The Geometry and Cohomology of Some Simple Shimura Varieties. (AM-151), Volume 151 9781400837205

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AnnaIs af Mathematics Studies Number 151

The Geometry and Cohomology of Some Simple Shimura Varieties by

Michael Harris and

Richard Taylor

With an appendix by Vladimir G. Berkovich

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

2001

Copyright @2001 by Princeton University Press

Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540

ln the United Kingdom: Princeton University Press, 3 Market Place,

Woodstock, Oxfordshire OX20 lSY

All llights Reserved

The Armals of Mathematics Studies are edited by John N. Mather and Elias M. Stein

ISBN 0-691-09090-4 (cloth) ISBN 0-691-09092-0 (pbk.)

The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed.

Printed on acid-free paper.

00

www.pup.princeton.edu

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

10 9 8 7 6 5 4 3 2 1

(Pbk.)

To Béatrice and Christine

Contents 1

Introduction Acknowledgements

15

I

Preliminaries 1.1 General notation 1.2 Generalities on representations .. Admissible representations of GLg 1.3 1.4 Base change . . . . . . . . . . . . . 1.5 Vanishing cycles and formal schemes 1.6 Involutions and unitary groups . . 1.7 Notation and running assumptions

17

II

Barsotti-Tate groups 11.1 Barsotti-Tate groups 11.2 Drinfeld leveI structures

59 59 73

III

Some simple Shimura varieties III. 1 Characteristic zero theory III. 2 Cohomology . . . . III. 3 The trace formula. III.4 Integral models

89 89

IV

V

17 21 28 37

40 45 51

94 105 108

Igusa varieties IV.1 Igusa varieties of the first kind IV.2 Igusa varieties of the second kind

121

Counting Points V.1 An application of Fujiwara's trace formula. V.2 Honda-Tate theory V.3 Polarisations I . V.4 Polarisations II . .

149 149

vii

121 133

157 163 168

viii

V.5 V.6 VI

Some local harmonic analysis The main theorem .. . . . .

Automorphic forms VI. 1 The Jacquet-Langlands correspondence . VI. 2 Clozel's base change . . . . . . . . . . .

182 191 195

195 198

VII Applications VII.1 Galois representations . . . . . VII.2 The local Langlands conjecture

217

Appendix. A result on vanishing cydes by V. G. Berkovich

257

Bibliography

261

Index

269

217 233

Introduction This book has twin aims. On the one hand we prove the local Langlands conjecture for G Ln over a p-adic field. On the other hand in many cases we are able to identify the action of the decomposition group at a prime of bad reduction on the l-adic cohomology of the "simple" Shimura varieties studied by Kottwitz in [K04]. These two problems go hand in hand. The local Langlands conjecture is one of those hydra-like conjectures which seems to grow as it gets proved. However the generally accepted formulation seems to be the following (see [He2]). Let K be a finite extension of Qp. Fix a non-trivial additive character 'li : K -+ ex. We will denote the absolute value on K which takes uniformisers to the reciprocal of the number of elements in the residue field by I IK. We will let W K denote its Weil group. Recall that local class field theory gives us a canonical isomorphism ArtK : K

X

-+ W~b.

(Normalised so that geometric Frobenius elements correspond to uniformisers.) The local Langlands conjecture provides some sort of description of the whole of W K in the sarne spirit. We willlet Irr(GLn(K)) denote the set of isomorphism classes or irreducible admissible representations of GLn(K) over e (or what comes to the sarne thing: irreducible smooth representations). If [1TI] E Irr(GL n1 (K)) and [1T2] E Irr(GL n2 (K)) then there is an L-factor L(1TI x 1T2,S) and an epsilon factor E(1TI x 1T2, S, 'li) associated to the pair 1TI, 1T2 (see for instance [JPSS]). On the other hand let WD RePn (WK) denote the set of isomorphism classes of n-dimensional Frobenius semi-simple Weil-Deligne representations of the Weil group, W K, of K over te. By a Frobenius semi-simple Weil-Deligne representation of WK over e we mean a pair (r,N) where r is a semi-simple representation of WK on a finite dimensional complex vector space V, which is trivial on an open subgroup, and an element N E End c(V) such that r(a)Nr(a)-1

= IArt i/(a)IKN 1

INTRODUCTION

2

for alI a E WK. Again if [(r, N)] E WDRePnl (WK) then there is an L-factor L«r, N), s) and an epsilon factor f«r, N), s, w) associated to (r, N) (see for instance [Tat2] and section VII.2 of this book for the precise normalisations we are using). By a local Langlands correspondence for K we shall mean a collection of bijections

for every n

~

1 satisfying the following properties.

1. If 7r E Irr(GL 1 (K» then recK(7r) = 7r o Art

Ii.

2. If [7rd E Irr(GL n1 (K» and [7r2] E Irr(GL n2 (K» then

and

3. If [7r] E Irr(GLn(K» and X E Irr(GL 1 (K» then recK(7r Q9 (X o det» = recK(7r)

Q9

recK(X).

4. If [7r] E Irr(GLn(K» and 7r has central character X then detrecK(7r) = recK(X). 5. If [7r] E Irr(GLn(K» then recK(7r V) = recK(7r)V (where V denotes contragredient) . Henniart showed (see [He5]) that there is at most one set of bijections reCK with these properties. The commonest formulation of the local Langlands conjecture for G Ln is the following theorem. Theorem A A local Langlands correspondence reCK exists for any finite extension K jQp.

However it seems to us that one would really like more than this simple existence theorem. On the one hand it would be very useful if one had some sort of explicit description of this map reCK. Our methods shed no light on this. One might well hope that the methods of Bushnell, Henniart and Kutzko will lead to an explicit version of this theorem. On the other hand one would also like to know that the local reciprocity map reCK is compatible with global reciprocity maps whenever the global map is known

INTRODUCTION

3

to existo Our methods do not resolve this latter question but they do shed considerable light on it. For instance in the cases considered by Clozel in [CU] we settle this question affirmatively up to semisimplification. (We do not identify the two N's.) Maybe a remark on the history of this problem is in order. The existence ofrecKhrr(GL 1 (K)) with the desired properties follows from local class field theory (due originally to Hasse [Has]), but this preceded the general conjecture. The key generalisation to n > I is due to Langlands (see [Lan]), who formulated some much more wide ranging, if less precise, conjectures. The formulation in the form described here, with its emphasis on epsilon factors of pairs, seems to be due to Henniart (see [He2]). Henniart's formulation has the advantage that there is at most one such correspondence, but as remarked above it limits somewhat the scope of Langlands' original desiderata. The existence of recKhrr(G L2(K)) with the desired properties was established by Kutzko ([Ku]), following earlier partial work by a number of people. The existence of recKhrr( GL 3(K)) with almost alI the desired properties was established by Henniart ([HeI]). ln particular, his correspondence had enough of these properties to characterise it uniquely. Both the work of Kutzko and Henniart relied on a detailed classification of alI elements of lrr( G Ln (K)). These methods have since been pushed much further, but to date have not provided a construction of reCK which demonstrably has the desired properties on lrr(GLn(K)) for any n > 3. ln the case of completions of functions fields of transcendence degree I over finite fields, the corresponding theorem was proved by Laumon, Rapoport and Stuhler ([LRS]). We willlet Cusp (GLn(K)) denote the subset ofIrr(GLn(K)) consisting of equivalence classes of supercuspidal representations. Let RePn (WK) denote the subset of WDRePn (WK) consisting of equivalence classes of pairs (r,N) with N = O. AIso let lrrn(WK) denote the subset of RePn(WK) consisting of equivalence classes of pairs (r, O) with r irreducible. It follows from important work of Zelevinsky [Ze] that it suffices to construct bijections

with the properties listed above (see [He2].) ln a key breakthrough, Henniart [He4] showed that there did exist bijections

which preserved conductors and were compatible with twists by unramified characters. He was however unable to show that these bijections had enough of the other desired properties to characterise them uniquely. The usefulness of this result is that it allows one to use counting arguments, for

4

INTRODUCTION

instance any injection Cusp (GLn(K)) n (see [BHK] and [Har3]). A posteriori we can show that recÍ< = reCK. Since the distribution of a preliminary version of this work, but before the distribution of the final version, Henniart [He6] has given a much simpler proof of theorem A by making much cleverer use of the non-Galois automorphic induction of [Har3] and of his own numericallocal Langlands theorem [He4]. He does not need the a priori construction of a map reCK compatible with some instances of the global correspondence, and thus he is able to by-pass alI the main results in this book. For the reader interested only in theorem A his is clearly the better proof. None the less we believe the results of this book are still important as they establish many instances of compatibility between the global and local correspondences. Let us now explain our construction of maps

To this end choose a prime I =I- p and fix an isomorphism C ~ Qf. Let k denote the residue field of K. For any 9 ~ 1 there is, up to isomorphism,

INTRODUCTION

5

a unique one-dimensional formal OK-module EK,g/k ac of OK-height g. Then End OK (EK,g) I8iz Q ~ D K,g, the division algebra with centre K and Hasse invariant l/g. The functor which associates to any Artinian local OK-algebra A with residue field k ac the set of isomorphism classes of deformations of EK,g to A is prorepresented by a complete noetherian local O K algebra RK,g with residue field k ac • (ln fact RK,g is a formal power series ring in 9 - 1 variables over the ring of integers of the completion of the maximal unramified extension of K.) We willlet EK,g denote the universal deformation of EK,g over RK,g. (ln the case 9 = 1 one just obtains the base change to the ring of integers of the completion of the maximal unramified extension of K of any Lubin-Tate formal OK module over K.) Drinfeld showed that for any integer m ~ O there is a finite flat RK,g-algebra RK,g,m over which EK,g has a universal Drinfeld leveI pm_ structure. We will consider the direct limit over m of the formal vanishing cycle sheaves of Spf RK,g,m with coefficients in «}tc. This gives a collection {Wk,l,g} of infinite-dimensional «}tc vector spaces with natural admissible actions of the subgroup of GLg(K) x D~,g x WK consisting of elements b, 15, 0') such that

For any irreducible representation p of D~,g we set Wk,I,g(P)

= Horn O~K

,9

(p, wk,l,g)'

This becomes an admissible GLg(K) x WK-module. ln the case 9 = 1 we have wk I 1 = (O) for i > O, while it follows from the theory of Lubin-Tate formal ~~ups (see [LT]) that w'k 11 (p) = «}tc with an action of K X x WK via p-1 x (p o Art 1/) (see sectio~ '3.4 of [Car3]). To describe Wk,I,9(P) in greater generality we must recall that Deligne, Kazhdan and Vigneras (see [DKV]) and Rogawski ([Rog2]) have given a bijection between irreducible representations of D ~,g and (quasi-)square integrable irreducible admissible representations of GLg(K) characterised by a natural character identity (see section 1.3). This generalises work of J acquet and Langlands in the case 9 = 2 so we will denote the correspondence p f-t JL (p). Carayol essentially conjectured ([Car3]) that if JL (p) is supercuspidal then - 1 (p) ~ JL (p)V x recK(JL (p) l8i wgK,l,g

1

det 1(1- g )/2).

We do not quite prove this (though it may be possible by our methods to do so). However, motivated by Carayol's conjecture, our first main theorem is the following. To state it let ['li K,g (p)) denote the virtual representation ",~.:::-1 (_I)i[wi ( _I)g-l L.."t_O K,l,g (p)).

INTRODUCTION

6

Theorem B If 7r is an irreducible supercuspidal representation of G Lg (K) then there is a (true) representation

such that in the Grothendieck group

ln the case n = 1 Lubin-Tate theory alIows one to identify

rl(7r) = 7r- 1 o Art [/. We use this theorem to define

by the formula

lt will also be convenient for us to extend ri to alI irreducible admissible representations of GLg(K) as folIows. If 7r is an irreducible admissible representation of G L 9 (K), then we can find positive integers 91,···, 9t which sum to 9 and irreducible supercuspidal representations 7ri of G Lgi (K) such that 7r is a subquotient of n-lnd (7r1 x ... x 7rt), where we are using the usual normalised induction (see section 1.2). Then we set t

rl(7r) = EBrl(7ri) 01Art [/I(gi-g)/2. i=l

This is welI defined and

for some N. Our second key result is that ri is compatible with many instances of the global Langlands correspondence. The folIowing theorem strengthens a theorem of Clozel [CU] (in which he only identifies [R(II)lwF y ] for all but finitely many places y, and specifically for none of the bad places).

