Period Spaces for p-divisible Groups (AM-141), Volume 141 9781400882601

Citation preview

Annals of Mathematics Studies

Number 141

Period Spaces forp-divisible Groups by

M. Rapoport and Th. Zink

PRINCETON UNIVERSITY PRESS

PRINCETON, NEW JERSEY 1996

Copyright © 1996 by Princeton University Press ALL R IG H T S R ESE R V E D T h e Annals o f M athematics Studies are edited by Luis A. Caffarelli, John N . Mather, and Elias M . Stein P rinceton U niversity Press books are printed on acid-free paper and m eet the guidelines for perm anence and durability o f the C om m ittee on Production G uidelines for Book Longevity o f the Council on Library Resources Printed in the U nited States o f America by Princeton Academic Press 10 9 8 7 6 5 4 3 2 1

Library o f Congress Cataloging-in-Publication Data A CIP catalog record for this book is available from the Library o f Congress IS B N 0-601-02 782-X IS B N 0 -691-02781-1 (pbk.)

T h e publisher would like to acknowledge the authors o f this volum e for providing the camera-ready copy from which this book was printed

C ontents In tro d u ctio n ...................................................................................................

vii

1. p—ad ic sy m m e tric d om ains ................................................................

3

2. Q u a si-iso g e n ies o f p -d ivisib le G roups .........................................

49

3. M o d u li sp aces o f p -d iv isib le grou p s .............................................

69

A p p en d ix : N orm al form s o f la ttic e chains ....................................... 131 4. T h e form al H ecke c o r r e s p o n d e n c e s ................................................ 197 5. T h e p erio d m orp h ism and th e rigid —a n a ly tic coverings . .. 229 6. T h e jp-adic u n iform ization o f S him ura varieties .......................273 B ib lio g r a p h y .....................................................................................................317 In d ex ....................................................................................................................323

Introduction Let E be a p-adic field and let QE be the complement of all ^-ratio n al hy­ perplanes in the projective space P d_1. This is a rigid-analytic space over E equipped with an action of GLd(E). Drinfeld [Dr 2] has constructed a system of unramified coverings QE of to which the action of GLd( E) is lifted. These covering spaces are interesting for at least two reasons. Firstly, these spaces can be used to p-adically uniformize the rigid-analytic spaces corresponding to Shim ura varieties associated to certain unitary groups. This uniform ization looks formally very similar to the complex uniformization by the open unit ball which gives rise to these Shimura varieties. Sec­ ondly, Drinfeld conjectured th at the ^-adic cohomology groups with compact supports H cl {£ld E £ i= p, give a realization of all supercuspidal representations of GLd( E) which would give a construction of these repre­ sentations analogous to the construction of the discrete series representa­ tions of semi-simple Lie groups through L2-cohomology (Griffiths, Schmid, Langlands, ...). In the present work we generalize Drinfeld’s construction to other p-adic groups. This construction is based on the moduli theory of p-divisible groups of a fixed isogeny type. The moduli spaces obtained in this way are formal schemes over the ring of integers O e whose generic fibres yield rigid-analytic spaces generalizing QE . The covering spaces are then ob­ tained by trivializing the Tate modules of the universal p-divisible groups over these formal schemes. Furthermore, we show how these spaces may be used to uniformize (an open part of !, see below) the rigid-analytic spaces associated to general Shimura varieties. We will also exhibit a rigid-analytic period m ap from the covering spaces to one of the p-adic symmetric spaces associated to the p-adic group. vii

viii

IN TR O D U C TIO N

Before describing in some more detail our main results we sketch the back­ ground of the problems considered here and our m otivation. The subject of p-adic uniformization starts with the paper of Mumford [M2] which was inspired by T ate’s work on the uniformization of elliptic curves with mul­ tiplicative reduction over a discretely valued field. In this paper Mumford introduced the one-dimensional formal scheme ClE and showed th a t an al­ gebraic curve with completely split reduction over SpecOE is uniformized by a suitable subset of 0 E . Cherednik [Ch] discovered th a t Shim ura curves associated to quaternion algebras which ramify at the prime p adm it a padic uniformization in the sense of Mumford by the whole of 0?E . Drinfeld [Dr2] subsequently gave an algebro-geometric proof using the m oduli the­ ory of p-divisible groups. In his paper Drinfeld formulates for any d > 2 a moduli problem of p-divisible groups and shows th at it is representable by the formal scheme 0,E , a higher-dimensional analogue of M um ford’s for­ mal scheme which had been introduced by Deligne and Mustafin [Mu]. For higher-dimensional Shim ura varieties a p-adic uniformization by 0 E is pos­ sible only in rare cases, comp, theorem IV below (comp. (6.50), cf. also [Rl]). For instance, the naive hope th at if the group giving rise to the Shim ura variety is anisotropic at the place p one should have uniformiza­ tion at the places of the Shimura field lying above p turns out to be quite false, as was first observed by Langlands [La]. Indeed, the special fibre of the Shim ura variety usually is not totally degenerate, comp. [Zl], [Rl]. A closely related observation is th at there may be infinitely many isogeny classes in the special fibre. This explains why our theorem III below which applies to a general Shimura variety exhibits a uniformization only of the tubular neighbourhood of a fixed isogeny class. Another m otivation for us was Drinfeld’s conjecture on the ^-adic coho­ mology groups of Od E . This conjecture was made more precise by Carayol [Ca]. His version also involves an action of the multiplicative group of the division algebra D with center E and invariant 1/d on the covering space Qe . It roughly states th at the resulting action of the triple prod­ uct W e x D x x GLd(E), where W e denotes the Weil group of E, is a Langlands correspondence. There recently has been a flurry of activity con­ cerning this conjecture (we mention the work of H. Carayol, of G. Faltings, of A. Genestier and of M. Harris). Carayol [Ca] also pointed out th a t a sim­ ilar conjecture can be made in the case where the Drinfeld moduli problem is replaced by the formal deformation problem of Lubin and Tate. Shortly

IN TR O D U C TIO N

IX

after Kottwitz form ulated a very elegant recipe for such correspondences for arbitrary reductive p-adic groups, cf. [R2], §5. From this point of view our construction of the covering spaces yields the rigid-analytic spaces for which his recipe should describe their Aadic cohomology. The conjecture of K ottwitz is the analogue in this purely local context of the global problem of determ ining the reciprocity laws describing the correspondence between autom orphic representations and ^-adic representations of Galois groups of num ber fields defined by Shimura varieties [Ko2]. The third m otivation for us was to elucidate in this context the role of p-adic period morphisms. This subject starts with Dwork’s investigation (comp. [Kal]) of the formal deformation space of an ordinary elliptic curve (comp, also [Ka 3], [DI]). His period morphism 7r maps the open unit disc D to the affine line A 1 and is given by the famous formula 7r*(r) = log q where q = T + 1 in term s of the coordinates r on A 1 resp. T on D. It de­ scribes the variation of the Hodge filtration of the deformed elliptic curve. Grothendieck [Gr2] introduced a new point of view through his rigidity the­ orem for p-divisible groups up to isogeny. The rigid-analytic point of view (which was present in Dwork’s original work) was re-introduced by Gross and Hopkins [HG2] when they defined a period m apping in the case of the formal deformation space of a supersingular elliptic curve. Their period m orphism m aps the open unit disc to the projective line. Although it can­ not be expressed in term s of elementary functions, a great deal is known about it, comp. [HG2], [Yu]. In the general case the period morphism maps one of our covering spaces to a Grassmann variety and describes the varia­ tion of the Hodge filtration induced by the universal p-divisible group. Our construction of it is closest in spirit to Grothendieck’s approach. In addi­ tion, the question of determining the image of a period m orphism touches on one of the fundam ental open problems in the domain of p-adic cohomol­ ogy, namely the conjectures of Fontaine [Fo2]. They constitute the p-adic analogue of R iem ann’s theorem characterizing the classical periods coming from abelian varieties. Assuming his conjectures to hold it turns out th at the image is a p-adic symmetric space, either in the more elementary sense as defined by Fontaine’s condition, or as defined by van der P ut and Voskuil [PV] through geometric invariant theory (they are identical, as proved by Totaro). We will now give an overview of our main results. We will first describe the moduli problems of p-divisible groups and the represent ability theorem which

X

IN TR O D U C TIO N

yields the formal schemes generalizing above. Next we will describe the covering spaces and the rigid-analytic period morphism. Finally we shall explain our non-archimedean uniformization theorems for Shim ura varieties. To formulate our represent ability theorem we introduce some notations. We fix a prime number p. If O is a complete discrete valuation ring of unequal characteristic ( 0,p) we denote by N il po the category of locally noetherian schemes S over Spec O such th at the ideal sheaf p • Os is locally nilpotent. We denote by S the closed subscheme defined by p - O s . A locally noetherian formal scheme over S p f O will be identified with the set-valued functor on N i lp o it defines. A morphism X —►y of formal schemes is called locally formally of finite type if the induced morphism T red —» Tred between their underlying reduced schemes of definition is locally of finite type. In w hat follows we call a quasi-isogeny between p-divisible groups X and Y over a scheme S £ N i l p z p an isogeny m ultiplied by a power of 1/p. The m oduli problems of p-divisible groups which we want to consider are of two types. The type (EL) will param etrize p-divisible groups with endomorphisms and with level structures within a fixed isogeny class. The type (PEL) will param etrize p-divisible groups with polarizations, endomor­ phisms and level structures within a fixed isogeny class. The m oduli prob­ lems depend on certain rational and integral data which we now formulate in both cases in a simplified form where the level structures are absent. Let L be an algebraically closed field of characteristic p and let W ( L ) be its ring of W itt vectors. Let K q = Ko(L) = W ( L ) Q and let a be the Frobenius autom orphism of K q. C ase (EL): The rational data consists of a 4-tuple (5 , V, 6, p), where B is a finite-dim ensional semi-sim ple algebra over Qp and V a finite left 5 -m o d u le. Let G = G L b (V) (algebraic group over Qp). Then b is an element of G ( K q). The final datum p is a homomorphism G m —►G k defined over a finite extension K of K q. Let V ® q p K — 0 Vi be the corresponding eigenspace decomposition and V 2K — 0 z>j V% the associ­ ated decreasing filtration. We require th at the filtered isocrystal over K , (F ® q p K o, 6(id(g)cr), V£), is the filtered isocrystal associated to a p-divisible group over Spec O k ([G rl], [Fol], [Me]). The integral data consists of a maxim al order O b in B and an O b -lattice A in P . C ase (PEL): In this case we assume p ^ 2. The rational data are given by a

IN T R O D U C T IO N

XI

6-tu p le (B , *, V, ( , ), b,p). Here B and V are as before. Furthermore, B is endowed with an anti-involution * and V is endowed with a non-degenerate alternating bilinear form ( , ) : V q p V —>Qp such that (dv, v') — (v, d*v'), d G B. The remaining d ata are as before relative to the algebraic group G over Qp whose values in a Qp-algebra R are G(R) = {g E G L B ( V ® R); (gv, gvf) = cfoX*, t/), c{g) G R x }. We require th at the rational data define the filtered isocrystal associated to a p-divisible group over Spec O k endowed with a polarization (= symmetric isogeny to its dual). The integral data are as before. We assume th at O b is stable under * and th at A is self-dual with respect to the alternating form (,)• In either case let E be the field of definition of the conjugacy class of p, a finite extension of Qp contained in K. Let E = E . K q, with ring of integers Oft. The representability theorem in rough outline may then be formulated as follows (3.25). T h eo rem I We fix data of type (EL) or (PEL). Let X be a p-divisible group with action of Ob over Spec L with associated isocrystal isomorphic to (V^Qp Ko, 6(id , T*) semi-stable if for every subobject (V7, J7'*) ^ (0) we have W ) < »(V). Therefore, (V, is weakly admissible if and only if it is sem i-stable and f i( V) = 0. The following proposition is the analogue of the canonical filtration of H arder-N arasim han-Q uillen-Tjurin in the context of vector bundles. The proof of this proposition is almost word - for - word the same ([HN]), the m ain point being th at for a m orphism of filtered isocrystals V' V which induces an isomorphism of the underlying vector spaces we have n( V' ) < f i ( V ), and will therefore be om itted. P r o p o sitio n 1.4 (Faltings) Let V = (V, be a filtered isocrystal over K . Then V possesses a unique decreasing filtration by subobjects V # para­ metrized by Q, called its canonical filtration, with the following property.. Let K“ + = slope a.

yl3- Jf va+ * V a > then V a/ V a+ is semi-stable of H N -

Furthermore, i f V a ^ V, then p ( v a) > K V ) . In particular, V is semi-stable if and only if its associated canonical filtra­ tion is trivial. R em ark s 1.5 (i) It is obvious th at any morphism of filtered isocrystals over K is strictly compatible with the canonical filtrations. The canonical filtration is also compatible with passage to the dual and with the formation of the tensor product of two filtered isocrystals. This last fact follows from the theorem of Faltings mentioned above. (ii) We recall the definition of the Tate object l(n ), n E Z. In the context of isocrystals, l ( n ) - ( K 0, pni

kj

.

( 1 -2)

D e fin itio n 1.18 Let G be a reductive group. We call the pair (/i, b) admis­ sible, if one of the following equivalent conditions is fullfilled: (i) For any Qp -rational representation V of G the filtered isocrystal I ( V ) is admissible. (ii) There is a faithful Qp-rational representation of G, such that Z ( V ) is admissible.

p -A D IC S Y M M E T R IC DOM AINS

13

We make the same definition for weakly admissible. Proof: If V is a faithful representation, then any Qp-rational representation appears as a direct sum m and of V®n V ®m. Hence the equivalence of the conditions follows from the following facts (Fontaine [Fo2]). A direct sum of filtered isocrystals is admissible, iff each sum m and is admissible. A tensor product of admissible filtered isocrystals is admissible. The same is true for weakly admissible filtered isocrystals, but the last fact is then a theorem of Faltings (cf. (1.3)). □ 1.19 Let (//, b) be an admissible pair in a reductive algebraic group G. Con­ sider Fontaine’s functor T from the category of admissible filtered isocrys­ tals over K to the category of crystalline representations of the Galois group G a l ( K / K ). We denote by T v the composite of T with the natural forget­ ful functor to the category of finite-dimensional Qp-vector spaces. Let 1Z£V(G) be the category of finite dimensional rational representations of G over Q p . Then the composite of T v with the functor (1.1) defines a fibre functor T v o l : 1Z£V(G) — ►(Q p —vector spaces). Let Ver be the standard fibre functor. Then H om (Fer, T VX) is a right torsor under the group G and hence defines a cohomology class: c l s ( v , b ) € H \ Q p ,G). We have a conjecture to compute G is a connected reductive group, component of the L-group of G. be the group Gal(Qp/ Q p). Then diagram:

this cohomology class in the case th at as follows. Denote by G the connected We denote the center by Z(G). Let T Kottwitz [Ko2] defines a com m utative

H \ QP, G ) ----------------^ B ( G )

K

X* { Z { G ) v )t0r ------------. X * ( Z ( G ) r ).

