A constructive approach to p-adic deformations of arithmetic homology

Transcription

1 A constructive approach to p-adic deformations of arithmetic homology Avner Ash 1 Boston College Chestnut Hill, MA Avner.Ash@bc.edu February 17, 2009 Abstract Let Γ be an arithmetic group contained in a semigroup S that commensurates Γ and let H be the corresponding Hecke algebra. Given a p-adic analytic family of Banach S-modules varying over an affinoid space Ω and an H-eigenclass ζ in the homology of Γ with coefficients in one member of the family, we give an explicit construction of a p-adic analytic deformation of ζ along some Zariski closed subspace V of Ω. We assume that the S-action is constant modulo p, that some element u S acts completely continuously on the family, and that ζ is ordinary for the Hecke operator ΓuΓ. Given a p-ordinary Hecke eigenclass in the homology of Γ with coefficients in a finite-dimensional rational representation, our construction yields a family of homology eigenclasses with coefficients in (infinite dimensional) p-adic Banach modules with p-adic weights k. If k is classical, one also obtains homology eigenclasses with coefficients in the finite dimensional highest weight module corresponding to k. A sufficient condition is given for these homology classes to be non-p-torsion. We describe a parallel series of results for cohomology. We give a lower bound for the dimension of V. We describe an example for GL(3)/Q where we can prove that V is at least two dimensional and parametrizes non-p-torsion cohomology classes. 1 The author wishes to thank the National Science Foundation for support of this research through NSF grant DMS Mathematics Subject Classification: Primary 11F33, Secondary 11F75. Keywords: p-adic, deformations, arithmetic, homology, cohomology, eigenvariety 1

2 1 Introduction Let p denote a rational prime number and K a finite extension of Q p. Let ϖ be a uniformizer in K. In this paper we study the homology of an arithmetic group Γ with coefficients in a p-adically analytic family of K-Banach modules that modulo ϖ is a constant family. These modules are parametrized by points k in a p-adic rigid-analytic affinoid space Ω. We write D k for the module with parameter k and D 0 k for the closed unit ball in D k. Our main results, contained in Theorem 35 together with Corollary 37, may be summarized as follows: Let k 0 Ω(K) and ζ 0 H (Γ, D 0 k 0 ) be an ordinary Hecke eigenclass. We obtain a finite number of rigid analytic functions on Ω whose zero-locus is a Zariski-closed subspace V Ω such that for each k V there exists a nonzero homology class ζ k H (Γ, D 0 k ) such that ζ k 0 = ζ 0, and for every k V, the reduction of ζ k modulo ϖ is a Hecke eigenclass whose Hecke eigenvalues are congruent to those of ζ 0 modulo ϖ. Under the additional hypothesis of Theorem 39, the classes ζ k will be nonp-torsion classes. This extra hypothesis holds for example if the degree equals the virtual cohomological dimension of Γ and Γ has no elements of order p. A priori V might consist only of the single point k 0. We obtain a lower bound for the dimension of V computed as follows. Let O be the ring of integers in K. Let α denote the system of Hecke eigenvalues on ζ 0 reduced modulo ϖ. Let d be the dimension over O/ϖ of the generalized α-eigenspace of H 1 (Γ, D 0 k 0 /ϖd 0 k 0 ). Then the codimension of V in Ω is at most d. In Theorem 35 we actually input a chain z 0 which modulo ϖ becomes a cycle in C (Γ, D 0 k 0 /ϖd 0 k 0 ) but z 0 itself need not be a cycle. However, if z 0 is not itself a cycle, the subspace V could be empty. If conditions could be found which imply that V is nonempty, this would provide a bridge from mod p homology to homology in characteristic 0. Our main results require three hypotheses. We work with a class of groups more general than arithmetic groups. In Section 2, we state Hypothesis 2, which is the group-theoretic assumption about Γ that we need. We also give a formula used to lift Hecke operators to the chain level. To prove our main results, we work directly with cycles and use a variant of the Bockstein construction. This requires the use of orthonormal (ON) bases for the Banach spaces D k. Families of Banach modules, their ON bases, and constancy modulo ϖ are covered in Sections 3 and 4. In Section 5 we state Hypotheses 19 and 21. They may be paraphrased by saying that there is a Hecke operator that acts completely continuously on the chain level and whose eigenvalue on ζ 0 is a p-adic unit. In this section, we construct Hida-type idempotents on the chain level. In Sections 6 we define and construct certain special kinds of ON bases for the chains C (D k ). We use these to derive the main results of this paper in Section 7. In Section 8 we show how our set-up applies when Γ is an arithmetic Q-group split at p. We describe briefly certain modules of distributions, based on ideas 2

