Cohomology of Drinfeld modular varieties. Part I. Geometry, counting of points and local harmonic analysis 9780521470605, 0521470609

Table of contents :
Contents
Preface
1 Construction of Drinfeld modular varieties
2 Drinfeld A-modules with finite characteristic
3 The Lefschetz numbers of Hecke operators
4 The fundamental lemma
5 Very cuspidal Euler-Poincare functions
6 The Lefschetz numbers as sums of global
elliptic orbital integrals
7 Unramified principal series representations
8 Euler-Poincare functions as pseudocoefficients
of the Steinberg representation
Appendices
A. Central simple algebras
B. Dieudonne's theory : some proofs
C. Combinatorial formulas
D1 Representations of unimodular, locally compact, totally discontinuous, separated, topological groups
References
Index

Citation preview

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS : 41 EDITORIAL BOARD D.J.H. GARLING, T. TOM DIECK, P. WALTERS

COHOMOLOGY OF DRINFELD MODULAR VARIETIES, PART I

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 06:30:53, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162

Already published 1 W.M.L. Holcombe Algebraic automata theory 2 K. Petersen Ergodic theory 3 P.T. Johnstone Stone spaces 4 W.H. Schikhof Ultrametric calculus 5 J.-P. Kahane Some random series of functions, 2nd edition 6 H. Cohn Introduction to the construction of class fields 7 J. Lambek & P.J. Scott Introduction to higher-order categorical logic 8 H. Matsumura Commutative ring theory 9 C.B. Thomas Characteristic classes and the cohomology of finite groups 10 M. Aschbacher Finite group theory 11 J.L. Alperin Local representation theory 12 P. Koosis The logarithmic integral I 13 A. Pietsch Eigenvalues and s-numbers 14 S.J. Patterson An introduction to the theory of the Riemann zeta-function 15 H.J. Baues Algebraic homotopy 16 V.S. Varadarajan Introduction to harmonic analysis on semisimple Lie groups 17 W. Dicks & M. Dunwoody Groups acting on graphs 18 L. J. Corwin & F.P. Greenleaf Representations of nilpotent Lie groups and their applications 19 R. Fritsch & R. Piccinini Cellular structures in topology 20 H Klingen Introductory lectures on Siegel modular forms 21 P. Koosis The logarithmic integral II 22 M.J. Collins Representations and characters of finite groups 24 H. Kunita Stochastic flows and stochastic differential equations 25 P. Wojtaszczyk Banach spaces for analysts 26 J.E. Gilbert & M.A.M. Murray Clifford algebras and Dirac operators in harmonic analysis 27 A. Prohlich & M.J. Taylor Algebraic number theory 28 K. Goebel & W.A. Kirk Topics in metric fixed point theory 29 J.F. Humphreys Reflection groups and Coxeter groups 30 D.J. Benson Representations and cohomology I 31 D.J. Benson Representations and cohomology II 32 C. Allday & V. Puppe Cohomological methods in transformation groups 33 C. Soule et al Lectures on Arakelov geometry 34 A. Ambrosetti Sz G. Prodi A primer of nonlinear analysis 35 J. Palis & F. Takens Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations 36 M. Auslander, I. Reiten & S. Smalo Representation theory of Artin algebras 37 Y. Meyer Wavelets and operators 38 C. Weibel An introduction to homological algebra 39 W. Bruns & J. Herzog Cohen-Macaulay rings 40 V. Snaith Explicit Brauer induction 41 G. Laumon Cohomology of Drinfeld modular varieties I 42 E.B. Davies Spectral theory and differential operators 43 J. Diestel, H. Jarchow Sz A. Tonge Absolutely summing operators 44 P. Mattila Geometry of sets and measures in Euclidean spaces 45 R. Pinsky Positive harmonic functions and diffusion 46 G. Tenenbaum Introduction to analytic and probabilistic number theory 50 I. Porteous Clifford algebras and the classical groups Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 06:30:53, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162

Cohomology of Drinfeld Modular Varieties, Part I Geometry, counting of points and local harmonic analysis Gerard Laumon Universite Paris-Sud

CAMBRIDGE UNIVERSITY PRESS Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 06:30:53, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521470605 © Cambridge University Press 1996 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1996 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Laumon, Gerard. Cohomology of Drinfeld modular varieties/Gerard Laumon. p. cm. - (Cambridge studies in advanced mathematics: 41) Includes bibliographical references and index. Contents: pt. 1. Geometry, counting of points, and local harmonic analysis ISBN 0 521 47060 9 (pt. 1) 1. Drinfeld modular varieties. 2. Homology theory. I. Title. II. Series. QA251.L287 1996 512'.24 - dc20 94-27643 CIP ISBN 978-0-521-47060-5 hardback Transferred to digital printing 2009 Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables and other factual information given in this work are correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter.

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 06:30:53, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162

Contents Preface

ix

1. Construction of Drinfeld modular varieties

1

(1.0) Notations

1

(1.1) Endomorphisms of the additive group

1

(1.2) Drinfeld modules

3

(1.3) Level structures

5

(1.4) Modular varieties

7

(1.5) Deformation theory

9

(1.6) Hecke algebras, correspondences

12

(1.7) Hecke operators

15

(1.8) Comments and references

18

2. Drinfeld ^4-modules with finite characteristic

19

(2.0) Notations

19

(2.1) Isogenies

19

(2.2) Isogeny classes of Drinfeld modules

22

(2.3) Tate modules of a Drinfeld module

27

(2.4) Dieudonne modules

31

(2.5) Dieudonne module of a Drinfeld module

35

(2.6) First description of an isogeny class

43

(2.7) Isogeny classes as double coset spaces

47

(2.8) Comments and references

50

3. The Lefschetz numbers of Hecke operators

51

(3.0) Introduction

51

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 06:30:52, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162

CONTENTS

(3.1) The Lefschetz numbers of correspondences

51

(3.2) Counting of fixed points

52

(3.3) Where the orbital integrals come in

56

(3.4) Transfer of conjugacy classes

60

(3.5) Transfer of Haar measures

66

(3.6) The Lefschetz numbers as sums of twisted orbital integrals

72

(3.7) Comments and references

74

4. The fundamental lemma

75

(4.0) Introduction

75

(4.1) Satake isomorphism

76

(4.2) Base change homomorphism

83

(4.3) Orbital integrals

87

(4.4) Twisted orbital integrals

93

(4.5) Main theorem

98

(4.6) The elliptic case

102

(4.7) The general case

114

(4.8) Non-closed orbital integrals

117

(4.9) Comments and references

128

5. Very cuspidal Euler—Poincare functions

129

(5.0) Introduction

129

(5.1) The function /

130

(5.2) Kottwitz's functions

133

(5.3) Elliptic orbital integrals of /

135

(5.4) if-invariant constant terms of /

142

(5.5) The function / is very cuspidal

152

(5.6) Non-elliptic orbital integrals of /

156

(5.7) Comments and references

156

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 06:30:52, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162

CONTENTS

vii

6. The Lefschetz numbers as sums of global elliptic orbital integrals

158

7. Unramified principal series representations

160

(7.0) Introduction

160

(7.1) Parabolic induction and restriction

160

(7.2) Cuspidal representations

165

(7.3) Principal series representations

168

(7.4) Unramified principal series representations

178

(7.5) Spherical representations

187

(7.6) Comments and references

191

8. Euler-Poincare functions as pseudocoefficients of the Steinberg representation

192

(8.0) Introduction

192

(8.1) The Steinberg representation

192

(8.2) Main theorem

207

(8.3) Some easy vanishing results

208

(8.4) Cohomological interpretation of tr?r(/)

213

(8.5) Unitarizable representations

220

(8.6) Proof of Howe and Moore's criterion of non-unitarizability

226

(8.7) Comments and references

248

Appendices A. Central simple algebras

249

(A.0) Central simple algebras

249

(A.I) Bicommutant theorem

250

(A.2) Central simple algebras over local fields

251

(A.3) Central simple algebras over function fields

253

(A.4) Comments and references

255

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 06:30:52, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162

ii

CONTENTS

B. Dieudonne's theory : some proofs

256

(B.I) Proof of (2.4.5)

256

(B.2) Proof of (2.4.6)

268

(B.3) Proof of (2.4.11)

272

(B.4) Comments and references

280

C. Combinatorial formulas

281

(CO) Introduction

281

(C.I) g-binomial coefficients

281

D. Representations of unimodular, locally compact, totally discontinuous, separated, topological groups 284 (D.0) Introduction

284

(D.I) Smooth representations of H

284

(D.2) Admissible representations of H

290

(D.3) Induction and restriction

292

(D.4) Cuspidal representations of H

311

(D.5) Injective and projective objects in Reps(H); cohomology

322

(D.6) Unitarizable representations

327

(D.7) Decomposition of representations into tensor products

334

(D.8) Comments and references

336

References

337

Index

341

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 06:30:52, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162

Preface

Shimura varieties are quasi-projective varieties over number fields that are associated with reductive groups and some other data. Similarly Drinfeld modular varieties are quasi-projective varieties over function fields that are associated with reductive groups and some other data. The cohomology of Shimura varieties is related to automorphic forms over number fields. In the same way, the cohomology of Drinfeld modular varieties is related to automorphic forms over function fields. The study of the relations between the cohomology of Drinfeld modular varieties and automorphic forms over function fields is the main theme of this book. The Drinfeld modular varieties that I will consider were discovered by Drinfeld in 1973. They depend on the following data: a function field F with a finite field of constants, a place oo of F, a positive integer d and a non-zero ideal / of the ring A of functions in F which are regular outside oo. The Drinfeld modular variety corresponding to (F, oo, d, /) is denoted by Mf. The underlying reductive group is GL& over F. As the classical modular curve X(N) (N a positive integer) parametrizes the elliptic curves with a level-iV-structure, Mf parametrizes the so-called elliptic modules of rank d (I will use the terminology Drinfeld modules of rank d) with a level-/-structure. As X(N) is affine and smooth of pure relative dimension 1 over Spec(Z[l/iV]) as long as N > 3, Mf is affine and smooth of pure relative dimension d— 1 over Spec(A) — V(I) as long as / ^ A. As one can define the Hecke correspondence Tp on X(N) for each prime p which doesn't divide N, Drinfeld has defined the Hecke correspondences Tlx (i = 1,..., d— 1) for each place x ^ oo of F which doesn't divide /. When / varies the affine schemes Mf can be organized into a projective system. The limit Md of this project ive system is an affine scheme over F. On M d , Drinfeld has defined an action of the adelic group GLd{A°°) where A°° = F ®A A is the ring of adeles of F outside oo. For each / as before we have

F ®A Mf = MdjKf where Kf° is the compact open subgroup Kei(GLd(A) ->

GLd(A/IA))

of GZ/d(A°°) and for each place x ^ oo of F which doesn't divide / and for each i = 1,..., d, the Hecke correspondence T%x on F (gu Mf is induced by Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:38:25, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.001

x

DRINFELD MODULAR VARIETIES

the action of I ^x

GLd(Fx) c GLd(A°°) 1/

on Md (wx is a uniformizer of F at x and is repeated i times on the diagonal). The cohomology of Drinfeld modular varieties that I will study in these notes is the ^-adic etale cohomology with compact supports H*(F®F

Md,®£)

for some fixed prime £ and for some fixed algebraic closures Q^ and F of Q^ and F respectively. On this cohomology, Gal(F/F) and GLd(A°°) act and these two actions commute. MAIN PROBLEM.