14

CHAPTER 1

The left vertical arrow is an isomorphism. P r o p o s itio n tive group G cocharacter (i its restriction

1.20 Assume that the derived group of the connected reduc­ is simply connected. Let (//, 6) be an admissible pair. The defines in a canonical way a character of Z(G). to Z ( G ) V by (fi. Then we have:

We denote

cls(//, b) = K(b) — (fi1. Here we consider the left hand side as an element of X* (Z(G)^ors. We conjecture th at this proposition holds without the assum ption th at the derived group is simply connected. It shows th at for a fixed b the invariant cls(/i, 6) depends only on the conjugacy class of //, if it is defined. Before proving proposition (1.20) we note th at for a torus T over Q p all admissible pairs may be described in an elementary way: P r o p o s itio n 1.2 1 The following conditions for a pair ((i, b) with respect to the torus T are equivalent: (i) (/i, 6) is weakly admissible (ii) (i — v is orthogonal to all Qp -rational characters of T . Here we view v as an element of X*(T) Q. (Hi) For any Qp -rational character \ o f T , we have ordp x(b) = < X,/i > . (iv)

(/i, b) is admissible.

Proof: Clearly the conditions (ii) and (iii) are equivalent. The first condition implies the third. Indeed, let V be the one-dimensional representation given by x- Then the isocrystal TV = ( V 0 Ko, bcr) is isotypic of slopeordp(%(&)). The only non zero weight space of Vk is Vk j for j =< y, (i >. Next we have to show th at (iii) implies (i). Let V be an irreducible repre­ sentation of T. Let N = ® N a c*GQ

p -A D IC S Y M M E T R IC DO M AIN S

15

be the decomposition of the associated isocrystal into isotypic components. We need to verify: ^ a dim N a = y ^ i dim Vk ,i . a

(1.3)

The characters of T appearing in Vk form an orbit under the Galois group G al(Q p /Q p). Let x be a particular character of this orbit and N m x be the product of elements of this orbit. Clearly the right hand side of (1.3) is

Y

T(F) ^ T ( Q p).

16

CHAPTER 1

By local class field theory its restriction to the units in F defines a p-adic representation K(p) of G a l ( K o / K 0F) on V. This representation is crys­ talline and hence the image of some admissible filtered isocrystal N(V(f i)) = (N, 0 there exists a finite subset S C T ( F ) with

|J Wt(e) = (J Ht(e). t eT( F)

tes

□

p -A D IC S Y M M E T R IC DOM AINS

25

P r o p o sitio n 1.34 For every e > 0, X ( C p) \ n t (e),

t € T(F)

is an admissible open subset of X k hence so is also X € = X ( C p) \

(J n t (e), teT(F)

a finite intersection of subsets of the previous kind. On X ( C p) \ { J teT^F^ H t there is a structure of admissible open subset of X k (considered as a rigid space over K ) characterized by the fact that for any sequence c\ > 62 . . . —» 0 the covering

x ei c x £2c ... is an admissible covering. Proof: For the first assertion it suffices to establish the following fact. Let f a ( X 0, . . . , X n), a E A, be a finite set of homogeneous polynomials. Then the subset of P n , {x E P n ; \fa (x)\ > e, some a} is an admissible open. This follows easily from the fact th at for i — 0 , . . . , n the intersection of this set with the set of points x E P n(Cp) where the unim odular representative x satisfies |x2-| = 1 is obviously isomorphic to the following admissible open subset of the closed polydisc in A n : f

,

Xq

e, some a £ A> . J a

. . . —►0

X ( C p) \ W t (eO,t = 1, 2, . . . is an admissible open covering of X \ %t. To show the second assertion, let / : Y —►X k be a m orphism of an affinoid variety Y into X k such th at f ( Y ) C X ( C p) \ ( J h a v e to see th at / factors through X €i for suitable i. But by the above remark, for every t E T ( F ) there exists a minim al i(t) such th at

26

CHAPTER 1

f ( Y ) C X ( C p ) \ H t (ei{t)). By the remarks preceding lem m a (1.33) the function t j—►i(t) is continuous. Since T ( F ) is compact it follows th at it assumes its m aximum, which proves the assertion. □ 1.35 Let G be an algebraic group over Qp . We fix a conjugacy class of cocharacters fi : G m —» G and the corresponding flag variety E over E, cf. (1.31). Let A'0 = W ( F p) q and fix an element b £ G(Ie E s . Proof: We first prove (ii). Let V be a faithful Qp-rational representation of G. The set of conditions on E * £ E ( K ) to be weakly admissible is param etrized by the set of ^ -sta b le subspaces V( C Vo = V K q. Let

p -A D IC S Y M M E T R IC DOM AINS

27

V, = V ® Q ps and equip Vs with the cr-linear operator = 6(idy 0 o'). Then (Vb, X ] A r k ( VA)A

*

Noting th at the left hand side is equal to oo rk(J"' fl V') —N ■rk(V ') i= - N

for some N 0, we see th at R is indeed a closed subscheme defined by conditions of Schubert type. Furthermore, the above considerations show th a t the set of weakly admissible filtrations in JFi(Cp) is given by

^ ( C p; r = ^ i(cp)\ (J

Ht.

* G T ( Q P)

Applying proposition (1.34) we see th at this is an admissible open subset of QPs defined over Qp*. The result follows from intersecting Jri {Cp )wa with T C T \ 0 q p E. The last assertion of (ii) is obvious. We now turn to the proof of (i). The case when b is decent follows at once from (ii). Let G be connected. Then by Kottwitz [Kol] there exists g E G(Ko) such th at bf = gba(g )~1 is decent. Identifiying :F&(Cp)™a with g ZFb'{CP)wa, the result for b' implies the assertion for 6. Let G\ — GL (V). Then G is a closed subgroup of G\ and b induces a cr-conjugacy class bi which is decent since G\ is connected. Let b\ = 9ibcr{gi)~l be a decent element in 6i and let C T \ 0 # E be the corresponding admissible open subset of weakly admissible filtrations. Then T wa = { T 0 £ E) fl g \ T ^ a is an admissible open subset of T %e E, as required. The last assertion of (i) is trivial. □

p -A D IC S Y M M E T R IC DOM AINS

29

1.37 In the rest of this chapter we will consider T wa = as an admissi­ ble open subset of T g with the rigid analytic structure given by proposition (1.36). We call this subset the p-adic symmetric space or p-adic period do­ main associated to the triple formed by G , the conjugacy class {p} of p and the element b £ G(Ko). (We refer to (5.45) for the problem of lowering the field of definition to E). We are now in a position to state our basic conjecture on the existence of local systems on T ™a . Here we adopt the fol­ lowing definition of a local system in Qp-vector spaces on a rigid-analytic space X (it was suggested to us by J. de Jong). Recall (cf. e.g. [SS]) th at the big etale site in the category of rigid spaces has as coverings morphisms Y —►X which are etale and such th at there exists an admissible covering of Y by afhnoids such th at its image is an admissible covering of X . Lo­ cally constant sheaves in Z /p n-m odules are defined exactly like in algebraic geometry, and so are sm ooth Zp-sheaves. However, the category of local systems of Qp-vector spaces is defined starting with sm ooth Zp-sheaves, tensoring the Horn groups with Q p and then imposing descent with respect to etale coverings. (We mention th at this last condition is autom atic in the algebraic case, if the base is norm al.) In particular, a local system in Q p-vector spaces on X is not necessarily defined by a sm ooth Zp-sheaf on X , but merely by a sm ooth Zp-sheaf over an etale covering Y of X . The local systems in Qp-vector spaces on X form a 0 -category and every point x £ X defines a fibre functor in the category of finite Qp-vector spaces. Before stating our conjecture we mention th at an etale surjective morphism y X factors in a unique way as Y —►X ' —►X such th at Y —> X ' is an etale covering and X ' —» X is etale and a bijection on points (this fact was com m unicated to us by J. de Jong). We conjecture the existence of an etale bijective morphism ( J ^ ay —► and of a 0 -functor from the category 1Z£V{G) to the category of local systems in Qp-vector spaces on (lF%ay with the following property: Let (p , b) be a weakly admissible pair of G and J 7* £ J ^ a( K) the corresponding point. Then the pair (p,b) is admissible and the fibre functor on 1Z£V{G) which associates to a representation the fibre in X* of the corresponding local system is isomorphic to the fibre functor considered in (1.19) with respect to (p,b),

T v o T : 7Z£V(G) — ►(Qp-vector spaces).

CHAPTER 1

30

We note th at the conjecture th at every weakly admissible pair (/i, 6) is admissible is due to Fontaine. The validity of this conjecture would imply the existence of interesting etale coverings of ( T way . We shall exhibit in cases related to p-divisible groups plausible candidates for these coverings, cf. (5.34). For further remarks on the tower of etale coverings comp. (5.53) 1.38 We now introduce the algebraic groups G over Qp which will occupy us in this paper. We will distinguish two cases. The first case will be related to classifying p-divisible groups with given endomorphisms and level struc­ tures, the second one will be related to classifying p-divisible groups with given endomorphisms and polarization and level structures. Accordingly we will nam e these cases (EL) resp. (PEL). We will fix data of the following type. C ase (EL): Let Let Let Let

F B V G

be a finite direct product of finite field extensions of Qp . be a finite central algebra over F. be a finite dimensional 5-m odule. — G L b ( V ) considered as an algebraic group over Qp .

Case(PEL): Let F, B, V be as in case (EL). Let ( , ) be a nondegenerate alternating Qp-bilinear form on V. Let 6 i ^ 6* be an involution on B which satisfies: (bv, w) = (v, b*w),

v, w E V.

Let G be the algebraic group over Qp , whose points with values in a Q p-algebra R are given by: G(R) = {g £ G L s ( V R); (gv, gw) - c(g)(v, w),

c(g) £ R}.

We denote by F q the elements of F , which are fixed by the involution *. Let b E G(Ko) where again A"o = Ko(Fp). Then we obtain an isocrystal associated to the natural representation of G on V, N ( V ) = V ® K o, 7) is simple.

□

In this case J is the multiplicative group of the central division algebra with invariant = 1 (mod d) over F. R em ark 1.49 We mention briefly some variants of example (1.44). For simplicity we take F — Qp. (i) For the first variant we consider a central simple algebra of dimension d2 over Qp, with invariant = s(m od d). We again take V to be a free D module of rank 1. We fix an integer r, 0 < r < d — 1 which is a m ultiple of s modulo d,

p -A D IC S Y M M E T R IC D O M AIN S

43

i •s = r + j •d . As element b E G(Ko) = (D opp 0 Aro)x we take b = ir-j - IT, where II E D is as before. Then the corresponding isocrystal (N , K0 N

=

0 iV c

^

=

0^C

and = N{0ai n(A ^) = N^oa- i. In this case • II preserves the sum m and Ao corresponding to the chosen embedding and induces there a a linear endomorphism 82 > . . . > 6a > 0 with Y ,# = js . i= 1

We consider D -invariant filtrations with T aJrl — (O),^70 = V 0 Qp and dim T z — Sl • d, i = 1, . . . , a. We may analyze this example in the same way as the preceding ones and obtain the following results. The Qp-variety T may be identified with the variety of incomplete flags on the standard vector space Q^, (0) C T a C T a~ l C . - . C ^ C C / with dim T l — 8Z, i = 1, . . . , a. The subset of weakly admissible elements over K is characterized by the following condition: For any rational subspace W C Qp we have a

^ 2 dim { P n ( W K )) < (j s / d ) • dim W .

2—1 In this case J ~ GLd. 1.50 In the preceding examples the element 6 was basic. We now give one example (of type (EL)) where b is not basic. Let B = Qp and let V = Q^n * We denote the canonical base by e i , . . . , n and let V- = span < e i , . . . , en >, V+ = span < en+i , . . e2n > • As the element b E G(Ao) = GL2n(Ko) we fake b — p • id^_ ® idy+ . Then the slope decomposition of the isocrystal ( N y) has the form N = No ® N\ with N 0 = V + ® K 0 i N 1 = V - ® K 0. We consider the space T — Grassn (I/) of subspaces T of dimension n. It is then easy to see th at T is weakly admissible if and only if

p -A D IC SY M M E T R IC D O M AIN S

45

r n ( v + ® K ) = (o). In other words, T™0, may be identified with the big cell defined by V+, an open algebraic subvariety of F . It may be identified with the variety of splittings of the exact sequence of vector spaces, 0 -+ V+ -► V -+ VL -+ 0, i.e. with an affine space of dimension n 2 = dim Horn (V I, V+). In this case J is the Levi subgroup of G, J ( Q P) = GL(V+) x G L (V -). 1.51 We now wish to relate the weakly admissible subset to Geometric Invariant Theory. We assume th at G is connected. We fix a conjugacy class of cocharacters p and the corresponding projective algebraic variety F over E. We fix an invariant inner product on the Lie algebra of a maximal torus of G and use it to interpret p as a conjugacy class of a character. To this character there is associated an ample line bundle C on F which is homogeneous under the derived group G jfr , after perhaps replacing C by a positive tensor power. Let b £ G (K q) be such th at the set of weakly admissible points i n F g is non-empty. Let J be the corresponding algebraic group over Qp. Then J Pi Gder is a subgroup of J defined over Qp which will be denoted by J der. Then J i er is a subgroup of G dg r and hence acts on (F , £ )$ . For any maxim al Qp-split torus T C J der let F b(T )ss C be the set of points which are sem i-stable for the restriction to Tp, of the action of Jj?er on (fF, £)%. For any finite extension K of E , let F l s(K ) = f ) F b( T y s(I C) is, under the identifica­ tion of J with Gp(Vo, < , > ) a positive tensor power of the n -th exterior power of Vq. Choose a basis of Vo, • • •) with < e,*, e_*> < e i,e j>

= 1, i = 1, . . . , n = 0,

i ^ -j.

Let T be the diagonal torus and let A E X* (T) be a 1-param eter subgroup, A(t)ei — f ' e i ,

i E {±1 , . . . , ±n } ; r_; = - r 2-.

We investigate the Mumford criterion in case A lies in the positive Weyl chamber, i.e. ri > r2 > .. . > rn > 0. Let I C { ± 1 , . . . , ±n} and let L i = span {e2-; i E 1}

p -A D IC SY M M E T R IC D O M AIN S

47

be the subspace spanned by the corresponding basis vectors. Then L j is totally isotropic if I D (—1) = 0. Consider the self-dual standard flag fixed by T, (0) C L i C L 2 C . . . C L n C T _n C . . . C i _ 2 C L - i = V0 with L{

=

L{

Let C VK be a Lagrangian subspace and let I (To) be the set of jum ps in the intersection of To with the standard flag, i.e. p

e I (To) & 3 v e To n (L^ 0 K )

with v $ preceding member of the chain. Then it follows easily for the action of T on the corresponding points in the Grassmannian,

\im\(t)-F0 = Li(t„). Furtherm ore, A(t) operates on the fibre in L/(;f0) of the homogeneous line bundle defined by the n -th exterior power representation through the char­ acter £ r M, p G I(To)- Therefore the point corresponding to To satisfies M um ford’s criterion with respect to T, A if and only if Y r^ ° p£l(Fo) By convexity this condition then holds for all A in the closure of the pos­ itive Weyl chamber. It suffices to check this condition on the extrem al 1-param eter subgroups i.e. the fundam ental co-weights, A* := ( n = .. . = n - 1, ri+1 = . . . = rn = 0),

1 < i < n.

