensional graded Q -vector spaces, and hence to the category of nite dimensional representations of Gm ; Q(n) corresponds to the nth power of the standard representation. This induces a group homomorphism between the tannakian fundamental groups

π1 (T (X)) → π1 (pure Tate motives) ∼ = Gm . Since the category of pure Tate motives is a full subcategory and every subobject of a pure Tate motive is pure, this morphism is surjective (cf. [13, Proposition 2.21a]), and the properties of the weight ltration imply the unipotence of its kernel UX , thus we have (51). The Lie algebra tX of π1 (T (X)) is an extension 0 → uX → tX → Q → 0 where uX is pronilpotent. This tX is called the motivic Lie algebra of X . We shall see that the knowledge of the cohomologies of uX (as Gm -modules) is equivalent

192


to the knowledge of the extension groups Ext•T (X) (Q(0), Q(m)) for all m in the next section.

5.1. Extension groups in a tannakian category. We start with a general

setting. Let K be a eld of characteristic zero. Let G be a proalgebraic group (in this paper a proalgebraic group means an ane proalgebraic group) over K , or equivalently, an ane group scheme over K (cf. [13]). A G -module V is a (possibly innite dimensional) K -vector space with algebraic G -action (cf. [30]). The category of G -modules is abelian with enough injectives, and hence we have the cohomology groups

H m (G, V ) := Extm G (K, V ) dened as the extension groups, where K denotes the trivial representation. The right hand side has an interpretation as Yoneda's extension groups, i.e., as the set of equivalence classes of m-step extensions (see [45]). Since each G module is locally nite [30, 2.13], every m-step extension representing an element of Extm G (K, V ) can be replaced by an equivalent extension consisting of nite dimensional modules when V is nite dimensional. Thus, the right hand side does not change when the category of G -modules is replaced by the category of nite dimensional G -modules. Let T be a neutral tannakian category over K with a ber functor ω , and let G be its tannakian fundamental group with base point ω . Since T is isomorphic to the category of nite dimensional G -modules, we have the following.

Let T be a neutral tannakian category and G be its tannakian fundamental group . Then , for any object V , we have

Lemma 5.1.

∼ m Extm T (K, V ) = H (G, V ).

¤

Suppose that G is an extension

1→U →G→R→1 of proalgebraic groups over K . Then, for any G -module V , we have the LyndonHochschild-Serre spectral sequence (cf. [30, 6.6 Proposition]):

E2s,t = H s (R, H t (U, V )) ⇒ H s+t (G, V ). If R is a reductive algebraic group, then every R-module is completely reducible. Consequently, H s (R, V ) vanishes for s ≥ 1 for all V , and

H m (G, V ) ∼ = H 0 (R, H m (U, V )). If, in addition, the action of G on V factors through R, then one has an R-module isomorphism H m (U , V ) ∼ = H m (U, K) ⊗ V.


Moreover, if we assume that U is prounipotent, then its Lie algebra u is a projective limit ∼ lim u/n u= ←−

n

of nite dimensional nilpotent Lie algebras. It has a topology as a projective limit, where each u/n is viewed as a discrete topological space. m Let V be a continuous u-module over K. The continuous cohomology Hcts (u,V ) m is dened as the extension group Ext (K, V ) in the category of continuous um m (u). It is easy to show that (u, K) by Hcts modules. We denote Hcts m (u) ∼ H m (u/n), Hcts = lim −→ n

where H m (u/n) can be computed as the cohomology of the complex of cochains

Hom(Λ• (u/n), K). The following is standard. Proposition 5.2. Let u be a pronilpotent Lie algebra , and let H1 (u) denote the abelianization of u. Then 1 H1 (u) ∼ (u), K). = Hom(Hcts

If H 2 (u) = 0, then u is free . It is also well known that the category of U -modules is equivalent to the category of continuous u-modules. Hence we have m H m (U, K) ∼ (u). = Hcts

Putting this together, we have the following.

Suppose that 1 → U → G → R → 1 is a short exact sequence of pro-algebraic groups over a eld K of characteristic zero . Assume that R is a reductive algebraic group , and that U is a prounipotent group . Let u be the Lie algebra of U . If V is an R-module , considered as a G -module , then

Theorem 5.3.

m H m (G, V ) ∼ (u) ⊗ V )R . = (Hcts

Consequently , we have the R-module isomorphism M m (H m (G, Vα ) ⊗ Vα∗ ), Hcts (u) ∼ = α

where {Vα } is a set of representatives of the isomorphism classes of irreducible R-modules , and ( )∗ denotes Hom( , K).

194


5.2. Deligne's conjecture. By applying Theorem 5.3 to (51), we have Gm ∼ m Extm T (X) (Q(0), Q(n)) = [Hcts (uX ) ⊗ Q(n)]

and Gm -module isomorphisms M m Hcts (uX ) ∼ Extm = T (X) (Q(0), Q(n)) ⊗ Q(−n),

(52)

n∈Z

where uX is the Lie algebra of UX , the prounipotent radical of π1 (T (X), ω). By a weight argument, each extension on the right hand side vanishes if n ≤ m − 1. Postulate 4.1 says that these Ext groups should be the motivic cohomology groups of X , and Theorem 3.1 says that they should be isomorphic to the Adams eigenspaces of the K -groups of X :

Assume the existence of a category T (XF,S ) of mixed Tate motives over XF,S with properties (i )(v ) as in Section 4. Suppose that Postulate 4.1 holds for all n ≥ 1. Let UXF,S be the unipotent radical of π1 (T (XF,S ), ω), and uXF,S be its Lie algebra . Then there is a natural Gm -module isomorphism M 1 (uXF,S ) ∼ K2n−1 (XF,S ) ⊗Z Q(−n), Hcts = Proposition 5.4.

n≥1 m (uXF,S ) = 0 whenever m ≥ 2. Moreover , the exactness of GrW and Hcts • implies that M H1 (GrW K2n−1 (XF,S )∗ ⊗ Q(n) • uXF,S ) = n≥1

and that H m (GrW • uXF,S ) = 0 when m > 1.

It follows from this that GrW • uXF,S is isomorphic to the free Lie algebra generated W by H1 (Gr• uXF,S ). Let us assume that there is a category T (XF,S ) satisfying (i)(v) in Section 4. Then Postulate 4.1 is equivalent to the following conjecture of Deligne: (i) [14, 8.2.1] For the category T (XF,S ) of mixed Tate motives smooth over XF,S one has a natural isomorphism

Conjecture 5.5 (Deligne).

Ext1T (XF,S ) (Q(0), Q(n)) ∼ = K2n−1 (XF,S ) ⊗ Q for all n, which is compatible with the Chern mappings. (ii) [14, 8.9.5] The group π1 (T (XF,S )) is an extension of Gm by a free prounipotent group. Note that by Denition 2.1, Theorem 3.1 and the isomorphism (52), (i) is equivalent to Postulate 4.1 for m = 1, and that (ii) is equivalent to Postulate 4.1 for m ≥ 2.