— Describe the virtual Q^[Gal(F/F) x GLd(A°°)]-

modules in terms of automorphic representations of GLd over F. The method that I will follow to attack this problem was initiated by Ihara and Langlands in the case of classical modular curves and greatly extended by Langlands, Kottwitz and many other mathematicians in the case of Shimura varieties. It was applied by Drinfeld himself to solve the above main problem for d = 2. Roughly speaking, it can be described in the following way. The first step is to use the Grothendieck-Lefschetz trace formula to express the trace of the above virtual module in terms of a number of fixed points. The second step is to compare this number of fixed points with the geometric side of a suitable Arthur-Selberg trace formula. The third step is to compute explicitly the spectral side of this ArthurSelberg trace formula. Finally, the identification of the cohomological side of the GrothendieckLefschetz trace formula with the spectral side of the Arthur-Selberg trace formula gives the desired description of our virtual module. In pursuing this program for Shimura varieties, one is faced with two main difficulties. The first one is related to the unstable conjugacy classes in a general reductive group. The second one is related to the non-compactness of the Shimura varieties. Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:38:25, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.001

PREFACE

xi

Fortunately the first difficulty does not occur for Drinfeld modular varieties: in GLd all the conjugacy classes are stable. From this point of view, Drinfeld modular varieties are much easier to study than Siegel modular varieties of principally polarized abelian varieties of dimension g (except if g = 1). But the second difficulty does occur: Drinfeld modular varieties are non-compact. This has two consequences. First of all we cannot apply directly the Grothendieck-Lefschetz trace formula. Secondly, the ArthurSelberg trace formula is very complicated in this case. Following ideas of Deligne and Kazhdan, I will show how to overcome this difficulty in the second volume of this book. Now let me briefly describe the content of the first volume of this book. In the first chapter I review Drinfeld's definitions of the Drinfeld modules of rank d over a base scheme, of the characteristic of such a module and of the level-/-structures on such a module. Then I recall the construction and the basic properties of Drinfeld modular varieties. Finally I recall Drinfeld's construction of the Hecke operators on these varieties. In the second chapter I review Drinfeld's description of the set of the Drinfeld modules of rank d for a given finite characteristic. This is completely analogous to the Honda-Tate theory for elliptic curves in finite characteristic. Using this description of the set of points of the Drinfeld modular varieties in finite characteristic one obtains a formula for the number of fixed points of the product of a Frobenius power and a Hecke correspondence acting on this set. This formula is given in terms of twisted orbital integrals. This is the aim of chapter 3. One cannot directly compare the formula for the number of fixed points with the geometric side of the Arthur-Selberg trace formula. An important step in this comparison is to replace the twisted orbital integrals by ordinary orbital integrals. This is the purpose of chapters 4 and 5. In chapter 4 I recall the statement and Drinfeld's proof of a particular case of the so-called fundamental lemma. I also review some important properties of the orbital integrals for GLd over a non-archimedean local field. In chapter 5 I recall Kottwitz's construction of Euler-Poincare functions and the computation of their orbital integrals. I also introduce the new notion of very cuspidal function and I show (with the help of Waldspurger) that a suitable linear combination / of Kottwitz's Euler-Poincare functions is very cuspidal. The notion of very cuspidal function does not play an important role in the first volume of this book but will be crucial in the second volume. Putting together the results of the previous chapters, one gets that the above number of fixed points is equal to the elliptic part of the ArthurSelberg trace formula. This is done in chapter 6. Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:38:25, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.001

xii

DRINFELD MODULAR VARIETIES

The last two chapters are devoted to a review of some fundamental results in local harmonic analysis which will be needed in the second volume of this book. In chapter 7 I recall the construction and the main properties of the unramified principal series representations over a non-archimedean local field. In particular, I give the classification of the spherical representations and compute their traces. In chapter 8, I prove that the very cuspidal function / which has been introduced in chapter 5 is a pseudo-coefficient of the Steinberg representation. In fact, let 717 (/ C {l,...,d— 1}) be the irreducible constituents of the -|

ley

induced representation from the standard Borel subgroup B to GLd of 6B ' (6B is the modulus character of B). Then I prove that for all / C A, that tr TT(/) = 0 for each smooth irreducible representation n of GLd which is not isomorphic to one of the TT/'S and that among the TT/'S only the trivial representation TTA and the Steinberg representation TT0 are unitarizable. There are four appendices. In appendix A I review the basic facts that are needed on central division algebras. In appendix B I have included the proofs of the theorems of Dieudonne theory in equal characteristic that I have used in chapter 2. Appendix C contains the proofs of combinatorial formulas that I use in chapter 4 (proof of (4.6.1)). In the long appendix D I recall the basic results about smooth representations of locally compact, totally discontinuous, separated, topological groups. This book originates from two graduate courses, one given during the fall of 1989 at the University of Minnesota and the other one during the spring of 1991 at Caltech. I am grateful to the Department of Mathematics of the University of Minnesota and especially to W. Messing and S. Sperber for their support and their kind hospitality during the fall of 1989. I thank the colleagues and the students who attended my lectures. I am grateful to the Department of Mathematics of Caltech and especially to D. Ramakrishnan for their kind hospitality and their support during the spring of 1991. The graduate course that I gave at Caltech was part of a special program organized jointly by D. Ramakrishnan and D. Blasius, H. Hida, J. Rogawski from UCLA. I thank them as well as the other mathematicians who attended my course. Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:38:25, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.001

PREFACE

xiii

I am indebted to D. Kazhdan and R. Kottwitz. In 1986 they lectured on similar subjects. Kazhdan's course at Harvard University dealt with the cohomology of Drinfeld modular varieties with values in certain local systems. The main topic of Kottwitz's course at the Universite de Paris-Sud was the computation of the zeta function of Shimura varieties following Langlands' method. I have been strongly influenced by both courses and by the numerous discussions that I had with D. Kazhdan and R. Kottwitz. In particular, I have organized my course in the same way as R. Kottwitz. I owe a special acknowledgment to J.-L. Waldspurger who helped me to complete the proof of the main theorem of chapter 5. I also had a lot of very useful discussions with him. Thanks are also due to several other colleagues, in particular H. Carayol, L. Clozel, G. Henniart, M. Rapoport and J.-P. Wintenberger for many stimulating discussions and their help in solving some difficulties. Special thanks go to M. Bonnardel and M. Le Bronnec who typed the manuscript with care while being introduced to T^X and to the editors who did a beautiful job.

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:38:25, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.001

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:38:25, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.001

Construction of Drinfeld modular varieties

(1.0) Notations Let p be a prime number and let X be a smooth, projective and connected curve over the finite field JFp = Z/pZ. We will denote by 77 the generic point of X and by F = K(TJ) its function field; we will identify the set of closed points \X\ of X with the set of places of F. For each x G |X|, we will denote by Fx the completion of F at the place x, by x : Fx —»Z U {00} the discrete valuation of F x , by Ox C F x the corresponding valuation ring, by wx a prime element of Ox (x(wx) = 1), by K(X) = Ox/(wx) the residue field of Ox and by deg(x) the degree of K(X) over 1FP (K(X) has pde&(x) elements). Once for all we fix a place 00 of F. The open subset X — {00} of X is affine; let A be its ring of regular functions, i.e. A = {a e F I x{a) > 0, Vx E X - {00}}. For each a € A — {0}, we have deg(oo)oo(a) = - dimF (A/(a)).

(1.1) Endomorphisms of the additive group Let k be a ring of characteristic p. The ring End((Ga)fc) of endomorphisms of the fc-group scheme (fja,k — Spec(fe[t]) is canonically isomorphic to the non-commutative polynomial ring k[r] with the commutation rule r - a = oP - r

(Va G k)

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:42:12, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.002

2

DRINFELD MODULAR VARIETIES

(see [Or]); each a G k is identified with the endomorphism d) are nilpotent, (ii) there exists one and only one ty such that fio = 1 and 6n = 0 for all (n > d). Proof: If i\)~x = X^Tn7"71? t h e n we have n

«-= E ^-C i+j+k=n

and fad? 7^ is nilpotent (resp. invertible) as long as j > 0 or i > d or k > 0 (resp. i — d and j = k = 0). Part (i) follows immediately from this remark. Let / be the ideal of k generated by the an's (n > d) and the /?n's (n > 0). Then there exists an integer s > 1 such that Is = 0. By induction we can (and we will) assume that 5 = 2 (replace successively (k,I) by (/c// 2 ;/// 2 ), 4

2

4

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:42:12, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.002

1. CONSTRUCTION OF DRINFELD MODULAR VARIETIES

3

Now to prove the uniqueness of ij) it is sufficient to prove that ip = 1 when an = 0 for all n > d and Sn = 0 for all n > d. By induction we can assume that /3n = 0 for all n > N and we need to prove that /3/v = 0 (N is any integer > 0). But we have

by equating the coefficients of rN+d, hence f3N = 0 (£d/3j£ G / p d C / 2 = (0) and ad is invertible), hence tp = 1. To prove the existence of ^ it is sufficient to prove that, if an = 0 for all n> N and some integer N > d, the coefficient of r n in )(

is zero for all n> N. But this is an obvious computation as aN

(l -

r^V1

= 1+

aN

rN-d D

REMARK

(1.1.3). — If A; is a field, we have a map deg : End(Ca,fc) -> N U {-oo}

satisfying the properties deg((p + ip)< max(deg , for all y?, ij) G End(Ga?fc), and defined by

if ad ^ 0 and deg(0) = —oo.