Therefore the above condition for all A in the closure of the positive Weyl chamber is equivalent to the condition

48

CHAPTER 1

Looking back at the definition of the flag and recalling th at (—I(^Fo)) H /(JFo) = 0 this condition is in turn equivalent to d im p S fl Li) < i/2,

i = 1 , . . . , n.

Since any totally isotropic subspace of Vq is conjugate under J ( Q P) to Li for suitable i (1 < i < n) and since J rSS( K ) is stable under the action of J ( Q p) it follows th at the points in ^ r5S(A") satisfy the condition appearing in the statem ent of proposition (1.52). Conversely, since all m axim al split tori and all 1-param eter subgroups may be conjugated under J ( Q P) into (T, A) as above with A in the closure of the positive Weyl chamber it follows th at all points satisfying the condition in proposition (1.52) lie in ZFSS(K). 1.54 We now discuss the other examples. In example (1.44) the group J is GL(V/) acting on the appropriate Grassmannian. The corresponding subset JrSS(K ) was determined in [PV], 2.8.2. and was found to be described by exactly the same condition as the one appearing in proposition (1.45). Something analogous applies to the last example of (1.49) (use [PV], 2.8.1.). The first example of (1.49) and example (1.47) are also com patible with theorem (1.52) Indeed, in these cases the group J is an inner form of G anisotropic modulo center so th at the condition describing !FSS(K ) is empty. It follows th at J rSS( li) = ^F(K) = JrU)a(K ). Finally, the non-basic example (1.50) may be treated in exactly the same way as (1.53) above.

2. Quasi—isogenies of p-divisible groups In this chapter we will define a moduli space for the quasi-isogenies of a given p-divisible group X. This moduli space will be a formal scheme over the W itt vectors. 2.1 Let us review some basic facts on formal schemes in the form needed here. Consider a preadmissible topological ring A . Let { X a } be a set of ideals of A th a t form a fundam ental system of neighbourhoods of 0. Then we define a contravariant functor Sp f A on the category of schemes S p f A (Z ) — lim Horn (Z , Spec A / X a ) a

This is a local functor, i.e. a sheaf for the Zariski topology on the category of quasicompact quasiseparated schemes. If the ring (A ,X a ) is adic, we will call S p f A an affine formal scheme. A local functor which has a covering by open subfunctors which are affine formal schemes, is called a formal scheme. If A is a preadmissible ring we may consider the category N i l p \ of schemes over S p f A. Then the category of formal schemes over S p f A is a full subcategory of the category of set valued sheaves on N i lp \ . 2.2 We call a morphism X —►y of formal schemes to be of finite type, etale, lisse, etc., if for any scheme Z and any morphism Z —►y the fibre product X X y Z is a scheme and X X y Z —►Z is of finite type, etale, sm ooth etc. in the usual sense. Let X be a formal scheme. Then there is a unique morphism X red —►X , where X red is a reduced scheme, such th at for any reduced scheme Z the following m ap is bijective 49

CHAPTER 2

50

Horn(Z, X red) — ►Horn(Z, X ) . X is called locally noetherian, if it is locally isomorphic to S p f A, where A is a noetherian adic ring. D e fin itio n 2.3 Let X and y be formal schemes that are locally noetherian. A morphism X —►y is called formally of finite type if X red 3^red is of finite type. The notion formally locally of finite type is defined in the same way. 2.4 If A is an adic noetherian ring and X is an affine noetherian formal scheme over S p f A, such th at X —►S p f A is formally of finite type, then there is a A-algebra A of finite type with a preadmissible topology {Xa }m L) Gtm+ 1 ,

Gt-m-}-l T

— Q>m

Q U ASI-ISO G EN IES OF p-D IV ISIB L E GROUPS

51

Moreover a i / a 2 is a A-module of finite type. We conclude by [EGA] loc. cit. th a t lim A / a m is an adic ring. □ We add a few remarks on isogenies of p-divisible groups. By a p-divisible group X over a scheme S we mean a Barsotti-Tate group in the sense of Grothendieck (see Messing [Me] ). D e fin itio n 2.6 A morphism f : X —+ Y of p-divisible groups over S is called an isogeny, iff f is an epimorphism of f.p.p.f. sheaves whose kernel is representable by a finite locally free group scheme. If S G N i l p z p, the kernel of an isogeny is of rank a power of p. If the rank is constant and equal to p h we call h the height of the isogeny. We have a converse to this definition. P r o p o sitio n 2.7 Let X be a p-divisible group over a scheme S. Let H be a finite locally free group scheme over S and H X a monomorphism over S. Then the f.p.p.f. sheaf X / H is a p-divisible group. Proof: Clearly the m ultiplication by p : X / H —►X / H is an epimorphism and X / H is a p-torsion group. We have to verify th at the kernel of the m ultiplication by p is representable by a finite locally free group scheme over S. Let us denote the kernel of multiplication by pn on X by X [n ]. By Oort and Tate [OT] we have th at H is a closed subscheme of X [n ] for big n (compare EGA IV 8.11.5). Hence for big n we get an exact sequence: 0 -+ H

X[n]

(X/H)[n] - + H - + 0 .

One knows th at the quotient X [ n ] /H is a finite locally free group scheme and th a t an extension of finite locally free group schemes in the category of f.p.p.f.-sheaves is again a finite locally free group scheme (see Grothendieck [Gr2]). Hence (X /H )[n) is a finite locally free group scheme. We finish the proof by writing the exact sequence: 0-

( X /H ) [ 1] -

(X /H )[ n + 1] —►(X/H)[n] - 0.

□ The m ultiplication by p on a p-divisible group is by definition an isogeny. It follows th a t the group Horns (X, Y ) is a torsion free Zp-module. Let us denote by Hom5 (X, Y ) the Zariski sheaf of germs of homomorphisms.

52

CHAPTER 2

D efin itio n 2.8 Let X and Y be p-divisible groups over a scheme S. A quasi-isogeny is a global section f of the sheaf Homs (X< Y) & Q, suck that any point of S has a Zariski neighbourhood, where pnf is an isogeny for a suitable natural number in. Let us denote by Q i s g s ( X , Y ) the group of quasi-isogenies from X to Y . Quasi-isogenies of p-divisible groups have the following well-known rigidity property. Let S f C S be a closed subscheme, such th at the defining sheaf of ideals J is locally nilpotent. Assume moreover th at p is locally nilpotent on S. Then the canonical homomorphism Qisgs ( X , Y ) — +Qisgs'(Xs',Ys>)

(2.1)

is bijective (Drinfeld [Dr2]). P r o p o sitio n 2.9 Let a : X —>Y be a quasi-isogeny of p-divisible groups over a scheme S. Consider the functor: F (T ) = {(f) G Horn(T ,S ) \ Y[n\ is the zero morphism. To show th at this last property is representable by a closed subscheme we prove: L em m a 2.10 Let a : M —►C be a morphism of Os-modules, on a scheme S. Assume that C is finite and locally free. Then the functor F (T ) = { G Hom(T, S) |

*a = 0}

is representable by a closed subscheme of S. Proof: We have an isomorphism H om (A f,£) = Hom(A^ 0 £*, Os)>

C = Horn(C ,O s ).

The m orphism a corresponds to a : M. 0 £* —►Os • The ideal Image a defines the desired closed subscheme.

Q U ASI-ISO G EN IES OF p-D IV ISIB L E GROUPS

53

2.11 Let X be a p-divisible over F p . Let W = W ( F p) be the ring of W itt vectors. We will define the functor of quasi-isogenies of X , and show th at it is representable by a formal scheme. Although we are only interested in the case of the field F p, it will be essential for the proofs to allow other perfect fields L of characteristic p. In this context we set W = W (L). We denote by a the absolute Frobenius autom orphism of W. The category N ilp w is the category of schemes S over Spec W such th at p is locally nilpotent on S. A scheme S E N ilp w may be viewed as a formal scheme with ideal of definition pO s • We denote by S the closed subscheme of S defined by the sheaf of ideals p O s • By the universal property of W itt vectors (Grothendieck[Gr2]) it is equivalent to give a morphism S —►S p f W or to give a morphism S Spec L . 2.12 We consider isocrystals over L (1.1). Our notation will differ a little from the first chapter. We write F = for the Frobenius morphism. We do this because we also need the Verschiebung V = p • $ - 1 . We define the dimension of an isocrystal N by the formula dim N = ordpdet V. We recall th at an isocrystal N is isoclinic of slope A E Q, if there is a W(L)~ lattice M C TV, such th at F 3M = prM , where s > 0 and r are integers, such th a t A — r j s. If TV is isoclinic of slope A we have the relation dim TV = (1 —A)height TV. We will call a sublattice M C N a crystal if it is stable under F and V. D e fin itio n 2.13 An isocrystal (AT, F) over L is called decent, if the vector space N is generated by elements n satifying an equation F sn = prn for some integers r and s > 0. R e m a rk s 2.14 Let us write N = V 0 W { L ) q for some Qp-vector space V. Let G = GL{V) considered as an algebraic group over Qp. Then we get a 0. As above we denote the height of N by h. Since N contains a crystal, we have s > r > 0. We may assume th at the field V fixed by crs contains L. The operator a = p s~rV ~ s acts on N (S)pv(L) W{L' ) (£)W(Lt^ W ( P ) = N p by the Frobenius autom orphism r over W( L ') on the last factor. The crystal M — M -f r(M ) -f . . . + r h~1(M) is by the previous proposition of the form M f W ( P ) . Since M + t ( M ) C V ~ 5(M ), we conclude th at length (M + t M ) / M < sd im N . Iterating this we get length M / M < (h — 1 ) 5 dim N . This number depends only on N (and L). In the general case let s be a common m ultiple of the dom inators of the slopes of N . Again we assume th at the fixed field V of a s contains L.

CHAPTER 2

56

Let TV = No 0 . . . 0 N t be the isotypic decomposition of N . We order the sum m ands such th at the slopes decrease: ro s

n_ s

rt_ s'

Since N is decent, we way assume th at F sn = pVin for each n E N . We prove by induction on the number of isotypic components th a t there exists an M f as claimed in the proposition, with the additional property th at it is a direct sum of isoclinic crystals. Let N > = N \ 0 . . . 0 Nt be the direct sum and M> C N >}p = N y ®w(L) W ( P ) be the image of M by the projection N p —> N >tp. We obtain an exact sequence 0 — ►M q — ►M — ►M y — ►0,

(2.2)

where the kernel is an isoclinic crystal Mo C Notp- By induction assum ption and the isoclinic case we may assume th at Mo and M y are obtained by the base change from crystals over L f and moreover th at M y is a direct sum of isoclinic crystals. If sequence (2.2) would split as an extension of crystals, we would obtain M by base change from a crystal M f over L ' . This would prove the proposition. Therefore it suffices to show th at the following is true. L e m m a 2.19 After push-out by pr° : Mo —►Mo the exact sequence (2.2) splits as a sequence of crystals. Proof: Let W = W {P ) and W[F] be the non-commutative polynomial ring (Fw = a(w )F). From the exact sequence 0 — ►M 0 — ►iV0 — ►No/Mo — ►0 we obtain an isomorphism Homiy[F](M > , N 0/ M 0) ~ E x t^ [F](M > , M 0) Hence the extension (2.2) corresponds to a W -linear homomorphism a : M y — ►No/Mo, which commutes with F. By assumption M> is generated as a W -module by elements m satisfying some equation F sm = prim } i = 1 , . . . , T Let

Q U ASI-ISO G EN IES OF p-D IV ISIB L E GROUPS

57

Of(m) == z be the image and z E No a representative. We assume th at k E Z is chosen such th at pk z — w £ Mo, but pk~ l z £ M q . By the equation F sm = pr*ra we obtain th at = pViw mod pkM q. On the other hand we have F s M q — pv°Mq. For k > r o we would get pViw = 0 mod pr°Mo which contradicts ro > r*. Hence we have & < ro, i.e. pr°Qf(m) = 0. This proves the lem m a and the proposition. R em ark s 2.20 (i) The hypothesis th at N be decent is indeed necessary for the conclusion of (2.18), as the following example shows. Let N = Qp with the standard basis ei, e2. We define the Verschiebung by the requirement V ei = ei, V e 2 = ae2, where a is a unit in Zp. Then (N, F) is decent, iff a is a root of unity. Let n be a positive integer and c G W ( F p) a unit. Consider the lattice in N 0 JT(Fp): M = W ( F p)(ex + p - nce2) + W ( F p)e2. Then M is a crystal, iff V M = M , i.e. cr(e)a = e (m odpn). An element satisfying this equality always exists. Let s > 0. It is easy to see th at the smallest lattice M s of the form M fs 0 p , W (F P) containing M is M s = W ( F p)(Cl + p~nte 2) + W (F p) p - n (e ~ (75(e))e2 + W ( F p)e2. We note th at e —cr5(e) = (7s(e)(as — 1). If A is not decent, we have for any s th at m s = ordp(a5 — 1) < 00. Hence we obtain for n > m s M s = W (F p)(ei + p -« 6 e 2) + W ( F p) p - n+m° e2. We get th at for any s the indices of M in M s can become arbitrarily large for a suitable choice of M , i.e. n.

CHAPTER 2

58

(ii) The content of proposition (2.18) may be interpreted as a statem ent about the B ruhat-Tits building of the general linear group over a p-adic field. As such it can be generalized to any connected reductive group over a p-adic field, comp. [RZ], [Rou]. 2.21 Having done these preparations we will write Ad as a union of repre­ sentable subfunctors. The last step is to exhibit Zariski open sets, which remain stable in this union. Let us start by giving an alternative definition of the functor Ad. Let X be a lifting of the p-divisible group X to S pf W( L) . Then a point of Ad with values in S E Nilpw(L) ls given by the following data: 1. A p-divisible group X on S. 2. A quasi-isogeny g : X s —►X of p-divisible groups on S. 2.22 We define the closed subfunctor Adn of Ad by the condition th a t pn g is an isogeny. The functor Adn is representable by the p-adic completion of a scheme locally of finite type over S. Indeed, Adn is a union of open and closed subfunctors Adn’m, which are given by the condition th at pn g is an isogeny of height m. To give such an isogeny is the same thing as to give a finite locally free group scheme G C X[m] 5 , which is of height m. Hence we see th at the functor Adn,m is representable by the p-adic completion of a closed subscheme of a Grassmanian variety associated to the algebra of functions on X(m ) . This proves the representability of Adn . In the sense of Zariski sheaves we have Ad = lim Adn . To prove the theorem we need still another representation of Ad as a union of representable subfunctors. To do this we define for any field extension P of L a quasi-metric on the set Ad(P). D e fin itio n 2.23 Let a : X —►Y be a quasi-isogeny of p-divisible groups over P. We define q(a) = heightpna, where n is the smallest integer such that pna is an isogeny. If P f is a field extension of P we have q(a) = q(ap'). L em m a 2.24 Let a : X —►Y be an isogeny of p-divisible groups over a scheme S. For any integer c the set of points s E S such that q(as) < c is closed.