Consequences of Deligne's conjecture. Deligne's conjecture suggests re-

strictions on the action of Galois groups on pro-` completions of fundamental groups of curves. Here is a sketch of how this should work. As in the beginning of Section 4, there should be a Betti realization functor

realB : T (XF,S ) → {Q -vector spaces} to the category of Q -vector spaces, and an `-adic realization functor

real` : T (XF,S ) → {`-adic GF -modules}, to the category of the Q ` -vector spaces with a continuous GF -action. The Galois modules should be unramied outside S∪[`], where [`] denotes the set of primes of F over `. We choose realB as our ber functor ω . Let ω` denote the functor real` which forgets the GF -action. Conjecturally, there is a comparison isomorphism

ω ⊗ Q` ∼ = ω` , so we shall identify these two. Dene T (XF,S )⊗Q ` to be the tannakian category whose objects are the same as those of T (XF,S ) and whose hom-sets are those of T (XF,S ) tensored with Q ` . The `-adic realization functor induces a functor

real` : T (XF,S ) ⊗ Q ` → {`-adic GF -modules}

(53)

(by an abuse of notation we denote it by real` again), and by forgetting the Galois action a ber functor ω` : T (XF,S ) ⊗ Q ` → VecQ ` (under a similar abuse of notation). Through the comparison isomorphism, it is easy to show that

π1 (T (XF,S ) ⊗ Q ` , ω` ) ∼ = π1 (T (XF,S ), ω) ⊗ Q ` . The following is closely related to the Tate conjecture on Galois modules. The realization functor real` in (53) is fully faithful, and its image is closed under taking subobjects. Postulate 5.6.

The rst condition is that every Galois compatible morphism comes from a morphism of motives up to extension of scalars, and the second condition is that every Galois submodule arises as an `-adic realization of a motive. We shall see that this postulate follows from Deligne's Conjecture 5.5 and our Theorem 9.2 (see Corollary 9.4). Every element of GF gives an automorphism of the forgetful ber functor of the category of GF -modules (i.e. forgetting the Galois action), and hence an automorphism of ω` . Thus we have a homomorphism

GF → π1 (T (XF,S ) ⊗ Q ` , ω` )(Q ` ) ∼ = π1 (T (XF,S ), ω)(Q ` ).

(54)

In addition, the GF -action on the `-adic realization of any (pro)object M of T (XF,S ) factors through π1 (T (XF,S )) ⊗ Q ` via the morphism (54).

Postulate 5.6 is equivalent to the statement that the above morphism (54) has Zariski dense image . Proposition 5.7.

196


Let G denote the tannakian fundamental group of the category nite dimensional `-adic GF -modules. By [13, Prop. 2.21a], the conditions Postulate 5.6 are equivalent to the surjectivity of G → π1 (T (XF,S ) ⊗ Q ` , ω` ). is a general fact that the image of GF → G(Q ` ) is Zariski dense. Proof.

of in It ¤

Assuming Postulate 5.6, the Zariski density of the image of (54) implies that for any object M of T (XF,S ), the Zariski closure of the image of GF in Aut(M ) • on GF should be a quotient of π1 (T (XF,S )) ⊗ Q ` . We can dene a ltration JM (which depends on M ) by m GF := the inverse image of Wm Aut M. JM

The image of the Galois group GF (µ`∞ ) of F (µ`∞ ) in π1 (T (XF,S ), ω` ) will lie in its prounipotent radical and should be Zariski dense in it. The exactness of GrW • will then imply that µ ¶ M m GrJM GF ⊗Z ` Q ` m0 I` GQ is 12 free up to GrI` GQ . The Lie algebra of the Zariski closure of the Galois group of Q(µ`∞ ) in the automorphism group of the `-adic unipotent com− → pletion of π1 (P 1 − {0, 1, ∞}, 01) is a prounipotent Lie algebra freely generated by elements z3 , z5 , z7 , . . ., where zm has weight −2m. Conjecture 5.9 (Goncharov).

Deligne's Conjecture 5.8 above and the generation part of Goncharov's conjecture are proved in [23]. A brief sketch of their proofs is given in Section 10. Modulo technical details, the main point is the computation of the tannakian fundamental group of the candidate for T (XF,S ) ⊗ Q ` given in the next section.

Polylogarithms revisited. Assuming the existence of tSpec Z (i.e. the Lie algebra of π1 (T (Spec Z))), we can give another interpretation of the ber P− of the → 01 polylogarithm local system. Being a motive over Spec Z , it is a tSpec Z -module. Note that since M W−1 P− = Q(n), → 01 n≥1

a direct sum of Tate motives (no nontrivial extensions), the restriction of the tSpec Z -action on W−1 P− to uSpec Z is trivial and [uSpec Z , uSpec Z ] annihilates → 01 P− . Since P− is an extension of Q(0) by W−1 P− , this implies that there is a → → → 01 01 01 homomorphism ¡ ¢ ψ : tSpec Z /[uSpec Z , uSpec Z ] ⊗ Q(0) −→ P− → 01 such that the diagram action

/ P− → nn6 01 n n nnn quotient nnnψ n n ² n tSpec Z /[uSpec Z , uSpec Z ] tSpec Z ⊗ P− → 01

commutes. By comparing graded quotients, it follows that ψ is an isomorphism

∼t P− /[uSpec Z , uSpec Z ] → 01 = Spec Z of motives over Spec Z .

198


6. `-adic Mixed Tate Modules over XF,S In this section, we describe a candidate for the category of `-adic realizations of objects and morphisms of T (XF,S ). This is essentially the category constructed by Deligne and Beilinson in their unpublished manuscript [2]. It is purely Galoisrepresentation theoretic, and requires no postulates. For technical reasons, we assume that S contains [`], the set of primes over `. This condition will be removed in Section 11. By a nite dimensional GF,S -module, we shall mean a nite dimensional Q ` -vector space on which GF,S acts continuously. We dene the category T` (XF,S ) of `-adic mixed Tate modules which are smooth over XF,S to be the category whose objects are nite dimensional GF,S modules M that are equipped with a weight ltration

· · · ⊆ Wm−1 M ⊆ Wm M ⊆ Wm+1 M ⊆ · · · of M by GF,S -submodules. The weight ltration satises: (i) all odd weight graded quotients of M vanish: GrW 2m+1 M = 0; W (ii) GF,S acts on its 2mth graded quotient Gr2m M via the (−m)th power of the cyclotomic character, (iii) the intersection of the Wm M is trivial and their union is all of M . Morphisms are Q ` -linear, GF,S -equivariant mappings. These will necessarily preserve the weight ltration, so that T` (XF,S ) is a full subcategory of the category of GF,S -modules. The category T` (XF,S ) is a tannakian category over Q ` with a ber functor ω 0 that takes an object to its underlying Q ` -vector space. We shall denote the tannakian fundamental group of this category by A`F,S := π1 (T` (XF,S ), ω 0 ). Every element of GF,S acts on ω 0 , which induces a natural, continuous homomorphism

ρ : GF,S → A`F,S (Q ` ). This has Zariski-dense image as T` (XF,S ) is a full subcategory of the category of GF,S -modules, closed under taking subobjects (cf. [13, Proposition 2.21a]).

Relation to mixed Tate motives over XF,S . As explained in Section 4,

the existence of a category T (XF,S ) of mixed Tate motives over XF,S satisfying (i)(v) in Section 4 implies the existence of an `-adic realization functor

real` : T (XF,S ) ⊗ Q ` → T` (XF,S ). This will induce a morphism of tannakian fundamental groups

A`F,S = π1 (T` (XF,S ), ω 0 ) → π1 (T (XF,S ), ω) ⊗ Q ` . The main result of [23] may be regarded as saying that A`F,S = π1 (T` (XF,S ), ω 0 ) is isomorphic to the conjectured value of the Q ` -form π1 (T (XF,S ), ω) ⊗ Q ` of the motivic fundamental group of XF,S . We shall explain this in Section 9.