Theorem C Suppose that L is a CM field and that II is a cuspidal automorphic representation of G Lg (AL ) satisfying the following conditions:

• lIoo has the same infinitesimal character as some algebraic representation over IC of the restriction of scalars from L to Q of GL g,

INTRODUCTION

7

• and for some finite place x of L the representation llx is square integrable. Then there is a continuous representation

such that for any finite place y of L not dividing l we have

Moreover for each finite place y of F the representation lly is tempered.

ln the case n = 2 and K = Qp and F+ = Q both theorems B and C were essentially proved by Deligne in his beautiful letter [De2]. (The argument was completed by Brylinski [Bry].) Carayol [Car2] generalised Deligne's method to essentially prove both theorems B and C in the general n = 2 case. We will simply generalise Deligne's approach to n > 2. The combination of theorems B and C, Henniart's numericallocal Langlands theorem [He4], and the non-Galois automorphic induction of [Har3] suffice to prove theorem A. Both theorems B and C follow without great difficulty from an analysis of the bad reduction of certain Shimura varieties. We will next explain this analysis. Unfortunately we must first establish some notation. Let E denote an imaginary quadratic field in which p splits: p = UU C • Let F+ denote a totally real field of degree d and set F = EF+. Fix a place w of F above u. Let B be a division algebra with centre F such that • the opposite algebra BOP is isomorphic to B Q9E,c E; • B is split at w;

• at any place x of F which is not split over F+, Bx is split; • at any place x of F which is split over F+ either Bx is split or Bx is a division algebra, • if n is even then 1 + dn/2 is congruent modulo 2 to the number of places of F+ above which B is ramified. Let n denote [B : F]1/2. We can pick a positive involution of the second kind * on B (Le. *IF = c and tr B/Q(XX*) > O for all nonzero x E B). If j3 E B*=-l then we willlet • G denote the algebraic group with G(Q) the subgroup of elements x E (BOP)X so that x*j3x = v(x)j3 for some v(x) E QX, • v: G

---+ Gm the corresponding character,

INTRODUCTION

8 • G1 the kernel of v, • and ( , ) the pairing on B defined by

(x,y)

= (trF/Q o trB/F)(x;3y*).

We can and will choose ;3 such that • G is quasi-split at alI rational primes x which do not split in E • and G 1(IR) ~ U(l, n - 1) x U(O, n)[F+:Ql-1. If U C G(AOO) is an open compact subgroup we will consider the following moduli problem. If 8 is a connected F -scheme and s is a closed geometric point of 8 then we consider equivalence classes of quadruples (A, À, i, where

m

• A is an abelian scheme of dimension [F+ : Q]n 2 ;

• À: A -t AV is a polarisation; • i : B 1 the varieties J&h2 m 8 do not occur in the reduction of any Shimura variety. They seem tb ~aturally exist only in characteristic p. The idea is now that over the "pro-object" lim+-B J&'!1 m 8 we have an isomorphism gO ~ ~Fw,n-h and Ri-\.I!f1(. denote the stabiliser of À in N, let H>. = HO N>. and let WR,>. denote the À-weight space. Thus dim WR,>. = a. Suppose that n EN>.. Let 0 1 = {À}, O 2 , .•• , Os denote the orbits of weights of T on WR under (n). Let WR,i denote the sum of the J.L weight

CHAPTERL PRELIMINARIES

28

spaces for p, E Di. If Xl, ... ,X,. are the eigenvalues of n on WR,i and if t E T then any eigenvalue X of nt on W R,i satisfies Xb

= X~

II p,(t)b(!L) , J.'EOi

for some j E {1, ... , u}. Here b is a positive integer (for example the order of n in NIT) and the b(p,) are non-negative integers with EJ.' b(p,) = b. They depend on n but not on t. Suppose that for any i > 1 we have À -::fi

(L b(p,)p,)lb. J.'EO;

(This would be true for instance if À lies at a vertex of the convex hull of the set of weights of T on WR.) ln this case, if n had two distinct eigenvalues YI, Y2 on W R,>', then for generic t E T the element nt would have eigenvalues yIÀ(t) and Y2À(t) on WR,>. but on no other WR,i (with i > 1). That would imply that a = dim W R ,>. ~ 2a, a contradiction. We conclude that in this case À extends to a homomorphism À : N>. -t Gm such that N>. acts on W R,>. by À. Now suppose that À does lie at a vertex of the convex hull of the set of weights of T on W R and that À is minimal with respect to B amongst this set of weights. Let X (resp. Y) denote the sum of the HO isotypical components of W R corresponding to irreducible representations of HO with minimal weight À (resp. any conjugate under N of À). Then Y ~ Ind~tX. On the other hand if Xl, ... ,X a is a basis of W R,>. then X ~ E9~=1 (Bxú as an H>.-module. We conclude that

as H -modules. Arguing recursively we see that WR module U. The lemma follows. O

1.3

~

U a for some H-

Admissible representations of G Lg

Let O be an algebraically closed field of characteristic O and of cardinality equal to that of C. Let K be a finite extension of Qp. Suppose that V lO is a vector space and that 7r : GLg(K) -t Aut (V) is an irreducible admissible representation with central character 'l/J7r. We will call 7r supercuspidal if for any v E V and f in the smooth dual of V the function GLg(K) -t O which sends X I-----t

f (xv)

1.3. ADMISSIBLE REPRESENTATIONS OF G Lg

29

is compactly supported modulo the centre K X of GLg(K). Choose an embedding of fields ~ : n y C. We will call 7r square integrable if for any v E V and f in the smooth dual of V the function GLg(K)/ K X -+ IR which sends

is integrable. It follows from Zelevinsky's classification [Ze] that this definition is independent of the choice of~. We will say that a representation 7r is ~-preunitary if there is a pairing ( , ) from V x V to te such that

• (avI +V2,V3)

= ~(a)(VI,V3) + (V2,V3) for all a E n and VI,V2,V3

E V,

• (VI, V2) = c(V2, VI) for ali VI, V2 E V (where c denotes complex conjugation),

• (v, v) > O for ali non-zero

V

E V,

• (7r(X)VI,7r(X)V2) = 1~(1/1,..(detx))12/g(VI,V2) for all VI,V2 E V and x E GLg(K). Rogawski ([Rog2]) and Deligne, Kazhdan and Vigneras (see [DKV]) have shown the existence of a unique bijection, which we will denote JL, from irreducible admissible representations of Dr,g to square integrable irreducible admissible representations of GLg(K) such that if p is an irreducible admissible representation of Dr,g then the character XJL (p) of JL (p) satisfies

= O if'Y E GLg(K) is regular semi-simple but not elliptic,

•

XJL(p)(-y)

•

GLg(K) is regular semi-simple and elliptic and if 8 is an element of Dr,g with the sarne characteristic polynomial as 'Y. XJL (p) (-y)

= (-I)g-ltr p(8) if'Y

E

If 7r is a square integrable irreducible admissible representation of the group GLg(K) with central character 1/1,.. then Deligne, Kazhdan and Vigneras also show the existence of a function 'P,.. E Cgo(GLg(K),1/1;I), which we will call a pseudo-coefficient for 7r, with the following properties. (We always use associated measures on inner forms of the sarne group. See for instance page 631 in [K05], where they are called compatible measures.)

• tr7r('P,..)

= vol(Dr,g/K

X ).

• Suppose that GL g1

X •••

x GLg,

=L C P

C GLg

CHAPTER I. PRELIMINARIES

30

is a Levi component of a parabolic subgroup of GLg. Suppose also that for i = 1, ... , s we are given a square integrable irreducible admissible representation 1I"i of GLg; (K) such that 1I"i '!- 11" and such that 1/J1r l •. . 1/J1r. = 1/J1r' Then

GLg(K) trn-Ind P(K) (11"1

X •••

x 1I"s)(cp1r)

= o.

• If "I E GLg(K) is a non-elliptic regular semi-simple element then O~Lg(K)(cp1r) = O.

• If "I E GLg(K) is an elliptic regular semi-simple element and if 8 E D~,g has the sarne characteristic polynomial as "I then

(See section A.4 of [DKV], especially the introduction to that section and subsection A.4.l .) Lemma 1.3.1 Let 11" be a square integrable representation of GLg(K) and let CP1r be a pseudo-coefficient for 11" as above. 1. If'Y E GLg(K) is a non-elliptic semi-simple element then O~Lg(K)(cp1r)

= O.

GLg(K) is an elliptic semi-simple element and if 8 E D~,g has the same characteristic polynomial as "I then

2. If'Y E

Proof: Consider the first parto Let T be a maximal torus containing "I. Then

o= L r u(t)O~:g(K)(cp1r) u

where u runs over a set of representatives of the unipotent conjugacy classes in ZGLg("{)(K), where ru denotes the Shalika germ associated to u and where t is any regular element of T sufficiently close to "I. Then homogeneity ([HC], theorem 14(1» teUs us that O = r1(t)O~Lg(K)(cp1r)

I.30 ADMISSIBLE REPRESENTATIONS OF G Lg

31

for any regular t E T sufficiently dose to "(o By [Rogl], rI (t) is not identically zero near "( and the first part of the lemma followso Consider now the second parto Let T be an elliptic maximal torus in GLg(K) containing "(o We can and will also think of T C D~,go Then we can take 6 to be "( E T C D~,go For t a regular element of T sufficiently dose to "( we have u

where u runs over a set of representatives of the unipotent conjugacy dasses in ZGL g("()(K) and where r u denotes the Shalika germ associated to Uo Again using homogeneity ([He], theorem 14(1)) we see that (-l)g-l vol (D~ /T)trJL -1 (1I"V) (6) ,g

= OGLg(K) ( > O there exists a homomorphism

ípn : A' - t Bco

1.6. INVOLUTIONS AND UNITARY GROUPS

45

with

Fix such an n, which we also suppose greater than N + t. Note that as A' /0 is finitely generated . E (F+) x. (Note that G Àa!3o

=

Now fix

arising from some {3 as in the lemma (for some choice of T). We will write simply # for the corresponding involution on BOP ® ADO, and G and G 1 for the corresponding groups over ADO. If T : F+ '-t IR then ( , ) has a well-defined extension

with invariants (1, n - 1) at T and (O, n) at alI other infinite places. Thus we get an involution #T on BOP ® A (up to equivalence) and groups GT and G T,1 over A By the above lemma we see that ( , )T can be taken to arise from some (non-canonical) {3T E B*=-1. Even the pairing ( , )!3T need not be canonical, there are # ker 1 (Q, G!3T) choices for it. If n is even then ker 1 (Q, G!3T) = (O), while if n is odd then ker 1 (Q, G!3T) = ker((F+)X jQx N F/ F + (FX) -+ A;+ JAX NF+/F(A;' )). ln either case the natural map ker1 (Q, G!3T) -+ H 1(Q, PG!3T) is trivial. (See page 400 of [Ko3] for both these assertions.) Thus the involution #!3T on BOP up to equivalence depends only on T and the groups G!3T and G!3T,1 also depend only on T and not on the choice of (3T. Thus we will denote them #r> G T and G T ,1. If R is an E-algebra then GT(R) can be identified with the set of pairs (g1,g2) E (BOP ®E R) x (BOP ®E,c R)

such that

Thus we have

CHAPTER 1. PRELIMINARIES

56

where

and inversely

(g, v)

f-----t

(g, Vg-#T).

ln particular we get an isomorphism RS~(GT xQ E) ~ RS~(Gm) x HBop,

where HBop /Q is the algebraic group defined by

HBop (R)

= (BOP ®Q R) x .

Suppose that x is a place of Q which splits as x = yyC in E. Then the choice of a place ylx allows us to consider Qx ..:t Ey as an E-algebra and hence to identify

G(Qx)

~ (B~P)X

x

Q;.

ln particular, we get an isomorphism r

G(Qp) ..:t

Q; x II (B~)X, i=l

which sends 9 to (V(g),gl, ... ,gr). We will often let r

(go,gl, ... ,gr) E

Q;

x II(B~)X i=l

denote a typical element of G(Qp). Similarly we will decompose a typical element 9 E G(AOO) as (ga,),xoj.pX (gp,O,gwp ... ,gwJ with gx E G(Qx), gp,o E and gw, E B~. j or as gP x gp,o x gw X g;', where gP = (gx )x#, gw = gWl and g;' = (gw2' ... ,gwJ. We will let G(AOO,W) denote the subgroup of G(AOO) consisting of elements with gp,o = 1 and gw = 1. Similarly if 11" is an irreducible admissible representation of G(AOO) over an algebraically closed field of characteristic O we may decompose it 11" ~ 11"P ® 11"P ~ 11"P ® 11"p,o ® 11"p,l ® ... ® 11"p,r ~ 11"P ® 11"p,o ® 11"w ® 11";' ~ 11"w ® 11"p,o ® 11"w. Note that 11"p,O = 1/J'Ir IEx. (where we recall that 1/J'Ir is the character of (A~) x which is the central ~haracter of 11"). Fix a maximal order Ai = OBw. in Bw. for each i = 1, ... , r. Our pairing ( , ) gives a perfect duality'between Vw; and Vwf . Let A( C Vw:' denote the dual of Ai C Vw.. Then if

Q;

A=

r

r

i=l

i=l

EB Ai EB EB A~ C V ®Q Qp,

I. 7. NOTATION AND RUNNING ASSUMPTIONS

57

we see that A is a Zp-Iattice in V ®Q IQp and that the pairing ( , ) on V restricts to give a perfect pairing A x A -+ Zp. There is a unique maximal Z(ptorder OB C B such that 0Ê = OB and OB,w, = OB w, for i = 1, ... , r. Then OB,p equals the set of elements of Bp which carry A into itself. On the other hand the stabiliser of A in G(lQp) •

IS

'7IX

fú p

X

ni=1 vBw.· r

,'"IX

Fix an isomorphism OBw ~ Mn(OF,w)' Composing this with the transpose map t we also get an isomorphism O~w ~ Mn(OF,w)' Moreover we get an isomorphism éA 1 ~ (OF,w)v,

(See the end of section 1.1 for the definition of é.) The action of an element 9 E Mn(OF,w) ~ (O~J on this module is via right multiplication by l. We will write All as an abbreviation for éA 1 . We get an identification r

A ~ ((OF,w ® All ) EB (OF,w ® All)V) EB EB(Ai EB Ai). i=2

Under this identification (gO, gl , ... , gr) E G (lQp) acts as r

((1 ® gd EB go(l ® g1 1)V) EB EB(9i EB go(gi 1)V). i=2

Let ç denote a representation of G on a ~c -vector space W~. We will often assume that ç is irreducible. This will always be the case from section Ill.30nwards. Fix a square root

of 1 IK : K X -+ ~x , i.e. fix a square root of p/K in iQj'c. If fK is even we assume that this square root is chosen to be p/K /2. AIso choose z : iQj'c ~ C, such that z o 1 11/2 is valued in R;o' We apologise for making such an ugly choice. The reader will see that alI our main results are independent of the choice of z, but it would require a lot of extra notation to make the proofs free of such a choice. Some of our main results do involve the choice of 1 11/2, but in each case this choice is involved in more than one place and alI that matters is that the sarne choice is made at each place.