Q U ASI-ISO G EN IES OF p-D IV ISIB L E GROUPS

59

Proof: We prove th at the set of points s E S', such th at ^(o;5) > c is open. The function q does not change if we m ultiply a by a power of p. Therefore we may assume th at a s is an isogeny, but p ~ l a is not an isogeny. Then there is a neighbourhood U of s, such th at p ~ l a t is not an isogeny for t E U . Let nt be the smallest integer, such th at pntat is an isogeny. Hence nt is nonnegative for t E U . Assuming th at height at is a constant function on U we find the result c < q(as) = height a s = height at < height pnta t = q(at ). D e fin itio n 2.25 Let a : X —» Y be a quasi-isogeny of p-divisible groups over P. We define d(a) = q(a) + q(a ~ 1). For two points of M ( P ) we define

d((x,e),(x',e')) = d(s'e-1) . If ra_|_ is the smallest integer such th at pm+a is an isogeny and ra_ is the smallest integer such th at pm~ a ~ 1 is an isogeny, we have d(a) = (ra+ + ra_)heightX . C o ro lla ry 2.26 Lemma 2.24 holds with q replaced by d. Because d((X, g), (X , p g )) = 0 the function d is not quite a m etric on M ( P ) . To get a m etric, we consider for k E Z the subfunctor M.[k) C Ad of quasiisogenies of height k. We set: height X —1

M =

]J

M{h).

h—0 It is easily checked th at the function d of definition (2.25) is a metric on M (P ). The proposition 2.18 may be reformulated as follows: P r o p o s itio n 2.27 There is a natural number c and a finite extension V of L, such that for any perfect field P containing V , and any point X E M ( P ) there is a point Y E M ( L ') , such that d (X ,Y p ) < c . We define for a natural number c a subfunctor M c of M consider the subfunctor A4(h) of quasi-isogenies of height h. We define height X —1

M =

]J

h—0

M (h).

CHAPTER 2

60

It is enough to show th at this functor is representable. Since m ultiplication of an isogeny by p does not change the value of the function d , M c ( S ) = {(X , g)

e M (S);

d(gs) < c

for

s E S}.

L e m m a 2.28 The functor M c is representable by a formal scheme, which is locally formally of finite type over Spf W (L ). Proof: Let M c(h) be the open and closed subfunctor of M c th at consists of points (X , q) such th at height g = h. Then M c is a disjoint union

M c = J J M c{h). hez The m ultiplication of g by p defines an isomorphism M c(h) —►M c(h + height X ). Therefore it is enough to show th at the following functor is representable by a formal scheme formally of finite type height X - 1

M e=

U h-0

We consider the functor M.™ = M n f ) M c- The functor is represented by the completion of the scheme M n along the closed set of points s G M n given by the conditions d(gs) < c and 0 < height gs < height X. Hence it is represented by a formal scheme formally of finite type over S p f W {L). Let (X ,g ) be a point of M c with values in a field P. Then p ~ xg is not an isogeny, because otherwise we would have height g = height p -f height p ~ l g > height X. Hence the smallest integer ra+, such th at pm+g is an isogeny must be nonnegative. Since height g~l = —height g, we have the inequalities —height X < height g~x < 0. Again we conclude th at the smallest integer m _ , such th at pm~ g~l is an isogeny, is nonnegative. By the remark after definition (2.25) we conclude th at

Q U ASI-ISO G EN IES OF p-D IV ISIB L E GROUPS

61

ra+ + ra_ < c/height X. Hence ra+ is bounded by c/height X. This implies th at we have an equality for n > c/height X CMc)red = ( M " +1)red. The equality follows because a quasi-isogeny a : X —►Y of p-divisible groups over a reduced scheme S is an isogeny, iff a s is an isogeny for each point s G S (see 2.9). We fix an affine open subscheme U C (M™)red for large n. For n big we get an affine open formal subscheme S p f R n of , whose underlying set is U . Hence we have a projective system of surjective maps of adic rings Rn +1

y Rri'

Let R be the projective limit. We write R n = R / a n . Let J be the inverse image of the ideal of definition in some R n . In order to show th a t M c = lim is a formal scheme, we have to prove th at the ring R is J -adic. Since R n is J -adic we may write R = lim R/a.n -f J m. The lim it is taken independently over all n, m. We claim th at for fixed m the following descending sequence stabilizes . . . a n + J m D a n+i + J m D . . . Indeed, let X n be the universal p-divisible group on S p f R n . Then X = lim X n defines a p-divisible group on R / 3 m for each m. We get g : X r / j m —► X by lifting the existing quasi-isogeny for m = 1. By the definition of representable we get for a suitable N a unique m ap Rjy —» R / J m th at induces the point (X, g). For any n > N the composite map Rn —►R n —►R / j m

R n / 3 mRn

has to be the canonical one. This implies th at the first arrow induces an isomorphism R n/ J mR n -+ R n / J mR jy. We conclude th at the descending sequence of ideals stabilizes.

62

CHAPTER 2

By proposition (2.5) we conclude th at R is an adic ring. This completes the proof of lem m a (2.28). C o ro lla ry 2.29 The functor M c is representable by a formal scheme for­ mally of finite type over S p f W (L ). The associated reduced scheme (M c)red, which is the subscheme defined by the largest ideal of definition, is projective. Proof: Indeed, for n > c/height X we have (Mc)red = (Mc)redBut the right hand side is a closed subscheme of ]J A4n’m, where nh < m < (n + 1)/* with h = height X. This follows because for each geometric point (X , g) of the right hand side height pn g = n height X -f height g < (n -f 1) height X.

□ Proof of theorem 2.16: Let c and V as in proposition (2.27). It is enough to show th at M is representable over L '. As in the proof of lem m a (2.28), we see th at it is equivalent to show, th at the subfunctor M is representable. Obviously the proposition (2.27) remains valid for the functor M . Let a be an integer. For a point (Y, y : Xp/ —* Y ) of A i { V ) we denote by M a{Y) C M a the closed subset of points s E A4a , such th a t d (X s ,Y s) < c, where X denotes the universal p-divisible group over Ada . It is easily seen by the triangular inequality, th at M a{X) — 0> if d(Xi,/, Y) > a + c. Let U[ be the open formal subscheme of A4a, whose underlying set is the complement of U •A^ ( y )VeM(L'),d(XL,,Y)>f Note th at the last union is finite, because Ada+C(L') is finite by the last corollary. Claim: If a > f + c we have U£ =

.

First we show this equality for the underlying sets. Let Z E U{+x (P ) a point with values in some field P. We have to show d(X p, Z) < a. By proposition (2.18) there exists a point Y E M ( L ‘) such th at d ( Y p ,Z p ) < c. But by the

Q U A SI-ISO G EN IES OF p-D IV ISIB L E GROUPS

63

definition of U^+1 it follows th at d ( X p ,Y p ) < f . Hence d (X p , Z) < / + c < a. The equality of formal schemes follows because A4a is the completion of A^a+i along the closed subset Image X i a —* «A/fa+i. Indeed, this implies th a t U{ is the completion of U*+1 along the closed subset E//+1. Hence the claim follows. We set U* = U[ for any a > f + c. Clearly Uf —► is an open immersion of formal schemes of finite type. We have M. = |J^ U* f because any point s of M such th at d(X.s, X s) < f — c is contained in the open set U *. Indeed, if s is in the complement of U *, there is a Y E M ( L ') , such th a t d ( X s , Ys) < c and d(X s ,Ys) > / . Hence we get the contradiction d (X s ,X s) > f — c. The theorem is proved. □ 2.30 We call a subset T C M. bounded if there is a natural number N such th at d ( X t , X t ) < N for each point t E T. We call a subset T C M bounded if there is an N , such th at for each point t £ T represented by (X t , Qt) we have height Qt < N ,

d(X t, X t) < N .

By the proof of the theorem (2.16) we see th at a subset of M. is bounded, iff it is contained in one of the sets U *. Since {U^)red is contained in the projective scheme (A4a)red for a suitable number a, we see th at {U^)red is quasiprojective. We obtain the following: C o ro lla ry 2.31 For a locally closed subscheme T of M , the following con­ ditions are equivalent: (i) T is bounded. (ii) T is quasicompact. (Hi) T is quasiprojective. Indeed for the proof it is enough to note th at M is the union of the quasi­ compact open subsets UU P r o p o s itio n 2.32 Any irreducible component of the scheme M red is pro­ jective.

64

CHAPTER 2

Proof: Let us first show th at an irreducible component is quasiprojective. By the last corollary it is enough to verify th at an irreducible component is bounded. Let rj be the general point of the component. By corollary (2.26) we have for each point of the irreducible component the inequality d ( K t , X t ) < d(X.TJyX r}). This shows quasiprojectivity. To finish the proof it suffices to show th at the valuative criterion for proper­ ness is satisfied for our irreducible component C oiMred- Let R be a discrete valuation ring over L with field of fractions Q. Suppose we have a Q- valued point (X , g) of C. To extend this point to Spec R we may replace g by png and hence assume th at g is an isogeny. Then Ker g C X g[n] for some n. Taking the scheme theoretic closure we extend Ker g to a finite flat group scheme H C X /^n]. Then qr : X r —►X. r / H is the desired extension of(x,e).

□

2.33 Let us denote by J ( Q P) the group of quasi-isogenies of X. There is a n atural right action of J(Q P) on the formal scheme M , (X ,Q )~ (X ,e

ot),

T€J(QP).

We will give conditions for the existence of the quotient of M by the action of a discrete subgroup of J ( Q P). P r o p o s itio n 2.34 Let T C 0, such th at p ~r F s acts identically on the C artier module Mo of Xo. We may assume Lo = Fp*. Let M be the Cartier module of X. Then Mo are the invariants of the operator p _ rF s . Hence any endomorphism of X maps Mo to itself.

Q U ASI-ISO G EN IES OF p-D IV ISIB L E GROUPS

65

This remark shows th a t we may suppose th at L is a finite field. Since Aired is locally of finite type the conditions U ynU = Q

and

{U(L))y

fi

U(L) =

0

a r e e q u iv a le n t .

The points of U(L) correspond to subcrystals M of the isocrystal M w (L) W ( L ) q . Since U is quasicompact and therefore bounded there are integers a, 6 E Z such th at p aM C M C pbM for

each

M

E

U (L ). If j M

G

U(L) for

s o m e 7 G T, w e g e t t h e in e q u a lit ie s

p a~bM C 7 ~ 1M C pb~aM . Hence if 7 runs through (2.3) the set of JT(T)-lattices {7 - 1 M } is finite. Since the set of elements of J ( Q P) C E n d w (L ) M), where O b is the unique m aximal order of D. We consider D as a subalgebra of M n (D) via the diagonal embedding. The normalizer of M n (Oo) in M n (D )x is D x • M u ( O d ) x • Indeed, consider an x G M n (D ) of th at normalizer. Then xO p C D n is an Mn (0£>)-lattice. Hence it is enough to show th at any Mn (O o)-lattice in D n has the form dOp 69

C H A P TE R 3

70

for some d E D x . This is well-known for n — 1. For general n it follows from the so called M orita equivalence (easy exercise), th at asserts th at the functor from the category of O ^-m odules to the category of Mn ( 0 Jo )-modules given by w «— ►

®od

w

is an equivalence of categories. If we fix a prime element n of O #, we may reformulate our second condition as follows: 2’) For any lattice A E C the lattices I I ^ A belong to C. Indeed any element 6 of the normalizer of O b has the form b — Uku, where k is some integer and u is a unit in O b - Hence bA = HkA. We call b a maximal element of the normalizer, iff k = 1. It is equivalent to say th at b £ O b and th at bOs is a maximal two-sided ideal in O b Fix some lattice Ao £ C. By the property 2’) it is enough to know the lattices between n _1Ao and Ao th at lie in C to recover the whole chain. Hence to give a chain C of lattices is equivalent to giving a finite set {Ao, . . . , Ar _i } of Mn ( 0 z>)-lattices in V, such th at Ao ^ Ai ^

^ Ar _i

n _1Ao

The number r th at appears here is called the period of the chain. We see th at C = {A2-};gz, where the A* for i not in the inter vail [0, r —1] are defined by the condition Ai —f — H Ai . 3.3 Next we consider the case of a semisimple algebra B. It is a product of simple algebras B = B \ x . . . x B rn . There are m aximal orders O b { of jB* , i = 1 , . . . , m, such th at O b = O b 1 x . . . x Osm • We get a corresponding decomposition of V,

M OD ULI SPACES OF p-D IV ISIB L E GROUPS

71

V — V\ 0 • • • 0 Vm .

Moreover, each O ^-lattice A C V may be w ritten in a unique way: A — Ai (B • • • 0 Am , where A; C Vi is an 0 # . -lattice. Let us call A, the i^1 projection of A and denote it by priA. D e fin itio n 3.4 A set C of O b -lattices A C V, is called a multichain, iff there exists for each i — 1 , . . . , m a chain of OBi-lattices Li in Vi, such that L consists of the Os-lattices for which pri A £ Li for i = 1 , . . . , m. 3.5 Let T be a Zp-scheme, such th at p is locally nilpotent on T. We are going to define the notion of a m ultichain of O b ® zp (Tr-modules of type (.L ). A typical m ultichain on T will be { A®z p O t } where A £ L. Let us fix a notation. Assume th at b £ B x is in the normalizer of O b. Then conjugation by b~l defines an isomorphism O b — ►O b

x

\— ►b~l xb

Let M be a O b 0 O r-m odule. We denote by M b the module obtained via restriction of scalars with respect to this isomorphism. Then m ultiplication by 6 induces a homomorphism b : M b — ►M . Let us begin with the case, where B is simple. We consider the chain L as a category with inclusions as morphisms. D e fin itio n 3.6 A chain of O b ® z p O t -modules of type (L ) on T is a func­ tor A i— ►M a from the category L to that of O b ® O t -modules. Moreover, for each b £ B x in the normalizer of O b , a periodicity isomorphism 9b : M A b ^

M bA

is given. We require that the following conditions are satisfied:

CHAPTER 3

72

1. Locally on T there exist isomorphisms of O b 0 O t -modules M \ ~ A 0 z p Ot • 2. I f b is maximal (cf.(3.2)) and A, A' E C are such that 6A C A' C A, we have an isomorphism of O b / hOs 0 O t -modules locally on T M \ / qa,a'(M \ i) — A /A / 0 O t

•

Here £a,A' : —► M a denotes the homomorphism that corresponds by functoriality to the inclusion A' C A. 3.

The periodicity isomorphisms are functorial, i.e. for any inclusion A' C A the following diagram is commutative: 9b

M bA,

QbA.bA'

QA,A

The

satisfy the cocycle condition: Mb1b2A

&2\&i

qbi yb2 K

a

f. For each b E B x , which is in the normalizer of O b the composition MA b is multiplication by b.

MbA

Ma

M O D U LI SPAC ES OF p-D IV ISIB L E GROUPS

73

Let us reformulate this definition more explicitly. As above we represent B as a m atrix algebra over a division algebra D in such a way th at O b — M u (O d )- We fix a prime element II £ O b - Then we may represent £ as a chain of 0 # -lattices in V ,i £ Z ,

• • • C A* C A*+i C • • •

such th at A*_r = IIA* for some fixed natural number r and any i £ Z. We m ay reformulate our definition as follows: C o ro lla ry 3.7 A chain of O b ® O t -modules of type (£) on T is an indexed set of O b 0 O t -modules such that

M i-r =

m /1,

ie

z.