It is interesting to note that we have not restricted to GF,S -modules of geometric origin as Deligne would like to. So one consequence of our result is that, if Deligne's Conjecture 5.5 is true, then all objects and morphisms of T (XF,S ) will be of geometric origin.

7. Weighted Completion of Pronite Groups In this and the subsequent two sections we will sketch how to compute the tannakian fundamental group π1 (T` (XF,S ), ω 0 ) of the category of `-adic mixed Tate modules smooth over XF,S , which was dened in Section 6. It is convenient to work in greater generality.5 Suppose that R is a reductive algebraic group over Q ` and that w : Gm → R is a central cocharacter that is, its image is contained in the center of R. It is best to imagine that w is nontrivial as the theory of weighted completion is uninteresting if w is trivial. Suppose that Γ is a pronite group and that a homomorphism ρ : Γ → R(Q ` ) has Zariski dense image and is continuous where we view R(Q ` ) as an `-adic Lie group. By a weighted Γ-module with respect to ρ and w we shall mean a nite dimensional Q ` -vector space with continuous Γ-action together with a weight ltration

· · · ⊆ Wm−1 M ⊆ Wm M ⊆ Wm+1 M ⊆ · · · by Γ-invariant subspaces. These should satisfy: (i) the intersection of the Wm M is 0 and their union is M , (ii) for each m, the representation Γ → Aut GrW m M should factor through ρ and a homomorphism φm : R → Aut GrW M , m (iii) GrW m M has weight m when viewed as a Gm -module via w

φm

Gm −−−−→ R −−−−→ Aut GrW m M. That is, Gm acts on GrW m M via the mth power of the standard character. The category of weighted Γ-modules consists of the Γ-equivariant morphisms between weighted Γ-modules. These morphisms automatically preserve the weight ltration and are strict with respect to it; that is, the functor GrW • is exact. One can show that the category of weighted Γ-modules is tannakian, with ber functor ω 0 given by forgetting the Γ-action. The weighted completion of Γ with respect to ρ : Γ → R(Q ` ) and w : Gm → R is the tannakian fundamental group of the category of weighted Γ-modules with respect to ρ and w.

Definition 7.1.

5 We may generalize further: weighted completion and its properties in this section are unchanged even if we replace Q ` by an arbitrary topological eld of characteristic zero and Γ by an arbitrary topological group.

200


Denote the weighted completion of Γ with respect to ρ and w by G . There is a natural homomorphism Γ → G(Q ` ) which has Zariski dense image as we shall see below. This denition diers from the one given in [23, Section 5], but is easily seen to be equivalent to it. (See below.) In particular, we can apply it when:

• Γ is GF,S , • R is Gm and w : Gm → Gm takes x to x−2 , • ρ is the composite of the `-adic cyclotomic character χ` : GF,S → Z × ` with × the inclusion Z × ,→ Q . ` ` In this case, the category of weighted Γ-modules is nothing but the category of mixed Tate modules T` (XF,S ). Recall that we denote the corresponding weighted completion by A`F,S := π1 (T` (XF,S ), ω 0 ).

Equivalence of denitions. Here we show that the denition of weighted completion given in [23] agrees with the one given here. Suppose that G is a linear algebraic group over Q ` which is an extension

1→U →G→R→1 of R by a unipotent group U . Note that H1 (U ) is naturally an R-module, and therefore a Gm -module via the given central cocharacter w : Gm → R. We can decompose H1 (U ) as a Gm -module: M H1 (U ) = H1 (U )n n∈Z

where Gm acts on H1 (U )n via the nth power of the standard character. We say that G is a negatively weighted extension of R if H1 (U )n vanishes whenever n ≥ 0. Given a continuous homomorphism ρ : Γ → R(Q ` ) with Zariski dense image, we can form a category of pairs (˜ ρ, G), where G is a negatively weighted extension of R and ρ˜ : Γ → G(Q ` ) is a continuous homomorphism that lifts ρ. Morphisms in this category are given by homomorphisms between the Gs that respect the projection to R and the lifts ρ˜ of ρ. The objects of this category, where ρ˜ is Zariski dense, form an inverse system. Their inverse limit is an extension

1→U →G→R→1 of R by a prounipotent group. There is a natural homomorphism ρˆ : Γ → G(Q ` ), which is continuous in a natural sense. It has the following universal mapping property: if ρ˜ : Γ → G(Q ` ) is an object of this category, then there is a unique homomorphism φ : G → G that commutes with the projections to R and with the homomorphisms ρ˜ : Γ → G(Q ` ) and ρˆ : Γ → G(Q ` ). In [23], the weighted completion is dened to be this inverse limit. The equivalence of the two denitions follows from the following result.


The inverse limit dened above is naturally isomorphic to the weighted completion of Γ relative to ρ and w.

Proposition 7.2.

Proof. Denote the inverse limit by G and by M = M(ρ, w) the category of weighted Γ-modules with respect to ρ : Γ → R(Q ` ) and w. We will show that M is the category of nite dimensional G -modules, from which the result follows. Suppose that M is an object of M. Then the Zariski closure of Γ in Aut M is an extension 1 → U → G → R0 → 1

of a quotient of R by a unipotent group. Here R0 is the Zariski closure of the image of Γ in Aut GrW • M . Because the action of Γ on each weight graded quotient factors through ρ, and because Gm acts on the mth weight graded quotient of M with weight m, it follows that this is a negatively weighted extension of R0 . Pulling back this extension along the projection R → R0 , we obtain a negatively weighted extension e→R→1 1→U →G

e ` ) that lifts both ρ and the of R and a continuous homomorphism Γ → G(Q 0 homomorphism Γ → R (Q ` ). By the universal mapping property of G , there e compatible with the projections to R and is a natural homomorphism G → G e ` ). Thus every object of M the homomorphisms from Γ to G(Q ` ) and G(Q is naturally a G -module. It is also easy to see that every morphism of M is G -equivariant. Conversely, suppose that M is a nite dimensional G -module. Composing with the natural homomorphism ρˆ : Γ → G(Q ` ) gives M the structure of a Γmodule. In [23, Sect. 4], it is proven that every G -module has a natural weight ltration with the property that the action of G on each weight graded quotient factors through the projection G → R and that Gm acts with weight m on the mth weight graded quotient. It follows that M is naturally an object of M. Since G -equivariant mappings are naturally Γ-equivariant, this proves that M is naturally the category of nite dimensional G -modules, which completes the proof. ¤

8. Computation of Weighted Completions Suppose that Γ, R, ρ : Γ → R(Q ` ) and w : Gm → R are as above. The weighted completion G of Γ is controlled by the low-dimensional cohomology • groups Hcts (Γ, V ) of Γ with coecients in certain irreducible representations V of R. If one knows these cohomology groups, as we do in the case of GF,S , one can sometimes determine the structure of the weighted completion. These cohomological results are stated in this section. The weighted completion of Γ with respect to ρ and w is an extension

1→U →G→R→1

202


where U is prounipotent. Now we are in the situation of Theorem 5.3. Denote the Lie algebra of U by u. Since u is a G -module by the adjoint action, the • natural weight ltration on u induces one on Hcts (u). By looking at cochains, it is not dicult to see that if u = W−N u for some N > 0, then m Wn Hcts (u) = 0 if n < N m.

