p-adic uniformization of unitary Shimura varieties

Yakov Varshavsky

doi:10.1007/BF02698861

p-adic uniformization of unitary Shimura varieties

Varshavsky, Yakov 2007-08-30 00:00:00 by YAI~OV VARSHAVSKY Introduction Let F C PGUa_I.I(R) ~ be a torsion-free cocompact lattice. Then F acts on the unit ball Ba-IC C g-1 by holomorphic automorphisms. The quotient F\B d-1 is a complex manifold, which has a unique structure of a complex projective variety X r (see [Sha, Ch. IX, w 3]). Shimura had proved that when F is an arithmetic congruence subgroup, X r has a canonical structure of a projective variety over some number field K (see [Del] or [Mil]). For certain arithmetic problems it is desirable to know a description of the reduction of X r modulo w, where w is some prime of K. In some cases it happens that the projective variety X r has a p-adic uniformization. By this we mean that the K~-analytic space (X r | Kw) an is isomorphic to A\~ for some p-adic analytic sym- metric space f2 and some group A, acting on f~ discretely. Then a formal scheme structure on A\f~ gives us an 0Kw-integral model for X r | Kw" Cherednik was the first who obtained a result in this direction. Let F be a totally real number field, and let B/F be a quaternion algebra, which is definite at all infinite places, except one, and ramified at a finite prime v of F. Then Gherednik proved in [Ch2] that the Shimura curve corresponding to B has a p-adic uniformization by the p-adic upper half-plane f/~o, constructed by Mumford (see [Mum 1]), when the subgroup defining the curve is maximal at v. Gherednik's proof is based on the method of elliptic elements, developed by Ihara in [111]. The next significant step was done by Drinfel'd in [Dr2]. First he constructed cer- tain covers of f2 2 (see below). Then, when F = Q, he proved the existence of a p-adic uniformization by some of his covers for all Shimura curves, described in the previous paragraph, without the assumption of maximality at v. The basic idea of Drinfel'd's proof was to invent some moduli problem, whose solution is the Shimura curve as well as a certain p-adically uniformized curve, showing, therefore, that they are isomorphic. Developing Drinfel'd's method, Rapoport and Zink (see [RZ1, Ra]) obtained some higher-dimensional generalizations of the above results. In this paper we generalize Gherednik's method and prove that certain unitary Shimura varieties and automorphic vector bundles over them have a p-adic unifor- mizafion. Our results include all previously known results as particular cases. 58 YAKOV VARSHAVSKY We now describe our work in more detail. Let F be a totally real number field of degree g over Q,, and let K be a totally imaginary quadratic extension of F. Let D and D in~ be central simple algebras of dimension d 2 over K with involutions of the second kind ~ and s respectively over F. Let G := GU(D, ~) and Gint := GU(D~nt, ~lnt) be the corresponding algebraic groups of unitary similitudes (see Definition 2.1.1 and Notation 2.1.2 for the notation). Let v be a non-archimedean prime of F that splits in K, let w and ~ be the primes of K that lie over v, and let 0o 1 be an archimedean prime of F. Suppose that D ~nt | Kw has Brauer invariant 1/d, that D | Kw ~- Matd(Kw), and that the pairs (D, ~) | F~ and (D ~t, ~int)| F, are isomorphic for all primes u of F, except v and 0ox. Assume also that ~ is positive definite at all archimedean places F~ol---IlL of F, that is that G(Fo01) - GUa(R ) for all i = 1, . .., g, and that the signature of s at 0ol is (d -- 1, 1), so that Gin~(F~) - GUa_I,I(R). Let Av I and Av f'* be the ring of finite adeles of F and the ring of finite adeles of F without the v-th component respectively. Set E' := F~ � G(Avf'~), and fix a central simple algebra Dw over K~ of dimension d s with Braner invariant lid. Then ') ~� E' ~ D. X and G(A~) = GLa(K~) x E'. In particular, the group GLa(K~) acts naturally on G(A~) by left multiplication. Let f2 a be the Drinfel'd's (d- 1)-dimensional upper half-space over K~ cons- Kw tructed in [Drl], and let { 2]~a"~}~su{0} be the projective system of 6tale coverings of f~ constructed in [Dr2]. This system is equipped with an equivariant action of the ~X � group GLd(Kw) x D~ such that if T~ denotes the n-th congruence subgroup of ~fi~, then we have T \E a' ~ ~ E d' ~ for all m >t n (see 1.3 1 and 1.4.1 for our notation and n\ K w ~ K w conventions, which differ from those of Drinfel'd). Denote by Gl~t(F)+ the set of all d~ (D~t) x such that d.~ht(d) is a totally positive element of F. Choose an embedding K ,-+ C, extending 0ol : F c.+ R. It defines us an embedding G~t(F)+ ~ GUa_I,I(R)~ = Aut(Ba-~). Choose finally an embedding of K~ into C, extending that of K. For each compact and open subgroup S of E' and each non-negative integer n let Xs, ~ be the weakly-canonical model over K~ of the Shimura variety corresponding to the complex analytic space (T, X S)\[Bd-1 � Gi~t(A~)]/Gt~t(F)+ and to the morphism h: S -+G ~"t| described in 3.1.1 (see Definition 3.1.12 and Remark 3.1.13 for the definitions). The experts might notice that our h is not the one usually used in moduli problems of abelian varieties. Let Vs, ~ be the canonical model of the automorphic vector bundle on Xs, ~ (see [Mil, III] or the last paragraph of the proof of Proposition 4.3.1 for the definitions), corresponding to the complex analytic space (T, x S)\[~(WtW| C)~" X GI"t(A~)]/G~t(F)+ (see 4.1.1 for the necessary notation). p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 59 Let Ps, ~ be the canonical model of the standard principal bundle over Xs, (see [Mil, III] or Corollary 4.7.2 for the definitions), corresponding to the complex analytic space (T~ � S)\[B a-~ � (PG~t| ~ � G~t(A~)]/G'~t(F)+ (see 4.1.1 for the necessary notation). Main Theorem. -- For each compact and open subgroup S of E' and each n e N u { 0 } we have isomorphisms of K~-analytic spaces: a) ,tXs,./a" ~ GLa(Kw)\[Z~" � (S\G(A~)/G(F))]; b) (Vs,.)an -% GLa(Kw)\[[3~o ' .(W ~) � (S\G(Arl)/G(F))] (see 4.1.1 for the necessary notation), where the group GL~(K,o ) acts on ~, ,,(W ~") as the direct factor of (G| Kw) (K.,), corresponding to the natural embedding K ~ Kw; c) (Ps,,)~-% [GLa(Kw)\[Z~ � ((PG| ~n � (S\G(A~)))/G(F)]] TM (see 4.1.1 for the definition of the twisting ( )t,,), where the group GLe(K~) acts trivially on (PG | Kw) ~. These isomorphisms commute with the natural projections for S 1 Q S2, n 1 ) n 2 and with the action of G~t(As - D~ X F x X G(Ar~:~). The idea of the proof is the following. Consider the p-adic analytic varieties ~s, of the right hand side of a) of the Main Theorem. They form a projective system and each of them has a natural structure Ys, ~ of a projective variety over K w. Kurihara proved in [Ku] that for every torsion-free cocompact lattice F C PGLa(Kw) the Chern numbers of F\~ are proportional to those of the (d- 1)-dimensional projective space and that the canonical class of F\~K~ is ample. The result of Yau (see [Ya]) then implies that B a-x is the universal covering of each connected component of the complex analytic space (Ys,.| C) ~n for all sufficiently small S e ~'(E) and all embeddings K w r C. It is technically better to work with the inverse limit of the Ys, ~'s equipped with the action of the group Gi~t(A~) -~ D~ ~� � E' on it rather then to work with each Ys, separately. Generalizing the ideas of Cherednik [Ch2] we prove that there exists a subgroup A C GUa_~,~(R) ~ � Gmt(A~) such that (Ys,.| C)~n~ (T. � S)\(B d-~ � G'~t(A~r))/A for all compact open subgroups S C E' and all n ~ N w { 0 }. Using Margulis' theorem on arithmeticity we show that the groups A and Gint(F)+ are almost isomorphic modulo centers. More precisely, we show that (Ys, ,, | C) ~n is isomorphic to a finite covering of (Xs,, | (])an" Using Kottwitz' results [Ko] on local Tamagawa measures we find that the volumes of (Ys,~| ~ and (Xs,.Qx~ C) ~n are equal. It follows that the varieties Ys,~| C and XS,.| C are isomorphic over C. Comparing the action of the Galois group on the set of special points on both sides we conclude that Ys, ~ and Xs,. are actually isomorphic over K w. 60 YAKOV VARSHAVSKY Notice that if one considers only Shimura varieties corresponding to subgroups which are maximal at w, then the use of Drinfel'd's covers in the proof of the p-adic uniformization is very minor. (We use them only for showing that the p-adically uni- formized Shimura varieties have Brauer invariant 1/d at w; that probably can be done directly.) In this case the proof would be technically much easier but contain all the essential ideas. The proof of the p-adic uniformization of standard principal bundles is similar. In addition to the above considerations it uses the connection on principal bundles. Using the ideas from [Mil, III] we show that the p-adic uniformization of standard principal bundles implies the p-adic uniformization of automorphic vector bundles. In fact Tannakian arguments show (see [DM]) that these statements are equivalent. This paper is organized as follows. In the first section we introduce certain cons- tructions of projective systems of projective algebraic varieties, give their basic properties and do other technical preliminaries. In the second section we give two basic examples of such systems. Then we for- mulate and prove the complex version of our Main Theorem for Shlmura varieties. The third and the forth sections are devoted to the proof of the theorem on the p-adic uniformization of Shimura varieties and of automorphic vector bundles respectively. Our proof appears to be very general. That is starting from any reasonable p-adic symmetric space, whose quotient by an arithmetic cocompact subgroup is algebraizable, we find Shimura varieties uniformized by it. For example, in another work ([Va]) we extend our results to Shimura varieties uniformized by the product of Drinfel'd's upper half-spaces. Hence it would be interesting to have more examples of such p-adic sym- metric spaces. Our result on the p-adic uniformization of automorphic vector bundles is not complete, because we prove the p-adic uniforrnization only under the assumption that the center acts trivially. In fact our proof of the complex version of the theorem works also in the general case, but to get an isomorphism over K w one should understand better the action of the Galois group on the set of special points. After this work was completed, it was pointed out to the author that Rapoport and Zink have recently obtained similar results concerning the uniformizafion of Shimura varieties by completely different methods (see [RZ2]). Notation and conventions 1) For a group G let Z(G) be the center of G, let PG :----- G/Z(G) be the adjoint group of G, and let G d~ be the derived group of G. 2) For a Lie group or an algebraic group G let G o be its connected component of the identity. 3) For a totally disconnected topological group E let ~'(E) be the set of all compact and open subgroups of E, and let E dl~ be the group E with the discrete topology. p-ADIG UNIFOlZMIZATION OF UNITARY SHIMURA VARIETIES 61 4) For a subgroup F of a group G let Comma(F ) be the commensurator of F in G. 5) For a subgroup F of a topological group G let F be the closure of F in G. 6) For a set X and a group G acting on X let X G be the set of all elements of X fixed by all g e G. 7) For a set X, a subset Y of X and a group G acting on X let Staba(Y ) be the set of all elements of G mapping Y into itself. 8) For an analytic space or a scheme X let T(X) be the tangent bundle on X. 9) For a vector bundle V on X and a point x e X let V~ be the fiber of V over x. 10) For an algebra D let D ~ be the opposite algebra of D. 11) For a finite dimensional central simple algebra D over a field let SD � be the subgroup of D � consisting of elements with reduced norm 1. 12) For a number field F and a finite set N of finite primes of F let /k~ be the ring of finite adeles of F, and let/k~ :N be the ring of finite adeles of F without the com- ponents from N. 13) For a field extension K/F let R~/r be the functor of the restriction of scalars from K to F. 14) For a natural number n let I s be the n � n identity matrix and let B"C C" be the n-dimensional complex unit ball. 15) For a scheme X over a field K and a field extension L of K write XT. or X | L instead of X � s~ K Spec L. 16) For an analytic space X over a complete non-archimedean field K and a for a complete non-archimedean field extension L of K let X @x L be a field extension from K to L. (A completion sign will be omitted in the case of a finite extension.) 17) By a p-adic field we mean a finite field extension of O~ for some prime number p. Let C~ be the completion of the algebraic closure of O~. 18) By a p-adic analytic space we mean an analytic space over a p-adic field in the sense of Berkovich [Be1]. 19) For an affinoid algebra A let d/(A) be the affinoid space associated to it. Acknowledgements First of all the author wants to thank Professor R. Livn6 for formulating the problem, for suggesting the method of the proof and for his attention and help during all stages of the work. He also corrected an earlier version of this paper. I am also grateful to Professor V. Berkovich for his help on p-adic analytic spaces, to Professor J. Rogawsld for the reference to [C1], to Professor Th. Zink for his corrections and interest, and to the referee for his suggestions. The work forms part of the author's Ph.D. Thesis in the Hebrew University of Jerusalem, directed by Professor R. Livn6. The revision of the paper was done while the author enjoyed the hospitality of the Institute for the Advanced Study at Princeton and was supported by the NSF grant DMS 9304580. 62 YAKOV VARSHAVSKY CONTENTS 1. Basic definitions and constructions ......................................................... 1.1. General preparations ................................................................. 1.2. GAGA results ....................................................................... 66 1.3. First construction .................................................................... 68 1.4. Drinfel'd's covers .................................................................... 72 1.5. Second construction .................................................................. 1.6. Relation between the p-adic and the real constructions .................................. 1.7. Elliptic elements ..................................................................... 1.8. Euler-Poincar6 measures and inner twists ............................................... 1.9. Preliminaries on torsors (= principal bundles) ........................................... 2. First Main Theorem ...................................................................... 87 2.1. Basic examples ...................................................................... First Main Theorem ................................................................. 9O 2.2. 2.3. Computation of Q(TrAd) ............................................................ 93 2.4. Proof of arithmetieity ................................................................ 95 2.5. Determination of H .................................................................. 97 2.6. Completion of the proof .............................................................. 100 3. The theorem on the p-adic uniformization .................................................. 102 3.1. Technical preliminaries ............................................................... 3.2. Theorem on the p-adic uniformization ................................................. 4. p-adic uniformization of automorphic vector bundles ......................................... 107 4. I. Equivariant vector bundles ........................................................... 107 4.2. Equivariant torsors ................................................................... 4.3. Connection between the Main Theorems ............................................... 110 4.4. Reduction of the problem ............................................................. 4.5. Proof of density ..................................................................... 113 4.6. Use of rigidity ...................................................................... 116 4.7. Rationality question .................................................................. 117 References ................................................................................. 118 1. BASIC DEFINITIONS AND CONSTRUCTIONS 1.1. General preparations Definition 1.1.1. -- A locally profinite group is a locally compact totally disconnected topological group. In such a group E, the set ~'(E) forms a fundamental system of neighbourhoods of the identity element, and f] S = { 1 }. s ~ ,~'(E) Lemma 1.1.2. -- Let E be a locally profinite group, and let X be a separated topological space with a continuous action E � X ~ X of E. For each S e ~'(E), set X s :----- S\X. Then { Xs }s is a projective system and X ~ lira X s . <s Proof. -- [Mil, Ch. II, Lem. 10.1]. [] This lemma motivates the following definition. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 63 Definition 1.1.3. -- Let X be a separated scheme over a field L, let E be a locally profinite group, acting L-rationally on X. We call X an (E, L)-scheme (or simply an E-scheme if L is clear or is not important) if for each S ~ o~-(E) there exists a quotient X s := S\X, which is a projective scheme over L, and X- li m X s . The following remarks show that E-schemes are closely related to projective systems of projective schemes, indexed by o~(E). Remark 1.1.4. -- If X is an E-scheme or merely a topological space with a conti- nuous action of E, then for each gee and each S, Teo~'(E) with SDgTg -~ we have a morphism Ps,~(g) : XT -+ Xs, induced by the action of g on X and satisfying the following conditions: a) Ps, s(g)-----Id ifgeS; b) Ps,T(g) o pT,~(h) = ps,~(gh); c) if T is normal in S, then 9T, T defines the action of the finite group S/T on X~, and X s is isomorphic to the quotient of X. r by the action of S/T. Remark 1.1.5. -- Conversely, suppose that for each S e ~-(E) there is given a scheme Xs, and for each g e E and each S, T ~ o~'(E) with S D gTg -1, there is given a morphism Ps,T(g):XT-+Xs, satisfying the conditions a)-c) of 1.1.4. Then for each T C S there is a map Ps,T(1) : X T -+ Xs, which is finite, by condition c). In this way we get a projective system of schemes and we can form an inverse limit scheme X := lira Xs. Then there is a unique action of E on X such that for each g e E and <s each S e ~'(E) the action of g on X induces an isomorphism pgso_l,s : Xs-% Xgsg_l. It follows from c) that X s -% S\X for each S e ~-(E). Definition 1.1.6. -- Let E be a topological group, which is isomorphic to E under an isomorphism qb : E -~ E. We say that an (E, L)-scheme X is ~-equivariantly isomorphic to an (F,, L)-scheme X if there exists an isomorphism q~ : X -% .~ of schemes over L such that for each g s E we have q0 o g ----- 4)(g) o q0. If in addition E = E and q0 is the identity, then we say that q0 is an isomorphism of (E, L)-schemes. Definition 1.1.7. -- Let L~/L1 be a field extension. We say that an (E, L1)-scheme X is an LJLl-descent of an (E, L,)-scheme Y if the (E, L2)-schemes Xz~ and Y are isomorphic. Suppose from now on that E is a noncompact locally profinite group. Notation 1.1.8. -- For a topological group G and a subgroup I" C G � E let pr o and pr~. be the projection maps from F to G and E respectively. Set I" o := pro(r), F~ := pr~,(P) and r s := pr (r (G x S)) for each S e #'(E). For each y s r set % := prQ(y) and y~ := pr~(y). 64 YAKOV VARSHAVSKY Lemma 1.1.9. -- Let F C G x E be a cocompact lattice. Suppose that pr G is injective. Then for each S E ~'(E) we have the following: a) [ s\E/r I< oo; b) [F G : Fs] = oo; c) F s is a cocompact lattice of G; d) F G C CommG(Fs). Proof. -- a) Since the double quotient (G x S)\(G x E)/F ~ S\E/F~. is compact and discrete, it is finite. b) The group E is noncompact, therefore [SkE I = oo. Hence, by a), :s n = I s\sr I = oo. But F a = prG(F ) = prG(pr~l(FE)), and likewise F s = prG(pr~l(F~ n S)). Since pr o is injective, we are done. c) The group F is a cocompact lattice in G X E, hence F n (G x S) is a cocompact lattice in G x S, and the statement follows by projecting to G (see [Shi, Prop. 1.10]). d) Let y e F, and set S' = y~. SyE 1 e ~-(E). Then y(Fn(G � S))y-~=Fn(G � S'). But S n S' ~o~-(E) is a subgroup of finite index in both S and S', hence YG Ps Y~l n F s = F sns, is a subgroup of finite index in both P s and YG Ps Y~l" [] Suppose that d >/ 2 and take G equal to PGLa(Kw) for some p-adic field Kw or to PGUa_I,~(R) ~ We shall call these the p-adic and the real (or the complex) cases respectively. Proposition 1.1.10. q Under the assumptions of Lemma 1.1.9 we have: a) r G D Gd~ b) pr~ is injective; c) for each S e ~-(E), the group r s is an arithmetic subgroup of G in the sense of Margulis (see [Ma, p. 292]); d) if S e ~-(E) is sufficiently small, then the subgroup rasa_~ is torsion-free for each a e E. Proof. -- a) For each S e ~'(E), F s is cocompact in G and [FG: Fs] = oo. It follows that F G is a closed non-discrete cocompact subgroup of G. Therefore its inverse image n-l(FG) in SU a_ 1,1(R) (resp. SLa(Kw) ) is also closed, non-discrete and cocompact, hence by [1Via, Ch. II, Thin. 5.1] it is all ofSUa_l,l(R ) (resp. SLa(Kw) ). This completes the proof. b) Set F 0 := pro(Ker prE). This is a discrete (hence a closed) subgroup of G, which is normal in F o. Therefore it is normal in F G D G a~ It follows that each y e F 0 must commute with some open neighborhood of the identity in G aer, hence F 0 is trivial. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 65 c) is a direct corollary of [Ma, Ch. IX, Thm. 1.14] by b)-d) of Lemma 1.1.9. d) (compare the proof of [Ch. 1, Lem. 1.3]). Choose an S e~(E), then F s C G is a cocompact lattice. Lemma 1.1.11. -- The torsion elements of F s comprise a finite number of conjugacy classes in F s. We first complete the proof of the proposition assuming the lemma. Let ax, ..., a~ E E be representatives of double classes FE\E/S (use Lemma 1.1.9, a)). For each i = 1, ..., n let M~ C F~i sa~l be a finite set of representatives of conjugacy classes of torsion non-trivial elements of Fois~Tt. Then the image of all non-trivial torsion elements of F~is~rl under the natural injection j, : PalS~7~ --% F n (G � a~ Sa~ -1) ,-+ a~ Sa~ -1 --% S is contained in the set X, ={s.j,(M3.s-l[s ~S}, which is compact and does not contain 1. Hence there exists T e .~'(E) not intersecting any of the X,'s. By taking a smaller subgroup we may suppose that T is a normal subgroup of S. Since all the j~'s are injective, the subgroup F~iT,:I =ji-I(T) is torsion-free for each i = 1, ..., n. For each ace there exist i~{1,...,n}, seS and y~F such that a=y~.a~s. Hence the subgroup F~r~_t ~- r c~ (G � aTa- 1) = F c~ ('to Gy~- 1 � y~ a, Ta~- 1 y~ 1) 7(F r~ (G � a~ Tab-l)) 7 -1 _-_ Falr~Zx is torsion-free. O Proof (of the lemma). -- The group G acts continuously and isometrically on some complete negatively curved metric space Y. Indeed, in the real case Y = B d-1 with the hyperbolic metric. In the p-adic case Y is a geometric realization (see [Br, Ch. I, appendix]) of the Bruhat-Tits building A of SLa(K~, ). This is a locally finite simplicial complex of dimension d- 1 which can be described as follows. Its vertices are the equivalence classes of free ~Kw-submodules of rank d of the vector space K~, where M and N are said to be equivalent when there exists a z K~, such that M = aN. The distinct vertices A1, As, 9 9 At form a simplex when there exist for them representa- tive lattices M1, M~, ..., Mk, such that Mx D Ms D ... 9 M~ D rr For more informa- tion see [Mus, w 1] or [Br, Ch. V, w 8]. The geometric realization Y of A has a canonical metric, that makes Y a complete metric space with negative curvature (see [Br, Ch. VI, w 3]). Moreover, the natural action of PGLa(K~) on the set of vertices of A can be (uniquely) extended to the tim- plicial, continuous and isometric action on Y. Now the Bruhat-Tits fixed point theorem (see [Br, Ch. VI, w 4, Thm. 1]) implies that any compact subgroup of G has a fixed point on Y. In particular, any torsion element of G has a fixed point on Y. Notice that in the p-adic case it then stabilizes the minimal simplex, containing the fixed point. 66 YAKOV VARSHAVSKY Conversely, the stabilizer in G of each point of Y is compact. In the real case this is true, since the group PGUd_I,x(R) ~ acts transitively on B e-1 and the group K =-StabBd-l(0)- Ud_x(R ) is compact. In the p-adic case the group PGLe(Kw) acts transitively on the set of vertices, and the stabilizer of the equivalence class of ~w C K~ is PGLd(O~w), hence it is compact. Since the stabilizer in G of any point y e Y must stabilize the minimal simplex ~ containing y, it must permute the finitely many vertices of a, so that it is also compact. It follows that the stabilizer of any point of Y in r s is compact and discrete, hence it is finite. To finish the proof of the lemma in the real case we note that for each x ~ B d- ~ there exists an open neighbourhood U, of x such that P, := {g ~ Us I g(U,) n U,, 0 } ={g e F s I g(x) = x} is finite (see [Shi, Prop. 1.6 and 1.7]). The space rs\B e-x is compact, hence there exist a finite number of points xl, x2, . .., x~ of B e-a, such that I's(lJ~= 1 U~i) --- B e-x. If y is a torsion element of Fs, then it fixes some point of B e-a. By conjugation we may assume that it fixes a point in some U~i , therefore y is conjugate to an element of the finite set [.J'~=~ F,i. In the p-adic case we first assert that A has only a finite number of equivalence classes of simplexes under the action of Ps- Since A is locally finite, it is enough to prove this assertion for vertices. The group G acts transitively on the set of vertices, and G = P s 9 K for some compact set K C G. Hence if v is a vertex of A, then K. v is a compact and discrete (because the set of all vertices of A is a discrete set in Y) subset of Y, and our assertion follows. Now the same considerations as in the real case complete the proof. [] 1.2. GAGA results In what follows we will need some GAGA results. Let L be equal to K w in the p-adic case and to C in the complex case. We will call both the complex and the p-adic (L-)analytic spaces simply (L-)analytic spaces. Recall that for each scheme X of locally finite type over L and each coherent sheaf F on X a certain L-analytic space X ~n and a coherent analytic sheafF ~n on X ~ can be associated (see [Bel, Thm. 3.4.1] in the p-adic case and [SGA1, Exp. XII] in the complex one). Theorem 1.2.1. -- Let X be a projective L-scheme. The functor F ~ F ~n from the category of coherent sheaves on X to the category of coherent analytic sheaves on X "n is an equivalence of categories. Proof. -- In the complex case the theorem is proved in [Sel, w 12, Thm. 2 and 3], in the p-adic one the proof is the same. One first shows by a direct computation that the p-adic analytic and the algebraic cohomology groups of P" coincide. Next, one concludes from Kiehl's theorem (see [Bel, Prop. 3.3.5]) that the cohomology group p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 67 of an analytic coherent sheaf on P" is a finite-dimensional vector space. Now the argu- ments of Serre's proof in the complex case hold in the p-adic case as well. See [Be1, 3.4] for the relevant definitions and basic properties. [] Corollary 1.2.2. ~ a) If X is an algebraic variety over L and X' is a compact L-analytic subvariety of X, then X' is a proper L-algebraic subvariety of X. b) The functor which associates to a proper L-scheme X the analytic space X ~ is fully faithful. Proof. -- Serre's arguments (see [Sel, w 19, Prop. 14 and 15]) hold in both the complex and the p-adic cases. [] Corollary 1.2.3. -- Let X be a projective L-scheme. The functor X' ~ (X') an induces an equivalence between: a) the category of vector bundles of finite rank on X and the category of analytic vector bundles of finite rank on X~; b) the category of finite schemes over X and the category of finite L-analytic spaces over X ~, if L is a p-adic field. Proof. -- a) To prove the statement we first notice that the category of vector bundles of finite rank is equivalent to the category of locally free sheaves of finite rank. In the algebraic case this is proved in [Ha, II, Ex. 5.18]. In the analytic case the proof is similar. Now the corollary would follow from the theorem if we show that locally free analytic sheaves of finite rank correspond to locally free algebraic ones. The analytic structure sheaf is faithfully flat over the algebraic one (see [Sel, w 2, Prop. 3] and [Bel, Thm. 3.4.1]). Therefore the statement follows from the fact that an algebraic flat coherent sheaf is locally free (see [Mi2, Thm. 2.9]). b) We first show that the correspondence (v:Y-+X) ~q~.(0y) (resp. (~ : Y -+ X ~n) ~ ~.(0~)) gives an equivalence between the category of finite schemes (resp. analytic spaces) over X (resp. X ~n) and the category of coherent 0 x -- (resp. Ox~ --)algebras. In the algebraic case this is proved in [Ha, II, Ex. 5.17]. In the analytic case the proof is exactly the same, because a finite algebra over an affinoid algebra has a canonical structure of an affinoid algebra (see ['Bel, Prop. 2.1.12]). [] Remark 1.2.4. R If X' is finite over X, then it is projective over X, therefore if, in addition, X is projective over Kw, then X' is also projective over K w. Corollary 1.2.5. -- Let X and Y be projective L-schemes, and let W and V be algebraic vector bundles of finite ranks on X and Y respectively. Then for each analytic map of vector bundles f: W an --> V an covering some map f: X ---> Y there exists a unique algebraic morphism g : W ---> V such that g~n =~ 68 YAKOV VARSHAVSKY Proof -- By definition, 37factors uniquely as W ~" Y> V ~" x y.. X ~" _-__ (V x r X) ~" oroj> V~" Corollary 1.2.3 implies that there exists a unique g' : W -+ V x y X such that (g')~" = ~". Set g := proj o g'. For the uniqueness observe that if h : W -+ V satisfies h " = ~, then it covers f Hence h factors as W h' p~oj> > V � y X V. Since 3 ~ and g' are unique, we have h' = g' and h= g. [] Remark 1.2.6. -- Using the results and ideas of [SGA1, Exp. XII] one can replace in the above results the assumption of projectivity by properness. We now introduce two constructions of E-schemes which are basic for this work. 1.3. First construction 1.3.1. Let D~ be an open Kw-analytic subset of (~ ~)~, obtained by removing from (PaK~I)" the union of all the Kw-rational hyperplanes (see [Bel] and [Be3] for the definition and basic properties of analytic spaces). It is called the (d -- 1)-dimensional Drinfel'd upper half-space over K~ (see also [Drl, w 6]). Then f~d is the genetic fiber Kw of a certain formal scheme ~a over OK~, constructed in [Mus, Ku], generalizing [Muml]. The group PGLd(K~) acts naturally on D~. (It will be convenient for us to consider pa-~ as the set of lines in A a and not as the set of hyperplanes, as Drinfel'd does. Therefore our action differs by transpose inverse from that of Drinfel'd.) Moreover, this action naturally extends to the 0~: -linear action of PGI-,d(K,o ) on ~. Further- more, PGLa(K~) is the group of all formal scheme automorphisms of ~a xw over OK~ (see [Mus, Prop. 4.2]) and of all analytic automorphisms of f~a over K~ (see [Be2]). Though the action of PGLd(Kw) on f~d is far from being transitive, we have the Kw following Lemma 1.3.2. -- There is no non-trivial closed analytic subspace of D~QK~ " C~, invariant under the subgroup := ~ e Ka~ -~ C PGLd(K,~ ). u= \- pil Proof. M Suppose that our lemma is false. Let Y be a non-trivial U-invariant closed analytic subset of f~a xw @~w C~. Then dim Y < dim ~d w (~Xw C~ = d -- 1. Choose a regular point y a Y(C~) (the set of regular points is open and non-empty). Then dim Tu(Y) = dim Y < d -- 1. Next we identify D~ ~)Kw C~ with an open t..% j by the map (za:... :z a)~-~ z~, . Then analytic subset of tAa-l,,, ( ...,za~ 1] \Zd Zd / p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 69 U~(z) = z + ~- for every z e k,~ -1 and every ~ e M-~(K~). In particular, y + Y e Y for every ~ e Aa-~(K.), contradicting the assumption that dim Tu(Y) < d- 1. [] Recall also that the group PGUd_I,I(R) ~ acts transitively on B d-1 and that it is the group of all analytic (holomorphic) automorpkisms of B a-a (see [Ru, Thm. 2.1.3 and 2.2.2]). In what follows we will need the notion of a pro-analytic space. Definition 1.3.3. -- A pro-analytic space is a projective system { X~ }~ei of analytic spaces such that for some ~0 e I all transition maps q0~ : X~ -+ X~; ~ 1> e i> ~0 are Etale and surjective. Definition 1.3.4. -- By a point of X:={X~}~e I we mean a system {Xe}aEi, where x, is a point of X~ for all e e I and %~(x~) = x~ for all ~/> ~ in I. For a point x=(x=}~e I ofX={X~}~e Ilet T.(X) :--= ( v = ( v. }~e~ I v~ e T.~(X.), d~.~(v~) = v~, for all fi i> e in I } be the tangent space of x in X. Definition 1.3.5. -- Let X = { X~ }~eI and Y = { Y~ }~ea be two pro-analytic spaces. To give a pro-analytic morphism f: X-,--Y is to give an order-preserving map a:I ~J, whose image is cofinal in J, and a projective system of analytic morphisms f~ : X~ -+ YoI~" A morpkism f is called #tale if there exists a0 e I such that for each >/ ~0 the morphism f~ is 6tale. Construction 1.3.6. -- Suppose that PC G � E satisfies the conditions of Lemma 1.1.9. We are going to associate to I' a certain (E, L)-scheme. Let X ~ be B a- 1 in the real case and f~d in the p-adic one. Consider the L-analytic Kw space .~ := (X ~ � Edi~)/I ', where IF' acts on X ~ � E di~ by the natural right action: (x, g) y := (y~ 1 x, gya). Then E acts analytically on X by left multiplication. Proposition 1.3.7. -- For each S e o~'(E) the quotient S\X = S\(X ~ � E)/I' exists and has a natural structure of a projective scheme X s over L. Proof. -- First take S ~ o~(E) satisfying part d) of Proposition 1.1.10. Then S\'-K has I S\E/I'E[ < oo connected components, each of them is isomorphic to Pas~_t\X ~ for some a e E. By c), d) of Proposition I. t. 10, each l',s~_l is a torsion-free arithmetic cocompact lattice of G. By [Ski, Prop. 1.6 and 1.7], [Sha, Ch. IX, 3.2] in the real case and by [Mus] or [Ku] in the p-adic one, each quotient I',s,_l\X ~ exists and has a unique structure of a projective algebraic variety over L. Therefore there exists a projective scheme X s over L such that X~s ~ ~ S\X 70 YAKOV VARSHAVSKY Take now an arbitrary S e o~(E). It has a normal subgroup T e o~(E) which satisfies part d) of Proposition 1.1.10. The finite group S/T acts on T\X ~ X~ n by analytic automorphisms and S\I~ ~ (S/T)\X~". Corollary 1.2.2 implies that the analytic action of S/T on X~ " defines an algebraic action on X T and that the projective scheme X s := (S/T)\X~ (the quotient exists by [Mum2, w 7]) satisfies (Xs) ~" -_ S\X. Moreover, the same corollary implies also that the algebraic structure on S \X is unique. [] For all geE and all S,T e o~-(E) with S DgTg -~ we obtain by Remark 1.1.4 analytic morphisms ps,~(g):X~ n -+Xh ". They give us by Corollary 1.2.2 uniquely determined algebraic morphisms Ps, ~(g) : XT -+ Xs, which provide us by Remark 1.1.5 an (E, L)-scheme X := lim X s. Proposition 1.3.8. -- a) There exists the inverse limit X an of the X~='s in the category of L-analytic spaces, which is isomorphic to X. b) Stabr(X ~ � { 1 }) = Ft. c) Let X o be the connected component Of X such that X~" D X ~ � { 1 } (note that X ~ � { 1 } is a connected component of X ~, and that the analytic topology is stronger then the Zariski topology). Then StabE(Xo) ---- I'~.. d) The group E acts faithfully on X. e) For each x ~ X the orbit E.x is (geometrically) Zariski dense. In particular, E acts transitively on the set of geometrically connected components of X. f) For each S e ~'(E) satisfying part d) of 1.1.10, the map X -+ X s is dtale; g) For each embedding K~ ~ C and each S e ~'(E) as in c), B a-1 is the universal covering of each connected component of (Ks, c)a" in the p-adic case and of X~ in the complex one. Proof. -- a) We start from the following Lemma 1.3.9. -- a) Let H be a torsion-free discrete subgroup of G. Then the natural projection X~ II\X ~ is an analytic (topological) covering. b) For each x e X ~ the stabilizer of x in G is compact. Proof. -- a) follows from [Ski, Prop. 1.6 and 1.7] in the real case and from [Be2, Lem. 4 and 6] in the p-adic one. b) By [Dr2, w 6] there exists a PGLa(K~)-equivariant map from f~e to the Kw Bruhat-Tits building A of SLd(K~) , thus it suffice to show the required property for stabilizers of points in A and B e. 1. This was done in the proof of Lemma 1.1.11. [] The lemma implies that for each sufficiently small S e o~-(E) tile analytic space X~ admits a covering by open analytic subsets U~ satisfying the following condition: for each i and each subgroup S D T e ~'(E) the inverse image 0~-I(U,) of U, under the natural projection pz:X~n-+ X~ n splits as a disjoint union of analytic spaces, each of them isomorphic to U, under p~. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 71 Now we will define a certain L-analytic space X" associated to X. As a set it is the inverse limit of the underlying sets of the X~n's. To define an analytic structure on X" consider subsets V~ C X an such that for some (hence for every) sufficiently small S ~'(E), the natural projection z% :X~n--> X~ n induces a bijection of V~ with an open analytic subset ns(V~) of X~ n, described in the previous paragraph. Provide then such a V~ with an analytic structure by requiring that ns : V~ -+ ~s (V~) is an analytic isomorphism. Then the analytic structure of the V~'s does not depend on the choice of the S's, and there exists a unique L-analytic structure on X ~" such that each V~ is an open analytic subset of X ~n. By the construction, X" is the inverse limit of the X~n's in the category of L-analytic spaces. Hence there exists a unique E-equivariant analytic map ~ : X --> X ~ such that ~ Ra n r~s for each S e ~'(E) the natural projection X -+ X~" factors as X --~ --> X~ n , where by ns we denote the natural projection. It remains to show that n is an isomorphism. For each S ~ ~'(E) satisfying part d) of Proposition 1.1.10, the natural projection X~ Ps\X ~ is a local isomorphism, hence the projections X ~ X s and n are local isomorphisms as well. The map ~s o ~ is surjective, hence for each x ~ X "n there exists a point y ~ such that ~zs(x ) -- rc s o n(y). Therefore, ~(y) = sx for some s ~ S. Since n is E-equi- variant, we conclude that rc(s-l(y)) = x. Hence n is surjective. Suppose that n(Yl) = n(Y2) for someyl,y 2 ~ .X. Let (Xx, gx) and (x2, g~) be their representatives in X ~ � E. Then for each S ~ ~'(E) there exist s ~ S and y ~ I ~ such that xl = y~ ~ (x2) and gl = sg, Yr.. Such ya's belong to the set { g ~ G [ g(Xx) = X 2 } (~ I~l SOl, which is compact (by the lemma) and discrete, hence finite. Therefore we can choose sufficiently small S ~ ~(E) such that gl y~l g~-~ ~ s ~ S must be equal to 1. This means that Yl =Y~. Thus 7: is a surjective, one-to-one local isomorphism, hence it is an isomorphism. b) is clear. c) For each S e o~'(E) let Ys be the connected component of X s such that Y;" is the image of X ~ � { 1 } C X ~" under the natural projection ~s : X" --> X~ =. Then X 0=limYs. It follows that gee satisfies g(X0) =X 0 if and only if g(Ys) =Ys <S for each S zoO'(E) if and only if X ~ x {g} C S(X ~ x 1) P for each S e o~'(E) if and only ifgeSF~, for each Scot'(E) if and only ifge [7 SFr = F--~. S ~ ~'~ E) d) If g ~ E acts trivially on X, then it acts trivially on X an ~ (X ~ � E a~c)/F. By b), g = yz for some y ~ F, and Yo acts trivially on X ~ Since pr~ is injective, y -- g = 1. e) Let Y be the Zariski closure of E.x. Then Y is E-invariant and, therefore, yan C~ (X ~ � is a closed IMnvariant analytic subspace of X ~ � X ~ By Proposition 1.1.10 a), it is Ga~Mnvariant. Since G a~" acts transitively on X ~ in the real case and by Lemma 1.3.2 in the p-adic one, Y" c~ (X ~ � { 1 }) has to be all of X ~ � { 1 }. It follows that Y = X. 72 YAKOV VARSHAVSKY f) holds, since the projection ~s:X~n-+ X~ " is a local isomorphism (see the proof of a)). g) The real case is clear, the p-adic case is deep. It uses Yau's theorem (see [Ku, Rem. 2.2.13]). [] Remark 1.3.10. -- The functorial property of projective limits implies that X *n satisfies the functorial properties of analytic spaces associated to schemes (see [Bel, Thm. 3.4.1] or [SGA1, Exp. XII, Thm. 1.1]). Lemma 1.3.11. ~ Let F C G � E and X be as above, let E' be a compact normal subgroup of E, and let F' C G � (E'\E) be the image of I' under the natural projection. Then we have the following: a) the map ~ : F -+ F' # an isomorphism; b) P' satisfies the conditions of Lemma 1.1.9; c) the quotient E'\X exists and is isomorphic to the (E'\E, L)-scheme corresponding to P'. Proof. -- a) The composition map I' -~ F' ~-~ G is injective, therefore 9 is an isomorphism and prG: F'-+ G is injective. b) F' is clearly cocompact. Let U � S C G � (E'\E) be an open neighbourhood of the identity with a compact closure. Then q~-~(U � S) is an open neighbourhood of the identity of G � E with a compact closure. It follows that ~-I(U � S) n F is finite, thus (U � S) (3 F' is also finite. Hence P' is discrete. c) Since E' is compact and normal, we have E' S = SE' ~ oq~(E) for each S E #'(E). Hence E'\X := lim XE, s is the required quotient. Next we notice that for each <--.