Moreover there is a O b 0 O t -homomorphism of degree one g : M i — ►M i +1 the following conditions are satisfied. 7. We Aaue isomorphisms of O b 0 Or-modules locally on T: ~ A*/Aj_i 0

Mi ~ A,- (g)

•

The map

er

: M,-_r = M/ 1 — ►Mi

is the multiplication by II. We note th a t the condition 1) does not claim any functoriality in A*. It just says th a t is locally on T a free Op 0 0T-m odule of the same rank as the O p-m odule At* (i.e. d im ^ F ) and M i / g ( M i - i ) is a free O d / U O d 0 O t ~ module of the same rank as the O d / R O d -vector space A»/A*_i. 3.8 We will also consider chains {M *}^z where we replace the condition 1) by the weaker conditions, th at Mi is locally on T a free O d 0 0T-m odule, and th a t M i / g ( M i - 1 ) is locally on T a free O d / R O d 0 0T-m odule. Then we speak ju st of a chain of O d 0 Ox-modules on T without fixing a type. The type (£ ) enters in the definition 3.6 only via the ranks of the modules above.

74

C H APTER 3

W e n o te t h a t th e M o r it a e q u iv a le n c e in d u c e s a b ije c tio n b e tw e e n c h a in s o f

O b 0 0 T - n i o d u l e s a n d c h a i n s o f O d 0 O t ~k l o d u l e s .

3.9 Let us return to the general case, where B need not to be simple. We consider the decomposition into simple algebras. Let £ be a m ultichain of (9j3-lattices in V. We denote by Ci the chain of 0 # . -lattices in Vi, which is the projection of C, Ci = { p n A | A G £}. D efin itio n 3.10 A multichain of O b ® z p Or-modules on T of type (£ ) is a set { M \ , . . . , M m ] , where M i is a chain of O b { ® z p O t -modules of type (A).

If A 6 C has the decomposition A = Ai © • • • ® Am , A; e Ci m we write M \ = 0 A/a*, where M \ i is defined by the chain M i . Again A »—► i =l M \ is a functor from C to the category of O b ® zp C^r-modules. Moreover for any b E B x th at normalizes O b we have a periodicity isomorphism eb : M A b ^ M

bA,

such th at the following diagram is commutative for 6A C A: „ M iA

Q A ,bA

Ma T h eo rem 3.11 Let {M a } be a multichain of O b ® O t -modules of type (£) on a Zp-scheme T, where p is locally nilpotent. Then locally for the Zariski topology on T the multichain {M a } is isomorphic to {A (g) (9t}ag£I f {M'a } is a second multichain of type (£) on T, then the following functor on the category of T-schemes

M O D U LI SPAC ES OF p-D IV ISIB L E GROUPS

T f h— Isom ({M a

0 T'}, {M' a

75

Ot >})

is representable by a smooth affine scheme over T . We rem ark th at by the M orita equivalence it is enough to prove this theorem in the case, where B = D is a division algebra. We refer to the appendix to this chapter for the proof and first formulate a similar theorem in the presence of a polarization. Whenever we consider the polarized case we will make the blanket assumption that p / 2 . 3.12 We fix data (5, 5 , V, ( , )) of type (PEL), cf. (1.36). Let * denote the involution on B . We let Ob be a maxim al order of B invariant under *. If W is a right 5-m odule, we define a left 5-m odule by restriction of scalars * . b — ►B opp . W ith this convention the dual vector space V * = HomQp(V, Qp) is a left 5-m odule and ( , ) induces an isomorphism of 5-m odules ip : V — ►V*. In the same way for an O ^-lattice A in V, A* = Homzp(A ,Z p) is a left O ^-m odule. The image of A* by the map A* — ►V*

V

is the dual lattice with respect to ( , ). We will denote it by A* as well. D e fin itio n 3.13 A multichain £ of lattices in V is called selfdual, if A £ £ implies A* £ £ . D e fin itio n 3.14 Let £ be a selfdual multichain of lattices in V . A polar­ ized multichain of O b ® z p OT-modules on the scheme T of type (£) is a multichain of O b zp Or-modules {Ma} of type (£) together with perfect O t -linear pairings Fa ' M a

x

M a*

— ►O t

such that the following conditions are satisfied

76

CHAPTER 3 1. £ \{am , m') = £h(m, a*ra'), m G M a, m! G Ma*, a G O b 2. £ \ (m, m ’) — —£\+ (m ; , m), m G MA, m ' G Ma* . Ze2 Ai C A 2 be lattices in C. Then £A1(m,eAi,AZn ) = £A2(eA2,A1m ,n ), m G MA, , n G MA. . Let b G £ x 6e in the normalizer of O b - We set b = ( 6- 1 )* so that for a lattice A we have the relation ( 6A)* = 6A*. We consider for a lattice A g £ the periodicity isomorphisms eb : M l ^ U M iA

e-b : m { . — * M-bA. = M (bA). .

Then we have the relation

£A(rn1, m 2) = £bA(0bmiJ6~bm 2), mi G MA, m 2 G M A*. 3.15 On the selfdual chain C we may consider the functor A 1— ►M A = M = Homzp(M A*, O t )> We have a periodicity map on this functor de­ fined by the diagram (m a ) ‘ = ( M i y ^

(M bA. y = M(*6A). = M bA.

One verifies with little pain th at {Ma} is a multichain of O b ® modules of type (£). Let us call MA the dual chain. We may restate the definition of a polarized multichain in this set-up more elegantly: A polarized multichain over the scheme T of type (C) is a multichain { M a } of type (£) together with an antisymmetric isomorphism of multichains {Ma } — {Ma }. The analogue of theorem (3.11) in the polarized case is the following theo­ rem. For the proof we again refer to the appendix to this chapter. T h eo rem 3.16 Let C be a self dual multichain of Os-laPices in V. Let T be a Z p-scheme, where p is locally nilpotent. Let {M a } be a polarized multichain of O b O zp OT-modules of type (£). Then locally for the etale

M O D U LI SPAC ES OF p-D IV ISIB L E GROUPS

77

topology on T the polarized multichain {Ma} is isomorphic to the polarized multichain {A 0 O t} Moreover, if { M A f } is a second polarized multichain of type (£) on T then the functor of isomorphisms of polarized multichains on the category o f T schemes T' Isom ({Ma 0 O t /}, {M a' 0 Ot >}) is representable by a smooth affine scheme over T . 3.17 We will now define moduli problems of p-divisible groups, th at are variants of the problem in chapter 2. Our starting point is one of the following two situations: C ase (EL): We fix (E, B, V) as in (1.38), and a maximal order O b in B . Let G be the corresponding algebraic group over Qp.

C ase (PEL): We fix (E, B , V ,{ , )) as in (1.38), and a m axim al order Ob in B fixed by the involution *. Let G be the corresponding alge­ braic group over Qp.

To define the variants of our functor M. we need a replacement for X. In term s of the group G this is given by an admissible pair, cf. (1.18). D e fin itio n 3.18 A set of data for moduli of p-divisible groups in the case (EL) relative to an algebraically closed field L of characteristic p is a tuple: (E ,5 ,O B ,F ,6 ,/i,£ ).

Here (E, B , O b ,V ) are the data of case (EL). We denote by G the associated reductive algebraic group over Qp. Let us denote by Ko the quotient field of W (L ). The datum b is an element of G(Ko). The next datum is a cocharacter p : G m — ►G

78

CHAPTER 3

that is defined over a finite field extension K of K q. Finally C is a multi­ chain of O b -lattices in V. We require, that the following conditions are fullfilled. (i) The pair ( 6, fi) is admissible, cf(1.18). (ii) The isocrystal (V 0 Ko,ba) has slopes in the interval [0,1]. (Hi) The weight decomposition ofV K with respect to the cocharacter p contains only the weights 0 and 1: v ® k

= v0 e v 1.

In the case (PEL) we have in addition to the data above the nondegenerate antisymmetric pairing (, ) on V that induces an involution * on O b • We require that the multichain C is self dual. The multiplier of the corresponding group G is denoted by c. Let us denote by v the slope morphism associated to b. In addition to the conditions above we require: (iv) The character cv : D —►G m is the character x i that corresponds to the rational number 1. Let us fix the set of data (EL) respectively (PEL). We consider two sets of d ata ( 6, p , £ ) and (&', / / , £ ') to be equivalent, iff b and 6' are in the same cr-conjugacy class, p and p ' are conjugate over a suitable finite extension K n of A'o, and there exists a bijection A A' between the chains C and C , such th at for any pair Ai and A2 lengthoF A i / A 2 = lengthoF Ai/A'2. Moreover in the case (PEL) we require th at the bijection A A' commutes with taking the dual lattice. Note th at we do not require the equivalence of the pairs ( 6,/i) and ( 6', /m u ') in the sense of definition (1.23). 3.19 Let us make a few comments on this definition. a) By the crystal associated to a p-divisible group X over L we mean the Lie algebra of the universal extension of some lifting of X to W (L ). It is canonically isomorphic to the Cartier module of X . The condition (ii) above says th at (N, F) = {V 0 Ko,ba) is the isocrystal of some p-divisible group X over L.

M O D U LI SPAC ES OF p-D IV ISIB L E GROUPS

79

The conditions (i) - (iii) are satisfied if there is a p-divisible group X over the ring of integers O k of K , such th at its reduction X l modulo the maxim al ideal is equipped with a quasi-isogeny X —►X l , and such th at the following condition is satisfied. In general let X be a p-divisible group over a base scheme S', where p is locally nilpotent. Then we denote by M ( X ) the Lie algebra of the universal extension of X . In our case where S = Spec O k the definition of M ( X ) makes sense because O k is a p-adic ring. The given quasi-isogeny allows us to identify M(X) Q with the A"-vector space N 0 # o K . Indeed we have a quasi-isogeny x x Spec

l

Spec O k I p — ►X x Spj o K Spec 0 K / p

th a t lifts the quasi-isogeny X —►X l - This induces a quasi-isogeny between the values of the crystals associated to the p-divisible groups at the divided power thickening Spec O k / p S p f O k • We get the desired identification (comp, also (5.15)). The condition is th a t under this identification the canonical filtration on the universal extension 0 — F i l 1 -> M ( X ) 0 Q -> L ie X 0 Q -> 0 coincides with the filtration given by p 0

Vi

V 0 K

V/Vi -> 0.

Conversely one expects th at the existence of an X with the properties above follows from the conditions (i) - (iii). If this is false it could happen th at the m oduli functors we are going to define are empty for some of the data of definition (3.18). b) The condition th at ( 6, p) is admissible implies th at for each character x of G th a t is defined over Qp, we have
= ordp x ( 6),

(cf.(1.21)). If we take for % fhe determ inant of an element g 6 G (Q P) acting on the Qp-vector space V detQp : G — ►G m,

CHAPTER 3

80

we get the equality: dim Vi = ordp detjf 0( 6; V Ho)■ If we take in the case (PEL) for x the multiplier c, we get from the condition (iv) th at < p, c >= 1 . This implies th at the subspaces Vo and V\ are isotropic with respect to the pairing obtained on V ® K by extension of scalars. We also note th at conditions (i) and (iii) imply condition (ii). 3.20 We recall (cf. (1.38)) th at the isocrystal N is equipped with an action of B. In the case (PEL) it is also equipped with an alternating bilinear form of isocrystals, ip : N ® N — ►1(1). Indeed since L is algebraically closed and since ordp c(b) = 1, we find u E W ( L ) X such th at c( 6) = p u ~ 1a(u) and put xp(vyv') = u (v , v'). If we choose another u , we change ip by a factor from . We call the set of bilinear forms on iV a Qp-homogeneous formal polarization. The form xp defines a polarization on the p-divisible group X , i.e. an anti­ sym m etric quasi-isogeny A : X —►X. The isogeny class of the pair (X, A) is well defined by the data of definition (3.18). Let E denote the Shim ura field, i.e. the field of definition of the conjugacy class of p, cf. (1.31). We denote by E the complete unramified extension of E with residue class field L, which is contained in K . We define a functor on the category N ilpog th at is associated to the d ata of definition (3.18). For a scheme S in N ilpog we will denote by S the closed subscheme of S defined by the sheaf of ideals p O s . The structure morphism

. We see th a t N is a free k 0 P-m odule, iff the dimension of the P-vectorspace N ^ is independent of (j). This follows in the same way as in the proof of lem m a (3.24) from the following commutative diagram:

CHAPTER 3

88

o

_♦

0

_►M 'a -2U

— ► Lie*XA - * 0

M*

I N a

M'+

— ► L ie + X v

-►0

I _____ ►

i

i

0

0

T h eo rem 3.25 The functor M is representable by a form al scheme, which is formally locally of finite type over S p f Og. Proof: We start with the representable functor M of theorem (2.16), for our X at hand. In fact the theorem is applicable, because over an algebraically closed field any isocrystal is decent. Let ( X, g) E M ( S ) be a point. We transport the action of B on X by quasi-isogenies via g to an action of B on X by quasi-isogenies. Let M o be the subfunctor of M , where O b acts by isogenies. This is clearly a closed subfunctor and therefore representable. We have an obvious morphism of functors: j :M —

J J M oA

It is enough to show th at this morphism is representable. We know (2.9) th a t the condition, th at a quasi-isogeny of p-divisible groups over a scheme S is an isogeny, is representable by a closed subscheme. Hence the condi­ tion th at Ob acts on X \ , the conditions (iii) and (v), and the condition th at g \\A is an isogeny for any two neighbours A C A' is relatively repre­ sentable with respect to j. The condition (iv) is clearly representable. In the presence of condition (iv), condition (i) is autom atic and condition (ii) is equivalent to condition (ii bis) prescribing the degree of certain isogenies. This is obviously repre­ sentable by an open subscheme. 3.26 Next we will consider the problem of determining the local equations of the formal scheme given by definition (3.21). We reduce this to a problem of linear algebra by constructing a local model, comp. [Rl].

M O D U LI SPACES OF p-D IV ISIB L E GROUPS

89

Let us start with a set of d ata (B, F, Ob, V, p, C) in the case (EL). In the case ( P E L ) we have in addition a nondegenerate antisym metric Qp-pairing ( ,

)onF.

The cocharacter // is given over the field K . Let E C K be the Shimura field. Let us define a functor M/oc on the category of O#-schemes. D e fin itio n 3.27 A p o i n t o f M loc wit h values in an O E - s c h e m e S is given by the f o l l o w i n g d a t a . 1. A functor from the category C to the category of O b ® Os-modules on S: A i— >tA,

A EC.

2. A morphism of functors (pA : A 0 Zp Os — ►LvWe require that the following conditions are satisfied: (i) t A is a finite locally free Os-module. For the action of O b on t A we have the following identity of polynomial functions d e to s (a ; t A) = d e t^ (a ; Vo),

a E Ob •

(ii) The morphisms p>A are surjective. (Hi) In the case (P E L ) the composite of the following maps is zero for each A:

t\

(A ® Os)* S A 0 O s ^

tk .

Clearly the functor M/oc is represented by a closed subscheme in a product of Grassmannians. We write M/oc for M/oc oE 0&Let us introduce a sm ooth covering of the formal scheme M . D e fin itio n 3.28 Let Af be the contravariant set-valued functor on the cat­ egory Nilpog, such that a point of Af with values in S G Nilp0& is given by the following data 1. A point (X a ,Qa ) o f M ( S ) .

C HAPTER 3

90 2.

An isomorphism of (polarized) multichains 7a ■M a

A ®Zp O s ■

Here M \ denotes the value of the crystal of X \ on S. 3.29 The sm ooth formal group

P(S) = Aut ({A®z, Os}) acts on Af via the data 2): P - ( X a,£>a,7a) = (A a, 6a ,P7a ),

p E V (S ).