(81)

Let Vα be an irreducible R-module. Since w is central in R, the Gm -action commutes with the R-action, so Schur's Lemma implies that there is an integer n(α) such that Gm acts on Vα via the n(α)th power of the standard character. This is the weight of Vα as a G -module. Now (81) and Theorem 5.3 imply

H m (G, Vα ) = 0 if n(α) > −N m. Note that always u = W−1 u. Suppose that V is an `-adic Γ-module, i.e., a Q ` -vector space with continuous • Γ-action. We shall need the continuous cohomology Hcts (Γ, V ), which is dened as the cohomology of a suitable complex of continuous cochains as in [42, Sect. 2]. A G -module V can be considered as an `-adic Γ-module through ρˆ : Γ → G(Q ` ). There is a natural group homomorphism m Φm : H m (G, V ) → Hcts (Γ, V )

for each m ≥ 0. Let {Vα }α be as in Theorem 5.3. These are considered as Γ-modules via ρ. The following theorem is our basic tool for computing u when the appropriate i continuous cohomology groups Hcts (Γ, Vα ) are known for i = 1, 2. Theorem 8.1.

For m = 1, 2, the mappings Φm dened above satisfy :

1 (i) Φ1 : H 1 (G, Vα ) → Hcts (Γ, Vα ) is an isomorphism if n(α) < 0; 2 2 2 (ii) Φ : H (G, Vα ) → Hcts (Γ, Vα ) is injective .

This and Theorem 5.3 imply the following, by using (81) and the comment following it. Corollary 8.2.

(i) There is a natural R-equivariant isomorphism M 1 1 (Γ, Vα ) ⊗ Vα∗ . (u) ∼ Hcts Hcts = {α:n(α)≤−1}

1 (ii) If N is an integer such that Hcts (Γ, Vα ) = 0 for 0 > n(α) > −N , then there is a natural R-equivariant inclusion M 2 2 Φ : Hcts (u) ,→ Hcts (Γ, Vα ) ⊗ Vα∗ . {α:n(α)≤−2N }

This is proved in [23]. Below we shall give another more categorical proof, similar to that in [2]. Corollary 8.3.

1 (i) If Hcts (Γ, Vα ) = 0 whenever n(α) < 0, then u = 0.

TANNAKIAN FUNDAMENTAL GROUPS ASSOCIATED TO GALOIS GROUPS 203 2 (ii) Let N be as in Corollary 8.2. If Hcts (Γ, Vα ) = 0 whenever n(α) ≤ −2N , then u is free as a pronilpotent Lie algebra . ¤

In the proof of Theorem 8.1 we shall use Yoneda extensions. Let V be a nite dimensional `-adic Γ-module. For each m ≥ 0, dene

Extm Γ (Q ` , V ) to be the m-th Yoneda extension group in the category of nite dimensional `-adic Γ-modules, where Q ` denotes the trivial Γ-module. For each m ≥ 1, there is a natural homomorphism m Extm Γ (Q ` , V ) → Hcts (Γ, V ),

which, by Theorem A.6, is an isomorphism when m = 1, and injective when m = 2. There is an exact functor from the category of weighted G -modules to the category of `-adic Γ-modules. It induces morphisms between the extension groups, and hence homomorphisms

Ψm : H m (G, V ) → Extm Γ (Q ` , V ),

m ≥ 0.

The homomorphisms Φm above factor through these: Φm

H m (G, V )

/ Extm Γ (Q ` , V ) m

Ψ

* m / Hcts (Γ, V ).

In fact, this is one of several equivalent ways to dene the natural mappings Φm . 8.1. In view of Theorem A.6, it suces to prove that Ψ1 is an isomorphism, and that Ψ2 is injective. Since the functor from the category of weighted Γ-modules to the category of Γ-modules is fully faithful, a 1-step extension of weighted Γ-modules splits if it splits as an extension of Γ-modules. This establishes the injectivity of Ψ1 . To prove surjectivity of Ψ1 , we dene a natural weight ltration on each Γmodule extension E of Q ` by Vα . Simply set W0 E = E and Wn(α) E = Vα . Since n(α) < 0, this makes E a weighted Γ-module. To prove that Ψ2 is injective, we need to show that if a 2-step extension Proof of Theorem

1 → Vα → E2 → E1 → Q ` → 1

(82)

lies in the trivial class of extensions of Γ-modules, then it also lies in the trivial class of extensions of weighted Γ-modules. If n(α) ≥ −1, then H 2 (G, Vα ) = 0, and there is nothing to prove. Thus we may assume n(α) ≤ −2. Since Wm is an exact functor, we may apply W0 to (82) to obtain another 2-step extension, without changing the extension class. Then, taking GrW 0 , we have a short exact sequence W 0 → GrW 0 E2 → Gr0 E1 → Q ` → 0

204


of R-modules. Since R is reductive, this has a splitting Q ` ,→ GrW 0 E1 . Taking the inverse images of this copy of Q ` along E2 → E1 → GrW E in E2 and in 1 0 E1 , we obtain a 2-step extension

0 → Vα → E20 → E10 → Q ` → 0 equivialent to (82) satisfying W−1 E20 = E20 and W0 E10 = E10 . Using the dual argument, we may assume that (82) satises W0 E1 = E1 , W−1 E2 = E2 , Wn(α)−1 E2 = 0, and Wn(α) E1 = 0. By Yoneda's characterization [45, p. 575] of trivial m-step extensions, the extension (82) represents the trivial 2-step extension class as Γ-modules if and only if there is a Γ-module E and exact sequences

0 → E2 → E → Q ` → 0 and 0 → Vα → E → E1 → 0 which are compatible with the existing mappings Vα ,→ E2 and E1 ³ Q ` . To establish the injectivity of Ψ2 , it suces to prove that E is a weighted Γ-module. But E has the weight structure W0 E = E and W−1 E = E2 . This completes the proof of Theorem 8.1. ¤ × Suppose that Γ = Z × ` , that R = Gm /Q ` and that ρ : Z ` ,→ Gm (Q ` ) = is the natural inclusion. Take w to be the inverse of the square of the standard character. (With this choice, representation theoretic weights coincide with the weights from Hodge and Galois theory.) In this example we compute the weighted completion of Z × ` with respect to ρ and w . Note that

Example 8.4.

Q× `

1 Hcts (Z × ` , Q ` (n)) = 0,

for all non-zero n ∈ Z , where Q ` (n) denotes the nth power of the standard representation of Gm . It has weight −2n under the central cocharacter. Corollary 8.3 tells us that the unipotent radical U of the weighted comple× tion of Z × ` is trivial, so that the weighted completion of Z ` with respect to ρ × is just ρ : Z ` → Gm (Q ` ). More generally, if Γ is an open subgroup of Z × ` , × then the weighted completion of Γ, relative to the restriction Γ → Q ` of the homomorphism ρ above and the same w, is simply Gm . Example 8.5. Let Mg be the moduli stack of genus g curves over Spec Z . Suppose that there is a Z[1/`]-section x : Spec Z[1/`] → Mg . We allow tangential sections, and then such x exist for all g . Let x ¯ : Spec Q → Spec Z[1/`] → Mg be a geometric point on the generic point of x. Let Cx¯ be the curve corresponding to x ¯. There is a short exact sequence of algebraic fundamental groups

1 → π1 (Mg ⊗ Q, x ¯) → π1 (Mg ⊗ Q, x ¯) → GQ → 1,

(83)

ˆ g of the mapping where the left group is isomorphic to the pronite completion Γ class group Γg of a genus g surface. We x such an isomorphism. We have the


natural representation

π1 (Mg ⊗ Q, x ¯) → Aut Hé1t (Cx¯ , Q ` ).