•

We have a ring homomorphism (1.1.4)

d : End((Ga,fc) ^ k

defined by

(1.2) Drinfeld modules Let S be a scheme of characteristic p and let d be an integer > 0. Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:42:12, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.002

4

DRINFELD MODULAR VARIETIES

DEFINITION (1.2.1) (Drinfeld). — A Drinfeld A-module of rank d over S is a pair (E,(aA)(*(i)), Vz G (I-i/A)*, VA - 1 , . . . , I, (c) the polynomial /

(t-L(i))eH0(

J] i£(I-1/A)d

is exactly the g.c.d. of the polynomials

n=0

for A — 1 , . . . , s. The theorem is now obvious.

D

If J C / are non-zero ideals of A, we have an obvious map of fibered categories

ri,j :Mdj^MJ and a commutative diagram

Mdj

e

e

LEMMA (1.4.2). — The map of fibered categories

ri,j : Mj -+ A4?|(X - ({oo} U V(J))) is representable, finite, etale and Galois with Galois group the kernel of the reduction modulo I homomorphism GLd{A/J)

->

GLd(A/I).

Proof : We have a right action of GLd{A/J) on Mdj : g € GLd(A/J) maps (E, X- ({oo} U V(I)) is smooth

of pure

relative

dimension

d—1.

Proof: Thanks to lemma (1.4.2), it is enough to consider the case I = A. To prove the theorem in this case we will use deformation theory. Let k be a ring of characteristic p and let ip : A —• k[r] be a Drinfeld A-module of rank d over k with characteristic 9 = d o (p : A —* k. We will assume for simplicity that (p is standard. Let O be a thickening offc,i.e. a ring of characteristic p endowed with an ideal m such that m2 = (0) and O/m = k. We are looking for Drinfeld A-modules (of rank d) over O

fi:A-> O[r] such that the reduction modulo m of is precisely m[r] be the Fp-bilinear map denned by T(a1,a2)=i/>(a1)il>{a2)-tl;(a1a2) for each a1, a2 G A. Then T satisfies the Hochschild cocycle condition a1 -T(a2,a3)-T(a1a2,a3)+T(a\a2a3)-T(a\a2)-a3

=0

for all o} ,a2 ,o? e A where m[r] = {^anrn

e O[r]\an G m,Vn > 0}

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:42:12, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.002

10

DRINFELD MODULAR VARIETIES

is viewed as an (A, A)-bimodule in the following way: a . P{T) • b = ^{a)P{r)^{b) for each a,b e A and P(r) G m[r] (as m2 = (0), this (A, ^4)-bimodule structure is well-defined and independent of the choice of -0 &nd we can replace ^(a) by d o ^(a) in its definition). Moreover, if we replace ip by another lifting ip + t where t : A —> m[r] is any JFp-linear map, T has to be replaced by T + 6t where 6t is the Hochschild coboundary (6t)(a\a2) = a1 • t(a2) - t(ala2) + t(al) - a2 for each a1, a2 G A. In other words, the obstruction to lifting

o}a2) and where the (A, A)-bimodule m[r] morphism A(S>wp A —> A (a}®a2 — is viewed as a (left) (^4®FP A)-module in the usual way. But by Grothendieck's duality (see [Ha](7.2))

for each integer n > 2 and in fact A(A,m[r])

= (TA/Fp

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:42:12, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.002

1. CONSTRUCTION OF DRINFELD MODULAR VARIETIES

11

where TA/WP — Hom^(fi^ / F , A) is viewed as a (right) ( ^ 4 0 F P A)-module via the augmentation map A 0jpp A —> A. Already this proves that the DeligneMumford stack M.\ is smooth over JFp of pure dimension d. Indeed, let k be a field of characteristic p and let cp : A —• /c[r] be a Appoint of A^^, then the tangent space of A^^ at tp is canonically isomorphic to T

A/wp ®A®¥pA k[r)

where the (^4, A)-bimodule structure of k[r] is given by a • P(T) - b = 0(a)P(T) X — {00} at some point

A,e k

induced by d : k[r] —> k (0 : A —* k is the characteristic of (p). But this map is clearly surjective and the theorem is proved. •

(1.6) Hecke algebras, correspondences Let A be the ring of adeles of F with its usual topology. Then we have A = F( where A°° = and, if we set

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:42:12, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.002

1. CONSTRUCTION OF DRINFELD MODULAR VARIETIES

13

the subring of A O = e>oo x O°° is a maximal compact subring. For each non-zero ideal / of A we will denote by Kf° the compact open subgroup Kev(GLd(O°°) —» GLdiO^/O00!)) of GLd(A°°) (the map is the reduction modulo / map) and we will consider the Hecke algebra (over Q) rii =cc {GLd(A

)//KI )

of (locally constant) functions with compact support f°° ' GLd(A°°) -> Q which are invariant by left and right translations by elements of Kf°; the product of two functions /f° and f%° in Hi is the convolution product denned by

(/r *

[ JGLd(A°°)

for each g°° G GLd(A°°), where dh°° is the Haar measure on GLd(A°°) normalized by In fact a basis of Wj° is given by the characteristic functions

of the double classes Kfg°°Kf GLd{A°°) and if

(g°° G GLd(A°°))

and if g?,g?

G

N ISOQ ^OO T^OO _,OO TSOQ

^1 9l AI 92 ^1

TT

TSOO ^.OOT^OO

— J_[ ^ 7 ^3 -^7 ' 71=3

we have iV ^—>

n=3

where is a positive integer (n = 3 , . . . , N). Let f°° 6 Hf and ar 6 |X| - ({oo} U V(I)), we will say that x is good for f°° if Supp(/O°) C GLn(Ox) x GLnCA00--) c GLn(A°°). Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:42:12, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.002

14

DRINFELD MODULAR VARIETIES

Obviously, for a given /°° G Hf, almost all the x G \X\ - ({00} U V(I)) are good for /°° and, for a given x G \X\ - ({oc} U V(I)), the subset

H?'x C Uf with elements the /°°'s such that x is good for /°° is a Q-sub-algebra; moreover as a Q-algebra T~C^'X is isomorphic to C

(same definition as before but replace A°° by A°°x,...). Let 5 be a noetherian scheme and let Y —• S be a separated 5-scheme of finite type. A geometric correspondence of Y over S is an isomorphism class of commutative diagrams of schemes

where c2 is finite and etale and c\ is proper. If (Z, c) and (Z',d) are two geometric correspondences of y over S (c = (ci,C2),c; = (c^c^)) we can form the correspondences

(z, c) + (z', d) = (zu z', (Cl n c;, c2 n 4 and (Z,C) • (Z',*/) = (Z7 X^ f y |Cl Z, (Ci O (Z' Xy C l ),C 2 O ( ^ Xy Z)))

of Y on 5. Let be the abelian group generated by the monoid of geometric correspondences of Y on S with the above addition law. The multiplication law obviously induces a ring structure on Corr(F/5)z- We will denote by [Z, c] the class of the geometric correspondence (Z, c) in Corr(Y/S)z (the zero element is [0, c] and [y, (zdy,zdy)] is a unit for this ring). Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:42:12, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.002

1. CONSTRUCTION OF DRINFELD MODULAR VARIETIES

15

The Q-algebra Corr(y/5) = Q ® will be called the Q-algebra of correspondences of Y on S. (1.7) Hecke operators For each non-zero ideal / of A, I ^ A, and for each x £ \X\ — ({oo} U V(I)) we will define Q-algebra homomorphisms (1.7.1)

Hf - Corr(M^/r7)^,

where Mfv is the generic fiber of Mf -^U X - ({oo} U V(I)), and

(1.7.2)

HTX

£

where MfXx is the restriction of Mf to Xx = Spec(Ox) -+ X-({oo}UV(I)), with the following compatibility: the diagram of Q-algebra homomorphisms

(1.7.3)

commutes (77 Xx and Mf is the fiber of 0 : Mf —• X — ({00} U V(I)) at 77^). In particular, we will get by restricting (1.7.2) to x a Q-algebra homomorphism (1.7.4) where M/^ is the fiber of 9 : Mf -> X - ({00} U F(/)) at x. In fact,' let g°° £ GLd(A°°) and let A c |X| - {00} be a finite subset containing V(I) and all x £ \X\ - ({oo}U^(7)) such that (g°°)x (£ GLd(Ox). We will construct a geometric correspondence of Mf on X — ({00} U A). It will induce the image of lK^g^K00 € 7Y/° by (1.7.1) when it is restricted to Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:42:12, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.002

16

DRINFELD MODULAR VARIETIES

rj and, for each x G \X\ — ({00} U A), the image of ix^g^Kj0 £ W^°'x by (1.7.2) when it is restricted to Xx. In a first step we assume in addition that (goc)~1 is a matrix with coefficients in O00. Then we can choose a non-zero ideal J of A such that J C / , V(J) C A, the kernel of the map