Chapter II

Barsotti-Tate groups lI.l

Barsotti-Tate groups

For the definition of a Barsotti-Tate group over a scheme S we refer the reader to section 2 of chapter l of [Me]. Suppose that S is a OK scheme, then by a Barsotti-Tate OK-module H/S we shall mean a Barsotti-Tate group H / S together with an embedding OK Y End (H). (We remark that ring morphisms are assumed to send the multiplicative identity to itself.) We calI a Barsotti-Tate OK-module H ind-etale if the underlying BarsottiTate group is ind-etale (see example 3.7 of chapter l of [Me)). There is an equivalence of categories between ind-etale Barsotti-Tate OK-modules and finite, torsion free lisse etale OK-sheaves on S (see example 3.7 of chapter l of [Me)). If S is connected we define the height of a Barsotti-Tate OK-module H to be the unique integer h(H) such that H[p~l has rank q'lo we can find reduced closed subschemes S[h] C S such that 1. S[h] :J S[h-l]; 2. the codimension of any component of S[h-l] in any component of S[h] which contains it is at most 1; 3. for any geometric point s of S we have that s lies in S[h] if and only if #H[P](k(s)) ~ p[K:Qp]h;

4. on S(h)

= S[h]_S[h-l] there is a short exact sequence of Barsotti-Tate

OK-module

(O) ---t HO ---t H ---t H et ---t (O) where HO is a formal Barsotti- Tate O K -module and where H et is an ind-etale Barsotti-Tate OK-module of height h.

11.1. BARSOTTI-TATE GROUPS

61

Proo/: By proposition 4.9 of [Me] it suffices to show that for 9 E Z2: 0 we can find closed subschemes 8~ C 8 such that 1. 8~ J 8~_1;

2. if s is a geometric point of 8 then s lies in 8~ if and only if #H[P] (k( s)) is less than or equal to p9; 3. the codimension of any component of which contains it is at most 1.

8~_1

in any component of

8~

The question is local on 8 so we may assume that 8 = Spec R for a noetherian ring R. We may further assume that 8 is reduced. By a simple inductive argument it suffices in fact to show that if for any geometric point s of 8 we have #H[P](k(s)) ~ p9 then we can find a reduced closed subscheme 8' C 8 such that 1. a geometric point s of 8 lies in 8' if and only if #h[P](k(s))

< p9;

2. any irreducible component of 8' has codimension at most one in any irreducible component of 8 containing it. Finally we may assume that 8 is in fact integral. We now follow the arguments of page 97 of [O]. Let 1i / 8 denote the locally free sheaf Lie (H[P]V) = Lie H V (the equality here follows from remark 3.3.20 of chapter II of [Me] because p = O on 8). For any geometric point s of 8 there is a canonical perfect pairing between 1i~.=1 and Hs[P](k(s)). (This seems to be well known, but we know of no reference for the statement in exactly this form, so we will sketch the proof. On page 138 of [MuI] we see that we can identify 1is with Horn (Hs[P], Ga ) and that V* then becomes identified with the map 4J I-t 4J o Fr*. We get a pairing

Hs[P](k(s)) x

1is

X

4J

x

---+ I------t

k(s)

4J o x,

where 4J E Horn (Hs[P],Ga ) and 4J o x E Ga(k(s)) = k(s). We see that it restricts to a pairing

Hs[P](k(s))

X

1i~.=1

---+ IFp .

If 4J o x = O for alI x E Hs[P](k(s)) then 4J factors through the local ring of Ga at O. If moreover 4J o Fr* = 4J then we see that 4J = O. Thus our pairing gives an injection

To show this is in fact an isomorphism one can count orders. Suppose that #Hs[P](k(s)) = ph. Then we have an embedding JL~ y Hs[PjV and so an

CHAPTER II. BARSOTTI-TATE GROUPS

62

embedding Lie JL~ g' (If 8 E ODK'9 then the push forward of (EK,g,J) along 8 : RK,g -t RK,g is (EK,g,J 08).) We willlet RK,g denote RK,g XW(k(PK)) k(fPK)ac. Set Ho = ~k,g X (KjOK)h a compatible Barsotti-Tate OK-module over k(VK)ac. Let THo denote its Tate module, i.e.

THo = Horn OK (KjOK, Ho(k(VK)ac)) ~ O~. Now consider the functor fram Artinian local OK-algebras with residue field k(VK)ac to sets which sends A to the set of isomorphism classes of pairs (H,j) where HjSpecA is a compatible Barsotti-Tate OK-module and j : Ho ~ H XA k(fPK)ac. This functor is again pro-represented, this time by Horn (THo, EK,g). By Horn (THo, EK,g) we mean the RK,g-formal scheme such that for any Artinian local RK,g algebra S we have

Noncanonically we have Horn (THo, EK,g) ~ E~,g, where the fibre product is taken over Spf RK,g' We also have, again noncanonically,

The universal deformation of Ho over Horn (T Ho, EK,g) is then the extension of EK,g by (KjOK)h classified by the tautological class in

where S = Hom(THo,EK,g). (See proposition4.5 of [Dr] and its praof.) Lemma 11.1.3 Suppose that Sjk(VK)ac is reduced of finite type. Suppose

also that HjS is a one-dimensional compatible Barsotti-Tate OK-module. Suppose moreover that over S there is an exact sequence of Barsotti- Tate OK-modules

where Het has constant height h and HO has constant height g. Let s be a closed point of S and choose an isomorphism j : ~K,g ~ H~. 1. Then HO j S~ gives rise to a morphism S~ -t Spf RK,g which in fact factors through Spec k( v K )ac . 2. H j S~ gives rise to a morphism S~ -t Horn (T H s , EK,g) which in fact factors through Horn (TH s , ~K,g) C Horn (THs , EK,g).

lI.1. BARSOTTl-TATE GROUPS

65

Proof: The statements are easily seen to be equivalent. We will prove the first one. Write RK,g = Oí(m[[T2 , ... ,Tg]], let P be a minimal prime of O~,s and let k denote the field of fractions of the image RK,g --+ O~,s/ P. As S is reduced, it suffices to show that T 2 , •.• , T g map to O in k. Suppose noto For the rest of this proof we will use without comment the notation of [Dr]. We can arrange that EK,g corresponds to a morphism AOK = OK[g1,g2, .. . ]--+ RK,g which

• and sends gj to zero for 1 ~ j i in the above range.

< p/Kg

- 1 and j

i- p/K i -

1 for some

(See the proof of proposition 4.2 of [Dr].) Choose i minimal such that Ti does not map to zero in k. Then HO Xs k corresponds to a morphism AOK --+ k which sends gj to O for j = 1,2, ... ,p/K(i-1) - 2 and sends gpfK(i-l)_1 to something nonzero. Thus HO Xs k has height i - I < 9 (see the proof of proposition 1.6 of [Dr]). This contradicts the fact that H x s Spec k is a compatible formal Barsotti-Tate O K-module of height 9 (because H / S is a compatible formal Barsotti-Tate OK-module of height

g). O

Corollary 11.1.4 Suppose that S/k(VK)ac is a smooth scheme of finite type. Suppose that H / S is a one-dimensional compatible Barsotti- Tate O K -module of constant height g. Suppose moreover that for each closed point s of S the formal completion S; is isomorphic to the equidimensional universal formal deformation space of H s . Then for h = O, ... ,g -1 the 10cally closed subscheme S(h) = S[h] - S[h-1] of S is either empty or smooth of dimension h. If s is a closed point of S(h) and if j : L,K,g-h .:t H2 then we get an identification S; ~ Horn (T Hs, EK,g-h) xO K k(VK) and under this identification (S(h))~ C S; corresponds to Horn (T Hs, L,K,g-h) C Horn (THs,EK,g-h) xO K k(VK). Proof: Because the formal completion of S at any closed point is isomorphic to k(VK )ac[[T2, ... , Tg]], every component of S has dimension 9 - 1. We must have S = S[g-1]. Hence by lemma I1.1.1 every irreducible component of S[h] has dimension at least h. Thus the sarne is true for S(h). On the other hand by the previous lemma if s is any closed point of S(h) then the formal completion (S(h))~ corresponds to a sub-formal scheme of Horn (TH s , L,K,g-h) C Horn (TH s , EK,g-h). Thus we must have (S(h))~ ~ Horn (T Hs, L,K,g-h) and, assuming such a closed point exists, we have that S(h) is smooth at s of dimension h. O

CHAPTER II. BARSOTTI-TATE GROUPS

66

The functor on schemes S/k(VK)ac which sends S to Aut (EK,g[pí(J/S) is represented by a scheme Aut (EK,g[pí(]) offinite type over k(VK)ac. (To see this simply think of these automorphisms as maps on sheaves of Hopf algebras.) If ml > m2 there is a natural morphism

We willlet Aut 1 (EK,g [pí(]) denote the intersection of the scheme theoretic images of Aut (EK,g[pí(']) in Aut (EK,g[pí(]) as m' varies over integers greater than or equal to m. We see that the scheme theoretic image of the morphism Aut l(EK,g[p~+1])

----t

Aut (EK,g[PK])

is just Aut 1 (EK,g[pí(]). Lemma 11.1.5 Aut 1 (EK,g[pí(])/k(VK)ac is finite and

Aut l(EK,g[PKWed ~ (ODK'9/PKODK,g)X.

Proo!: Suppose first that Aut 1 (EK,g[pí(]) has an irreducible component Vm of dimension > o. Then we can find irreducible components Vm , of Aut 1 (EK,g[pí(']) for m' > m such that whenever mil ~ m' ~ m then Vm" maps to Vm' and is dominating. Let k(Vm') denote the function field of V~~d, so that whenever mil ~ m' ~ m we have k(Vm') '--t k(Vm,,). Let k be an algebraically closed extension field of k( v K )ac of uncountable transcendence degree. Then there are uncountably many maps k(Vm) '--t k and each can be extended into a compatible series of injections k(Vm' ) '--t k for m' > m. Thus Aut 1 (EK,g[pí(])(k) has uncountably many points which can be lifted compatibly to each Aut 1 (EK,g [pí(']) (k) with m' > m. This implies that the image of

is uncountably infinite. On the other hand it follows from proposition 1.7 of [DrJ that this image is just (ODK,g/pí(ODK,g)X, which is finite. This contradiction shows that Aut 1 (EK,g[pí(]) is zero dimensional. As each Aut 1 (E K,g[PK]) is zero dimensional and as for m' > m the morphism

is dominating we see that for m'

~

m

lI.1. BARSOTTl-TATE GROUPS

67

It follows that Aut 1 (E K,g[p1(])(k(VK)ac) equals the image of Aut (EK,gjk(VK)ac) ---+ Aut (EK,g[p1(l/k(VK)ac). Again by proposition 1.7 of [Dr] this is just (ODK'9jp1(ODK'9)X and so the lemma follows. O We remark that for m > 1 the scheme Aut (E K,g[p1(J) has dimension > O. By an explicit calculation with Dieudonne modules we checked in an earlier version of this work that Aut 1 (EK,g[p1(w ed coincides with the reduced subscheme of the image of Aut (EK,g[p~+lJ) -+ Aut (E K,g[p1(]). However we will not actually need that stronger result here, so we do not reproduce the argumento Now suppose that S is a reduced k(VK)ac-scheme and that HjS is a one-dimensional compatible formal Barsotti-Tate OK-module of constant height g. We want to investigate how far H differs from EK,g xSpeck(vK)acS. Consider the functor on S-schemes which sends T j S to the set of isomorphisms (over T) j : Ek,g[p1(]

X Spec k(VK)ac

T ---+ H[p1(] Xs T.

lt is easy to see that this functor is represented by a scheme X m (H j S) of finite type over S. (Think about j as a map of sheaves of Hopf algebras on T.) Then we define Ym(HjS) to be the intersection ofthe scheme theoretic images of the

Xm,(HjS) ---+ Xm(HjS) for m' ~ m. Finally we set J(m)(HjS) = Ym(Hjsyed. We will also let juniv denote the universal isomorphism

over J(m)(HjS). Thus Ym(HjS) and Jm(HjS) are finite type over S. If TjS is any scheme then Xm(HjT) = Xm(HjS) Xs T. If TjS is flat then

(because the formation of scheme theoretic image commutes with flat base change). We see that

J(m)(EK,gjk(VK)ac)

= Aut 1(EK,g [p1(])red ~ (ODK'9jp1(ODK'9)X.

ln fact if Sjk(VK)ac is any reduced scheme then

= (J(m) (EK,gjk(VK)ac) x s)red = ((ODK,9jp1(ODK,9)~yed = (ODK,9jp1(ODK,9)~' J(m)(EK,gjS)

CHAPTER II. BARSOTTI-TATE GROUPS

68

Each of the schemes Xm(HjS), Ym(HjS) and J(m)(HjS) has a natural right action of (ODK'9jPKODK,g)x. (8 E O~,g takes j to j 08.) If S = T xSpeck(VK) Speck(VK)ac for a reduced scheme Tjk(VK) and if H = Ho XT S for a compatible formal Barsotti-Tate OK-module HojT, then this action extends to one of Di > j compatible morphisms

gij : Xi

-+

lj over

• a continuous homomorphism g* : r -+ 6 such that for any i and any a E r we have gij o a = g*(a) o gij, • and a linear map g* : Wp' -+ Wp such that for alI a E

r

>> j

we have

p(a) o g* = g* o p'(g*(a)).