(84)

It is known that the image of (84) is isomorphic to GSpg (Z ` ), where GSpg denotes the group of symplectic similitudes of a symplectic module of rank 2g . By considering the action of the mapping class group on the Z/`Z homology ˆ g → Spg (Z/`). Let Γ`g be of the surface, we obtain a natural representation Γ ˆ g that also maps to Spg (Z/`) and such that the kernel the largest quotient of Γ of the induced mapping Γ`g → Spg (Z/`) is a pro-` group. One can construct a quotient

1 → Γ`g → Γarith,` → GQ,{`} → 1 g of the short exact sequence (83) such that the homomorphism (84) induces a homomorphism

ρ : Γarith,` → GSpg (Q ` ) g from (84). Dene the central cocharacter ω : Gm → GSpg by x → x−1 I2g . In [24] we show that the weighted completion Ggarith,` of Γarith,` is an extension g

Gg ⊗ Q ` → Ggarith,` → A`Q,{`} → 1, where Gg is the completion of Γg relative to the standard homomorphism ρ : Γg → Spg (Q), which is studied in [21] and for which a presentation is given in [22]. We expect that the left homomorphism is injective.

9. Computation of A`F,S In this section, we compute A`F,S , the tannakian fundamental group of the category of `-adic mixed Tate modules over XF,S . An equivalent computation was done by Beilinson and Deligne in [2]. We shall need the following result of Soulé [41] when ` is odd. The case ` = 2 follows from [40]. Recall that dn is dened in (31). Theorem 9.1 (Soulé

[41]). With notation as above ,

1 (GF,S , Q ` (n)) K2n−1 (XF,S ) ⊗ Q ` ∼ = Hcts

and hence

½ 1 dimQ ` Hcts (GF,S , Q ` (n))

=

d1 + #S dn

when n = 1, when n > 1.

2 In addition , Hcts (GF,S , Q ` (n)) vanishes for all n ≥ 2.

206


` Denote the unipotent radical of A`F,S by KF,S . We have the exact sequence ` 1 → KF,S → A`F,S → Gm → 1,

and the corresponding exact sequence of Lie algebras

0 → k`F,S → a`F,S → Q ` → 0. The Lie algebra a`F,S , being the Lie algebra of a weighted completion, has a natural weight ltration. Note that since w is the inverse of the square of the standard character, all weights are even. Thus the weight ltration of a`F,S satises

a`F,S = W0 a`F,S ,

` k`F,S = W−2 a`F,S and GrW 2n+1 aF,S = 0 for all n ∈ Z,

and we may take N = 2 in Corollary 8.2. The basic structure of A`F,S now follows from Corollary 8.2, Corollary 8.3 and Soulé's computation above. ` [23]). The Lie algebra GrW • kF,S is a free Lie algebra and there is a natural Gm -equivariant isomorphism

Theorem 9.2 (Hain-Matsumoto

1 Hcts (k`F,S ) ∼ =

∞ M

1 Hcts (GF,S , Q ` (n)) ⊗ Q ` (−n) ∼ = Q ` (−1)d1 +#S ⊕

n=1

M

Q ` (−n)dn ,

n>1

` where dn is dened in (31). Any lift of a graded basis of H1 (GrW • kF,S ) to a W ` ` graded set of elements of GrW • kF,S freely generates Gr• kF,S .

As a corollary of the proof, we have:

There are natural isomorphisms ( Q` when m = n = 0, ∼ H 1 (G , Q (n)) when m = 1 and n > 0, Extm F,S T` (XF,S ) (Q ` , Q ` (n)) = ` cts 0 otherwise .

Corollary 9.3.

Consequently , for all n ∈ Z , there are natural isomorphisms Ext1T` (XF,S ) (Q ` , Q ` (n)) ∼ = K2n−1 (Spec OF,S ) ⊗ Q ` .

Suppose that there is a category of mixed Tate motives T (XF,S ) with properties (i )(v ) as in Section 4. If Deligne's conjecture 5.5 is true , then , the image of the `-adic realization functor real` in (53) is equivalent to the category of weighted GF,S -modules . In particular , Postulate 5.6 follows . Corollary 9.4.

Proof. Deligne's conjecture 5.5 implies that π1 (T (XF,S )), the tannakian fundamental group, is an extension

1 → UXF,S → π1 (T (XF,S )) → Gm → 1,


where UXF,S is a free prounipotent group generated by K2n−1 (XF,S )∗ . This and Theorem 9.2 show that the natural map π1 (T` (XF,S )) → π1 (T (XF,S )) ⊗ Q ` is an isomorphism, and it follows that

real` : T (XF,S ) ⊗ Q ` → `-adic GF -modules is fully faithful and its image is equivalent to the category of weighted GF,S modules. ¤ Note that these theorems can be generalized to the case where S may not contain all the primes above `, see Section 11.

Another example. Suppose that S is a nite set of rational primes containing `. Suppose that F is a nite Galois extension of Q with Galois group G, which is unramied outside S . Dene ρ : GQ,S → Gm (Q ` ) × G by

ρ(σ) = (χ` (σ), f (σ)) where f : GQ,S → G is the quotient homomorphism and χ` is the `-adic cyclotomic character. Dene w : Gm → Gm × G by w : x 7→ (x−2 , 1). It is a central cocharacter. Denote the weighted completion of GQ,S with respect to ρ and w by GQ,S . Denote the set of primes in OF that lie over S ⊂ Spec Z by T . Proposition 9.5.

sequence

There is a natural inclusion ι : A`F,T → GQ,S and an exact ι

1 −−−−→ A`F,T −−−−→ GQ,S −−−−→ G −−−−→ 1. If {Vα } is a set of representatives of the isomorphism classes of irreducible representations of G, then {Q ` (m) £ Vα } is a set of representatives of the isomorphism classes of irreducible representations of Gm × G, where W £ V denotes the exterior tensor product of a representation W of Gm and V of G. Consider the restriction mapping Proof.

i i (GQ,S , Q ` (m) £ Vα ) → Hcts (GF,T , Q ` (m) £ Vα )G φ : Hcts

and the transfer mapping [42] i i ψ : Hcts (GF,T , Q ` (m) £ Vα )G → Hcts (GQ,S , Q ` (m) £ Vα ).

A direct computation on cocycles shows that φ ◦ ψ and ψ ◦ φ are both multiplication by the order of G, and are thus isomorphisms. i Therefore Hcts (GQ,S , Q ` (m) £ Vα ) vanishes if Vα is nontrivial, and it equals i Hcts (GF,T , Q ` (m)) if Vα is trivial. This shows that the unipotent radical of the completion GQ,S is isomorphic to that of A`F,T .