We see th at Af is a left P -torsor over AA and therefore representable by a formal scheme, which is of finite type over AA. There is a natural morphism Af ( X a ,Qa ,7 a )

— ► M ,oc 1— ► A ® zp Os

7 _1

M a — ►L ie^A ,

th at factors through the p-adic completion M /oc of M /oc. By Grothendieck and Messing [Me] the map Af — ►M loc is formally smooth. It is formally locally of finite type since Af and M /oc are formally locally of finite type over S p f Og. 3.30 Let us fix a closed point x E M . We identify its residue class field k ( x ) with the residue class field k = L of Og. We will consider sections of the sm ooth morphism Af — ►AA in a pointed etale neighbourhood (£/, y) of x. By definition of Af a section s over U is given by an isomorphism of (polarized) chains

7a : M a

O u — ►A G zp O u .

We are going to explain a condition on the section s th a t ensures th a t the composition

M O D U LI SPACES OF p-D IV ISIB L E GROUPS

U

91

AT — ►M 'oc

is formally etale. Consider a local artinian augmented /c-algebra A, such th at the square of the m axim al ideal of A is zero. Consider a morphism a : Spec A — ►U

(3.1)

which is concentrated in x. Let us denote by Ya for A £ £ the p-divisible groups on SpecA induced by the universal p-divisible groups X a on M . Let Ya = Y a X s Pec a Spec k be its reduction. Let N \ (respectively N \ ) be the Lie-algebra of the universal extension of Y a (respectively Y a ) . By the crystalline nature of N a we have a canonical isomorphism t

: N a ®k A

NA -

On the other hand the section s provides an isomorphism

7a • X a — ►A 0 z p A . D e fin itio n 3.31 We call a section s rigid of the first order in x, if for all algebras A as above and morphisms a as above the following diagram is commutative N a ® k A -X >

na

ACzpA where

= j A 0a

3.32 Any closed point x e M has an etale neighbourhood, such th at there is a section sa in this neighbourhood which is rigid of the first order in x. Indeed, let 2 be the maxim al ideal of definition of Oj#. Let M 2 be the closed subscheme of M defined by J 2. For any formal scheme X of finite type over M we will denote by X 2 the scheme X M 2. Since N M is a sm ooth morphism, it is enough to find an etale neighbourhood U2 —►M 2

CHAPTER 3

92

of x and a section s : U2 —+ M 2 which is rigid of the first order in x in an obvious sense. Since M 2 M 2 is a morphism of schemes of finite type it is enough to ask for the existence of a section over Spec , where O®*1 is the strict henselization of the local ring of M 2 at x. By Hensel’s lem m a it is enough to find a section over Spec Since any morphism (3.1) factors through the spectrum of the artinian ring A! — 0 ^ / m % + 7rO®*1, it is enough to construct a section over Spec A !, such th at the diagram of definition (3.31) is commutative. This is obvious. P r o p o s itio n 3.33 Lei x E M be a closed point and let s : (U,y) —> M be a section in a pointed etale neighbourhood (U,y) of x which is rigid of the first order. Then the composition U -^M

— ►M Ioc

is formally etale in a Zariski open neighbourhood of y in U . It follows th at any point of M has an etale neighbourhood, which is formally etale over M /oc. For the proof we need the following general result on formal smoothness contained in EGA. L e m m a 3.34 Let M —> S be a morphism of locally noetherian formal schemes, which is formally of finite type and formally smooth. Let M M be a closed subscheme of M defined by a coherent sheaf of ideals K, C Oj\f. Let x be a point of M , and y be its image in M . Then the composite M —►S of the morphisms above is formally smooth in a Zariski open neighbourhood of the point x, iff the map K./K? ® oM K(x ) — ”

K(y) .

induced by the universal derivation ([EGA] Ojv 20.5.11.2), is injective. Proof: Consider the standard exact sequence /C//C2

►

Om

►

0•

By [EGA] O jv 20.7.8 the condition th at M —►S is formally sm ooth in a neighbourhood of x is equivalent to the condition th at 5 is formally left invertible in a neighbourhood of x. The topological CV-m odule is

M OD ULI SPACES OF p-D IV ISIB L E GROUPS

93

formally projective [EGA] O jv 20.4.9. The topology on is the J'-adic topology, where J C Oj^f is some ideal of definition [EGA] Oj y 20.4.5. It follows th a t O m is a formally projective (9>f-module, th at carries the adic topology induced from O m • Let us denote by X the m aximal ideal of definition of O m • By [EGA] Oj y 19.1.9 the condition th at S is formally left invertible in a neighbourhood of x is equivalent to the condition th at /C//C2

O m /Z

Om / ?

► ^AT/5

is left invertible. One checks th at both modules are coherent modules over O m /% . Hence we conclude the proof of the lem m a by [EGA] 0 i v 19.1.12. C o ro lla ry 3.35 I f the map in (3.34) 25 an isomorphism, then Ai formally etale in a neighbourhood of x. Proof: This follows from the fact th at

S is

— (0) in a neighbourhood of x

in this case, [EGA] 0 i v 20.1.1. Proof of Proposition (3.33): First we show th at the map is formally smooth. We write X — U , y = M x j ^ U , Z — M /oc. Consider the diagram of formal schemes over T = S p f Og.

y

X

z

By [EGA] O iv 20.7.18 we have split exact sequences

0

►

0

►

p El\/ t

*

^ y /r

* ^ y /z

*

^y/r

* ^y/x

*

®

If we apply s* to the lower exact sequence, we get a canonical splitting

94 Let

CHAPTER 3 and q ( y ) = z . The m orphism Let us use the notation Q x j T ( x ) =

y = s(x)

k(x).

fibres. We claim th at the formal smoothness of th a t the map Z / t ( z ) ®K( z)

k (x

qs

qs

defines an inclusion k ( z ) —> k ( x ) for the geometric

is equivalent to the assertion

) -2-+ C l y / T ( y ) ® K(y)

k (x

)

Q}x / t ( x )

is injective. Indeed, consider the diagram

X/T

y / r i=»

S K /K 2

s * ny\ / z It is easy to see th at d induces an injection of the geometric fibres at x , iff fi does. Hence the claim follows from the lemma. Let us identify k , k ( x ) , /c(y), and k ( z ) . By duality it is enough to show th at Horno x (CIx / T , k ) — ►Hom o2 ( f l ^ T )K) is surjective. We set A = O x fx/™% + nOx,x and B = O z , z / m l + nOz,z- These are augmented artinian K-algebras, such th at the squares of the m axim al ideals are zero. Our assertion is th at the map Der*(A, k ) — 5- D er«(5, «) is surjective. This means th at any com m utative diagram

M O D U LI SPAC ES OF p-D IV ISIB L E GROUPS

95

such th a t the vertical arrows are concentrated in x , respectively z, adm its a diagonal arrow as indicated. Indeed the left vertical arrow is given by a chain {Xa} of p-divisible groups over k . The point y defines a rigidification of their crystals, i.e. an isomor­ phism of polarized chains M a - A 0 z p ac. Set I a = LieXA- Then the Hodge filtration defines the point z of the period space M loc

Ma

—

\

A zp ac /

h The right vertical arrow in the diagram above gives a lifting of the last map A

«[e] — ►U-

By the horizontal isomorphism above M a ~ A 0 z p Ac[e] is identi­ fied with the value of the crystal of X a at k [s }. Hence by Grothendieck and Messing we get a lifting X a of X a, such th at M a, the Lie algebra of the universal extension of X a, is identified with M a ac|e]. We get a rigidification Ma ~ M a 0« «[£:] A 0 z p ac[s] and hence an element of W(ac[£]). The point is, th at this is the image of the point {Xa} G M ( k [s ]) under s : M Af because s is rigid of the first order. It follows th a t the map on derivations is bijective. The formal etaleness follows im m ediately from corollary (3.35). □ We conjecture th at M is flat over S p f Og. By proposition (3.33) it is equivalent to ask whether the local model M /oc is flat over Op,. At the end of this chapter we review some examples which support this conjecture. 3.36 In the definition (3.18) of the moduli data we had assumed th at L is algebraically closed. We now want to consider an arbitrary sufficiently big perfect field L of characteristic p. We keep the notations of (3.18) except th a t we now impose an additional condition, namely th at b is decent, cf.

( 1.8).

96

CHAPTER 3

L e m m a 3.37 Assume that we are in case (PEL) and that b satisfies a decency equation with an integer s > 0, cf. (1.8). Assume that L contains the field F p*. Let K q = W ( L ) q p. Then there is a unit u £ W( L), such that the Ko-bilinear form w) = u (v, w),

v,w e N

is a polarization of the isocrystal N. Proof: As in the case where L is algebraically closed (comp. (3.19)) our conditions imply th at ordp c(6) = 1. We define Nm& £ G ( Q pS) by the equation (ba)s = (Nm b)crs . For v , w £ N we find the equation for the given symplectic form on V : ((&V, b, p, C), that satisfies the conditions (i) (ii) and (iii) of the definition (3.18). Moreover we require that b £ G ( K q) is decent, and that with respect to the inclusion of the

M O D U LI SPAC ES OF p-D IV ISIB L E GROUPS

97

Shimura field E C K , the residue class field of E is contained in L. This gives an OE-algzbra structure on L. In the case (PEL) we have in addition the nondegenerate antisymmetric pairing ( , ), that satisfies the conditions of definition 3.18. In this case we assume that L contains where s > 0 appears in a decency equation for b. Let E be the complete unramified extension of the Shim ura field E with residue class field L. By proposition (3.38) there is a Qp-homogeneously polarized p-divisible group X over L whose isocrystal is (N, Q£ VO- W ith this X the m oduli problem A4 of definition (3.21) makes sense over Og in this more general situation. C o ro lla ry 3.40 The functor M of definition (3.21) associated to the data of definition (3.39) is representable by a formal scheme which is formally locally of finite type over Og. 3.41 Our next aim is to define a completion of the formal scheme M of definition ( 3 .21 ) over the ring of integers O e of the Shimura field for any data of definition (3.18) over an algebraically closed field L. For the following proposition we start with data of definition (3.18) in case L is algebraically closed or with data of definition (3.40) if L is an arbitrary sufficiently large perfect field (in the latter case b is decent). Let s be any integer, such th at the m orphism sv factors through G m. We set

7s = p ‘ (sv(p))~1. This is a quasi-isogeny of height s dim X of the p-divisible group X, which lies in the group J ( Q P). Let T s = y f be the cyclic group generated by j s . Hence T s acts on the functor M .

P r o p o s itio n 3.42 Let A4S be the Zariski sheaf associated to the functor M { S ) / T b. Then A4S is representable by a formal scheme, which is locally of finite type over S p f Og.

98

CHAPTER 3

More explicitly let S' be a connected scheme in N i l p o e - Two data ( X c , q) and Qf) over S define the same point of M s(S), if for some integer k the quasi-isogeny qj * q' * 1 lifts to an isomorphism X A f —►X A , A £ C. Since j s is in the center of the algebra E n d X the group J ( Q P) continues to act on M sProof: We fix a member Ao of the lattice chain C. Let M ( n ) C M be the subfunctor where g \ 0 is a quasi-isogeny of height n. This is an open and closed subfunctor and M is a disjoint union: M — J J M{n). n

Let us first exclude the uninteresting case where X is etale. Since j s is a quasi-isogeny of height sd im X , the action of j s on M is homogeneous of a nonzero degree. We get an isomorphism: sdim X TT M ( n ) —►M s . n= 1 In the case where X is etale the quasi-isogeny y s is the identity. Hence the result is trivial in this case. R e m a rk 3.43 ; If the group X is etale, we are in the case (EL). Since our functor M is em pty unless dim X = dim Vi, we may assume th at the m orphism p is trivial. Assume th at L is a perfect field, and th a t b E G (W (L )q ) is a decent conjugacy class. Take a decent b E b. Then for a certain integer 5, we have: (ba)s = sv( p)as = pscr5,

b E G (Qps).

Since by Hilbert Satz 90 the cohomology group H 1(Qps / Q p , G (Q P*)) is trivial, we conclude th at p E b. By corollary 3.40 we find a formal scheme A4o over S p f Zp , such th at M o x S p f W ( L ) = M . The scheme Ado is a disjoint union of copies of S p f Zp. We have one copy for each chain of Ojg-lattices in V conjugate to C. This is the model of M over the integers Zpof the Shim ura field, which we would like to define in the general case. 3.44 We will now define some sort of descent data on the functors M resp. M s - Let us start with a more general setting.

M O D U LI SPACES OF p-D IV ISIB L E GROUPS

99

Let E be a finite extension of Qp. Let L be an algebraically closed extension of the residue class field /c of E. We denote by E the complete unramified extension of E with residue class field L. Let us denote by r G G a l ( E / E ) the Frobenius autom orphism . Let S = (5, q =- {£a} of M over a connected scheme S it suffices to define the integer

M O D U LI SPAC ES OF p-D IV ISIB L E GROUPS

105

n ( X c ,e) = < x ( X c , e ) , n > ,

(3.3)

n(g( Xc,Q)) = < wj(flf),n > + n (X £ ,e).

(3.4)

such th a t for g G J(Qp)

L e m m a 3.53 Let F be a field. Lei g : X —»■ Y a quasi-isogeny of p divisible Os-modules over an algebraically closed field P, which both satisfy the condition S.21 (iv). Then the height of g is an integral multiple of i ( B ) • f ( F / Q p), where f ( F / Q p) is the index of inertia. Proof: Let M resp. N denote the Cartier modules of X resp. Y . Using the decompositions M = 0 M ^, N = 0 A ^ from the proof of 3.24, we get maps induced by g M 4, ® Qp — ►iV^ Qp. The orders of the determ inants of these m aps are well defined. Since the m aps commute with V we conclude by 3.21 (iv), th at these orders are independent of (j). □ For the construction of x we may assume th at M is not empty. Then we may take for X a/>-divisible O ^-m odule, which satisfies 3.21 (iv). We fix a particular A and set n ( * £ , e) =

v"7vT height

gA G Z.

We have to verify the identity (3.4),

1

1

t{B)

i{B)

—T dT height Q x g ' 1 = < w j ( s ) , n > - - —

height gA .

This follows from i (B) < L0 J (g)1n > = ordp det(#; V 0 K 0) = height g. Next we consider the case (PEL), where F is a field. We may assume M. is not em pty and choose a point (X ^, g) over L. We may take X = X a for a fixed lattice A. We choose the polarization A : X —►X in such a way th at it induces an isomorphism X \ Xj^. Then we have height A = logp |A/A|.

CHAPTER 3

106

We have to define equivariant maps in the sense of (3.4) c , h : M — ►Z, such th at 2n = ((dimQpF )/i(B ))c . This amounts to the assertion th at 2 height gA is divisible by dimQpF . But we have

2 height gA = height gk + height gA - logp |A/A|. Hence the divisibility follows from 3.21 (v). Finally we consider the case F — F q x F q) B — D x D opp. We may assume X = Y x Y , where Y is an O ^-m odule and Y the dual O^opp-module. We fix a lattice of the form Ao 0 Ao C W 0 W* of the multichain C. For a point ( X c , g) of M. we have quasi-isogenies a : Y — > XAo, (3 : Y — ►X ^ q, such th at a x f3 — QAo§ Ao- The quasi-isogeny j3a is up to a constant in Qp an isomorphism. Then we define maps nn*,c: M — by n = —1/ i (D) • height a , n* = —1/ i ( D) • height /?, and S = ~ dim A' q 1 vW w height This gives the desired J ( Q p)-equivariant morphism h

: M — ►A.