Such a morphism

9 gives rise to a morphism

as folIows. Choose L and A' C W p' as above, and set A = g* A'. It suffices to define compatible maps jj* : g* CP',A' ,i' ---t Cp,A,i

for any i'

~

i. For this it suffices to give compatible maps

whenever we have a commutative diagram

u L

.!. X

V

.!.

--4 Y

with the vertical maps etale. Choose j(i') and j(i) as above with j(i) » j(i'). If an element of Cp',A',i'(V) is represented by f : ll'O(lj(i') Xy V) -+ A' IIi' then we define jj*(f) to be the composite

Now suppose we have morphisms

g:

(X, {Xi}, r, p) ---t (Y, {li}, 6, p')

CHAPTER III. SOME SIMPLE SHIMURA VARIETIES

96 and

h: (Y,{Yi},~,p') - t (Z,{Zd,~,p"). We define the composite

h o 9 to be

Then it is tedious but straightforward to check that

(h o 9)* =

ff* o g*(h*).

Choose a nested collection Ul,i of open compact subgroups of G(Q) with trivial intersection. If U is a sufficiently small open compact subgroup of G(A.c'°) we let Ui denote the set of U E U such that UI E UI,i and let UI denote the projection of U to G(Q). Recall that ç is a finite dimensional representation of G on a qC-vector space Wç. The collection (Xu , {XU,}, UI, ç) defines a lisse etale qc -sheaf Cd X u. It is canonically independent of the choice of Ul,i' If 9 E G(AOO) and U, Vare sufficiently small open compact subgroups of G(AOO) with g-1 V 9 C U then we have the morphism

9 = (g, {g}, C(gl), ç(gl))

: (Xv, {Xv;}, Vi, ç) - t (Xu , {Xu,}, UI, ç)

where c(91) denotes the conjugation v of sheaves on X v

I-t

gll vgl . Thus we get a morphism

ff* : g* Cç - t Cç, which we will simply denote by g. We see that

- * (gh)

= ff* o g*(h*).

We will set

If 9 E G(AOO) and U, Vare sufficiently small open compact subgroups of G(AOO) with g-IVg C U then we get a morphism

If V C U then we see that

III. 2. COHOMOLOGY

97

Thus Hi(X,LE,) becomes an admissible G(AOO)-module, in fact an admissiblejcontinuous G(AOO) x Gal(FacjF)-module. We willlet [H(X,LE,)] denote the virtual G(AOO) x Gal (Fac j F)-module

2) _l)n-l-i[H (X, LE,)] i

i

(see section I.2). We will also decompose

as rr runs over irreducible representations of G(AOO) and [RE,(rr)] is in the Grothendieck group of continuous Gal (Fac j F)-modules. To discuss the cohomology of these sheaves in more detail we will need a parametrisation of irreducible representations ~ of G over q-c. First recall that irreducible representations of GLn are classified by n-tuples ã = (al, ... ,an ) E zn with al ~ ... ~ ano We willlet X(tG~)+ denote the set of such n-tuples. The n-tuple ã corresponds to the irreducible representation with extremal weight sending a diagonal matrix with entries (h, ... , tn) to t~l ... t~n . Let Sd denote the standard representation of GLn, Le. the one parametrised by (0,0, ... ,1) E X(tG~)+. If ã E X(tG~)+ then we may find non-negative integers tã and mã and an idempotent cã E Q[Smã] such that the representation parametrised by ã can be given explicitly as

Now fix ao : E i

X Fac, C~)) griDDR(é~é(m~)Hi+mqAme x FÀc,Q;C(t~))) ®lQIjC®Q,F).,l®". Q;c

!:>i

gri+teé~é(m~)DDR(Hi+me (Ame X

!:>i

gri+teé~é(mç)

!:>i

(H~"ttme (Ame / F) ®F (FÀ ®IQI, Q;C)) ®lQIic®Q,F).,l®". gri+teéçé(m~)H~"ttme (Ame / F) ®F,,,. Q;c gri+tez(é~)é(mç)H~"ttme(Ame/F) ®F,t". C.

!:>i

Ff, Q;C)) ®lQIic®Q,F).,l®". Q;c

qc

CHAPTER III. SOME SIMPLE SHIMURA VARIETIES

100

More explicit computation of the groups

and

depends on comparison with the analytic theory. Choose T : F '-+ C. Using our isomorphism ~ : Qllc -=+ C the Qllc -sheaf C{ corresponds to a locally constant sheaf C!? j XU,,.(C). More precisely if is a representation of G,. over C then we define a locally constant sheaf C~?P jXu,,.(C) by

e

i.e. if 7r denotes the projection from G(AOO)jU x G,.(IR)jU,. to X u,,. and if U C XU,,.(C) is an open set then C~?P(U) is the set of locally constant functions

such that

fbx) = ,f(x) for all, E G,.(Q!) and all x E isomorphism

7r- 1 U.

The isomorphism l induces a functorial

ln fact the sheaves C~?P have a natural structure of a variation of mixed C Hodge structure. Let Q C G Ln denote the parabolic subgroup consisting of matrices with last row of the form (O, ... , O, *) and let Q,. C G,. x C denote the parabolic subgroup

,.'

,.'

where T' runs over embeddings F '-+ C such that T'IE = TIE but T' i:- T. Thus (G,. x C)jQ,. is isomorphic to (lpln-l)V, the Grassmanian of hyperplanes in the affine n-space V Q9F,,. C. There is a natural embedding 11.,. '-+ G,.(C)jQ,.(C) as an open subset, and this gives rise to the complex structure on 11.,.. It sends I to the I = -i subspace in V Q9F,,. C.

III. 2. COHOMOLOGY

101

Define a morphism wt : Gm -+ G r to be the composite of t H t- I with the inclusion Gm Y Z(G r ) coming from Q Y BOP. AIso define a morphism hdg: Gm -+ Qr which sends tI--t

(C

t~I)' 1, ... ,1)

I , ( lnO-I

E Gm x Q x

II GLn. r'

If Ç' is a representation of G r define a filtration Wm (W{I ) on W{I by setting Wm (W{I ) to be the sum over m' ~ m of the t H t m' weight spaces for Ç' owt. This induces a filtration W mL~?P of L~?P. If JL is a representation of Qr define a filtration FmWJt on WJt by setting FmWJt to be the sum over m' 2': m of the t H t m' weight spaces for JL o hdg. If JL is a representation of Qr then we obtain a coherent sheaf EJt over G r (C) / Qr (C) as the sheaf of holomorphic sections of the vector bundle

(Gr(C) x WJt)/Qr(C)

--t

Gr(C)/Qr(C),

where q : (g,x) H (gq,JL(q)-IX). If "( E Gr(Q) then "(*EJt is naturally isomorphic to EJt (as (g,x) H ("(g,x) gives a map ofvector bundles above 9 H "(g). Thus EJt descends to XU,r(C). The filtration FmWJt induces a filtration FmEJt of Ew If Ç' is a representation of G r over C we get an isomorphism E{I ~ L~?P ®c OXU,T(IC}

coming from the map of vector bundles over G r (C) / Q r (C): Gr(C)/Qr(C) Because W{I

®C,c

C

~

W{I (g, x)

X

(Gr(C) X WC)/Qr(C) (g, ç/(g)-IX).

--t 1----+

W({I)C as Gr(Q)-modules we also see that L{I

®C,c

C

~

L({')c

and that E({I)C ~ L~?P

®C,c OXU,T(IC}.

We make L~?P a variation of mixed C-Hodge structures by taking W mL~?P to be the (canonically split) weight filtration, F m E{I to be the Hodge filtration and defining -pn L~?P ®C,c O XU,T (IC) to be the subsheaf corresponding to FmE({I)c. ln particular Hi(Xu,r(C),L~?P) has a canonical mixed C-Hodge structure. Now suppose again that ç is an irreducible representation of G over Qfc. We see that

CHAPTER III. SOME SIMPLE SHIMURA VARIETIES

102

as locally constant sheaves which are variations of CHodge structures. ln particular we get isomorphisms (depending on the choice of ~) c:.; c:.; c:.;

griDDR,u(Hi(Xu x Fac,.c~)) gri+te~(ê~)ê(m~)H~tme (Ame / F) ®F,tu C gr~z(ê~)ê(m~)Hi+me (A:e (C), C(t~)) gr~Hi(Xu,tu(C), .c;?).

Moreover we see that the Hodge-de Rham spectral sequence

Hi(XU,tu(C),gr~(.c!? ® O~u .• ,,(C)))

::}

Hi+i(XU,tu(C),.c!?)

degenerates at E 1 (being a direct summand of the Hodge spectral sequence for Hi+i(A:e(C),C)). Note that gr~(.c!? ® O~u .• ,,(C)) is the complex with P1 ) oP (gr i-P(.ctoP) F t~ ® vXu .• ,, (C) ® HXU .• ,,(C)

in degree p. If we identify O~u .• ,,(C) with t:1\P(Lie (G T xC)/Lie QT)V

then this is just the grading from the filtration F m defined above on (' .ctop t~

®CG-,,"(Lie(GTxC)/LieQT)v,

ln [Fal] Faltings defines a subcomplex V'e .ctop oe "',~ C

t~ ®

Hxu .• ,,(C)'

which has the same cohomology and which is a direct summand as a filtered complexo Thus we get a degenerating spectral sequence

Hi(XU,tu(C), gr ~K:~) ::} H i (XU,tU (C), .c!eP) and hence isomorphisms

griDDR,u(Hi(Xu x Fac,.c~)) ~ Hi(XU,tu(C),gr~K:~). ln fact [Fal] defines K~~ as t:JLP(t~) with the above defined filtration F m for a certain representation IJ,P(Z~) of QtU modulo its unipotent radical N,u' lrreducible representations of

Qtu/Ntu ~ Gm x (GLn-l x GLt) x

II GLn T'

(where T' runs over embeddings F'--t C with are parametrised by tuples where

T'IE

= (zCT)IE but

T' "I-

ZCT)

III. 2. COHOMOLOGY

103

• bo, bl E Zj

• btu

E X(Gn-1 )+. m ' and

.... + • bT , E X(G~) . Let p

= ((1 -

n)/2, (3 - n)/2, ... , (n - 1)/2). AIso for p

= O, ... , n -

1 set

ln our case the explicit formulae in [Fal] (see in particular page 73) teU us that j.tP (ze) is irreducible and parametrised by

(ao (e, alE), wp(ã(e, alE)". Note that if ã(e, alE)".

wp(ã(e, alE)".

+ p) -

p, ã(e, aIE),-l T ' ) .

= (al, ... ,an) then

+ p) -

P= (aI, ... ,an-p-l,an-p+1

+ 1, ... ,an + l,a n_p -

p).

ln particular Kf~ contributes only to the p - ã(e, aIE)".,n-p - ao(e, alE)

graded piece. As ã(e, alE)". E X(G~)+ we see that p - ã(e, aIE)".,n-p ao(e, alE) strictly increases with p. ln particular if j = p- ã(e, aIE)".,n-pao (e, alE) for a necessarily unique p E {O, ... , n - I} then gr j DDR,,,.(Hi(Xu

X

Fac, .c~)) ~ Hi-P(Xu"".(C), &p.P (,~))

while for alI other j

Matsushima's formula gives us an isomorphism

~~ Hi(XU,T (C) , .c!(f)) ~

E9

7r 00

® Hi(Lie GT(IR) , UT , 7r 00 ® z(e))# ker 1 (IQI,G.,.),

where 7r runs over irreducible constituents of the space of automorphic forms for GT((Q)\G T (A), each taken with its multiplicity in the space of automorphic forms. (See [Ko4] page 655.) ln particular Hi(X, .c~) is a semi-simple G(Aoo)-module and we may decompose

11"

CHAPTER III. SOME SIMPLE SHIMURA VARIETIES

104

where 7r runs over irreducible representations of G(AOO) and where R~(7r) is a continuous finite dimensional representation of Gal (Fac / F). We have

[Rç (7r)] = ~) _1)n-l-i[R~(7r)]. i

Moreover R~ (7r) has dimension

where 7r 00 runs over irreducible representations of GT(IR) and where m T (7r) is the multiplicity of 7r in the space of automorphic forms on GT(Q)\GT(A). Similarly (see [Harl]) lim--+u Hi-P(XU,'IT (C), t:"p (.ç)) is isomorphic to

EB 7r

00

l8i Hi-P(Lie Q'IT' U. IT , 7r 00 l8i JLP(z~))kerl «(lI,G •.,.),

'Ir

where 7r runs over irreducible constituents of the space of automorphic forms on G.IT(Q)\G.IT (A), each taken with its multiplicity in the space of automorphic forms. Thus for any irreducible representation 7r of G(AOO) we have .