208


By functoriality of weighted completion, we have a homomorphism A`F,T → GQ,S which induces the isomorphism on the unipotent radical. The statement follows. ¤

10. Applications to Galois Actions on Fundamental Groups Let G` denote GQ,{`} . In this section, we sketch how our computation of the weighted completion of G` can be used to prove Deligne's Conjecture 5.8 about the action of the absolute Galois group GQ on the pro-` completion of the fundamental group of P 1 (C) − {0, 1, ∞}. Modulo a few technical details, which are addressed in [23], the proof proceeds along the expected lines suggested in Section 5.2 given the computation of A`F,S . We begin in a more general setting. Suppose that F is a number eld and that X is a variety over F . Set X = X ⊗ Q and denote the absolute Galois group of F by GF . Suppose that the étale cohomology group Hé1t (X, Q ` (1)) is a trivial GF -module. Let S be a set of nite primes of F , containing those above `. Suppose that X has a model X over Spec OF,S which has a base point section x : Spec OF,S → X (possibly tangential) such that (X , x) has good reduction outside S .6 Then the GF -action on the pro-` fundamental group π1` (X, x) factors through GF,S . Denote the `-adic unipotent completion of π1` (X, x) by P (see [23, Appendix A]) and its Lie algebra by p. The lower central series ltration of p gives it the structure of a pro-object of the category T` (XF,S ) of `-adic mixed Tate modules over XF,S . It follows that the GF -action on P induces a homomorphism

A`F,S → Aut P ∼ = Aut p and that the action of GF on P factors through the composition of this with the natural homomorphism GF → GF,S → A`F,S (Q ` ). One can show (see [23, ` Sect. 8]) that the image of GF (µ`∞ ) in A`F,S lies in and is Zariski dense in KF,S . 1 For the rest of this section, we consider the case where X = P − {0, 1, ∞}, − → F = Q , S = {`} and x is the tangential base point 01. Goncharov's conjecture [19, Conj. 2.1] (cf. the generation part of Conjecture 5.9) follows immediately, since kQ,{`} is generated by z1 , z3 , z5 , . . ., where zj has weight −2j . The image of z1 can be shown to be trivial. We are now ready to give a brief sketch of the proof of Conjecture 5.8. One can dene a ltration I`• on GQ similar to I`• using the lower central series L• P of P instead:

I`m GQ = ker{GQ → Out P/Lm+1 P} we mean here is that there is a scheme Xe, proper over Spec OF,S , and a divisor D in Xe which is relatively normal crossing over Spec OF,S such that X = Xe − D, and D does not intersect with x. In the tangential case, the tangent vector should be non-zero over each point of Spec OF,S . 6 What


where Lm P is the mth term of its lower central series. The lower central series of P is related to its weight ltration by

W−2m P = Lm P,

GrW 2m+1 P = 0.

There is a natural isomorphism (see [23, Sect. 10]) m ∼ [Grm I` GQ ] ⊗ Q ` = [GrI` GQ ] ⊗ Q ` .

Thus it suces to prove that [Grm I` GQ ]⊗Q ` is generated by elements s3 , s5 , s7 , . . ., where sj has weight −2j . As above, the homomorphism GQ → Out P factors through the sequence

GQ → GQ,{`} → AQ,{`} → Out P of natural homomorphisms. A key point ([23, Sect. 8]) is that the image of ` I` 1 GQ in KQ,` is Zariski dense. This and the strictness can be used to establish isomorphisms ` GrIm` GQ ⊗ Q ` ∼ = GrW −2m (im{kQ,` → OutDer p}) ∼ = im{GrW k` → GrW OutDer p} −2m Q,`

−2m

for each m > 0. ` Theorem 9.2 implies that GrW • kQ,` is freely generated by σ1 , σ3 , σ5 , . . . where W ` σ2i+1 ∈ Gr−2(2i+1) kQ,` . It is easy to show that the image of σ1 vanishes in W ` GrW • OutDer p. It follows that the image of Gr• kQ,` is generated by the images of σ3 , σ5 , σ7 , . . ., which completes the proof. Ihara proves the openness of the group generated by σ2i+1 in a suitable Galois group, see [28]. He also establishes the non-vanishing of the images of the σ2i+1 and some of their brackets in [27]. Remark 10.1.

11. When ` is not contained in S Let [`] denote the set of all primes above ` in OF . In this section, we generalize the denition of the category T` (XF,S ) of `-adic mixed Tate modules smooth over XF,S = Spec OF − S (see Section 6) to the case where S does not necessarily contain [`]. For this, we dene the category T` (XF,S ) of `-adic mixed Tate modules over XF,S to be the full subcategory of T` (XF,S∪[`] ) (dened in Section 6) consisting of the Galois modules which are crystalline at every prime p ∈ [`] − S . (Recall that an `-adic GF -module M is crystalline at a prime p of F if it is crystalline as GFp -module, where Fp is the completion of F at p and GFp is identied with the decomposition group of GF at p, see [16; 7] for crystalline representations.) It is known that the crystalline property is closed under tensor products, direct sums, duals, and subquotients [16], so that T` (XF,S ) is a tannakian category. Denote its tannakian fundamental group by A`F,S . We have a short exact

210


sequence

` 1 → KF,S → A`F,S → Gm → 1,

and the corresponding exact sequence of Lie algebras

0 → k`F,S → a`F,S → Q ` → 0. Let V be a GF,S -module. The nite part of the rst degree Galois cohomol1 1 (GF,S , V ) is dened in [7, (3.7.2)]. This corresponds ogy Hctsf (GF,S , V ) ⊂ Hcts to those extensions of Q ` by V as GF,S -modules, which are crystalline at ev1 ery prime in [`] outside S . By a remark on p. 354 in [7], Hctsf (GF,S , Q ` ) = × (OF,S ) ⊗Z ` Q ` , so its dimension is d1 + #S = r1 + r2 + #S − 1. Theorem 9.2 1 1 is generalized as follows, by replacing Hcts with Hctsf [23]. We shall give a categorical proof below. ` The Lie algebra GrW • kF,S is a free Lie algebra and there is a natural Gm -equivariant isomorphism

Theorem 11.1.

1 Hcts (k`F,S ) ∼ =

∞ M

1 Hctsf (GF,S , Q ` (n)) ⊗ Q ` (−n) ∼ = Q ` (−1)d1 +#S ⊕

n=1

M

Q ` (−n)dn ,

n>1

` where dn is dened in (31). Any lift of a graded basis of H1 (GrW • kF,S ) to a W ` W ` graded set of elements of Gr• kF,S freely generates Gr• kF,S .

There are natural isomorphisms ( Q` when m = n = 0, m 1 ∼ ExtT` (XF,S ) (Q ` , Q ` (n)) = Hctsf (GF,S , Q ` (n)) when m = 1 and n > 0, 0 otherwise . Consequently , for all n ∈ Z , there are natural isomorphisms Corollary 11.2.

Ext1T` (XF,S ) (Q ` , Q ` (n)) ∼ = K2n−1 (Spec OF,S ) ⊗ Q ` .

¤

This shows that T` (XF,S ) has all the properties of the category T (XF,S ) ⊗ Q ` , where T (XF,S ) is the category whose existence is conjectured by Deligne. In ` particular, GrW • kQ,∅ is free with generators σ3 , σ5 , . . .. Proof of Theorem

11.1. It suces to show that the natural mapping

1

1 (GF,S , Q ` (n)) Φ : H 1 (A`F,S , Q ` (n)) → Hctsf

is an isomorphism when n ≥ 1 and that the natural mapping 2 Φ2 : H 2 (A`F,S , Q ` (n)) → Hcts (GF,S , Q ` (n))

is injective when n ≥ 2. The proof is similar to that of Theorem 8.1. To show that Φ1 is an isomorphism, it suces to show that an extension E of Q ` by 1 (GF,S , Q ` (n)) is crystalline, which is Q ` (n) corresponding to an element of Hctsf well-known. So the rst assertion follows. We now consider the case of Φ2 . Set Vα = Q ` (n). We may assume n ≥ 2. It suces to show that E in the proof of Theorem 8.1 is crystalline provided


E1 and E2 are crystalline. But this follows from the next result, which will be proved below. Proposition 11.3.