- S ~ #'(E) the subgroup S := S\E' S belongs to ~'(E'\E) and that each T ~ ~'(E'\E) is of this form. Since X~ ~, s "~ = E' S\[X ~ � E]/r ~ = S\[X ~ � (E'\E)]/F', we are done. [] 1.4. Drlnfel'd's covers 1.4.1. Now we need to recall some Drinfel'd's results [Dr2] concerning covers of D.~w. (A detailed treatment is given in [BC] for d = 2 and in [RZ2] for the general case.) Let K w be as before and let D w be a central skew field over K w with invariant 1/d. Let 01) w C D w be the ring of integers. Fix a maximal commutative subfield K~ J of D~, unramified over K w. Let ~ e K w be a uniformizer and let Fr~ be the Frobenius auto- morphism of K~ ~ over K w. Then D w is generated by K~ and an element II with the following defining relations: H a = z~, II-a = Frw(a ) 9 H for each a e K~ I. Denote by d~' the ring of integers of the completion of the maximal unramified extension K ~ ~ w of K w. Drinfel'd had constructed a commutative formal group Y over ~w @ow ~ with an action of 0D~ on it. For a natural number n denote by r. the kernel of the homomorphism Y ---> Y. Let ~. := F. | K w be the generic fiber of F. and let ~._uaC ~. be the kernel of l-["n-~(= ~.-va). p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 73 Put Z xw a'" :=s __s and set T.:= 1 +~"O.~e~(Dw~). Then Ea'm~ " is an ~tale Oalois covering of ~a, x~ o := D~ a QK~ K~^ ~ with Galois group (09~/~) . � ~= 0gJT,. Za,- ~ Ea,,~-i We also denote 0 � by T 0. The action of ~ induces dtale covering maps ~. : x~ x~ D w giving a K~-pro-analytic space Za Z a'" tV � acts naturally on Za xw:={ x~ }." The group ~ x~, and we have Za'"x~ =~ T,\Z~w for each n ~N ~{ 0 }. Moreover, Drinfel'd had also constructed an action of the group GLa(K~) � D~ on Zd viewed as a pro-analytic ~K w , space over K,, which extends the action of O � and satisfies the following properties Dw (notice that our convention 1.3.1 differ from those of Drinfel'd): a) the diagonal subgroup { (k, k) ~ GLa(K~) � D~ ] k ~ K~ } acts trivially; b) GLa(K~) (resp. D~ x) acts on Y,~ = a ^ ^=* ~K | by the product of the natural action of PGLa(K~) on D~,, (resp. the trivial action on D~w) and the Galois action nr g ~ Fr~0 ~wlaet~' (resp. g ~-, Fr~ ~,aw(ar on K~. 1.4.2. In the case d = 1 Drinfel'd's coverings can be described explicitly. Let L be a p-adic field. Then, by property a) above, the action of L � � L � on E~ is determined uniquely by its restriction to the second factor. Denote by 0 L : L � -+ Gal(L~b/L) the Artin homomorphism (sending the uniformizer to the arithmetic Frobenius automorphism). Lemma 1.4.3. -- One has Z~ ~- jg(~',b), and the action of (1, l) ~{ 1 } � L � on Y.~ is given by the action of 0~.(/)-1 e Gal(L~b/L) on L ~b. Proof. -- This follows from the fact that Drinfel'd's construction for d = 1 is equivalent to the construction of Lubin-Tate of the maximal abelian extension of L (see, for example, [OF, Ch. VI, w 3]). [] 1.4.4. Let L be an extension of K. of degree d and of ramification index e. For every embeddings L ~ Matn(K~), L '-+ D~ (such exist by [OF, Ch. VI, w 1, App.]) and K~ ~ L "~ and for every n ~ N w { 0 } there exists a closed L-rational embedding i,:Y~' ~" ,-+ Z a'" X~, which is (L � � L� and commutes with the projec- t.ions n,. Moreover, i0 : ~[ (~,. t'.~' ,-+ D~ ~w K~ is the product of our embedding ~ ,-~ I'. ~ and a closed embedding i : ~ ~ ~, with image roa ~T,� (see [Dr2, \'~KwJ Prop. 3.1]). Taking an inverse limit we obtain an embedding Y : Z~ ,-~ ~a W w ~ Lemma 1.4.5. ~ Let H be a subgroup of RLtK~(G,, ) (K~) -_ L � Zariski dense in RLm, o(G,, ). Then Im "~ = { x ~ Y'~Kw I (l, l) x = x for every 1 ~ H }. Proof. -- Since for each l e H C L � the action of (l, l) on Y~ is trivial, and since is (L � � L� Im 7' is contained in the set of fixed points of (1, l), l ~ H. Conversely, if x ~ Z~ is fixed by all (l, l), l e H, then its image ~ e O~w under the natural projection P :xaxw ~ ~axw belongs to (D~) ~ : ~--x#r~ ~*.� i(O~). Since 0(ImT) = Imi, there exists yr such that 0(Y)=~(= p(x)). Recall that ~Ea � = D~\ x~. Therefore y = 8x for some 8 eD.. It follows that (l, 8lS-1)y =y 10 74 YAKOV VARSHAVSKY for each l ~H, hence also (1,818-1l-1)y =y. Since the covering Za~:~, ~ r .� a is 4tale, the group O� acts freely on Z a,:w. Therefore 8l 8-1 l-~ = 1 for each l ~ H. Hence 8 belongs to the centralizer of H in D x, so that to L x. It follows that X = 8-17 EL � .ImT' = Im'~. [] Proposition 1.4.6. -- For each n E N w { 0 } the group SD~ (~ T 1 acts trivially on the set 7: o of connected components of Z~ QK,, C~. Proof. -- Recall (see 1.4.4) that each maximal commutative subfield LC D~ gives us (after some choices) a closed L-rational L� embedding i s : Z~' ~" -+ Z a'" Let q/ be a connected component of Z~' ~" ~ C~. Take ~ ~ 7:0, ~K w 9 which contains i,(~t). Then, by Lemma 1.4.3, f is defined over t'. ~b and l(~ r) = (0L(/))-~(~ r) for each l~L � Fix a ~F 0 ~ n0, and let M be the field of definition of 5F 0. Then M 3 K~. Since the quotient D~ \ ~:w ~ D~w is geometrically connected, D x acts transitively on n 0. Since the action of D~ on T: 0 is K,,-rational, M is the field of definition of every f ~ n o. In particular, M is the closure of a Galois extension of Kw, and M C ~'ab for every extension L of K~, of degree d. Taking L be unramified we see that the group AutC~t(M) of continuous automorphisms of M over K~ is meta-abelian (----- extension of two abelian groups). Set H := { 8 ~ D~ � I there exists a s(8) s Aut K~ ~176 ~"-J such that 8(~0) = s(8)-l(fo)}. Then H is a group and a:H --> AuthOr(M) is a well-defined homomorphism. We claim that H = D~. Take a 8 e D~, then Kw[8 ] is a commutative subfield of D~,. Let L be a maximal commutative subfield of Dw containing 8. Then by (1.1), 8(~) = (0L(8))-l(gY) for some ~ ~0. Take 8' ~D~ such that gY = 8'(~Y0). Then (8')-188'(YCo) = (8')-~o (0L(8))-1o8'(~o) -~ (0L(8))-I(Y'o), so that (8')-188 ' ell. Thus each element of D~ is conjugate to some element of H. In particular, Z(D~) C H. Since T, acts trivially on Z ~, ~'" it is also contained in H. Hence H D T,-Z(D~) has a finite index in D~. Therefore our claim follows from the following Lemma 1.4.7. -- Let G be a group and let H be a subgroup of G of finite index. Suppose that G= [.J gHg-1. Then G----H. g~G/R Proof. -- Set K:= ['] gHg -a. Then K is a normal subgroup of G of e~G finite index, and G/K= [J g(H/K) g-a= [.j (g(H/K) g-l --{1}) u{1}. Hence o~GIH r I G/KI~< [ G/HI (] H/KI -- 1) + 1 = ]G/K] -- ]G/HI + 1, therefore G = H. [] Now the proposition follows from the fact that SD~ is the derived group of D~ x (see [PR, 1.4.3]) and that T1 n SD~ is the derived group of SD x (see [PR, 1.4.4, Thm. 1.9]). [] p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 75 1.5. Second construction Construction 1.5.1. -- Suppose that a subgroup F C GLa(Kw) � E satisfies the following conditions: a) Z(V) = Z(GLa(Kw) � E) c~ F; b) the subgroup Z(F) C Z(GLa(Kw) � E) is cocompact; c) PP C PGLa(K~) � PE satisfies the assumptions of Lemma 1.1.9 (this imply, in particular, that the closure of F is cocompact in GLa(Kw) � E); d) the intersection of Z(F) with Z(GLa(Kw) ) � { 1 } is trivial. We are going to associate to P a certain (D~ � E, Kw)-scheme. Z d E)/F. The group D~ � E acts on X by Consider the quotient X := ( K~ � the product of the natural action of D~ on Z a and the left multiplication by E. Kw Proposition 1.5.2. -- For each S e ~'(D~ � E) the quotient S\X = S\(E~ � E)/P has a natural structure of a Kw-analytic space, which has a unique structure X s of a projective scheme over K w. Proof. -- First take S = T, � S' for some n e N ~3 { 0 } and some sufficiently small S' e ~-(F) (to be specified later). Then S\X = S'\(Zx ~a,. � E)/P is a disjoint union of [ SkE/F~ [< oo (as in Lemma 1.1.9) quotients of the form F,s,~_~\X~" with a e E. Thus it remains to prove the statement for quotients d, F,s,~_~\ZK . For simplicity of notation we assume that a = 1. Set r ,,o := n pr (z(r)) = pr (r (Z(GLa(Kw)) � (Z(E) n S'))). First we construct the quotient d, Ps,,0\Exw. Assumptions a) and b) of 1.5.1 imply that the closure of Fs,,0 is cocompact in Z(GLa(Kw) ) -~ K~, hence valw(det(Fs,,0)) = dk Z for some k e N. Let K(~ ~) be the unique unramified extension of K w of degree dk', then Consider the natural ~tale projection 7:, : E g'~ Kw-+ Z d'~ xw-+ fig K~," Let {~gC'(A~) }~ex be an affinoid covering of D~. a Since the projection Ea'"x., -+ Za'~ is finite, each ~-l(,,g(Ai) ) C X~/~ is finite over the affinoid space d/(A~| ^~ " K~). Hence it is iso- morphic to an affinoid space ,/g(B,) for a certain K~-affinoid algebra Bi, finite over Ai 6x~ K ^ w . Since ~, is D~-invariant, we have a natural action of Fs,,0 on B,. Set C~ := B r"',~ Since an affinoid algebra is noetherian, we see that C~ is finite over the Kw-affinoid algebra A,. Hence C~ has a canonical structure of a Kw-affinoid algebra (see [Be1, Prop. 2.1.12]). Gluing together the Jg(C~)'s, we obtain a Kw-analytic space rs, ' \Z a, ~ finite and dtale over D~w. Ok K w Put g := S-Z(E)/Z(E) C PE. Then g ~ ~'(PE). To construct F \X a'- v\ Kw we observe p \X a," that the action of PF~ = Ps,,0\F s, on s,,0\ Kw covers its action on ~aK~. Suppose that S' is so small that S satisfies part d) of Proposition 1.1.10. Recall that 76 YAKOV VARSHAVSKY by Lemma 1.3.9 each x e ~, has an open analytic neighbourhood U. such that 7(U,) n U, # 0 for all -~ E PF~ -- { 1 } and, as a consequence, PP~\D~ is obtained by gluing the U~'s. Let ~. be the natural projection from I's, ' o\\~a'"x~ to ~.a For each y eFs, 0\E~ ~ set V~:=~a(U~.~.~). Then the quotient K~,-analytic space pF~\(Fs, ' 0\Zx~)a.~ = Ps,\\Y/'"Kw is obtained by gluing the V~'s. Since Fs,\\Za'"~ is a finite (and dtale) covering of PF~\~, which has a structure of a projective scheme over K~ by [Mus, Ku], P \~]a.. also has such a structure by S'\ K w Corollary 1.2.3 and the remark following it. Finally consider an arbitrary S ~ ~'(D x x E). It has a normal subgroup g of the form g = T, x S' with sufficiently small S'~ ~'(E), therefore to complete the proof we can use the same considerations as in the end of the proof of Propo- sition 1.3.7. [] The same argument as in Construction 1.3.6 gives us a (D x x E, Kw)-scheme X = lira X s. ~S Proposition 1.5.8. -- a) The kernel E 0 of the action of D~ � E on X is the closure of the subgroup Z(F) CZ(GLa(Kw)x E)=Z(D~ x x E) after the natural identification Z(GL~(K~)) ---- K~ ---- Z(D~). b) Let Eo be the closure of Z(P) in E, and let F'C PGL~(K~) x (Eo\E) be the image of F under the natural projection. Then F' satisfies the as~mptio~ of I_zmma I. I. 9. c) The quotient D~ \X exists and is isomorphic to the ~,o\E-scheme corresponding to F' by Construction 1.3.6. d) The quotient (D~ � Z(E))\X exists and is isomorphic to the (PE, K~)-scheme X' corresponding to PF by Construction 1.3.6. e) For each x e X the orbit (D~ x E)x is Zariski dense in X. f) For each sufficiently small S e ~'(E) and each n e N u { 0 } the map X --~ Xr. x s is ~tale, and B a-x is the universal covering of each connected component of (X~,� s,o) ~ for each embedding K, ~ C. In particular, the projective system X ~ := { X~" }T e ~D~ � E~, associated to X, is a Kw-pro-analytic space. Proof. -- a) Notice that g E E 0 if and only ifg acts trivially on X s (or, equivalently, on X~ n = S\(Z~w x E)/F) and normalizes S for each S e~'(D~ X E). For each 7 ~ Z(F) C Z(D~ x E) let y~ be the projection ofy to the first factor. Since (70 x y,~) acts trivially on Y/ xw, we have V([x, e]) = [V,~(x), y~, e] ,~ [(Vo � V,) (x), e] = [x, e] for each x e Y,~ and e e E, that is 7 acts trivially on each X~ ". Since 7 is central, it certainly normalizes S. This shows that the closure of Z(F) is contained in E 0. Conversely, suppose that some (gs, g,)eD~ � with gleD~ and g, e E belongs to E 0. Choose S' e~-(E) and n e Nu {0}. It suffice to show that (gl, g~) e (T, � S')Z(I'). Since (gl, g,) acts trivially on S'\(Z~" � E)/F, we have [gl(x), g~] '~ [x, 1] for each x ~ Z a'x~ ". This means that there exists an element y = y, e P such that gl(x) = 7~1(x) and g, ~ S' YE" Let x' be the projection of x to y a,0~,, and let x" p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 77 be its projection to fl d The group D~ acts trivially on ~e therefore 7~1(x '') = x" K w" K w ~ 9 Choose x so that no non-trivial element of PGLd(K~) fixes x", then YG belongs to Z(GLa(K~)) ~- K~. Assumption c) of 1.5.1 implies that Y ~ Z(F). Since gx(X) = "~G-I(x) = "~w(X), we conclude that g~-i y~(x)= x. Hence x'= (gi-ly~,)(x')= Fr~Wllaetlnlvwll(x'), so that g~-I y~ e ~� Since Z a'' is an &ale Galois covering of ]~a,0 with Galois group Dw" Kw ~w O~/T,, the equality (gi-~yw)(x)=x implies that gi-~y~ ~T,. It follows that (g~, g2) e (T~ � S')(y~, yz) C (T, � S') Z(I'), as claimed. b) The natural projection PGLa(K~) � (E0\E) -+ PGLa(K~) � PE induces an isomorphism F' -~ PP. Hence I" is discrete and has injective projection to PGLd(K~). It is cocompact, because so is 1" C PGLa(K~) � E. c) Notice first that for each open subgroup E 0 fi S C D x � E, compact modulo E0, the quotient S\X exists and is projective. Assumption b) of 1.5.1 implies that E 0 is cocompact in D~ � Z(E). Therefore for each S e o~-(D~ � E) the quotient D~ S\X = (D x E 0 S)\X = (D x � ~'o) S\X exists. Set g := (D; x Eo)k(D; � go) S ~ ~'(E0\E). Then (D; SkX) ~" = (D; x E0) S\[Z~:w x E]/r -~ gk[a~w x (E0kE)]/r', and the state- ment follows as in the proof of Lemma 1.3.11 c). d) follows from c) and Lemma 1.3.11 c). e) follows from c) and Proposition 1.3.8 e). f) Take T e ~@(PE) satisfying part d) of Proposition 1.1.10. Then there exists S e o~'(E) such that Z(E)\S.Z(E) = T. Since we have shown in the proof of Propo- sition 1.5.2 that X~r, � s is 6tale over T\X' for each n e N u { 0 }, the statement follows immediately from Proposition 1.3.8 f), g). [] Corollary 1.5.4. -- For each a s E the composition map po : z ~ -~ z ~ x { a } ~ (z~ x E'~)/r -, x ~" Kw Kw of pro-analytic spaces over K,~ is ~tale and one-to-one. Proof. -- The &aleness is clear. Let Xl and x~ be points of Ne such that BE w p~(xx) ---- p~(x~). Let ~ s PE' be projection of a, and let p; be the injection K~ -+ n~ x { ~- } ~ (a~ x (PE')d'~)/Pr ~ (X'? n. Then we conclude from the commutative diagram ~d, n Pa Xa n Kw ;~ Dro~ l IP r~ ~w , (x')"- P~ that xx and x2 have the same projection y ~ D~w. 78 YAKOV VARSHAVSKY Choose S e o~'(E) so small that the group PFasa_l is torsion-free (use Propo- sition 1.1.10 d)). Then no non-central element of P~sa-1 fixes y. For each n e N let n,,s be the projection Xan-+ (XTn� Then the image of 7r,, s o p~ is isomorphic to P,s~_~\Z~$. Hence there exists ~., ~ P~s~-i such that the projections of y,(xl) and x 2 to Z a'" coincide. Therefore ~%(y)=y, so that ~', e Z(GLd(Kw) ) = K~ It follows Kw that the sequence {-(, }, converges to some .( e K~, which satisfies y(x~) = x 2. Then (y, 1) e Z(D$ � E) fixes z := G(xl) = pa(x2). Since (y, 1) is central, it then fixes the whole (D$ � E)-orbit of z. Hence, by Proposition 1.5.3 c), it acts trivially on X. There- fore by Proposition 1.5.3 a), the element (y, 1) belongs to Z(P) C Z(GLa(Kw) � E). Assumption d) of 1.5.1 implies that y = 1, hence x~ = x~. [] 1.6. Relation between the p-adic and the real constructions The following proposition (and its proof) is a modification of Ihara's theorem (see [Ch2, Prop. 1.3]). It will allow us to establish the connection between the p-adic (1.3.6, 1.5.1) and the real (or complex) (1.3.6) constructions. Proposition 1.6.1. -- Let X be an (E, C)-scheme. Suppose that a) E acts faithfully on X; b) E acts transitively on the set of connected components of X; c) there exists S e ~'(E) such that the projection X -+ X s is dtale, and B d-1 is the universal covering of each connected component of X~s ~. Then X can be obtained from the real case of Construction 1.3.6. Remark 1.6.2. -- a) It follows from Proposition 1.3.8 that all the above conditions are necessary. b) Let X be an (E, C)-scheme and let E 0 be the kernel of the action of E on X. Then X is an (E0\E , C)-scheme with a faithful action of E0\E. Conversely, any (E0\E , C)- scheme can be viewed as an (E, C)-scheme with a trivial action of E 0. c) Let X be an (E, C)-scheme and let X 0 be a connected component of X. Put X' := U g(X0). Then X' is an (E, C)-scheme with a transitive action of E on the set gE~. of its connected components, and X is a disjoint union of such (E, C)-schemes. Remarks b) and c) show that assumptions a) and b) of the proposition are not so restrictive. Proof. -- We start the proof with the following Lemma 1.6.3. ~ Suppose that { X~ }~, ~ i is a projective system of complex manifolds such that the transition maps X~ ~ X~, where ~, ~ ~ I with ~ >1 ~, are analytic coverings. Then there exists a projective limit X of the X~'s in the category of complex manifolds. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 79 Proof. -- Choose an 0t e I. Cover X~ by open balls { U~ }~ea, and let r~ : X' ~ X~ be an analytic covering. Then the inverse image rc-~(U~) of each U~ is a disjoint union of analytic spaces, each of them isomorphic to U~ under z~. Hence the cons- truction of the projective limit from the proof of Proposition 1.3.8, a) can be applied. [] Now we return to the proof of the proposition. By assumption c), X~ n is a complex manifold for each sufficiently small S e ~'(E), and the natural covering X~ n ~ X~ n is 6tale (analytic) for each T C S in ~-(E). Therefore by the lemma there exists an analytic space X ~ := lim X~ n. Since X s is a complex projective scheme for each S e o~(E), the set of its connected components coincides with the set of connected components of X~ n. Hence assumption b) implies that the group E acts transitively on the set of connected components of X "n. Let M be a connected component of X ~n. Denote by F the stabilizer of M in E. Then I'~ acts naturally on M, and the transitivity statement above implies that X"~==_ (M x Ed'~)/r~. For each S ~ ~'(E) the analytic space X~" ~ S\(M � E)/Pz is compact. Therefore, as in the proof of Lemma 1.1.9, I S\E/lP~ I < oo and [P~ : P~ raS] = oo. Note that M s := (Pz n S)\M is a connected component of X~ ". Suppose that S satisfies condi- tion c); then the map M ~ M s is 6tale and B d-a is the universal covering of M s. Hence it is also the universal covering of M. It follows that Uz C Aut(M) can be lifted to I" R C Aut(B a-~) = PGUa_~,I(R) ~ The kernel A of the natural homomorphism zc:I" R -+ P~ is the fundamental group of M. Let Ps C PGU a_ 1, ~(R) ~ be the fundamental group of the compact analytic space M s , then r s is a cocompact lattice in PGUa_~,~(R) ~ satisfying P s = r~-x(PE ta S). It follows that [Ps : Fs] = [I'~ : P~ r3 S] = oo. Therefore, as in the proof of Propo- sition 1.1.10 a), we see that Pl~ is dense in PGUa_I,I(R) ~ The group A is discrete in PGUa_x,~(R) ~ and normal in I'R, thus it is trivial (compare the proof of Propo- sition 1.1.10 b)). In particular, M - B a-~ and z~ is an isomorphism. Put I-':={(T, rc(T))[u ~ � E. Since P s is discrete in PGUa_~,x(R) ~ so is I' in PGUa_x,x(R) ~ � E. Let K C PGUa_I,~(R) ~ be the stabilizer of 0 e B a-1. Then X~ ~ ~ S\(B a-1 � E)/F ----- (K � S)\(PGUa_~,I(ll) ~ � E)/F. Since K, S and X~ n are compact, P is cocompact in PGUa_x,I(ll) ~ � E. Since Ker(pra) equals the kernel of the action of E on X, the projection pr o is injective. This shows that P satisfies all the assumptions of Construction 1.3.6. [] Corollary 1.6.4. -- Choose an embedding K,o ~ C. Let X be an (E, Kw)-scheme obtained by the p-adic case of Construction 1.3.6 or an (E,, Kw)-scheme obtained by Construction 1.5.1. Then X c can be constructed by the real case of Construction 1.3.6. Proof. N This is an immediate consequence of Propositions 1.6.1, 1.3.8 and 1.5.3. [] 80 YAKOV VARSttAVSKY 1.7. EIHptic elements Definition 1.7.1. -- Suppose that a group G acts on a (pro-)analydc space (or a scheme) X. An element g e G is called elliptic if it has a fixed point x such that the linear transformation of the tangent space of x, induced by g, has no non-zero fixed vectors. In such a situation we call x an elliptic point of g. Lemma 1.7.2. -- Let )'1, )`2, ..., xe be the eigenvalues of some element g ~ GLa(L ) (with multiplicities). Let v ~ Pa-I(L) be one of the fixed points of g corresponding to )`l. Then -- are the eigenvalues of the linear transformation of the tangent space of v, induced )`I' )`I ' " " " ' )`I byg. Proof. -- Simple verification. [3 Proposition 1.7.3. -- The set of elliptic elements of PGUa_I.I(R) ~ with respect to its action on B a-1 and of PSLa(K~) with respect to its action on ~w is open and non-empty. Proof. ~ In the real case we observe that an element g := diag()u, )`2, .-., )`d) ~ PGUd-I,i(R) ~ fixes (0, 0, ..., 0) ~ B d-1. Therefore by Lemma 1.7.2, g is elliptic if k, 4: )`d for all i 4: d. It follows that the set of elliptic elements is non-empty. It is open, because if g has a fixed point in B a-1 corresponding to an eigenvalue of g appearing with multi- plicity 1, then the same is true in some open neighbourhood of g. In the p-adic case we start with the following Lemma 1.7.4. -- An element g e GLa(Kw) is elliptic (acting on ~Kw) if and only if its characteristic polynomial is irreducible over K w . Proof. -- Suppose that the characteristic polynomial Xg of g is irreducible over K w. Then g has d distinct eigenvalues. Let X be some eigenvalue of g, let v 4:0 be the eigen- vector ofg corresponding to )`, and let ~ ~ pd-1(~.~o) be the fixed point of g corresponding to v. By Lemma 1.7.2, the linear transformation of the tangent space of ~, induced by g, has no fixed non-zero vector. So it remains to be shown that ~ ~ D~. If ~ r D~w , then it lies in a K~-rational hyperplane. Therefore there exist elements ax, 9 9 aa ~ K,~, not all 0 (say a a :~ 0) such that (al, ..., aa).v = 0. We also know that (g -- )`I) v ---- 0. Let A be the matrix obtained from g -- )`I by replacing the last row by (al, ..., aa). Then Av = 0, so that det A----0. The determinant of A is a polynomial in )` of degree (d- 1) with coefficients in K~ with leading coefficient (-- 1)(d-l~aa + O. This contradicts the fact that the minimal polynomial of ), over K,~ has degree d. Suppose now that the characteristic polynomial )~g ofg equals the product fx" 9 9 9 "fk p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 81 of polynomials irreducible over K~ (k > 1). Consider the matrix fx(g). If fl(g) = 0, then the minimal polynomial mg of g divides fl. Hence each root of ~G, being a root of ms, is a root off1. Eachf has only simple roots, therefore f [fl for each i. Since fx is irreducible, all the f's are equal up to a constant. Hence Za = cfl k for some c e K~. In particular, each root of Xg is at least double. Lemma 1.7.2 then implies that g is not elliptic. Hence we can suppose thatf(g) 4= 0 for all i = I, 2, ..., k. Let X be an eigenvalue of g, let v be the eigenvector corresponding to X, and let ~ e pa-x(~) be the fixed point of g corresponding to v. Choose i s { 1, ..., k } such that X is a root of f. Then f(g) v =f(X) v = 0. The matrixf(g) ~ 0 has all its entries in K~, hence ~ lies in a K,,-rational hyperplane. Therefore g is not elliptic. [] Now we return to the proof of the proposition. Embed an extension L = K~(X) of K,0 of degree d in Mata(K,, ). Then X ~LxC GLa(K,~ ) has an irreducible cha- racteristic polynomial over K W. Therefore the set of elliptic elements of PGLd(Kw) is non-empty. It is open because by Krasner's lemma if g c GLn(K,o ) has a characte- ristic polynomial irreducible over K,,, then an)" g' e GLa(K,o), close enougL to g, has the same property (see [La, Gh. II, w 3, Prop. 4]). It follows that the set of elliptic elements of PSLa(Kw) is also open. For showing that it is non-empty observe that if an element g ~ PGLa(K,o ) is elliptic but ga is not elliptic, then by Lemma 1.7.2 the characteristic polynomial of any representative of ga in GLa(Kw) has at least two equal roots. Hence such a g belongs to some proper Zariski closed subset of PGL a. It follows that there exists an elliptic element g ~ PGLa(K~) such that ga is elliptic as well. Since g~ always belongs to PSLa(K~), we are done. [] Proposition 1.7.5. -- a) An element (g, 8) ~ GLd(K,~ ) � D~ is elliptic with respect to its action on X a (viewed as a pro-analytic space over K~,) if and only if the characteristic Kw polynomials of g and 8 are K~-irreducible and coincide. b) For every element g e GLa(K,~) elliptic with respect to its action on G~, there exists a 8 ~ D,~ such that (g, 8) is elliptic with respect to its action on y a Proof. -- a) Let x E X a be an elliptic point of (g, 8), and let ~ ~ fin be its image K w Kw Kw "--> D~. Since 0 is ~tale, it induces an isomorphism under the natural projection p : Z a a of tangent spaces (up to an extension of scalars). Hence g is elliptic with respect to its action on ~a By Lemma 1.7.4, g generates a maximal commutative subfield L : = K~ (g) K w ~ of Matd(K,~). Choose an embedding j : Kw(g ) ~ D~, (such exists by [CF, Ch. VI, w 1, App.]). It defines an L� embedding "/':Z~'-->Z a (see 1.4.1). We know that Kw ~- e (D.~) v'~ = p o z (ZL). In particular, there exists y E Y(Z~,) such that p(y) ----- 5(. Since T is L� the element (g,j(g)) ~ GLa(K~) � D~ fixes y. Using the � za =~a we havey-----d0x for some d 0eD~ fact that x ~ 0- ~(~) and that D~ \ x~, K~, Hence, the elements (g, doaj(g) do) E GLa(K~) � D,~ and d' := doaj(g) do 8 -~ ~ D~ fix x. Xl 82 YAKOV VARSHAVSKY In particular, d ~ D~ fixes some point (the projection of x) on Ea, ~w 0 (~w C~, therefore d"E ~� Since the Galois covering E ~ � d ~)� acts freely on Z d It fol- lows that d= 1, hence 8 = do~j(g) d o. This completes the proof of the implication " only if", because g ~ Matd(K~) andj(g) ~ D~ have the same characteristic polynomials. Conversely, suppose that the characteristic polynomials of g and 8 are K~-irreducible and coincide. Then the subfields K~(g)C Mata(K~) and K~0(8 ) C Dw have degree d over K~ and are isomorphic under the K~-isomorphism sending g to 8. Using this isomorphism we obtain embeddings of the field L := K~(g) into Mata(Kw) and into D~. These embeddings define (by 1.4.4) an (L � � L� embedding "~:~ ~-+ X d such that every point x E ~(X~) is fixed by all elements of the form Kw (l,l) eL � � L� GLd(K~) � D~. In particular, x is a fixed point of (g, 8). As before, the action of (g, 8) on the tangent space of x coincides with the action of g on the tangent space of ~-. Since ~ is an elliptic point ofg (by Lemma 1.7.4), x is an elliptic point of (g, ~). b) If an element g e GLa(K~) is elliptic, then by Lemma 1.7.4 it has an irreducible characteristic polynomial over K~. Therefore K~(g)C Matu(K~) is a field extension of K~ of degree d. Then for every embedding j of K~(g) into D~ the element (g,j(g)) is elliptic by a). [] 1.8. Euler-Poincar6 measures and inner twists Here we give a brief exposition of Kottwitz' result [Ko, w 1]. 1.8.1. Let L be a local field of characteristic 0, and let H be a connected reductive group over L. Serre [Se2] proved that there exists a unique invariant measure (called the Euler-Poincar6 measure) ~a on H(L) such that aa(1-'\H(L)) is equal to the Euler- Poincar6 characteristic ZE(I') of H*(r, Q) for every torsion-free cocompact lattice I" in H(L). In particular, aa(H(L))= 1 ff the group H(L) is compact. The Euler- Poincard measure is either always negative, always positive or identically zero. It is non-zero if and only if H has an anisotropic maximal L-toms. (A result of Kneser shows that in the p-adic case this happens if and only if the connected center of H is anisotropic.) 1.8.9.. Let G be an inner form of H. Choose an inner twisting p : H --~ G over I.. Choose a non-zero invariant differential form o~a of top degree on G. Set co~r := p*(c%). Using the fact that H is reductive, that the twisting is inner and that co G is invariant, we see that o~ is invariant, defined over L, and does not depend on p. Hence co G and o~u define invariant measures [ o~ I and I o~ E [ on G(L) and H(L) respectively (see [We2, 2.2]). Definition 1.8.3. -- The invariant measures ~t on H(L) and Ex' on G(L) are called compatible if there exists some c ~ R such that ~ = c[ ~a [ and ~' = c I ~ I" 1.8.4. -- Now suppose that H has an anisotropic maximal L-torus T, so that the Euler-Poincar~ measure Va on H(L) is non-trivial. (Notice that for semisimple p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 83 groups of type A, this assumption is satisfied automatically). Denote by I~rrl the absolute value of ~z H. Write N(T, H) for the finite set Ker[Ha(L, T) -+ Ha(L, H)] and write I N(T, H) ] for its cardinality. It is well known that T transfers to G, thus we can also consider the finite set N(T, G). Proposition 1.8.5 ([Ko, Thm. 1]). -- The invariant measure IN(T, H) 1-11 ~t n [ on H(L) is compatible with the invariant measure IN(T, G) I-1I [zol on G(L). Remark 1.8.6.- a) In the p-adic case, the sets N(T, H) and N(T, G) always have the same cardinality. b) In the real case, N(T, H) = f2(H(C), T(C))/f2(H(R), T(R)), where f2 stands for the Weyl group. In particular, [ N(diag, PGUa) [ = 1 and ] N(diag, PGU a_ 1. a) ] is d (resp. 1) if d~ 2 (resp. d-= 2). 1.9. Preliminaries on torsors (-~ principal bundles) Definition 1.9.1. -- Let G be an affine group scheme over a field L (resp. an L-analytic group), and let X be an L-scheme (resp. an L-analytic space). A G-torsor over X is a scheme (resp. an analytic space) T over X with an action G � T -+ T of G on T over X such that for some surjective 6tale covering X' -+ X the fiber product T � x X' is the trivial G-torsor over X' (that is isomorphic to G � X'). Remark 1.9.2. -- Since each 6tale morphism of complex analytic spaces is a local isomorphism, our definition in this case coincides with the classical one. Lemma 1.9.3. -- a) If T is a G-torsor over X, then the map ~T : G x T -+ T x x T (~(g, t) = (gt, t)) is an isomorphism. b) Let T and T' be two G-torsors over X and Y respectively. Then for each G-equivariant map f: T ~ T' the natural morphism T --~ T' � y X is an isomorphism. Proof. -- a) Since the problem is local for the 6tale topology on X (see [Mi2, Ch. I, Rem. 2.24] in the algebraic case, [Be3, Prop. 4.1.3] in the p-adic analytic and Remark 1.9.2 in the complex one), we may suppose that T is trivial. Then our morphism (g, (h, x)) ~ ((gh, x), (h, x)) is invertible. b) For trivial torsors the statement is clear. The general case follows as in a). [] Remark 1.9.4. -- By [Mi2, Ch. I, Rem. 2.24 and Prop. 3.26] our definition in the algebraic case is equivalent to the standard one. In particular, a G-torsor over X is affine and faithfully flat over X. Lemma 1.9.5. -- Let X be a separated scheme over a field L, let G and H be two affine group schemes over L, let T be a G-torsor over X, and let ~r : T -~ X be the natural projection. a) The functor ,~" ~ rc*~" defines an equivalence between the category of quasi-coherent sheaves on X and the category of G-equivariant quasi-coherent sheaves on T, that is, quasi-coherent sheaves on T with a G-action that lifts the action of G on T. 84 YAKOV VAKSHAVSKY b) The funetor Z F. Z x x T defines an equivalence between the following categories: i) the category of vector bundles of finite rank on X and the category of G-equivariant vector bundles of finite rank on T; ii) the category of H-torsors over X and the category of G-equivariant H-torsors over T; iii) (/f X is noetherian and regular) the category of P"-bundles on X and the category of G-equivariant P"-bundles on T. The quasi-inverse functor is "Z ~-* G\ Z. Proof. -- This is a consequence of a descent theory. a) Abusing notation we will write ~'X y, Y1 instead of p* o~" for every morphism P : Y1 ~ Y2 and every sheaf of modules o~" on Y,. Let o~- be a G-equivariant quasi- coherent sheaf on T. Define an isomorphism ~ : (~ x ~ T) x x T -% T x x (~ x ~ T) over T Xx T by the formula ?(f, gt, t) = (gt, g-l f, t) for all g ~ G, t E T and f~ if, (use Lemma 1.9.3). Then ? satisfies the descent conditions of [Mi2, Prop. 2.22]. Since T -+ X is affine and faithfully flat, there is a unique quasi-coherent sheaf o~" on X such that o~ ~ o*" X x T. Since the construction of descent is functorial (see [Mi2, 2.19]), we obtain an equivalence of categories. Notice that o*- ~ G\(o~- x x T). b) follows from a) in a standard way (use [Ha, II, Ex. 5.18, 5.17 and 7.10]). [] From now on we suppose that the reader is familiar with basic definitions of tensor categories (see [DM]). Notation 1.9.6. -- For a field L, an affine group scheme (resp. an analytic group) G over L and a scheme (resp. an analytic space) X over L: a) let HepL(G ) be the category of finite-dimensional representations of G over L; b) let r x be the category of vector bundles of finite rank on X; c) let Torx(G ) be the category of G-torsors over X. We will sometimes identify categories with the sets of their objects. Definition 1.9.7. -- Let L be a field and let G be an affine group scheme over L. A G-fiibre functor with values in a separated scheme (resp. analytic space) X over L is an exact faithful tensor functor from Hepr,(G) to r x. Remark 1.9.8. I If X ---- Spec R is affine, then g/be x is equivalent to the category of finitely generated projective modules over R, hence our definition is a global version of that of [DM, 3.1]. 1.9.9. -- Let T be a G-torsor over X, then by Lemma 1.9.5, the correspondence V ~-~ G\(V x T) defines a G-fibre functor with values in X. This correspondence defines a functor ~ from Torx(G ) to the category of G-fibre functors with values in X. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES Theorem 1.9.10. -- The functor ~ determines an equivalence between Torx(G ) and the category of G-fibre functors with values in X. Proof. -- The local version is [DM, Thm. 2.11 and 3.2]. The gluing works because X is separated. [] 1.9.11. Later on, we will use the following description of the quasi-inverse functor "~ ofv. Let ~q be a G-fibre functor with values in X. For each morphism no : To ~ X we define two tensor functors ~ql:V ~-~ V x T O and ~2 = no o ~q from ~epT,(G) to ~r Let ~(To, no):= Isom(~q,, ~ql) be the set of isomorphisms of tensor functors. The action of G on the first factor ofV x T O defines an action of G on ~ql, and afortiori defines an action of G on ~(To, no). Thus ~ is a functor from the category of schemes over X to the category of sets with a G-action. Theorem 1.9.10 says that this functor is representable by a G-torsor v(~q) over X (see [DM, Thm. 2.11 and 3.2] and their proofs). 1.9.12. Let T be a G-torsor over X. For each V~lepL(G) the identity map of T, viewed as a T-valued point of T, corresponds to a certain isomorphism r x T~ (G\(V x T))Xx T. Then ~v is the quotient of the G-equivariant isomorphism Id v x ~0T:V X G x T-~V x T x xT (for the diagonal action of G on the first two factors on both sides) by the action of G. Explicitely, ~v(V, t) = ([v, t], t). Proposition 1.9.18. -- Let L be equal to K~ or to C as in 1.3.1. Let X be a projective L-scheme, and let G be a linear algebraic group over L. The functor T ~ T ~ induces an equivalence between the category of G-torsors over X and the category of G~n-torsors over X ~. Proof. -- A quasi-inverse functor can be described as follows. Let ~:T -+ X ~ be a G~"-torsor. Then the map V ~-* G~\(V ~" x T) defines a G-fibre functor with values in X =. Since the correspondence described in Corollary 1.2.3 commutes with tensor products, the tensor categories r x and r are equivalent. Therefore Theorem 1.9.10 gives us an algebraic G-torsor n:T---> X. It remains to show that there exists a canonical isomorphism T-~T a". By the definition of T we have for each V e g~epi,(G ) a canonical isomorphism ~v : G~n\( V~ X T) ~ G~\(V ~ x Tan). We also have (as in 1.9.12) natural iso- morphisms T x V ~ --'% T Xx (G~\(T x V~")) mapping (t, v) to (t, It, v]). Hence each point t o of ~ defines canonical isomorphisms { to} X to} � � v (to, [t0, v]). Since t o defines a point of X ~ and therefore of X, it gives us by the universal property ofT (see 1.9.11) a point d/(t0) ~ T an, satisfying ~bv([to, v]) = [~(to) , v] for all V ~ 9tepL(G ). Taking V to be a faithful representation of G, we obtain that the map (of sets) d/:T ~ T ~n is G~-equivariant, therefore it is one-to-one and surjective. It remains 86 YAKOV VARSHAVSKY to show that the maps + and +-1 are analytic. Let us prove it, for example, for ~b. Let p : X' ~ X "~ be an 6tale surjective covering such that p*(T ~n) ~ G an � X'. By [Be3, Prop. 4.1.3] in the p-adic case and by Remark 1.9.2 in the complex one it will suffice to show that p* ~b : p* (T) -+ p* (T a") ~ Can� X' (or just its projection to the first factor W : p*(T) -+ G an) is analytic. Consider the map ~V-'Van X p*(T) proj) GanXWa n X p*(Y)] P"+';) G"~\[V ~ x o*(T~)] ~ G"n\(V " x G an x X') g V an x X' p~oj> Van. It is analytic, and satisfies ~v(V, t) = (~:'(t)) -1 v. Hence r~' is analytic as well. [] Corollary 1.9.14. -- Let X and Y be projective L-schemes, let G and H be algebraic groups over L, and let ~b : G -+ H be an algebraic group homomorphism over L. If T ~ Torx(G ) and S E Tory(H), then for any +-equivariant analytic map 37: T ~ -+ S an (that is, satisfying f~igt) = +(g) ff (t) for all g E G an and t ~ Tan), there is a unique algebraic morphism f: T -+ S such that fan _ 3~ Proof (compare the proof of Corollary 1.2.5). -- Sinceffis +-equivariant, it covers some algebraic morphismf: X ~ Y (use Corollary 1.2.2). Thereforefffactors through S an � ~ (S � Hence we may suppose, replacing S by S � that X = Y and that f is the identity. Consider the H-torsor H X T over T equipped with the following G-action: g(h,t) = (h~b(g)-l, gt) for all g eG, hell and t eT. By Lemma 1.9.5, there exists an H-torsor H � T := G\(H � T) over X. Let i be the composition of the embedding t ~ (1, t) of T into H X T with the natural projection to H � T. Then by the defi- nition, every ~b-equivariant algebraic morphism w:T ~ S factors as a composition of i with the unique H-equivariant map H x a T ---> T (defined by [h, t] ~ h~(t)). Therefore (H � H an � an is an H~-torsor over X an having the same functorial property. Now we are ready to prove our corollary. From the ~-equivariance off we conclude that it factors uniquely as f: ~ T an ---> ,a" (H x o T)a, d~ S~. By the proposition, 3 ~ has a unique underlying algebraic morphism f' : H x o T --> S. Set f: = f' o i. The uniqueness can be derived from the above considerations as in the proof of Corollary 1.2.5. [] Now we recall the notion and basic properties of connections on torsors (following [St, Ch. VI, w 1]). Definition 1.9.15. -- Let X be a smooth scheme or an analytic space, and let : P -+ X be a G-torsor. A connection on P is a G-equivariant vector subbundle af' of the tangent bundle T(P) of P such that ~'1~9 is an isomorphism .X~'~-% T=c~(X ) for each p ~ P. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 87 1.9.16. Starting from the isomorphism qh, : G � P-% P � P we obtain an isomorphism of tangent spaces (q~p), : Te(G ) � T~(P) -% Tr(P ) � T~(P) and an identification (X ~-* projl((~p),(X, 0)) of ~ := Lie(G) = T~(G) with the tangent space to the fiber through p ~ P. Therefore a connection 3r ~ on P gives us a canonical decompo- sition T~(P) = fr | for each p ~ P. Now considering the projection of T~(P) onto with kernel 3r for each p ~ P we get a certain f~-valued differential 1-form ~ = Y2(~), called the connection form of ~tt~ Definition 1.9.17. -- Let W be a connection on a G-torsor P, whose connection form is ~. Let h be the natural projection of T~(P) on 3~ for all p ~ P. The curvature of the connection 3(r ~ is the 2-form Dfl defined by ( X ^ Y I D~ ) := (h(X) ^ h(Y) [ d~ ). A connection with zero curvature is called flat. Remark 1.9.18. -- The trivial torsor P - G � X has a natural flat connection, consisting of vectors, tangent to X. We will call such a connection trivial. Lemma 1.9.19. -- Let X be a simply connected complex manifold, let 7~ : P -+ X be a G-torsor, and let ~ be a flat connection on P. Then there exists a unique decomposition P -% G � X such that ~ corresponds to the trivial connection on G � X. Proof --By [St, Ch. VII, Thm. 1.1 and 1.2], there exists a unique G-equivariant diffeomorphism q~ : P -% G � X over X which maps W to the trivial connection. Hence q~ induces complex isomorphism between tangent spaces T~(P) = ~@Yt~ and T,I~j(G � X) = fr | T.I~(X) for each p ~ P. In other words both q~ and q~-i are almost complex mappings between complex manifolds. [He, Ch. VIII, p. 284] then implies that q~ is biholomorphic. [] 2. FIRST MAIN THEOREM 2.1. Basic examples Definition 2.1.1. -- Let K[k be a quadratic field extension and let D be a central simple algebra over K. We say that e : D -+ D is an involution of the second kind over k if e(d 1 + d2) = e(dl) + e(d2) , ~(d 1 d2) = e(d2) ~(dl) for all dl, d 2 ~ D and tile restriction of ~ to K is the conjugation over k. Notation 2.1.2. -- For k, D and e as in Definition 2.1.1, let G = GU(D, e) be the algebraic group over k of unitary similitudes, that is G(R) = { d e (D | R) � I de(d) ~ R � } for each k-algebra R. Define the similitudes homomorphism G ~ G,, by x ~ xo~(x). Notice also that by the Skolem-Noether theorem the group G satisfies PG(L) = G(L)/Z(G(L)) for every field extension L of k. 2.1.3. First basic example. -- Let F be a totally real field of degree g over Q,, let K be a totally imaginary quadratic extension on F. Let D be a central simple algebra 88 YAKOV VARSHAVSKY of dimension d 2 over K with an involution of the second kind ~ over F. Set G :----- GU(D, a), and put D, := D | K, for each prime u of K. Let v be a (non-archimedean) prime of F that splits in K and let w and ~ be the primes of K that lie over v. Then D @F F~ _-_ D w | D~, and the projection to the first factor together with the similitude homomorpkism induce an isomorphism G(F,)--% D~ X F~. We identify G(F~) with D~ x F~ by this isomorphism. Suppose that D~-~ Mata(Kw). Identifying D. with Mata(K.) by some iso- morphism we identify G(F,) with GLa(Kw) � F~ x . Suppose that ~ is positive definite, that is G(F~oi)- GUa(R) for all archimedean completions Fooi~ R of F. Put E' := F~ x x G(Arf:*), then E' is a noncompact locally profinite group. Set r:= G(F) C G(A~) ---- GLa(K~) � E', embedded diagonally. Proposition 2.1.4.- The subgroup F C G(A~) = GLa(K,~ ) � E' satisfies the assumptions of Construction 1.5.1. Proof. -- a) is trivial. b) is true, because the closure of Z(F) _-_ K � is cocompact in Z(G(A~)) ~ (A1x) � c) Since PE'= PG(Ar 1;') and PG(F,)=~ PGLa(K,o), we have to show that PF(= PG(F)) is a cocompact lattice in PG(A~). Lemma 2.1.5. -- If H is an F-anisotropic group, then H(F) is a cocompact lattice in H(A~). Proof. -- See [PR, Thm. 5.5]. [] Since PG is anisotropic over each Fo~i, it is anisotropic over F. Hence by the lemma, PG(F) is a cocompact lattice in PG(Ar). The compactness of the PG(Fool)'s implies also that the projection of PG(F) to PG(A~) is a cocompact lattice as well (see [Shi, Prop. 1.10]). Observe also that the projection PG(F) -+ PG(F~) -~ PGLa(Kw) is injective. d) Sincc z(r) ,~ = K � and Z(G(A~)) ~- (Al~) � , we have to show that the inter- section of K � C (ASK) � with K~ � { 1 } is trivial. This can be shown either by the direct computation or using the relation between global and local Artin maps (see [CF, Ch. VII, Prop. 6.2]). [] Fix a central skew field D~ over K~ with invariant 1/d. Set E =: D,, � E', then Construction 1.5.1 gives us an (E, K~)-scheme X corresponding to F. 2.1.6. Second basic example. -- By Brauer-Hasse-Noether theorem (see [Wel, Ch. XIII, w 6]) there exists a unique central skew field D t"t over K which is locally isomorphic to D at all places of K except w and ~ and has Brauer invariant l/d at w. By Landherr theorem (see [Sc, Gh. 10, Thm. 2.4]), D mt admits an involution of the second kind over F. Fix an embedding o%:K ~-+ C. It induces an archimedean com- pletion Foo, of F, and we have the following p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 89 Proposition 2.1.7. ~ a) There exists an involution of the second kind o~ mt of D ~ over F such that: i) the pairs (D, a) | F, and (D mr, 0trot)| r F, are isomorphic at all places u of F, except v and ool; ii) the signature of (D ~t, ~mt) at oo~ /s (d -- 1, 1). b) The group G ~t := GU(D lnt, ~t~,) is determined uniquely (up to an isomorphism) by conditions i), ii) of a). Proof. ~ a) follows from [G1, (2.2) and the discussion around it] as in [C1, Prop. 2.3]. b) follows immediately from [Sc, Ch. 10, Thm. 6.1]. [] Let G "t be as in the proposition. Then embedding oo~ defines an isomorphism Dmt | Kool--~ Mata(C), and we identify pGt~t(F~o~) with PGUa_a,~(R ) by the induced isomorphism. Set G~t(F)+ := Grit(F) n G~"t(F=~) ~ Then G~t(F)+ = G~(F) if d> 2, and [Gt"t(F) : G~nt(F)+] = 2 if d = 2. Set E ~t := G~t(A~) and let Eo~nt C E t"t be the clo- sure of Z(Glnt(F))C E t~t. Embed diagonally G~"t(F) into Gt"t(Foo~) � E t~ and define I 't"t to be the image of Grit(F)+ under the natural projection to PGmt(Foo~) � (Emt/E~ont) = PGUa_~,~(R)� Proposition 2.1.8. -- The subgroup F '~t is a cocompact lattice in PGU,_ ,,~ (R)~ � (E~/E'o"~) and it has an injective projection to the first factor. Proof. -- Notice that the natural projection E lnt /E o ~.t ~E"VZ(E ~"t) = PE '~ induces an isomorphism I~Int--~-PGI~*(F)+CPGUa_I,x(R)~215 PE lnt and that the group Z(E~t)/Eo ~t-~ (A~)� � is compact. Therefore it will suffice to prove that PGlnt(F) is a cocompact lattice with an injective projection to the first factor of PGI"t(F~o~) x PE fnt. This can be proved by exactly the same considerations as in the proof of Proposition 2.1.4, c). [] By the proposition, I ~n~ satisfies the assumptions of Construction 1.3.6, so it determines an (Elnt/Eo ~t, C)-scheme ~t~t, which can be regarded as an (E ~t, C)-scheme with a trivial action of El0 ~t. Remark 2.1.9. -- For each S ~ ~-(E ~t) we have the following isomorphisms (~V~smt)= ~ S\[B d-1 x (E'n*/Eo~t)]/r ~' _--_ (S.Z(Glnt(F))\[B a-1 X G'n*(A~)]/G~t(F)+ (S.Z(G~nt(F))\[B d-1 x G'nt(A~)]/G'n*(F)+ -_ S\[B '-1 x G'nt(A~)]/G'~t(F)+. (Emt/E~on~). 90 YAKOV VARSHAVSKY 2.2. First Main Theorem Definition 2.2.1. -- An isomorphism r -% E mt is called admissible if it is a product of G(A~ :~) -% G~t(A~v:~), induced by some A~v:~-linear algebra isomorphism D | Av s;* -% D*~t | Av f'* (compare Proposition 2.1.7), and the composition map ~X :K D~ � F, -% (D~ t) � � F~ --~ G*"t(F~), constructed from some algebra isomorphism D~ -% D~t | K~ as in 2.1.3. 2.2.2. Fix a field isomorphism C -% C~, whose composition with embedding oo 1 : K ,-+ C (chosen in 2.1.6) is the natural embedding K r K~ ,-+ C~. Identifying C with C~ by means of this isomorphism we can view, in particular, K~ as a subfield of C. First Main Theorem 2.2.3. -- For some admissible isomorphism (P:E -% E ~at there exists a r isomorphism fvfrom the (E, C)-scheme X c to the (E ~t, C)-scheme ~t. 2.2.4. Let E 0 be the kernel of the action of E on X, and put E := E/E 0. By Corollary 1.6.4 there exists a subgroup A C PGUa_I.~(R) ~ � E such that the (E, C)- scheme X c corresponds to A by the real case of Construction 1.3.6. By Proposition 1.5.3, each admissible isomorphism r -% E ~t satisfies q)(E0) = ~ol~/nt- Hence r induces an isomorphism ~ : E -% Ei"t/E0.~t Theorem 2.2.5. -- There exists an admissible isomorphism r ~ E ~nt and an inner automorphism e? of PGUd_I. 1 such that (~ � U~) (A) = F int. Lemma 2.2.6. -- Theorem 2.2.5 implies the First Main Theorem. Proof. -- Theorem 2.2.5 implies that there exists a r analytic iso- morphism 9~:(Xc) ~n -% (~int)an. From the (I)-equivariance we obtain analytic iso- morphism 9~, s : (Xs, c) ~n -% ~ for each S e ~-(E). Corollary 1.2.2 provides us with an algebraic isomorphism fr s : Xs, c --% satisfying (arc, s) ~n ~ 3~, s. Taking their inverse limit we obtain a q)-equivariant isomorphismfe :---= limfe, s : Xc --~ Xmt- [] Thus we have reduced our First Main Theorem to a purely group-theoretic statement. For proving it we need to know more information about A. First we introduce some auxiliary notation. 2,2.7. Let A'CPGUd_I,I(R)~ � PE and A"CPGUd_I,I(R)~ � PE' be the images of A under the natural projections. Since the groups Eo\Z(E ) and Eo\D~ x .Z(E) are compact, Lemma 1.3.11 shows that subgroups A' and A" correspond by the real case of Construction 1.3.6 to the (PE, C)-scheme X~ := Z(E)\X c and to the (PE', C)- scheme X~' := (D~ � Z(E))\X c respectively. The same lemma implies also that the natural projections A -+ A' and A -+ A" are isomorphisms. Let E' 0 be the image of E 0 under the canonical projection to E'. Let F' be the -~olsl~'i~t ~'t~lnt~olS~J p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 91 image of V under the projection GLa(K,) x E' -+ GLa(Kw) x (E0\E'). Then, by Pro- position 1.5.3 c), the group F' corresponds by the p-adic case of Construction 1.3.6 ~X to the (E0\E' , K,)-scheme X'" := Dw\X. Recall also that, by Proposition 1.5.3 d), the (PE',Kw)-scheme X"----(I~ x Z(E))\X is obtained from the subgroup PF C PGLa(K~) >< PE' by the p-adic case of Construction 1.3.6. For each subset | of A, A' or A" (resp. of F, F' or PF) we denote by O= (resp. | and | its projections to the first and to the second factors respectively (compare 1.1.8). Our next task is to establish the connection between A and F. The next key proposition is the modifications of [Ch2, Prop. 2.6]. In it we apply Ihara's technique of elliptic elements to relate elements in A and in F. Proposition 2.2.8. -- For each 8 e A with elliptic projection 8~ e PGU a_ 1,1(R) o, there exist y ~ F and YD ~ D,~ ~� with (YG, YD) ~ GLa(K~) x D~ elliptic (with respect to its action on EK~ )a and a representative ~' = (~o, ~'E) e GUe_I,I(R) ~ � E of 8 satisfying the following conditions: a) the elements (YD, YE) and "~r. are conjugate in E; b) the characteristic polynomials of ~o~ and YG are equal. ~� GL~(K~) � D~ Conversely, for each y ~ F and YD ~ Dw with (Yo, 7D) ~ ~ � elliptic, there exist 8 ~ A with elliptic projection 80~ ~ PGUa_I, 1(11)~ and a representative~ e GUa_I,I(R) 0 � E of 8 satisfying conditions a) and b). Proof. -- If an element 8~0 ~A= is elliptic, then 8| has a fixed elliptic point P on B a-1. The action of 8= on B a-1 coincides with the action of 8 E on B a-1 ~ B a-1 X { 1 } C (B a-1 X E)/A ~ (Xc) an, therefore P, viewed as a point of (Xc) an (or of X(C)), is an elliptic point of 8~. Using the isomorphism C -% C~, chosen above, P can be considered as a point of X(C~), hence as a point of the p-adic pro-analytic space X a". There exists an element g e E such that the point P' := g(P) lies in ~ := pl(EKw) in the notation of Corollary 1.5.4. Let = be the natural projection X---> X"'. Choose a representative ~'e E of t lit an g 8E g-1 ~ ~. Since ~' fixes P, it fixes the projection P" := =(P') ~ (Xc~) . Hence stabilizes the connected component D~,, GK~ C~ � { 1 } C (X~;') an containing P". By Proposition 1.5.3 c), the image of ~ under the canonical projection E-+E0\E' belongs to the projection of F' to E'0\E'. We can therefore choose y e F whose projection to E'0\E' coincides with that of ~'eE. Therefore ~,y~l belongs to D~, X E 0 -~ D~ E 0. Hence there exists a y, ~ D w such that F(y{ 1, .~1) ~ E0" It follows that (YD, YE) ~ E is also a representative of g 8 E g-1. The action of (YD, YE) on the tangent space of P' e ~ is conjugate to the action of 8 E on the tangent space of P, therefore P' is an elliptic point of (YD, YE)" Since P1 is dtale, one-to-one (use Corollary 1.5.4) and D, x r-equivariant, the action of (YD, YE) on the tangent space of P' m ~/ coincides with the action of (YG, 7i)) on the tangent 92 YAKOV VARSHAVSKY space of ptl(P ') e Yfl Therefore p~--l(p,) is an elliptic point of (Ya, YD). It follows Kzo " that the action of (Ya, YD) on the tangent space of p-l(p,) e Na is conjugate to the K w action of 8~o on the tangent space of P e B a-x. Using the 6talness of the projection Nd a%-+~ we conclude from Lemma 1.7.2 that there exists a representative 8~o E GU d_ 1, x(R)~ of 8ao such that the characteristic polynomials of 8~ and YG are equal. Hence 7 := (8~o, g-l(yD, "f~)g) is the required representative of 8. The proof of the opposite direction is very similar, but much easier technically. If an element (u YD) s s o � D~ is elliptic, then it has an elliptic point QeEa Kto 9 Hence Q' := pl(Q) e X ~ is an elliptic point of (YD, Y~) e E. Hence Q' can be considered as a point of the complex analytic space (Xc)an___ (Bd-1 � ~)/A. Choose a representative (x,g)eBa-1 � E of Q'. Then the element g(YD, Yv.)g-1 e E fixes Q" :=g(Q')eBa-1 x {1}, hence it stabilizes the connected component B d-~ x{1}C(Xc) ~. It follows that the image ofg(yD, YE) g- ~ under the projection of E to E belongs to A E. The rest of the proof is exactly the same as in the other direction. [] Corollary 9..9..9. -- For each 8 e A with elliptic projection 8oo e PGUa_~4(R)~ , there exists a representative 7= x x x o(A; -~ such that a) if we view K as a subset of C, of K~, and of K | A~ :~ respectively, then the characteristic polynomials of Too, 7~ and 7::" have their coefficients in K and coincide; b) 9~ and the similitude factor of~::" belong to F, viewed as a subset Of F{ and of (A~: ~) � respec- tively, and coincide. Proof. -- Take y and 7 as in the proposition. Then the statement follows from Proposition 1.7.5. [] Proposition 2.2.10. -- We have the inclusion A-~E D (SD~ x n Tx) x P(Gd~r(AFt'")). Proof. -- Let X' o be the connected component of X~ such that (Xo) a" D B a- a � { 1 }. -7 Then by Proposition 1.3.8 c), A S = Stabe~,(X0). Proposition 1.4.6 implies that the group SD~ n T 1 acts trivially on the set of connected components of X~, therefore it remains to show only that A~' D p(Gd~'(A~;')). To prove it we first observe that by the strong approximation theorem (see, for example, [Ma, Ch. II, Thm. 6.8]), the closure PG(F) of PG(F) in PE' = PG(A~ ;~) contains P(Gd~ So the proposition follows from tp Lemma 2.9..11. -- We have Az = PG(F). p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 93 Proof. -- Proposition 1.3.8 c) we see that A~ is the stabilizer of the connected component Yoo of X~' such that (Yo~) ~ ~ B a-a � { 1 } and PG(F) is the stabilizer of the connected component Y~ of X~; such that (Y~)~") f~a ~% � { 1 }. Since the group PE' acts transitively on the set of geometrically connected components of X", the sub- -'-77, --77, groups A E and PG(F) are conjugate in PE'. Since A E contains P(Ga~(As it is normal. So we are done. [] 2.3. Computation of Q(Tr Ad) In the next subsection a field Q(Tr Ad) (generated by the traces of the adjoint representation) will be a field of definition of a certain algebraic group. Remark 2.8.1. -- If g e GLa, then by direct computation we obtain that Tr Ad g = Trg.Tr(g-a). Hence for g E PGL a we have Tr Adg = Tr~'-Tr(~ -1) -- 1 for each representative ~ e GLa of g. co 1 Proposition 2.3.2. -- We have Q(Tr Ad Aoo) = F ~ R. Proof. -- It follows from Proposition 2.2.8, Proposition 1.7.5 and Remark 2.3.1 that Q(Tr Ad 8= I 8= ~ as is elliptic) = Q(Tr Ad YG [ YG ~ PPa C PGLa(K~) is elliptic). Let F' be the last-named field. Then F' C F, since PI' = PG(F) and since PG is an alge- braic group defined over F. It follows from the weak approximation theorem that for each non-archimedean prime u 4: v of F, the closure of the projection to PG(F~) of the set { e Prlv is elliptic} contains an open non-empty subset of PG(Fu). (Recall that the closure of Pr a in PGLa(Kw) contains PSLa(Kw) by Proposition 1.1.10, and that the set of elliptic elements of PSLa(K~) is open and non-empty by Proposition 1.7.3.) Therefore F' is dense in each non-archimedean completion F, of F for u 4= v. Thus F' splits completely in F at almost all places. Hence F' = F (see [La, Ch. VII, w 4, Thm. 9]). This part of the proof is completely identical with Cherednik's proof of [Ch2, Prop. 2.7]. Now we want to prove that Q(Tr Ad A~o ) = Q(Tr Ad ~ [ 8o0 ~ A~o is elliptic). Since the group PGUa_I, 1 is absolutely simple, the representation Ad : PGUa_~,~(R ) ~ GL(Lie(PGUd_~,~(R)) ) -- GLa_~(R ) is absolutely irreducible. Therefore our statement is a consequence of the following general Lemma 2.3.3. -- Let ? be an absolutely irreducible algebraic representation of PGU d_ 1,1 and let A be a dense subgroup of PGUd_I,I(R) ~ Then Q(Tr(o(A))) = Q(Tr p(8) [ 8 e A is elliptic). 94 YAKOV VARSHAVSKY Proof. -- Let L be the last-named field. If g e PGUa_I, a(R) ~ is elliptic and g' is not elliptic for some r e Z -- { 0 }, then by Lemma 1.7.2, g belongs to some Zarisld closed proper subset of PGUe_I, ~. Therefore for each N e N, there exists an open subset WC PGUd_I,I(R) ~ such that for g e W and r e Z satisfying 1 ~< [r[.< N, the element g' is elliptic. Choose g e W. By the continuity of multiplication, there exists an open neighbourhood U C W of g such that for gl, 9 9 -, gk e U, and nl, ..., n k e Z, satisfying nl+ ... +nk4: 0, [n~[ + ... +[nk[~< N, the element gi'~.....g~k is elliptic. Take N ----- 6m 2, where m is the dimension ofp. Since PGUa_I,I(R) ~ is a connected real Lie group, it is generated by U. The subgroup A is dense in PGUa_I,~(R) ~ by Proposition 1.1.10, therefore A n U generates the group 7~ (see [Ma, Ch. IX, I_,em. 3.3]). Since the restriction of p to the Zariski dense subgroup A is absolutely irreducible, Burnside's theorem (see [Wa, vol. II, Ch. XVII, 130]) implies that ~ := dimR(Spana(p(A)) ) = m ~. Set ~0 := { 1 } C A, and for each positive integer n set ~-:--{g~.....g~klg~eAoo ~g, ln~[ -t- ... +]n~l~< n}CA. Denote dim~(Span,(0(A")) ) by ~,. Since A = O 7~", we have 1 =~0~< ~1~ .-.~< ~,~< ...~< ~=sup~,. Cb Moreover, if~, = ~, + 1 for some n, then ~ = N, + 1 ..... N. Therefore ~,,~ _ 1 ---- m~. Hence there exist elements ~ e ~m"-a, i = 1, ..., m z such that { p(~) }~ constitute a basis for Mat,,~(ll). Choose any g e A ~ U and take ~ := g"~+a ~i. Then{ 0(8~) }~ still constitutes a basis for Matm~(ll ). Each ~ is of the form g~'~.....g~, where the gi's belong to Ac~Uandthen~'ssatisfyn~-k ... +n~>/2 and [n~] +In~[ + ... +[n~[~< 2m ~. In particular, each 8~ is elliptic, therefore Tr p(8~) e L. Lemma 2.3.4. ~ If for some ~ e PGUa_I,I(R) ~ the elements ~ are elliptic for all i = 1, ..., m ~, then O(~) can be written as a linear combination of the p(~)'s with coeJficients in L. Proof. -- Let el, ..., em~ be the dual basis of { p(S~) }~ relative to the bilinear form (x,y) ~ Tr(xy). If 8 is as in the lemma, then Tr O(~)= Tr(p(8)p(8,))eL for all i = 1,..., m 2. Hence p(8) can be written as a linear combination of the e~'s with coefficients in L. Therefore it is enough to prove that each e, can be written as a linear combination of the p(3~)'s with coefficients in L. The last condition is equivalent to the condition that each p(8~) can be written as a linear combination of the e~'s with coefficients in L. Thus, as we mentioned above, to complete the proof it is enough to show that each 8, satisfies the conditions of the lemma. This follows directly from the definition of the ~i's and of U. [] The choice of the ~,'s assures that for every ~ e A n (U w U-1) the elements ~8,~ p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 95 are elliptic for all i----- 1, ..., m 2. Therefore the above lemma implies that p(8) can be written as a linear combination of the p(8~)'s with coefficients in L. The set U t3 generates the group &, hence for every 8 ~ &, the linear transformation p(8) can be written as a polynomial in the p(8~)'s with coefficients in L. For any i,j, k ~{ 1, ..., m 2 } the elements 8~ 8j 8 k are elliptic, therefore by the lemma each p(8~ 8~)= p(Si)p(~j) can be written as a linear combination of the 9(Sk)'s with coefficients in L. Hence every polynomial in the p(8~)'s with coefficients in L, can be written as a linear combination of the p(8~)'s with coefficients in L. In particular, this is true for each p(8) with ~ ~ &. Hence Q(Tr p(A)) C L. [] Corollary 2.3.5. -- Suppose that a subgroup A C A~o is Zariski dense in PGU d_ 1,1 and that Ao~ C Commmua_l,~(R)(A). Then Q(Tr Ad A) ---- Q(Tr Ad Ao~) (= F). Proof. -- Set L:----- Q (Tr Ad A), then there exists an L-form V of Lie (PGU d_ 1,1 (R)) preserved by Ad A (see [Ma, Ch. VIII, Prop. 3.22]). Take any 8 ~ A| Then some sub- group of finite index A' of T~ satisfies 8A' 8 -1 C A, hence (Ad 8) (Ad A') (Ad 8)-1(V) ---- V. Since the subgroup A' is also Zariski dense in PGUa_~,I, Burnside's theorem implies that Ad A' generates End V as an L-vector space. Therefore (Ad 8) (End V) (Ad 8) -1 C End V. In other words, Ad(AdS)(EndV)= End V. Let H be the Zariski closure of Ad&C GL(V). Then H is an L-form of AdPGUa_I,1, hence LieHC EndV is an L-form of Lie(Ad PGUd_I,1). In particular, Lie H = End V n Lie(Ad PGUd_I,1) , therefore Ad(AdA~o)(LieH)= Lie H. Since PGUa_I, 1 is adjoint, the homomor- phism ad := Ad. : Lie PGUd_I, 1 -+ Lie(Ad PGUa_I,1) is an isomorphism. Therefore V:= ad-l(LieH) is an L-form of Lie(PGUd_I,1) and AdA~oC GL(~). It follows that Q (Tr Ad Ao~) C L. [] 2.4. Proof of arithmeticity 2.4.1. Consider the subgroup A'C PGUa_I,I(R) ~ � PEC PGUd_I,I(R ) � PE, defined in 2.2.7. For a finite place u of F let G, be PGF, for u 4: v and PGLI(Dw), viewed as an algebraic group over F, ~- Kw, for u = v. In what follows it wiU be also convenient to introduce a formal symbol oo and to write Foo instead of R and Goo instead of PGU d_ 1,1 (the algebraic group over Fo0 ~ R). Let 1V[ be a finite set of non-archimedean primes of F, containing v for simplicity of notation. Set ~I := M woo and choose S e ~'(PG(A~:~)). For each subset M' of M, denote I[ G,(F,) by GM,. Denote also the projection of A'n (G~ � S) to u~M' by A s . Let A~ (resp. A s ) be the projection of A s to Go0(F~o) (resp. to GM). For u ~ M and 8 ~ A s denote the projection of 8 to G,(F,) by 8,. 96 YAKOV VARSHAVSKY Definition 9..4.9+. -- A lattice F C: G~m is called irreducible if for every proper non- empty subset M' C M the subgroup (F n GM, ) (F n G-~m_m. ) is of infinite index in P (compare [Ma, p. 133]). Definition 9,. 4.3. -- We say that a lattice P of G~M has property (QD') if the closure of FGo~(Foo) in C~M has finite index. Remark 9..4.4. -- Since the group PGU a_ 1, t is isotropic over R, it follows from [Ma, p. 290, Rem. (v)] that ff F has property (Q.D'), then it has property (Q.D) in the sense of Margulis (see [Ma, p. 289]). Proposition 9..4.5. -- The subgroup A s C G~M is a finitely generated cocompact irreducible lattice, which is of infinite index in Commo~(A s) and has property (Q.D'). Proof.- Observe that PGUa_I,I(R)x PE = ~ X PG(/k~ :M) and that A' is a cocompact lattice in G~M � PG(Av I:M) having injective projection to PGUa_~.I(R), hence to G~m. It follows from Lemma 1.1.9 that A s C ~ is a cocompact lattice, which is of infinite index in ComnM~(AS). By Proposition 2.2.10 the closure of G~o(Foo)A' in ~ � PG(AIv :'+) contains G~o(F~o) � (S~ n T1) � P(G~er(A~:*)). Hence the closure of Goo(Foo)A' in ~ contains Go~(F| � (SD~ n TI) � II P(GaCr(F,)). .~M--{v} In particular, A s has property (QD'). Let M' be a non-empty subset of M. Then A s n G~s, ={1}, because the projection of A' to PGUa_I,t(R) is injective. Suppose that A s is not irreducible, then [A s : (A s r3 ~-M')] < oo. Hence A" : (G+(F+) zX +) n Cm_,+,] < oo. p(Gae'(F.)) and Since G+o(F+o)A s D Go~(F~o) x (SD~ n TI) x I-[ u 9 {t,} (G+(F+)AS) n G,-~M_w C G~_M, , we get a contradiction. Since A s is a cocompact lattice in C_~M , it is finitely generated (see [Ma, Ch. IX, 3.1 (v)]). rq 2.4.6. Now we are going to use the results of Margulis (see [Ma]). By [Ma, Ch. VIII, Prop. 3.22], there exists a basis in Lie(PGUd_I.I(R)) such that all transformations in Ad Aoo are written in this basis as matrices with entries in Q(TrAdA~) = FC Foo 1 ~ R. Define a homomorphism 9:G~o -+GLa~_t rational over R by assigning to g 9 Goo the matrix of Ad g in the above basis. It follows that ~(A,~) C GLa*_~(F ). Let H be the Zariski closure of ~(A~o); then H is an algebraic group, defined over F and 9(A~o ) C H(F). Since A~o is Zariski dense in Goo and since the group Go -----PGUa_I,1 is adjoint, ~ induces an isomorphism PGUa_I, x -% Hire01. In particular, H is an F-form of PGUd_I, x. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 95 are elliptic for all i----- 1, ..., m 2. Therefore the above lemma implies that p(8) can be written as a linear combination of the p(8~)'s with coefficients in L. The set U t3 generates the group &, hence for every 8 ~ &, the linear transformation p(8) can be written as a polynomial in the p(8~)'s with coefficients in L. For any i,j, k ~{ 1, ..., m 2 } the elements 8~ 8j 8 k are elliptic, therefore by the lemma each p(8~ 8~)= p(Si)p(~j) can be written as a linear combination of the 9(Sk)'s with coefficients in L. Hence every polynomial in the p(8~)'s with coefficients in L, can be written as a linear combination of the p(8~)'s with coefficients in L. In particular, this is true for each p(8) with ~ ~ &. Hence Q(Tr p(A)) C L. [] Corollary 2.3.5. -- Suppose that a subgroup A C A~o is Zariski dense in PGU d_ 1,1 and that Ao~ C Commmua_l,~(R)(A). Then Q(Tr Ad A) ---- Q(Tr Ad Ao~) (= F). Proof. -- Set L:----- Q (Tr Ad A), then there exists an L-form V of Lie (PGU d_ 1,1 (R)) preserved by Ad A (see [Ma, Ch. VIII, Prop. 3.22]). Take any 8 ~ A| Then some sub- group of finite index A' of T~ satisfies 8A' 8 -1 C A, hence (Ad 8) (Ad A') (Ad 8)-1(V) ---- V. Since the subgroup A' is also Zariski dense in PGUa_~,I, Burnside's theorem implies that Ad A' generates End V as an L-vector space. Therefore (Ad 8) (End V) (Ad 8) -1 C End V. In other words, Ad(AdS)(EndV)= End V. Let H be the Zariski closure of Ad&C GL(V). Then H is an L-form of AdPGUa_I,1, hence LieHC EndV is an L-form of Lie(Ad PGUd_I,1). In particular, Lie H = End V n Lie(Ad PGUd_I,1) , therefore Ad(AdA~o)(LieH)= Lie H. Since PGUa_I, 1 is adjoint, the homomor- phism ad := Ad. : Lie PGUd_I, 1 -+ Lie(Ad PGUa_I,1) is an isomorphism. Therefore V:= ad-l(LieH) is an L-form of Lie(PGUd_I,1) and AdA~oC GL(~). It follows that Q (Tr Ad Ao~) C L. [] 2.4. Proof of arithmeticity 2.4.1. Consider the subgroup A'C PGUa_I,I(R) ~ � PEC PGUd_I,I(R ) � PE, defined in 2.2.7. For a finite place u of F let G, be PGF, for u 4: v and PGLI(Dw), viewed as an algebraic group over F, ~- Kw, for u = v. In what follows it wiU be also convenient to introduce a formal symbol oo and to write Foo instead of R and Goo instead of PGU d_ 1,1 (the algebraic group over Fo0 ~ R). Let 1V[ be a finite set of non-archimedean primes of F, containing v for simplicity of notation. Set ~I := M woo and choose S e ~'(PG(A~:~)). For each subset M' of M, denote I[ G,(F,) by GM,. Denote also the projection of A'n (G~ � S) to u~M' by A s . Let A~ (resp. A s ) be the projection of A s to Go0(F~o) (resp. to GM). For u ~ M and 8 ~ A s denote the projection of 8 to G,(F,) by 8,. 98 YAKOV VARSHAVSKY for each g ~ G.(F,). Hence for each 8 e A' we have Tr Ad(~ � -c) (8) ---- (Tr Ad(8~); ..., t%(Tr Ad(8,)), ...) (Fo~ 1; ..., FI~, ...). Recall that (9 � (A')C H(F), hence TrAd((~ � v)A')C F. On the other hand, Corollary 2.2.9 implies that Tr Ad(8) e F C F~I � A~ for each 8 e A' with elliptic 8~o. In particular, for such ~'s we have Tr Ad(~) = Tr Ad(~,) e F for each u. Since we showed in the proof of Proposition 2.3.2 that O_..(Tr Ad(8oo) I 8oo is elliptic) = F, we conclude from the above that the restriction of each r :F,--~ FII,~ to F is the identity. Since each r is continuous, the claim follows. [] 2.5.3. Next we will show that in the case d > 2 we have F' =- K. Indeed, if a prime u of F splits in K, then PG'(F~)- G,(F~)~-PD~ for some central simple algebra D. over F.. It follows that u splits in F'. By [La, Ch. VII, w 4, Thin. 9], F' ---- K. As we mentioned before, we may take F' = K also in the case d---- 2. Proposition 2.5.4. -- The map v induces a continuous isomorphism PE--% H(A~). Proof. -- Since PE "~ = PD~ ~ � � PE' and H(A~) =~H(F,) � HtA :;*~, F j, we need only to show that v* : 1-I,** G,(F,) --~ 1-I,,~ H(F,) induces a continuous isomorphism PE' -~ H(A~;*). First we claim that v ~ induces a continuous map from A'C PE' to H(A~:*). In fact, let a sequence { 8. }. C A~ converge to g e PE'. Then the sequence { 8, 8~-~_ 1 }, converges to 1. Therefore for each S e~'(PE') there exists N seN such that ~ 8:-~ 1 ~ A~' ('~ S (hence ~*(8. 8~-~1) ~ -c'(A~ C~ S)) for all n t> N s. Since -r'(A~ n S) is commensurable with H(0F) , it is contained in a compact subset of H(A~:*). Therefore the sequence {-r"(8. 8.+1) -1 }.C H(AIv :~) has a limit point. Let h be some limit point of { ~(8. 8~,-~1) )., and let { ~*(8-i 8-1-i+ 1) }, be a subsequence, converging to h. Then for each prime u ~ v of F we have 8 -1 .(lim = h, = lim %((8,, .,+1).) ---- (8,, ,,+1).) 1, ~oo because % is continuous. It follows that 1 is the only limit point of {-~(8, 8~,-~1)}., therefore the sequence { v"(8.) v ~t~- .+1) 1 }. converges to 1. Now by similar arguments we see that the sequence { v"(8,) }, converges to v"(g) e H(A~:"). Moreover, the same arguments also imply that if we show that -~"(PE') = H(A~;"), then the continuity of ~ and of (v") -1 will follow automatically. Observe that for each non-archimedean place u we have G(F,)der = Gde'(F,) (resp. G'(F.)a~ (G')aer(F.)) (see [PR, 1.3.4 and Thm. 6.5] in the anisotropic and [PR, Thin. 7.1 and 7.5] in the isotropic cases respectively). Therefore v" induces an isomorphism of derived groups rl.,, P(Gder(F.)) ~ I-[.,~ P((G')de'(F.)). p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 99 By Proposition 2.2.10, A~ D p(Gd~*(A~:*)) = PG(A~ :~) n IIu** p(Gd~'(F,)). Hence by the facts shown above, "r~(p(Gd~ C PG'(A~:') n II P((G')d~'(F.)) = P((G')d~'(A~;*)). In particular, "~([I~,~ p(Gd~'(d)s,))) = II~,~ v,(P(Ga~r(d)~.))) C P((G')de'(A~:~)). It follows that %(P(Ga~'(d~F.))C P((G')aor(o~.)) for almost all u 4= v. Since each % is algebraic, the subgroups %(P(Gd~ and P((G')d~r(d)F.)) are conjugate (hence equal) for almost all u ~e v. It follows that I:~(P(Gae*(A~;*))) = P((G')d~r(As Therefore to complete the proof it will suffice to show that PG(As :~) (resp. PG'(A~;")) is the normalizer of P(Ga~ ;*)) in the product II~, ~ PG(F~), and similarly for PG'. Since PG(As is the restricted topological product of the PG(F~)'s with respect to the PG(d~F~)'s , it remains to show that the normalizer of p(Gdor(d~F~)) in PG(F~) is PG(d)s. ) for almost all u. This can be done by direct calculation. [] We will use the same letter -~ to denote the isomorphism between PE and PG' (As ~). 2.5.5. Notice that a regular function t := Tra/det on GL a defines a function on PGLa. Moreover, an algebraic automorphism t~ of PGLa is inner if and only if it satisfies t o ~b = t. Therefore there is a unique choice of an algebra D' defining G' (see 2.5.1) such that the function t' := Trn/det on PG', defined by the natural embedding G'(F) ~ D', satisfies t' o q~ = t. Proposition 2.5.6. -- We have D' ~ D i"t, G' ~ G ~t and x is induced by some admissible isomorphism. Proof. -- By Corollary 2.2.9, for each 8 cA' with elliptic 8~o we have t(8| = t(8~) = t(8 s:") ~K. Since (9 � "r (8) E PG'(F) C PG'(F~o, x A), we have t((9 x "~) 8) s K. By our assumption, t(q)(8oo)) = t(8~o) for all 8 s A'. Hence for each 8 cA' with elfiptic 800 we have t(%(8,))---t(8,) for each non-archimedean prime u ofF. Recall that the algebraic isomorphism % : PG(F,) -+ PG'(F,) for u v is induced either by an F, dinear isomorphism D| F,-%D'| F~ or by an F,-linear iso- morphism D| s F. --% (D')~174 s Fu, composed with an inverse map (g ~_,g-a). In the first case we have t(%(g.))= t(g,) for all g~, ~ G,(F,,), and in the second one t(%(g~)) = t(g~ ~) for all g~ ~ G~(F~). To exclude the second possibility we need to show the existence of a 8 ~ A' with elliptic 8~ such that t(~o0) + t(8;~). Since the condition t(g) = t(g -~) is Zariski closed and non-trivial and since the closure of all elliptic elements of A~0 e PGUa_I,I(R) ~ contains an open non-empty set, we are done. It follows that D' is locally isomorphic to D ~nt at every non-archimedean place of K, except possibly at w and ~, and that the map -d : PG(A~ ;~) --% PG'(A~ ~") is induced 100 YAKOV VARSHAVSKY by some admissible isomorphism. To prove the statement for the v-component we copy the above proof replacing PG(F,) by PGLx(D~) and D | F, by D~ 9 ~o,, Since D' and D ~ are locally isomorphic at all places, they are isomorphic. We showed before that PG'(Foo~)--PGUa_I,I(R) and that for each i=2, ...,g the group PG'(F~ol) is compact and, therefore, is isomorphic to PGUa(R ). Propo- sition 2.1.7 b) then implies that G' - G t~. [3 From now on we identify G' with G tnt. 2.6. Completion of the proof Our next task is to prove the following Proposition 2.6.1. -- We have (~ � "~) (A') = pG~t(F)+. Proof. -- First observe that (q~ X v) (A')~ = q~(Ao~) C ~(PGUa_I,I(R) ~ = PG'nt(ro,,) ~ therefore (q~ � v) (A') C pG'nt(F)+ and (q~ � -~) (A") C PG~t(F)+. Since the projection of PG~t(F) to PGmt(F~ol) � PG~t(A~ ;') is injective, it remains to show that (2.1) [PGt~t(F) : PG~nt(F)+] = [PGmt(F) : (~0 � -~) (A")]. We are going to use of Kottwitz' results described in 1.8. Recall that PG ~t is an inner form of PG. Let o~eo and co~i~t be non-zero invariant differential forms of top degree on PG and PG ~nt respectively, connected with one another by some inner twist as in 1.8.2. They define invariant measures [~Po [ and I o~mi~t [ on PG(F~) and PG~t(Fu) for every completion F. of F and product measures on PG(Ar) and PG~"t(AF) respectively (see [We2, Ch. 2]). It follows from Weil's conjecture on Tama- gawa numbers and from Ono's result (see Ono's appendix to [We2]) that (9,.9,) [ I (PG~t(AF)/PGt~t(F)) = I c~ [ (PG(AF)/PG(F)). Lemma 2.6.2. -- Let A and B be locally compact groups, let S be a compact and open subgroup of A and let P be a lattice in A � B with injective projection to B. Then for every right invariant measures ~a on A and ~ on B we have (~A � ~B) ([A � B]/r) = ~.(S). ~([(S\A) � B]/F). Proof. -- Let F a be the projection of F to A. Choose representatives { a~ }i e i of the double classes S\A/F a. For each i e I let F~ be the projection of the subgroup (ai-1 Sa, � B) c~ F to B. Then P~ is a lattice in B, therefore there exists a measurable subset U, of B such that B is the disjoint union II U, 7. Since P has an injective YE Fi projection to B, we have A � B= II H (Sa � U~)y. Then ([Za � ([A � B]/r) = = = AS). Z � V,) = � B]/r). [] 0')pGint p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 101 By the lemma, for each S e o~'(PG(As the left hand side of (2.2) is equal to (2.3) I/ I o~mt~t [ (pG'nt(Fo~)) 9 [ O~mmt [ (PGt~t(F,)). [ OJmmt I ('r'(S)) 9 I I x and the right hand side of (2.2) is equal to (2.4) II I o~m I (PG(FOO,)). I (ol, o I(S). [ o~m [ (S\PG(A~)/PG(F)). i=1 By definition, [ o~mt, t ] (PG~'t(F~,)) = [ ~ [ (PG(Fo~i)) for each i = 2, ..., d and [ orbit I (e(S)) = ] to m I (S) for each S ~ ~'(PG(/k~:v)). Since the expressions of (2.3) and (2.4) are equal, Proposition 1.8.5 and Remark 1.8.6 imply that (~'.5) ~pGIni; (e(S)\[PG~t(F~) � pG'nt(As Fool. = d. ~m~o(S\PG(As (" q- " was added to multiply the left hand side by 2 when d = 2). If S is sufficiently small, then for each a e PG(/kr s;*) the group a -~ San PG(F) is torsion-free by Proposition 1.1.10. Let Y~-ts~ be the projective variety over K~, such that Y~ls~ ~ = ( a-1 Sa c~ PG(F) )~\ fP x~" By Kurihara's result (see [Ku, Thm. 2.2.8]) ca_l(Ty~_l~,) = )~lg(a -1 Sa N PG(F)).ca_I(Tpa-x), where ca_l(Ty~_ls~ ) (resp. ca_l(Tre-1)) is the (d- 1)-st Chern class of the tangent bundle of Y~-~s~ (resp. pa-1). Notice that ca_l(Tra-~ ) = d, hence ca_l(Tya._lsa) = d. V%o,~((a -~ Sa c~ PG(F)),\PG(F,)). (Y )an~ A" \B a-l, we have Since a-lSa, C ~ a -1 8a ca-~ (TY~-I s,) = Ca -' (Tc~,- ~ s~' c) = Z~,(A~_x " s.kB a-1 ) (see for example [BT, Prop 9 11.24 and (20.10.6)]). The last expression is equal to Z'A" " 1~(a-ls~t) = ~pGUd_ 1 t(A'a'_Zsa\PC-.Ua_I,X(R)). This shows that for each a ~ PG(A~;') we have a. -1 sa n PG(F))v\PG(L)) = so\PGUa_ 1, l(g)). Summing this equality for a running over a set of representatives of double classes in S\PG(A~:~)/PG(F), we obtain that a. ~m.~(S\PG(A~)/PG(F)) = ~xmee_1, I(S\[PGUa- 1,1(11) x PG(A~: ~)]/A"). Since the right hand side of the last expression is equal to ~l:,Gint ('rv(S)\[PG~t(F~ol) � PG't(As � "r *) (A")), FaO 1 we conclude (2.1) from (2.5). [] 102 YAKOV VARSHAVSKY 2.6.3. By Proposition 2.5.6 there exists an admissible isomorphism @ : E --% E ~"t, inducing the isomorphism x : PE ~ PE int. Choose 8 e A with elliptic ~| z A~o and Tr Ad(8o~) # -- 1. Choose its representative~ ~ GUd_~,x(R) ~ � E as in Corollary 2.2.9. Then (TRY')(Tr~'-~) zK � Let ~' be the projection of~' to PGUd_I,~(R)~ � E. Set := (V X @) (~'), and let ~r be its projection to E. By the definition of admissible maps, Tr(~) ~ K x. Let ~ be the image of 8 in A', then ~ := (v � 9) (]) belongs to PG~t(F)+. Let y' ~ G~nt(F)+ be some representative , , = ~ -1 K � of ~, then ~1 ~E ~ Z(Eint) 9 Therefore ~1 "~E (Tryr) (Tr y~,) ~ = Z(Glnt(F)). Thus ~x and y~ have equal projections to PGUa_~,~(R)~215 (Ei=VEi"~), hence x r The condition { 8o~ is elliptic and Tr Ad(8| ~ -- 1 } is open and non-empty, therefore the above 8's generate the whole group A ~_ A o (see [Ma, Ch. IX, Lem. 3.3]). It follows that (V x ~)(A)C F ~"~. Since the projection z~:P int -->PG~t(F)+ is an isomorphism, Proposition 2.6.1 implies that (9 x ~)(A)= F i~. This completes the proof of Theorem 2.2.5 and of the First Main Theorem. 3. THE THEOREM ON THE p-ADIC UNIFORMIZATION The First Main Theorem implies that for some admissible isomorphism r : E --~ E i"t there exists a @-equivariant C-rational isomorphism f~:X 0 -~ ~l~t. Therefore for some C/K~-descent X i"t of the (E i"t, C)-scheme .~t, fr induces a K.-rational iso- morphism X--% X i"t. To describe X lnt we need some preparations, following [Dell (see also [Mill). 3.1. Techn;cal prel;minaries In this subsection we recall basic notions related to Shimura varieties and give their explicit description in the cases we are interested in. 3.1.1. First we realize ,~f,t as a Shimura variety. Set H l"t := RF/QG ~t. Then H ~t is a reductive group over Q such that Hi"t(A I) = Glnt(A~) and Hi"t(R) = rI;=l Gint(Fo~i). Put S := Re/~ G,~ and let h be a homomorphism S---> H~ t such that for each z E C � ~ S(R) we have h(z) = (diag(1, 1, z/~)-~; Id; ;Ia) ~ I] G~t/F ~=1 using the identification of Gint(Fooi) with GUa_I,X(R ) chosen in 2.1.6. Then the conjugacy class M ~nt of h in Hint(R) is isomorphic to B a- 1 if d > 2 and to C -- R if d ---- 2. Then the pair (H l"t, M ~t) satisfies Deligne's axioms (see [Del, 1.5 and 2.1] or [Mil, II, 2.1]), and the Shimura variety Mc(H i"t, Mint), corresponding to (H ~t, Mint), is isomorphic to ,~t. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 3.1.2. For each pair (Hint, M mr) as above there is a number field E(H lnt, M) 'nt C C, called the reflex field of (H int, Mint), which is defined as follows (compare [Del, 1.2, 1.3 and 3.7]). The group Homc(Sc, (G~)c) is a free abelian group of rank 2 with generators z and f such that ff i : S(R) ~ S(C) is the natural inclusion, then for each weC � -_~S(C) we have zoi(w) = z and ~-oi(w) =~. Let r:(Gm) c~S c be the algebraic homomorphism such that (z ~ fq) o r(x) = x ~. Then E(H mr, M mr) is the field , h of definition of the conjugacy class of the composition map r": (G,,)c ~ S c --~ Proposition 3.1.3.- We have E(H mr, M mr) = K, the latter being viewed as a subfield of C through the embedding 0% chosen in 2.1.6. Proof. -- Note that Hint(C) is naturally embedded into GLa(C) 2g so that each factor corresponds to an embedding of K into C. Supposing that the first and the second factors corresponds to our fixed embedding and to its complex conjugate respectively we have r"(z) = (diag(1, ..., 1, z-a); diag(1, ..., 1, z); Id; ...; Id) for each z e C � Therefore the reflex field E(H int, M mr) contains K C C. On the other hand, the Skolem-Noether theorem implies that for each a e Autm(C ) the homo- morphism a(r") is conjugate to r". This implies the assertion. [] 3.1.4. Let T C H ~t be a maximal torus of H ~nt, defined over Q, such that some conjugate h'e IV[ ~t of h in H~t(R) factors through Tx~. Then we have a natural embedding i T : Me(T, h') ,-+ Mc(H int, MEt), where Mc(T , h') is the Shimura variety corresponding to (T, h'). Since T is commutative, the reflex field F~ := E(T, h') of r h ~ T! p (T, h') is the field of definition of the morphism r": (G,,)c-+ Sr T c. Hence defines a morphism of algebraic groups over Q * ~ R~,tQ(T ) T. r,:E T := R~TtQ(G,,) e~lQ'"" ~/Q> Notice that E T D E(H ~t, M~"t). Let 0~T be the Artin isomorphism of global class field theory sending the uniformizer to the arithmetic Frobenius automorphism. Let ~ : Gal(E~b/F_@ ~ T(AI)/T(Q) be the composition map 0-~ * 0 * * Gal(E~b/ET) ~> ET(R ) \ET(A)/ET(Q) r' proj > T(R)~ --~ T(M)/T(Q). For each E' D E(H lnt, M lnt) we denote the composition map Oal(E~ b.E'/E') ~, GaI(E~b/ET) x~> T(A')/T(Q) by ~,~,. 104 YAKOV VARSHAVSKY Lemma 3.1.5. -- Each maximal toms T of H mr, defined over Q,, is equal to the inter- section of H ~"t with Rz/Q G,, for a unique maximal commutative subfield L of D tn~. (In such a situation we will call T an L-toms.) In this case, ~.t induces a nontrivial automorphism of L, and the subgroup T(Q.) c L � = RL/K G,,(K) is Zariski dense in RLm G,,. Proof. ~ Let L be the subalgebra of D I"t spanned over K by T(Q) C Hint(Q) C D t"t, then L is a commutative subfield and T(Q,)C H~t(Q)t~ RL/Q G,,(Q,). Since T is connected and O is infinite and perfect, the subgroup T(Q,) is Zariski dense in T (see [Bo, Ch. V, Cor. 13.3]). It follows that T C Hmt n Rr4Q G,,. Since T is maximal, L have to be maximal and T = H ~t n R~/Q G,~. For each g ~T(Q) we have ~t(g) ~g-aF� C T(Q.), so that ~t~t(L) = L. To prove the last assertion we observe that there exists a maximal F-rational subtorus T' of G ~t such that T = RF/Q(T' ). Then the subgroup T(Q)= T'(F) is Zariski dense in T~ - R~m G,, � (G,,)K. Hence its projection to RT,m G,~ is also Zariski dense. [] 3.1.6. Now we want to calculate the reflex field E T. Observe that L | C C D ~t | C ~ Mata(C) 2g. Possibly after a conjugation we may assume that L | C is the subalgebra of diagonal matrices of Mata(C) 2g. Then each diagonal entry of each of the 2g copies of Mata(C ) corresponds to an embedding of L into C, and the map r": (G,~)c -+ T c is as follows: r"(z) ----- (diag(1, ..., l, z-a); diag(1, ..., 1, z); Ia; ... ; Ia). Let q be the embedding L ~ C, corresponding to the right low entry of the first matrix, then the right low entry of the second matrix corresponds to the embedding ~1 := tz o 0t ~at. Now we embed L into C via q. Proposition 3.1.7. -- We have E T = Ls C, and r': E~--~ T is characterized by r'(l) = l-a-~'~t(l)for each l eEl(Q.)C L � Proof. -- As was noted before, E T D E(H t"t, itnt). Hence by Proposition 3.1.3, ETDK. By the definition, ,(r"(z))= r"(,(z)) for each , e AutE,(C), hence the group Auto(C) must stabilize q, so that E T D L. Finally, it is clear that r" is defined over LC C. For each l e E T = L we have r'(t) = N~./Q(diag(1, ..., 1, l-~); diag(1, ..., 1, l); Ia; ... ; I~) = l-l-~t(l). [] Set L, := L | K~ C Din t := D ln~ | K~. Since Di~ ~ is a division algebra, L,, is a field extension of Kw of degree d, and L~ = L. K~. Lemma 3.1.8. ~ The following relations hold: a) Er.K ~ ---- L~; b) E~b.K,o = (L,~) ~. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VAI~IETIES Proof. -- a) was proved above. b) The group GaI(E~b-K,~/Ex.K,o) is abelian, hence L,oCE~b.K~CL~. By the class field theory, the composition of the canonical projections Gal((L,o)~b/Lw) -+ Gal(E~ b. Kw/Ex-Kw) ~ Gal(E~b/E~) ~ Gal(L~b]L) is injective (use, for example, [CF, Ch. VII, Prop. 6.2]). Therefore we have the required equality. [] Proposition 3.1.9. -- For each l ~ L~ C (D~ t) � the element ~t � Gi.t(A~,.) _ H~t(Af) (l-1,1,1) ~(D,~) � r~ x belongs to T(AQ, and its equivalence class in T(AI)/T(Q) # Xr, Kw(0Lw(l)). Proof. -- The statement follows immediately from the explicit formulas of Pro- position 3.1.7 using the connection between local and global Artin maps. [] Definition 3.1.10. -- A point x ~Mc(H t"~, Mmt)(C) is called (T-)special if x ~ iT(Mc(T, h) (C)). Remark 3.1.11. -- The group T(A I) acts naturally on the set of T-special points and the group T(Q) acts on it trivially. Hence by continuity the closure T(Q) C T(A f) acts trivially on the set of T-special points, therefore the action of T(AQ/T(Q) on it is well-defined. Definition 3.1.12. -- Let K' D E(H ~n~, Mint) be a subfield of C. A C/K'-descent of the (Hint(Af),C)-scheme Mc(H t~, M ~t) is called weakly-canonical if for each "[~ ab, T~t maximal torus T ~-+ H ~t as above, each T-special point x is defined over ~T -~, and for each ~ e Gal(E~rb.K'/Ez.K ') we have ,(x)= ~,K,(~) (x). Remark 3.1.13. -- Our definition of the canonical model coincides with that of [Mi3], which differs from those of [De2] and [Mil] (see the discussion in [Mi3, 1.10]). The seeming difference (by sign) between our reciprocity map and that of [Mi3] is due to the fact that we consider left action of the adelic group whereas Milne considers right action. Proposition 3.1.14. -- For each field K' satisfying E(H ~t, M l~t) C K' C C there exists a unique (up to an isomorphism) weakly-canonical C/K'-descent of the (H~t(Af), C)-scheme Mc(H l~t, 1V[in~). Proof. -- Uniqueness is proved in [Del, 5.4], for the existence see [Del, 6.4] or [Mil, II, Thm. 5.5]. [] By Proposition 3.1.3, we have E(H ~t, M ~t) = K C K~C C (in our conven- tion 2.2.2). Hence by Proposition 3.1.14, the (E mr, C)-scheme ,X~* has a unique weakly-canonical C/Kw-descent X mr. 14 106 YAKOV VARSHAVSKY 3.2. Theorem on the F-adic un;form;zation Now we are ready to formulate our Second Main Theorem 3.2.1. -- For each admissible isomorphism ~ : E --% E int there exists a ~-equivariant isomorphism fo from the (E, K~)-scheme X to the (E i"t, K~)-scheme X int. Corollary 3.2.2. -- After the identification of E with E ~t by means of rb we have for each S e o~-(E) of the form T, x S', where S' e ~'(E'), an isomorphism of K,~-analytic spaces %: (Xs~t)~n--~GLa(K~)\(Z~, " X (S'\G(A~)/G(F))). These isomorphisms commute with the natural projections for T D S and with the action of E -~ E ~nt. Proof (of the Second Main Theorem) : Step 1. -- We want to prove that for (I) and fo as in the First Main Theorem, the C/K,-descent of IK t"t corresponding to X is weakly-canonical. For tiffs we have to show that for each maximal torus T ~ H int as in 3.1.4 and each T-special point x ----f~(y) e Mc(H i"t, M t') (C) : ~l~t we have: a) y ~ X(C~) is defined over E~b-K~; b) a(y) = q)-l(Xm,,,(,))(y) for each a e Gal(ESb.KJET.K,). By Proposition 3.1.9, Lemma 3.1.8 and the definition of admissible map, it will suffice to show that when L~ is embedded into D~ by means of the isomorphism Dtnt __% ~ from Definition 2 2.1 we have (3.1) i) every pointy ~X(C~), fixed by (I)-I(T(O_..)), is rational over (L,)ab; X "~X "~X q~'SX El. ii) 0L.(1 )(y) =l-l(y) for each leL. CD. =D. � X Let (x, a) e Z a K~ � E' be a representative ofy e X(C~). Then (~(x), a) is a repre- sentative of a(y) for each (not necessarily continuous) ~ e Autr~(C~). Recall that for each embedding L.'--~ Mata(K~) there exists an (L~ � L~)-equivariant L.-rational embedding ~ : E 1~,~ ~ Z~. Proposition 3.2.3. -- There exists an embedding L~ ~ Mata(K~) such that the image of the corresponding T : Z 1~ "-+ Z a~ contains x. Proof. -- Let x' e ft a be the projection of x. Then Kw y' := [(x', a)] m (b~\X) a" - (O~, � (E')d'~)/G(F) Z(G(F)) (use Proposition 1.5.3) is the projection of y. Since q~-I(T(Q)) stabilizes y, it also stabilizes y', therefore the projection of a -1 q)-l(T(O_..)) a to E' is contained in G(F) Z(G(F))C E'. In other words, for each teT(Q)C D ~nt we have pr~,(a -1 q)-l(t) a) = g. z for some g e G(F) and some z e Z(G(F)). p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES Since 9 is induced by some algebra isomorphism D(A~ ;~) -% Dint(A~;'), we have Trt= Tr(a-l(I)-l(t)a)= (Trg) z. Therefore for t's with non-zero trace we get z ----- (Tr t)(Trg) -1 ~ K � C (AIK:~'~) � This means that pr~,(a -1 (I)-l(t) a) E G(F) C E'. As the set of all t's in T(Q) C D int with Tr t 4= 0 generates L C D ~"t as an algebra, the map l ~-~ pr~,(a -~ (I)-X(l) a) defines embeddings L ,-+ D and L~ ~ D | K~ ~ Mata(K~). This shows that pr~,(a -~ ag-~(T(Q)) a) C G(F) C E', so that a 'O-'(T(Q))aCD, � I'~CD. X =E. Hence (I)-I(T(Q)) preserves p~(Ea~)C (X%) ~ (in the notation of Corollary 1.5.4). Moreover, it follows from the definition of the embeddings L~ '-+ Dw and L~ ~ Mata(K~) that for each teT(Q) C L � C L~ the image of a-~a~-~(t) as D$ � F~ under the canonical map D~ ~ � � Ft.--% D~ ~ � � F GC D~ ~ � � GL~(K~) is equal to (t, t). Since y is a fixed point q)-a(T(Q)), we conclude from the above that (t, t) (x) ----- x for every t e T(Q). Noticing that T(Q) is Zariski dense in R~ m G,, by Lemma 3.1.5 and that RLm G,, | K~ -~ RL~/x ~ G,,, Lemma 1.4.5 completes the proof. [] Since 2" is (L~ � L~)-equivariant and L~-rational, the proposition together with Lemma 1.4.3 imply (3.1). In other words, we have proved that for some admissible isomorphism r : E -% E ~nt there exists a q~-equivariant K~-linear isomorphism fr :X-%X ~"t. Step 2. -- Let tF be another admissible isomorphism E--% E ~t. The definition of admissibility together with the theorem of Skolem-Noether imply that tF o r ~ : E ~"t --% E ~"t is an inner automorphism, so that there exists gv e Ein* such that tF o (I)-X(g) ----_ g~,ggyr ~ for all g ~ E ~t. Take f,r : X ~ Then for each g ~ E we have fv o g = g~ ofo o g = g,v o O(g) ofo = (g,r o O(g) o g~v ~) o (g,v ofo) = (XF o ~-a) (O(g)) of~ = XF(g) of~,, that is f,r is a ~-equivariant isomorphism. This completes the proof of the Second Main Theorem. [] 4. p-ADIC UNIFORMIZATION OF AUTOMORPHIC VECTOR BUNDLES In the previous section we proved that the Shimura varieties corresponding to the pairs (H ~t, M ~t) have p-adic uniformization. Our next task is to show the analogous result for automorphic vector bundles. 4.1. Equivariant vector bundles 4.1.1. Set H :-~ Rr/QG. Then for some algebraic group H over K. we have natural isomorphisms H~w ~ GLg � H, PHKw ----- PGL d � PH and PH~tKw ~ PGLI(D~t) � PH, where the first factors correspond to the natural embed- 108 YAKOV VARSHAVSKY ding F ~ K~. Using these decompositions let PHKw acts on I~K~ ~ through the natural action of the frst factor and the trivial action of the second one, and let H(K.), PH(Kw) and PHt"t(R) ~ ~ PGU a ~ I(R) ~ � PGUa(R) ~-~ act similarly on Z a on ~a and _ , Kw ' K w on g a-~ respectively. Let ~ be the natural embedding Ba-~,-+ (l~c-~) ~, and let ~w (resp. ~,,, ~) be the composition of the natural projection E~ -+ D~ (resp. Ea'~x~ -+ D~)a and the natural embedding f~a ~ (i~x~)~. Let 7: e K~ be a uniformizer, let H be an element of GLa(K.) satisfying ~a = 7:, and let H' e PGLa(K~) be the projection of ~I. Set H := (II', l) e PGLa(K,o ) � PH(K,o ) _-- PH(Kw). Let K~ ~ be the unramified field extension of K. of degree & Since the Brauer invariant of _~D ~nt is 1/d, the group PH is isomorphic to the quotient of PHK~ | K~ a~ by the (d) equivalence relation Fr(x) ,~ II -a xII, where Fr e Gal(K~/K.) is the Frobenius auto- morphism. For each scheme Y over K~ on which PHx~ acts K~-rationally define a twist yt,, := (Fr(x) ~ II -a x)\Y | __~K ~d~ . Then Y | K~' -~ ytw | K~a~ and the natural action of PH t~t x,, on it is K~-rational. Let W be a PHx -equivariant vector bundle on P~I, that is a vector bundle on P~ 1 equipped with an action of the group PHKw , lifting its action on I~K~ 1 . Then (I~ -l~tw and ~((W~W) ~) (resp. (Wt~,p tw) is a PH~tw-equivariant vector bundle on ~ K~ J , [3:(wa"), ~o,,~(wa~)) is a PHmt(R) ~ (resp. H(K,o)-)equivariant analytic vector bundle on B a- 1 (resp. Z ~, a :Cx~). a,, For each S e o~-(E) (resp. S e o~'(Emt)) consider a double quotient ~' * tw an V s := S\[~;(W =) � E']/F (resp. V~" := S\[~R(Wc ) � Eat]/Fmt). Proposition 4.1.2.- For each S e~'(E) (resp. S e o~'(Emt)) Vs (resp. V~ nt) has a natural structure of an affine scheme V s over X s (resp. ~mt ~mt~ Moreover, V s (resp. ~s t) is --s over ~s J" a vector bundle on K s (resp. ~nt) if S is sufficiently small. Proof. -- We give the proof in thep-adic case. The complex case is similar, but easier. I) First we take S of the form T, x S' with sufficiently small S' e #'(E'). Then Vs is a finite disjoint union of quotients of the form Fas,a_x\~*,(W ~n) with some a ~ E'. Since the projection za~2-+O~w factors through each r,s,,_l,0\z~2 (in the notation of the proof of Proposition 1.5.2), the quotient F~s,~_L0\~:,,(W ~n) is d,~, naturally an analytic vector bundle on F.s,~ Now (as in the proof of Proposi- tion 1.5.2) the quotient vector bundle Pro a_xk(ros,._x,0k ;,.(w o)) ros,o_lk ;,.(w on P~s' a- I\Z~ is obtained by gluing. For the algebraization we use Corollary 1.2.3 a). II) For each T e o~(E) there exists a normal subgroup of the form S = T, � S', where S' e o~-(E') is sufficiently small. Then by the same considerations as in Propo- sition 1.3.7, V T can be defined as (T/S)\V s (using Corollary 1.2.3 a)). ~"tKw p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 109 III) Suppose that Vst and Vs, , constructed in I) and II), are vector bundles on Xs~ and Xs, respectively for S~ff S, in o~-(E). Then the natural morphism f: Vs~-+ Vs~ � Xs~ of vector bundles on Xs~ induces an isomorphism on each fiber. Hence it is an isomorphism. IV) Suppose that T C S in oq~(E) and that V s is a vector bundle on X s. Choose a normal subgroup So e o~-(E) of T such that Vs0 is a vector bundle on Xs~ Then Vw = (T/So)\Vso ~ (T/So)\Vs � Xso ~ Vs � ((T/So)\Xs0) -~ Vs � X~, so V~ is a vector bundle on X~. [] 4.1.8. Choose S eo~(E) (resp. S Eo~-(E~"t)) sufficiently small. Then V s (resp. ~i.t~ is a vector bundle on X s (resp. ~l.t X s) Thus V:=V s � (resp. --S J ~t := ~,t � ~int) is a vector bundle on X (resp. iKmt). By Step III) of the proof, V (resp. ~t) does not depend on S. Each g e E (resp. g e E at) defines an iso- morphism V~ --% V~o_~ (resp. (~t)an _~ "~an'~/. Therefore by Corollary 1.2.3 a), g defines an isomorphism V s ~V0sg_x (resp. -si'Ji"t ~ The product of this isomorphism and the action of g on X (resp. ~t~t) gives us an isomorphism g : V = V s � X --% V0so_~ � X = V (resp. g : V l"t __~ ~t). Thus we have cons- tructed an algebraic action of E (resp. of E ~nt) on V (resp. ~mt), satisfying S\V -~ V s for all S e~'(E) (resp. S\ ~t =~--si'~mt for all S ~o~'(E~t)). Moreover, V =limV s and ~t = lira ~t. s By [Mil], there exists a unique canonical model V ~nt of ~mt over K~ (the definition of the canonical model will be explained in the last paragraph of the proof of Proposition 4.3.1) such that V ~t is an E~nt-equivariant vector bundle on X i't. Our main task is to prove the following Third Main Theorem 4.1.4. -- For any admissible isomorphism qb:E-% E t~t, each isomorphism f~ from the First or the Second Main Theorem can be lifted to a ~-equivariant iso- morphism f,D, v : V -% V int. We will prove this theorem, using standard principal bundles (----- torsors) (see [Mil, Ch. III, w 3]). 4.2. Equivariant torsors 4.2.1. For each S e~(E) (resp. S e~'(Ei"t)) consider the double quotient -s :-- S\[ Ba-~ � (PHc "t)~" � n!nt]/rmt) . Ps :-- S\[Zaw � (PHK~) " � E']/F (resp. V,.t Proposition 4.2.2. -- For each S E o~-(E) (resp. S e o~'(Ei"t)) Ps (resp. P~s nt) has a natural structure of an affine scheme Ps over X s (resp. ~i,t_s over -~s~"t~/" Moreover, Ps is a PHKw- torsor over X s (resp. ~i,t_s is a PHio"t-torsor over Y2s mt if S is sufficiently small). -+~--osg-~J'~i~t kVoSo--lJ[~lnt 110 YAKOV VARSHAVSKY The proof is almost identical to that of Proposition 4.1.2 (using Proposition 1.9.13 and Lemma 1.9.3 instead of Corollary 1.2.3 a) and arguments of step III) respec- tively). [] 4.2.3. Arguing as in 4.1.3 and using Corollary 1.9.14 we obtain an E-equi- variant PHx-torsor P = < limP s over X (resp. an Eint-equivariant PH~"t-torsor ~t =l<im~nt over ~L,t). By [Mil, III, Thm. 4.3], there exists a unique canonical model pint of ~i~t over K~ (the definition will be explained in Corollary 4.7.2) such that pi.t is an E~n~-equivariant PH~-torsor over X ~nt. Let ~:P-+X and ~n~: p~n~ _+X~n~ be the natural projections. Denote also the natural projection from the PH~-torsor p~w to X by ~tw. Fourth Main Theorem 4.2.4. -- For any admissible isomorphism 9 : E -~ E ~nt, each isomorphism fr from the First or the Second Main Theorems can be lifted to a ~-equivariant isomorphism fr p : p~w _~ p~,t of PH~t-torsors. 4.3. Connection between the Main Theorems Proposition 4.3.1. -- The Fourth Main Theorem implies the third one. Proof. ~ Consider the pro-analytic maps ~'O : [Z a~, x (PHK~) a" � (E,)dL~]/F .__> ~/Pa-t~"- x~ , and (~,)~,, : [Ba-1 � (pH~.t)~n � (E~.t)d~]/F~nt _+ (p~--l)an given by ~'(x, g, e) = g~(x) and (~')~nt(x, g, e) = g~c(x). Then 7' (resp. (~,)~nt) is (PHx~) ~"- (resp. (PH~nt) a"-) equivariant and commutes with the action of E (resp. E~"t). Hence it defines an equivariant analytic map ~:pan_+ ~-K~ J tl~- a~. (resp. ~i,t: (pi,t)an .__> (p~-,)a,). Proposition 4.3.2. -- There exists a unique algebraic morphism o:P---> ~K: 1 (resp. '~int : ~int ._>. p~-l) such that p~" ~-~ (resp. (~mt)~n ~ ~t). Proof. -- We prove the statement for p (in the second case the proof is exactly the same). We have to show that the graph Gr(~)C P~ � t~- ~-K~ l~n J corresponds to ~ an ~pa-l~an be the morphism an algebraic subscheme. For each S ~ ~'(E) let ps:Ps ~ ~_Kw J . ~ an ~1~ -l~an it remains to show induced by 7. Since Gr(~) = <hmGr(ps) C (timPs) x ~ K~ J , S S an d -- 1 an that the graph Gr(ps)C Ps � (PKw) corresponds to a unique algebraic subvariety for each S sufficiently small. Take S so small that X s is smooth, then by Lemma I. 9.5 b) there exists a quotient d--1 Qs :-PHK~\(Ps � Px~ ) by the diagonal action of PHi. Moreover, Qs is a Pa-l-bundle on Xs, hence it is projective over K~. Let ~:Ps � ~1 _+ Q.s be the p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 111 natural projection. Since Os is (PHK~)~"-equivariant, Gr(os) is invariant under the diagonal action of (PHKw) ~". Therefore the quotient Q:= (PH~%)~\Gr(~s) is a closed analytic subspace of Q.~, so that it is algebraic (see Corollary 1.2.2). It follows that its inverse image 0~-~(Q.) = Gr(ps) is also algebraic. The uniqueness is clear. [] Claim 4.3.3. -- The map ~int is the only (PH~'~t) ~" X Ei"~-equivariant analytic map from (~ln~)a, to (Pao-~)~". Proof. -- Let p' : (P~)~" --> (1~ - ~)~ be any such map. Composing it with the natural (PHi"t) ~ � E*"t-equivariant projection [B a-~ x (PHi't) ~ x (g'~176 (P'~176 and, identifying a complex analytic space with the set of its C-rational points, we obtain a PHI"t(C) � Emt-equivariant analytic map ~": [B ~-' x pH~t(C) x (~i"<)~"1/r"'-~ (~-')*. Let Po be the restriction of p to B a-1 -% B d-~ � { 1 } � { 1 }. Then p"(x, g, e) = gpo(x) for all x e B e- x, g e (PH~nt) ~ and e ~ E l~t. Therefore ypo(X) = po(yX) for all y E F ~t and x e B a-1. Since the subgroup F i~t is dense in PGUa_I,I(R) ~ we obtain by conti- nuity that ypo(x)= po(yx) for all 3" e PGUa-~,~(R) ~ and x eB ~-~. In particular, for the origin 0 e B ~-1 we get Stab~,ov~_~,#~)o(0 ) C Stab~g~_x,x~)o(po(0)). The subgroup Stabmv~_,,~(mo(0 ) stabilizes precisely one point (0 : ... : 0 : 1) e pa-x(C) if d> 2 and two points (0: 1) and (1 9 0) in pI(C) if d = 2. The case po(0) = (1 : 0) is impossible, because identifying PX(C) with C = C ~ oo by (x :y) ~-~ x/y we would get in this case 9o(z) = 1/~ for all z e B x, contradicting the analyticity of ~0- We conclude that ~o(0) = (0 : ... :0: 1). Hence ~o = ~ and ~' = ~r. [] 4.3.4. Next we show that the map '~lnt : p~t ___> tpa-l~tw ~AKw /c is Kw-rational. Recall that the map ~-Int is PH~Lequivariant and that the actions of the group PH ~r t~ w on both --1 tw pint and (~w) are Kw-rational. Therefore for each a e Aut(C/K~) the analytic map a(~tnt) ~" is (PHi"t) a" � Et"t-equivariant, hence it coincides with (~t)a. = ~t. By the uniqueness of the algebraic structure, a(~ 'l"t) = ~,~.t. It follows that ~t defines a PH Kw l~t � E~t'eq uivariant map pl~t : pt~t __~ tpa-~w~ ~-xw ; 9 K~ X E-equivariant map ~tw : ptw __> ~ ~w / " Notice also that ~ defines a PH ~t rpa-lXt, Suppose that the Fourth Main Theorem holds, then Lemma 4.3.5. ~ We have pi~t ofo, p = pt~. Proof. -- By the claim, ~mt _ (p~.t)~. is equal to (p~w o (fo, P)o-1)~"- From the uniqueness of algebraic structures we conclude that p~.t= p~o (fr Now we descent to K~ as in 4.3.4. [] 112 YAKOV VARSHAVSKY It follows from the definitions that p*(W) ~ r~*(V) (hence p*(W) tw-~ (rctw)*(V)) and (~t)*(W~) ~ (r~t')*(Vm~). Lemma 1.9.5 allows us to define V i"t by the requirement that (~nt)*(V~"t) ~ (p~"t)*(Wtw). (By the definition, this is the canonical model of V~"~ on X~"t.) Lemma 4.3.5 implies that fr can be lifted to the qS-equivariant isomor- phism p*(W) tw---_ (ptw)*(Wt*)-% (pmt)*(Wt*), commuting with the PH~t-action. This gives us the PH~t-equivariant isomorphism (r~tw)*(V)-~ (rd"t)*(Vi"t). Hence the Third Main Theorem follows from Lemma 1.9.5. [] Remark 4.3.6. -- Tannakian arguments can be used to show (see Theorem 1.9.10 and the discussion around it) that the Third Main Theorem implies the Fourth one. We will not use tiffs implication. 4.4. Reduction of the problem 4.4.1. Now we start the proof of the Fourth Main Theorem. For simplicity of nota- tion we identify E with E ~t by means of 9 and X with X i"t by means of re. Recall that Ps, c is a PHe-torsor over Xs, r for all sufficiently small S e o~'(E), hence (Ps, c) ~" is a (PHc)~-torsor over (Xs, c) ~ and (Pc)an-= (Ps,r ~ � c~. (Xe) a~ is a (PHc)a"-torsor int int disc int over (Xo) ~_-__ [B a-~ � (E /E 0) ]/PI' .SetY:= (na~)-a(Ba-, � (Pc) an . Then Y is a (PHc)a"-torsor over B a-~. Recall that E 0 = -0V~t acts trivially on P, hence (Pc) an-- (Y � (E~VE~n~)di'~ ~*. Proposition 4.4.2. -- There exists a homomorphism j : PF ~t -+ PH (C) and an isomorphism int tnt disc lnt (Pc) an --% (B a-a � (PHc) = � (E /E o ) )/Pr such that (x, h, g) y = (V~o 1 x, hj(y), gy~.) for all x e B a-a, h ~ (PHc) a", g e Eint/E~ t and y ~ PF t~t. Proof. -- The proposition asserts that there exists a decomposition y _~ Bd-* � (PHc) ~ such that the group P1 "i"t acts on B a-a � (PHc) an by the product of actions on factors. The trivial connection on y a~:w � (PHEw)~ -~ ZaKw is I" � DX-invariant, there- fore it defines a natural E-invariant flat connection ~ on the (PH~)"~-torsor Z a (PH~w)~ Ed~sc]/F Z a [ K~ � � over [ K~ � Ed~]/I'" Since for all sufficiently small S coW-(E) the projection (Y~ � (PHxw)a~ � Edi*)/P _+p~n is 6tale, it induces an isomorphism of tangent spaces up to an extension of scalars. Hence ~ induces a flat connection o~ s on P~n. By the definition, o~ s is a (PHKw)aMnvariant analytic vector subbundle of (T,~) a~, therefore Lemma 1.9.5 and Corollary 1.2.3 imply the existence of a unique flat connection J~f~s on Ps such that ~s ~ ~. Since the projection ~s :P ~ Ps is ~tale, o~f s defines a unique flat connection oY' on P satisfying (~s).(~) = o~ s. Moreover, W is E-equivariant and does not depend on S. The connection ~ determines flat connections o%,0 c on Pc and (~Y'c) a~ on (Pc) ~. p-ADIG UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 113 Let 5(f' be the restriction of (o%fc) ~ to Y. Then ~f" is a PFmt-invariant fiat connection on the (PHc)~-principal bundle Y over the simply connected complex manifold B a-a. By Lemma 1.9.19, there exists a decomposition Y-~ B a-a � (PHc) ~n such that the corresponding action of PF ~"t on B a-x � (PI-Ie) ~ preserves the trivial connection. For each y G PP~ let ~ : B d- ~ � (PHc) ~ --~ (PHc) ~ be the analytic map such that y(x, h) -= ('r(x),~(x, h)) for all x e B a-a and h e (PHr ~. Since the action of PF ~"t preserves the trivial connection, we have O'~/Ox =- 0 for each y e PF ~"t. Hence analytic ~ 's depend only on h. Since the action of PF t~t commutes with the action of (PHc) a", we have ~(h) = h~(1) for all h e (PHc) ~n and y e PF ~t. Therefore the map y ~-~(1) -~ is the required homomorphism. [] Theorem 4.4.3. -- There exists an inner isomorphism (= inner twisting) ~c : PHc -~ PH~ I~ such that j o (I) c : PF ~ = pGi~(F)+ -+ pHm~(c) ~ PGi~(C | F) is induced by the natural (diagonal) embedding F ~ C | F ~- C ~. Remark 4.4.4. -- Algebraization considerations as in Lemma 2.2.6 (using Propo- sition 1.9.13 instead of Corollary 1.2.2) show that Theorem 4.4.3 implies the existence of a ~-equivariant isomorphism P~ -~ ~t, lifting J~. 4.5. Proof of density To prove Theorem 4.4.3 we will use Margulis' results. For this we first show that the subgroup j(PF ~) is sufficiently large. We start with the following technical Lemma 4.5.1. -- Let n and d be positive integers. For each i ----- 1, ..., n we denote by pr~ the projection to the i-th factor. a) Let fr 9 9 ff , be Lie algebras, and let ~ be an ideal in the Lie algebra fr ---- 1-I"~=~ f~. Then 3#' ~ l-l~=~[pr~ ~f, ~f~]. b) Let A be a subgroup of PGLa(C)% Suppose that pr,(A) is infinite for every i = 1, ..., n. If A :---- CommmLacc~,(A ) is Zariski dense in (PGLa)" , then the same is true for A. c) If a subgroup A C PGUa(R)" is Zariski dense (in (PGUa)"), then it is dense. Proof. -- a) If x = (xl, ..., x,) e II~= 1 fg~ belongs to ~o, taken [x,y~] = (0, ..., [x~,y,], ..., 0) = [pr~ x,y~] eaf for all 2; e ~,- b) Let J be the Zariski closure of A in (PGLa)" , then 8J ~-1 c~ j is an algebraic subgroup of finite index in J for each 8 e A. Hence ~j0 8-1 ____ j0. In particular, the subgroup AdA stabilizes LieJ~ IAe(PGLa)". Since A is Zariski dense in (PGLa)" , the Lie algebra Lie j~ = LieJ is an ideal in Lie(PGLa)". By our assumption, pq(J) is an infinite algebraic group for each i = 1, ..., n, therefore pr~(LieJ) :# 0 is an ideal 15 114 YAKOV VARSHAVSKY in a simple Lie algebra Lie(PGLa). Therefore a) implies that LieJ = Lie(PGLa)". Since the group (PGLa) ~ is connected, J -- (PGLd)" r Let M be the closure of A in PGUa(R ) % Then M is a Lie subgroup of the Lie group PGUa(R)% Hence Lie IV[ is an Ad M-invariant subspace of LIe(PGUa(R))% Since the adjoint representation is algebraic, Lie M is an ideal in Lie(PGUa(R))% Since M is compact, it has a finite number of connected components. Hence M ~ is also Zariski dense, therefore it is not contained in PGUa(R)~-I x { 1 } x PGU~(R) ~-~ for any i = 1, ..., n. It follows that Lie M ---- Lie M ~ is not contained in Lie(PGUa(R)) '-1 x { 0 } � Lie(PGUa(R)) ~-', so that pq(Lie M) 4= 0. Now the assertion follows exactly in the same way as in b). [] Proposition 4.5.2. -- The subgroup j(PFint) is Zariski dense in PH c. Proof. -- Let G'C PH c be the Zariski closure ofj(PFmt). Then := (B ~-1 � (G') an X (PG'nt(A~:v))di*)/Ps Int is a pGint(A~:~)-invariant (G')~"-subtorsor of thc (PHc)~-torsor (pgpn = x n -~ [B ~-~ X (Pno) ~ X over (X~')~n~ [Bd-t X (PG~t(A~'*))dI~]/PI '~t. Hence by Proposition 1.9.13 there it exists an algebraic G'-subtorsor R of Pc such that R ~n ~ R. Using our identification of Cf with C, we obtain a closed analytic subspace (Rcp)= c (p~)= " = (D.~,, a | ^ C~, X (PH%) ~ x (PE')d~)/Pr. Recall that Pr = PH(O) is naturally embedded into PH(C~). Lemma 4.5.3. -- The subgroup generated by the elements of Pr with elliptic projections to PGLa(K~) is Zariski dense in PHcp. Proof. -- The subgroup of PGLa(K~) generated by the set of all elliptic elements is open and normal, because a conjugate of an elliptic element is elliptic. Hence it contains PSLa(K~). The subgroup PF o n PSLa(K~) is dense in PSLn(K,, ). Therefore, by [Ma, Oh. IX, Lem. 3.3], the subgroup of PF a generated by all elliptic elements of PI' o contains PF a n PSLa(K~). In particular, it has finite index in PF o = PH(Q). Since PH is connected, the statement follows from [Bo, Oh. V, Cor. 18.3]. [] If G'# PHr then by the lemma there exists Y e PP ~:ith elliptic pro- jection to PGUa_~.x(R) whose image V~ ~PH(C~) does not belong to G'(C~). Let xeD~| X{1}C (X~'p) ~" be an elliptic point of y~,ePE', and let ~' be an arbitrary point of ~ c# , over x. = tR ~n rR ~ lying Then y~(Z) y~(~) is another point of ~ c# , lying over x. Hence "G must belong to G'(C~), contradicting to our choice of y. [] p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 115 Recall that we defined in Proposition 4.3.2 the algebraic PH~:~ � E-equivariant pa-a Identify PH e with (PGLa) ~ in such a way that the first factor map p:P-+ Kw " corresponds to the embedding 0% : K '-+ ~. Denote by j~ : PF~"~ -+ PGLa(C) the composition of j with the projection pr~ of PGLa((]) ~ to its h-th factor. Denote also by pr ~ the projection of PGLa(E) ~ to the product of all factors except the i-th. We will some- times identify PF ~"* with its projection PF~ ~ C PGUa_~,~(R) ~ Proposition 4.5.4. -- The subgroup j~(PF ~"~) is not relatively compact in PGLe((I). Proof. -- If not, then jl(PF ~t) is contained in some maximal compact subgroup of PGLd(C ) (see for example [PR, Prop. 3.11]). After a suitable conjugation we may assume that jl(PF ~"t) C PGU~(R). By Proposition 4.5.2, j~(PF ~t) is Zariski dense in PGLd, hence it is infinite. Therefore, by Lemma 4.5.1, jl(PF ~t) is dense in PGUd(R ). Consider the map p : P(13) --> pd-l((]) and its restriction P0 to B ~-1 � { 1 } ~ B ~-~. Then, as in the proof of Claim 4.3.3, po:B e-x -+ l~a-l(E) satisfies po(,(x) =Jx(Y) p0(x) for all x e B ~-1 and y e PF int. The group PGUe(R) acts transitively on I~-I(G), hence o0(B a-l) is dense in Pa-I(C). Now we want to prove thatj~ : PF ~ ---> PGUa(R ) can be extended to a continuous homomorphism .~ : PGUa_~,~(R) ~ ---> PGUa(R ). For each g e PGUe_a,~(R) ~ choose a sequence { 3', },C PF~"*C PGUa_~,~(R) ~ converging to g. Since PGUa(R ) is compact, there exists a subsequence { y,,}~C {y,}, such that {j~(y,~)}, converges to some a ~ PGUa(R). Then po(gx) = lim po(u = (limjx(~,,,)) Oo(X) = apo(X) for all x e B a-a. It follows that a = a(g) depends only on g, since po(B ~-1) is dense in Pa-I(C) and since the group PGLa(G ) acts faithfully on 1 ~- 1((I). In particular, a(g) does not depend on the choice of{ y, }, and a(g) = limj~(%). It follows that jx := a is the required extension. Since PGUa_~,I(R) ~ is simple and j~(PF i"~) is dense, ~ must be injective and surjective. Hence is it an isomorphism, a contradiction. [] Proposition 4.5.5. -- For each i = 1, ..., g the homomorphism Ji : PFt"t -+ PGLa(C) is injective. Proof. -- Suppose that for some i the subgroup Ai := Ker(ji) is non-trivial. Then A~ is a normal subgroup of PP'nt ~ PP'~t C PGUd_I,I(R) ~ Hence the closure of (A~)~ is a non-trivial normal subgroup of a simple group PGUa_I,I(R) ~ Therefore the pro- jection (A~)~ is dense in PGU a_ 1, I(R) ~ Hence there exists an element ~ e A~ with elliptic projection ~ e PGUa_I,I(R) ~ Therefore the element (j(~), ~) e PGLa(C) g � PE' has a fixed point [y, g, el e P"(C~) = (~2~w(13~) � PGLa(C~) g � PE')/PF. Hence (g-lj(~) g, e-~ ~, e) stabilizes [y, 1, 1] e P"(Cv). It follows that e -1 ~ e = yg for some y e PF = PH(O) and g-~j(~) g e PH(C~) is the image of y. Hence jk(~) 4:1 for all k, contradicting to our assumption. [] 116 YAKOV VARSHAVSKY 4.6. Use of rigidity Now we are going to use the following theorem of Margulis [Ma, Ch. VII, Thm. 5.6]. Theorem 4.6.1. -- Let L be a local field, let J be a connected absolutely simple adjoint L-group, and let A be a finite set. For each o~ e A let k~ be a local field and let G~ be an adjoint absolutely simple k~-isotropic group. Set G := I-[ G,(k~). Let F be an irreducible lattice in G and let A be a subgroup of Comrno(F ). Suppose that rank G := Z rankk~ G~/> 2. ezG A If the image of a homomorphism v:A--~J(L) is Zariski dense in J and not relatively compact in J(L), then there exists a unique ~ ~ A, a continuous homomorphism 0 : k~ --~ L and a unique O-algebraic isomorphism ~ : G~ -~J such that v(x) = ~q(0(pr~(X))) for all X e A. 4.6.2. We use the notation of 2.4.1 with A' = PF int. Take any M and S such that rank C~M >1 2. Then by Proposition 2.4.5, F := A s is an irreducible lattice in G-~M. We will try to apply Theorem 4.6.1 in the following situation. Take G = C~M , A be the projection of A' to G-~M , L = C, J = (PGLa) c and v be the homomorphism Ji: PF~nt -+ PGLa(C) for some i ~{ 1, ..., g }. Consider first i = 1. By Proposition 4.5.2 and Proposition 4.5.4, v = 3"1 satisfies the conditions of Theorem 4.6.1, hence there exists an algebraic isomorphism B1 : (PGUa_I.1) c--% (PGLa) c such that jl(~') = Bx(y~) for all ~, e PF ~t. Now take i >/ 2. Suppose that j~(PF int) is not relatively compact. Then using again Proposition 4.5.2 we conclude from Theorem 4.6.1 that there exists an algebraic isomorphism ~: (PGUa_~,~)c--~ (PGLa) c suck that j~(~)= ~(~%o) for all ~, ~PF ~t. In particular, j(PF ~) is not Zarisld dense in (PGLd) g. This contradicts to Proposition 4.5.2. Therefore after a suitable conjugation we may assume that j~(PF ~t) C PGUa(R ) for all i = 2, ..., g. It follows that up to an algebraic automorphism of (PGLa) g, j(PF 'nt) C PGUa_,,I(R ) � PGUd(R) g-' -~ PHmt(R) and that 3"1 is the natural embedding PG~t(F)+ ~-+PGmt(Foo~). Therefore j together with the natural embedding PGmt(F)+ ~-~ PG~t(A~ TM) embed PF t"t into PGUa_,,~(R) ~ � PGU~(R) ~-~ X PG~'~(A~:~). Lemma 4.6.3. -- The closure of the projection of PF "t to PGUa(R) ~-~ � PGmt(A~ :~) contains PGUn(R) ~ x P((G"t)~(A~:*)). Proof. -- Let (goo,g/) be an element of PGUa(R) ~ x P((G~t)d~r(A~:~)), let U C PGUa(R) ~-~ be an open neigkbourhood of goo, and let S ~'(PG~t(A~;*)). We have to show that PF ~"t n (PGUa_ ~, ~(R) � U � g[ S) 4: El. By the strong approxi- int Y; v marion theorem there exists a 7 e PF i~t whose projection to PG (A~) belongs to gr S. Let y' be the projection of y-~ to PGU~(R) ~-x. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 117 Since j(PF ln~) belongs to the commensurator of A s :=j(PF~ nt) in PGLa(C) ~, Proposition 4.5.2, Proposition 4.5.5 and Lemma 4.5.1, b), c) imply that p?(As) is dense in PGUa(R) ~-~. It follows that there exists 8 e PF ~ whose projection to PGUa(R) ~-1 � PGt~*(A~ ;~) belongs to y'U � S. Then ~'~ belongs to PF t~t c~ (PGUa_I,i(R) ~ � U � g~S). [] 4.6.4. Now we proceed as in the proof of Theorem 2.2.5. Let M and S be as in 2.4.1, and let PF~ nt be the projection of PF(~] := pF~n~ c~ PGUa(R) g-x � G~ � S to PGUa(R) g-x � G~M. The proof of Proposition 2.4.5 holds in our case, hence PF~ "t is arithmetic. It follows that there exists a permutation ~ of the set { 2, ..., g } such that for every i = 2,...,g there exists a unique algebraic isomorphism r,: PGr~oi--% PGU d satisfying r~(y) =jo(~)(y) for each y e--(s). pp~t In particular, a and the r~'s do not depend on M and S. Since Pr = 0 we then have r,(y) =Jo(,)(Y)for all i e{ 2, ...,g} M, S and y e PF ~"t. This shows the existence of an algebraic isomorphism r which will satisfy Theorem 4.4.3 if we show that it is inner. But this can be immediately shown by the stan- dard argument using elliptic elements and function t defined in 2.5.5 (compare for example the proofs of Proposition 2.5.6 and Proposition 4.5.5). 4.7. Rationality question Consider the (PH~')=-torsor (Pm')~=~ [B a-~ � (PH~r = � (E~~ ~"~ over (,~mt)=. As in the p-adic case, it has a canonical flat connection 3/amt. The same considerations as in the p-adic case (see the proof of Proposition 4.4.2) show that there exists a unique connection ~t on ~tnt such that (9~Int) ~n _~ ~mt It follows from the proofs of Proposition 4.4.2 and Theorem 4.4.3 that (j~. P).(~c) = ~lnt. Lemma 4.7.1. -- If an analytic automorphism ~0:(~int)an .~. (~mt)~. commutes with the action of (pH~t)~" � E lnt, preserves ~,~lnt and induces the identity map on (~tnt)~ __ (pHc)~.\(~.t)~n, then *? is the identity. Proof. -- Recall that (~mt)~n ~ [Ba-1 X (pH~nt) =n � (E~nt/E~t)d~]/Pr ~t. Since ~o induces the identity map on (.~i.t)~n, there exists a holomorphic map + : B a-~ ---> (PH~nt) an such that ~[x, 1, 1] = [x, +(x), 1] for all x e B a-x. Since ~0 preserves ~tnt, we have O~?/Ox - O. Hence + is a constant, say a. Then ~0[x, h, e] = [x, ha, e] for all x e B a-l, h e (PH~t) ~n and e e E mr. In particular, V[x, 1, 1] = r 1 x,j(y), y~] = [y~l x,j(y)a, 2"r.] = [x,j(y) aj(~t) -1, 1] for all ~, e pplnt. Therefore j(y) aj(y) -1 = a for all Y e Flnt" Since PF mt is Zariski dense in PH~ t, a= 1. [] 118 YAKOV VARSHAVSKY Corollary 4.7.9.. __ The torsor ~mt has a unique Eint-equivariant structure p,nt of a PH~tw-torsor over X ~nt such that there exists a connection ~,~t on p~nt satisfying =~ ;Tdint" (pint is called the canonical model of'P l~t over Xint.) Proof. -- The existence is proved in [Mil, III, w 3]. Suppose that P' and P" are two structures satisfying the above conditions. Let f: Pc ~ pint_% Pc be the natural isomorphism. For each. e AUtKw(C ) set q~. := .(f)-a of. Then the automorphism (~%)~" of (P~)= - (~int)an satisfies the assumptions of the lemma. Hence (q~o)= is the identity, so that ~(f) =f for all . e Autx~(C ). It follows that P' = P". [] To finish the proof of the Fourth (and the Third) Main Theorem it remains to show that the homomorphism j~, p:p~w-%--CP~nt is K~-linear. Since = = this follows from the lemma by the same considerations as the corollary. REFERENCES [BC] J.-F. BOUTOT et H. CARAYOL, Uniformisation p-adique des courbes de Shimura : les th6or6mes de Cherednik et de Drinfel'd, Astdrisque 196-197 (1991), 45-156. [Bel] V. G. B~RKOVlCH, Spectral Theory and Analytic Geometry over Non-Archimedean Fields, American Mathematical Society, Providence, Rhode Island, 1990. [Be2] V. G. BERKOVmH, The Automorphism Group of the Drinfeld Half-Plane, C.-R. Acad. Sci. Paris, Set. I Math. 89.1 (1995), 1127-1132. V. G. BERKOVmrI, ]~tale Cohomology for Non-Archimedean Analytic Spaces, Publ. Math. IHES 78 [Be3] (1993), 5-161. [13o] A. BOREL, Linear Algebraic Groups, Springer, 1991. [Br] K. S. BROWN, Buildings, Springer, 1989. [BT] R. Boat and L. W. Tu, Differential Forms in Algebraic Topology, Springer, 1982. [CF] J. W. S. 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MILNE (eds.), Academic Press (1990), 283-414. J. MILI~, l~tale Cohomology, Princeton Univ. Press, 1980. [M~2] [Mi3] J. MXLN~, The Points on a Shlmura Variety Modulo a Prime of Good Reduction, in Zeta Functions of Picard Modular Surfaces, R. LANGLANDS and D. (eds.), Univ. Montreal (1992), 151-253. [Muml] D. MtrMFORD, An Analytic Construction of Degenerating Curves over Complete Local Rings, Compositio Math. 24 (1972), 129-174. [Mum2] D. MUMFORD, Abelian Varieties, Oxford Univ. Press, 1970. [Mus] G. A. Nonarchimedean Uniformization, Math. USSR Sbornik 34 (1978), 187-214. V. PLATONOV and A. RAPIIqCHUK, Algebraic Groups and Number Theory, Academic Press, 1994. [PR] M. RAPOPORT, On the Bad Reduction of Shimura Varieties, in Automorphic Forms, Shimura Varieties and IRa] L-Functions, Vol. 2, L. CLOZ~L and J. MXLNE (eds.), Academic Press (1990), 253-321. [RZ1] M. RAPOPORT and Th. ZINK, l~ber die Lokale Zetafunktion yon Shimura VarietAten, Invent. Math. 68 (1982), 21-101. [RZ2] M. RAPO~ORT and Th. ZINK, Period Spaces for p-Divisible Groups, Annals of Math. Studies 141, Princeton University Press, 1996. [Ru] W. RumN, Function Theory in the Unit Ball of C ", Springer, 1980. [Sc] W. SC~a~LAU, Quadratic and Hermitian Forms, Springer-Verlag, 1985. [SGAI] A. Sdminaire de GJom~trie algdbrique I, Revgtements dtales et groupe fondamental, Lecture Notes in Math. 224, Springer, 1971. [Sha] L R. SHAFA~WCH, Basic Algebraic Geometry, Springer, 1974. [Shi] G. Sm~RA, Introduction to the Arithmetic Theory of Automorphir Functions, Princeton, Princeton Univ. Press, [Sel] J.-P. SERRE, G~omfitrie algfibrique et g~omfitrie analytique, Ann. Inst. Fourier 6 (1956), 1-42. [SeZ] J.-P. SERR~, Cohomologie des groupes diserets, in Prospects in Mathematics, Ann. of Math. Studies 70, Princeton Univ. Press (1971), 77-169. [St] S. STERNBERO, Lectures on Differential Geometry, Prentice Hall, Inc. Englewood Cliffs, NJ, 1964. [Ti] J. Trrs, Classification of Algebraic Semisimple Groups, Proc. 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Publisher: Springer Journals