.

1

dimgr J DDR,IT(Rê(7r)) = #ker (Q,G.IT ) L'lr oo m.lT (z(7r) l8i 7r dim Hi-P(Lie Q'IT' U. IT , 7r00 l8i JLP(z~)) (0 )

if j = p-ã(~, aIE)IT,n-p-ao(~, alE) for a necessarily unique p E {O, ... , nI}, and

otherwise. Here again 7r00 runs over irreducible representations of G.IT(IR) and m'lT (7r) denotes the multiplicity of 7r is the space of automorphic forms on G.IT(Q)\G.IT(A). Summarising the above discussion we get the following proposition. Proposition 111.2.1 Let ~ be an irreducible representation of G over Qfc . Recall that we have fixed an isomorphism Z : Qfc ..; cc.

1. We have a decomposition

'Ir

where 7r runs over irreducible representations G(AOO) over Qfc and where R~(7r) is a finite dimensional continuous representation of the Galois group Gal (Fac / F).

III. 3. THE TRACE FORMULA 2. For any

T :

105

F y .', i', (71P )', aD are equivalent if there exists a prime to p isogeny {) : A -+ A' and"( E Z~) such that {) carries >. to "(>.', i to i', rjP to (71P )', and ai to a~. Again XUp,m(S, s) is canonically independent of s so we obtain a functor from connected, locally noetherian OF,w-schemes to sets. We extend it to alllocally noetherian OF,w-schemes by setting XUP,m(ll Si) = TI XUP,m(Si). On locally noetherian Fw-schemes we have natural isomorphisms XUp,m ~ X~p(m) ~ XUP(m)' If m1 = O then it is known that this functor is represented by a projective scheme XUP,m/OF,w. (Representability and quasi-projectivity follow as on page 391 of [K03] or as in section 5.3 of [Carl]. Properness follows from the valuative criterion as in section 5.5 of [Carl], the point being that if A is an abelian variety of dimension dn 2 with an action of an order in B over the field of fractions of a DVR and if A has semistable reduction then A has good reduction (otherwise the toric part of the reduction has too small a dimension to have an action of an order in B). The leveI structure then extends uniquely to the Neron model à of A, because Ã[n] is etale over the DVR for n supported on W2, . .. ,Wr and the primes not dividing p (use the fact that LieA[wi]OO = (O) for i > 1).) Hence by II.2.I, this functor is represented for alI m by a projective scheme XUP,m/OF,w. We have a canonical isomorphism

The inverse system of the XUP,m/OF,w again has an action of G(AOO). The action of 9 E G(AOO,P) just sends (A, >., i, 71P , ai) to (A, >., i, 71P o 9, ai). The action of (90,91, ... ,9r) E G(ijp) is slightly trickier to describe. To do so let us suppose that for each i ~ 1 we have the following integrality conditions -1

• 9i

",op EVB,Wi'

-1

• 90 9i E

",op VB,Wi'

I

•

mi-mi Wi 9i

E

//'lop VB,Wi'

Under these assumptions we will define a morphism (9i) : XUp,m ~ XUP,m "

It will send (A,.x, i, rjP, ai) to (A/ (C EB Cl.. ),pval p(go) >., i, rjP, ai o 9i), where

• C1 C cA[W~l] is the unique closed subscheme for which the set of a1(x) with x E 91All/All is a complete set ofsections;

CHAPTER III. SOME SIMPLE SHIMURA VARIETIES

110

• for i > 1, Ci = ai(giAi/Ai); • C

= (onF,w 00

F,UJ

C 1) EB IJ:7z=2 ffir C,z C A[u-va1p(go)1',

• C1. is the annihilator of C C A[u-va1p(go)1 inside A[(uc)-valp(go)l under the À- Weil pairing;

• pvalp(go)À is the polarisation A/(C EB C1.) -+ (A/(C EB C1.))V which makes the following diagrarn commute

AV t

(A/(C EB C1.))V; • ai o g1 : w~m~ Au/ An -+ (éA[wFl/Cd(S) is the homomorphism making the following diagrarn commute mi

---+

éA[wFl/C1(S)

w~m~ glAn/g1An

---+

mi (éA[wFlIC1)[W 1 1](S)

w1m1 An/g1 An

---+

(éA[wrn 11/C1)(S)

w1m1 An/ An

~

éA[wrn 11(S);

W~

1

An/ An

t t

t

t t

t

• for i > 1, ai o gi : w~m; Ai/ Ai -+ A[wi"'l/Ci is the homomorphism making the following diagram commute I

w~mi Ai/Ai

t

w~m; giAd giAi

t

---+

A (wi"'lICi

t

---+ (A[wi"'l/Ci)[w~;l ~

t

W i-miA i / gi Ai

~

A(W~illCi

Wi-miA i /A i

-4

Q'

A[w~il·

t

t

It is tedious but straightforward to check that this does define an action. We see that (p-2 ,p-1, ... ,p-1) acts in the sarne way as p E G(AOO,P) and so acts invertibly on the inverse system. Thus this definition can be extended to the whole oí G(Qp). We also see that on the generic fibre (Le. over Fw) this definition (when it makes sense) agrees with the action previously defined. (A less tedious argument is to first note that this definition

111

IlI.4. INTEGRAL MODELS

coincides with the previously defined action on the generic fibre and then use the fact that the generic fibre is Zariski dense in XUp,m to check the first two assertions. That the generic fibre is indeed dense follows at once from lemma 111.4.1 below.) We next establish some important pieces of notation. We will let AI XUp,m denote the universal abelian variety. We write simply g I XUp,m for IBA. If s is a closed geometric point of XUp,m we willlet h(s) denote the height of g~t. We willlet XUp,m denote the reduction XUp,m xSpecOF.w Speck(w). We will let X~!,m denote the reduced closed subscheme of X UP,m which is the closure of the set of closed geometric points s with h(s) :S h. We will also let -(h) -[h] -[h-I] XUp,m = XUp,m - XUP,m' The action of G(AOO) on the inverse system of the XUp,m takes the in-(h)

verse system of locally closed subschemes X UP m to itself (because they are defined in an invariant manner). ' Lemma 111.4.1 Throughout this lemma we suppose that UP is sufficiently small. Let m, mi and mil be r-tuples of non-negative integers with mI = O and m~ = mi for i> 1. Let s be a closed point of X UP,m XSpeck(w) k(w)ac and fix an isomorphism g~ -=+ ~Fw,n-h(s)'

1. The formal completion of XUp,m xSpecOF.w Spec Opll< at s is isomorphic to the universal formal deformation space for lhe Barsotti- Tate OF,w-module gs. Thus we get an identification (XUp,m

XSpecOF.w

Spec Op::,,)~ ~ Horn (TQs, ~Fw,n-h(s));

while (X~p(~~ xSpeck(w) Spec k(w)ac)~ is identified to the closed formal subscheme Horn (Tgs, ~Fw,n-h(s)) C Horn (Tgs, ~Fw,n-h(s))' -(h)

2. XUP,m/SpecOF,w is smooth. Moreover each XUp,m/Speck(w) is either empty or smooth of dimension h. 3. The closed points of XUP,m' xSpeck(w) k(w)ac above sare in natural bijection with the surjective homomorphisms

8: W-m~Al1/All ~ g~t[wm~l(k(s)). We will write S8 for the point corresponding to 8. Then we can identify the formal completion of XUp,m' xSpecOF.w SpecOp::" at S8 with Horn (w-m~Tgs, ~Fw,n-h(s))

XSpf RFw.n-h(s)

Spf RFw,n-h(s),m~,

CHAPTER III. SOME SIMPLE SHIMURA VARIETIES

112

such that the morphism

corresponds to the natural morphism Horn (w-m~ Tgs, EFw,n-h(s))

X Spf RFw.n-h(s)

Spf RFw,n-h(s),m~

.J..

Horn (Tgs, EFw,n-h(s))'

-(h(s)) Moreover the formal completion of XUp,m' at Só corresponds to the closed formal subscheme Horn (w-m~Tgs, EFw,n-h(s)) inside Horn (w-m~Tgs, EFw,n-h(s))

XSpf RFw.n-h(s)

Spf RFw,n-h(s),m~'

4. XUp,m' /OF,w is regular and ftat. (h) /k()' 5. X UP,m' W ~s smoo th an d th e morp h'~sm X(h) UP,m' -t X(h). UP,m M fi nz"t e and ftat of degree #GLn(OF,w/Wm~ )/#GLn_h(OF,w/Wm~). 6. Suppose that (UP)" C UP and that for all i we have the natural morphism

m~' ~ m~.

Then

X(UP)/I,m/l -----+ XUp,m' is finite and ftat of degree [UP : (UP)"] I1~=1 #GLn(OF,w';Wr;-:' OF,w.)/ (I1~=1 #GLn(OF,w';Wr;-:OF,w.)). If m~

= m~

then this morphism is in fact etale.

Proof: First of alI it is standard that (XUp,m xSpecOF.w SpecOFnr)~ is the formal deformation space for (r + 2)-tuples deforming (As, Às, ai,s) (where i > 1). By the Serre-Tate theorem this is the sarne as deformations of the (r+2)-tuple (As [POO], Às, is, ai,s)' As À : As[u OO ] ..:::t As[(uC)OO] we see that this is the sarne as deformations of the (r + l)-tuple (As [u oo ], is, ai,s). As As[wF] is ind-etale for i > 1 it has a unique deformation over any Artinian local ring with residue field k(s) as does ai,s' Thus we need only consider deformations of the pair (As[w OO ], is). As OB,w ~ Mn(OF,w) this is the sarne as deformations of gs = €As[w OO ] with its OF,w-action. This proves the first assertion of the lemma. The rest of the first part of the lemma follows from the discussion before lemma 11.1.3 and from corollary 11.1.4. The second part of the lemma follows from the first.

i;,

IlI.4. INTEGRAL MODELS

113

The first assertion of the third part of the lemma follows from lemma II.2.1. The second assertion follows from the discussion proceeding lemma -(h(s)) 11.2.4. The scheme X UP m' can be constructed as the reduced subscheme of -(h(s)) -(h(s)) A • the fi bre product of XUp,m and XUP,m' over XUp,m. Thus (X UP,m' )S5 IS the reduced formal subscheme of the fibre product over Horn (TÇ}s, ~Fw,n-h(s)) of Horn (TÇ}s, ~Fw,n-h(s)) and Horn (w-m~ TÇ}s, ~Fw ,n-h(s))

XSpf RFw,n-h(s)

(Here we make use of lemma 11.1.6.) Hence formal subscheme of Horn (w-m~ TÇ}s, ~Fw,n-h(s))

Spf RFw,n-h(s),m~ .

(xtp(~~, )~5 is the reduced

X Spf RFw,n-h(s)

Spf RFw,n-h(s),m~,

i.e. Horn (w-m~TÇ}s, ~Fw,n-h(s))' The fourth part now follows on applying proposition 4.3 of [Dr] because both these properties can be detected on formal completions at closed points. (If A is a noetherian local ring with maximal ideal m then dimA~ = dimA, m/m2 . ; m A /(m A )2 and A~/A is faithfully flat.) As for the fifth part, finiteness follows from lemma II.2.1. Smoothness and flatness follow from the computation of the formal completions. The degree can also be computed on formal completions: suppose that s is a closed point of xtj m x Spec k( w )ac . The number of closed points of

xtj,m' x Spec k( w )ac ~bove s is the number of surjective homomorphisms from (OF,w/Wm~)n to (OF,w/wm~)h. If Sõ is one of these points the degree

of (xtj,m' x Spec k( w )ac)~5 over (xtj,m x Spec k( w )ac)~ is the rank of h [m'] -(h) -(h). ~Fw,n-h W 1 . Thus the degree of XUP,m' over XUp,m IS (#OF,w/Wm~ )h(n-h)(#GLn(OF,w/Wm~ )/#GLn(OF,W/Wm~ )õ)

= #GLn(OF,w/Wm~ )/#GLn_h(OF,w/Wm~).

=

We can divide the proof of the sixth part into two cases: the case where = m~ and the case where UP = (UP)" and m~' = m~ for i > 1. ln the second of these two cases it is standard that the morphism is etale of the stated degree. ln the first case it follows from lemma II.2.4. D

m~

(We remark that one can use the results of Drinfeld's paper [Dr] to show that in fact if mI = O then X~~ m is smooth. We will not give details here as we will not need this resulto Ít seems to us an interesting question whether this remains true for mI > O.) The universal abelian variety A extends over XUp,m' If m = O then it is smooth over OF,w' We have seen in section III.2 that Ri(1T) is a Tate

114

CHAPTER III. SOME SIMPLE SHIMURA VARIETIES

twist of a direct summand of the cohomology of Am for suitable m. Thus we deduce the following lemma.