3.54 In the end of this chapter we discuss some examples. Drinfeld[Dr2] first considered a functor M in the case (EL) in the following situation (compare 1.44). Let B = D be a central division algebra over F with invariant 1/d. Let V — D considered as a D-module. In the following we will also use the right D-module structure on V. It gives us an identification G (Q P) = ( D opp) x . We keep the notation F, 7r, n , r, e : F —* Qp from (1.44). Since the invariant is 1/d we have Ud = ir and the ring of integers in D is O d = Op[n],

na? = r(a?)n, x G Op.

M O D U LI SPAC ES OF p-D IV ISIB L E GROUPS

107

In this example L will be the field F p, which we identify with the residue class field of Qp. We write W = W ( L ) for the W itt vectors. Then K 0 C C p , the completion of Qp. Let K / K q be a finite Galois extension contained in C p , such th a t s (F) C A". Then we have the decompositions D ® qp K

=

n

*

^

r):F—>K

V ® Q p I
K

We take for Vo C V ® qp K a A-vectorspace of dimension d, which is in­ variant by the action of D (from the left), and such th at Vo C Ve. For Vi C V ® q p K we take any complementary space invariant by the action of D. Let /i : G m —» G be the cocharacter with weight decomposition V &>qp K = V o ® V i . It is the cocharacter p defined under (1.44). Finally we will define the structure of a crystal on O d p W, such th at the induced isocrystal is (K, q(a ; Liea A ) for a E Op opi}a Os . This is a function on W a which is by definition 1 if Lie^X = 0. Then the left hand side of 3.21 (iv) is the product of the qa . Hence the condition is

M OD ULI SPACES OF p-D IV ISIB L E GROUPS

111

rip“ Since the functions p a do not vanish everywhere there are constants k a £ (E ) x such th at qa = k apa . The degree of qa is the dimension of LieaX . Hence we get r ,„ kOs . Then the equation qa = k ap a implies th at this homomorphism is ea . This completes the verification th at X is a special formal O ^-m odule. 3.59 We may thus replace the condition 3.21 (iv) in the definition of the functor M by the condition th at X is a special formal 0jr>-module. In [Dr2] Drinfeld considered the subfunctor given by the condition th at q is of height zero. It is obvious th at the notion of a special formal O d-m odule makes sense over any O^-scheme S. If i : O d E n d X is a special formal O ^-m odule then the action of i ( f ) on Lie X for f E O f coincides with the action of e(f)eO E - O s . Let S be the spectrum of an algebraically closed field L. We denote by M the r — W p(L)-crystal associated to X . Since O d acts on M , we have on M the structure of an O d 0 o F,e W p(L)-module. We choose an embedding e : O p — * O p — >WF (L)

(3.6)

which extends e. Let Mi = | m £ M ; i ( f ) m =

| .

Then we have M = ©Af,-. The operator ^(n), which we denote also by n , is PTF(T)-linear and homo­ geneous of degree 1 ,

CHAPTER 3

112

H : Mi — ►M i+1 ,

n d = 7r ;

while V is homogeneous of degree 1 and r -1 -linear V : Mi — ►Mi+1 ,

V(wm) = r _ 1 (ti;)Vm, w G W>(Zr).

The lTjp(L)-modules M* are free, and M i/V M ,-.i are 1-dimensional Lvector spaces. The length of the lTp(L)-m odule M,-/IIM,-_i is indepen­ dent of i. Since we are interested in special formal O d ~modules th at are isogenous to X, we will assume th at this length is 1 or equivalently th at rankjyF(£)M; = d, i.e. F-height X = d2. The isogeny class of X is uniquely determined by the r — A/p(L)-isocrystal (Mo 0 Qp, V - 1II). L e m m a 3.60 Over an algebraically closed field L any two special formal Op-modules of F-height d 2 are isogenous. The group J ( Q P) defined by (3.22) is isomorphic to GLd(F). Proof: By a theorem of Dieudonne there is a unique isotypic isocrystal of slope zero and height d. Therefore it suffices to see th at (Mo 0 Qp, V _ 1 II) is isotypic of slope zero. Consider the maps induced by m ultiplication with i(H),

n : Mi/VMi _! —

Mi+1/VMi.

Since Ud = 7r = 0 in A, we obtain th at there is an index i , such th a t the above map is zero. We give a definition before finishing the proof. D e fin itio n 3.61 Let X be a special formal O d -module over an O^-scheme S. We have a decomposition of the tangent space of X LieX =

0 Lie**. i£Z / dZ

Here i ( f ) for f G F acts on Liel{X) via the homomorphism r ~ le : Op —» ° E ^ O S. We call the index i critical for X if the map

n : Ue*X is zero.

— ►Liei+1X

M O D U LI SPAC ES OF p-D IV ISIB L E GROUPS

113

For S = Spec L we have LielX — M i j V M { - \ . We have seen th at there is a critical index i for X . In this case IIM; C V\M*, and since both modules have the same index in Mi we obtain IIM* = VM{. Hence Mi C Mo ® Qp is a lattice stable by V ~ 1U. Therefore Mo ® Qp is isotypical of slope zero.

□

The fact th at there is only one isogeny class is the reason the formal scheme M is p-adic, as we are going to prove now. Let R be a complete noetherian local ring of characteristic p, which is equipped with a Og-algebra structure. Then we have an injection Fp —> O g / p O g —►R. Let us denote by L the residue class field of R. P r o p o s itio n 3.62 Let X be a special formal Oj^-module over R. Then any quasi-isogeny M extends uniquely to a quasi-isogeny X # —►X . Before proving this we note: C o ro lla ry 3.63 In the Drinfeld example M is a p-adic formal scheme lo­ cally of finite type over S p f Og. Proof: It is enough to show by (2.2) th at Z — M. XsPfOg Spec 0 %/ p Og is a scheme. Since Z is a formal scheme, it is enough to verify th at a sheaf of ideals of definition is locally nilpotent. Consider a point z of Z and let R be the completion of the local ring Oz,z • The special formal O ^-m odule X given over S p f O z )Z extends to a special formal Op-m odule on S p e c O z yz and hence on Spec R, which we also denote by X . By the proposition there is a unique quasi-isogeny X # —> X , which extends the given quasi-isogeny over the closed point. By definition of M we get a m orphism S p e c R —►S p f Oz,z, such th at the following diagram is com m utative Spec R — ►S p f Oz,z

But this means th at an ideal of definition X C O z yz is nilpotent in R. Therefore X is nilpotent itself, and the corollary is proved. □

114

CHAPTER 3

3.64 For the proof of proposition (3.62) we shall use Cartier theory (Drinfeld [Dr2], Zink [Z2]). Let R be an Op-algebra. We denote by E p = E the C artier ring of R relative to Op and a prime element w E O f - Then E p is the set of all formal sums £

v ’m f ' ,

i,j> 0

where i , j are integers and aij E R. One requires th at for fixed i only finitely m any aij are non-zero. One has the relations F[a] = [ag]F, [a]V = V [a?], where q is the num ber of elements in the residue class field ac of O f- Furtherm ore E p is a Op-algebra. The structure morphism maps a (q —1 ) ^ root of unity £ E O f to [£] E E r . Moreover we have F V = 7r and oo

[a] + [b] = [a -j- 6] -f ^ V*[Pj(a, 6)]F*, 2= 1 where the Pi are universal polynomials. The category of formal Op-modules (compare Definition (3.57)) over R is equivalent to the category of reduced Cartier modules. Let us assume th at we are given a homomorphism Op R th at extends the structure morphism O f R- Then E # is a 0 ^-algebra. The category of special formal Op-m odules over R is equivalent to the category of triples (M, M,-, II), where M is a reduced Ep-m odule, M = ©z€Z/d Mi is a grading of the abelian group M and II : M —> M is a E p-m odule homomorphism. One requires th at the following conditions are satisfied. The operators [a], V , F, II act homogeneously on M and have the degrees deg[a] = 0 , d e g V = l , d e g F = —l , d e g l l = 1. We have Jld = 7r. The /^-modules M i / V M i - \ are locally free of rank 1. We note th at the decomposition M /V M = 0

M i j V M i _!

i£Z/d

is exactly the decomposition in definition (3.61) for the corresponding spe­ cial formal Op-m odule X . Assume th at the i^-modules are free for i E Z/d. Let raz* E My i E Z /d , be a V-basis of M such th at m; E Mi. Then the elements IIm 2 may be uniquely expressed as follows

M O D U LI SPAC ES OF p-D IV ISIB L E GROUPS

lira* = ^ 2 V n [a;,n]rai_n+ 1 , n> 0

i

€ Z /d .

115

(3.7)

Conversely for any set of elements a2-n £ R , n > 0, i £ Z / d such th at Yiiez/d ~ ^ there is a unique (M, M*, II) with a V-basis mj, such th at the equations above are satisfied. Let X be a special formal O r -module over R and assume th at the index i £ Z / d is critical for X (3.61). Then we have IIM* C VM* and because V is injective we get an operator

u

= V ^ n : Mi — ►Mi.

Let R ' be a ir/algebra. We denote by X r > the special formal O ^-m odule obtained by base change, and by M r • its Cartier module. W ith this assum ption we have the following L e m m a 3.65 (Drinfeld): For any n > 0 the functor which associates to a R-algebra R f the set of invariants of the operator U

_

r>

((MR.)i/Vnd(MR,)i)U

is representable by a scheme etale over Spec R. Proof: Since the question is local on S p e c R we may assume th at a V-basis exists. Any element of (MR>)i/Vnd(MRi)i has a unique representation n d —l

y : V s [£5]ra;_5, 5=

x s £ R f.

(3.8)

0

This identifies the functor JR! ( M ^ y V nd(M ^ )i with the affine space and Uwith an endomorphism of A ^ . Hence the functor in (3.65) is representable by a scheme of finite presentation over Spec R. Weshow by the infinitesimal criterion th at this scheme is etale. Let R* —►R " be a surjective homomorphism of R -algebras with nilpotent kernel a. We have to show th at the map

(.MR,)i/Vnd(MRI)i -» (M * « ),y v n. But for given t € N we have therefore clear for large t.

E V t M. The desired equation is □

Let us denote by r)f[n] the etale scheme given by Lemma (3.65) on SpecR. It commutes with base change R —>R!\ rk Rl M = n f [«] X S pe c R S p e c R ' .

Furtherm ore r)f[n] has an O ^-m odule structure given by th at of M . If X is of the m inimal possible F-height d2, we see by the case of an algebraically closed field th at rjf[n] is locally for the etale topology isomorphic to the constant scheme associated to (O f / ^ O f Y • 3.66 We are now ready to prove proposition (3.62). Let us start with the case th at there is an index i critical for X . By assumption F-height X = F-height X = d2. Hence rjf[n] is a finite scheme locally isomor­ phic to {Of / ^ O f Y as a scheme with O^-action. Since we have a quasiisogeny X l this scheme is constant over Spec L and hence over Spec R. Let 71 , . . . , jd G lim rjf [n] be a O f -module basis. Then 71 , . . . , 7d G n

lim M / V ndM = My and we have n 7; = V 7 n

The C artier module of the special formal O ^-m odule X. r is given by the equations nm ^ = V m ^, k E Z / d in the sense of (3.7). We get a m ap of Cartier modules

Mr mk

—►M

1— ► V k~tj k,

k — i ,i + 1 , • • •, i + d - 1

Using the fibre criterion for isogenies (Zink [Z3]) we see th at the m orphism of formal groups X r —►X induced by this map is an isogeny of height d ( d - 1 ).

M OD ULI SPACES OF p-D IV ISIB L E GROUPS

117

If we m ultiply this isogeny X # —>X by a suitable quasi-isogeny of X , we get the desired lifting of X ^ Xl. Let us now consider the general case. Then S = Spec R is the union of the closed subsets Si C S, where the index i is critical. Let us more generally consider two p-divisible groups X and Y over a noetherian scheme S, where p is locally nilpotent. By the rigidity property (see after definition (2.8)) it makes sense to speak of a quasi-isogeny from X to Y over a closed subset TcS. L e m m a 3.67 Let S be a union of finitely many closed subsets S = Si U . . . U S r . Assume we are given quasi-isogenies Ua is an open immersion. Let us denote by EF the union of the formal schemes Ua with respect to the open immersions defined above. P r o p o sitio n 3.70 The formal scheme EF is separated, over O f Proof: A scheme X over O f is separated if there exists an open covering {Ui}i£i, such th at for all i , j £ I the canonical immersions Ui O Uj —> Ui X o F Uj are closed. Since we have Ua H Ur = Uadt by definition of S, we need to verify th at the canonical morphism Uaht Ua X of Ur is a closed immersion. We leave this to the reader. □ 3.71 There is a left action of GLd(F) on E F d . An element g £ GLd(F) transform s a S'-valued point : rj%—> to C. The units in the center of GLd(F) act trivially. The projection B x Z —►Z induces a natural morphism 'E d

v rj

•-F

*

where Z denotes the constant formal scheme over S p f O f associated to Z. Following Drinfeld we denote the fibre over 0 of this morphism by ClF . We note th a t the fibres over different connected components n : S p f O f ~ * Z are all canonically isomorphic. Indeed for a simplex A £ B x Z the functor Ua , depends only on the projection of A to B. Therefore we obtain a canonical isomorphism Ep — ►Cip x Z.

CHAPTER 3

120

The additive group Z acts via the second factor on EF . The action of m G Z takes a point rjik —►Cik to the point {r^fc+m —►£ ifc+m}, where ?7ifc+m = rjik, = £*fc. We call this the translation by m. T h e o r e m 3.72 (Drinfeld): There is an isomorphism of formal schemes M.

►d F

X

Of S p /

O^ ,

where the morphism S p f Op —►S p f O f is given by q. For a suitable iso­ morphism GLd(F) ~ J(Qp) this map is equivariant. The Weil descent datum on M gives on the right hand side the composite of the canoni­ cal Weil descent datum and translation by 1. The translation by m on Sjp x Spf o F S p f 0 ^ induces on M. the morphism which associates to a point (X , g) the point ( X nm, H~m g), where n _m here denotes the morphism X —►X nm defined by (3.20). For the proof we refer to Drinfeld (loc. cit.). 3.73 We will give here a few comments on the proof which we willuse later. Let us denote by M the r — W f (Fp) crystal of X, M = O d ®Of ^ f ( F p ) . Note th at there is a unique isomorphism W f (Fp) — Op th at induces the given O ^-algebra structure on Fp, and such th at 0F

WF(fp) —

Op

is the embedding s. We use this isomorphism to identify W f (Fp) and Op. W ith respect to the choice of the extension e of e to Op (3.6) we have the decomposition M = 0 M ,- . We may write M i = W f { Fp)

^T-ie.Op O

d

•

Any index i £ Z / d is critical with respect to M , i.e. V M j operator V _1II : M* —»• M ,• is given by the formula

= IIM 2. The

M OD ULI SPACES OF p-D IV ISIB L E GROUPS

0 x) = r(w) (g>11x11

V

121

w E Wf (F j>), a? £ Oz? .