Copyright: Copyright © 1998 by Publications Mathematiques de L’I.H.E.S
Subject: Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
ISSN: 0073-8301
eISSN: 1618-1913
DOI: 10.1007/BF02698861
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Abstract

by YAI~OV VARSHAVSKY Introduction Let F C PGUa_I.I(R) ~ be a torsion-free cocompact lattice. Then F acts on the unit ball Ba-IC C g-1 by holomorphic automorphisms. The quotient F\B d-1 is a complex manifold, which has a unique structure of a complex projective variety X r (see [Sha, Ch. IX, w 3]). Shimura had proved that when F is an arithmetic congruence subgroup, X r has a canonical structure of a projective variety over some number field K (see [Del] or [Mil]). For certain arithmetic problems it is desirable to know a description of the reduction of X r modulo w, where w is some prime of K. In some cases it happens that the projective variety X r has a p-adic uniformization. By this we mean that the K~-analytic space (X r | Kw) an is isomorphic to A\~ for some p-adic analytic sym- metric space f2 and some group A, acting on f~ discretely. Then a formal scheme structure on A\f~ gives us an 0Kw-integral model for X r | Kw" Cherednik was the first who obtained a result in this direction. Let F be a totally real number field, and let B/F be a quaternion algebra, which is definite at all infinite places, except one, and ramified at a finite prime v of F. Then Gherednik proved in [Ch2] that the Shimura curve corresponding to B has a p-adic uniformization by the p-adic upper half-plane f/~o, constructed by Mumford (see [Mum 1]), when the subgroup defining the curve is maximal at v. Gherednik's proof is based on the method of elliptic elements, developed by Ihara in [111]. The next significant step was done by Drinfel'd in [Dr2]. First he constructed cer- tain covers of f2 2 (see below). Then, when F = Q, he proved the existence of a p-adic uniformization by some of his covers for all Shimura curves, described in the previous paragraph, without the assumption of maximality at v. The basic idea of Drinfel'd's proof was to invent some moduli problem, whose solution is the Shimura curve as well as a certain p-adically uniformized curve, showing, therefore, that they are isomorphic. Developing Drinfel'd's method, Rapoport and Zink (see [RZ1, Ra]) obtained some higher-dimensional generalizations of the above results. In this paper we generalize Gherednik's method and prove that certain unitary Shimura varieties and automorphic vector bundles over them have a p-adic unifor- mizafion. Our results include all previously known results as particular cases. 58 YAKOV VARSHAVSKY We now describe our work in more detail. Let F be a totally real number field of degree g over Q,, and let K be a totally imaginary quadratic extension of F. Let D and D in~ be central simple algebras of dimension d 2 over K with involutions of the second kind ~ and s respectively over F. Let G := GU(D, ~) and Gint := GU(D~nt, ~lnt) be the corresponding algebraic groups of unitary similitudes (see Definition 2.1.1 and Notation 2.1.2 for the notation). Let v be a non-archimedean prime of F that splits in K, let w and ~ be the primes of K that lie over v, and let 0o 1 be an archimedean prime of F. Suppose that D ~nt | Kw has Brauer invariant 1/d, that D | Kw ~- Matd(Kw), and that the pairs (D, ~) | F~ and (D ~t, ~int)| F, are isomorphic for all primes u of F, except v and 0ox. Assume also that ~ is positive definite at all archimedean places F~ol---IlL of F, that is that G(Fo01) - GUa(R ) for all i = 1, . .., g, and that the signature of s at 0ol is (d -- 1, 1), so that Gin~(F~) - GUa_I,I(R). Let Av I and Av f'* be the ring of finite adeles of F and the ring of finite adeles of F without the v-th component respectively. Set E' := F~ � G(Avf'~), and fix a central simple algebra Dw over K~ of dimension d s with Braner invariant lid. Then ') ~� E' ~ D. X and G(A~) = GLa(K~) x E'. In particular, the group GLa(K~) acts naturally on G(A~) by left multiplication. Let f2 a be the Drinfel'd's (d- 1)-dimensional upper half-space over K~ cons- Kw tructed in [Drl], and let { 2]~a"~}~su{0} be the projective system of 6tale coverings of f~ constructed in [Dr2]. This system is equipped with an equivariant action of the ~X � group GLd(Kw) x D~ such that if T~ denotes the n-th congruence subgroup of ~fi~, then we have T \E a' ~ ~ E d' ~ for all m >t n (see 1.3 1 and 1.4.1 for our notation and n\ K w ~ K w conventions, which differ from those of Drinfel'd). Denote by Gl~t(F)+ the set of all d~ (D~t) x such that d.~ht(d) is a totally positive element of F. Choose an embedding K ,-+ C, extending 0ol : F c.+ R. It defines us an embedding G~t(F)+ ~ GUa_I,I(R)~ = Aut(Ba-~). Choose finally an embedding of K~ into C, extending that of K. For each compact and open subgroup S of E' and each non-negative integer n let Xs, ~ be the weakly-canonical model over K~ of the Shimura variety corresponding to the complex analytic space (T, X S)\[Bd-1 � Gi~t(A~)]/Gt~t(F)+ and to the morphism h: S -+G ~"t| described in 3.1.1 (see Definition 3.1.12 and Remark 3.1.13 for the definitions). The experts might notice that our h is not the one usually used in moduli problems of abelian varieties. Let Vs, ~ be the canonical model of the automorphic vector bundle on Xs, ~ (see [Mil, III] or the last paragraph of the proof of Proposition 4.3.1 for the definitions), corresponding to the complex analytic space (T, x S)\[~(WtW| C)~" X GI"t(A~)]/G~t(F)+ (see 4.1.1 for the necessary notation). p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 59 Let Ps, ~ be the canonical model of the standard principal bundle over Xs, (see [Mil, III] or Corollary 4.7.2 for the definitions), corresponding to the complex analytic space (T~ � S)\[B a-~ � (PG~t| ~ � G~t(A~)]/G'~t(F)+ (see 4.1.1 for the necessary notation). Main Theorem. -- For each compact and open subgroup S of E' and each n e N u { 0 } we have isomorphisms of K~-analytic spaces: a) ,tXs,./a" ~ GLa(Kw)\[Z~" � (S\G(A~)/G(F))]; b) (Vs,.)an -% GLa(Kw)\[[3~o ' .(W ~) � (S\G(Arl)/G(F))] (see 4.1.1 for the necessary notation), where the group GL~(K,o ) acts on ~, ,,(W ~") as the direct factor of (G| Kw) (K.,), corresponding to the natural embedding K ~ Kw; c) (Ps,,)~-% [GLa(Kw)\[Z~ � ((PG| ~n � (S\G(A~)))/G(F)]] TM (see 4.1.1 for the definition of the twisting ( )t,,), where the group GLe(K~) acts trivially on (PG | Kw) ~. These isomorphisms commute with the natural projections for S 1 Q S2, n 1 ) n 2 and with the action of G~t(As - D~ X F x X G(Ar~:~). The idea of the proof is the following. Consider the p-adic analytic varieties ~s, of the right hand side of a) of the Main Theorem. They form a projective system and each of them has a natural structure Ys, ~ of a projective variety over K w. Kurihara proved in [Ku] that for every torsion-free cocompact lattice F C PGLa(Kw) the Chern numbers of F\~ are proportional to those of the (d- 1)-dimensional projective space and that the canonical class of F\~K~ is ample. The result of Yau (see [Ya]) then implies that B a-x is the universal covering of each connected component of the complex analytic space (Ys,.| C) ~n for all sufficiently small S e ~'(E) and all embeddings K w r C. It is technically better to work with the inverse limit of the Ys, ~'s equipped with the action of the group Gi~t(A~) -~ D~ ~� � E' on it rather then to work with each Ys, separately. Generalizing the ideas of Cherednik [Ch2] we prove that there exists a subgroup A C GUa_~,~(R) ~ � Gmt(A~) such that (Ys,.| C)~n~ (T. � S)\(B d-~ � G'~t(A~r))/A for all compact open subgroups S C E' and all n ~ N w { 0 }. Using Margulis' theorem on arithmeticity we show that the groups A and Gint(F)+ are almost isomorphic modulo centers. More precisely, we show that (Ys, ,, | C) ~n is isomorphic to a finite covering of (Xs,, | (])an" Using Kottwitz' results [Ko] on local Tamagawa measures we find that the volumes of (Ys,~| ~ and (Xs,.Qx~ C) ~n are equal. It follows that the varieties Ys,~| C and XS,.| C are isomorphic over C. Comparing the action of the Galois group on the set of special points on both sides we conclude that Ys, ~ and Xs,. are actually isomorphic over K w. 60 YAKOV VARSHAVSKY Notice that if one considers only Shimura varieties corresponding to subgroups which are maximal at w, then the use of Drinfel'd's covers in the proof of the p-adic uniformization is very minor. (We use them only for showing that the p-adically uni- formized Shimura varieties have Brauer invariant 1/d at w; that probably can be done directly.) In this case the proof would be technically much easier but contain all the essential ideas. The proof of the p-adic uniformization of standard principal bundles is similar. In addition to the above considerations it uses the connection on principal bundles. Using the ideas from [Mil, III] we show that the p-adic uniformization of standard principal bundles implies the p-adic uniformization of automorphic vector bundles. In fact Tannakian arguments show (see [DM]) that these statements are equivalent. This paper is organized as follows. In the first section we introduce certain cons- tructions of projective systems of projective algebraic varieties, give their basic properties and do other technical preliminaries. In the second section we give two basic examples of such systems. Then we for- mulate and prove the complex version of our Main Theorem for Shlmura varieties. The third and the forth sections are devoted to the proof of the theorem on the p-adic uniformization of Shimura varieties and of automorphic vector bundles respectively. Our proof appears to be very general. That is starting from any reasonable p-adic symmetric space, whose quotient by an arithmetic cocompact subgroup is algebraizable, we find Shimura varieties uniformized by it. For example, in another work ([Va]) we extend our results to Shimura varieties uniformized by the product of Drinfel'd's upper half-spaces. Hence it would be interesting to have more examples of such p-adic sym- metric spaces. Our result on the p-adic uniformization of automorphic vector bundles is not complete, because we prove the p-adic uniforrnization only under the assumption that the center acts trivially. In fact our proof of the complex version of the theorem works also in the general case, but to get an isomorphism over K w one should understand better the action of the Galois group on the set of special points. After this work was completed, it was pointed out to the author that Rapoport and Zink have recently obtained similar results concerning the uniformizafion of Shimura varieties by completely different methods (see [RZ2]). Notation and conventions 1) For a group G let Z(G) be the center of G, let PG :----- G/Z(G) be the adjoint group of G, and let G d~ be the derived group of G. 2) For a Lie group or an algebraic group G let G o be its connected component of the identity. 3) For a totally disconnected topological group E let ~'(E) be the set of all compact and open subgroups of E, and let E dl~ be the group E with the discrete topology. p-ADIG UNIFOlZMIZATION OF UNITARY SHIMURA VARIETIES 61 4) For a subgroup F of a group G let Comma(F ) be the commensurator of F in G. 5) For a subgroup F of a topological group G let F be the closure of F in G. 6) For a set X and a group G acting on X let X G be the set of all elements of X fixed by all g e G. 7) For a set X, a subset Y of X and a group G acting on X let Staba(Y ) be the set of all elements of G mapping Y into itself. 8) For an analytic space or a scheme X let T(X) be the tangent bundle on X. 9) For a vector bundle V on X and a point x e X let V~ be the fiber of V over x. 10) For an algebra D let D ~ be the opposite algebra of D. 11) For a finite dimensional central simple algebra D over a field let SD � be the subgroup of D � consisting of elements with reduced norm 1. 12) For a number field F and a finite set N of finite primes of F let /k~ be the ring of finite adeles of F, and let/k~ :N be the ring of finite adeles of F without the com- ponents from N. 13) For a field extension K/F let R~/r be the functor of the restriction of scalars from K to F. 14) For a natural number n let I s be the n � n identity matrix and let B"C C" be the n-dimensional complex unit ball. 15) For a scheme X over a field K and a field extension L of K write XT. or X | L instead of X � s~ K Spec L. 16) For an analytic space X over a complete non-archimedean field K and a for a complete non-archimedean field extension L of K let X @x L be a field extension from K to L. (A completion sign will be omitted in the case of a finite extension.) 17) By a p-adic field we mean a finite field extension of O~ for some prime number p. Let C~ be the completion of the algebraic closure of O~. 18) By a p-adic analytic space we mean an analytic space over a p-adic field in the sense of Berkovich [Be1]. 19) For an affinoid algebra A let d/(A) be the affinoid space associated to it. Acknowledgements First of all the author wants to thank Professor R. Livn6 for formulating the problem, for suggesting the method of the proof and for his attention and help during all stages of the work. He also corrected an earlier version of this paper. I am also grateful to Professor V. Berkovich for his help on p-adic analytic spaces, to Professor J. Rogawsld for the reference to [C1], to Professor Th. Zink for his corrections and interest, and to the referee for his suggestions. The work forms part of the author's Ph.D. Thesis in the Hebrew University of Jerusalem, directed by Professor R. Livn6. The revision of the paper was done while the author enjoyed the hospitality of the Institute for the Advanced Study at Princeton and was supported by the NSF grant DMS 9304580. 62 YAKOV VARSHAVSKY CONTENTS 1. Basic definitions and constructions ......................................................... 1.1. General preparations ................................................................. 1.2. GAGA results ....................................................................... 66 1.3. First construction .................................................................... 68 1.4. Drinfel'd's covers .................................................................... 72 1.5. Second construction .................................................................. 1.6. Relation between the p-adic and the real constructions .................................. 1.7. Elliptic elements ..................................................................... 1.8. Euler-Poincar6 measures and inner twists ............................................... 1.9. Preliminaries on torsors (= principal bundles) ........................................... 2. First Main Theorem ...................................................................... 87 2.1. Basic examples ...................................................................... First Main Theorem ................................................................. 9O 2.2. 2.3. Computation of Q(TrAd) ............................................................ 93 2.4. Proof of arithmetieity ................................................................ 95 2.5. Determination of H .................................................................. 97 2.6. Completion of the proof .............................................................. 100 3. The theorem on the p-adic uniformization .................................................. 102 3.1. Technical preliminaries ............................................................... 3.2. Theorem on the p-adic uniformization ................................................. 4. p-adic uniformization of automorphic vector bundles ......................................... 107 4. I. Equivariant vector bundles ........................................................... 107 4.2. Equivariant torsors ................................................................... 4.3. Connection between the Main Theorems ............................................... 110 4.4. Reduction of the problem ............................................................. 4.5. Proof of density ..................................................................... 113 4.6. Use of rigidity ...................................................................... 116 4.7. Rationality question .................................................................. 117 References ................................................................................. 118 1. BASIC DEFINITIONS AND CONSTRUCTIONS 1.1. General preparations Definition 1.1.1. -- A locally profinite group is a locally compact totally disconnected topological group. In such a group E, the set ~'(E) forms a fundamental system of neighbourhoods of the identity element, and f] S = { 1 }. s ~ ,~'(E) Lemma 1.1.2. -- Let E be a locally profinite group, and let X be a separated topological space with a continuous action E � X ~ X of E. For each S e ~'(E), set X s :----- S\X. Then { Xs }s is a projective system and X ~ lira X s . <s Proof. -- [Mil, Ch. II, Lem. 10.1]. [] This lemma motivates the following definition. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 63 Definition 1.1.3. -- Let X be a separated scheme over a field L, let E be a locally profinite group, acting L-rationally on X. We call X an (E, L)-scheme (or simply an E-scheme if L is clear or is not important) if for each S ~ o~-(E) there exists a quotient X s := S\X, which is a projective scheme over L, and X- li m X s . The following remarks show that E-schemes are closely related to projective systems of projective schemes, indexed by o~(E). Remark 1.1.4. -- If X is an E-scheme or merely a topological space with a conti- nuous action of E, then for each gee and each S, Teo~'(E) with SDgTg -~ we have a morphism Ps,~(g) : XT -+ Xs, induced by the action of g on X and satisfying the following conditions: a) Ps, s(g)-----Id ifgeS; b) Ps,T(g) o pT,~(h) = ps,~(gh); c) if T is normal in S, then 9T, T defines the action of the finite group S/T on X~, and X s is isomorphic to the quotient of X. r by the action of S/T. Remark 1.1.5. -- Conversely, suppose that for each S e ~-(E) there is given a scheme Xs, and for each g e E and each S, T ~ o~'(E) with S D gTg -1, there is given a morphism Ps,T(g):XT-+Xs, satisfying the conditions a)-c) of 1.1.4. Then for each T C S there is a map Ps,T(1) : X T -+ Xs, which is finite, by condition c). In this way we get a projective system of schemes and we can form an inverse limit scheme X := lira Xs. Then there is a unique action of E on X such that for each g e E and <s each S e ~'(E) the action of g on X induces an isomorphism pgso_l,s : Xs-% Xgsg_l. It follows from c) that X s -% S\X for each S e ~-(E). Definition 1.1.6. -- Let E be a topological group, which is isomorphic to E under an isomorphism qb : E -~ E. We say that an (E, L)-scheme X is ~-equivariantly isomorphic to an (F,, L)-scheme X if there exists an isomorphism q~ : X -% .~ of schemes over L such that for each g s E we have q0 o g ----- 4)(g) o q0. If in addition E = E and q0 is the identity, then we say that q0 is an isomorphism of (E, L)-schemes. Definition 1.1.7. -- Let L~/L1 be a field extension. We say that an (E, L1)-scheme X is an LJLl-descent of an (E, L,)-scheme Y if the (E, L2)-schemes Xz~ and Y are isomorphic. Suppose from now on that E is a noncompact locally profinite group. Notation 1.1.8. -- For a topological group G and a subgroup I" C G � E let pr o and pr~. be the projection maps from F to G and E respectively. Set I" o := pro(r), F~ := pr~,(P) and r s := pr (r (G x S)) for each S e #'(E). For each y s r set % := prQ(y) and y~ := pr~(y). 64 YAKOV VARSHAVSKY Lemma 1.1.9. -- Let F C G x E be a cocompact lattice. Suppose that pr G is injective. Then for each S E ~'(E) we have the following: a) [ s\E/r I< oo; b) [F G : Fs] = oo; c) F s is a cocompact lattice of G; d) F G C CommG(Fs). Proof. -- a) Since the double quotient (G x S)\(G x E)/F ~ S\E/F~. is compact and discrete, it is finite. b) The group E is noncompact, therefore [SkE I = oo. Hence, by a), :s n = I s\sr I = oo. But F a = prG(F ) = prG(pr~l(FE)), and likewise F s = prG(pr~l(F~ n S)). Since pr o is injective, we are done. c) The group F is a cocompact lattice in G X E, hence F n (G x S) is a cocompact lattice in G x S, and the statement follows by projecting to G (see [Shi, Prop. 1.10]). d) Let y e F, and set S' = y~. SyE 1 e ~-(E). Then y(Fn(G � S))y-~=Fn(G � S'). But S n S' ~o~-(E) is a subgroup of finite index in both S and S', hence YG Ps Y~l n F s = F sns, is a subgroup of finite index in both P s and YG Ps Y~l" [] Suppose that d >/ 2 and take G equal to PGLa(Kw) for some p-adic field Kw or to PGUa_I,~(R) ~ We shall call these the p-adic and the real (or the complex) cases respectively. Proposition 1.1.10. q Under the assumptions of Lemma 1.1.9 we have: a) r G D Gd~ b) pr~ is injective; c) for each S e ~-(E), the group r s is an arithmetic subgroup of G in the sense of Margulis (see [Ma, p. 292]); d) if S e ~-(E) is sufficiently small, then the subgroup rasa_~ is torsion-free for each a e E. Proof. -- a) For each S e ~'(E), F s is cocompact in G and [FG: Fs] = oo. It follows that F G is a closed non-discrete cocompact subgroup of G. Therefore its inverse image n-l(FG) in SU a_ 1,1(R) (resp. SLa(Kw) ) is also closed, non-discrete and cocompact, hence by [1Via, Ch. II, Thin. 5.1] it is all ofSUa_l,l(R ) (resp. SLa(Kw) ). This completes the proof. b) Set F 0 := pro(Ker prE). This is a discrete (hence a closed) subgroup of G, which is normal in F o. Therefore it is normal in F G D G a~ It follows that each y e F 0 must commute with some open neighborhood of the identity in G aer, hence F 0 is trivial. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 65 c) is a direct corollary of [Ma, Ch. IX, Thm. 1.14] by b)-d) of Lemma 1.1.9. d) (compare the proof of [Ch. 1, Lem. 1.3]). Choose an S e~(E), then F s C G is a cocompact lattice. Lemma 1.1.11. -- The torsion elements of F s comprise a finite number of conjugacy classes in F s. We first complete the proof of the proposition assuming the lemma. Let ax, ..., a~ E E be representatives of double classes FE\E/S (use Lemma 1.1.9, a)). For each i = 1, ..., n let M~ C F~i sa~l be a finite set of representatives of conjugacy classes of torsion non-trivial elements of Fois~Tt. Then the image of all non-trivial torsion elements of F~is~rl under the natural injection j, : PalS~7~ --% F n (G � a~ Sa~ -1) ,-+ a~ Sa~ -1 --% S is contained in the set X, ={s.j,(M3.s-l[s ~S}, which is compact and does not contain 1. Hence there exists T e .~'(E) not intersecting any of the X,'s. By taking a smaller subgroup we may suppose that T is a normal subgroup of S. Since all the j~'s are injective, the subgroup F~iT,:I =ji-I(T) is torsion-free for each i = 1, ..., n. For each ace there exist i~{1,...,n}, seS and y~F such that a=y~.a~s. Hence the subgroup F~r~_t ~- r c~ (G � aTa- 1) = F c~ ('to Gy~- 1 � y~ a, Ta~- 1 y~ 1) 7(F r~ (G � a~ Tab-l)) 7 -1 _-_ Falr~Zx is torsion-free. O Proof (of the lemma). -- The group G acts continuously and isometrically on some complete negatively curved metric space Y. Indeed, in the real case Y = B d-1 with the hyperbolic metric. In the p-adic case Y is a geometric realization (see [Br, Ch. I, appendix]) of the Bruhat-Tits building A of SLa(K~, ). This is a locally finite simplicial complex of dimension d- 1 which can be described as follows. Its vertices are the equivalence classes of free ~Kw-submodules of rank d of the vector space K~, where M and N are said to be equivalent when there exists a z K~, such that M = aN. The distinct vertices A1, As, 9 9 At form a simplex when there exist for them representa- tive lattices M1, M~, ..., Mk, such that Mx D Ms D ... 9 M~ D rr For more informa- tion see [Mus, w 1] or [Br, Ch. V, w 8]. The geometric realization Y of A has a canonical metric, that makes Y a complete metric space with negative curvature (see [Br, Ch. VI, w 3]). Moreover, the natural action of PGLa(K~) on the set of vertices of A can be (uniquely) extended to the tim- plicial, continuous and isometric action on Y. Now the Bruhat-Tits fixed point theorem (see [Br, Ch. VI, w 4, Thm. 1]) implies that any compact subgroup of G has a fixed point on Y. In particular, any torsion element of G has a fixed point on Y. Notice that in the p-adic case it then stabilizes the minimal simplex, containing the fixed point. 66 YAKOV VARSHAVSKY Conversely, the stabilizer in G of each point of Y is compact. In the real case this is true, since the group PGUd_I,x(R) ~ acts transitively on B e-1 and the group K =-StabBd-l(0)- Ud_x(R ) is compact. In the p-adic case the group PGLe(Kw) acts transitively on the set of vertices, and the stabilizer of the equivalence class of ~w C K~ is PGLd(O~w), hence it is compact. Since the stabilizer in G of any point y e Y must stabilize the minimal simplex ~ containing y, it must permute the finitely many vertices of a, so that it is also compact. It follows that the stabilizer of any point of Y in r s is compact and discrete, hence it is finite. To finish the proof of the lemma in the real case we note that for each x ~ B d- ~ there exists an open neighbourhood U, of x such that P, := {g ~ Us I g(U,) n U,, 0 } ={g e F s I g(x) = x} is finite (see [Shi, Prop. 1.6 and 1.7]). The space rs\B e-x is compact, hence there exist a finite number of points xl, x2, . .., x~ of B e-a, such that I's(lJ~= 1 U~i) --- B e-x. If y is a torsion element of Fs, then it fixes some point of B e-a. By conjugation we may assume that it fixes a point in some U~i , therefore y is conjugate to an element of the finite set [.J'~=~ F,i. In the p-adic case we first assert that A has only a finite number of equivalence classes of simplexes under the action of Ps- Since A is locally finite, it is enough to prove this assertion for vertices. The group G acts transitively on the set of vertices, and G = P s 9 K for some compact set K C G. Hence if v is a vertex of A, then K. v is a compact and discrete (because the set of all vertices of A is a discrete set in Y) subset of Y, and our assertion follows. Now the same considerations as in the real case complete the proof. [] 1.2. GAGA results In what follows we will need some GAGA results. Let L be equal to K w in the p-adic case and to C in the complex case. We will call both the complex and the p-adic (L-)analytic spaces simply (L-)analytic spaces. Recall that for each scheme X of locally finite type over L and each coherent sheaf F on X a certain L-analytic space X ~n and a coherent analytic sheafF ~n on X ~ can be associated (see [Bel, Thm. 3.4.1] in the p-adic case and [SGA1, Exp. XII] in the complex one). Theorem 1.2.1. -- Let X be a projective L-scheme. The functor F ~ F ~n from the category of coherent sheaves on X to the category of coherent analytic sheaves on X "n is an equivalence of categories. Proof. -- In the complex case the theorem is proved in [Sel, w 12, Thm. 2 and 3], in the p-adic one the proof is the same. One first shows by a direct computation that the p-adic analytic and the algebraic cohomology groups of P" coincide. Next, one concludes from Kiehl's theorem (see [Bel, Prop. 3.3.5]) that the cohomology group p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 67 of an analytic coherent sheaf on P" is a finite-dimensional vector space. Now the argu- ments of Serre's proof in the complex case hold in the p-adic case as well. See [Be1, 3.4] for the relevant definitions and basic properties. [] Corollary 1.2.2. ~ a) If X is an algebraic variety over L and X' is a compact L-analytic subvariety of X, then X' is a proper L-algebraic subvariety of X. b) The functor which associates to a proper L-scheme X the analytic space X ~ is fully faithful. Proof. -- Serre's arguments (see [Sel, w 19, Prop. 14 and 15]) hold in both the complex and the p-adic cases. [] Corollary 1.2.3. -- Let X be a projective L-scheme. The functor X' ~ (X') an induces an equivalence between: a) the category of vector bundles of finite rank on X and the category of analytic vector bundles of finite rank on X~; b) the category of finite schemes over X and the category of finite L-analytic spaces over X ~, if L is a p-adic field. Proof. -- a) To prove the statement we first notice that the category of vector bundles of finite rank is equivalent to the category of locally free sheaves of finite rank. In the algebraic case this is proved in [Ha, II, Ex. 5.18]. In the analytic case the proof is similar. Now the corollary would follow from the theorem if we show that locally free analytic sheaves of finite rank correspond to locally free algebraic ones. The analytic structure sheaf is faithfully flat over the algebraic one (see [Sel, w 2, Prop. 3] and [Bel, Thm. 3.4.1]). Therefore the statement follows from the fact that an algebraic flat coherent sheaf is locally free (see [Mi2, Thm. 2.9]). b) We first show that the correspondence (v:Y-+X) ~q~.(0y) (resp. (~ : Y -+ X ~n) ~ ~.(0~)) gives an equivalence between the category of finite schemes (resp. analytic spaces) over X (resp. X ~n) and the category of coherent 0 x -- (resp. Ox~ --)algebras. In the algebraic case this is proved in [Ha, II, Ex. 5.17]. In the analytic case the proof is exactly the same, because a finite algebra over an affinoid algebra has a canonical structure of an affinoid algebra (see ['Bel, Prop. 2.1.12]). [] Remark 1.2.4. R If X' is finite over X, then it is projective over X, therefore if, in addition, X is projective over Kw, then X' is also projective over K w. Corollary 1.2.5. -- Let X and Y be projective L-schemes, and let W and V be algebraic vector bundles of finite ranks on X and Y respectively. Then for each analytic map of vector bundles f: W an --> V an covering some map f: X ---> Y there exists a unique algebraic morphism g : W ---> V such that g~n =~ 68 YAKOV VARSHAVSKY Proof -- By definition, 37factors uniquely as W ~" Y> V ~" x y.. X ~" _-__ (V x r X) ~" oroj> V~" Corollary 1.2.3 implies that there exists a unique g' : W -+ V x y X such that (g')~" = ~". Set g := proj o g'. For the uniqueness observe that if h : W -+ V satisfies h " = ~, then it covers f Hence h factors as W h' p~oj> > V � y X V. Since 3 ~ and g' are unique, we have h' = g' and h= g. [] Remark 1.2.6. -- Using the results and ideas of [SGA1, Exp. XII] one can replace in the above results the assumption of projectivity by properness. We now introduce two constructions of E-schemes which are basic for this work. 1.3. First construction 1.3.1. Let D~ be an open Kw-analytic subset of (~ ~)~, obtained by removing from (PaK~I)" the union of all the Kw-rational hyperplanes (see [Bel] and [Be3] for the definition and basic properties of analytic spaces). It is called the (d -- 1)-dimensional Drinfel'd upper half-space over K~ (see also [Drl, w 6]). Then f~d is the genetic fiber Kw of a certain formal scheme ~a over OK~, constructed in [Mus, Ku], generalizing [Muml]. The group PGLd(K~) acts naturally on D~. (It will be convenient for us to consider pa-~ as the set of lines in A a and not as the set of hyperplanes, as Drinfel'd does. Therefore our action differs by transpose inverse from that of Drinfel'd.) Moreover, this action naturally extends to the 0~: -linear action of PGI-,d(K,o ) on ~. Further- more, PGLa(K~) is the group of all formal scheme automorphisms of ~a xw over OK~ (see [Mus, Prop. 4.2]) and of all analytic automorphisms of f~a over K~ (see [Be2]). Though the action of PGLd(Kw) on f~d is far from being transitive, we have the Kw following Lemma 1.3.2. -- There is no non-trivial closed analytic subspace of D~QK~ " C~, invariant under the subgroup := ~ e Ka~ -~ C PGLd(K,~ ). u= \- pil Proof. M Suppose that our lemma is false. Let Y be a non-trivial U-invariant closed analytic subset of f~a xw @~w C~. Then dim Y < dim ~d w (~Xw C~ = d -- 1. Choose a regular point y a Y(C~) (the set of regular points is open and non-empty). Then dim Tu(Y) = dim Y < d -- 1. Next we identify D~ ~)Kw C~ with an open t..% j by the map (za:... :z a)~-~ z~, . Then analytic subset of tAa-l,,, ( ...,za~ 1] \Zd Zd / p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 69 U~(z) = z + ~- for every z e k,~ -1 and every ~ e M-~(K~). In particular, y + Y e Y for every ~ e Aa-~(K.), contradicting the assumption that dim Tu(Y) < d- 1. [] Recall also that the group PGUd_I,I(R) ~ acts transitively on B d-1 and that it is the group of all analytic (holomorphic) automorpkisms of B a-a (see [Ru, Thm. 2.1.3 and 2.2.2]). In what follows we will need the notion of a pro-analytic space. Definition 1.3.3. -- A pro-analytic space is a projective system { X~ }~ei of analytic spaces such that for some ~0 e I all transition maps q0~ : X~ -+ X~; ~ 1> e i> ~0 are Etale and surjective. Definition 1.3.4. -- By a point of X:={X~}~e I we mean a system {Xe}aEi, where x, is a point of X~ for all e e I and %~(x~) = x~ for all ~/> ~ in I. For a point x=(x=}~e I ofX={X~}~e Ilet T.(X) :--= ( v = ( v. }~e~ I v~ e T.~(X.), d~.~(v~) = v~, for all fi i> e in I } be the tangent space of x in X. Definition 1.3.5. -- Let X = { X~ }~eI and Y = { Y~ }~ea be two pro-analytic spaces. To give a pro-analytic morphism f: X-,--Y is to give an order-preserving map a:I ~J, whose image is cofinal in J, and a projective system of analytic morphisms f~ : X~ -+ YoI~" A morpkism f is called #tale if there exists a0 e I such that for each >/ ~0 the morphism f~ is 6tale. Construction 1.3.6. -- Suppose that PC G � E satisfies the conditions of Lemma 1.1.9. We are going to associate to I' a certain (E, L)-scheme. Let X ~ be B a- 1 in the real case and f~d in the p-adic one. Consider the L-analytic Kw space .~ := (X ~ � Edi~)/I ', where IF' acts on X ~ � E di~ by the natural right action: (x, g) y := (y~ 1 x, gya). Then E acts analytically on X by left multiplication. Proposition 1.3.7. -- For each S e o~'(E) the quotient S\X = S\(X ~ � E)/I' exists and has a natural structure of a projective scheme X s over L. Proof. -- First take S ~ o~(E) satisfying part d) of Proposition 1.1.10. Then S\'-K has I S\E/I'E[ < oo connected components, each of them is isomorphic to Pas~_t\X ~ for some a e E. By c), d) of Proposition I. t. 10, each l',s~_l is a torsion-free arithmetic cocompact lattice of G. By [Ski, Prop. 1.6 and 1.7], [Sha, Ch. IX, 3.2] in the real case and by [Mus] or [Ku] in the p-adic one, each quotient I',s,_l\X ~ exists and has a unique structure of a projective algebraic variety over L. Therefore there exists a projective scheme X s over L such that X~s ~ ~ S\X 70 YAKOV VARSHAVSKY Take now an arbitrary S e o~(E). It has a normal subgroup T e o~(E) which satisfies part d) of Proposition 1.1.10. The finite group S/T acts on T\X ~ X~ n by analytic automorphisms and S\I~ ~ (S/T)\X~". Corollary 1.2.2 implies that the analytic action of S/T on X~ " defines an algebraic action on X T and that the projective scheme X s := (S/T)\X~ (the quotient exists by [Mum2, w 7]) satisfies (Xs) ~" -_ S\X. Moreover, the same corollary implies also that the algebraic structure on S \X is unique. [] For all geE and all S,T e o~-(E) with S DgTg -~ we obtain by Remark 1.1.4 analytic morphisms ps,~(g):X~ n -+Xh ". They give us by Corollary 1.2.2 uniquely determined algebraic morphisms Ps, ~(g) : XT -+ Xs, which provide us by Remark 1.1.5 an (E, L)-scheme X := lim X s. Proposition 1.3.8. -- a) There exists the inverse limit X an of the X~='s in the category of L-analytic spaces, which is isomorphic to X. b) Stabr(X ~ � { 1 }) = Ft. c) Let X o be the connected component Of X such that X~" D X ~ � { 1 } (note that X ~ � { 1 } is a connected component of X ~, and that the analytic topology is stronger then the Zariski topology). Then StabE(Xo) ---- I'~.. d) The group E acts faithfully on X. e) For each x ~ X the orbit E.x is (geometrically) Zariski dense. In particular, E acts transitively on the set of geometrically connected components of X. f) For each S e ~'(E) satisfying part d) of 1.1.10, the map X -+ X s is dtale; g) For each embedding K~ ~ C and each S e ~'(E) as in c), B a-1 is the universal covering of each connected component of (Ks, c)a" in the p-adic case and of X~ in the complex one. Proof. -- a) We start from the following Lemma 1.3.9. -- a) Let H be a torsion-free discrete subgroup of G. Then the natural projection X~ II\X ~ is an analytic (topological) covering. b) For each x e X ~ the stabilizer of x in G is compact. Proof. -- a) follows from [Ski, Prop. 1.6 and 1.7] in the real case and from [Be2, Lem. 4 and 6] in the p-adic one. b) By [Dr2, w 6] there exists a PGLa(K~)-equivariant map from f~e to the Kw Bruhat-Tits building A of SLd(K~) , thus it suffice to show the required property for stabilizers of points in A and B e. 1. This was done in the proof of Lemma 1.1.11. [] The lemma implies that for each sufficiently small S e o~-(E) tile analytic space X~ admits a covering by open analytic subsets U~ satisfying the following condition: for each i and each subgroup S D T e ~'(E) the inverse image 0~-I(U,) of U, under the natural projection pz:X~n-+ X~ n splits as a disjoint union of analytic spaces, each of them isomorphic to U, under p~. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 71 Now we will define a certain L-analytic space X" associated to X. As a set it is the inverse limit of the underlying sets of the X~n's. To define an analytic structure on X" consider subsets V~ C X an such that for some (hence for every) sufficiently small S ~'(E), the natural projection z% :X~n--> X~ n induces a bijection of V~ with an open analytic subset ns(V~) of X~ n, described in the previous paragraph. Provide then such a V~ with an analytic structure by requiring that ns : V~ -+ ~s (V~) is an analytic isomorphism. Then the analytic structure of the V~'s does not depend on the choice of the S's, and there exists a unique L-analytic structure on X ~" such that each V~ is an open analytic subset of X ~n. By the construction, X" is the inverse limit of the X~n's in the category of L-analytic spaces. Hence there exists a unique E-equivariant analytic map ~ : X --> X ~ such that ~ Ra n r~s for each S e ~'(E) the natural projection X -+ X~" factors as X --~ --> X~ n , where by ns we denote the natural projection. It remains to show that n is an isomorphism. For each S ~ ~'(E) satisfying part d) of Proposition 1.1.10, the natural projection X~ Ps\X ~ is a local isomorphism, hence the projections X ~ X s and n are local isomorphisms as well. The map ~s o ~ is surjective, hence for each x ~ X "n there exists a point y ~ such that ~zs(x ) -- rc s o n(y). Therefore, ~(y) = sx for some s ~ S. Since n is E-equi- variant, we conclude that rc(s-l(y)) = x. Hence n is surjective. Suppose that n(Yl) = n(Y2) for someyl,y 2 ~ .X. Let (Xx, gx) and (x2, g~) be their representatives in X ~ � E. Then for each S ~ ~'(E) there exist s ~ S and y ~ I ~ such that xl = y~ ~ (x2) and gl = sg, Yr.. Such ya's belong to the set { g ~ G [ g(Xx) = X 2 } (~ I~l SOl, which is compact (by the lemma) and discrete, hence finite. Therefore we can choose sufficiently small S ~ ~(E) such that gl y~l g~-~ ~ s ~ S must be equal to 1. This means that Yl =Y~. Thus 7: is a surjective, one-to-one local isomorphism, hence it is an isomorphism. b) is clear. c) For each S e o~'(E) let Ys be the connected component of X s such that Y;" is the image of X ~ � { 1 } C X ~" under the natural projection ~s : X" --> X~ =. Then X 0=limYs. It follows that gee satisfies g(X0) =X 0 if and only if g(Ys) =Ys <S for each S zoO'(E) if and only if X ~ x {g} C S(X ~ x 1) P for each S e o~'(E) if and only ifgeSF~, for each Scot'(E) if and only ifge [7 SFr = F--~. S ~ ~'~ E) d) If g ~ E acts trivially on X, then it acts trivially on X an ~ (X ~ � E a~c)/F. By b), g = yz for some y ~ F, and Yo acts trivially on X ~ Since pr~ is injective, y -- g = 1. e) Let Y be the Zariski closure of E.x. Then Y is E-invariant and, therefore, yan C~ (X ~ � is a closed IMnvariant analytic subspace of X ~ � X ~ By Proposition 1.1.10 a), it is Ga~Mnvariant. Since G a~" acts transitively on X ~ in the real case and by Lemma 1.3.2 in the p-adic one, Y" c~ (X ~ � { 1 }) has to be all of X ~ � { 1 }. It follows that Y = X. 72 YAKOV VARSHAVSKY f) holds, since the projection ~s:X~n-+ X~ " is a local isomorphism (see the proof of a)). g) The real case is clear, the p-adic case is deep. It uses Yau's theorem (see [Ku, Rem. 2.2.13]). [] Remark 1.3.10. -- The functorial property of projective limits implies that X *n satisfies the functorial properties of analytic spaces associated to schemes (see [Bel, Thm. 3.4.1] or [SGA1, Exp. XII, Thm. 1.1]). Lemma 1.3.11. ~ Let F C G � E and X be as above, let E' be a compact normal subgroup of E, and let F' C G � (E'\E) be the image of I' under the natural projection. Then we have the following: a) the map ~ : F -+ F' # an isomorphism; b) P' satisfies the conditions of Lemma 1.1.9; c) the quotient E'\X exists and is isomorphic to the (E'\E, L)-scheme corresponding to P'. Proof. -- a) The composition map I' -~ F' ~-~ G is injective, therefore 9 is an isomorphism and prG: F'-+ G is injective. b) F' is clearly cocompact. Let U � S C G � (E'\E) be an open neighbourhood of the identity with a compact closure. Then q~-~(U � S) is an open neighbourhood of the identity of G � E with a compact closure. It follows that ~-I(U � S) n F is finite, thus (U � S) (3 F' is also finite. Hence P' is discrete. c) Since E' is compact and normal, we have E' S = SE' ~ oq~(E) for each S E #'(E). Hence E'\X := lim XE, s is the required quotient. Next we notice that for each <--.- S ~ #'(E) the subgroup S := S\E' S belongs to ~'(E'\E) and that each T ~ ~'(E'\E) is of this form. Since X~ ~, s "~ = E' S\[X ~ � E]/r ~ = S\[X ~ � (E'\E)]/F', we are done. [] 1.4. Drlnfel'd's covers 1.4.1. Now we need to recall some Drinfel'd's results [Dr2] concerning covers of D.~w. (A detailed treatment is given in [BC] for d = 2 and in [RZ2] for the general case.) Let K w be as before and let D w be a central skew field over K w with invariant 1/d. Let 01) w C D w be the ring of integers. Fix a maximal commutative subfield K~ J of D~, unramified over K w. Let ~ e K w be a uniformizer and let Fr~ be the Frobenius auto- morphism of K~ ~ over K w. Then D w is generated by K~ and an element II with the following defining relations: H a = z~, II-a = Frw(a ) 9 H for each a e K~ I. Denote by d~' the ring of integers of the completion of the maximal unramified extension K ~ ~ w of K w. Drinfel'd had constructed a commutative formal group Y over ~w @ow ~ with an action of 0D~ on it. For a natural number n denote by r. the kernel of the homomorphism Y ---> Y. Let ~. := F. | K w be the generic fiber of F. and let ~._uaC ~. be the kernel of l-["n-~(= ~.-va). p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 73 Put Z xw a'" :=s __s and set T.:= 1 +~"O.~e~(Dw~). Then Ea'm~ " is an ~tale Oalois covering of ~a, x~ o := D~ a QK~ K~^ ~ with Galois group (09~/~) . � ~= 0gJT,. Za,- ~ Ea,,~-i We also denote 0 � by T 0. The action of ~ induces dtale covering maps ~. : x~ x~ D w giving a K~-pro-analytic space Za Z a'" tV � acts naturally on Za xw:={ x~ }." The group ~ x~, and we have Za'"x~ =~ T,\Z~w for each n ~N ~{ 0 }. Moreover, Drinfel'd had also constructed an action of the group GLa(K~) � D~ on Zd viewed as a pro-analytic ~K w , space over K,, which extends the action of O � and satisfies the following properties Dw (notice that our convention 1.3.1 differ from those of Drinfel'd): a) the diagonal subgroup { (k, k) ~ GLa(K~) � D~ ] k ~ K~ } acts trivially; b) GLa(K~) (resp. D~ x) acts on Y,~ = a ^ ^=* ~K | by the product of the natural action of PGLa(K~) on D~,, (resp. the trivial action on D~w) and the Galois action nr g ~ Fr~0 ~wlaet~' (resp. g ~-, Fr~ ~,aw(ar on K~. 1.4.2. In the case d = 1 Drinfel'd's coverings can be described explicitly. Let L be a p-adic field. Then, by property a) above, the action of L � � L � on E~ is determined uniquely by its restriction to the second factor. Denote by 0 L : L � -+ Gal(L~b/L) the Artin homomorphism (sending the uniformizer to the arithmetic Frobenius automorphism). Lemma 1.4.3. -- One has Z~ ~- jg(~',b), and the action of (1, l) ~{ 1 } � L � on Y.~ is given by the action of 0~.(/)-1 e Gal(L~b/L) on L ~b. Proof. -- This follows from the fact that Drinfel'd's construction for d = 1 is equivalent to the construction of Lubin-Tate of the maximal abelian extension of L (see, for example, [OF, Ch. VI, w 3]). [] 1.4.4. Let L be an extension of K. of degree d and of ramification index e. For every embeddings L ~ Matn(K~), L '-+ D~ (such exist by [OF, Ch. VI, w 1, App.]) and K~ ~ L "~ and for every n ~ N w { 0 } there exists a closed L-rational embedding i,:Y~' ~" ,-+ Z a'" X~, which is (L � � L� and commutes with the projec- t.ions n,. Moreover, i0 : ~[ (~,. t'.~' ,-+ D~ ~w K~ is the product of our embedding ~ ,-~ I'. ~ and a closed embedding i : ~ ~ ~, with image roa ~T,� (see [Dr2, \'~KwJ Prop. 3.1]). Taking an inverse limit we obtain an embedding Y : Z~ ,-~ ~a W w ~ Lemma 1.4.5. ~ Let H be a subgroup of RLtK~(G,, ) (K~) -_ L � Zariski dense in RLm, o(G,, ). Then Im "~ = { x ~ Y'~Kw I (l, l) x = x for every 1 ~ H }. Proof. -- Since for each l e H C L � the action of (l, l) on Y~ is trivial, and since is (L � � L� Im 7' is contained in the set of fixed points of (1, l), l ~ H. Conversely, if x ~ Z~ is fixed by all (l, l), l e H, then its image ~ e O~w under the natural projection P :xaxw ~ ~axw belongs to (D~) ~ : ~--x#r~ ~*.� i(O~). Since 0(ImT) = Imi, there exists yr such that 0(Y)=~(= p(x)). Recall that ~Ea � = D~\ x~. Therefore y = 8x for some 8 eD.. It follows that (l, 8lS-1)y =y 10 74 YAKOV VARSHAVSKY for each l ~H, hence also (1,818-1l-1)y =y. Since the covering Za~:~, ~ r .� a is 4tale, the group O� acts freely on Z a,:w. Therefore 8l 8-1 l-~ = 1 for each l ~ H. Hence 8 belongs to the centralizer of H in D x, so that to L x. It follows that X = 8-17 EL � .ImT' = Im'~. [] Proposition 1.4.6. -- For each n E N w { 0 } the group SD~ (~ T 1 acts trivially on the set 7: o of connected components of Z~ QK,, C~. Proof. -- Recall (see 1.4.4) that each maximal commutative subfield LC D~ gives us (after some choices) a closed L-rational L� embedding i s : Z~' ~" -+ Z a'" Let q/ be a connected component of Z~' ~" ~ C~. Take ~ ~ 7:0, ~K w 9 which contains i,(~t). Then, by Lemma 1.4.3, f is defined over t'. ~b and l(~ r) = (0L(/))-~(~ r) for each l~L � Fix a ~F 0 ~ n0, and let M be the field of definition of 5F 0. Then M 3 K~. Since the quotient D~ \ ~:w ~ D~w is geometrically connected, D x acts transitively on n 0. Since the action of D~ on T: 0 is K,,-rational, M is the field of definition of every f ~ n o. In particular, M is the closure of a Galois extension of Kw, and M C ~'ab for every extension L of K~, of degree d. Taking L be unramified we see that the group AutC~t(M) of continuous automorphisms of M over K~ is meta-abelian (----- extension of two abelian groups). Set H := { 8 ~ D~ � I there exists a s(8) s Aut K~ ~176 ~"-J such that 8(~0) = s(8)-l(fo)}. Then H is a group and a:H --> AuthOr(M) is a well-defined homomorphism. We claim that H = D~. Take a 8 e D~, then Kw[8 ] is a commutative subfield of D~,. Let L be a maximal commutative subfield of Dw containing 8. Then by (1.1), 8(~) = (0L(8))-l(gY) for some ~ ~0. Take 8' ~D~ such that gY = 8'(~Y0). Then (8')-188'(YCo) = (8')-~o (0L(8))-1o8'(~o) -~ (0L(8))-I(Y'o), so that (8')-188 ' ell. Thus each element of D~ is conjugate to some element of H. In particular, Z(D~) C H. Since T, acts trivially on Z ~, ~'" it is also contained in H. Hence H D T,-Z(D~) has a finite index in D~. Therefore our claim follows from the following Lemma 1.4.7. -- Let G be a group and let H be a subgroup of G of finite index. Suppose that G= [.J gHg-1. Then G----H. g~G/R Proof. -- Set K:= ['] gHg -a. Then K is a normal subgroup of G of e~G finite index, and G/K= [J g(H/K) g-a= [.j (g(H/K) g-l --{1}) u{1}. Hence o~GIH r I G/KI~< [ G/HI (] H/KI -- 1) + 1 = ]G/K] -- ]G/HI + 1, therefore G = H. [] Now the proposition follows from the fact that SD~ is the derived group of D~ x (see [PR, 1.4.3]) and that T1 n SD~ is the derived group of SD x (see [PR, 1.4.4, Thm. 1.9]). [] p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 75 1.5. Second construction Construction 1.5.1. -- Suppose that a subgroup F C GLa(Kw) � E satisfies the following conditions: a) Z(V) = Z(GLa(Kw) � E) c~ F; b) the subgroup Z(F) C Z(GLa(Kw) � E) is cocompact; c) PP C PGLa(K~) � PE satisfies the assumptions of Lemma 1.1.9 (this imply, in particular, that the closure of F is cocompact in GLa(Kw) � E); d) the intersection of Z(F) with Z(GLa(Kw) ) � { 1 } is trivial. We are going to associate to P a certain (D~ � E, Kw)-scheme. Z d E)/F. The group D~ � E acts on X by Consider the quotient X := ( K~ � the product of the natural action of D~ on Z a and the left multiplication by E. Kw Proposition 1.5.2. -- For each S e ~'(D~ � E) the quotient S\X = S\(E~ � E)/P has a natural structure of a Kw-analytic space, which has a unique structure X s of a projective scheme over K w. Proof. -- First take S = T, � S' for some n e N ~3 { 0 } and some sufficiently small S' e ~-(F) (to be specified later). Then S\X = S'\(Zx ~a,. � E)/P is a disjoint union of [ SkE/F~ [< oo (as in Lemma 1.1.9) quotients of the form F,s,~_~\X~" with a e E. Thus it remains to prove the statement for quotients d, F,s,~_~\ZK . For simplicity of notation we assume that a = 1. Set r ,,o := n pr (z(r)) = pr (r (Z(GLa(Kw)) � (Z(E) n S'))). First we construct the quotient d, Ps,,0\Exw. Assumptions a) and b) of 1.5.1 imply that the closure of Fs,,0 is cocompact in Z(GLa(Kw) ) -~ K~, hence valw(det(Fs,,0)) = dk Z for some k e N. Let K(~ ~) be the unique unramified extension of K w of degree dk', then Consider the natural ~tale projection 7:, : E g'~ Kw-+ Z d'~ xw-+ fig K~," Let {~gC'(A~) }~ex be an affinoid covering of D~. a Since the projection Ea'"x., -+ Za'~ is finite, each ~-l(,,g(Ai) ) C X~/~ is finite over the affinoid space d/(A~| ^~ " K~). Hence it is iso- morphic to an affinoid space ,/g(B,) for a certain K~-affinoid algebra Bi, finite over Ai 6x~ K ^ w . Since ~, is D~-invariant, we have a natural action of Fs,,0 on B,. Set C~ := B r"',~ Since an affinoid algebra is noetherian, we see that C~ is finite over the Kw-affinoid algebra A,. Hence C~ has a canonical structure of a Kw-affinoid algebra (see [Be1, Prop. 2.1.12]). Gluing together the Jg(C~)'s, we obtain a Kw-analytic space rs, ' \Z a, ~ finite and dtale over D~w. Ok K w Put g := S-Z(E)/Z(E) C PE. Then g ~ ~'(PE). To construct F \X a'- v\ Kw we observe p \X a," that the action of PF~ = Ps,,0\F s, on s,,0\ Kw covers its action on ~aK~. Suppose that S' is so small that S satisfies part d) of Proposition 1.1.10. Recall that 76 YAKOV VARSHAVSKY by Lemma 1.3.9 each x e ~, has an open analytic neighbourhood U. such that 7(U,) n U, # 0 for all -~ E PF~ -- { 1 } and, as a consequence, PP~\D~ is obtained by gluing the U~'s. Let ~. be the natural projection from I's, ' o\\~a'"x~ to ~.a For each y eFs, 0\E~ ~ set V~:=~a(U~.~.~). Then the quotient K~,-analytic space pF~\(Fs, ' 0\Zx~)a.~ = Ps,\\Y/'"Kw is obtained by gluing the V~'s. Since Fs,\\Za'"~ is a finite (and dtale) covering of PF~\~, which has a structure of a projective scheme over K~ by [Mus, Ku], P \~]a.. also has such a structure by S'\ K w Corollary 1.2.3 and the remark following it. Finally consider an arbitrary S ~ ~'(D x x E). It has a normal subgroup g of the form g = T, x S' with sufficiently small S'~ ~'(E), therefore to complete the proof we can use the same considerations as in the end of the proof of Propo- sition 1.3.7. [] The same argument as in Construction 1.3.6 gives us a (D x x E, Kw)-scheme X = lira X s. ~S Proposition 1.5.8. -- a) The kernel E 0 of the action of D~ � E on X is the closure of the subgroup Z(F) CZ(GLa(Kw)x E)=Z(D~ x x E) after the natural identification Z(GL~(K~)) ---- K~ ---- Z(D~). b) Let Eo be the closure of Z(P) in E, and let F'C PGL~(K~) x (Eo\E) be the image of F under the natural projection. Then F' satisfies the as~mptio~ of I_zmma I. I. 9. c) The quotient D~ \X exists and is isomorphic to the ~,o\E-scheme corresponding to F' by Construction 1.3.6. d) The quotient (D~ � Z(E))\X exists and is isomorphic to the (PE, K~)-scheme X' corresponding to PF by Construction 1.3.6. e) For each x e X the orbit (D~ x E)x is Zariski dense in X. f) For each sufficiently small S e ~'(E) and each n e N u { 0 } the map X --~ Xr. x s is ~tale, and B a-x is the universal covering of each connected component of (X~,� s,o) ~ for each embedding K, ~ C. In particular, the projective system X ~ := { X~" }T e ~D~ � E~, associated to X, is a Kw-pro-analytic space. Proof. -- a) Notice that g E E 0 if and only ifg acts trivially on X s (or, equivalently, on X~ n = S\(Z~w x E)/F) and normalizes S for each S e~'(D~ X E). For each 7 ~ Z(F) C Z(D~ x E) let y~ be the projection ofy to the first factor. Since (70 x y,~) acts trivially on Y/ xw, we have V([x, e]) = [V,~(x), y~, e] ,~ [(Vo � V,) (x), e] = [x, e] for each x e Y,~ and e e E, that is 7 acts trivially on each X~ ". Since 7 is central, it certainly normalizes S. This shows that the closure of Z(F) is contained in E 0. Conversely, suppose that some (gs, g,)eD~ � with gleD~ and g, e E belongs to E 0. Choose S' e~-(E) and n e Nu {0}. It suffice to show that (gl, g~) e (T, � S')Z(I'). Since (gl, g,) acts trivially on S'\(Z~" � E)/F, we have [gl(x), g~] '~ [x, 1] for each x ~ Z a'x~ ". This means that there exists an element y = y, e P such that gl(x) = 7~1(x) and g, ~ S' YE" Let x' be the projection of x to y a,0~,, and let x" p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 77 be its projection to fl d The group D~ acts trivially on ~e therefore 7~1(x '') = x" K w" K w ~ 9 Choose x so that no non-trivial element of PGLd(K~) fixes x", then YG belongs to Z(GLa(K~)) ~- K~. Assumption c) of 1.5.1 implies that Y ~ Z(F). Since gx(X) = "~G-I(x) = "~w(X), we conclude that g~-i y~(x)= x. Hence x'= (gi-ly~,)(x')= Fr~Wllaetlnlvwll(x'), so that g~-I y~ e ~� Since Z a'' is an &ale Galois covering of ]~a,0 with Galois group Dw" Kw ~w O~/T,, the equality (gi-~yw)(x)=x implies that gi-~y~ ~T,. It follows that (g~, g2) e (T~ � S')(y~, yz) C (T, � S') Z(I'), as claimed. b) The natural projection PGLa(K~) � (E0\E) -+ PGLa(K~) � PE induces an isomorphism F' -~ PP. Hence I" is discrete and has injective projection to PGLd(K~). It is cocompact, because so is 1" C PGLa(K~) � E. c) Notice first that for each open subgroup E 0 fi S C D x � E, compact modulo E0, the quotient S\X exists and is projective. Assumption b) of 1.5.1 implies that E 0 is cocompact in D~ � Z(E). Therefore for each S e o~-(D~ � E) the quotient D~ S\X = (D x E 0 S)\X = (D x � ~'o) S\X exists. Set g := (D; x Eo)k(D; � go) S ~ ~'(E0\E). Then (D; SkX) ~" = (D; x E0) S\[Z~:w x E]/r -~ gk[a~w x (E0kE)]/r', and the state- ment follows as in the proof of Lemma 1.3.11 c). d) follows from c) and Lemma 1.3.11 c). e) follows from c) and Proposition 1.3.8 e). f) Take T e ~@(PE) satisfying part d) of Proposition 1.1.10. Then there exists S e o~'(E) such that Z(E)\S.Z(E) = T. Since we have shown in the proof of Propo- sition 1.5.2 that X~r, � s is 6tale over T\X' for each n e N u { 0 }, the statement follows immediately from Proposition 1.3.8 f), g). [] Corollary 1.5.4. -- For each a s E the composition map po : z ~ -~ z ~ x { a } ~ (z~ x E'~)/r -, x ~" Kw Kw of pro-analytic spaces over K,~ is ~tale and one-to-one. Proof. -- The &aleness is clear. Let Xl and x~ be points of Ne such that BE w p~(xx) ---- p~(x~). Let ~ s PE' be projection of a, and let p; be the injection K~ -+ n~ x { ~- } ~ (a~ x (PE')d'~)/Pr ~ (X'? n. Then we conclude from the commutative diagram ~d, n Pa Xa n Kw ;~ Dro~ l IP r~ ~w , (x')"- P~ that xx and x2 have the same projection y ~ D~w. 78 YAKOV VARSHAVSKY Choose S e o~'(E) so small that the group PFasa_l is torsion-free (use Propo- sition 1.1.10 d)). Then no non-central element of P~sa-1 fixes y. For each n e N let n,,s be the projection Xan-+ (XTn� Then the image of 7r,, s o p~ is isomorphic to P,s~_~\Z~$. Hence there exists ~., ~ P~s~-i such that the projections of y,(xl) and x 2 to Z a'" coincide. Therefore ~%(y)=y, so that ~', e Z(GLd(Kw) ) = K~ It follows Kw that the sequence {-(, }, converges to some .( e K~, which satisfies y(x~) = x 2. Then (y, 1) e Z(D$ � E) fixes z := G(xl) = pa(x2). Since (y, 1) is central, it then fixes the whole (D$ � E)-orbit of z. Hence, by Proposition 1.5.3 c), it acts trivially on X. There- fore by Proposition 1.5.3 a), the element (y, 1) belongs to Z(P) C Z(GLa(Kw) � E). Assumption d) of 1.5.1 implies that y = 1, hence x~ = x~. [] 1.6. Relation between the p-adic and the real constructions The following proposition (and its proof) is a modification of Ihara's theorem (see [Ch2, Prop. 1.3]). It will allow us to establish the connection between the p-adic (1.3.6, 1.5.1) and the real (or complex) (1.3.6) constructions. Proposition 1.6.1. -- Let X be an (E, C)-scheme. Suppose that a) E acts faithfully on X; b) E acts transitively on the set of connected components of X; c) there exists S e ~'(E) such that the projection X -+ X s is dtale, and B d-1 is the universal covering of each connected component of X~s ~. Then X can be obtained from the real case of Construction 1.3.6. Remark 1.6.2. -- a) It follows from Proposition 1.3.8 that all the above conditions are necessary. b) Let X be an (E, C)-scheme and let E 0 be the kernel of the action of E on X. Then X is an (E0\E , C)-scheme with a faithful action of E0\E. Conversely, any (E0\E , C)- scheme can be viewed as an (E, C)-scheme with a trivial action of E 0. c) Let X be an (E, C)-scheme and let X 0 be a connected component of X. Put X' := U g(X0). Then X' is an (E, C)-scheme with a transitive action of E on the set gE~. of its connected components, and X is a disjoint union of such (E, C)-schemes. Remarks b) and c) show that assumptions a) and b) of the proposition are not so restrictive. Proof. -- We start the proof with the following Lemma 1.6.3. ~ Suppose that { X~ }~, ~ i is a projective system of complex manifolds such that the transition maps X~ ~ X~, where ~, ~ ~ I with ~ >1 ~, are analytic coverings. Then there exists a projective limit X of the X~'s in the category of complex manifolds. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 79 Proof. -- Choose an 0t e I. Cover X~ by open balls { U~ }~ea, and let r~ : X' ~ X~ be an analytic covering. Then the inverse image rc-~(U~) of each U~ is a disjoint union of analytic spaces, each of them isomorphic to U~ under z~. Hence the cons- truction of the projective limit from the proof of Proposition 1.3.8, a) can be applied. [] Now we return to the proof of the proposition. By assumption c), X~ n is a complex manifold for each sufficiently small S e ~'(E), and the natural covering X~ n ~ X~ n is 6tale (analytic) for each T C S in ~-(E). Therefore by the lemma there exists an analytic space X ~ := lim X~ n. Since X s is a complex projective scheme for each S e o~(E), the set of its connected components coincides with the set of connected components of X~ n. Hence assumption b) implies that the group E acts transitively on the set of connected components of X "n. Let M be a connected component of X ~n. Denote by F the stabilizer of M in E. Then I'~ acts naturally on M, and the transitivity statement above implies that X"~==_ (M x Ed'~)/r~. For each S ~ ~'(E) the analytic space X~" ~ S\(M � E)/Pz is compact. Therefore, as in the proof of Lemma 1.1.9, I S\E/lP~ I < oo and [P~ : P~ raS] = oo. Note that M s := (Pz n S)\M is a connected component of X~ ". Suppose that S satisfies condi- tion c); then the map M ~ M s is 6tale and B d-a is the universal covering of M s. Hence it is also the universal covering of M. It follows that Uz C Aut(M) can be lifted to I" R C Aut(B a-~) = PGUa_~,I(R) ~ The kernel A of the natural homomorphism zc:I" R -+ P~ is the fundamental group of M. Let Ps C PGU a_ 1, ~(R) ~ be the fundamental group of the compact analytic space M s , then r s is a cocompact lattice in PGUa_~,~(R) ~ satisfying P s = r~-x(PE ta S). It follows that [Ps : Fs] = [I'~ : P~ r3 S] = oo. Therefore, as in the proof of Propo- sition 1.1.10 a), we see that Pl~ is dense in PGUa_I,I(R) ~ The group A is discrete in PGUa_x,~(R) ~ and normal in I'R, thus it is trivial (compare the proof of Propo- sition 1.1.10 b)). In particular, M - B a-~ and z~ is an isomorphism. Put I-':={(T, rc(T))[u ~ � E. Since P s is discrete in PGUa_~,x(R) ~ so is I' in PGUa_x,x(R) ~ � E. Let K C PGUa_I,~(R) ~ be the stabilizer of 0 e B a-1. Then X~ ~ ~ S\(B a-1 � E)/F ----- (K � S)\(PGUa_~,I(ll) ~ � E)/F. Since K, S and X~ n are compact, P is cocompact in PGUa_x,I(ll) ~ � E. Since Ker(pra) equals the kernel of the action of E on X, the projection pr o is injective. This shows that P satisfies all the assumptions of Construction 1.3.6. [] Corollary 1.6.4. -- Choose an embedding K,o ~ C. Let X be an (E, Kw)-scheme obtained by the p-adic case of Construction 1.3.6 or an (E,, Kw)-scheme obtained by Construction 1.5.1. Then X c can be constructed by the real case of Construction 1.3.6. Proof. N This is an immediate consequence of Propositions 1.6.1, 1.3.8 and 1.5.3. [] 80 YAKOV VARSttAVSKY 1.7. EIHptic elements Definition 1.7.1. -- Suppose that a group G acts on a (pro-)analydc space (or a scheme) X. An element g e G is called elliptic if it has a fixed point x such that the linear transformation of the tangent space of x, induced by g, has no non-zero fixed vectors. In such a situation we call x an elliptic point of g. Lemma 1.7.2. -- Let )'1, )`2, ..., xe be the eigenvalues of some element g ~ GLa(L ) (with multiplicities). Let v ~ Pa-I(L) be one of the fixed points of g corresponding to )`l. Then -- are the eigenvalues of the linear transformation of the tangent space of v, induced )`I' )`I ' " " " ' )`I byg. Proof. -- Simple verification. [3 Proposition 1.7.3. -- The set of elliptic elements of PGUa_I.I(R) ~ with respect to its action on B a-1 and of PSLa(K~) with respect to its action on ~w is open and non-empty. Proof. ~ In the real case we observe that an element g := diag()u, )`2, .-., )`d) ~ PGUd-I,i(R) ~ fixes (0, 0, ..., 0) ~ B d-1. Therefore by Lemma 1.7.2, g is elliptic if k, 4: )`d for all i 4: d. It follows that the set of elliptic elements is non-empty. It is open, because if g has a fixed point in B a-1 corresponding to an eigenvalue of g appearing with multi- plicity 1, then the same is true in some open neighbourhood of g. In the p-adic case we start with the following Lemma 1.7.4. -- An element g e GLa(Kw) is elliptic (acting on ~Kw) if and only if its characteristic polynomial is irreducible over K w . Proof. -- Suppose that the characteristic polynomial Xg of g is irreducible over K w. Then g has d distinct eigenvalues. Let X be some eigenvalue of g, let v 4:0 be the eigen- vector ofg corresponding to )`, and let ~ ~ pd-1(~.~o) be the fixed point of g corresponding to v. By Lemma 1.7.2, the linear transformation of the tangent space of ~, induced by g, has no fixed non-zero vector. So it remains to be shown that ~ ~ D~. If ~ r D~w , then it lies in a K~-rational hyperplane. Therefore there exist elements ax, 9 9 aa ~ K,~, not all 0 (say a a :~ 0) such that (al, ..., aa).v = 0. We also know that (g -- )`I) v ---- 0. Let A be the matrix obtained from g -- )`I by replacing the last row by (al, ..., aa). Then Av = 0, so that det A----0. The determinant of A is a polynomial in )` of degree (d- 1) with coefficients in K~ with leading coefficient (-- 1)(d-l~aa + O. This contradicts the fact that the minimal polynomial of ), over K,~ has degree d. Suppose now that the characteristic polynomial )~g ofg equals the product fx" 9 9 9 "fk p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 81 of polynomials irreducible over K~ (k > 1). Consider the matrix fx(g). If fl(g) = 0, then the minimal polynomial mg of g divides fl. Hence each root of ~G, being a root of ms, is a root off1. Eachf has only simple roots, therefore f [fl for each i. Since fx is irreducible, all the f's are equal up to a constant. Hence Za = cfl k for some c e K~. In particular, each root of Xg is at least double. Lemma 1.7.2 then implies that g is not elliptic. Hence we can suppose thatf(g) 4= 0 for all i = I, 2, ..., k. Let X be an eigenvalue of g, let v be the eigenvector corresponding to X, and let ~ e pa-x(~) be the fixed point of g corresponding to v. Choose i s { 1, ..., k } such that X is a root of f. Then f(g) v =f(X) v = 0. The matrixf(g) ~ 0 has all its entries in K~, hence ~ lies in a K,,-rational hyperplane. Therefore g is not elliptic. [] Now we return to the proof of the proposition. Embed an extension L = K~(X) of K,0 of degree d in Mata(K,, ). Then X ~LxC GLa(K,~ ) has an irreducible cha- racteristic polynomial over K W. Therefore the set of elliptic elements of PGLd(Kw) is non-empty. It is open because by Krasner's lemma if g c GLn(K,o ) has a characte- ristic polynomial irreducible over K,,, then an)" g' e GLa(K,o), close enougL to g, has the same property (see [La, Gh. II, w 3, Prop. 4]). It follows that the set of elliptic elements of PSLa(Kw) is also open. For showing that it is non-empty observe that if an element g ~ PGLa(K,o ) is elliptic but ga is not elliptic, then by Lemma 1.7.2 the characteristic polynomial of any representative of ga in GLa(Kw) has at least two equal roots. Hence such a g belongs to some proper Zariski closed subset of PGL a. It follows that there exists an elliptic element g ~ PGLa(K~) such that ga is elliptic as well. Since g~ always belongs to PSLa(K~), we are done. [] Proposition 1.7.5. -- a) An element (g, 8) ~ GLd(K,~ ) � D~ is elliptic with respect to its action on X a (viewed as a pro-analytic space over K~,) if and only if the characteristic Kw polynomials of g and 8 are K~-irreducible and coincide. b) For every element g e GLa(K,~) elliptic with respect to its action on G~, there exists a 8 ~ D,~ such that (g, 8) is elliptic with respect to its action on y a Proof. -- a) Let x E X a be an elliptic point of (g, 8), and let ~ ~ fin be its image K w Kw Kw "--> D~. Since 0 is ~tale, it induces an isomorphism under the natural projection p : Z a a of tangent spaces (up to an extension of scalars). Hence g is elliptic with respect to its action on ~a By Lemma 1.7.4, g generates a maximal commutative subfield L : = K~ (g) K w ~ of Matd(K,~). Choose an embedding j : Kw(g ) ~ D~, (such exists by [CF, Ch. VI, w 1, App.]). It defines an L� embedding "/':Z~'-->Z a (see 1.4.1). We know that Kw ~- e (D.~) v'~ = p o z (ZL). In particular, there exists y E Y(Z~,) such that p(y) ----- 5(. Since T is L� the element (g,j(g)) ~ GLa(K~) � D~ fixes y. Using the � za =~a we havey-----d0x for some d 0eD~ fact that x ~ 0- ~(~) and that D~ \ x~, K~, Hence, the elements (g, doaj(g) do) E GLa(K~) � D,~ and d' := doaj(g) do 8 -~ ~ D~ fix x. Xl 82 YAKOV VARSHAVSKY In particular, d ~ D~ fixes some point (the projection of x) on Ea, ~w 0 (~w C~, therefore d"E ~� Since the Galois covering E ~ � d ~)� acts freely on Z d It fol- lows that d= 1, hence 8 = do~j(g) d o. This completes the proof of the implication " only if", because g ~ Matd(K~) andj(g) ~ D~ have the same characteristic polynomials. Conversely, suppose that the characteristic polynomials of g and 8 are K~-irreducible and coincide. Then the subfields K~(g)C Mata(K~) and K~0(8 ) C Dw have degree d over K~ and are isomorphic under the K~-isomorphism sending g to 8. Using this isomorphism we obtain embeddings of the field L := K~(g) into Mata(Kw) and into D~. These embeddings define (by 1.4.4) an (L � � L� embedding "~:~ ~-+ X d such that every point x E ~(X~) is fixed by all elements of the form Kw (l,l) eL � � L� GLd(K~) � D~. In particular, x is a fixed point of (g, 8). As before, the action of (g, 8) on the tangent space of x coincides with the action of g on the tangent space of ~-. Since ~ is an elliptic point ofg (by Lemma 1.7.4), x is an elliptic point of (g, ~). b) If an element g e GLa(K~) is elliptic, then by Lemma 1.7.4 it has an irreducible characteristic polynomial over K~. Therefore K~(g)C Matu(K~) is a field extension of K~ of degree d. Then for every embedding j of K~(g) into D~ the element (g,j(g)) is elliptic by a). [] 1.8. Euler-Poincar6 measures and inner twists Here we give a brief exposition of Kottwitz' result [Ko, w 1]. 1.8.1. Let L be a local field of characteristic 0, and let H be a connected reductive group over L. Serre [Se2] proved that there exists a unique invariant measure (called the Euler-Poincar6 measure) ~a on H(L) such that aa(1-'\H(L)) is equal to the Euler- Poincar6 characteristic ZE(I') of H*(r, Q) for every torsion-free cocompact lattice I" in H(L). In particular, aa(H(L))= 1 ff the group H(L) is compact. The Euler- Poincard measure is either always negative, always positive or identically zero. It is non-zero if and only if H has an anisotropic maximal L-toms. (A result of Kneser shows that in the p-adic case this happens if and only if the connected center of H is anisotropic.) 1.8.9.. Let G be an inner form of H. Choose an inner twisting p : H --~ G over I.. Choose a non-zero invariant differential form o~a of top degree on G. Set co~r := p*(c%). Using the fact that H is reductive, that the twisting is inner and that co G is invariant, we see that o~ is invariant, defined over L, and does not depend on p. Hence co G and o~u define invariant measures [ o~ I and I o~ E [ on G(L) and H(L) respectively (see [We2, 2.2]). Definition 1.8.3. -- The invariant measures ~t on H(L) and Ex' on G(L) are called compatible if there exists some c ~ R such that ~ = c[ ~a [ and ~' = c I ~ I" 1.8.4. -- Now suppose that H has an anisotropic maximal L-torus T, so that the Euler-Poincar~ measure Va on H(L) is non-trivial. (Notice that for semisimple p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 83 groups of type A, this assumption is satisfied automatically). Denote by I~rrl the absolute value of ~z H. Write N(T, H) for the finite set Ker[Ha(L, T) -+ Ha(L, H)] and write I N(T, H) ] for its cardinality. It is well known that T transfers to G, thus we can also consider the finite set N(T, G). Proposition 1.8.5 ([Ko, Thm. 1]). -- The invariant measure IN(T, H) 1-11 ~t n [ on H(L) is compatible with the invariant measure IN(T, G) I-1I [zol on G(L). Remark 1.8.6.- a) In the p-adic case, the sets N(T, H) and N(T, G) always have the same cardinality. b) In the real case, N(T, H) = f2(H(C), T(C))/f2(H(R), T(R)), where f2 stands for the Weyl group. In particular, [ N(diag, PGUa) [ = 1 and ] N(diag, PGU a_ 1. a) ] is d (resp. 1) if d~ 2 (resp. d-= 2). 1.9. Preliminaries on torsors (-~ principal bundles) Definition 1.9.1. -- Let G be an affine group scheme over a field L (resp. an L-analytic group), and let X be an L-scheme (resp. an L-analytic space). A G-torsor over X is a scheme (resp. an analytic space) T over X with an action G � T -+ T of G on T over X such that for some surjective 6tale covering X' -+ X the fiber product T � x X' is the trivial G-torsor over X' (that is isomorphic to G � X'). Remark 1.9.2. -- Since each 6tale morphism of complex analytic spaces is a local isomorphism, our definition in this case coincides with the classical one. Lemma 1.9.3. -- a) If T is a G-torsor over X, then the map ~T : G x T -+ T x x T (~(g, t) = (gt, t)) is an isomorphism. b) Let T and T' be two G-torsors over X and Y respectively. Then for each G-equivariant map f: T ~ T' the natural morphism T --~ T' � y X is an isomorphism. Proof. -- a) Since the problem is local for the 6tale topology on X (see [Mi2, Ch. I, Rem. 2.24] in the algebraic case, [Be3, Prop. 4.1.3] in the p-adic analytic and Remark 1.9.2 in the complex one), we may suppose that T is trivial. Then our morphism (g, (h, x)) ~ ((gh, x), (h, x)) is invertible. b) For trivial torsors the statement is clear. The general case follows as in a). [] Remark 1.9.4. -- By [Mi2, Ch. I, Rem. 2.24 and Prop. 3.26] our definition in the algebraic case is equivalent to the standard one. In particular, a G-torsor over X is affine and faithfully flat over X. Lemma 1.9.5. -- Let X be a separated scheme over a field L, let G and H be two affine group schemes over L, let T be a G-torsor over X, and let ~r : T -~ X be the natural projection. a) The functor ,~" ~ rc*~" defines an equivalence between the category of quasi-coherent sheaves on X and the category of G-equivariant quasi-coherent sheaves on T, that is, quasi-coherent sheaves on T with a G-action that lifts the action of G on T. 84 YAKOV VAKSHAVSKY b) The funetor Z F. Z x x T defines an equivalence between the following categories: i) the category of vector bundles of finite rank on X and the category of G-equivariant vector bundles of finite rank on T; ii) the category of H-torsors over X and the category of G-equivariant H-torsors over T; iii) (/f X is noetherian and regular) the category of P"-bundles on X and the category of G-equivariant P"-bundles on T. The quasi-inverse functor is "Z ~-* G\ Z. Proof. -- This is a consequence of a descent theory. a) Abusing notation we will write ~'X y, Y1 instead of p* o~" for every morphism P : Y1 ~ Y2 and every sheaf of modules o~" on Y,. Let o~- be a G-equivariant quasi- coherent sheaf on T. Define an isomorphism ~ : (~ x ~ T) x x T -% T x x (~ x ~ T) over T Xx T by the formula ?(f, gt, t) = (gt, g-l f, t) for all g ~ G, t E T and f~ if, (use Lemma 1.9.3). Then ? satisfies the descent conditions of [Mi2, Prop. 2.22]. Since T -+ X is affine and faithfully flat, there is a unique quasi-coherent sheaf o~" on X such that o~ ~ o*" X x T. Since the construction of descent is functorial (see [Mi2, 2.19]), we obtain an equivalence of categories. Notice that o*- ~ G\(o~- x x T). b) follows from a) in a standard way (use [Ha, II, Ex. 5.18, 5.17 and 7.10]). [] From now on we suppose that the reader is familiar with basic definitions of tensor categories (see [DM]). Notation 1.9.6. -- For a field L, an affine group scheme (resp. an analytic group) G over L and a scheme (resp. an analytic space) X over L: a) let HepL(G ) be the category of finite-dimensional representations of G over L; b) let r x be the category of vector bundles of finite rank on X; c) let Torx(G ) be the category of G-torsors over X. We will sometimes identify categories with the sets of their objects. Definition 1.9.7. -- Let L be a field and let G be an affine group scheme over L. A G-fiibre functor with values in a separated scheme (resp. analytic space) X over L is an exact faithful tensor functor from Hepr,(G) to r x. Remark 1.9.8. I If X ---- Spec R is affine, then g/be x is equivalent to the category of finitely generated projective modules over R, hence our definition is a global version of that of [DM, 3.1]. 1.9.9. -- Let T be a G-torsor over X, then by Lemma 1.9.5, the correspondence V ~-~ G\(V x T) defines a G-fibre functor with values in X. This correspondence defines a functor ~ from Torx(G ) to the category of G-fibre functors with values in X. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES Theorem 1.9.10. -- The functor ~ determines an equivalence between Torx(G ) and the category of G-fibre functors with values in X. Proof. -- The local version is [DM, Thm. 2.11 and 3.2]. The gluing works because X is separated. [] 1.9.11. Later on, we will use the following description of the quasi-inverse functor "~ ofv. Let ~q be a G-fibre functor with values in X. For each morphism no : To ~ X we define two tensor functors ~ql:V ~-~ V x T O and ~2 = no o ~q from ~epT,(G) to ~r Let ~(To, no):= Isom(~q,, ~ql) be the set of isomorphisms of tensor functors. The action of G on the first factor ofV x T O defines an action of G on ~ql, and afortiori defines an action of G on ~(To, no). Thus ~ is a functor from the category of schemes over X to the category of sets with a G-action. Theorem 1.9.10 says that this functor is representable by a G-torsor v(~q) over X (see [DM, Thm. 2.11 and 3.2] and their proofs). 1.9.12. Let T be a G-torsor over X. For each V~lepL(G) the identity map of T, viewed as a T-valued point of T, corresponds to a certain isomorphism r x T~ (G\(V x T))Xx T. Then ~v is the quotient of the G-equivariant isomorphism Id v x ~0T:V X G x T-~V x T x xT (for the diagonal action of G on the first two factors on both sides) by the action of G. Explicitely, ~v(V, t) = ([v, t], t). Proposition 1.9.18. -- Let L be equal to K~ or to C as in 1.3.1. Let X be a projective L-scheme, and let G be a linear algebraic group over L. The functor T ~ T ~ induces an equivalence between the category of G-torsors over X and the category of G~n-torsors over X ~. Proof. -- A quasi-inverse functor can be described as follows. Let ~:T -+ X ~ be a G~"-torsor. Then the map V ~-* G~\(V ~" x T) defines a G-fibre functor with values in X =. Since the correspondence described in Corollary 1.2.3 commutes with tensor products, the tensor categories r x and r are equivalent. Therefore Theorem 1.9.10 gives us an algebraic G-torsor n:T---> X. It remains to show that there exists a canonical isomorphism T-~T a". By the definition of T we have for each V e g~epi,(G ) a canonical isomorphism ~v : G~n\( V~ X T) ~ G~\(V ~ x Tan). We also have (as in 1.9.12) natural iso- morphisms T x V ~ --'% T Xx (G~\(T x V~")) mapping (t, v) to (t, It, v]). Hence each point t o of ~ defines canonical isomorphisms { to} X to} � � v (to, [t0, v]). Since t o defines a point of X ~ and therefore of X, it gives us by the universal property ofT (see 1.9.11) a point d/(t0) ~ T an, satisfying ~bv([to, v]) = [~(to) , v] for all V ~ 9tepL(G ). Taking V to be a faithful representation of G, we obtain that the map (of sets) d/:T ~ T ~n is G~-equivariant, therefore it is one-to-one and surjective. It remains 86 YAKOV VARSHAVSKY to show that the maps + and +-1 are analytic. Let us prove it, for example, for ~b. Let p : X' ~ X "~ be an 6tale surjective covering such that p*(T ~n) ~ G an � X'. By [Be3, Prop. 4.1.3] in the p-adic case and by Remark 1.9.2 in the complex one it will suffice to show that p* ~b : p* (T) -+ p* (T a") ~ Can� X' (or just its projection to the first factor W : p*(T) -+ G an) is analytic. Consider the map ~V-'Van X p*(T) proj) GanXWa n X p*(Y)] P"+';) G"~\[V ~ x o*(T~)] ~ G"n\(V " x G an x X') g V an x X' p~oj> Van. It is analytic, and satisfies ~v(V, t) = (~:'(t)) -1 v. Hence r~' is analytic as well. [] Corollary 1.9.14. -- Let X and Y be projective L-schemes, let G and H be algebraic groups over L, and let ~b : G -+ H be an algebraic group homomorphism over L. If T ~ Torx(G ) and S E Tory(H), then for any +-equivariant analytic map 37: T ~ -+ S an (that is, satisfying f~igt) = +(g) ff (t) for all g E G an and t ~ Tan), there is a unique algebraic morphism f: T -+ S such that fan _ 3~ Proof (compare the proof of Corollary 1.2.5). -- Sinceffis +-equivariant, it covers some algebraic morphismf: X ~ Y (use Corollary 1.2.2). Thereforefffactors through S an � ~ (S � Hence we may suppose, replacing S by S � that X = Y and that f is the identity. Consider the H-torsor H X T over T equipped with the following G-action: g(h,t) = (h~b(g)-l, gt) for all g eG, hell and t eT. By Lemma 1.9.5, there exists an H-torsor H � T := G\(H � T) over X. Let i be the composition of the embedding t ~ (1, t) of T into H X T with the natural projection to H � T. Then by the defi- nition, every ~b-equivariant algebraic morphism w:T ~ S factors as a composition of i with the unique H-equivariant map H x a T ---> T (defined by [h, t] ~ h~(t)). Therefore (H � H an � an is an H~-torsor over X an having the same functorial property. Now we are ready to prove our corollary. From the ~-equivariance off we conclude that it factors uniquely as f: ~ T an ---> ,a" (H x o T)a, d~ S~. By the proposition, 3 ~ has a unique underlying algebraic morphism f' : H x o T --> S. Set f: = f' o i. The uniqueness can be derived from the above considerations as in the proof of Corollary 1.2.5. [] Now we recall the notion and basic properties of connections on torsors (following [St, Ch. VI, w 1]). Definition 1.9.15. -- Let X be a smooth scheme or an analytic space, and let : P -+ X be a G-torsor. A connection on P is a G-equivariant vector subbundle af' of the tangent bundle T(P) of P such that ~'1~9 is an isomorphism .X~'~-% T=c~(X ) for each p ~ P. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 87 1.9.16. Starting from the isomorphism qh, : G � P-% P � P we obtain an isomorphism of tangent spaces (q~p), : Te(G ) � T~(P) -% Tr(P ) � T~(P) and an identification (X ~-* projl((~p),(X, 0)) of ~ := Lie(G) = T~(G) with the tangent space to the fiber through p ~ P. Therefore a connection 3r ~ on P gives us a canonical decompo- sition T~(P) = fr | for each p ~ P. Now considering the projection of T~(P) onto with kernel 3r for each p ~ P we get a certain f~-valued differential 1-form ~ = Y2(~), called the connection form of ~tt~ Definition 1.9.17. -- Let W be a connection on a G-torsor P, whose connection form is ~. Let h be the natural projection of T~(P) on 3~ for all p ~ P. The curvature of the connection 3(r ~ is the 2-form Dfl defined by ( X ^ Y I D~ ) := (h(X) ^ h(Y) [ d~ ). A connection with zero curvature is called flat. Remark 1.9.18. -- The trivial torsor P - G � X has a natural flat connection, consisting of vectors, tangent to X. We will call such a connection trivial. Lemma 1.9.19. -- Let X be a simply connected complex manifold, let 7~ : P -+ X be a G-torsor, and let ~ be a flat connection on P. Then there exists a unique decomposition P -% G � X such that ~ corresponds to the trivial connection on G � X. Proof --By [St, Ch. VII, Thm. 1.1 and 1.2], there exists a unique G-equivariant diffeomorphism q~ : P -% G � X over X which maps W to the trivial connection. Hence q~ induces complex isomorphism between tangent spaces T~(P) = ~@Yt~ and T,I~j(G � X) = fr | T.I~(X) for each p ~ P. In other words both q~ and q~-i are almost complex mappings between complex manifolds. [He, Ch. VIII, p. 284] then implies that q~ is biholomorphic. [] 2. FIRST MAIN THEOREM 2.1. Basic examples Definition 2.1.1. -- Let K[k be a quadratic field extension and let D be a central simple algebra over K. We say that e : D -+ D is an involution of the second kind over k if e(d 1 + d2) = e(dl) + e(d2) , ~(d 1 d2) = e(d2) ~(dl) for all dl, d 2 ~ D and tile restriction of ~ to K is the conjugation over k. Notation 2.1.2. -- For k, D and e as in Definition 2.1.1, let G = GU(D, e) be the algebraic group over k of unitary similitudes, that is G(R) = { d e (D | R) � I de(d) ~ R � } for each k-algebra R. Define the similitudes homomorphism G ~ G,, by x ~ xo~(x). Notice also that by the Skolem-Noether theorem the group G satisfies PG(L) = G(L)/Z(G(L)) for every field extension L of k. 2.1.3. First basic example. -- Let F be a totally real field of degree g over Q,, let K be a totally imaginary quadratic extension on F. Let D be a central simple algebra 88 YAKOV VARSHAVSKY of dimension d 2 over K with an involution of the second kind ~ over F. Set G :----- GU(D, a), and put D, := D | K, for each prime u of K. Let v be a (non-archimedean) prime of F that splits in K and let w and ~ be the primes of K that lie over v. Then D @F F~ _-_ D w | D~, and the projection to the first factor together with the similitude homomorpkism induce an isomorphism G(F,)--% D~ X F~. We identify G(F~) with D~ x F~ by this isomorphism. Suppose that D~-~ Mata(Kw). Identifying D. with Mata(K.) by some iso- morphism we identify G(F,) with GLa(Kw) � F~ x . Suppose that ~ is positive definite, that is G(F~oi)- GUa(R) for all archimedean completions Fooi~ R of F. Put E' := F~ x x G(Arf:*), then E' is a noncompact locally profinite group. Set r:= G(F) C G(A~) ---- GLa(K~) � E', embedded diagonally. Proposition 2.1.4.- The subgroup F C G(A~) = GLa(K,~ ) � E' satisfies the assumptions of Construction 1.5.1. Proof. -- a) is trivial. b) is true, because the closure of Z(F) _-_ K � is cocompact in Z(G(A~)) ~ (A1x) � c) Since PE'= PG(Ar 1;') and PG(F,)=~ PGLa(K,o), we have to show that PF(= PG(F)) is a cocompact lattice in PG(A~). Lemma 2.1.5. -- If H is an F-anisotropic group, then H(F) is a cocompact lattice in H(A~). Proof. -- See [PR, Thm. 5.5]. [] Since PG is anisotropic over each Fo~i, it is anisotropic over F. Hence by the lemma, PG(F) is a cocompact lattice in PG(Ar). The compactness of the PG(Fool)'s implies also that the projection of PG(F) to PG(A~) is a cocompact lattice as well (see [Shi, Prop. 1.10]). Observe also that the projection PG(F) -+ PG(F~) -~ PGLa(Kw) is injective. d) Sincc z(r) ,~ = K � and Z(G(A~)) ~- (Al~) � , we have to show that the inter- section of K � C (ASK) � with K~ � { 1 } is trivial. This can be shown either by the direct computation or using the relation between global and local Artin maps (see [CF, Ch. VII, Prop. 6.2]). [] Fix a central skew field D~ over K~ with invariant 1/d. Set E =: D,, � E', then Construction 1.5.1 gives us an (E, K~)-scheme X corresponding to F. 2.1.6. Second basic example. -- By Brauer-Hasse-Noether theorem (see [Wel, Ch. XIII, w 6]) there exists a unique central skew field D t"t over K which is locally isomorphic to D at all places of K except w and ~ and has Brauer invariant l/d at w. By Landherr theorem (see [Sc, Gh. 10, Thm. 2.4]), D mt admits an involution of the second kind over F. Fix an embedding o%:K ~-+ C. It induces an archimedean com- pletion Foo, of F, and we have the following p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 89 Proposition 2.1.7. ~ a) There exists an involution of the second kind o~ mt of D ~ over F such that: i) the pairs (D, a) | F, and (D mr, 0trot)| r F, are isomorphic at all places u of F, except v and ool; ii) the signature of (D ~t, ~mt) at oo~ /s (d -- 1, 1). b) The group G ~t := GU(D lnt, ~t~,) is determined uniquely (up to an isomorphism) by conditions i), ii) of a). Proof. ~ a) follows from [G1, (2.2) and the discussion around it] as in [C1, Prop. 2.3]. b) follows immediately from [Sc, Ch. 10, Thm. 6.1]. [] Let G "t be as in the proposition. Then embedding oo~ defines an isomorphism Dmt | Kool--~ Mata(C), and we identify pGt~t(F~o~) with PGUa_a,~(R ) by the induced isomorphism. Set G~t(F)+ := Grit(F) n G~"t(F=~) ~ Then G~t(F)+ = G~(F) if d> 2, and [Gt"t(F) : G~nt(F)+] = 2 if d = 2. Set E ~t := G~t(A~) and let Eo~nt C E t"t be the clo- sure of Z(Glnt(F))C E t~t. Embed diagonally G~"t(F) into Gt"t(Foo~) � E t~ and define I 't"t to be the image of Grit(F)+ under the natural projection to PGmt(Foo~) � (Emt/E~ont) = PGUa_~,~(R)� Proposition 2.1.8. -- The subgroup F '~t is a cocompact lattice in PGU,_ ,,~ (R)~ � (E~/E'o"~) and it has an injective projection to the first factor. Proof. -- Notice that the natural projection E lnt /E o ~.t ~E"VZ(E ~"t) = PE '~ induces an isomorphism I~Int--~-PGI~*(F)+CPGUa_I,x(R)~215 PE lnt and that the group Z(E~t)/Eo ~t-~ (A~)� � is compact. Therefore it will suffice to prove that PGlnt(F) is a cocompact lattice with an injective projection to the first factor of PGI"t(F~o~) x PE fnt. This can be proved by exactly the same considerations as in the proof of Proposition 2.1.4, c). [] By the proposition, I ~n~ satisfies the assumptions of Construction 1.3.6, so it determines an (Elnt/Eo ~t, C)-scheme ~t~t, which can be regarded as an (E ~t, C)-scheme with a trivial action of El0 ~t. Remark 2.1.9. -- For each S ~ ~-(E ~t) we have the following isomorphisms (~V~smt)= ~ S\[B d-1 x (E'n*/Eo~t)]/r ~' _--_ (S.Z(Glnt(F))\[B a-1 X G'n*(A~)]/G~t(F)+ (S.Z(G~nt(F))\[B d-1 x G'nt(A~)]/G'n*(F)+ -_ S\[B '-1 x G'nt(A~)]/G'~t(F)+. (Emt/E~on~). 90 YAKOV VARSHAVSKY 2.2. First Main Theorem Definition 2.2.1. -- An isomorphism r -% E mt is called admissible if it is a product of G(A~ :~) -% G~t(A~v:~), induced by some A~v:~-linear algebra isomorphism D | Av s;* -% D*~t | Av f'* (compare Proposition 2.1.7), and the composition map ~X :K D~ � F, -% (D~ t) � � F~ --~ G*"t(F~), constructed from some algebra isomorphism D~ -% D~t | K~ as in 2.1.3. 2.2.2. Fix a field isomorphism C -% C~, whose composition with embedding oo 1 : K ,-+ C (chosen in 2.1.6) is the natural embedding K r K~ ,-+ C~. Identifying C with C~ by means of this isomorphism we can view, in particular, K~ as a subfield of C. First Main Theorem 2.2.3. -- For some admissible isomorphism (P:E -% E ~at there exists a r isomorphism fvfrom the (E, C)-scheme X c to the (E ~t, C)-scheme ~t. 2.2.4. Let E 0 be the kernel of the action of E on X, and put E := E/E 0. By Corollary 1.6.4 there exists a subgroup A C PGUa_I.~(R) ~ � E such that the (E, C)- scheme X c corresponds to A by the real case of Construction 1.3.6. By Proposition 1.5.3, each admissible isomorphism r -% E ~t satisfies q)(E0) = ~ol~/nt- Hence r induces an isomorphism ~ : E -% Ei"t/E0.~t Theorem 2.2.5. -- There exists an admissible isomorphism r ~ E ~nt and an inner automorphism e? of PGUd_I. 1 such that (~ � U~) (A) = F int. Lemma 2.2.6. -- Theorem 2.2.5 implies the First Main Theorem. Proof. -- Theorem 2.2.5 implies that there exists a r analytic iso- morphism 9~:(Xc) ~n -% (~int)an. From the (I)-equivariance we obtain analytic iso- morphism 9~, s : (Xs, c) ~n -% ~ for each S e ~-(E). Corollary 1.2.2 provides us with an algebraic isomorphism fr s : Xs, c --% satisfying (arc, s) ~n ~ 3~, s. Taking their inverse limit we obtain a q)-equivariant isomorphismfe :---= limfe, s : Xc --~ Xmt- [] Thus we have reduced our First Main Theorem to a purely group-theoretic statement. For proving it we need to know more information about A. First we introduce some auxiliary notation. 2,2.7. Let A'CPGUd_I,I(R)~ � PE and A"CPGUd_I,I(R)~ � PE' be the images of A under the natural projections. Since the groups Eo\Z(E ) and Eo\D~ x .Z(E) are compact, Lemma 1.3.11 shows that subgroups A' and A" correspond by the real case of Construction 1.3.6 to the (PE, C)-scheme X~ := Z(E)\X c and to the (PE', C)- scheme X~' := (D~ � Z(E))\X c respectively. The same lemma implies also that the natural projections A -+ A' and A -+ A" are isomorphisms. Let E' 0 be the image of E 0 under the canonical projection to E'. Let F' be the -~olsl~'i~t ~'t~lnt~olS~J p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 91 image of V under the projection GLa(K,) x E' -+ GLa(Kw) x (E0\E'). Then, by Pro- position 1.5.3 c), the group F' corresponds by the p-adic case of Construction 1.3.6 ~X to the (E0\E' , K,)-scheme X'" := Dw\X. Recall also that, by Proposition 1.5.3 d), the (PE',Kw)-scheme X"----(I~ x Z(E))\X is obtained from the subgroup PF C PGLa(K~) >< PE' by the p-adic case of Construction 1.3.6. For each subset | of A, A' or A" (resp. of F, F' or PF) we denote by O= (resp. | and | its projections to the first and to the second factors respectively (compare 1.1.8). Our next task is to establish the connection between A and F. The next key proposition is the modifications of [Ch2, Prop. 2.6]. In it we apply Ihara's technique of elliptic elements to relate elements in A and in F. Proposition 2.2.8. -- For each 8 e A with elliptic projection 8~ e PGU a_ 1,1(R) o, there exist y ~ F and YD ~ D,~ ~� with (YG, YD) ~ GLa(K~) x D~ elliptic (with respect to its action on EK~ )a and a representative ~' = (~o, ~'E) e GUe_I,I(R) ~ � E of 8 satisfying the following conditions: a) the elements (YD, YE) and "~r. are conjugate in E; b) the characteristic polynomials of ~o~ and YG are equal. ~� GL~(K~) � D~ Conversely, for each y ~ F and YD ~ Dw with (Yo, 7D) ~ ~ � elliptic, there exist 8 ~ A with elliptic projection 80~ ~ PGUa_I, 1(11)~ and a representative~ e GUa_I,I(R) 0 � E of 8 satisfying conditions a) and b). Proof. -- If an element 8~0 ~A= is elliptic, then 8| has a fixed elliptic point P on B a-1. The action of 8= on B a-1 coincides with the action of 8 E on B a-1 ~ B a-1 X { 1 } C (B a-1 X E)/A ~ (Xc) an, therefore P, viewed as a point of (Xc) an (or of X(C)), is an elliptic point of 8~. Using the isomorphism C -% C~, chosen above, P can be considered as a point of X(C~), hence as a point of the p-adic pro-analytic space X a". There exists an element g e E such that the point P' := g(P) lies in ~ := pl(EKw) in the notation of Corollary 1.5.4. Let = be the natural projection X---> X"'. Choose a representative ~'e E of t lit an g 8E g-1 ~ ~. Since ~' fixes P, it fixes the projection P" := =(P') ~ (Xc~) . Hence stabilizes the connected component D~,, GK~ C~ � { 1 } C (X~;') an containing P". By Proposition 1.5.3 c), the image of ~ under the canonical projection E-+E0\E' belongs to the projection of F' to E'0\E'. We can therefore choose y e F whose projection to E'0\E' coincides with that of ~'eE. Therefore ~,y~l belongs to D~, X E 0 -~ D~ E 0. Hence there exists a y, ~ D w such that F(y{ 1, .~1) ~ E0" It follows that (YD, YE) ~ E is also a representative of g 8 E g-1. The action of (YD, YE) on the tangent space of P' e ~ is conjugate to the action of 8 E on the tangent space of P, therefore P' is an elliptic point of (YD, YE)" Since P1 is dtale, one-to-one (use Corollary 1.5.4) and D, x r-equivariant, the action of (YD, YE) on the tangent space of P' m ~/ coincides with the action of (YG, 7i)) on the tangent 92 YAKOV VARSHAVSKY space of ptl(P ') e Yfl Therefore p~--l(p,) is an elliptic point of (Ya, YD). It follows Kzo " that the action of (Ya, YD) on the tangent space of p-l(p,) e Na is conjugate to the K w action of 8~o on the tangent space of P e B a-x. Using the 6talness of the projection Nd a%-+~ we conclude from Lemma 1.7.2 that there exists a representative 8~o E GU d_ 1, x(R)~ of 8ao such that the characteristic polynomials of 8~ and YG are equal. Hence 7 := (8~o, g-l(yD, "f~)g) is the required representative of 8. The proof of the opposite direction is very similar, but much easier technically. If an element (u YD) s s o � D~ is elliptic, then it has an elliptic point QeEa Kto 9 Hence Q' := pl(Q) e X ~ is an elliptic point of (YD, Y~) e E. Hence Q' can be considered as a point of the complex analytic space (Xc)an___ (Bd-1 � ~)/A. Choose a representative (x,g)eBa-1 � E of Q'. Then the element g(YD, Yv.)g-1 e E fixes Q" :=g(Q')eBa-1 x {1}, hence it stabilizes the connected component B d-~ x{1}C(Xc) ~. It follows that the image ofg(yD, YE) g- ~ under the projection of E to E belongs to A E. The rest of the proof is exactly the same as in the other direction. [] Corollary 9..9..9. -- For each 8 e A with elliptic projection 8oo e PGUa_~4(R)~ , there exists a representative 7= x x x o(A; -~ such that a) if we view K as a subset of C, of K~, and of K | A~ :~ respectively, then the characteristic polynomials of Too, 7~ and 7::" have their coefficients in K and coincide; b) 9~ and the similitude factor of~::" belong to F, viewed as a subset Of F{ and of (A~: ~) � respec- tively, and coincide. Proof. -- Take y and 7 as in the proposition. Then the statement follows from Proposition 1.7.5. [] Proposition 2.2.10. -- We have the inclusion A-~E D (SD~ x n Tx) x P(Gd~r(AFt'")). Proof. -- Let X' o be the connected component of X~ such that (Xo) a" D B a- a � { 1 }. -7 Then by Proposition 1.3.8 c), A S = Stabe~,(X0). Proposition 1.4.6 implies that the group SD~ n T 1 acts trivially on the set of connected components of X~, therefore it remains to show only that A~' D p(Gd~'(A~;')). To prove it we first observe that by the strong approximation theorem (see, for example, [Ma, Ch. II, Thm. 6.8]), the closure PG(F) of PG(F) in PE' = PG(A~ ;~) contains P(Gd~ So the proposition follows from tp Lemma 2.9..11. -- We have Az = PG(F). p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 93 Proof. -- Proposition 1.3.8 c) we see that A~ is the stabilizer of the connected component Yoo of X~' such that (Yo~) ~ ~ B a-a � { 1 } and PG(F) is the stabilizer of the connected component Y~ of X~; such that (Y~)~") f~a ~% � { 1 }. Since the group PE' acts transitively on the set of geometrically connected components of X", the sub- -'-77, --77, groups A E and PG(F) are conjugate in PE'. Since A E contains P(Ga~(As it is normal. So we are done. [] 2.3. Computation of Q(Tr Ad) In the next subsection a field Q(Tr Ad) (generated by the traces of the adjoint representation) will be a field of definition of a certain algebraic group. Remark 2.8.1. -- If g e GLa, then by direct computation we obtain that Tr Ad g = Trg.Tr(g-a). Hence for g E PGL a we have Tr Adg = Tr~'-Tr(~ -1) -- 1 for each representative ~ e GLa of g. co 1 Proposition 2.3.2. -- We have Q(Tr Ad Aoo) = F ~ R. Proof. -- It follows from Proposition 2.2.8, Proposition 1.7.5 and Remark 2.3.1 that Q(Tr Ad 8= I 8= ~ as is elliptic) = Q(Tr Ad YG [ YG ~ PPa C PGLa(K~) is elliptic). Let F' be the last-named field. Then F' C F, since PI' = PG(F) and since PG is an alge- braic group defined over F. It follows from the weak approximation theorem that for each non-archimedean prime u 4: v of F, the closure of the projection to PG(F~) of the set { e Prlv is elliptic} contains an open non-empty subset of PG(Fu). (Recall that the closure of Pr a in PGLa(Kw) contains PSLa(Kw) by Proposition 1.1.10, and that the set of elliptic elements of PSLa(K~) is open and non-empty by Proposition 1.7.3.) Therefore F' is dense in each non-archimedean completion F, of F for u 4= v. Thus F' splits completely in F at almost all places. Hence F' = F (see [La, Ch. VII, w 4, Thm. 9]). This part of the proof is completely identical with Cherednik's proof of [Ch2, Prop. 2.7]. Now we want to prove that Q(Tr Ad A~o ) = Q(Tr Ad ~ [ 8o0 ~ A~o is elliptic). Since the group PGUa_I, 1 is absolutely simple, the representation Ad : PGUa_~,~(R ) ~ GL(Lie(PGUd_~,~(R)) ) -- GLa_~(R ) is absolutely irreducible. Therefore our statement is a consequence of the following general Lemma 2.3.3. -- Let ? be an absolutely irreducible algebraic representation of PGU d_ 1,1 and let A be a dense subgroup of PGUd_I,I(R) ~ Then Q(Tr(o(A))) = Q(Tr p(8) [ 8 e A is elliptic). 94 YAKOV VARSHAVSKY Proof. -- Let L be the last-named field. If g e PGUa_I, a(R) ~ is elliptic and g' is not elliptic for some r e Z -- { 0 }, then by Lemma 1.7.2, g belongs to some Zarisld closed proper subset of PGUe_I, ~. Therefore for each N e N, there exists an open subset WC PGUd_I,I(R) ~ such that for g e W and r e Z satisfying 1 ~< [r[.< N, the element g' is elliptic. Choose g e W. By the continuity of multiplication, there exists an open neighbourhood U C W of g such that for gl, 9 9 -, gk e U, and nl, ..., n k e Z, satisfying nl+ ... +nk4: 0, [n~[ + ... +[nk[~< N, the element gi'~.....g~k is elliptic. Take N ----- 6m 2, where m is the dimension ofp. Since PGUa_I,I(R) ~ is a connected real Lie group, it is generated by U. The subgroup A is dense in PGUa_I,~(R) ~ by Proposition 1.1.10, therefore A n U generates the group 7~ (see [Ma, Ch. IX, I_,em. 3.3]). Since the restriction of p to the Zariski dense subgroup A is absolutely irreducible, Burnside's theorem (see [Wa, vol. II, Ch. XVII, 130]) implies that ~ := dimR(Spana(p(A)) ) = m ~. Set ~0 := { 1 } C A, and for each positive integer n set ~-:--{g~.....g~klg~eAoo ~g, ln~[ -t- ... +]n~l~< n}CA. Denote dim~(Span,(0(A")) ) by ~,. Since A = O 7~", we have 1 =~0~< ~1~ .-.~< ~,~< ...~< ~=sup~,. Cb Moreover, if~, = ~, + 1 for some n, then ~ = N, + 1 ..... N. Therefore ~,,~ _ 1 ---- m~. Hence there exist elements ~ e ~m"-a, i = 1, ..., m z such that { p(~) }~ constitute a basis for Mat,,~(ll). Choose any g e A ~ U and take ~ := g"~+a ~i. Then{ 0(8~) }~ still constitutes a basis for Matm~(ll ). Each ~ is of the form g~'~.....g~, where the gi's belong to Ac~Uandthen~'ssatisfyn~-k ... +n~>/2 and [n~] +In~[ + ... +[n~[~< 2m ~. In particular, each 8~ is elliptic, therefore Tr p(8~) e L. Lemma 2.3.4. ~ If for some ~ e PGUa_I,I(R) ~ the elements ~ are elliptic for all i = 1, ..., m ~, then O(~) can be written as a linear combination of the p(~)'s with coeJficients in L. Proof. -- Let el, ..., em~ be the dual basis of { p(S~) }~ relative to the bilinear form (x,y) ~ Tr(xy). If 8 is as in the lemma, then Tr O(~)= Tr(p(8)p(8,))eL for all i = 1,..., m 2. Hence p(8) can be written as a linear combination of the e~'s with coefficients in L. Therefore it is enough to prove that each e, can be written as a linear combination of the p(3~)'s with coefficients in L. The last condition is equivalent to the condition that each p(8~) can be written as a linear combination of the e~'s with coefficients in L. Thus, as we mentioned above, to complete the proof it is enough to show that each 8, satisfies the conditions of the lemma. This follows directly from the definition of the ~i's and of U. [] The choice of the ~,'s assures that for every ~ e A n (U w U-1) the elements ~8,~ p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 95 are elliptic for all i----- 1, ..., m 2. Therefore the above lemma implies that p(8) can be written as a linear combination of the p(8~)'s with coefficients in L. The set U t3 generates the group &, hence for every 8 ~ &, the linear transformation p(8) can be written as a polynomial in the p(8~)'s with coefficients in L. For any i,j, k ~{ 1, ..., m 2 } the elements 8~ 8j 8 k are elliptic, therefore by the lemma each p(8~ 8~)= p(Si)p(~j) can be written as a linear combination of the 9(Sk)'s with coefficients in L. Hence every polynomial in the p(8~)'s with coefficients in L, can be written as a linear combination of the p(8~)'s with coefficients in L. In particular, this is true for each p(8) with ~ ~ &. Hence Q(Tr p(A)) C L. [] Corollary 2.3.5. -- Suppose that a subgroup A C A~o is Zariski dense in PGU d_ 1,1 and that Ao~ C Commmua_l,~(R)(A). Then Q(Tr Ad A) ---- Q(Tr Ad Ao~) (= F). Proof. -- Set L:----- Q (Tr Ad A), then there exists an L-form V of Lie (PGU d_ 1,1 (R)) preserved by Ad A (see [Ma, Ch. VIII, Prop. 3.22]). Take any 8 ~ A| Then some sub- group of finite index A' of T~ satisfies 8A' 8 -1 C A, hence (Ad 8) (Ad A') (Ad 8)-1(V) ---- V. Since the subgroup A' is also Zariski dense in PGUa_~,I, Burnside's theorem implies that Ad A' generates End V as an L-vector space. Therefore (Ad 8) (End V) (Ad 8) -1 C End V. In other words, Ad(AdS)(EndV)= End V. Let H be the Zariski closure of Ad&C GL(V). Then H is an L-form of AdPGUa_I,1, hence LieHC EndV is an L-form of Lie(Ad PGUd_I,1). In particular, Lie H = End V n Lie(Ad PGUd_I,1) , therefore Ad(AdA~o)(LieH)= Lie H. Since PGUa_I, 1 is adjoint, the homomor- phism ad := Ad. : Lie PGUd_I, 1 -+ Lie(Ad PGUa_I,1) is an isomorphism. Therefore V:= ad-l(LieH) is an L-form of Lie(PGUd_I,1) and AdA~oC GL(~). It follows that Q (Tr Ad Ao~) C L. [] 2.4. Proof of arithmeticity 2.4.1. Consider the subgroup A'C PGUa_I,I(R) ~ � PEC PGUd_I,I(R ) � PE, defined in 2.2.7. For a finite place u of F let G, be PGF, for u 4: v and PGLI(Dw), viewed as an algebraic group over F, ~- Kw, for u = v. In what follows it wiU be also convenient to introduce a formal symbol oo and to write Foo instead of R and Goo instead of PGU d_ 1,1 (the algebraic group over Fo0 ~ R). Let 1V[ be a finite set of non-archimedean primes of F, containing v for simplicity of notation. Set ~I := M woo and choose S e ~'(PG(A~:~)). For each subset M' of M, denote I[ G,(F,) by GM,. Denote also the projection of A'n (G~ � S) to u~M' by A s . Let A~ (resp. A s ) be the projection of A s to Go0(F~o) (resp. to GM). For u ~ M and 8 ~ A s denote the projection of 8 to G,(F,) by 8,. 96 YAKOV VARSHAVSKY Definition 9..4.9+. -- A lattice F C: G~m is called irreducible if for every proper non- empty subset M' C M the subgroup (F n GM, ) (F n G-~m_m. ) is of infinite index in P (compare [Ma, p. 133]). Definition 9,. 4.3. -- We say that a lattice P of G~M has property (QD') if the closure of FGo~(Foo) in C~M has finite index. Remark 9..4.4. -- Since the group PGU a_ 1, t is isotropic over R, it follows from [Ma, p. 290, Rem. (v)] that ff F has property (Q.D'), then it has property (Q.D) in the sense of Margulis (see [Ma, p. 289]). Proposition 9..4.5. -- The subgroup A s C G~M is a finitely generated cocompact irreducible lattice, which is of infinite index in Commo~(A s) and has property (Q.D'). Proof.- Observe that PGUa_I,I(R)x PE = ~ X PG(/k~ :M) and that A' is a cocompact lattice in G~M � PG(Av I:M) having injective projection to PGUa_~.I(R), hence to G~m. It follows from Lemma 1.1.9 that A s C ~ is a cocompact lattice, which is of infinite index in ComnM~(AS). By Proposition 2.2.10 the closure of G~o(Foo)A' in ~ � PG(AIv :'+) contains G~o(F~o) � (S~ n T1) � P(G~er(A~:*)). Hence the closure of Goo(Foo)A' in ~ contains Go~(F| � (SD~ n TI) � II P(GaCr(F,)). .~M--{v} In particular, A s has property (QD'). Let M' be a non-empty subset of M. Then A s n G~s, ={1}, because the projection of A' to PGUa_I,t(R) is injective. Suppose that A s is not irreducible, then [A s : (A s r3 ~-M')] < oo. Hence A" : (G+(F+) zX +) n Cm_,+,] < oo. p(Gae'(F.)) and Since G+o(F+o)A s D Go~(F~o) x (SD~ n TI) x I-[ u 9 {t,} (G+(F+)AS) n G,-~M_w C G~_M, , we get a contradiction. Since A s is a cocompact lattice in C_~M , it is finitely generated (see [Ma, Ch. IX, 3.1 (v)]). rq 2.4.6. Now we are going to use the results of Margulis (see [Ma]). By [Ma, Ch. VIII, Prop. 3.22], there exists a basis in Lie(PGUd_I.I(R)) such that all transformations in Ad Aoo are written in this basis as matrices with entries in Q(TrAdA~) = FC Foo 1 ~ R. Define a homomorphism 9:G~o -+GLa~_t rational over R by assigning to g 9 Goo the matrix of Ad g in the above basis. It follows that ~(A,~) C GLa*_~(F ). Let H be the Zariski closure of ~(A~o); then H is an algebraic group, defined over F and 9(A~o ) C H(F). Since A~o is Zariski dense in Goo and since the group Go -----PGUa_I,1 is adjoint, ~ induces an isomorphism PGUa_I, x -% Hire01. In particular, H is an F-form of PGUd_I, x. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 95 are elliptic for all i----- 1, ..., m 2. Therefore the above lemma implies that p(8) can be written as a linear combination of the p(8~)'s with coefficients in L. The set U t3 generates the group &, hence for every 8 ~ &, the linear transformation p(8) can be written as a polynomial in the p(8~)'s with coefficients in L. For any i,j, k ~{ 1, ..., m 2 } the elements 8~ 8j 8 k are elliptic, therefore by the lemma each p(8~ 8~)= p(Si)p(~j) can be written as a linear combination of the 9(Sk)'s with coefficients in L. Hence every polynomial in the p(8~)'s with coefficients in L, can be written as a linear combination of the p(8~)'s with coefficients in L. In particular, this is true for each p(8) with ~ ~ &. Hence Q(Tr p(A)) C L. [] Corollary 2.3.5. -- Suppose that a subgroup A C A~o is Zariski dense in PGU d_ 1,1 and that Ao~ C Commmua_l,~(R)(A). Then Q(Tr Ad A) ---- Q(Tr Ad Ao~) (= F). Proof. -- Set L:----- Q (Tr Ad A), then there exists an L-form V of Lie (PGU d_ 1,1 (R)) preserved by Ad A (see [Ma, Ch. VIII, Prop. 3.22]). Take any 8 ~ A| Then some sub- group of finite index A' of T~ satisfies 8A' 8 -1 C A, hence (Ad 8) (Ad A') (Ad 8)-1(V) ---- V. Since the subgroup A' is also Zariski dense in PGUa_~,I, Burnside's theorem implies that Ad A' generates End V as an L-vector space. Therefore (Ad 8) (End V) (Ad 8) -1 C End V. In other words, Ad(AdS)(EndV)= End V. Let H be the Zariski closure of Ad&C GL(V). Then H is an L-form of AdPGUa_I,1, hence LieHC EndV is an L-form of Lie(Ad PGUd_I,1). In particular, Lie H = End V n Lie(Ad PGUd_I,1) , therefore Ad(AdA~o)(LieH)= Lie H. Since PGUa_I, 1 is adjoint, the homomor- phism ad := Ad. : Lie PGUd_I, 1 -+ Lie(Ad PGUa_I,1) is an isomorphism. Therefore V:= ad-l(LieH) is an L-form of Lie(PGUd_I,1) and AdA~oC GL(~). It follows that Q (Tr Ad Ao~) C L. [] 2.4. Proof of arithmeticity 2.4.1. Consider the subgroup A'C PGUa_I,I(R) ~ � PEC PGUd_I,I(R ) � PE, defined in 2.2.7. For a finite place u of F let G, be PGF, for u 4: v and PGLI(Dw), viewed as an algebraic group over F, ~- Kw, for u = v. In what follows it wiU be also convenient to introduce a formal symbol oo and to write Foo instead of R and Goo instead of PGU d_ 1,1 (the algebraic group over Fo0 ~ R). Let 1V[ be a finite set of non-archimedean primes of F, containing v for simplicity of notation. Set ~I := M woo and choose S e ~'(PG(A~:~)). For each subset M' of M, denote I[ G,(F,) by GM,. Denote also the projection of A'n (G~ � S) to u~M' by A s . Let A~ (resp. A s ) be the projection of A s to Go0(F~o) (resp. to GM). For u ~ M and 8 ~ A s denote the projection of 8 to G,(F,) by 8,. 98 YAKOV VARSHAVSKY for each g ~ G.(F,). Hence for each 8 e A' we have Tr Ad(~ � -c) (8) ---- (Tr Ad(8~); ..., t%(Tr Ad(8,)), ...) (Fo~ 1; ..., FI~, ...). Recall that (9 � (A')C H(F), hence TrAd((~ � v)A')C F. On the other hand, Corollary 2.2.9 implies that Tr Ad(8) e F C F~I � A~ for each 8 e A' with elliptic 8~o. In particular, for such ~'s we have Tr Ad(~) = Tr Ad(~,) e F for each u. Since we showed in the proof of Proposition 2.3.2 that O_..(Tr Ad(8oo) I 8oo is elliptic) = F, we conclude from the above that the restriction of each r :F,--~ FII,~ to F is the identity. Since each r is continuous, the claim follows. [] 2.5.3. Next we will show that in the case d > 2 we have F' =- K. Indeed, if a prime u of F splits in K, then PG'(F~)- G,(F~)~-PD~ for some central simple algebra D. over F.. It follows that u splits in F'. By [La, Ch. VII, w 4, Thin. 9], F' ---- K. As we mentioned before, we may take F' = K also in the case d---- 2. Proposition 2.5.4. -- The map v induces a continuous isomorphism PE--% H(A~). Proof. -- Since PE "~ = PD~ ~ � � PE' and H(A~) =~H(F,) � HtA :;*~, F j, we need only to show that v* : 1-I,** G,(F,) --~ 1-I,,~ H(F,) induces a continuous isomorphism PE' -~ H(A~;*). First we claim that v ~ induces a continuous map from A'C PE' to H(A~:*). In fact, let a sequence { 8. }. C A~ converge to g e PE'. Then the sequence { 8, 8~-~_ 1 }, converges to 1. Therefore for each S e~'(PE') there exists N seN such that ~ 8:-~ 1 ~ A~' ('~ S (hence ~*(8. 8~-~1) ~ -c'(A~ C~ S)) for all n t> N s. Since -r'(A~ n S) is commensurable with H(0F) , it is contained in a compact subset of H(A~:*). Therefore the sequence {-r"(8. 8.+1) -1 }.C H(AIv :~) has a limit point. Let h be some limit point of { ~(8. 8~,-~1) )., and let { ~*(8-i 8-1-i+ 1) }, be a subsequence, converging to h. Then for each prime u ~ v of F we have 8 -1 .(lim = h, = lim %((8,, .,+1).) ---- (8,, ,,+1).) 1, ~oo because % is continuous. It follows that 1 is the only limit point of {-~(8, 8~,-~1)}., therefore the sequence { v"(8.) v ~t~- .+1) 1 }. converges to 1. Now by similar arguments we see that the sequence { v"(8,) }, converges to v"(g) e H(A~:"). Moreover, the same arguments also imply that if we show that -~"(PE') = H(A~;"), then the continuity of ~ and of (v") -1 will follow automatically. Observe that for each non-archimedean place u we have G(F,)der = Gde'(F,) (resp. G'(F.)a~ (G')aer(F.)) (see [PR, 1.3.4 and Thm. 6.5] in the anisotropic and [PR, Thin. 7.1 and 7.5] in the isotropic cases respectively). Therefore v" induces an isomorphism of derived groups rl.,, P(Gder(F.)) ~ I-[.,~ P((G')de'(F.)). p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 99 By Proposition 2.2.10, A~ D p(Gd~*(A~:*)) = PG(A~ :~) n IIu** p(Gd~'(F,)). Hence by the facts shown above, "r~(p(Gd~ C PG'(A~:') n II P((G')d~'(F.)) = P((G')d~'(A~;*)). In particular, "~([I~,~ p(Gd~'(d)s,))) = II~,~ v,(P(Ga~r(d)~.))) C P((G')de'(A~:~)). It follows that %(P(Ga~'(d~F.))C P((G')aor(o~.)) for almost all u 4= v. Since each % is algebraic, the subgroups %(P(Gd~ and P((G')d~r(d)F.)) are conjugate (hence equal) for almost all u ~e v. It follows that I:~(P(Gae*(A~;*))) = P((G')d~r(As Therefore to complete the proof it will suffice to show that PG(As :~) (resp. PG'(A~;")) is the normalizer of P(Ga~ ;*)) in the product II~, ~ PG(F~), and similarly for PG'. Since PG(As is the restricted topological product of the PG(F~)'s with respect to the PG(d~F~)'s , it remains to show that the normalizer of p(Gdor(d~F~)) in PG(F~) is PG(d)s. ) for almost all u. This can be done by direct calculation. [] We will use the same letter -~ to denote the isomorphism between PE and PG' (As ~). 2.5.5. Notice that a regular function t := Tra/det on GL a defines a function on PGLa. Moreover, an algebraic automorphism t~ of PGLa is inner if and only if it satisfies t o ~b = t. Therefore there is a unique choice of an algebra D' defining G' (see 2.5.1) such that the function t' := Trn/det on PG', defined by the natural embedding G'(F) ~ D', satisfies t' o q~ = t. Proposition 2.5.6. -- We have D' ~ D i"t, G' ~ G ~t and x is induced by some admissible isomorphism. Proof. -- By Corollary 2.2.9, for each 8 cA' with elliptic 8~o we have t(8| = t(8~) = t(8 s:") ~K. Since (9 � "r (8) E PG'(F) C PG'(F~o, x A), we have t((9 x "~) 8) s K. By our assumption, t(q)(8oo)) = t(8~o) for all 8 s A'. Hence for each 8 cA' with elfiptic 800 we have t(%(8,))---t(8,) for each non-archimedean prime u ofF. Recall that the algebraic isomorphism % : PG(F,) -+ PG'(F,) for u v is induced either by an F, dinear isomorphism D| F,-%D'| F~ or by an F,-linear iso- morphism D| s F. --% (D')~174 s Fu, composed with an inverse map (g ~_,g-a). In the first case we have t(%(g.))= t(g,) for all g~, ~ G,(F,,), and in the second one t(%(g~)) = t(g~ ~) for all g~ ~ G~(F~). To exclude the second possibility we need to show the existence of a 8 ~ A' with elliptic 8~ such that t(~o0) + t(8;~). Since the condition t(g) = t(g -~) is Zariski closed and non-trivial and since the closure of all elliptic elements of A~0 e PGUa_I,I(R) ~ contains an open non-empty set, we are done. It follows that D' is locally isomorphic to D ~nt at every non-archimedean place of K, except possibly at w and ~, and that the map -d : PG(A~ ;~) --% PG'(A~ ~") is induced 100 YAKOV VARSHAVSKY by some admissible isomorphism. To prove the statement for the v-component we copy the above proof replacing PG(F,) by PGLx(D~) and D | F, by D~ 9 ~o,, Since D' and D ~ are locally isomorphic at all places, they are isomorphic. We showed before that PG'(Foo~)--PGUa_I,I(R) and that for each i=2, ...,g the group PG'(F~ol) is compact and, therefore, is isomorphic to PGUa(R ). Propo- sition 2.1.7 b) then implies that G' - G t~. [3 From now on we identify G' with G tnt. 2.6. Completion of the proof Our next task is to prove the following Proposition 2.6.1. -- We have (~ � "~) (A') = pG~t(F)+. Proof. -- First observe that (q~ X v) (A')~ = q~(Ao~) C ~(PGUa_I,I(R) ~ = PG'nt(ro,,) ~ therefore (q~ � v) (A') C pG'nt(F)+ and (q~ � -~) (A") C PG~t(F)+. Since the projection of PG~t(F) to PGmt(F~ol) � PG~t(A~ ;') is injective, it remains to show that (2.1) [PGt~t(F) : PG~nt(F)+] = [PGmt(F) : (~0 � -~) (A")]. We are going to use of Kottwitz' results described in 1.8. Recall that PG ~t is an inner form of PG. Let o~eo and co~i~t be non-zero invariant differential forms of top degree on PG and PG ~nt respectively, connected with one another by some inner twist as in 1.8.2. They define invariant measures [~Po [ and I o~mi~t [ on PG(F~) and PG~t(Fu) for every completion F. of F and product measures on PG(Ar) and PG~"t(AF) respectively (see [We2, Ch. 2]). It follows from Weil's conjecture on Tama- gawa numbers and from Ono's result (see Ono's appendix to [We2]) that (9,.9,) [ I (PG~t(AF)/PGt~t(F)) = I c~ [ (PG(AF)/PG(F)). Lemma 2.6.2. -- Let A and B be locally compact groups, let S be a compact and open subgroup of A and let P be a lattice in A � B with injective projection to B. Then for every right invariant measures ~a on A and ~ on B we have (~A � ~B) ([A � B]/r) = ~.(S). ~([(S\A) � B]/F). Proof. -- Let F a be the projection of F to A. Choose representatives { a~ }i e i of the double classes S\A/F a. For each i e I let F~ be the projection of the subgroup (ai-1 Sa, � B) c~ F to B. Then P~ is a lattice in B, therefore there exists a measurable subset U, of B such that B is the disjoint union II U, 7. Since P has an injective YE Fi projection to B, we have A � B= II H (Sa � U~)y. Then ([Za � ([A � B]/r) = = = AS). Z � V,) = � B]/r). [] 0')pGint p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 101 By the lemma, for each S e o~'(PG(As the left hand side of (2.2) is equal to (2.3) I/ I o~mt~t [ (pG'nt(Fo~)) 9 [ O~mmt [ (PGt~t(F,)). [ OJmmt I ('r'(S)) 9 I I x and the right hand side of (2.2) is equal to (2.4) II I o~m I (PG(FOO,)). I (ol, o I(S). [ o~m [ (S\PG(A~)/PG(F)). i=1 By definition, [ o~mt, t ] (PG~'t(F~,)) = [ ~ [ (PG(Fo~i)) for each i = 2, ..., d and [ orbit I (e(S)) = ] to m I (S) for each S ~ ~'(PG(/k~:v)). Since the expressions of (2.3) and (2.4) are equal, Proposition 1.8.5 and Remark 1.8.6 imply that (~'.5) ~pGIni; (e(S)\[PG~t(F~) � pG'nt(As Fool. = d. ~m~o(S\PG(As (" q- " was added to multiply the left hand side by 2 when d = 2). If S is sufficiently small, then for each a e PG(/kr s;*) the group a -~ San PG(F) is torsion-free by Proposition 1.1.10. Let Y~-ts~ be the projective variety over K~, such that Y~ls~ ~ = ( a-1 Sa c~ PG(F) )~\ fP x~" By Kurihara's result (see [Ku, Thm. 2.2.8]) ca_l(Ty~_l~,) = )~lg(a -1 Sa N PG(F)).ca_I(Tpa-x), where ca_l(Ty~_ls~ ) (resp. ca_l(Tre-1)) is the (d- 1)-st Chern class of the tangent bundle of Y~-~s~ (resp. pa-1). Notice that ca_l(Tra-~ ) = d, hence ca_l(Tya._lsa) = d. V%o,~((a -~ Sa c~ PG(F)),\PG(F,)). (Y )an~ A" \B a-l, we have Since a-lSa, C ~ a -1 8a ca-~ (TY~-I s,) = Ca -' (Tc~,- ~ s~' c) = Z~,(A~_x " s.kB a-1 ) (see for example [BT, Prop 9 11.24 and (20.10.6)]). The last expression is equal to Z'A" " 1~(a-ls~t) = ~pGUd_ 1 t(A'a'_Zsa\PC-.Ua_I,X(R)). This shows that for each a ~ PG(A~;') we have a. -1 sa n PG(F))v\PG(L)) = so\PGUa_ 1, l(g)). Summing this equality for a running over a set of representatives of double classes in S\PG(A~:~)/PG(F), we obtain that a. ~m.~(S\PG(A~)/PG(F)) = ~xmee_1, I(S\[PGUa- 1,1(11) x PG(A~: ~)]/A"). Since the right hand side of the last expression is equal to ~l:,Gint ('rv(S)\[PG~t(F~ol) � PG't(As � "r *) (A")), FaO 1 we conclude (2.1) from (2.5). [] 102 YAKOV VARSHAVSKY 2.6.3. By Proposition 2.5.6 there exists an admissible isomorphism @ : E --% E ~"t, inducing the isomorphism x : PE ~ PE int. Choose 8 e A with elliptic ~| z A~o and Tr Ad(8o~) # -- 1. Choose its representative~ ~ GUd_~,x(R) ~ � E as in Corollary 2.2.9. Then (TRY')(Tr~'-~) zK � Let ~' be the projection of~' to PGUd_I,~(R)~ � E. Set := (V X @) (~'), and let ~r be its projection to E. By the definition of admissible maps, Tr(~) ~ K x. Let ~ be the image of 8 in A', then ~ := (v � 9) (]) belongs to PG~t(F)+. Let y' ~ G~nt(F)+ be some representative , , = ~ -1 K � of ~, then ~1 ~E ~ Z(Eint) 9 Therefore ~1 "~E (Tryr) (Tr y~,) ~ = Z(Glnt(F)). Thus ~x and y~ have equal projections to PGUa_~,~(R)~215 (Ei=VEi"~), hence x r The condition { 8o~ is elliptic and Tr Ad(8| ~ -- 1 } is open and non-empty, therefore the above 8's generate the whole group A ~_ A o (see [Ma, Ch. IX, Lem. 3.3]). It follows that (V x ~)(A)C F ~"~. Since the projection z~:P int -->PG~t(F)+ is an isomorphism, Proposition 2.6.1 implies that (9 x ~)(A)= F i~. This completes the proof of Theorem 2.2.5 and of the First Main Theorem. 3. THE THEOREM ON THE p-ADIC UNIFORMIZATION The First Main Theorem implies that for some admissible isomorphism r : E --~ E i"t there exists a @-equivariant C-rational isomorphism f~:X 0 -~ ~l~t. Therefore for some C/K~-descent X i"t of the (E i"t, C)-scheme .