' O r ) - 1 : (F/A)d - (FA4)d is contained in (J" 1 /^-)^ and the image of (J~1/A)d (I^/Ay. Let us denote by

by t (^ o o )~ 1 contains

ci : Mjf|(X - ({00} U A)) -> M/|(X - ({00} U A)) the restriction of the morphism 77, j considered in (1.4). We define another morphism c2 : Mdj\{X - ({00} U A)) -> Mf\(X - ({00} U A)) n

by the following procedure. Let S —> X — ({00} U A) be a morphism of schemes and let (E,(p,t) G MJ(S) be of characteristic 9. We can consider the finite and etale S'-subscheme in .A-modules i(Ke]f O r ) " 1 ) * )

cEjCE

and we can take the quotient

in the category of S'-schemes in A-modules, i.e. the 5-group scheme E' is endowed with a ring homomorphism A ^U End(£')(1.7.5) (Drinfeld). — (E\ GLd(A/I))

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:42:12, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.002

18

DRINFELD MODULAR VARIETIES

and a right action of this kernel on Mjf, see (1.4.2)). The morphism Z\ is obviously invariant by the right action of Kf\(Kf n ((tf00)"1 Kf g°°)) and it is easy to see that this is the same for ?2. So by taking the quotient with respect to this right action we get the required geometric correspondence

(Mj\(X - ({00} u A)))/(K?\(Kf° n (GT)- 1 /^ 0 0 )))

Mf\(X - ({00} U A))

Mf\(X - ({00} U A))

e X-({oo}UA) . This geometric correspondence is clearly independent of the choice of J. Moreover, if we replace g°° by a~1g°° where a G A — {0} C F x is viewed as a central matrix in GLd(F) C GLd(A°°), we get the same geometric correspondence as for g°°. For a general g°° G GLd(A°°), we can always find some a G Fx such that a(p oo )~ 1 is a matrix with coefficients in O00. Then we associate to g°° the geometric correspondence constructed above for a~lg°°. It is easy to see that this geometric correspondence depends only on the double class K^g^Kf0 and the set A. We will denote it

(I.7.6)

Mf\(X - ({oc} U A))

e\

Mf\(X - ({00} U A))

/e

X-({oo}uA) . REMARK (1.7.7). - I f ^ ° ° G ^ = GLd{O°°), we can take A = V(I). The morphisms c\ and cnTn. n=0

If t; Gfe[r]there exist unique w and t;7 in k[r] such that deg(^') < deg(u) and 7; = w o u + i;r. If v\Ker(u) = 0, then v'\ Ker(tz) = 0 and v7 = 0

D

Let (E, (p) and (£", y?7) be Drinfeld ^4-modules of rank d over k. DEFINITION (2.1.2) (Drinfeld). — An isogeny from (E,(p) to (E',(p') is a non-zero homomorphism of group schemes over k

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.003

2. DRINFELD A-MODULES WITH FINITE CHARACTERISTIC

21

such that (f'(a) ou = uo (p{a) for each a E A. If we choose isomorphisms of group schemes over k, E = Ca,fc5 E' = Ga?/c, an isogeny u from (E,ip) to (E',(p') becomes an element of k[r] and has a degree and a height. Those two integers are clearly independent of the chosen isomorphisms and are called the degree of the isogeny, deg(-u), and the height of the isogeny, h(u). (2.1.3). — For any isogeny u from (E,(p) to (E',. Then D is the centralizer of F in k(r) or what is the same the centralizer of F(II) in fe(r). As F(II) contains the center IFP(II) of fe(r), F(U) is exactly the center F of D and [k(r):D] = [F(U):Fp(U)} (see (A. 1.3)). Moreover, as II is algebraic over F (dimi? k(r) < +oo), part (i) is now proved, i.e. _ F = F(U) = F[U}. As k(r) is totally ramified over IFP(II) at II = oo (resp. II = o), the same is true for F over JFP(II) and D over F. More precisely there exists a unique place oo (resp. 5) of F dividing the place II = oo (resp. U = o) of JFP(II), and Fob &]=; D (resp. F$ 0 ^ D) is a division algebra central over Fob (resp. Fo)In fact, let £>: A:((r-1))x —> TL (resp. o: k((r))x —> Z) be defined by (resp.

if ajv 7^ 0. Then there exists an integer e(66 /oo) (resp. e(o /oj) > 1, such that ^ _ _ oo (a) = e(66 /oo)oo(a), Va G F ^ , and (resp.

[*(r):F]=e(/oo)[fc:«(oo)] ^ _ § (o) = e(o /5)TO(O), V o G F~,

and

_

[fc(r):F]=e(o/S)[fc:/c(S)]). Therefore as (resp.

oo (a) = — ddeg(oo)oo(a), V a G A (6 (a) > o) 4=^ (o(a) > o), V a e A),

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.003

2. DRINFELD A-MODULES WITH FINITE CHARACTERISTIC

25

the place oo (resp. 5) divides the place oo (resp. 6) of F. Obviously, we have x(U) = 0 for each place x ^ oo, o of F. Let us compute |n|oo. If e(oo/oo) is the ramification index at oo of F over F, we have IJJI

=

p-deg(oo)6b(II)/e(6b/oo)

^

But e(6o /oo)oo(n) =6o (II) = —rdeg(o) and e(66 /oo)e(oo/oo) = ddeg(oo) (for each a E A we have oo (a) = —ddeg(oo)oo(a)). So we have oo (II) = —rdeg(o)e(oo/oo)/ddeg(oo) and This finishes the proof of part (ii). Next let us check that [F : F] divides d and that [D:F] = (d/[F : F})2. The integer — So (II) is also the ramification index at oo of F over F P (II). Consequently, if we set deg(oo) = [K(5O) : F p ], we have

[F : Fp(n)] = -S5(n)deg(55) = as

V

-^°1[F : F]

_ [F : F] = e(oo/oo) deg(oo)/deg(oo).

But [F:Fp(U)] = [k(r):D] divides and the quotient is [D : F]. Hence the assertion follows. As k(r) is split over F P (II) outside II = oo and II = o, D is split over F outside oo and o. The division algebra F ^ ®p D is the centralizer of Foo in k{{T~1)). As the invariantjrf k^r'1)) overJF p ((n^)) js -l/[fc((r" 1 )) : F p ((n- 1 ))], the invariant of F^ ®~D is - l / f i ^ ^-D : ^6b] (see (A.2.4)). So part (iii) is proved. Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.003

26

DRINFELD MODULAR VARIETIES

Finally let (F, II) be a Weil (F, oo, o)-pair of rank d over k. Let us prove the existence of an embedding F *-> k(r) of Fp(II)-algebras. It is sufficient to check that [F : JFP(II)] divides [k(r) : Fp(II)] 1 / 2 and that F does not split over IFp(II) at the places II = oo and II = o, i.e. where k(r) is ramified (see (A.3.3)). But oo(II) < 0 (resp. o(H) > 0) so oo (resp. 5) divides the place II = oo (resp. II = 6) of JFP(II). Moreover 2; (II) = 0 for all places x ^ oo, o of F. So oo (resp. o) is the only place of F dividing the place II = oo (resp. II = 6) of IFp(II) and F does not split over IFP(II) at the place II = oo (resp. II = 6). As before |n|oo = ql/d implies

and, as [F : F] divides d by hypothesis, [F : FP(H)] divides r deg(o) = [k{r) : V2

Let us choose one embedding F c-> k(r). Let A be the integral closure of A in F, i.e. A = {a e F\ x(a) > 0 for each place x ^ oo of F}. Using an inner automorphism of fc(r) we can modify the embedding in such way that it maps A into the maximal order k[r] of k(r) (A is contained in some maximal order of k(r) and two maximal orders of k(r) are conjugate in k(r)). Let

0 and x(Il) = 0 for all places x ^ oo, o of F (oo is the unique place of F over oo; if X is the smooth projective model of F over IFp, Pic— (IFP) is a finite group, so some positive power of the line bundle (9~(deg(o)oo — deg(oo)o) is trivial and there exists always a II G F as before). To each Weil (F, oo, o)-pair (F, II) of rank d over a finite extension of K(O) we can attach an (F, oo, o)-type of rank d, (F', S7), in the following way. Let F ' be the intersection of the intermediate fields F C F[n m ] C F Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.003

2. DRINFELD A-MODULES WITH FINITE CHARACTERISTIC

27

(m e TL, m > 1) and let o' be the place of F 7 induced by o. The pair (F7, of) obviously satisfies the above conditions (a) and (b). Let II' £ F' with oo'(n') < 0, o7(II7) > O^and x'(W) = 0 for all places x' ^ So7, o7 of F 7 (oo7 is the unique place of F 7 over oo, i.e. the place induced by oo). There exists some integer ra > 1 such that F' = F[Um] and obviously 5o7(IIm) < 0, d/(Urn) > 0 and z'(II m ) = 0 for all places xf ^ So7, o7 of F 7 . So we can find two integers n, n7 > 1 such that

x 7 (n 7 n / /n m n ) = o for all places x' of F 7 . But this means that II7n /Wnn is a non-zero constant in F 7 , i.e. a root of unity. Increasing n and n7 if necessary we can assume that n 7 n 7 n m n = 1. Then we have the inclusions

FcF[n7n'] cF[n7] c f ' c F [ n m n ] c F . Therefore F[n7] = F1 and (F7, o7) is an (F, oo, o)-type of rank d. To each Drinfeld A-module of rank d over K(O) with characteristic ^4 —» AC(O) «(o), (F, ip), we can attach an (F, oo, o)-type of rank d in the following way. There exists a finite extension k of K(O) contained in K(6) such that {E, 1, (r, s) = 1). (ii) T/ie Fo-algebra A r)S of endomorphisms of the simple object (NriS1 fr^s) is "the" central division algebra over Fo with invariant —s/r e Q/Z.