Let 0 −→ V −→ U −→ Q ` (1)n −→ 0

be a short exact sequence of crystalline `-adic representations of GFp . Assume that V is a successive extension of direct sums of a nite number of copies of Q ` (r) with r ≥ 2. Then , for any extension 0 −→ U −→ E −→ Q ` −→ 0

of `-adic representations of GFp , E is crystalline if and only if its pushout by the surjection U → Q ` (1)n is crystalline . Let U be E2 as in the proof of Theorem 5.3. Since W−2 E2 = E2 , U is an extension of Q ` (1)n for some n. Since m ≥ 2, the pushout of E along U → Q ` (1)n is a quotient of E1 , and hence is crystalline. Thus the proposition says that E is crystalline, which completes the proof of Theorem 11.1. ¤ Proposition 11.3 follows from the following two lemmas. Lemma 11.4.

Let 0 → V1 → V2 → V3 → 0

be a short exact sequence of crystalline `-adic representations of GFp . Then we have a long exact sequence 0 → H 0 (GFp , V1 ) → H 0 (GFp , V2 ) → H 0 (GFp , V3 ) 1 1 1 → Hctsf (GFp , V1 ) → Hctsf (GFp , V2 ) → Hctsf (GFp , V3 ) → 0.

¤

This follows from [7, Cor. 3.8.4].

Let V be a crystalline `-adic representation of GFp . If V is a 1 1 successive extension of Q ` (r) (r ≥ 2), then Hctsf (GFp , V ) = Hcts (GFp , V ). Lemma 11.5.

The proof is by induction on the dimension of V . In the case dim(V ) = 1, this is well-known (loc. cit. Example 3.9). Assume dim V = n ≥ 2 and the claim is true for n − 1. By assumption, there exists an exact sequence of `-adic representations of GFp : Proof.

0 −→ V 0 −→ V −→ Q ` (r) −→ 0 for some integer r ≥ 2 such that V 0 satises the assumption of the lemma. By Lemma 11.4, we have the following commutative diagram whose two rows are

212


exact:

0

- H 1 (GFp , V 0 ) - H 1 (GFp , V ) - H 1 (GFp , Q ` (r)) ctsf ctsf ctsf

0

? 1 - Hcts (GFp , V 0 )

? 1 - Hcts (GFp , V )

- 0

? 1 - Hcts (GFp , Q ` (r))

The right vertical arrow is an isomorphism and the left one is also an isomorphism by the induction hypothesis. Hence the middle one is also an isomorphism. ¤ 11.3. By Lemma 11.4, we have the following commutative diagram whose two rows are exact:

Proof of Proposition

0

- H 1 (GFp , V ) - H 1 (GFp , U ) - H 1 (GFp , Q ` (1)n ) ctsf ctsf ctsf

0

? 1 - Hcts (GFp , V )

? 1 - Hcts (GFp , U )

- 0

? 1 - Hcts (GFp , Q ` (1)n )

and the left vertical arrow is an isomorphism by Lemma 11.5. Hence the right square is cartesian. ¤

Appendix: Continuous Cohomology and Yoneda Extensions In this appendix we prove a result about the relation between continuous cohomology and Yoneda extension groups in low degrees. It is surely well known, but we know of no reference. Suppose that K is a topological eld, and Γ a topological group. A continuous Γ-module is a Γ-module V , where V is a nite dimensional K -vector space. The action Γ → GL(V ) is required to be continuous, where GL(V ) is given the topology induced from that of K . Denote by C(Γ, K) the category of nite dimensional continuous Γ-modules. Since any K -linear morphism between nite dimensional vector spaces is continuous, this is an abelian category. For continuous Γ-modules A and B , dene Ext•Γ (A, B) to be the graded group of Yoneda extensions of B by A in the category C(Γ, K). For a continuous Γ-module A, one also has the continuous cohomology groups • Hcts (Γ, A) dened by Tate [42], which are dened using the complex of continuous cochains.

If A is a continuous Γ-module , there is a natural isomorphism 2 1 ∼ (Γ, A). = Hcts (Γ, A) and a natural injection Ext2Γ (K, A) ,→ Hcts

Theorem A.6.

Ext1Γ (K, A)

It is well known that an extension 0 → A → E → K → 0 in C(Γ, K) gives a continuous cocycle f : Γ → A by choosing a lift e ∈ E of 1 ∈ K and

Proof.


dening f (σ) = σ(e) − e. Conversely, for a given continuous cocycle f , we may dene continuous Γ-action on A ⊕ K by σ : (a, k) 7→ (σ(a) + kf (σ), k). These are mutually inverse, which establishes the rst claim. To prove the second claim, we rst dene a K -linear mapping 2 ϕ : Ext2Γ (K, A) → Hcts (Γ, A)

as follows. For c ∈ Ext2Γ (K, A), choose a 2-fold extension 0 → A → E2 → E1 → K → 0 that represents it. By [45], c is the image under the connecting homomorphism

δ : Ext1Γ (K, E2 /A) → Ext2Γ (K, A) of the class c˜ of the extension 0 → E2 /A → E1 → K → 0. We shall construct ϕ so that the diagram

/ Ext1 (K, E2 /A) Γ

Ext1Γ (K, E2 ) '

δ

/ Ext2 (K, A) Γ ϕ

ψ '

² / H 1 (Γ, E2 /A) cts

² 1 Hcts (Γ, E2 )

δcts

² / H 2 (Γ, A) cts

commutes, where the rows are parts of the standard long exact sequences constructed in [45] and [42, Sect. 2]. Dene ϕ(c) to be δcts (ψ(˜ c)). To prove ϕ(c) is well-dened, it suces to show that two 2-fold extensions that t into a commutative diagram

0 −−−−→ A −−−−→ E20 −−−−→ E10 −−−−→ K −−−−→ 0

° ° °

  y

  y

° ° °

0 −−−−→ A −−−−→ E2 −−−−→ E1 −−−−→ K −−−−→ 0 2 (Γ, A). But this follows from the functoriality of the give a same element of Hcts • connecting homomorphism for Hcts , i.e., the commutativity of 1 2 Hcts (Γ, E20 /A) −−−−→ Hcts (Γ, A)

  y

° ° °

1 2 Hcts (Γ, E2 /A) −−−−→ Hcts (Γ, A).

The K -linearity of ϕ is easily checked. Finally, the injectivity of ϕ follows from the fact that for each extension as above, ϕ is injective on the image of the ¤ connecting homomorphism δ : Ext1Γ (K, E2 /A) → Ext2Γ (K, A). Note that one may dene m Extm Γ (K, A) → Hcts (G, A)

by induction on m in the same way.

214


Acknowledgements We would like to thank Marc Levine for clarifying several points about motivic cohomology and Owen Patashnick for his helpful comments on the manuscript. We are indebted to Kazuya Kato and Akio Tamagawa for pointing out a subtlety regarding continuous cohomology related to Theorem A.6, and to Takeshi Tsuji for the proof of Proposition 11.3. We would also like to thank Sasha Goncharov for pointing out the existence and relevance of the unpublished manuscript [2] of Beilinson and Deligne, and Romyar Shari for correspondence of Galois cohomology when ` = 2. Finally, we would like to thank the referee for doing a very thorough job and for many useful comments.