~t, fr induces a K.-rational iso- morphism X--% X i"t. To describe X lnt we need some preparations, following [Dell (see also [Mill). 3.1. Techn;cal prel;minaries In this subsection we recall basic notions related to Shimura varieties and give their explicit description in the cases we are interested in. 3.1.1. First we realize ,~f,t as a Shimura variety. Set H l"t := RF/QG ~t. Then H ~t is a reductive group over Q such that Hi"t(A I) = Glnt(A~) and Hi"t(R) = rI;=l Gint(Fo~i). Put S := Re/~ G,~ and let h be a homomorphism S---> H~ t such that for each z E C � ~ S(R) we have h(z) = (diag(1, 1, z/~)-~; Id; ;Ia) ~ I] G~t/F ~=1 using the identification of Gint(Fooi) with GUa_I,X(R ) chosen in 2.1.6. Then the conjugacy class M ~nt of h in Hint(R) is isomorphic to B a- 1 if d > 2 and to C -- R if d ---- 2. Then the pair (H l"t, M ~t) satisfies Deligne's axioms (see [Del, 1.5 and 2.1] or [Mil, II, 2.1]), and the Shimura variety Mc(H i"t, Mint), corresponding to (H ~t, Mint), is isomorphic to ,~t. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 3.1.2. For each pair (Hint, M mr) as above there is a number field E(H lnt, M) 'nt C C, called the reflex field of (H int, Mint), which is defined as follows (compare [Del, 1.2, 1.3 and 3.7]). The group Homc(Sc, (G~)c) is a free abelian group of rank 2 with generators z and f such that ff i : S(R) ~ S(C) is the natural inclusion, then for each weC � -_~S(C) we have zoi(w) = z and ~-oi(w) =~. Let r:(Gm) c~S c be the algebraic homomorphism such that (z ~ fq) o r(x) = x ~. Then E(H mr, M mr) is the field , h of definition of the conjugacy class of the composition map r": (G,,)c ~ S c --~ Proposition 3.1.3.- We have E(H mr, M mr) = K, the latter being viewed as a subfield of C through the embedding 0% chosen in 2.1.6. Proof. -- Note that Hint(C) is naturally embedded into GLa(C) 2g so that each factor corresponds to an embedding of K into C. Supposing that the first and the second factors corresponds to our fixed embedding and to its complex conjugate respectively we have r"(z) = (diag(1, ..., 1, z-a); diag(1, ..., 1, z); Id; ...; Id) for each z e C � Therefore the reflex field E(H int, M mr) contains K C C. On the other hand, the Skolem-Noether theorem implies that for each a e Autm(C ) the homo- morphism a(r") is conjugate to r". This implies the assertion. [] 3.1.4. Let T C H ~t be a maximal torus of H ~nt, defined over Q, such that some conjugate h'e IV[ ~t of h in H~t(R) factors through Tx~. Then we have a natural embedding i T : Me(T, h') ,-+ Mc(H int, MEt), where Mc(T , h') is the Shimura variety corresponding to (T, h'). Since T is commutative, the reflex field F~ := E(T, h') of r h ~ T! p (T, h') is the field of definition of the morphism r": (G,,)c-+ Sr T c. Hence defines a morphism of algebraic groups over Q * ~ R~,tQ(T ) T. r,:E T := R~TtQ(G,,) e~lQ'"" ~/Q> Notice that E T D E(H ~t, M~"t). Let 0~T be the Artin isomorphism of global class field theory sending the uniformizer to the arithmetic Frobenius automorphism. Let ~ : Gal(E~b/F_@ ~ T(AI)/T(Q) be the composition map 0-~ * 0 * * Gal(E~b/ET) ~> ET(R ) \ET(A)/ET(Q) r' proj > T(R)~ --~ T(M)/T(Q). For each E' D E(H lnt, M lnt) we denote the composition map Oal(E~ b.E'/E') ~, GaI(E~b/ET) x~> T(A')/T(Q) by ~,~,. 104 YAKOV VARSHAVSKY Lemma 3.1.5. -- Each maximal toms T of H mr, defined over Q,, is equal to the inter- section of H ~"t with Rz/Q G,, for a unique maximal commutative subfield L of D tn~. (In such a situation we will call T an L-toms.) In this case, ~.t induces a nontrivial automorphism of L, and the subgroup T(Q.) c L � = RL/K G,,(K) is Zariski dense in RLm G,,. Proof. ~ Let L be the subalgebra of D I"t spanned over K by T(Q) C Hint(Q) C D t"t, then L is a commutative subfield and T(Q,)C H~t(Q)t~ RL/Q G,,(Q,). Since T is connected and O is infinite and perfect, the subgroup T(Q,) is Zariski dense in T (see [Bo, Ch. V, Cor. 13.3]). It follows that T C Hmt n Rr4Q G,,. Since T is maximal, L have to be maximal and T = H ~t n R~/Q G,~. For each g ~T(Q) we have ~t(g) ~g-aF� C T(Q.), so that ~t~t(L) = L. To prove the last assertion we observe that there exists a maximal F-rational subtorus T' of G ~t such that T = RF/Q(T' ). Then the subgroup T(Q)= T'(F) is Zariski dense in T~ - R~m G,, � (G,,)K. Hence its projection to RT,m G,~ is also Zariski dense. [] 3.1.6. Now we want to calculate the reflex field E T. Observe that L | C C D ~t | C ~ Mata(C) 2g. Possibly after a conjugation we may assume that L | C is the subalgebra of diagonal matrices of Mata(C) 2g. Then each diagonal entry of each of the 2g copies of Mata(C ) corresponds to an embedding of L into C, and the map r": (G,~)c -+ T c is as follows: r"(z) ----- (diag(1, ..., l, z-a); diag(1, ..., 1, z); Ia; ... ; Ia). Let q be the embedding L ~ C, corresponding to the right low entry of the first matrix, then the right low entry of the second matrix corresponds to the embedding ~1 := tz o 0t ~at. Now we embed L into C via q. Proposition 3.1.7. -- We have E T = Ls C, and r': E~--~ T is characterized by r'(l) = l-a-~'~t(l)for each l eEl(Q.)C L � Proof. -- As was noted before, E T D E(H t"t, itnt). Hence by Proposition 3.1.3, ETDK. By the definition, ,(r"(z))= r"(,(z)) for each , e AutE,(C), hence the group Auto(C) must stabilize q, so that E T D L. Finally, it is clear that r" is defined over LC C. For each l e E T = L we have r'(t) = N~./Q(diag(1, ..., 1, l-~); diag(1, ..., 1, l); Ia; ... ; I~) = l-l-~t(l). [] Set L, := L | K~ C Din t := D ln~ | K~. Since Di~ ~ is a division algebra, L,, is a field extension of Kw of degree d, and L~ = L. K~. Lemma 3.1.8. ~ The following relations hold: a) Er.K ~ ---- L~; b) E~b.K,o = (L,~) ~. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VAI~IETIES Proof. -- a) was proved above. b) The group GaI(E~b-K,~/Ex.K,o) is abelian, hence L,oCE~b.K~CL~. By the class field theory, the composition of the canonical projections Gal((L,o)~b/Lw) -+ Gal(E~ b. Kw/Ex-Kw) ~ Gal(E~b/E~) ~ Gal(L~b]L) is injective (use, for example, [CF, Ch. VII, Prop. 6.2]). Therefore we have the required equality. [] Proposition 3.1.9. -- For each l ~ L~ C (D~ t) � the element ~t � Gi.t(A~,.) _ H~t(Af) (l-1,1,1) ~(D,~) � r~ x belongs to T(AQ, and its equivalence class in T(AI)/T(Q) # Xr, Kw(0Lw(l)). Proof. -- The statement follows immediately from the explicit formulas of Pro- position 3.1.7 using the connection between local and global Artin maps. [] Definition 3.1.10. -- A point x ~Mc(H t"~, Mmt)(C) is called (T-)special if x ~ iT(Mc(T, h) (C)). Remark 3.1.11. -- The group T(A I) acts naturally on the set of T-special points and the group T(Q) acts on it trivially. Hence by continuity the closure T(Q) C T(A f) acts trivially on the set of T-special points, therefore the action of T(AQ/T(Q) on it is well-defined. Definition 3.1.12. -- Let K' D E(H ~n~, Mint) be a subfield of C. A C/K'-descent of the (Hint(Af),C)-scheme Mc(H t~, M ~t) is called weakly-canonical if for each "[~ ab, T~t maximal torus T ~-+ H ~t as above, each T-special point x is defined over ~T -~, and for each ~ e Gal(E~rb.K'/Ez.K ') we have ,(x)= ~,K,(~) (x). Remark 3.1.13. -- Our definition of the canonical model coincides with that of [Mi3], which differs from those of [De2] and [Mil] (see the discussion in [Mi3, 1.10]). The seeming difference (by sign) between our reciprocity map and that of [Mi3] is due to the fact that we consider left action of the adelic group whereas Milne considers right action. Proposition 3.1.14. -- For each field K' satisfying E(H ~t, M l~t) C K' C C there exists a unique (up to an isomorphism) weakly-canonical C/K'-descent of the (H~t(Af), C)-scheme Mc(H l~t, 1V[in~). Proof. -- Uniqueness is proved in [Del, 5.4], for the existence see [Del, 6.4] or [Mil, II, Thm. 5.5]. [] By Proposition 3.1.3, we have E(H ~t, M ~t) = K C K~C C (in our conven- tion 2.2.2). Hence by Proposition 3.1.14, the (E mr, C)-scheme ,X~* has a unique weakly-canonical C/Kw-descent X mr. 14 106 YAKOV VARSHAVSKY 3.2. Theorem on the F-adic un;form;zation Now we are ready to formulate our Second Main Theorem 3.2.1. -- For each admissible isomorphism ~ : E --% E int there exists a ~-equivariant isomorphism fo from the (E, K~)-scheme X to the (E i"t, K~)-scheme X int. Corollary 3.2.2. -- After the identification of E with E ~t by means of rb we have for each S e o~-(E) of the form T, x S', where S' e ~'(E'), an isomorphism of K,~-analytic spaces %: (Xs~t)~n--~GLa(K~)\(Z~, " X (S'\G(A~)/G(F))). These isomorphisms commute with the natural projections for T D S and with the action of E -~ E ~nt. Proof (of the Second Main Theorem) : Step 1. -- We want to prove that for (I) and fo as in the First Main Theorem, the C/K,-descent of IK t"t corresponding to X is weakly-canonical. For tiffs we have to show that for each maximal torus T ~ H int as in 3.1.4 and each T-special point x ----f~(y) e Mc(H i"t, M t') (C) : ~l~t we have: a) y ~ X(C~) is defined over E~b-K~; b) a(y) = q)-l(Xm,,,(,))(y) for each a e Gal(ESb.KJET.K,). By Proposition 3.1.9, Lemma 3.1.8 and the definition of admissible map, it will suffice to show that when L~ is embedded into D~ by means of the isomorphism Dtnt __% ~ from Definition 2 2.1 we have (3.1) i) every pointy ~X(C~), fixed by (I)-I(T(O_..)), is rational over (L,)ab; X "~X "~X q~'SX El. ii) 0L.(1 )(y) =l-l(y) for each leL. CD. =D. � X Let (x, a) e Z a K~ � E' be a representative ofy e X(C~). Then (~(x), a) is a repre- sentative of a(y) for each (not necessarily continuous) ~ e Autr~(C~). Recall that for each embedding L.'--~ Mata(K~) there exists an (L~ � L~)-equivariant L.-rational embedding ~ : E 1~,~ ~ Z~. Proposition 3.2.3. -- There exists an embedding L~ ~ Mata(K~) such that the image of the corresponding T : Z 1~ "-+ Z a~ contains x. Proof. -- Let x' e ft a be the projection of x. Then Kw y' := [(x', a)] m (b~\X) a" - (O~, � (E')d'~)/G(F) Z(G(F)) (use Proposition 1.5.3) is the projection of y. Since q~-I(T(Q)) stabilizes y, it also stabilizes y', therefore the projection of a -1 q)-l(T(O_..)) a to E' is contained in G(F) Z(G(F))C E'. In other words, for each teT(Q)C D ~nt we have pr~,(a -1 q)-l(t) a) = g. z for some g e G(F) and some z e Z(G(F)). p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES Since 9 is induced by some algebra isomorphism D(A~ ;~) -% Dint(A~;'), we have Trt= Tr(a-l(I)-l(t)a)= (Trg) z. Therefore for t's with non-zero trace we get z ----- (Tr t)(Trg) -1 ~ K � C (AIK:~'~) � This means that pr~,(a -1 (I)-l(t) a) E G(F) C E'. As the set of all t's in T(Q) C D int with Tr t 4= 0 generates L C D ~"t as an algebra, the map l ~-~ pr~,(a -~ (I)-X(l) a) defines embeddings L ,-+ D and L~ ~ D | K~ ~ Mata(K~). This shows that pr~,(a -~ ag-~(T(Q)) a) C G(F) C E', so that a 'O-'(T(Q))aCD, � I'~CD. X =E. Hence (I)-I(T(Q)) preserves p~(Ea~)C (X%) ~ (in the notation of Corollary 1.5.4). Moreover, it follows from the definition of the embeddings L~ '-+ Dw and L~ ~ Mata(K~) that for each teT(Q) C L � C L~ the image of a-~a~-~(t) as D$ � F~ under the canonical map D~ ~ � � Ft.--% D~ ~ � � F GC D~ ~ � � GL~(K~) is equal to (t, t). Since y is a fixed point q)-a(T(Q)), we conclude from the above that (t, t) (x) ----- x for every t e T(Q). Noticing that T(Q) is Zariski dense in R~ m G,, by Lemma 3.1.5 and that RLm G,, | K~ -~ RL~/x ~ G,,, Lemma 1.4.5 completes the proof. [] Since 2" is (L~ � L~)-equivariant and L~-rational, the proposition together with Lemma 1.4.3 imply (3.1). In other words, we have proved that for some admissible isomorphism r : E -% E ~nt there exists a q~-equivariant K~-linear isomorphism fr :X-%X ~"t. Step 2. -- Let tF be another admissible isomorphism E--% E ~t. The definition of admissibility together with the theorem of Skolem-Noether imply that tF o r ~ : E ~"t --% E ~"t is an inner automorphism, so that there exists gv e Ein* such that tF o (I)-X(g) ----_ g~,ggyr ~ for all g ~ E ~t. Take f,r : X ~ Then for each g ~ E we have fv o g = g~ ofo o g = g,v o O(g) ofo = (g,r o O(g) o g~v ~) o (g,v ofo) = (XF o ~-a) (O(g)) of~ = XF(g) of~,, that is f,r is a ~-equivariant isomorphism. This completes the proof of the Second Main Theorem. [] 4. p-ADIC UNIFORMIZATION OF AUTOMORPHIC VECTOR BUNDLES In the previous section we proved that the Shimura varieties corresponding to the pairs (H ~t, M ~t) have p-adic uniformization. Our next task is to show the analogous result for automorphic vector bundles. 4.1. Equivariant vector bundles 4.1.1. Set H :-~ Rr/QG. Then for some algebraic group H over K. we have natural isomorphisms H~w ~ GLg � H, PHKw ----- PGL d � PH and PH~tKw ~ PGLI(D~t) � PH, where the first factors correspond to the natural embed- 108 YAKOV VARSHAVSKY ding F ~ K~. Using these decompositions let PHKw acts on I~K~ ~ through the natural action of the frst factor and the trivial action of the second one, and let H(K.), PH(Kw) and PHt"t(R) ~ ~ PGU a ~ I(R) ~ � PGUa(R) ~-~ act similarly on Z a on ~a and _ , Kw ' K w on g a-~ respectively. Let ~ be the natural embedding Ba-~,-+ (l~c-~) ~, and let ~w (resp. ~,,, ~) be the composition of the natural projection E~ -+ D~ (resp. Ea'~x~ -+ D~)a and the natural embedding f~a ~ (i~x~)~. Let 7: e K~ be a uniformizer, let H be an element of GLa(K.) satisfying ~a = 7:, and let H' e PGLa(K~) be the projection of ~I. Set H := (II', l) e PGLa(K,o ) � PH(K,o ) _-- PH(Kw). Let K~ ~ be the unramified field extension of K. of degree & Since the Brauer invariant of _~D ~nt is 1/d, the group PH is isomorphic to the quotient of PHK~ | K~ a~ by the (d) equivalence relation Fr(x) ,~ II -a xII, where Fr e Gal(K~/K.) is the Frobenius auto- morphism. For each scheme Y over K~ on which PHx~ acts K~-rationally define a twist yt,, := (Fr(x) ~ II -a x)\Y | __~K ~d~ . Then Y | K~' -~ ytw | K~a~ and the natural action of PH t~t x,, on it is K~-rational. Let W be a PHx -equivariant vector bundle on P~I, that is a vector bundle on P~ 1 equipped with an action of the group PHKw , lifting its action on I~K~ 1 . Then (I~ -l~tw and ~((W~W) ~) (resp. (Wt~,p tw) is a PH~tw-equivariant vector bundle on ~ K~ J , [3:(wa"), ~o,,~(wa~)) is a PHmt(R) ~ (resp. H(K,o)-)equivariant analytic vector bundle on B a- 1 (resp. Z ~, a :Cx~). a,, For each S e o~-(E) (resp. S e o~'(Emt)) consider a double quotient ~' * tw an V s := S\[~;(W =) � E']/F (resp. V~" := S\[~R(Wc ) � Eat]/Fmt). Proposition 4.1.2.- For each S e~'(E) (resp. S e o~'(Emt)) Vs (resp. V~ nt) has a natural structure of an affine scheme V s over X s (resp. ~mt ~mt~ Moreover, V s (resp. ~s t) is --s over ~s J" a vector bundle on K s (resp. ~nt) if S is sufficiently small. Proof. -- We give the proof in thep-adic case. The complex case is similar, but easier. I) First we take S of the form T, x S' with sufficiently small S' e #'(E'). Then Vs is a finite disjoint union of quotients of the form Fas,a_x\~*,(W ~n) with some a ~ E'. Since the projection za~2-+O~w factors through each r,s,,_l,0\z~2 (in the notation of the proof of Proposition 1.5.2), the quotient F~s,~_L0\~:,,(W ~n) is d,~, naturally an analytic vector bundle on F.s,~ Now (as in the proof of Proposi- tion 1.5.2) the quotient vector bundle Pro a_xk(ros,._x,0k ;,.(w o)) ros,o_lk ;,.(w on P~s' a- I\Z~ is obtained by gluing. For the algebraization we use Corollary 1.2.3 a). II) For each T e o~(E) there exists a normal subgroup of the form S = T, � S', where S' e o~-(E') is sufficiently small. Then by the same considerations as in Propo- sition 1.3.7, V T can be defined as (T/S)\V s (using Corollary 1.2.3 a)). ~"tKw p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 109 III) Suppose that Vst and Vs, , constructed in I) and II), are vector bundles on Xs~ and Xs, respectively for S~ff S, in o~-(E). Then the natural morphism f: Vs~-+ Vs~ � Xs~ of vector bundles on Xs~ induces an isomorphism on each fiber. Hence it is an isomorphism. IV) Suppose that T C S in oq~(E) and that V s is a vector bundle on X s. Choose a normal subgroup So e o~-(E) of T such that Vs0 is a vector bundle on Xs~ Then Vw = (T/So)\Vso ~ (T/So)\Vs � Xso ~ Vs � ((T/So)\Xs0) -~ Vs � X~, so V~ is a vector bundle on X~. [] 4.1.8. Choose S eo~(E) (resp. S Eo~-(E~"t)) sufficiently small. Then V s (resp. ~i.t~ is a vector bundle on X s (resp. ~l.t X s) Thus V:=V s � (resp. --S J ~t := ~,t � ~int) is a vector bundle on X (resp. iKmt). By Step III) of the proof, V (resp. ~t) does not depend on S. Each g e E (resp. g e E at) defines an iso- morphism V~ --% V~o_~ (resp. (~t)an _~ "~an'~/. Therefore by Corollary 1.2.3 a), g defines an isomorphism V s ~V0sg_x (resp. -si'Ji"t ~ The product of this isomorphism and the action of g on X (resp. ~t~t) gives us an isomorphism g : V = V s � X --% V0so_~ � X = V (resp. g : V l"t __~ ~t). Thus we have cons- tructed an algebraic action of E (resp. of E ~nt) on V (resp. ~mt), satisfying S\V -~ V s for all S e~'(E) (resp. S\ ~t =~--si'~mt for all S ~o~'(E~t)). Moreover, V =limV s and ~t = lira ~t. s By [Mil], there exists a unique canonical model V ~nt of ~mt over K~ (the definition of the canonical model will be explained in the last paragraph of the proof of Proposition 4.3.1) such that V ~t is an E~nt-equivariant vector bundle on X i't. Our main task is to prove the following Third Main Theorem 4.1.4. -- For any admissible isomorphism qb:E-% E t~t, each isomorphism f~ from the First or the Second Main Theorem can be lifted to a ~-equivariant iso- morphism f,D, v : V -% V int. We will prove this theorem, using standard principal bundles (----- torsors) (see [Mil, Ch. III, w 3]). 4.2. Equivariant torsors 4.2.1. For each S e~(E) (resp. S e~'(Ei"t)) consider the double quotient -s :-- S\[ Ba-~ � (PHc "t)~" � n!nt]/rmt) . Ps :-- S\[Zaw � (PHK~) " � E']/F (resp. V,.t Proposition 4.2.2. -- For each S E o~-(E) (resp. S e o~'(Ei"t)) Ps (resp. P~s nt) has a natural structure of an affine scheme Ps over X s (resp. ~i,t_s over -~s~"t~/" Moreover, Ps is a PHKw- torsor over X s (resp. ~i,t_s is a PHio"t-torsor over Y2s mt if S is sufficiently small). -+~--osg-~J'~i~t kVoSo--lJ[~lnt 110 YAKOV VARSHAVSKY The proof is almost identical to that of Proposition 4.1.2 (using Proposition 1.9.13 and Lemma 1.9.3 instead of Corollary 1.2.3 a) and arguments of step III) respec- tively). [] 4.2.3. Arguing as in 4.1.3 and using Corollary 1.9.14 we obtain an E-equi- variant PHx-torsor P = < limP s over X (resp. an Eint-equivariant PH~"t-torsor ~t =l<im~nt over ~L,t). By [Mil, III, Thm. 4.3], there exists a unique canonical model pint of ~i~t over K~ (the definition will be explained in Corollary 4.7.2) such that pi.t is an E~n~-equivariant PH~-torsor over X ~nt. Let ~:P-+X and ~n~: p~n~ _+X~n~ be the natural projections. Denote also the natural projection from the PH~-torsor p~w to X by ~tw. Fourth Main Theorem 4.2.4. -- For any admissible isomorphism 9 : E -~ E ~nt, each isomorphism fr from the First or the Second Main Theorems can be lifted to a ~-equivariant isomorphism fr p : p~w _~ p~,t of PH~t-torsors. 4.3. Connection between the Main Theorems Proposition 4.3.1. -- The Fourth Main Theorem implies the third one. Proof. ~ Consider the pro-analytic maps ~'O : [Z a~, x (PHK~) a" � (E,)dL~]/F .__> ~/Pa-t~"- x~ , and (~,)~,, : [Ba-1 � (pH~.t)~n � (E~.t)d~]/F~nt _+ (p~--l)an given by ~'(x, g, e) = g~(x) and (~')~nt(x, g, e) = g~c(x). Then 7' (resp. (~,)~nt) is (PHx~) ~"- (resp. (PH~nt) a"-) equivariant and commutes with the action of E (resp. E~"t). Hence it defines an equivariant analytic map ~:pan_+ ~-K~ J tl~- a~. (resp. ~i,t: (pi,t)an .__> (p~-,)a,). Proposition 4.3.2. -- There exists a unique algebraic morphism o:P---> ~K: 1 (resp. '~int : ~int ._>. p~-l) such that p~" ~-~ (resp. (~mt)~n ~ ~t). Proof. -- We prove the statement for p (in the second case the proof is exactly the same). We have to show that the graph Gr(~)C P~ � t~- ~-K~ l~n J corresponds to ~ an ~pa-l~an be the morphism an algebraic subscheme. For each S ~ ~'(E) let ps:Ps ~ ~_Kw J . ~ an ~1~ -l~an it remains to show induced by 7. Since Gr(~) = <hmGr(ps) C (timPs) x ~ K~ J , S S an d -- 1 an that the graph Gr(ps)C Ps � (PKw) corresponds to a unique algebraic subvariety for each S sufficiently small. Take S so small that X s is smooth, then by Lemma I. 9.5 b) there exists a quotient d--1 Qs :-PHK~\(Ps � Px~ ) by the diagonal action of PHi. Moreover, Qs is a Pa-l-bundle on Xs, hence it is projective over K~. Let ~:Ps � ~1 _+ Q.s be the p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 111 natural projection. Since Os is (PHK~)~"-equivariant, Gr(os) is invariant under the diagonal action of (PHKw) ~". Therefore the quotient Q:= (PH~%)~\Gr(~s) is a closed analytic subspace of Q.~, so that it is algebraic (see Corollary 1.2.2). It follows that its inverse image 0~-~(Q.) = Gr(ps) is also algebraic. The uniqueness is clear. [] Claim 4.3.3. -- The map ~int is the only (PH~'~t) ~" X Ei"~-equivariant analytic map from (~ln~)a, to (Pao-~)~". Proof. -- Let p' : (P~)~" --> (1~ - ~)~ be any such map. Composing it with the natural (PHi"t) ~ � E*"t-equivariant projection [B a-~ x (PHi't) ~ x (g'~176 (P'~176 and, identifying a complex analytic space with the set of its C-rational points, we obtain a PHI"t(C) � Emt-equivariant analytic map ~": [B ~-' x pH~t(C) x (~i"<)~"1/r"'-~ (~-')*. Let Po be the restriction of p to B a-1 -% B d-~ � { 1 } � { 1 }. Then p"(x, g, e) = gpo(x) for all x e B e- x, g e (PH~nt) ~ and e ~ E l~t. Therefore ypo(X) = po(yX) for all y E F ~t and x e B a-1. Since the subgroup F i~t is dense in PGUa_I,I(R) ~ we obtain by conti- nuity that ypo(x)= po(yx) for all 3" e PGUa-~,~(R) ~ and x eB ~-~. In particular, for the origin 0 e B ~-1 we get Stab~,ov~_~,#~)o(0 ) C Stab~g~_x,x~)o(po(0)). The subgroup Stabmv~_,,~(mo(0 ) stabilizes precisely one point (0 : ... : 0 : 1) e pa-x(C) if d> 2 and two points (0: 1) and (1 9 0) in pI(C) if d = 2. The case po(0) = (1 : 0) is impossible, because identifying PX(C) with C = C ~ oo by (x :y) ~-~ x/y we would get in this case 9o(z) = 1/~ for all z e B x, contradicting the analyticity of ~0- We conclude that ~o(0) = (0 : ... :0: 1). Hence ~o = ~ and ~' = ~r. [] 4.3.4. Next we show that the map '~lnt : p~t ___> tpa-l~tw ~AKw /c is Kw-rational. Recall that the map ~-Int is PH~Lequivariant and that the actions of the group PH ~r t~ w on both --1 tw pint and (~w) are Kw-rational. Therefore for each a e Aut(C/K~) the analytic map a(~tnt) ~" is (PHi"t) a" � Et"t-equivariant, hence it coincides with (~t)a. = ~t. By the uniqueness of the algebraic structure, a(~ 'l"t) = ~,~.t. It follows that ~t defines a PH Kw l~t � E~t'eq uivariant map pl~t : pt~t __~ tpa-~w~ ~-xw ; 9 K~ X E-equivariant map ~tw : ptw __> ~ ~w / " Notice also that ~ defines a PH ~t rpa-lXt, Suppose that the Fourth Main Theorem holds, then Lemma 4.3.5. ~ We have pi~t ofo, p = pt~. Proof. -- By the claim, ~mt _ (p~.t)~. is equal to (p~w o (fo, P)o-1)~"- From the uniqueness of algebraic structures we conclude that p~.t= p~o (fr Now we descent to K~ as in 4.3.4. [] 112 YAKOV VARSHAVSKY It follows from the definitions that p*(W) ~ r~*(V) (hence p*(W) tw-~ (rctw)*(V)) and (~t)*(W~) ~ (r~t')*(Vm~). Lemma 1.9.5 allows us to define V i"t by the requirement that (~nt)*(V~"t) ~ (p~"t)*(Wtw). (By the definition, this is the canonical model of V~"~ on X~"t.) Lemma 4.3.5 implies that fr can be lifted to the qS-equivariant isomor- phism p*(W) tw---_ (ptw)*(Wt*)-% (pmt)*(Wt*), commuting with the PH~t-action. This gives us the PH~t-equivariant isomorphism (r~tw)*(V)-~ (rd"t)*(Vi"t). Hence the Third Main Theorem follows from Lemma 1.9.5. [] Remark 4.3.6. -- Tannakian arguments can be used to show (see Theorem 1.9.10 and the discussion around it) that the Third Main Theorem implies the Fourth one. We will not use tiffs implication. 4.4. Reduction of the problem 4.4.1. Now we start the proof of the Fourth Main Theorem. For simplicity of nota- tion we identify E with E ~t by means of 9 and X with X i"t by means of re. Recall that Ps, c is a PHe-torsor over Xs, r for all sufficiently small S e o~'(E), hence (Ps, c) ~" is a (PHc)~-torsor over (Xs, c) ~ and (Pc)an-= (Ps,r ~ � c~. (Xe) a~ is a (PHc)a"-torsor int int disc int over (Xo) ~_-__ [B a-~ � (E /E 0) ]/PI' .SetY:= (na~)-a(Ba-, � (Pc) an . Then Y is a (PHc)a"-torsor over B a-~. Recall that E 0 = -0V~t acts trivially on P, hence (Pc) an-- (Y � (E~VE~n~)di'~ ~*. Proposition 4.4.2. -- There exists a homomorphism j : PF ~t -+ PH (C) and an isomorphism int tnt disc lnt (Pc) an --% (B a-a � (PHc) = � (E /E o ) )/Pr such that (x, h, g) y = (V~o 1 x, hj(y), gy~.) for all x e B a-a, h ~ (PHc) a", g e Eint/E~ t and y ~ PF t~t. Proof. -- The proposition asserts that there exists a decomposition y _~ Bd-* � (PHc) ~ such that the group P1 "i"t acts on B a-a � (PHc) an by the product of actions on factors. The trivial connection on y a~:w � (PHEw)~ -~ ZaKw is I" � DX-invariant, there- fore it defines a natural E-invariant flat connection ~ on the (PH~)"~-torsor Z a (PH~w)~ Ed~sc]/F Z a [ K~ � � over [ K~ � Ed~]/I'" Since for all sufficiently small S coW-(E) the projection (Y~ � (PHxw)a~ � Edi*)/P _+p~n is 6tale, it induces an isomorphism of tangent spaces up to an extension of scalars. Hence ~ induces a flat connection o~ s on P~n. By the definition, o~ s is a (PHKw)aMnvariant analytic vector subbundle of (T,~) a~, therefore Lemma 1.9.5 and Corollary 1.2.3 imply the existence of a unique flat connection J~f~s on Ps such that ~s ~ ~. Since the projection ~s :P ~ Ps is ~tale, o~f s defines a unique flat connection oY' on P satisfying (~s).(~) = o~ s. Moreover, W is E-equivariant and does not depend on S. The connection ~ determines flat connections o%,0 c on Pc and (~Y'c) a~ on (Pc) ~. p-ADIG UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 113 Let 5(f' be the restriction of (o%fc) ~ to Y. Then ~f" is a PFmt-invariant fiat connection on the (PHc)~-principal bundle Y over the simply connected complex manifold B a-a. By Lemma 1.9.19, there exists a decomposition Y-~ B a-a � (PHc) ~n such that the corresponding action of PF ~"t on B a-x � (PI-Ie) ~ preserves the trivial connection. For each y G PP~ let ~ : B d- ~ � (PHc) ~ --~ (PHc) ~ be the analytic map such that y(x, h) -= ('r(x),~(x, h)) for all x e B a-a and h e (PHr ~. Since the action of PF ~"t preserves the trivial connection, we have O'~/Ox =- 0 for each y e PF ~"t. Hence analytic ~ 's depend only on h. Since the action of PF t~t commutes with the action of (PHc) a", we have ~(h) = h~(1) for all h e (PHc) ~n and y e PF ~t. Therefore the map y ~-~(1) -~ is the required homomorphism. [] Theorem 4.4.3. -- There exists an inner isomorphism (= inner twisting) ~c : PHc -~ PH~ I~ such that j o (I) c : PF ~ = pGi~(F)+ -+ pHm~(c) ~ PGi~(C | F) is induced by the natural (diagonal) embedding F ~ C | F ~- C ~. Remark 4.4.4. -- Algebraization considerations as in Lemma 2.2.6 (using Propo- sition 1.9.13 instead of Corollary 1.2.2) show that Theorem 4.4.3 implies the existence of a ~-equivariant isomorphism P~ -~ ~t, lifting J~. 4.5. Proof of density To prove Theorem 4.4.3 we will use Margulis' results. For this we first show that the subgroup j(PF ~) is sufficiently large. We start with the following technical Lemma 4.5.1. -- Let n and d be positive integers. For each i ----- 1, ..., n we denote by pr~ the projection to the i-th factor. a) Let fr 9 9 ff , be Lie algebras, and let ~ be an ideal in the Lie algebra fr ---- 1-I"~=~ f~. Then 3#' ~ l-l~=~[pr~ ~f, ~f~]. b) Let A be a subgroup of PGLa(C)% Suppose that pr,(A) is infinite for every i = 1, ..., n. If A :---- CommmLacc~,(A ) is Zariski dense in (PGLa)" , then the same is true for A. c) If a subgroup A C PGUa(R)" is Zariski dense (in (PGUa)"), then it is dense. Proof. -- a) If x = (xl, ..., x,) e II~= 1 fg~ belongs to ~o, taken [x,y~] = (0, ..., [x~,y,], ..., 0) = [pr~ x,y~] eaf for all 2; e ~,- b) Let J be the Zariski closure of A in (PGLa)" , then 8J ~-1 c~ j is an algebraic subgroup of finite index in J for each 8 e A. Hence ~j0 8-1 ____ j0. In particular, the subgroup AdA stabilizes LieJ~ IAe(PGLa)". Since A is Zariski dense in (PGLa)" , the Lie algebra Lie j~ = LieJ is an ideal in Lie(PGLa)". By our assumption, pq(J) is an infinite algebraic group for each i = 1, ..., n, therefore pr~(LieJ) :# 0 is an ideal 15 114 YAKOV VARSHAVSKY in a simple Lie algebra Lie(PGLa). Therefore a) implies that LieJ = Lie(PGLa)". Since the group (PGLa) ~ is connected, J -- (PGLd)" r Let M be the closure of A in PGUa(R ) % Then M is a Lie subgroup of the Lie group PGUa(R)% Hence Lie IV[ is an Ad M-invariant subspace of LIe(PGUa(R))% Since the adjoint representation is algebraic, Lie M is an ideal in Lie(PGUa(R))% Since M is compact, it has a finite number of connected components. Hence M ~ is also Zariski dense, therefore it is not contained in PGUa(R)~-I x { 1 } x PGU~(R) ~-~ for any i = 1, ..., n. It follows that Lie M ---- Lie M ~ is not contained in Lie(PGUa(R)) '-1 x { 0 } � Lie(PGUa(R)) ~-', so that pq(Lie M) 4= 0. Now the assertion follows exactly in the same way as in b). [] Proposition 4.5.2. -- The subgroup j(PFint) is Zariski dense in PH c. Proof. -- Let G'C PH c be the Zariski closure ofj(PFmt). Then := (B ~-1 � (G') an X (PG'nt(A~:v))di*)/Ps Int is a pGint(A~:~)-invariant (G')~"-subtorsor of thc (PHc)~-torsor (pgpn = x n -~ [B ~-~ X (Pno) ~ X over (X~')~n~ [Bd-t X (PG~t(A~'*))dI~]/PI '~t. Hence by Proposition 1.9.13 there it exists an algebraic G'-subtorsor R of Pc such that R ~n ~ R. Using our identification of Cf with C, we obtain a closed analytic subspace (Rcp)= c (p~)= " = (D.~,, a | ^ C~, X (PH%) ~ x (PE')d~)/Pr. Recall that Pr = PH(O) is naturally embedded into PH(C~). Lemma 4.5.3. -- The subgroup generated by the elements of Pr with elliptic projections to PGLa(K~) is Zariski dense in PHcp. Proof. -- The subgroup of PGLa(K~) generated by the set of all elliptic elements is open and normal, because a conjugate of an elliptic element is elliptic. Hence it contains PSLa(K~). The subgroup PF o n PSLa(K~) is dense in PSLn(K,, ). Therefore, by [Ma, Oh. IX, Lem. 3.3], the subgroup of PF a generated by all elliptic elements of PI' o contains PF a n PSLa(K~). In particular, it has finite index in PF o = PH(Q). Since PH is connected, the statement follows from [Bo, Oh. V, Cor. 18.3]. [] If G'# PHr then by the lemma there exists Y e PP ~:ith elliptic pro- jection to PGUa_~.x(R) whose image V~ ~PH(C~) does not belong to G'(C~). Let xeD~| X{1}C (X~'p) ~" be an elliptic point of y~,ePE', and let ~' be an arbitrary point of ~ c# , over x. = tR ~n rR ~ lying Then y~(Z) y~(~) is another point of ~ c# , lying over x. Hence "G must belong to G'(C~), contradicting to our choice of y. [] p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 115 Recall that we defined in Proposition 4.3.2 the algebraic PH~:~ � E-equivariant pa-a Identify PH e with (PGLa) ~ in such a way that the first factor map p:P-+ Kw " corresponds to the embedding 0% : K '-+ ~. Denote by j~ : PF~"~ -+ PGLa(C) the composition of j with the projection pr~ of PGLa((]) ~ to its h-th factor. Denote also by pr ~ the projection of PGLa(E) ~ to the product of all factors except the i-th. We will some- times identify PF ~"* with its projection PF~ ~ C PGUa_~,~(R) ~ Proposition 4.5.4. -- The subgroup j~(PF ~"~) is not relatively compact in PGLe((I). Proof. -- If not, then jl(PF ~t) is contained in some maximal compact subgroup of PGLd(C ) (see for example [PR, Prop. 3.11]). After a suitable conjugation we may assume that jl(PF ~"t) C PGU~(R). By Proposition 4.5.2, j~(PF ~t) is Zariski dense in PGLd, hence it is infinite. Therefore, by Lemma 4.5.1, jl(PF ~t) is dense in PGUd(R ). Consider the map p : P(13) --> pd-l((]) and its restriction P0 to B ~-1 � { 1 } ~ B ~-~. Then, as in the proof of Claim 4.3.3, po:B e-x -+ l~a-l(E) satisfies po(,(x) =Jx(Y) p0(x) for all x e B ~-1 and y e PF int. The group PGUe(R) acts transitively on I~-I(G), hence o0(B a-l) is dense in Pa-I(C). Now we want to prove thatj~ : PF ~ ---> PGUa(R ) can be extended to a continuous homomorphism .~ : PGUa_~,~(R) ~ ---> PGUa(R ). For each g e PGUe_a,~(R) ~ choose a sequence { 3', },C PF~"*C PGUa_~,~(R) ~ converging to g. Since PGUa(R ) is compact, there exists a subsequence { y,,}~C {y,}, such that {j~(y,~)}, converges to some a ~ PGUa(R). Then po(gx) = lim po(u = (limjx(~,,,)) Oo(X) = apo(X) for all x e B a-a. It follows that a = a(g) depends only on g, since po(B ~-1) is dense in Pa-I(C) and since the group PGLa(G ) acts faithfully on 1 ~- 1((I). In particular, a(g) does not depend on the choice of{ y, }, and a(g) = limj~(%). It follows that jx := a is the required extension. Since PGUa_~,I(R) ~ is simple and j~(PF i"~) is dense, ~ must be injective and surjective. Hence is it an isomorphism, a contradiction. [] Proposition 4.5.5. -- For each i = 1, ..., g the homomorphism Ji : PFt"t -+ PGLa(C) is injective. Proof. -- Suppose that for some i the subgroup Ai := Ker(ji) is non-trivial. Then A~ is a normal subgroup of PP'nt ~ PP'~t C PGUd_I,I(R) ~ Hence the closure of (A~)~ is a non-trivial normal subgroup of a simple group PGUa_I,I(R) ~ Therefore the pro- jection (A~)~ is dense in PGU a_ 1, I(R) ~ Hence there exists an element ~ e A~ with elliptic projection ~ e PGUa_I,I(R) ~ Therefore the element (j(~), ~) e PGLa(C) g � PE' has a fixed point [y, g, el e P"(C~) = (~2~w(13~) � PGLa(C~) g � PE')/PF. Hence (g-lj(~) g, e-~ ~, e) stabilizes [y, 1, 1] e P"(Cv). It follows that e -1 ~ e = yg for some y e PF = PH(O) and g-~j(~) g e PH(C~) is the image of y. Hence jk(~) 4:1 for all k, contradicting to our assumption. [] 116 YAKOV VARSHAVSKY 4.6. Use of rigidity Now we are going to use the following theorem of Margulis [Ma, Ch. VII, Thm. 5.6]. Theorem 4.6.1. -- Let L be a local field, let J be a connected absolutely simple adjoint L-group, and let A be a finite set. For each o~ e A let k~ be a local field and let G~ be an adjoint absolutely simple k~-isotropic group. Set G := I-[ G,(k~). Let F be an irreducible lattice in G and let A be a subgroup of Comrno(F ). Suppose that rank G := Z rankk~ G~/> 2. ezG A If the image of a homomorphism v:A--~J(L) is Zariski dense in J and not relatively compact in J(L), then there exists a unique ~ ~ A, a continuous homomorphism 0 : k~ --~ L and a unique O-algebraic isomorphism ~ : G~ -~J such that v(x) = ~q(0(pr~(X))) for all X e A. 4.6.2. We use the notation of 2.4.1 with A' = PF int. Take any M and S such that rank C~M >1 2. Then by Proposition 2.4.5, F := A s is an irreducible lattice in G-~M. We will try to apply Theorem 4.6.1 in the following situation. Take G = C~M , A be the projection of A' to G-~M , L = C, J = (PGLa) c and v be the homomorphism Ji: PF~nt -+ PGLa(C) for some i ~{ 1, ..., g }. Consider first i = 1. By Proposition 4.5.2 and Proposition 4.5.4, v = 3"1 satisfies the conditions of Theorem 4.6.1, hence there exists an algebraic isomorphism B1 : (PGUa_I.1) c--% (PGLa) c such that jl(~') = Bx(y~) for all ~, e PF ~t. Now take i >/ 2. Suppose that j~(PF int) is not relatively compact. Then using again Proposition 4.5.2 we conclude from Theorem 4.6.1 that there exists an algebraic isomorphism ~: (PGUa_~,~)c--~ (PGLa) c suck that j~(~)= ~(~%o) for all ~, ~PF ~t. In particular, j(PF ~) is not Zarisld dense in (PGLd) g. This contradicts to Proposition 4.5.2. Therefore after a suitable conjugation we may assume that j~(PF ~t) C PGUa(R ) for all i = 2, ..., g. It follows that up to an algebraic automorphism of (PGLa) g, j(PF 'nt) C PGUa_,,I(R ) � PGUd(R) g-' -~ PHmt(R) and that 3"1 is the natural embedding PG~t(F)+ ~-+PGmt(Foo~). Therefore j together with the natural embedding PGmt(F)+ ~-~ PG~t(A~ TM) embed PF t"t into PGUa_,,~(R) ~ � PGU~(R) ~-~ X PG~'~(A~:~). Lemma 4.6.3. -- The closure of the projection of PF "t to PGUa(R) ~-~ � PGmt(A~ :~) contains PGUn(R) ~ x P((G"t)~(A~:*)). Proof. -- Let (goo,g/) be an element of PGUa(R) ~ x P((G~t)d~r(A~:~)), let U C PGUa(R) ~-~ be an open neigkbourhood of goo, and let S ~'(PG~t(A~;*)). We have to show that PF ~"t n (PGUa_ ~, ~(R) � U � g[ S) 4: El. By the strong approxi- int Y; v marion theorem there exists a 7 e PF i~t whose projection to PG (A~) belongs to gr S. Let y' be the projection of y-~ to PGU~(R) ~-x. p-ADIC UNIFORMIZATION OF UNITARY SHIMURA VARIETIES 117 Since j(PF ln~) belongs to the commensurator of A s :=j(PF~ nt) in PGLa(C) ~, Proposition 4.5.2, Proposition 4.5.5 and Lemma 4.5.1, b), c) imply that p?(As) is dense in PGUa(R) ~-~. It follows that there exists 8 e PF ~ whose projection to PGUa(R) ~-1 � PGt~*(A~ ;~) belongs to y'U � S. Then ~'~ belongs to PF t~t c~ (PGUa_I,i(R) ~ � U � g~S). [] 4.6.4. Now we proceed as in the proof of Theorem 2.2.5. Let M and S be as in 2.4.1, and let PF~ nt be the projection of PF(~] := pF~n~ c~ PGUa(R) g-x � G~ � S to PGUa(R) g-x � G~M. The proof of Proposition 2.4.5 holds in our case, hence PF~ "t is arithmetic. It follows that there exists a permutation ~ of the set { 2, ..., g } such that for every i = 2,...,g there exists a unique algebraic isomorphism r,: PGr~oi--% PGU d satisfying r~(y) =jo(~)(y) for each y e--(s). pp~t In particular, a and the r~'s do not depend on M and S. Since Pr = 0 we then have r,(y) =Jo(,)(Y)for all i e{ 2, ...,g} M, S and y e PF ~"t. This shows the existence of an algebraic isomorphism r which will satisfy Theorem 4.4.3 if we show that it is inner. But this can be immediately shown by the stan- dard argument using elliptic elements and function t defined in 2.5.5 (compare for example the proofs of Proposition 2.5.6 and Proposition 4.5.5). 4.7. Rationality question Consider the (PH~')=-torsor (Pm')~=~ [B a-~ � (PH~r = � (E~~ ~"~ over (,~mt)=. As in the p-adic case, it has a canonical flat connection 3/amt. The same considerations as in the p-adic case (see the proof of Proposition 4.4.2) show that there exists a unique connection ~t on ~tnt such that (9~Int) ~n _~ ~mt It follows from the proofs of Proposition 4.4.2 and Theorem 4.4.3 that (j~. P).(~c) = ~lnt. Lemma 4.7.1. -- If an analytic automorphism ~0:(~int)an .~. (~mt)~. commutes with the action of (pH~t)~" � E lnt, preserves ~,~lnt and induces the identity map on (~tnt)~ __ (pHc)~.\(~.t)~n, then *? is the identity. Proof. -- Recall that (~mt)~n ~ [Ba-1 X (pH~nt) =n � (E~nt/E~t)d~]/Pr ~t. Since ~o induces the identity map on (.~i.t)~n, there exists a holomorphic map + : B a-~ ---> (PH~nt) an such that ~[x, 1, 1] = [x, +(x), 1] for all x e B a-x. Since ~0 preserves ~tnt, we have O~?/Ox - O. Hence + is a constant, say a. Then ~0[x, h, e] = [x, ha, e] for all x e B a-l, h e (PH~t) ~n and e e E mr. In particular, V[x, 1, 1] = r 1 x,j(y), y~] = [y~l x,j(y)a, 2"r.] = [x,j(y) aj(~t) -1, 1] for all ~, e pplnt. Therefore j(y) aj(y) -1 = a for all Y e Flnt" Since PF mt is Zariski dense in PH~ t, a= 1. [] 118 YAKOV VARSHAVSKY Corollary 4.7.9.. __ The torsor ~mt has a unique Eint-equivariant structure p,nt of a PH~tw-torsor over X ~nt such that there exists a connection ~,~t on p~nt satisfying =~ ;Tdint" (pint is called the canonical model of'P l~t over Xint.) Proof. -- The existence is proved in [Mil, III, w 3]. Suppose that P' and P" are two structures satisfying the above conditions. Let f: Pc ~ pint_% Pc be the natural isomorphism. For each. e AUtKw(C ) set q~. := .(f)-a of. Then the automorphism (~%)~" of (P~)= - (~int)an satisfies the assumptions of the lemma. Hence (q~o)= is the identity, so that ~(f) =f for all . e Autx~(C ). It follows that P' = P". 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Symp. Pure Math. 9, Amer. Math. Soc. (1966), 32-62. Y. p-Adie Uniformization of Unitary Shimura Varieties II, to appear in the Journal of Wa] Diff. Geom. [Wa] B. L. VAN DER WA~-RDEN, Modern Algebra, Frederick Ungar Publishing CO, 1950. A. WmL, Adkles and Algebraic Groups, Progress in Math. 23, Birkh~user, 1982. [Well [We2] A. WmL, Basic Number Theory, Springer, 1967. S.-T. YAu, Calabi's Conjecture and Some Results in Algebraic Geometry, Proc. Nat. Acad. Sci. USA 74 (1977), 1798-1799. Y* g*, Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada, Manuscrit re~u le 21 mars 1995, rgvisg le 29 dgcembre 1996. V~mSX-LaVSKY, GROTHENVI~-CK, MUSTAVXN, t~MAKRXSHNAN

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