•

A proof of (2.4.5) is given in (B.I). If k is arbitrary the category of Dieudonne i^-modules over k is obviously Fo-linear, artinian and noetherian. If k' is a field extension of k contained in K(O) we have an obvious base change functor from the category of Oomodules (resp. i^-modules) over k to the category of those over k'; we will denote it by k' (g)/- (—). If (JV, / ) is an indecomposable Dieudonne F0-module over k there exist a unique pair of integers (r, s) with r > 1 and (r, s) = 1 and a unique integer t > 1 such that K(O) (g)/- (N,f) is isomorphic to (-/Vr)S,/r)S)*. We will say that the rational number s/r is the slope of (AT, / ) . If (iV, / ) is an arbitrary Dieudonne Fo-module overfc,its slopes are the slopes of its indecomposable constituents. If A G Q is a slope of (TV, / ) , there is a canonical decomposition

such that all the slopes of (TV7, / ' ) are equal to A and all the slopes of (TV", / " ) are distinct from A; the multiplicity of the slope A of (N, f) is by definition the dimension of N' over F&. If (M, / ) is a Dieudonne Oo-module over k its slopes (resp. the multiplicity of a given slope) are (resp. is) the slopes (resp. the multiplicity of the given slope) of (Fo 0 (flja,fc has an obvious structure of a k-scheme in K(O)-vector spaces), (ii) as a k-scheme in K(O)-vector spaces, the dimension ofGn is nd for some integer d>0 which is independent of n {the order of Gn is p d ( ) d (iii) the sequence

of k-schemes

in

OQ-modules

is exact. The integer d of (ii) is called the rank of G. A morphism between two tuo-divisible fc-schemes in Oo-mod\iles is a morphism between the corresponding inductive systems offinitefc-schemes in (9o-modules. The category of Dieudonne 0o-modules over k and the category of i n divisiblefc-schemesin (9o-modules are both (9o-linear and exact. We will now construct an equivalence between these two (9o-linear and exact categories. Let us begin by reviewing a very particular case of Dieudonne's theory. On the one hand, if M is a finite dimensional fc-vector space and M —• M is a pdes(°)-linear map, we have a functor Gk(M,f) from the category of Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.003

34

DRINFELD MODULAR VARIETIES

(commutative, unitary and associative) fc-algebras to the category of K(O)vector spaces with (2.4.9)

Gk(MJ)(R) = {ge Uomk(M,R)\g(f(m)) = g(m)pdee(°\ Vm G M}.

It is easy to see that Gk(M,f) is in fact a finite fc-scheme in ^(o)-vector spaces of dimension dim^(M) which can be embedded in 0 (take a basis of M over k). On the other hand, if G is a finite kscheme in /^(o)-vector spaces which can be embedded in G^k for some integer N > 0, the group of homomorphisms offc-schemesin «(o)-vector spaces (2.4.10)

Mk(G) = Uom(G, M a pdes(°) -linear map, and the category of finite k-schemes in K(O)-vector spaces G which can be embedded in (G^fc for some integer N > 0 are exact and quasi-inverse one of the other. Moreover, if G = Gk(M,f), the dimension of G (over /^(o)) is equal to dim^(M) and G is connected (resp. etale) if and only if f is nilpotent (resp. an isomorphism). • A proof of (2.4.11) is given in (B.3). Then we can set (2.4.12)

Gk(MJ) = lim Gk(M/w^MJ

mod wnoM)

n

for any Dieudonne Oo-module (M, / ) over k and (2.4.13)

(Mk(G),fk(G)) = lim(Mk(Gn)Jk(Gn)) n

for any ^-divisible fc-scheme in (9o-modules G = lim Gn. n

COROLLARY (2.4.14). — The contravariant functors Gk and (Mk,fk) between the category of Dieudonne O0-modules over k and the category of wo-divisible k-schemes in O0-modules are Oo-linear, exact and quasi-inverse one of the other. Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.003

2. DRINFELD A-MODULES WITH FINITE CHARACTERISTIC

35

Moreover, if G = Gk(M, f), the rank of G is equal to the rank of (M,/) and the canonical decomposition

(see (2.4.6)) induces the canonical decomposition GcxkGet

G=

of G into its connected component Gc = Gk(Mc,fc) quotient Get = Gk(Met Jet).

and its maximal etale •

(2.5) Dieudonne module of a Drinfeld module Let k be afieldextension of n(o) contained in K(O) and let (£?, ip) be a Drinfeld c A-module of rank d over k with characteristic A —» K(O) — > k. Let Vo C A be the maximal ideal defining {o} in X — {oo} = Spec(A). For each integer n > 0, thefc-schemein ((9o/(a7™))-modules of ^"-division points (Oo = A-po) Evn C E r

o

has the following properties: (2.5.1). — (i) E-pn is finite of dimension nd over n(o). (ii) There exists a unique integer h > 0 such that hdeg(o)o(a) is the height of (p(a) for each a G A — {0} (see (2.1)) and the connected component Epn of E-pn has dimension nh over n(o). LEMMA

(iii) E-p-n can be embedded in 0. Proof: As E-pn c E, part (iii) is obvious. Let h(a) be the height of ip(a) for a G A — {0}. We have h(aid2) = h(a\) + h(a2) for all ai, 0 such that /i(a) = ATo(a) for all a G A - {0}. Now, if a G A — {0}, thefc-schemein (74/(a))-modules of a-division points Ea = ker((p(a) : E —• E) is finite of order p-^deg(cx))oo(a) a n d i t s c o n n e c t e d component Eca is of order ph^. Using the same argument as in (1.3.2) we get part (i) of the lemma and that the dimension of Ey>n over JFP is nN. But E^n is a A;-scheme in ft(o)vector spaces, so there exists an integer h > 0 such that N = /ideg(o) and the lemma is proved. • Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.003

36

DRINFELD MODULAR VARIETIES

We have an inductive system E-poo = (Ej>o

c

—• E-p2 °-> E-pz

eocro(ho).

So, if ft00'0 6 GLd(A°°'°) and ho G GLd(F^),

the double class

is a fixed point of Frob£ x(Mfo(g°°iO),c) if and only if there exist 6 G (D°w)x, A:00'0 G i^f'° and ko G GL d ((9^y) such that the following conditions are satisfied: \Nr(eo)aro(ho)

= 6hoko;

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:34, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.004

54

DRINFELD MODULAR VARIETIES

or, what is the same, if and only if there exists 8 G (Dopp)x following conditions are satisfied:

such that the

(hor^NrieoWho) G where we have set (3.2.4)

Nr(h'o)

= h'oao(h'o) • • •

aro-\tio)

for all h'o e GLd(F^). We will denote by Fo^r the unique extension of Fo of degree r which is contained in F-^-r and by Oo,r the ring of integers of Fo,r. LEMMA (3.2.5). — The map

25 surjective and the inverse image of the identity matrix by this map is GLd(Oo,r). Proof: See [Gr] (§3, Prop. 3). LEMMA (3.2.6). — For any /i°°'° e GLd(A°°>o) and any ho G the intersection

• GLd(F^),

inside GLd{Jk°°'°) x GLd{F^rr) is reduced to the identity matrix. Proof: Let 8 be in this intersection and let F' = F[8] C Dopp be the field generated by 8 over F. Let x' be a place of F' dividing a place x ^ oo, o of F. As gld(Ox) is compact, the restriction of x' to F'x D hxgld(Ox)h~1 is bounded below (Fx = Fx F Fr and hx is the x-component of /i°°'°). Therefore the set of integers {x'(8n) = nxf(8)\n G Z} is bounded below and x'{8) = 0. Similarly, x'{8) = 0 for all places of Ff dividing o. As FQO 0 F F' C (Foo ®F D)opp is also a field, we have proved that 8 is a constant (x'{8) = 0 for all places xf of F'), Finally let x G F ( / ) , then x(det(5 - 1)) > 0 so that x(NF,/F(8 - 1)) > 0 where NF>/F is the norm map of Ff over F and there exists a place x' dividing x such that x'(6~-l)>0. Therefore 8 = 1. • Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:34, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.004

3. THE LEFSCHETZ NUMBERS OF HECKE OPERATORS

55

Thanks to the lemmas (3.2.5) and (3.2.6), Lef r (/°°'°) ( — } is equal to the number of double classes (Dopp)x(h^°(K^°

n ((^ oo ' o )- 1 X / oo '^ oo '°)),

with ft00'0 G GLd(A°°'°) and ho G GLd(F^o))

such that

(ho)~1eoao(ho) G < zuo > and there exists 8 G [Dopp)x with the following properties: f (/i 0 0 ' 0 )" 1 ^ 0 0 ' 0 G

K^g00*0,

Moreover, for each such a double class, the conjugacy class of the corresponding 6 G (Dopp)x is well-defined: if we replace ft00'0 by 8ihoo>°koo>o and /io by (5i/iofcr with 81 G (Z) opp ) x , /c°°'° G 1*7°'° n ((0~'°)- 1 lTf ) 'V o>o ) and kr G GLd(Oo,r) then 5 must be replaced by opp)^x, let usfixan element h6o G GLd(F^) such that (/i^-^A^oK^) = 1 and let us set

7* = (kJ)-U(fcJ). It is easy to see that 7 £ G GLd(Fo,r) K ( 7 * ) = 7*). Then Lef r (/°°'°) ( ^ 5) is equal to the sum over r-admissible 8 G (Dopp)? of the number of double classes \v

)s ih

\ Ki

n

\\9 )

K

i 9

)),nonrGLd(Uo,r)\

with /i°°'° G GLd(&°°'°) and /i r G GLd(Fo,r) such that '0 G i*7°'V°' 0 ,

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:34, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.004

56

DRINFELD MODULAR VARIETIES

where (Dopp)* is the centralizer of 6 in (D

= {6f G (Dopp)x\6'6 opp x

)

= 66'}

and

>r = < wo > nGLd(Fo,r) = GLd(Oo,r)

.

GLd{OOtr).

\ 0

1/

But the map ^

(^/

n

\\9

)

K

i

9

))*-*h

K

i

from the set of classes satisfying (/i 0 0 ' 0 )- 1 ^ 0 0 ' 0 e

K^g00*0

to the set of classes satisfying

is clearly bijective. So we have proved PROPOSITION

(3.2.7). — For each (F,oo,o)-type (F,o), Lefr(/°°'°)(£— is

equal to the sum over r-admissible 6's in (Dopp)? classes

of the number of double

with h°°'° G GLd(A°°'°) and hr G GLd{Fo,r) such that (/i r )~ 1 7 r cr o (/i r ) G< tuo > r .