References [1] A. Beilinson: Higher regulators and values of L-functions (Russian), Current problems in mathematics, Vol. 24 (1984), 181238. [2] A. Beilinson, P. Deligne: Motivic Polylogarithms and Zagier's Conjecture, unpublished manuscript, 1992. [3] A. Beilinson, P. Deligne: Interprétation motivique de la conjecture de Zagier reliant polylogarithmes et régulateurs, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., 55, Part 2, Amer. Math. Soc., Providence, RI, 1994, 97121. [4] A. Beilinson, R. MacPherson, V. Schechtman: Notes on motivic cohomology, Duke Math. J. 54 (1987), 679710. [5] S. Bloch: Algebraic cycles and higher K -theory, Adv. in Math. 61 (1986), 267304. [6] S. Bloch: The moving lemma for higher Chow groups, J. Algebraic Geom. 3 (1994), 537568. [7] S. Bloch, K. Kato: L-Functions and Tamagawa Numbers of Motives, The Grothendieck Festschrift Volume I, Progress in Math. Vol.86, Birkhäuser, (1990), 333-400. [8] S. Bloch, I. Kriz: Mixed Tate motives, Ann. of Math. 140 (1994), 557605. [9] S. Bloch, A. Ogus: Gersten's conjecture and the homology of schemes, Ann. Sci. École Norm. Sup. (4) (1974), 181201. [10] A. Borel: Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4) 7 (1974), 235272 (1975). [11] A. Borel: Cohomologie de SLn et valeurs de fonctions zêta aux points entiers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), 613636. [12] A. Borel, J.-P. Serre: Le théorème de Riemann-Roch, Bull. Soc. Math. France 86 1958, 97136. [13] P. Deligne and J. Milne: Tannakian categories, in Hodge Cycles, Motives, and Shimura Varieties, (P. Deligne, J. Milne, A. Ogus, K.-Y. Shih editors), Lecture Notes in Mathematics 900, Springer-Verlag, 1982. [14] P. Deligne: Le groupe fondamental de la droite projective moins trois points, in Galois groups over Q (Berkeley, CA, 1987), 79297, Math. Sci. Res. Inst. Publ., 16, Springer, New York-Berlin, 1989.


[15] V. Drinfeld: On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal Q /Q ), (Russian) Algebra i Analiz 2 (1990), 149181; translation in Leningrad Math. J. 2 (1991), no. 4, 829860. [16] J.M. Fontaine: Sur certains types de représentations p-adiques du groupe de Galois d'un corps local: construction d'un anneau de Barsotti-Tate, Ann. of Math. 115 (1982), 529577. [17] H. Gillet: Riemann-Roch theorems for higher algebraic K -theory, Adv. in Math. 40 (1981), 203289. [18] A. Goncharov: Volumes of hyperbolic manifolds and mixed Tate motives, J. Amer. Math. Soc. 12 (1999), 569618. [19] A. Goncharov: Multiple ζ -values, Galois groups, and geometry of modular varieties, European Congress of Mathematics, Vol. I (Barcelona, 2000), 361392, Progr. Math., 201, Birkhäuser, 2001. [20] R. Hain: Classical polylogarithms, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., 55, Part 2, Amer. Math. Soc., Providence, RI, 1994, 342. [21] R. Hain: Completions of mapping class groups and the cycle C − C − , in Mapping Class Groups and Moduli Spaces of Riemann Surfaces, Contemp. Math. 150 (1993), 75105. [22] R. Hain: Innitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997), 597651. [23] R. Hain, M. Matsumoto: Weighted Completion of Galois Groups and Galois Actions on Fundamental Groups, Compositio Math., to appear, math.AG/0006158. [24] R. Hain, M. Matsumoto: Completions of Arithmetic Mapping Class Groups, in preparation. [25] H. Hiller: λ-rings and algebraic K -theory, J. Pure Appl. Algebra 20 (1981), 241 266. [26] Y. Ihara: Pronite braid groups, Galois representations and complex multiplications, Ann. of Math., 123 (1986), 43106.

[27] Y. Ihara: The Galois representation arising from P 1 − {0, 1, ∞} and Tate twists of even degree, in Galois groups over Q , Publ. MSRI, No. 16 (1989), Springer-Verlag, 299313. [28] Y. Ihara: Some arithmetic aspects of Galois actions in the pro-p fundamental group of P 1 − {0, 1, ∞}, Arithmetic fundamental groups and noncommutative algebra (Berkeley, CA, 1999), 247273, Proc. Sympos. Pure Math., 70, Amer. Math. Soc., 2002. [29] U. Jannsen: Mixed motives and algebraic K -theory, Lecture Notes in Mathematics, 1400, Springer-Verlag, Berlin, 1990. [30] J. Jantzen: Representations of Algebraic Groups, Pure and Applied Mathematics Vol.131, Academic Press, 1987. [31] C. Kratzer: λ-structure en K -théorie algébrique, Comment. Math. Helv. 55 (1980), 233254. [32] M. Levine: Tate motives and the vanishing conjectures for algebraic K -theory, Algebraic K -theory and algebraic topology (Lake Louise, AB, 1991), NATO Adv.

216


Sci. Inst. Ser. C Math. Phys. Sci., 407, Kluwer Acad. Publ., Dordrecht, 1993, 167 188. [33] M. Levine: Bloch's higher Chow groups revisited, K -theory (Strasbourg, 1992), Astérisque No. 226, (1994), 10, 235320. [34] M. Levine: Mixed motives, Mathematical Surveys and Monographs, 57, Amer. Math. Soc., Providence, RI, 1998. [35] M. Matsumoto: On the Galois image in derivation of π1 of the projective line minus three points, in Recent developments in the inverse Galois problem (Seattle, WA, 1993), Contemp. Math. 186 (1995), 201213. [36] J. Milnor: Introduction to algebraic K -theory, Annals of Mathematics Studies, No. 72. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. [37] D. Quillen: Finite generation of the groups Ki of rings of algebraic integers, Algebraic K -theory, I: Higher K -theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 179198. Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973. [38] D. Quillen: On the cohomology and K -theory of the general linear groups over a nite eld, Ann. of Math. 96 (1972), 552586. [39] D. Quillen: Higher algebraic K -theory, I, Algebraic K -theory, I: Higher K -theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85147. Lecture Notes in Math., Vol. 341, Springer, Berlin 1973. [40] J. Rognes and C. Weibel: Two-primary algebraic K -theory of rings of integers in number elds. (Appendix A by Manfred Kolster), J. Amer. Math. Soc., 13 (2000), 154. [41] C. Soulé: On higher p-adic regulators, Lecture Notes in Math. 854 (1981), 372401. [42] J. Tate: Relations between K2 and Galois Cohomology, Invent. Math., 30 (1976), 257274. [43] H. Tsunogai: On ranks of the stable derivation algebra and Deligne's problem, Proc. Japan Academy Ser. A 73 (1997), 2931. [44] V. Voevodsky, A. Suslin, E. Friedlander: Cycles, transfers, and motivic homology theories, Annals of Mathematics Studies, 143, Princeton University Press, 2000. [45] N. Yoneda: On Ext and exact sequences, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 507576. Richard Hain Department of Mathematics Duke University Durham, NC 27708-0320

[email protected]

Makoto Matsumoto Department of Mathematics Faculty of Science Hiroshima University Higashi-Hiroshima, 739-8526 Japan

[email protected]

!"$#$% &' ()* + ,-/.021334

57698;:=@?BADCE:=I R C E JRU 8>I E G CG g T E O 5:A 9v.

./

/

@/.

1 (

!#"%$&(' M l 0,4? Á *) ' ' )¿,+.-0/1Z) ÂmÂ+ ¾(2 '

N : 5 5{;L96 ; S 5\9

Galois Groups and Fundamental Groups (Mathematical Sciences Research Institute Publications)

Galois groups and fundamental groups

Galois groups and fundamental groups