The invariants A* C M,- of this operator consist exactly of the elements tu £ O f ,

0 gi(s) are constant of value pz-. Let (X, g) G A4^9u92\ F p). Then (X, g) is isomorphic to (X ,5 i x g 2 : X — + X),

CHAPTER 3

128

where gi € G L n (Qp) are representatives of gi £ G L n (Qp ) / G L n (Zp). There­ fore M.(9l,g2\ F p) consists of a single point. The rest of the argum ent is sim ilar to th at used in proposition (3.55), with the result of Lubin-Tate being replaced by the fact th at the universal formal deformation of (X, g) is represented by the formal torus T with character group Zn , cf. [DI]. More precisely ([DI], p. 131), let e\ , . . . , en be the standard basis of Ao- = Ao fl V_, en+i , . . . , e 2 n be the standard basis of Ao+ = Ao Pi V+. Let qij £ Horn (A0_, A0+), 2, j = l , . . . , n be the element which sends e* into e n + j and all other basis elements to zero. P u t Tij = qij —1. Then the universal deformation space is canonically isomorphic to S p f W ( F p)[[Tu , . . . , T nn}}. 3.82 We call a set of data (J9, F, O b >V, 6, £ ) of type (EL) unramified if B is a product of m atrix algebras over unramified extensions of Qp and if the mul­ tichain £ is a product of chains of lattices consisting of multiples (by powers of p) of a single lattice. In the case (PEL) we require in addition th at in each of the factor chains there is one member which is selfdual with respect to the given alternating form. In other words, in the unramified case the data £ is completely determined by giving a single O ^ -lattice A in V which in case (PEL) is supposed to be selfdual. In the unramified case the Shim ura field E associated to a set of data of our moduli problem, (F, F , O b , V, 6, //, £ ), is an unramified extension of Qp and hence E = Ko(L). An object of our m oduli problem over S £ N i l p o Ko is a pair (X, g) consisting of a p-divisible group with O b -action over S and a quasi-isogeny g :X

x S pec

l

S — ►X

x s S.

The conditions of (3.21) reduce in this case to the determ inant condition and, in case (PEL), to the condition th at the given polarization on X induce on X a multiple by a power of p of a principal polarization. The unramified case is considered by Kottwitz in [Ko3]. He shows by an application of the

M OD ULI SPACES OF p-D IV ISIB L E GROUPS

129

deform ation theory of Grothendieck-Messing th at the representing formal scheme M is formally sm ooth over S p f O k 0 ([Ko3],§5). In particular, in this case the flatness conjecture before (3.36) is obviously true.

A ppendix: N orm al forms o f lattice chains We will give the proofs of the theorems (3.11) and (3.16). We start with the proof of theorem (3.11). Clearly we may assume th at B = M n (D), where D is a central division algebra over a local field F. Moreover, we assume th at Ob — M ^ O d ) - We consider Od C Ob as a subalgebra by the diagonal embedding. We will fix a prime element II of D. Let £ = be the given chain of O ^-lattices in V. Then a chain o f Ob 0 O T - m o d u l e s o f t ype {£) on an affine scheme T = S p e c R is given by the following d ata (corollary 3.7): A sequence of Ob 0

zp

R-modules

U M i -1+ M i+l

•

i GZ ,

and for any i £ Z a periodicity isomorphism e : M i-r

Mf1

such that the following conditions are satisfied: 1. locally on S p e c R there exist isomorphisms of Ob 0 Mi ~ Ai ® R,

2. 9e = Oq,

zp

R-modules

M i / e { M i - 1) ~ Aj/Aj_i ® R

e r = n0.

131

132

A P P E N D IX TO C H A P T E R 3

R e m a rk s A .l The first condition says th at Mi is locally on S p e c R a free O d ® i^-module of the same rank as the D-module V. Let ac(D) = O d / H O d be the residue class field of D. Then the second condition says th at locally on S p e c R the ac(D) 0 /^-module Mi / g {Mi -\ ) is free of the same rank as the Ac(D)-vector space L e m m a A .2 For k < r the following sequence of O b ® zp R-modules is exact and splits locally on Spec R for any i £ Z

0 — M i l e k {Mi-k) — M i+1/ g k+1( M i_ k) — ♦ M i+1/ e(Mi) — 0 . Especially there exists locally on S p e c R isomorphisms M i / g k ( M i- k ) — A i / A i - k 0 R of O b 0 z p R-modules. Proof: We may assume th at B = D. Moreover we may assume th at the ac(Z>) 0 /^-modules Mi +i /g(Mi ) are free. Clearly the sequence of ac(D) 0 Rmodules is exact on the right and the surjection M i + i / e k+1( Mi -k) — ^ M i+1/ e(Mi) splits. Hence we get surjections M i / e k ( M i - k) © M i+1/ e(Mi) — . M i+1/ e k+1( M i - k ) . Hence by induction M i / g k {Mi^k) is the quotient of an ac(D) 0 z p i^-module Fifk which is locally on S p e c R free, and has the same rank as the ac(D)vector space Ai / Ai -kTo see th at M i / g k( Mi -k ) is locally on S p e c R a free ac(D) 0 z p R -module of the same rank as A i / A i - k we apply descending induction on k. For k = r this follows from our assumptions. Assuming by induction th at is locally on S p e c R free of the given rank, we obtain a surjection of projective modules of the same rank Fi,k © Mi +i / e( Mi ) — * Mi +i / gk+1 ( M i- k ) • This is then also injective. Hence Fitk —* Mi/Qk(M i- k ) is an isomorphism. The exactness on the left of the sequence asserted in the lem m a is immediate.

□

N O R M A L FO RM S OF L A T T IC E CHAINS Let us denote by Me- the Mi -

k {D)

k (D)

133

@ i?-module

Mi = M i / H M i =

C o ro lla ry A .3 For any integer i and any k such that 0 < k < r there is an exact sequence M i+k- r C : Mi -£■ M i+k Proof: Indeed by the lemma we have an injection: pk

Coker g

— Mi f M i ^ —v

* Mi+f~ / Mi+k—r = Mi+k

□ We obtain a trivial example of a chain of O b ® /^-modules if we tensor the chain C by R .

P r o p o s itio n A .4 Lei T be a scheme over Zp, such that p is locally nilpo­ tent on T. Let {Mi } be a chain of O b ® zp Or-modules of type (£). Then locally on T the chain {Mi} is isomorphic to C 0 R. Moreover, the functor on the category of T-schemes r

— A Aut ({M i ® R O t '})

is representable by a smooth group scheme over T. Proof: Let T = Spec R be affine. We may assume th at B — D is a division algebra. If N is an Oii-m odule we denote by N the J^-linear section s of the surjection Mi —+ M i / g ( M i - 1 ). Let Ui C Mi be the image of s. We may lift Ui to a direct sum m and Ui of the O d ® /^-module Mi. Clearly the Ui may be chosen to be periodic, i.e. for each i the morphism 9 induces an isomorphism 0 : Ui-r — u p . The m aps g induce an obvious map

134

A P P E N D IX TO C H A P T E R 3

o 0 Ui-k — ►M i . k=r—1

(A .l)

We claim th at this map is surjective. By Nakayam a’s lem m a we need to verify this modulo II. Let us denote by M- _k the image of M*_jb in Mi for the given i. We obtain a flag by direct summands 0 C M /_r+1 c • •• C M /_! C Mi ■

(A.2)

By the lem m a we have isomorphisms M i.kieiM i.i-!) ~

m

U /

m

U -

i

■

Hence the images of Ui-k in Mi define a splitting of the flag ( A.2). This shows th at (A .l) is a surjection mod n . Since (A .l) is a surjection of projective modules of the same rank it is an isomorphism. In term s of the £/* the map Mi —» Mt-+i looks as follows r —1

( J ) Ui-k k=0

r - 2

►( J ) Ui-kk=-1

On the sum m and Ui-k for k ^ r — 1 this map induces the identity to the corresponding sum m and on the right hand side. On t/s—r+i it induces the map n 0 : Ui- r+1 — ►£/*•+1. From this we see th at any two chains of type (£) are locally isomorphic, since locally on T the O d O (Tr-module Ui is free of the same rank as the Ac(D)-vector space A,-/A*_i. The representability of the functor of autom orphism s by a scheme of finite type over T is obvious. D e fin itio n A .5 We call the modules {Ui} a splitting of the chain {Mi}. They are characterized by the property that Ui is a direct summand of Mi such that Ui maps isomorphically to M i / g ( M i - i ) , and the Ui are periodic with respect to 6.

N O R M A L FO RM S OF L A T T IC E CHAINS

135

Let R —►S be a surjection with nilpotent kernel. Let {Vi} be a splitting of { M i } ® R S . Then any set of liftings of the V to direct sum m ands Ui of Mi which is periodic is a splitting for {Mi}. Therefore any given splitting {Vi} lifts. The formal smoothness of the functor A u t \s a consequence. Indeed, let a be an autom orphism of {Mi} ®r S. We find liftings Ui and U- of the splittings Vi respectively a(Vi). Then the isomorphism Vi —> a(Vi) lifts to an isomorphism Ui —+ Uj which is periodic. It gives the lifting of a to an autom orphism of {Mi}. This completes the proof of the proposition. □ A . 6 Let us now turn to the case of polarized chains (theorem 3.16). W ithout loss of generality we assume th at the invariants Fo of the involution * on F form a field. We do a case by case verification according to the following list. (I) F = F q x F o and the involution on F induces the obvious transposition. There is a central division algebra D over F q, such th at B = M n ( D) x M n (Dopp) and the involution on B is given by (du d2)* = {d2,di), (II) B = Mn (F ),

where

d \ )d2 G M n (D) = M n ( D) opp .

F = F0.

(III)

B = M n (F) and F / F q is a quadratic

extension.

(IV)

B = M n ( D ) where D is a quaternion

algebra overF and F = F q.

(We remark th a t on M n ( D) there are no involutions of the second kind). We note th a t the case where B = Q and £ is a m aximal selfdual chain was treated by de Jong [dJl]. A .7 Let us consider the case (I). We have the decomposition B ~ M n (D) x M n ( D opp).

(A.3)

136

A P P E N D IX TO C H A P T E R 3

We may choose the isomorphism (A.3) in such a way th at it takes the m ax­ imal order Ob to M u (Od) x M n { 0 0£ p). From (A. 3 ) we get a decomposition of the representation V : V = W®W. The spaces W and W are isotropic subspaces of V with respect to ( , ). Hence ( , ) puts these two spaces in duality W x W ~ ^ Q P. More precisely ( , ) identifies W with the dual space W* = HomQp(W, Qp) with its natural M n ( D opp)~action from the left. A multichain of Oj^-lattices in V is a pair £ , £ , where £ is a chain of M n ( O n )-lattices in W and £ is a chain of M n (0°pP)-lattices in W. The m ultichain is selfdual if and only if the two chains £ and £ are dual to each other. A polarized multichain of O b ® O t ~-modules on a scheme T of type (£, £) is a pair of chains, a chain of M h (O d ) 0 0T-m odules of type (£) and a chain of M n {0°pP) 0 C?T-modules of type (£), which are dual to each other. L e m m a A .8 The functor which associates to a polarized multichain {M\ } \ £ of type ( £ , £ ) the unpolarized chain { M a } a £jC ° f t]JPe (£) is an equivalence of categories. Proof: We have an obvious quasi-inverse functor: If A E £ and hence A* E £ , we put M a * = Homc»T ( M a , O t ) with the natural M n (0°pP) 0 z p 0T-m odule structure. We have to check th at M a * is of type £ . But this is obvious, since we know th at locally on T the chain { M a } is isomorphic to £ 0 R. □ This lem m a shows th at the case (I) of theorem (3.16) reduces to the unpo­ larized theorem (3.11). A .9 For the other cases we recall some basic facts on quadratic forms in the generality needed for the proof. Let R be a commutative unitary ring and 5 a unitary i?-algebra, which need not be commutative. Assume th at we are given an involution s i— ►s

N O R M A L FO RM S OF L A T T IC E CHAINS

137

of 5, i.e. a /^-algebra anti-isom orphism of order 2 on S. Let us assume th at 2 is invertible in R. Let M and N be left 5-modules and L a 5-bimodule. A sesquilinear form is a biadditive map $ : M x N — >L th a t satisfies the relations $(sm , n) = 3>(m, n)s 3>(ra, s n ) = s4>(m, n). We will say th at a 5-m odule is i7-locally free, if it is locally free with respect to the Zariski-topology on Spec R. We note th at we can view any right 5-m odule as a left5-m odule by restriction of scalars with respect to the involution. For example the right 5-m odule Horns - ( M , L ) becomes a left module by the rule (s X ® § C — ► T 05 Horn§ _ ( N , S ) j rsj R o m S- ( N , l ) ,

140

A P P E N D IX TO C H A P T E R 3

where the last isomorphism of left 5-modules maps i 0

Op r** We will see th at the chain Q is selfdual for a suitable inner product on W. P r o p o sitio n A. 16 Up to multiplication by a unit in 0 F there is a unique perfect bilinear form F : O nF x O nF — >0F that satisfies the equation F (x,y) = tx C y satisfies the equality (A.9). We set T = Op C F n . The dual lattice T* with respect to F = fF is the perfect pairing we are looking for. C o ro lla ry A .17 There is a perfect Zp -bilinear pairing : O nF x On F — * Zp that satisfies the equation (Ax,y) = (x,A*y),

x , y € 0 F , A e Mn ( 0 F) .

Any other pairing (j)f with this property is of the form '{x,y) = cf>(fx,y), for a suitable f E Op. Proof: Let ^ be a generator of the different ideal of F over Qp . Then the pairings and determine each other by the equation (/£> y) = t rF/

Q

y

)

■

□ P r o p o s itio n A. 18 Let M\ — Op 0 o F ^1 and M 2 = Op 0 o F N 2 be left Ob 0 R-modules which are projective and finite. Then there is a bijection between perfect R-bilinear forms £ : M i x M 2 — ►R , that satisfy the equation S (b * mi , ra2) = £( mi , 6m2),

beOb

and perfect O f 0 R-bilinear forms B : Ah x Ah — ►O f

R-

The forms B and £ determine each other uniquely by the equation £ (u i m , u 2 0 ^ 2) =

u 2# ( n i , n 2)),

riiENi, UiEOp.

144

A P P E N D IX TO C H A P T E R 3

We om it the easy proof. A slight modification of this proposition may be applied to our form ( , ) on V = F n 0 t W and provides us with a F-bilinear form ( on I f , which satisfies the equation:

(u 0 w, v! 0 w') = t > 0 let us denote by F - t the image of N - t in Nq/ttNo. We obtain a flag of O f /k O f 0 O t -modules, whose quotients are locally free on T, 0 C F _ r+1 C . . . C f 0 = No/it No-

(A .13)

A P P E N D IX TO C H A P T E R 3

148

Let Bo be the perfect pairing Bo modulo ir on N o / n N o .

L e m m a A .26 F - t is the orthogonal complement of F - r+t with respect to Bo. Proof: The orthogonal complement of F - r+t is the image of the following submodule of No in Fo: {no G N 0 ; Bo(Qt ( N ^ t ) , n 0) G 7rOf 0 T) =

{n0 G AT0 ;

e*(«o)) £ *’0 * ’ ®