• (3.3) Where the orbital integrals come in We will now give a formula for Lef r (/°°' O V~^ in terms of orbital integrals and twisted orbital integrals. For each r-admissible 6 in (Dopp)^ let GLd(A°°'°)6 be the centralizer of 6 in GLd(A°°'°) and let GL°°^(FO) be the cro-centralizer of 7* in GLd(Fo,r) (recall that GL°° 6 is an Fo-group and that GL°° AFO) is the set of hr G GLd(FOir) such that 6

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:34, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.004

3. THE LEFSCHETZ NUMBERS OF HECKE OPERATORS

57

We will denote by dh°°'° (resp. dhr) the Haar measure on GLd(A°°'0) (resp. GLd(Fo^r)) which is normalized by dh°°>° = v o l ^ ^ ' V / i 0 0 ' 0 ) = 1 /

7V-OO,O

(resp. /

dhr = vol(GL d (O o , r ), d/ir) = 1).

JGLd(Oo,r)

Let dti^'°

(resp. d/i^r) be an arbitrary but fixed Haar measure on

Let us recall that /°°'° E Wj°'° is the characteristic function of j(Oo,OgOQ,oj£°o,o £. (^^^(A 00 ' 0 ) and let us denote by /o, r E C?(GLd(Fo,r)//GLd{Oo,r)) the characteristic function of the double class < wo >rC GLd(Fo^r). We can introduce the orbital integral (3.3.1)

O«(/°°fO, dfc~'°) = /

fco>o((hoo>°)-'1

"~

rtX

and the twisted orbital integral (3.3.2) JGL°d^6(Fo)\GLd(Fo,r)

for each r-admissible 6 E (Dopp)?. As /°°'° and /OjT. are non-negative functions, these integrals make sense even if they are infinite. In fact we will see in (4.8.9) that they are absolutely convergent. We have the embeddings (see (2.7)) (Dopp)* C ((A°°'° ® F D)opp)*

c GL d (A°°'°) 6

and with the obvious notations, where i is the composite of the inclusion

and the isomorphism

given by ho \-> (h6o)~lhoh6o (we let the reader check the details). In particular, the quotient X

makes sense and we can endow it with the measure d/i^°'° x dhs,r divided by the counting measure d6f on (Dopp)£. The following proposition is an obvious corollary of (3.2.7): Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:34, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.004

58

DRINFELD MODULAR VARIETIES

(3.3.3). — For each (F,oo,o)-type (F,o), Lef r (/°°'°) ( — } is equal to the sum over r-admissible 6's in (Dopp)^ of the product of PROPOSITION

by (this product is obviously independent of the choices of the Haar measures dh^'° and As r > 0, Lefr(/°°'°) is finite and consequently, for each (F, oo, o)-type (F,5), Lef r (/°°'°),~~. is finite. This implies the following proposition that we will also check directly: (3.3.4). — For each r-admissible 8 e (Dopp)^ r > 0), the above embeddings PROPOSITION

(recall that

((A°°'° ®F D)opp)* and are both isomorphisms. In particular, the volume

is equal to vol((Dopp)Z\((A°°

0 F D)°")*

^

where d8'°° is the Haar measure on ((A°° End(iV,/)

(End(iV,/) is the cr^-centralizer of 6~1Nr(eo) in gld{F-^r-r)). We have the decompositions 5|o

x\o x'\x

7

(x is a place of F and x is a place of F'). Accordingly we have decompositions

x|o

x'jx

of Dieudonne Fo,r-modules over K(O). NOW (iV, / ) is purely of slope zero if and only if each (Nx',fx>) is purely of slope zero. But N$ = F^— where n £ = [h : Fo]d/[F : F] and the matrix of f£ is S£ € D°~pp C ^ ( ^ o ) if x ^ o and (Ss^Nries) € Df p C ^ ( i ^ ( o ) ) (n5 = ft) if x = J . So (Nx',fxt) is purely of slope zero if and only if x'{8) = 0 for each xf ^ of and {No',fof) is purely of slope zero if and only if o(det((^o)~1^Vr(£:o))) = 0, i.e. o/(6) = r[F'5, : Fo}/h[i^{df) : K(O)] (the details are left to the reader). Finally let us prove that the conditions x'(6) = 0 for all places x' ^ of of Fr dividing o and of(6) ^ 0 imply Fr D F. Let d be the restriction of GI to F' C F'. Then o;(5) 7^ 0 and o' is the unique place of F' dividing Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:34, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.004

60

DRINFELD MODULAR VARIETIES

d. Let 55' (resp. 00') be the unique place of F' (resp. F') dividing 00 (Foe ®F Fr C Foo ®F F' C (Foo ® F L>)opp are fields). If X' is the smooth projective model of Ff over JFp, Pic^/p (JFP) is a finite group. So there exists IT G F' such that 00'(IT) ^ 0, O'(IT) / o and Z'(IT) = 0 for all other places x1 of F'. We have So'(IT) ^ 0, o'(n') ^ 0 and z'(II') = 0 forjall other places of F1 if we consider II' as an element of F'. Now let U e F be such that oo(II) ^ 0, o(II) ^ 0 and x(II) = 0 for all other places x of F. Then F = Fpl^] for any integer N ^ 0 ((F,5) is an (F,oo,o)-type). But we can find non-zero integers N and N' such that

for all places x' of F': we have 00'(II) ^ 0, o ^ I I ) ^ 0 and x'(II) = 0 for all other places of F ' if we view II as an element of F'. So HrN' /IiN belongs to the finite field of constants in F' and multiplying N and Nr by a positive integer if necessary we can assume that TL'N'/nN = 1. In particular UN = U/N' e F ' and F = F[UN] is contained in Ff.

U

Proof of (3.3.4) : As 6 G (Dopp)£ is r-admissible it follows from (3.3.6) that F[6] D F and the desired equalities of centralizers are obvious. It is well known that the coset space F£(Dopp)x\((A F D)opp)x is opp x compact (see [We 1] (3.1.1)). So the coset space (D ) \((A°° ®F D)opp)x and its closed subspace (Dopp)x\((A°° 0 F D)opp)^ are compact. Therefore the Volume of (Dopp)^\((A°° ® F D)opp)^ for any measure is finite. • (3.4) Transfer of conjugacy classes Let 7 G GLd{F). Let us recall that 7 is said to be elliptic if the F-subalgebra F[7] of gld(F) generated by 7 is a field, i.e. if the minimal polynomial of 7 is irreducible over F. Let 7 G GLd(F) be elliptic and let F' = F[i\. Then 7 is said to be elliptic at the place 00 if and only if Foo ® F F' is a field, i.e. there exists only one place 00' of F ' dividing the place 00 of F. On the other hand, 7 is said to be r-admissible (at the place o) if and only if o(det 7) = r and there exists a place o' of Ff dividing the place o of F such that #'(7) = 0 for all the places All these properties depend only on the conjugacy class of 7 in Let 7 G GLd(F) be elliptic, elliptic at the place 00 and r-admissible at the place o; let F ' = Fpy] and let oc/ be the unique place of Ff dividing 00 and d be the unique place of F1 dividing o and such that 0^7) ^ o. We can Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:34, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.004

3. THE LEFSCHETZ NUMBERS OF HECKE OPERATORS

61

attach to 7 an (F, 00, o)-type (F,o) of rank d in the following way. There exists IT e F' such that OO'(IT) ^ 0, O'(IT) ^ 0 and z'(lT) = 0 for all other places of F' (see the proof of (3.3.6)). We set F=

P|

F[U/n] C Ff

and we let oo and o be the restriction to F of the places oc/ and ol of F' respectively. Using the same argument as before Corollary (2.2.3), we check that (F, 3) is an (F, oo, o)-type of rank d. Let D be "the unique" central division algebra over F with invariants -[F:F]/d

if x = 55,

[F : F)/d

if x = o,

0

otherwise.

As Fob ®z F' and F5 ®~ F ' are fields (there exists only one place of F' dividing 00 and only one place of F' dividing o : F^ 0 F F' is a field and if x' is a place of F1 dividing o we obviously have a?'(7) ^ 0) and as [F' : F] divides d/[F : F] ([F' : F] divides d), there exists at least one embedding of F-algebras and two such embeddings are conjugate in Dopp (see (A.3.3)). In particular, we get an element 6 G (Dopp)x (the image of 7 by such an embedding) which is well-defined up to conjugacy in (Dopp)x. Moreover, it follows from (3.3.6) that 6 is r-admissible at the place o. Let GLd{F\eii be a system of representatives in GLd{F) of the elliptic conjugacy classes. Then we have constructed a map (3.4.1)

{7 e GLd(F\eu\-i is elliptic at the place 00 and r-admissible at the place 0} S

€ (Dopp)^ \S is r-admissible at the place 0}

where (F, o) runs over a system of representatives of the isomorphism classes of (F, 00, o)-type of rank d and where, for each (F, o), (Dopp)^ has the same meaning as before. PROPOSITION

(3.4.2). — The map (3.4.1) is bijective.