3 of Glenn Stevens, which satisfy all of our hypotheses. In this case, the points of Ω correspond to weights, i.e. p-adic-valued characters k of the integral points of the maximal Q p -split torus T in G(Q p ). A comparison lemma stated in Section 8 gives the connection between the homology of Banach modules and the homology of the finite-dimensional representations. In Section 9 we show how to modify all the constructions, theorems and proofs so they apply to cohomology. In Section 10, we give an example for GL(3)/Q where we can prove that the space V of deformations is at least 2-dimensional. Our methods provide the ability, in principle, to compute V to any desired degree of accuracy. In this case, the weight space Ω is 3-dimensional and we know from [3] that there cannot be a 3-dimensional space of deformations and that modulo twisting, V contains at most finitely many classical points. The fact that there is a 2-dimensional space of deformations in this example can also be obtained by modifying an argument of Hida s in [14], as mentioned in the introduction to [3]. Similar results of Calegari and Mazur giving lower bounds on the dimension of the deformation space in the case of GL(2)/F, F an imaginary quadratic field, may be found in [12]. In Section 11 we prove two lemmas that may be of independent interest. The first shows how to extend Hida s method of forming idempotents to the case of a completely continuous operator on a Banach module. The second is a lemma of the lifting families of eigenvalues type required if we want to construct a family of Hecke eigenclasses over V. The subspace V of Ω in Theorem 35, which parametrizes the deformations we construct, can be thought of as part of the projection to weight space of an eigenvariety E parametrizing the set of all p-adic deformations of Hecke eigenclasses of finite slope. A point in E is a pair (k, θ) where θ parametrizes a Hecke eigenclass in H (Γ, D k ). When Γ is an arithemtic group, the projection E V is expected to be locally finite, so that E and V have locally the same dimension. There is presently no single agreed-upon definition of E. For two approaches, see [10] and [13]. Eric Urban [16] has made some conjectures on the dimension of E. We assume that the arithmetic group Γ is of an appropriate type, having p in its level. Let l be the rank of the torus T. We denote by D k an appropriate space of locally analytic distributions of weight k. Let θ be a system of Hecke eigenvalues occurring in the cohomology H (Γ, D k ) with k Hom cont (T (Z p ), C p ). A simple form of Urban s conjecture is the following: Conjecture 1 (Urban). Let x = (k, θ) be a point of the eigenvariety and assume that k is a classical weight. The irreducible components of the eigenvariety passing through x are all of dimension l δ if and only if there exist two non-negative integers a, b and a positive integer m such that (a) The θ-generalized eigenspace of H r (Γ, D k ) is nonzero if and only if a r b and its dimension is m ( b a r a), and (b) δ = b a. There are cases where one expects several irreducible components of the 3

4 eigenvariety of different dimensions. In those cases, Urban has an analogous conjecture. Our Theorem 35 may be viewed as giving some evidence for Urban s conjecture in the case where m = 1, assuming that the generalized eigenspace of the cohomology mod ϖ with Hecke eigenvalues equal to those of θ (mod ϖ) has minimal possible dimension. For in this case, the integer d in our main result will coincide with the integer δ in Conjecture 1. (Cf. Theorem 5.1, p. 101 of [7].) Thanks to David Pollack for many helpful conversations and for the computation mentioned in section 10. Special thanks to Glenn Stevens for his valuable comments on an earlier version of this paper. 2 Chains If an element g of a ring or a group acts on an element m of a module on the right, we write the action as m g. Consider a group G, a subsemigroup S of G and a subgroup Γ of S. Assume that for any s S, both Γ s 1 Γs and Γ sγs 1 have finite index in Γ. We let H denote the Hecke algebra of double cosets Γ\S/Γ with coefficients in Z. Multiplication is defined by convolution of the indicator functions on the double cosets. We assume that H is commutative. Then H acts on the right of the homology of Γ with coefficients in any Z-module M which is also a right S-module. Since we won t be varying Γ, we will denote the homology of Γ with coefficients in a Γ-module M by H (M). For more information on how Hecke operators are defined and how they act on homology and cohomology, see for example, [4], [5], or [6]. Hypothesis 2. There exists a resolution F Z p by free, finitely-generated right Z p [Γ]-modules. Fix such a resolution. Denote the chains with coefficients in any right Z p [Γ]- module M with respect to this resolution by C (M) = F Zp[Γ] M. Denote the boundary maps by. The following lemma is immediate: Lemma 3. For i 0, let {f ij } F i be a free basis of F i over Z p [Γ] and d j M. Write f ij = b λ bf i 1,b γ b with λ b Z p and γ b Γ. Then ( j f ij Γ d j ) = j,b λ b f i 1,b Γ d j γ 1 b in C i 1 (M). We fix once and for all a free basis {f ij } of F i over Z p [Γ]. Let r i denote the rank of F i over Z p [Γ]. This enables us to define the following, noncanonical, isomorphisms of Z p -modules: 4