Lemma 111.4.2 Suppose that 7r is an irreducible representation of G(AOO) with 7rp,O and 7rw unramified. If 1 = p then R~ (7r) is crystalline at w. If 1 =j:. p then R~ (7r) is unramified at w, and if a is an eigenvalue of R~ (7r) (Frob w ) then a is algebraic and for each embedding of a in C we have lal 2 = (#k(w))w(~) . Now and in the rest of the book we assume that p =j:. l. ln this case, the lisse Qic sheaf C~ can be defined over the whole of XUp,m in exactly the sarne manner it was defined over the generic fibre XUP(m). If 9 E G(AOO) maps XUp,m to X(Up)/,m/ then again ~(gl) induces a morphism of sheaves 9 : g* C~

----+ C~

over XUp,m. The next lemma will be proved in section V.4 below. (It follows from corollary V.4.5.) -(O)

Lemma 111.4.3 The scheme X UP,m is non-empty. As a first application of this lemma we have the following corollary. -(h)

Corollary 111.4.4 The scheme X UP,m for h pure dimension h.

= 0, ... ,n -

1 is smooth of

As a second application we now provide the postponed proof of lemma II.2.7. Proof of lemma II. 2. 7: Choose a totally real field F+ with a place w above p such that F;}; e:! K and choose an imaginary quadratic field E in which p splits. We may then choose u, B, *, ( , ) and Ai as in section 1.7 and such that dimF B = g2. AIso choose a sufficiently small open compact subgroup UP C G(AOO,P). Let

and let x be any closed point of

X~~,(O,o, ... ,O)

XSpeck(w) Speck(w)ac C X o.

(The existence of x follows from the last lemma.) That x and the collection of the X m have the asserted properties follows from lemma II1.4.1. O Now let c)i denote the vanishing cycles for XUp,m C XUp,m. Then we have a spectral sequence Hi(X UP,m

x k(w)ac, c)i 0 C~) =? Hi+i(XUp,m x

F!C, C~).

IIIA. INTEGRAL MODELS

115

(See lemma 1.5.2.) If (g,O') E G(ACXl) x WFw then we have a natural map (g, a) : (g x Frob~(i ®.c ç ) is an admissible PM (A00)_ module. Then the above map gives an isomorphism of G(Aoo) x W Fwmodules (BOP)X

. _

.

lnd PMw(Fw)H~(X M, cJ>3 ® .cç) ---t

lim

-+UP,m

H~(X~; m x Speck(w)ac,cJ>i ®.c ç ). '

ln particular H~(X~J,m x Spec k(w)ac, cJ>i ® .cç ) is an admissible G(Aoo) x W Fw -module.

CHAPTER III. SOME SIMPLE SHIMURA VARIETIES

118

Proof:Recall the Iwasawa decomposition (B~)X It follows that

= PM(Fw)(O~w)x.

surjects to

This gives rise to maps

which are compatible with the maps

H~(X~j,m x Speck(w)ac,cpi 0Cç ) 4lim-tup,m H~(X~j,m x Spec k(w)ac, cpi 0 Cç ). As each of the maps

is an isomorphism, the lemma will follow on passing to the limit as long as we can check that the map

is an isomorphism. Injectivity is straightforward. As for surjectivity any

f E Ind~:~1:)H~(XM,cpi 0 Cç) being locally constant factors through one of the finite quotients PM(OF,w/W m1 )\(O~w/wml)X. Then f will be in the image ofInd some UP and

(C)0p B,w

/wrní)x I

PM(C)F,w/W rn1 )

m'. O

. _ H~(Xup

m'

"

.

M X

Speck(w)ac,cpJ í2JC ç ) for

Putting together the analysis of this section we obtain the following proposition.

IlI.4. INTEGRAL MODELS

119

Proposition 111.4.8 For h = O, ... ,n -1 choose a direct summand M h C Al1 of mnk n - h. Suppose that for each O ::; h ::; n - 1, O ::; j ::; n - 1 and O ::; i ::; 2h the G Mh (Aoo )-module H~(X Mh' ipi Q9 Lç) is admissible. Then we have an equality of virtual G (A00) x W Fw -modules [H(X, Lç)Z;]

=L

(_I)n-Hi+iln d ~::l;w) [H~(X Mh' ipi

Q9

Lç)].

h,i,i

We willlet (xt:'m)'\ (resp. X{;p,m,M) denote the formal completion of -(h)

-

XUp,m along the locally closed subscheme XUP,m (resp. XUP,m,M). The comparison theorem of [Berk3] implies that ipilxg'J,,,. (resp. ipiIXup,,,.,M) coincides with the formal vanishing cycles for x~2,m C (X&~,m)'\ (resp.

XUP,m,M C X{;p,m,M) (see section 1.5).

ln terms of

g/x~2,m (resp.

9 / X UP ,m,M) the formal completion is completely characterised by the fol-

lowing useful universal property.

Lemma 111.4.9 Suppose that X is a locally noetherian formal scheme over OF,w and assume p = O on xred. Suppose also that H/X is a BarsottiTate OF,w-module and that "f is a Drinfeld w m1 -structure on H/X. Moreover suppose that we are given a morphism f : xred --+ x~2,m (resp. X UP ,m,M) under which 9 with its canonical Drinfeld w m1 -structure pulls back to Hlx.ed with the Drinfeld w m1 -structure "flxred. Then there is a unique extension of f to a morphism X --+ X{;p,m,M under which 9 with its canonical Drinfeld w m1 -structure pulls back to H and "f respectively.

1:

Proof: Let (A, À, i, fjP, ai) be the pullback to xred of the universal object over XUp,m' Exactly as in the first paragraph of the proof of lemma II1.4.1 we see that deformations of (A, À, i, fjP, ai) to X are in natural bijection with the deformations of (f* g, f* ad. Thus we have a unique deformation over X of (A, À, i, fjP, ai) which gives rise to (H, "f). Call it (A', >.', i', (fjP)/, aD. Thus we have a unique morphism X --+ XUp,m such that the universal (r + 4)-tuple pulls back to (A', >.', i', (fjP)/, aD. This morphism restricts on x red to f and so must factor through X{;p mM' We see that is also the uni que such morphism extending f und~r ~hich (9, ad pulls back to (H, "f). D

1:

1

Chapter IV

Igusa varieties ln our setting there seem to be two natural analogues of the familiar Igusa curves in the theory of elliptic modular curves. We will call these Igusa varieties of the first and second kind. When we refer to these Igusa varieties we will refer only to the analogue of the ordinary locus on the usual Igusa curves. We have not looked at the question of whether our Igusa varieties admit natural smooth compactifications, although we feel this is a natural and interesting questiono ln the case of elliptic modular curves, the Weil pairing on the p-divisible group of an elliptic curve allows one to identify these two kinds of Igusa variety.

IV.1

Igusa varieties of the first kind

ln this section we introduce the more naive notion of Igusa variety of the first kind in the context of the Shimura varieties we are studying. To this end fix an integer h in the range O h n-1. AIso if m = (mI, ... , m r ) E Z;'o then let m denote (O,m2,'" ,m r ). - By an Igusa variety of the first kind

:s :s

I(h) IX (h) UP,m UP,m

we shall mean the moduli space for isomorphisms aIet .. ( W -m, O F,w 10)h F,w -(h)

XUP,m

~ -+

n et [W mI] .

':J

Thus lu";,ml x~j,m is Galois (and in particular finite etale, but not necessarily connected) with Galois group GLh(OF,wlwm1). The morphism (h) IUp,m

-+

-(h) XUp,m

factors naturally through

121

(h) IUp,m'

I

if mI

< mI and mi I

=

CHAPTER IV. IGUSA VARIETIES

122

IiJ':'m

mi for i > 1. The inverse system of the has a natural action of G(AOO,P) x GLh(OF,w) x n~=2(Ol,wJx. Let (Z x GLh(Fw))+ denote the sub-semigroup of elements (e,g) E Z x GLh(Fw) such that ti7 w to the integral part of -e/(n - h) times is integral. Then the inverse system of the m has an action of G(AOO,P) xQ; x (Z XGLh(Fw))+ xn~=2(B!:r.)X extending that ofG(AOO,P) x GLh(OF,w) x n~=2(Ol,wJx. We leave the action of G(AOO,P) to the reader. First suppose that (go,e,g~t,gi) E Q; x (Z X GLh(Fw))+ x n~=2(B!:r.)X also satisfies ~ . 1 h -1 Oop d -1 Oop • lor t > we ave gi E B,w; an go gi E B,w;'

9

IiJ'j

• (g~t)-1 E Mh(OF,w) and golg~t E Mh(OF,w), • (n - h)w(go) ~ e ~ O,

.

> 1 we have w~;-m: gi E O~w" ,

• for i

ml-m~ et

gl E Mh ( OF,w ) .

• w1

Under these assumptions we will define a morphism t) ( go,e,gle,gi

: I(h) UP,m --+

I(h) UP,m"

It will send (A, À, i, 1f', a!t, ai) to (AI (CEElC.l ),pval p(go) À, i, 1f', a!t og~t, ai o gi), where we have set • C 1 C eA[w;nl) is the unique closed subscheme for which there is an exact sequence (O) --+ ker F-!1 c --+ C 1 --+ a~t(F~/O~,w[(g~t)-I]) --+ (O), (this makes sense as if d denotes the integral part of -e/(n - h) then ker F-!1 c :::) G~[Wd) and G~[wd) :::) a(F~/O~,w[(grt)-I]) (we are using the fact that (e,gd E (Z x GLh{Fw))+)); • for i

• C

> 1, Ci = ai(giAil Ai);

= (O~,w Q90F,w Cd EEl EB~=2 Ci

• C.l is the annihilator of C under the

À- Weil

C

C A[u-va1p(go));

A[u-va1p(go)) inside A[(uC)-Valp(go))

pairing;

is the polarisation AI(C EEl C.l) ---+ (AI(C EEl C.l))V which makes the following diagram commute

• pvalp(go)À

IV.l. IGUSA VARIETIES DF THE FIRST KIND

• a~tog~t : (W-;m~OF,W/OF,W)~(h)

123

-+ (eA[w1"']/Cd et is the homomor-

UP.m

phism making the following diagrarn commute

(w-;m~ OF,w/OF,w)h

---t

(eA[w1"'l/CI)et

---t

(eA[w1"'J/Cl)et[W~~ J

h /getOh W-1 m1 0 F,w 1 F,w

---t

(eA[w~lJ/Cltt

h /getOh W-1 m1 0 F,w 1 F,w

---t

eA[w~l Jet /art((Fw/OF,w)h[(g~t)-l])

(W 1m1 OF,w/OF,w)h

a ' ---4

t

F,w /getOh 1 F,w

w-m~getOh 1

1

t t t

t

t t t

e

eA[w~lJet;

• for i > 1, ai o gi : w;m; Ad Ai -+ A[wf'J/Ci is the homomorphism making the following diagrarn commute

w;m; Ai/Ai

---t

A[wf'J/Ci

wi •giAd giAi

---t

(A[wf'J/Ci)[w;n;J

Wi-miA i / gi Ai

---t

A[W;"i]/Ci

Wi-miAi /A i

--.!t

a·

A[W;"iJ.

-m'.

t t t

t t t

It is tedious but straightforward to check that this does define an action. We see that (p-2,p-r, ... ,p-l) acts in the sarne way as p E G(AOO,P) and so acts invertibly on the inverse system. Thus this definition can x (Z X GLh(Fw))+ X I1~=2(B~)x. (A be extended to the whole of less tedious argument is to use the compatibility described below with the action of G(AOO,P) x x FM(Fw) x I1~=2(B~)X on the inverse system of the XUP,m,M.) -/1 We will denote by Fr* the element

Q;

Q;

We see that 1.

--/1

Q; x Z x GLh(Fw) is generated by Q; x (Z X GLh(Fw))+ and Fr*

2. Fr*/1 :

IiJ'),m

-+

I~~,m

;

is just (Fr*)/1. (Note that according to the

definitions above Fr*/1 does take

IiJ'),m to itself.)

CHAPTER IV. IGUSA VARIETIES

124

Now fix j : Au ~ O},w with kernel M. This induces a homomorphism j* : PM(Fw) 9

~ f-----t

Z x GLh(Fw) (wodet(gIM),jogoj-I).

We will define a morphism j* : IiN,m -t

X UP ,m,M

such that

I(h) UP,m

~

XUp,m,M

-(h) XUp,m

(Fr*)h (n-h)"'l ----t

-(h) XUp,m

4-

4-

commutes. More precisely j* is the map which takes (A, À, i, rfP, a~t, ai) to . ;;;3J h (A (p"'l/l(n-h») ,A\ (p"'l/l(n-h») ,Z,'f/ ,aI, F m l!1(n-h) oa t.) were

(To see that aI is well defined and that it is a Drinfeld leveI structure use lemma 11.2.1.) Because

-

-(h)

XUp,m,M/XUp,m

is finite flat of degree

(see lemma lII.4.6), because XUp,m,M is smooth and hence normal (see lemma lII.4.6), and because the composite

is also finite flat of the sarne degree we see that j* is an isomorphism. Suppose that 9 E G(AOO,P) x Q; x PM(Fw) x I1~=2(B::r.)X and that j*(g) E G(AOO,P) x Q; x (Z X GLh(Fw))+ x I1~=2(B::r.)x. Suppose also that (UP)' :) g-IUPg. lffor each i we have mi »m~ then I(h) UP,m

4XUp,m,M

I (h)

(UP)',m'

4-

X(UP)',m',M

commutes. We now look at natural formal extensions of these 19usa varieties. ln particular we willlet (Ii;'J,m)" /(xi;'/rnY' denote the unique etale covering

Ii;'/m/ x~j,m (see [Berk2]). If t E Z::::o then we willlet (Ii;'J,m)"(t)/(I&':J,m)" denote the moduli space for Drinfeld w t _ structures on gO /(I&':J,m)A. with reduced subschemes

IV.l. IGUSA VARIETIES OF THE FIRST KIND

125

Lemma IV.1.I 1. The natural morphism (Ii;'/m)"(t) finite and ftat of degree #GLn_h(OF,w/Wt).