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:34, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.004

62

DRINFELD MODULAR VARIETIES

Proof: Let us construct an inverse map to (3.4.1). We start with ((F, 5), 8) and we fix an embedding of F-algebras F — gld(F) by choosing a basis of F over F (so that we can identify gld(F) with End j p(F d /f F:F ')). Any two such embeddings are conjugate. Then the centralizer gld(F)p of F in gld{F) can be identified with ^ , m . F j ( F ) and we have an injective map from the set of conjugacy classes in Dopp (a central division algebra over F of dimension (d/[F : F]) 2 ) into the set of conjugacy classes in gld{F)p (if 8X G Dopp, F ' = F[8X] C Dopp is a field of degree over F which divides d/[F : F] and, consequently, it can be embedded into gld{F)p ^ 9ld/[F:F](F))' L e t ^ F e GLd(F)p (the centralizer of F in GLd{F)) be a representative of the image of the conjugacy class of (5 by the above injective map. We can consider jp as an element of GLd(F). Then it is easy to see that jp is elliptic, elliptic at oo and r-admissible at o. So we can map ((F, 5), 8) to the representative 7 G GLd{F\eu of the conjugacy class of 7^ in GLd(F). ' • It follows from the definition of (3.4.1) that we have PROPOSITION (3.4.3). — Let 7 G GLd(F\en be elliptic at 00 and radmissible at 0. Let ((F, o),0

with the following property: let d = d\-\-- —\-dn be any partition of d (n and d i , . . . , dn are positive integers), let P be the corresponding standard parabolic subgroup of GLd with its standard Levi decomposition

and its canonical isomorphism M ^ GLdl x ... x GLdn (the unipotent radical of P is N and P contains the Borel subgroup of upper triangular matrices) and let Px

= mxnx e P{FX) = M{FX)N{FX)

such that there exists hx G GLd(Fx) with KXVxhx G KxgxKx. Then we have |x(det(mXj»))| < Cx for all i = 1,..., n where (mx,u • • • ,m x , n ) G GLdl(Fx) x • • • x GLdn{Fx) is the image of mx by the above canonical isomorphism. Moreover, if KxgxKx = Kx (i.e. gx G Kx), we can take Cx = 0. Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:34, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.004

3. THE LEFSCHETZ NUMBERS OF HECKE OPERATORS

65

Proof : Let d = d\ + • • • + dn be a partition of d and P = MN be the corresponding standard parabolic subgroup. We have the Iwasawa decomposition GLd(Fx) = P(FX)KX. So, on the one hand, we can decompose the Kx-double-class KxgxKx finite number of i^-classes

into a

where P{xj) = mWnW G P(FX) = M(FX)N(FX) for each j G J. On the other hand, in the statement of the lemma, we can replace the hypothesis "there exists hx G GLd(Fx) with h~1pxhx G KxgxKx1 by the hypothesis "px G KxgxKx" (the x(det(m x ^))'s depend only on the conjugacy class of px in P(FX)). Now P(Fx)nKx = P(OX), so the hypothesis "px G KxgxKx" implies that there exists j £ J such that x(det{mXti)) = x(det(m^])) for all i = 1,... ,n ((m^{,... ,ra^n) is the image of m^ in GLdx{Fx) x • • • x GLdn(Fx) by the canonical isomorphism) and we can take for Cx the maximum of sup{|x(det(mg))| |1 > i > n,j G J } when d~ d\-\

+ dn runs over the set of partitions of d.

•

Proof of (3.4.7) : Let F' = F[y] G ^/ d (^). If 7 is elliptic, F ' is a field. If 7 is elliptic at oo, Foo 0 F Ff is a field and there exists a unique place oo' of Ff dividing oo. If 7 is r-admissible at o, there exists a unique place d of F' dividing o and such that 0^(7) ^ 0; moreover o;(7) = rdeg(o)/deg(o'). Finally, if there exists ft00'0 G GLd(A°°>o) with (h

> )

j/i

> e Kj

g

> Kj

C KA

g

> KA

,

thanks to (3.4.8), we have deg(x / )|x / ( 7 )|
\x Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:34, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.004

66

DRINFELD MODULAR VARIETIES

1 0 7 = {lx')x'\x and a partition d = J2 (d/[F' ' F])[FX'

: F

x] 5 fixing an

x'\x

embedding of i^-algebras

for each x'\x, we get an element «7x', • • • ,7x'))x'|x

where 7^. is repeated d/[F' : F] times and M is the standard Levi subgroup of GLd associated to our partition of d; obviously 7 is conjugate to ((7z',---,7*'))z'|z i n GLd{Fx). Now let P(T) = T ^

+a ^ ' ^ -

1

+ • • • + a[F,:F] G

be the minimal polynomial of 7. It is not difficult to see that deg(x)x(ai) > i ini\

(r xf i^i) 1

lle(xf /x)\

^

\xf divides x > )

for all places x of F and each i = 1,...,[F' : F], where e(xf/x) is the ramification index of F'x, over Fx. Therefore, if 7 satisfies the hypotheses of the proposition, the degree [F1 : F] of its characteristic polynomial P(T) is bounded above by d and the divisors of the coefficients a i , . . . , a\p':F] of P(T) are bounded below. So P(T) can take only a finite number of distinct values and the proposition follows. •

(3.5) Transfer of Haar measures Let 7 G GLd(F)^eu be elliptic at 00 and r-admissible at o. Let ((F,o),5) be its image by (3.4.1). We will now give a formula for the volume

in terms of GLd(A)7 and ((F^ ® F D)opp)%. Let us fix a Haar measure d ^ on ((FQQ ® F D)opp)£. Let G^OO be the Haar measure on F£ normalized by vol((!?£>, dzoo) = 1. We set F' = F[«] = F[ 7 ] Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:34, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.004

3. THE LEFSCHETZ NUMBERS OF HECKE OPERATORS

67

(F' and F^ = Foo ®F F' are fields; F'x is at the same time the center of (D°w)% and the center of GLd(F)7; see (3.4.3)). As F^\((F OO ® F D)°™)* is compact, we have (3.5.1)

^

where d6'A = d ^ x d6foc and where both the numerator and the denominator are finite. We want to construct a Haar measure d^fA on GLd(A)7 such that (3.5.2)

vo\(F*(D°™)Z\((A®F

D)0™)*^

where dj' is the counting measure on GLd(F)1. Thanks to (3.4.3) we can identify (Dopp)£ with the central division algebra D' over F' with invariants 1/d' at oo', -1/d 7 at o' and 0 elsewhere (d7 = d/[Ff : F]) and gld(F)7 with gld'(Ff). So we have

and F*GLd(F)y\GLd(A)^ = F£GLd,(F')\GLd,(A'), where A' is the ring of adeles of F', we have Haar measures d6'A, = d6'A on (A'®F, D')x, dzoo on F£, dS' onD'x, &i on GLd,(F'), and we want to construct a Haar measure dj^, = dj^ on GLd>(A') such that (3.5.3)

vol(F*D'*\(A' ®F,

D')\^^)

= vol(F£GLd,(F')\GLd,(A'), 5 ^ 7 ) Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:34, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.004

68

DRINFELD MODULAR VARIETIES

As GLd'{F') is an inner twist of D/x as an F'-group scheme there is a general method to construct d^, (see [We 1] and [Ko 1] §1). Let us recall it. Let k be a field and let ks be a separable closure of k. Let G and H be two connected reductive k-group schemes. An inner twisting between G and H is an isomorphism of A;s-group schemes V> : ks ®fc G ^

ks ®k H

such that, for each a G Gal(fcs/fc), there exists at least one ga G G(ks) with

If there exists an inner twisting between G and H one says that G is an inner twist of H. If we fix such an inner twisting, then it induces an isomorphism of A;s-Liealgebras Lie(^) : ks ®k Lie(G) -^ ks ®fc Lie(H) with for each a G Gal(A:s/fc). Therefore, if we set n = dimfc G = di we get an isomorphism An Lie(V>) : ks 0fc An Lie(G) ^ ks ®fc An Li of 1-dimensional fcs-vector spaces with

i.e. an isomorphism A of 1-dimensional A:-vector spaces. If we replace tp by i\) o Int(p) for some p G G(fcs) this last isomorphism is not changed. Moreover we have an exact sequence of groups G(ks) - ^ Ant{ks 0fc G) -> r -> 1 where Aut(fcs 0^ G) is the group of automorphisms of the fcs-group scheme ks 0fc G and F is the group of outer automorphisms of the ks-group scheme ks H a n d is called the transfer of dh (from H to its inner twist G). If k is a function field with completion kv at the place v and ring of adeles A, we can transfer a Haar measure dh& on H(A) to an inner twist G of H. We decompose dh& as an absolutely convergent product

= T\dhv Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:44:34, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.004

70

DRINFELD MODULAR VARIETIES

of Haar measures dhv on H(kv). For each v, let dgv be the transfer of dhv from kv k H to its inner twist kv k G. For almost all v, kv ®k G is in fact isomorphic to kv (S>k H. Therefore the product of Haar measures

is also absolutely convergent and defines a Haar measure dg& on G(A) which is clearly independent of the decomposition dh& = J\ dhv. We call dg^ the V

transfer of dh& (from H to its inner twist G). Now, let dyA, be the transfer of d6A, from Dfx to its inner twist GLd'(Fr) (as an F'-group scheme). PROPOSITION (3.5.4). — For this choice of d^A, the equality (3.5.3) (and consequently (3.5.2) with dj'A = d^'A, ) holds.

Proof: Let UJQ be the usual volume form on GL^, i.e. UJG = det(7)~ d dr>m A drw A • • • A drfd'.d'-i A d^d',d> The Tamagawa measure of GLd> (A) is defined in the following way (see [We 1] (3.1)):

where q' is the number of elements of the field of constants in F', where g' is the genus of i7", where

is the convergence factor, where \LJG\X' is the Haar measure on GLd>(F'x,) associated to UJQ as above and where x1 runs over the set of places of JF" (if X' is the smooth projective model, ,(

Q

where do!x, is the Haar measure on Fx, which is normalized by vol(Ox,, da'x,) = 1 for each place x' of Ff). Let

GLd,{lk')1 = {{ixl) G GLd>(A')lX;deg(a;')a:'(det(7;0) = 0} then the Tamagawa number of GLd',F' is defined by r{GLd.iF,) = vo\(GLd>(F')\GLd 1 and we denote by kr "the" finite extension of k of degree r (kr is a finite field with qr elements) and by Fr the unramified extension of F with residue field extension kr over k. We denote by Or the ring of integers of Fr and by vr the discrete valuation of Fr; the restriction of vr to O C Or is v and vo G Or is a uniformizer of Or. Let G — GLd for a positive integer d. We set (K =G(O) c \Kr = G(Or) C and we denote by (H =C?(G(F)//K), \Hr=C^(G(Fr)//Kr) the corresponding Hecke algebras. K (resp. Kr) is a maximal compact subgroup of the topological group G(F) (resp. G(Fr)) and H (resp. Wr) is Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:48:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.005

76

DRINFELD MODULAR VARIETIES

the Q-vector space of if-bi-invariant (resp. /fr-bi-invariant) functions with compact supports / : G(F) - Q (resp. fr : G(Fr) - , Q); the product on H (resp. Hr) is given by the convolution product

f(h)r(h-1g)dh

(/'*/")(») = / )G{F) JG(F)

(resp.