-----t

(Ii;'/m)" is

2. (IiF/m)'\(t)/(I&'!J,m)'\(t) is the unique etale cover with reduced sub-

schemes Ii;'/m/ x~2,m' 3. (IiF/m)"(t) has the following universal property. Suppose that X is a locally noetherian OF,w-formal scheme and assume p = O on xred. Suppose also that 1í/X is a Barsotti-Tate OF,w-module and that we are given a morphism f : x red -+ Ii;') m under which g pulls back to 1íl x red. Then we have an exact seque~ce

over X, with 1í o formal and 1íet ind-etale. Suppose finally that 'Y is a Drinfeld w t -structure on 1í o/ X. Then there is a unique extension of f to a morphism X -+ (Ii;') m)" under which g pulls back to 1í

1:

and the canonical Drinfeld wt-st:Ucture on to 'Y.

go /(IiF) m)"(t)

pulls back

'

Proof: The first part follows from corollary 11.2.5. The second part follows because

From the definition of (I&'!J,m)"(t) the third part reduces to the special case t = O. ln this case, by lemma 111.4.9, we obtain a unique morphism X -+ (Ii;') m)" under which g pulls back to 1í. The third part ofthe lemma now follo~s from lemma 1.5.8. O We willlet (GLn-h(Fw) x GLh(Fw))+ denote the set ofpairs (gO,get) in GLn-h(Fw) x GLh(Fw) for which there exists a scalar a E F,; such that both ag et E Mh(OF,w) and (agO)-l E Mn-h(OF,w). This is a subsemigroup of GLn-h(Fw) x GLh(Fw). There is a natural homomorphism from

G(AOO,P) X

Q;

r

xGLn-h(Fw) X GLh(Fw) xII (B:)X i=2

to

G(AOO,P) X Q; X Z X GLh(Fw)

xII (B:)X r

i=2

CHAPTER IV. IGUSA VARIETIES

126 under which

We will denote this map g f-t [g]. Under this homomorphism

G(AOO,P)

X

r

Q;

X

(GLn-h(Fw)

X

GLh(Fw))+

X

II(B:)X i=2

is taken to

G(AOO,P)

X

rIJ;

r

X

(Z

X

GLh(Fw))+

X

II(B~~)x. i=2

If w is a uniíormiser in OF,w then we willlet

-/1(n-h) Fr*ro - (1 p- /1 (n-h) " w- 1 1, 1) -, in

G(AOO,P)

X

Q;

r

X

(GLn-h(Fw)

X

GLh(Fw))+

X

II(B:) X • i=2

Then

-/1(n-h) - /1 (n-h) [Fr;" ] = Fr* , and G(AOO,P) X Q; X GLn-h(Fw) X GLh(Fw) X n;=2(B:) X is generated as a semi-group by G(AOO,P) X X (GLn-h(Fw) X GLh(Fw))+ X n;=2(B~)X

Q;

--/1(n-h) and Fr;" .

The inverse system oíthe (Ii;'':'m)'\(t) has a natural action oí G(AOO,P) X X (GLn-h(Fw) X GLh(Fw))+ X n;=2(B:)X, which is compatible via [ ] with the action oí G(AOO,P) X x (Z X GLh(Fw))+ X n;=2(B~)X on the inverse system oí the Ii;') m. We willleave the action oí G(AOO,P) to the reader and describe the a~tion oí x (GLn-h(Fw) x GLh(Fw))+ x (B:) x. To this end suppose that

Q;

Q;

Q;

n;=2

(gO,g~,g~t,gi) E

Q;

r

X

(GLn-h(Fw)

X

GLh(Fw))+

and that d -1 Oop • Elor z• > 1 we h ave gi-1 E Oop B ,w', an go gi E B ,w",

X

II(B:)X, i=2

IV.l. IGUSA VARIETIES DF THE FIRST KIND

• (g~t)-l • gÜ1g~

m1• w1

E

E

Mh(OF,w),

Mn-h(OF,w),

E

,

Mh(OF,w) and gü 1gi t

127

gl E M h(O F,w,)

m I et

Choose a E Z~o maximally such that wag~t E Mh(OF,w) and (wag~)-l E Mn-h(OF,w)' Finally also suppose that

• wt-t'+ago E Mn-h(OF,w)' We will define a morphism

(gO,g~,g~t,gi) : (I&~,m)"(t) ----+ (IiJ'J,m,)"(t'), which extends

° et ,gi )] : I(h), [(gO,gl,gl

UP m

----+ I(h) UP ,m"

Let e 1 be the unique closed subscheme of Q[wmI]/IiJ'j m for which there is a short exact sequence '

(O) ----+ ker F-hw(detg~) ----+

e 1 ----+ a!t(F~/oi,w[(git)-l]) ----+ (O).

To define the desired extension (go,g~,g~t,gi) of [(go,g~,git,gi)] it suffices (by the universal property of (I&~ m' )"(t')) to specify a lifting g' of 9 /e 1

from I&hj m to (IiJ'j m)"(t) togeth~r with a Drinfeld wt'-level structure on (Ç')o.' , We now explain the construction of g' and the Drinfeld w t ' -structure on (Ç')o. To do so fix a uniformiser W of OF,w' Note that we have an embedding et

(Fw/OF,w)h[(gi t )-l] ~ get[w a] ----+ g/gO[w a]. (By an embedding we mean a compatible system of embeddings over each closed subscheme of (IiJ'j m)"(t).) We also have a Drinfeld wt-structure w-aa~ on (Ç/gO[wa])o. We set g' equal to the quotient of (ÇfÇO[w a]) by (w-aa~)(F~-h /O~~h[( w ag~)-l])

+ a!t( (Fw/OF,w)h[(gi t )-l]).

This does not depend on the choice of w. By corollary 11.2.5 we see that the composite of w-aa~ with wag~ : (w- t' OF,w/OF,w)n-h y

(W-tOF,w/OF,w)n-h /( (Fw/OF,w)n-h[( wa g~)-l])

CHAPTER N. IGUSA VARIETIES

128

gives a Drinfeld w t ' -structure on {9')o. This Drinfeld w t ' -structure is also independent of the choice of tu. It is tedious but straightforward to check that this does define an action. We see that (p-2 ,p-l, ... ,p-l) acts in the sarne way as p E G(AOO,P) and so acts invertibly on the inverse system. Thus this definition can be extended to the whole ofQ; x (GLn-h(Fw) xGLh(Fw))+ xI1:=2(B~)x. (A less tedious argument is to use the compatibility described below with the action of G(AOO,P) x CQ; x PM(Fw) x I1:=2(B~)X on the inverse system ~/1(n-h) (h) of the X{)p mM.) Note that Fr~ maps (Iup m)A(t) to itself and defines a lifti~g of (Fr*)/1(n-h) , which is analogous to' the canonicallifting of Frobenius in the theory of elliptic modular curves. Now fix homomorphisms jet : An ~ 0i,w and jO : An ~ Or;.-;h such that jO EB jet is an isomorphism. Let M = ker jet. These choices define a Levi component L(jo ,jet) C PM, i.e. the elements of PM which also preserve ker jO. They also induce an isomorphism (3·0 ,3·et) *.. L (j°,r) 9

~ t---t

GLn-h(Fw) x GLh(Fw)

(j0 o 9 o (j0)-l, jet o 9 o (jet)-l).

If tu is a uniformiser in OF,w, we will define a morphism

(3.0 ,)·et ,tu )* : (I(h) UP,m )A ( ml)

----t

XA UP,m,M

which extends the morphism ().et)* : I(h) UP,m ----t X UP,m,M·

To define such a morphism it suffices (by lemma IIIA.9) to specify a deformation of the pair

.et)/I(h) (g (p ffl1h (n-h» , Fml/1(n-h) Oaet UP,m l o)

°

(h) m )A( ml. ) As a deformatlOn . of g (pffl1!t(n-h» we take g/g [w m1 ]. to (Iup Then ~e have the identification gO[w m1 ] X get[w m1 ] ~ (x,y) t---t

(9/g0[w m1 ])[w m1 ] tu- m1 x+y.

As a deformation of F ml!t(n-h) o a~t o jet we take (a~ o jO) EB (a~t o jet). (Note that over Ii:),m we are identifying g(pffl 1h (n-h» and g/gO[w m1 ] so h) ( ffl 1h (n-h» . that FmI f 1 (n- : g --+ g p corresponds to the natural proJection g --+ g/gO[w m1 ].) Lemma IV.l.2

1. The morphism (j0, jet, tu)* is an isomorphism.

129

IV.I. IGUSA VARIETIES OF THE FIRST KIND

2. Suppose that 9 E G(AOO,P) x Q; XL(jo,j"t) XI1~=2(B~)X and suppose that (j0,jet)*(g) E G(AOO,P) x x (GLn-h(Fw) x GLh(Fw))+ X I1~=2(B~)x. Suppose also that (UP)' :J g-IUPg and that for each i we have mi > > m~. Then

Q;

(If~~)/,m/)"(mD

(Ii}'J,m)'\ (md .J..

.J..

X{)p,m,M

X(UP)' ,m',M

commutes, where the vertical maps are (j0, jet , ro)* . Proof: The second part is formal. To prove the first part we will verify that X{)p,m,M has the sarne universal property as (Ii:'J,m,M)"(md. That is we will show that if X is a locally noetherian OF,w-formal scheme, if p = O on x red , if li/X is a Barsotti-Tate OF,w-module, if f : x red -+ XUP,m,M is a morphism under which 9 pulls back to (1i/liO[w m1 ])IXred and if'"Y is a Drinfeld w m1 -structure on 11.°/ X, then there is a unique extension of f to a morphism X -+ (I&'!1 m)" under which 9 pulls back to 1i/1l0[w m1 ] and the canonical Drinfeld ~t-structure aI ° jO on 9° / X{)p,m,M pulls back to ro- m1 '"Y. To see that X{)p,m,M has this universal property we use lemma III.4.9 and note that there is a natural bijection between

1:

• Drinfeld wm1 -structures '"Y on 110 / X • and Drinfeld wm1 -structures 8: W- m1A ll/All -+ (1i/1i0[w m1 ])[w m1 ] over X which restrict to aI ° jet on xred. This bijection sends 8 to ro m1 8 ° (ll::mlM/M) and '"Y to ro- m1 '"Y 0 jO +ãi\ where ãi t : (W-m1All/All) """* li et [w m1 ] is the unique lifting over X ofthe pullback from X{)p,m,M to x red of aI : (W-m1All/An) """* 9[w m1 ]et. O We willlet cJ>i(t)/I&'!1,m XSpec k(w) Spec k(w)ac denote the formal vanishing cycles for I&'!1,m C (Ii:'J,m)"(t). (If t

= mI then it follows from the last

lemma that (Ii:') m)"(t) is isomorphic to the completion of a proper scheme of finite type ov~r OF,w along a locally closed subscheme of the special fibre. ln general (Ii}') m)"(t) is etale locally isomorphic to (I&'!1 m,)"(t) with

m~ = t.

Thus cJ>i(t)iI&~,m xSpeck(w) Speck(w)ac is well defin~d.) Note that, if UP :J (UP)' and for each i we have mi :::; m~, then the restriction of cJ>i(t)/Ii}'J,m xSpeck(w) Speck(w)ac to If~)"m' XSpeck(w) Spec k( w )ac is canonically isomorphic to

CHAPTER IV. IGUSA VARIETIES

130

(see [Berk3]). Suppose that x is a closed point of Ii:/m x Spec k( w )ac and suppose that

ix : ~n-h .:t 9~.

Then we obtain a natural map

i; : «I&':!,m)"(t) XSpfOF,w Spf Op~r)~ ----t Spf RFw,n-h,t, and hence a homomorphism

Lemma IV.1.3

(j;)* : q,t,l,n-h,t

.:t q>i(t)x.

Proo/: We will let Spf R(Çx) (resp. Spf Rt(Ç~)) denote the universal deformation space for 9x (resp. 9~ with its (unique) Drinfeld w t leveI structure). Then we have

«(Ii:/m)"(t) XSpfOF,w Spf Opnr)~ e:< Spf R(Çx) XSpf Ro(g2) Spf Rt(Ç~) XSpf ;Fw,n-h Spf RF""n-h,t.