^ * fr)(9r) = I

ftthrWiK

JG(Fr)

where dh (resp. dhr) is the Haar measure on G(F) (resp. G(Fr)) which is normalized by / dh = 1 (resp. /

dhr = 1).

JKr

JK

(4.1) Satake isomorphism Following Satake we will analyze the structure of the Hecke algebra H. Similar results will obviously hold for 7ir. It follows from the theorem of elementary divisors ([Bou] Alg. VII.6) that we have the Cartan decomposition (4.1.1) x

where A runs over the set {A = (Ai,..., Ad) € Zd|Aj > A2 > • • • > \d} and where we have set

(

wXl

0 \ eG(F).

0

Xd

w )

In particular, we get a basis of the Q-vector space H by considering the characteristic functions f\ of the double classes KzuxK. Let A = Q[y^5 l/y/o\ C ^- We will consider the right if-invariant function (4.1.2)

z : G(F) -> A[zuzi\

..

.,zd,z?\

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:48:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.005

4. THE FUNDAMENTAL LEMMA

77

defined in the following way. Let B C G be the Borel subgroup of upper triangular matrices; we have the Iwasawa decomposition ([We 2] Ch. II, §2, Thm. 1) (4.1.3)

G(F) = B(F)K , B(F) HK = B(O).

So it is enough to define the right #((9)-invariant function z\B(F). Let (4.1.4)

6B{F} : B(F) - /

C Qx

be the modulus character of B(F) (if db is any left or right Haar measure on J5(F), then 8B{F)(b') = d{b'bb'-l)/db for each V € S ( F ) ) ; in fact

where |a for each a € F. Let (4.1.5)

X*

: B(F) ->

be the quasi-character defined by

Then we set 4>z\B(F) = 1 /O

(it is obvious that 8g,F\ and Xz are right B(O)-invariant). Now the Satake transform oi f ^4) is defined by

(4.1.6)

r(z)= ff

t

JG G(F)

(recall that dg is the Haar measure on G(F) normalized by vol(iif, dg) — 1). If P C G is a standard parabolic subgroup (i.e. B C P C G) and if P = MN is its standard Levi decomposition (N is the unipotent radical of P and M c P i s the Levi subgroup which contains the torus T C B C G of diagonal matrices), we will endow M(F) and JV(F) with the Haar measures dm and dn normalized by vol(M((9),dra) = 1 and vol(A/"((9),dn) = 1 respectively. Then we have the following integration formula: (4.1.7)

( / f(g)dg = JG(F)

/ JM(F) JN(F)

/ ip(mnk)dk dn dm JK

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:48:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.005

78

DRINFELD MODULAR VARIETIES

for any locally constant function with compact supports

t®W, we can define its constant term along P , fp G by (4.1.9)

/p(m) = ^2F,(m) /

f(mn)dn

for each m G M(F), where (4.1.10)

« P ( F ) : P ( F ) -> /

is the modulus character of P(F). We also have a function (4.1.11)

M,z : M(F) -+ A[zuzi\

...,zd,

z~dx\

defined by M,z{m) = * P ( F ) ( m ) ^ ( m ) for each m G M(F) and the Satake transform of / G A) is defined by (4.1.12)

/v(*)= /

A®HM

(/ : M(F)

/H^WdrnG^i,^ 1 ,...^^- 1 ].

JM(F)

LEMMA (4.1.13). — T/ie Satake transformations

and and the constant term along P

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:48:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.005

4. THE FUNDAMENTAL LEMMA

79

are homomorphisms of A-algebras. Moreover the diagram

(-) v

A®HM

is commutative. Proof : This is an easy consequence of the integration formula (4.1.7) applied to B and P. • REMARK

(4.1.14). — For P = B and consequently for M = T, we have Q • /T,A

where /T,A is the characteristic function of the double class T(O)wxT{O) C T(F). Moreover, for each A, A7, A" G Z d , we have /T,A' * /T,A" = /T,A'+A"

and So in this case the Satake transformation comes from the isomorphism of Q-algebras which maps /T,A to zx. If P is given by the partition di + • • • + ds = d of d so that M = GLdl

x ... x GLds

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:48:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.005

80

DRINFELD MODULAR VARIETIES

we have obvious Q-algebra isomorphisms

n M = nGLdi ®'and

i and the Satake transformation

is the tensor product of the Satake transformations for GL^ (j = 1, • • •, s).

n LEMMA (4.1.15). — Let B = TU be the Borel subgroup of upper triangular matrices in G with T the maximal torus of diagonal matrices. Let 7 G T(F) be regular in G(F), i.e. 7 = diag(7i,... ,7^) with 7^ G Fx and 7^ ^ 7^/ for any i' ^ i". The centralizer G 7 0 / 7 in G is equal to T. Let dt = dg1 be the Haar measure on T(F) = G7(F) normalized by vol(T(0), dt) = l.LetfeH and let

O1(f,dg1)= I

fig-'ig)^-

(this converges absolutely as the orbit OG(7)(F) closed). Then

0/7 in G(F) is obviously

where

DT\G(t) = det(l - Ad(i- 1 ),Lie(T(F))\Lie(G(F))) for each t € T(F) (DT\G(j) ^ 0). Proof: Applying the integration formula (4.1.7) for P = B, we get

=f U(F)

But, as 7 is regular, the morphism of tx7-adic manifolds U(F) -> U(F) , u h-> 7 " 1 ^ - 1 7 ^ is bijective with constant Jacobian J( 7 ) = |det(l - Ad( 7 - 1 ),Lie(C/(F)))| ^ 0. Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:48:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.005

4. THE FUNDAMENTAL LEMMA

81

So by the change of variable XL -— 7

*-^

jU

we get 01{f,dg1)=

/ /

du ffru^jrrr.

As I^T\G(7)I =

SB{F){I)J{I)2

the lemma is proved. COROLLARY

•

(4.1.16). — The image of the Satake transformation

is contained in the invariants under the natural action of the symmetric group 1

1

Proof : If we consider a regular element 7 = diag(7i,... ,7^) G T(F) C G(F) we have thanks to (4.1.15). But if w G ®d and if we G(F) is the corresponding permutation matrix i ( 1 ) , . . . , 7 ™ - i ( d ) ) =w

7

So 0w.7(f,dgw^)

= 0 7 (/,d^ 7 )

(dg^.w = dt = dg^) and

Therefore we have and the corollary is proved.

•

THEOREM (4.1.17) (Satake). — The Satake transformation is an isomorphism of A-algebras ofH onto Alzi^z^1^... ,zd, z^1]®d.

Proof: Let A,/iGZ d with \i > - - > \d, f*i > - - > Hd and let a(A, fi) e A be the coefficient of z^ in fx(z)- The theorem follows easily from Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:48:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.005

82

DRINFELD MODULAR VARIETIES CLAIM

(4.1.18). — We have a(A,/i) = 0 unless \i < X for the partial order f Mi < Ai, Mi + M2 < Ai 4-A2, Mi + • • • + Md-i < Ai 4- • • • + A d _i, fii H

h Md = Ai H

h Ad,

and a(X,X)=qx> where - 1 2

d - 3 '

2

3-d ''"'

2

1-d '

2

and

2=1

Proof of the claim : It follows from (4.1.13) and (4.1.14) that

where

T(F). But _

n

so the problem is reduced to proving the following assertions: (1) if there exists u G U(F) (U is the unipotent radical of B, B = T.U) such that w u G Kzu K, then M < A; (2) if ue U(F), then wxu G KwxK if and only if u G U(O) = U(F) n i^. Let us prove these assertions. For any integer n > 1 and any g G GLn(F), we set Then it is easy to see that, for anyfci,fc2G GLn(O), we have

v(kigk2) = v(g) Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:48:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.005

4. THE FUNDAMENTAL LEMMA

83

and that, for any A E Z n with Ai > • • • > An, we have v(9) = An, v(A2g) = An_i 4- An,

if

(

wXl

0 \&GLn(F).

0

w>

So, if g£ G(F), we have g £ KwxK if and only if v(A2g) = Xd-i + Ad, . v(Adg) = Ai = + . . . + \d.

Now assertion (1) follows immediatly from the inequalities

(i = 1,2,..., d) with equality if i = d and assertion (2) is left to the reader.

• Another important consequence of the above claim is COROLLARY

(4.1.19). — The Satake transform of f\ E H where A =

( 1 , 0 , . . . , 0 ) is

Proof: We have < 6, A > = (d - l)/2 and the only fieZd such that fi < A is A itself.

with /xi > • • • > •

(4.2) Base change homomorphism Let 7i and T-Lr be as in (4.0). We can now define the base change homomorphism (4.2.1)

b:A

by requiring that

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:48:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.005

84

DRINFELD MODULAR VARIETIES

is the inclusion and

for each fr e Hr (see 4.1.17). Similarly, for each standard parabolic subgroup P = MN of G we have a base change homomorphism (4.2.2)

bM :A

If P is given by the partition d\-\ (4.2.3)

bM = bGLdi

h ds = d, we have 0 •••0

bGLds

with the notations of (4.1.14). LEMMA (4.2.4). — The diagram

Ar 0 Hr

Proof: This is an obvious consequence of (4.1.13).

•

Following Drinfeld (see [Kaz]) we will now give a complete description of the function

f = Kfr) when

i.e.

P R O P O S I T I O N ( 4 . 2 . 5 ) ( D r i n f e l d ) . — For each

integer

d>l,

let

Z be the function defined by

Downloaded from https://www.cambridge.org/core. University of Exeter, on 29 Dec 2017 at 05:48:03, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/CBO9780511666162.005

4. THE FUNDAMENTAL LEMMA

85

if 9 € gld(O) H GLd(F) and v(detg) = r, where p is the nullity (i.e. the dimension over k of the kernel) of the matrix ~g G gld(k) obtained by reducing g modulo vogld(O), and by

Md) = o otherwise (recall that r is a fixed positive integer). Then for each standard parabolic subgroup P = MN by the partition d\ + • • • + ds = d and for each m = (9i,---,9s)

e M(F) = GLdl(F)

x ••• x

of G = GLd given

GLds(F)

we have s

(