.:t Horn (T9x, EF""n-h)

As Horn (T9x, EF""n-h) and Spf RFw,n-h are formally smooth we see that Horn (T9x, EFw,n-h) XSpf RFw,n-h Spf RF""n-h,t

----t

Spf RF""n-h,t

induces an isomorphism on vanishing cycles (see lemma 1.5.6). The lemma follows. O The inverse system of sheaves

q>i(t)jI&':!,m xSpeck(w) Speck(w)ac has an action ofG(AOO,P) xQ; x (GLn-h(Fw) xGLh(Fw))+ XI1:=2(B::J:) X x WF", in the following sense. If r

(g,a) E G(AOO,P) x Q; x (GLn-h(Fw) X GLh(Fw))+ X II(B:;')X X WF", i=2 and if [g] : Ii:/m --t If:%), ,m' then for t

> > t' we get a natural map

(g, a) : ([g] X (Frob~«T))*)*q>(t')

q>(t)

x Spec k( w)ac . We' now wish to describe the action of (g, a) on stalks. Thus let x be a closed point of li:/m x Spec k( w )ac and let y = ([g] x (Frob~«T) )*)x, a

on

Ii;) m

----t

IV.I. IGUSA VARIETIES OF THE FIRST KIND

131

closed point of If~~)',m' x Speck(w)ac. Suppose also that j., : ~n-h -:'t 9~. If > t',

SpfR Fw,n-h,t

t

t

((Ii:';'m)"(t') XSpfOF,w Spf OF~r)~ commutes (the left vertical arrow being 9 x (Frob~(O"»)* and the right vertical arrow (g~, s'. For this it suffices to give compatible isomorphisms

over J&~),m,s' First note that a- 18 gives an isomorphism s: (Fro b*w )c-w(detÔ)~ ~ ~ I~ [ F-/l C ) . a -1 u: ~Fw,n-h -t ~Fw,n-h ~Fw,n-h a

AIso note that ~ ( 9 P ,gp,o, c, gw et , gWi )*gO . a .. gOlgo[ aF-c/l) -t

Thus for our isomorphism

« Frob*w )c-w(detó)E Fw,n-h )[WS')-::'t «gP , 9p,o, c, get w' 9Wi )*gO)[W

S ')

N.2. IGUSA VARIETIES OF THE SECOND KIND

135

we may simply take a- 1 8, followed by the map induced by the universal isomorphism

over J&"J,m,s' in turn followed by a. It is straightforward but tedious to check this is independent of the choice of a and does define an action. -/1 Note that the element (Fr* ,1) simply acts as (Fr*)/1. (So for instance on I&"J,m X Speck(w)ac it acts as (Fr*)/1 = (Fr*)/1 x (Frob~)-l.) We will be most interested in the part of this action which is an action of k(w)aC-schemes. To this end define

to be the set of elements

(8,')') E D;w,n-h x GLh(Fw) such that (w(det8),')') E (2 x GLh(Fw))+. AIso set

G(h) (AOO )

r

= G(AOO,P) x Q;

x D;w,n-h X GLh(Fw) X

II (B~)X i=2

and

G(h) (AOO )+ = G(AOO,P)

X

Q;

II (B~)x. r

X

(D;w,n-h

GLh(Fw))+

X

X

i=2

If we embed G(h) (AOO)+ into

G(AOO,P)

X

Q;

r

X

(2

X

GLh(Fw))+

X

II (B~)X

X

D;w,n-h

i=2

by sending (gP,gp,o,8,g';;,gw.) to (gP,gp,o,w(det8),g';;,gw" 8). ln this way G(h)(AOO) acts on the inverse system of the J&"J,m,s over k(w)ac. We note that this action is compatible with w o det : D;w,n-h --+ 2 and the action of

G(AOO,P)

X

Q;

r

X

(2

X

GLh(Fw))+

X

II (B~)X i=2

on the inverse system of the

Ii;:,m.

CHAPTER IV. IGUSA VARIETIES

136

If p is an irreducible admissible representation of D;'w,n-h over IQfc we

get a lisse etale sheaf :Fp/ J&':J,m,s coming from the restriction of p to

G(AOO,P) X 9z

ij such that for all b E B we have

À

o i(b)

for which there exist • an isomorphism V c>91Q> AOO'P ..:::t VP A of B c>91Q> Aoo'P-modules under which the standard pairing, ( , ), on V c>9 Aoo,p corresponds to a (AOO,p)X multiple of the À-Weil pairing on VP A, • an isogeny 1]~ : ~n-h ..:::t éA[wOO]O, • an isomorphism F~ ..:::t éVwA, and • for each j

> 1, an isomorphism Aj c>9z

p

ijp ..:::t VwjA of Bwj-modules.

Here, we call two triples (A, À, i) and (A', À', i') equivalent if there exists ---+ A' such that

')' E ijx and an isogeny a : A

152

CHAPTER V. COUNTING POINTS • ')' À

= a v o À' o a and

• aoi(b)=i/(b)oaforallbEB.

There is a natural map 71" : J(h)(k(w)ac) (A, À, i, rl, 'TJp,o, 'TJ~, 'TJ':v, 'TJWi)

----+

PIC(h)

I---t

(A, À, i).

The group G(h)(AOO) preserves and acts transitively on each fibre of 71". To describe the fibres more precisely we need a little more notation. Suppose [(A, À, i)] E PIC(h). Let = End ~(A),

• C(A,'\',i) • M(A,'\',i) • +(A,'\',i)

denote the centre of C(A,,\.,i) ,

denote the À-Rosati involution on

• H(A,'\',i)/ij

C(A,'\',i)'

and

denote the algebraic group such that, for any Q-algebra

R, we have H(A,'\',i)(R)

= {g E C(A,'\',i) 0Q R: gg+(A.À.i l E R X }.

We see that M(A,'\',i) is canonically determined by [(A, À, i)], and that both the pair (C(A,'\',i) ' +(A,'\',i») and the algebraic group H(A,'\',i) are determined by [(A, À, i)], but only up to H(A,'\',i) (ij)-conjugacy. The choice of some isomorphism V 0Q Aoo,p ~ VP A of B 0Q AOO,p_ modules under which the standard pairing, ( , ), on V o Aoo,p corresponds to a (AOO,P) x multiple of the À- Weil pairing on VP A, gives rise to an embedding H(A,'\',i) X

Aoo,p

'--t

G x Aoo,p.

Similarly the choice an isogeny ~n-h ----+ êA[wOO]O gives rise to a map

the choice of an isomorphism F~ ~ êVwA gives rise to a map C(A,'\',i),w

----+

Mh(Fw ),

and for each j > 1 the choice of an isomorphism of Bw;-modules Aj 0z p ijp ~ Vw; A gives rise to embeddings C(A,'\',i),w; '--t B!0.

Thus we get an embedding L(A,'\',i) : H(A,'\',i) (AOO ) '--t G(h) (AOO )

which is canonical up to G(h) (AOO )-conjugacy. The following lemma is easy.

V.l. AN APPLICATION OF FUJIWARA 'S TRACE FORMULA

153

Lemma V.1.2 As sets with right G(hLaction we have

The choice of t(A,À,i) (in its conjugacy class) and the isomorphism depend on the choice of a point in 7[-1 [( A, >., i)]. If this point is varied then there exists 9 E G(h) such that t(A,À,i) changes by conjugation by 9 and the isomorphism changes by left translation by g.

ln what follows we will often use z to denote an element of PlC(h), and will write Hz (resp. Mz, etc) for H(A,À,i) (resp. M(A,À,i) etc) for some (A, >., i) E z. Let cp E Cgo(G(h)(f~,CJo)+ jZ; X O~Fw,n_J. Any such cp can be written as a finite sum of the form cp =

L agchar

UV(m,O)gUV(m,O)

9

for some fixed UP and m (depending on cp). As always we can and will assume that UP is sufficiently small. By a fixed point of [UP(m, O)gUP(m, O)] we will mean a point x E J(h)(k(w)ac)j(UP(m, O) ngUP(m,O)g-l)

such that x = xg E J(h) (k(w)ac)jUP(m, O). This set appears to depend on g, not just on UP(m, O)gUP(m, O), but if we replace 9 by U1gU2 (with U1,U2 E UP(m, O)) the two sets are in natural bijection via x H Xu 11. (If U1gU2 = u~gu~, with u~,u~ E UP(m,O) as well, then x H X(U~)-l gives the sarne map.) We will denote this set defined up to canonical bijection Fix([UP(m, O)gUP(m, O)]). Suppose x is such a fixed point. Choose 9 E UP(m, O)gUP(m, O) and a point x E J(h)(k(w)ac) above x E J(h) (k(w)ac)jUP(m, O)

n gUP(m, O)g-l.

Then we see that

xg=xu for some u E UP(m, O). We have tr [UP(m, O)gUP(m, O)] I(:Fp ® Lç)x = tr (p ® ~)(gu-1) (see page 433 of [K03]). (We remark that the right hand side is indeed independent of the various choices. First if we replace x by xv for some v E UP(m, O) n gUP(m, O)g-l then gu- 1 is replaced by v- 1gu- 1v and so

154

CHAPTER V. COUNTING POINTS

the value of the trace is unchanged. Secondly if we replace 9 by U1gU2 and ii by ii(Ut}-1 then gu- 1 is replaced by u1gu-1ul1 and again the value of the trace is unchanged.) Again suppose that x E Fix([UP(m, O)gUP(m, O)]) and again choose 9 E UP(m, O)gUP(m, O) and ii E J(h)(k(w)ac) above

x E J(h) (k(wtC)jUP(m, O) n gUP(m, 0)g-1. Let z = 7r(ii). Then we can represent ii by an element y E G(h) (AOO), and we see that

yg = tz(a)yu for some a E Hz(Q) and some u E UP(m, O). We will show that the conjugacy class [a] of a in Hz(Q) depends only on x. We have to check independence of the following choices. • We could postmultiply a by an element of

But as Hz(lR) is compact modulo the centre this intersection is a finite group and so as UP(m, O) is sufliciently small we see that

• We could replace y by tz(b)yv with b E Hz(Q) and v E UP(m, O) n gUP(m,0)g-1. ln this case a is replaced by bab- 1 and u is replaced by v- 1U(g-1vg). • We could replace 9 by U1gU2 and y by yu 11 with U1,U2 E UP(m, O). Then a remains unchanged and u is replaced by UU2. • We could preconjugate tz by b E Hz(Q) and postconjugate by g' E G(h)(AOO) while replacing y by g'y. Then u is unchanged and a is replaced by b- 1ab. Thus we may write [a(x)] for this conjugacy class. Notice that tr [UP(m, O)gUP(m, O)]I(Fp 0 Lç) = tr (p 0 ~)(tz(a(x))), because gu- 1 = y-1tz(a)y. Now we ask the converse question: given a E Hz(Q) how many points x E Fix([UP(m, O)gUP(m, O)]) are there with [a (x)] = [a]? One may check that the answer is the cardinality of the double coset space

V.l. AN APPLICATION OF FUJIWARA'S TRACE FORMULA

155

where

x

= {y E G(h) (AOO )

:

y-1tz([a])y n gUP(m, O)

i- 0}.

If a,b E Hz(Q) with both y-1tz(a)y and y-1tz(b)y E gUP(m,O) then

tz(a- 1b)

E

y-1gUg- 1y and so (because UP is sufficiently small and HAIR)

is compact modulo its centre) we see that a = b. We deduce that the number of x E Fix([UP(m, O)gUP(m, O)]) with [a(x)] = [a] is also given by

#(tAZH z (a)(Q))\X' jUP(m, O) n gUP(m, 0)g-1) where

x' = {y E G(h) (A

OO )

:

y- 1tAa)y E gUP(m, O)}.

A similar argument shows that for any y E G(h) (AOO) we have

tAZH.(a) (Q))y(UP (m, O) n gUP(m, 0)g-1) = z (a)(Q) tz(b)y(UP(m, O) n gUP(m, 0)g-1),

= IlbEZH

and so the number of x E Fix([UP(m, O)gUP(m, O)]) with [a(x)] given by

= [a] is also

vol (UP(m, O) n gUP(m, 0)g-1 )-1 vol({y E tz(ZHz(a)(Q))\G(h) (AOO) : y- 1tAa)y E gUP(m, O)}), where we use any Haar measure on G(h)(AOO) and where we use a Haar measure on tz(ZH.(a)(Q)) which gives each point volume 1. This can be rewritten

where again the measure on tz(ZH.(a)(Q)) gives each point volume 1 and where the Haar measures on the other groups are arbitrary as long as they are chosen consistently for each occurrence of a given group. This appears to depend on the choice of 9 E [UP(m, O)gUP(m, O)]. Adding the formulas for 9 running over a set of representatives for UP(m, O)gUP(m, O)jUP(m, O) and dividing by

= [UP(m, O) : UP(m, O) n gUP(m, 0)g-1], E Fix([UP(m, O)gUP(m, O)]) with [a (x)] = [a]

#(UP(m, O)gUP(m, O)jUP(m, O)) we see that the number of x is also given by vol (UP(m, 0))-1

aM(WO), so that Do is isomorphic to the centraliser of M in BOP, and let to denote the involution on BOP = EndB(Wo) and on Do induced by ( , }o. Let HIJA /Q (resp. Go/Q) denote the reductive algebraic group such that for any Q-algebra R the R-points of HIJA are the set of 9 E Do 01Q1 R such that gto g E R X (resp. 9 E BOP 01Q1 R such that gto g E R X ). Thus HIJA C G o and we have a natural isomorphism

There is also an isomorphism 'IjJ : Ht-v x

rote ..:; HIJA x rote

such that

• over àoo,p the above natural isomorphism and 'IjJ difIer by an inner automorphism, and hence that • for any u E Gal (rote /Q), 'IjJ and u('IjJ) difIer by an inner automorphism (see lemma 1.6.2). AIso let