5 Definition 4. For any Γ-module M and any i 0, let ξ i : C i (M) M ri be defined by ξ i : f ij Γ m j (m j ). j We can lift the action of a single double coset in H from homology to cochains with coefficients in a right S-module M as follows. Note that if we do this for two different double cosets, we cannot expect the lifted actions on chains to commute. Let s S. Set = s 1 Γs, = sγs 1. Write Γ/Γ = t γ t(γ ). We define a new resolution of Z p by Z p [ ] modules which we denote by F. The underlying Z p -modules are the same as in F, but the action of δ is given by f δ = f sδs 1. We fix a homotopy equivalence τ : F F of Γ -modules, so that τ(f sδs 1 ) = τ(f) δ for any δ Γ. We now define s on C (M) by the formula (f Γ m) s = t τ(f γ t ) Γ m γ t s (1) and extending by linearity. Lemma 5. s is a well-defined Z p -linear chain map (i.e. commutes with ) and induces the action of the Hecke operator ΓsΓ on H (M). Proof. A simple computation. Definition 6. Set S to be the semigroup generated by the s for s S. Remark 7. If M is an A[S]-module for some Z p -algebra A, C (M) is an A- module where A acts on the tensor product F Zp[Γ] M through the second factor. Then and s are A-module maps. Moreover, suppose that M i is an A i [S]-module for some Z p -algebra A i, i = 1, 2 and that we have compatible homomorphisms of algebras f : A 1 A 2 and modules φ : M 1 M 2. Then we obtain an induced chain map φ : C (M 1 ) C (M 2 ) which is linear with respect to f and S-equivariant, by the formula f ij Γ m i f ij Γ φ(m i ). j j 3 ϖ-adic families of Banach modules Fix a finite extension K of Q p. Let O denote the ring of integers in K and fix a uniformizer ϖ O. We set F = O/ϖ. We normalize the absolute value on K so that ϖ = p 1. Let K a denote a fixed algebraic closure of K. Use to denote the norm on any complete K-algebra B. Let B 0 denote the closed unit ball in B and set B = B 0 /ϖb 0. For the definitions and conventions we use for Banach K-algebras and Banach modules over such algebras, see [11]. 5

6 Definition 8. If D is a left A-Banach module over a Banach K-algebra A and also a right S-module, for a semigroup S that acts via A-module Banach morphisms of operator norm 1, such that the A and S actions commute, then we will say that D is an A S module. Note that D 0 is an A 0 S module and D is an A S-module. If Ω is a reduced affinoid rigid-analytic space defined over K, we will simply say Ω is a K-affinoid. We denote its affinoid algebra by A Ω. By k Ω we mean k Ω(K a ). We let K(k) denote the field O Ω,k /m Ω,k which is a finite extension of K. Let O k denote the ring of integers in K(k), m k = (ϖ k ) its maximal ideal and F(k) = O k /m k the residue field, which is a finite field of characteristic p. If k 1, k 2 K a, we let K(k 1, k 2 ) denote the compositum of K(k 1 ) and K(k 2 ), and let O k1,k 2 denote its ring of integers. We shall always assume that Ω(K). Recall from [11] the definition of an ON (orthonormal) basis for a Banach A-module D, where A is a Banach K-algebra: A set of elements {d r } in D is such a basis if and only if any element d D can be written uniquely as a convergent sum d = r a rd r with a r A, a r 0 as r, and d equals the sup of the a r. In particular, d r = 1 and d r D 0 for all r. We say that D is ON-able if it has an ON basis. If {d r } is an ON basis for D over K, and B is any K-Banach module, then {1 d r } is an ON basis for B ˆ K D over B. Since any K-Banach space is ON-able over K, so is B ˆ K D over B, and (B ˆ K D) 0 = B 0 ˆ O D 0. We will also need the following lemma: Lemma 9. Let A be a Banach K-algebra and D a Banach A-module such that K = A = D. Then a subset S of D is an ON basis for D over A if and only if S D 0 and the image of S in D is a free basis of D over A. Proof. The lemma follows immediately from Lemma 1.1 of [11]. Definition 10. Let Ω be a K-affinoid, D a K-Banach space and S a semigroup. Set D Ω = A Ω ˆ K D with the obvious A Ω -Banach module structure, with A Ω acting on the left. A ϖ-adic family of S-modules over Ω of type D is an A Ω Smodule structure on the A Ω -Banach module D Ω. Let D Ω be a ϖ-adic family of S-modules over Ω of type D. Given k Ω, let ev k : A Ω K(k) denote evaluaton at k. We obtain the K(k) S-module D k := K(k) ˆ AΩ,ev k D Ω. We denote the resulting S-action on D k by d k s. For each k Ω, transitivity of tensor product gives a natural isomorphism D k K(k) K D as K(k)-Banach spaces, and we will identify the two sides via this isomorphism. In particular, if k Ω(K), we identify D k D as K-Banach spaces. For each k Ω and s S, D 0 k k s D 0 k. Set D k = D 0 k /ϖ kd 0 k. If y = lim t l d l D Ω = A Ω ˆ D, we write the specialization of y at k Ω as y(k) = lim t l (k)d l D k = K(k) D. Thus, in terms of the identifications just made, we have that (t d)(k) = 1 t d K(k) ˆ A Ω ˆ D. In this notation, ((1 d) s)(k) = d k s. 6

7 We extend specialization as follows to chains: Apply Remark 7 to the case A 1 = A Ω, A 2 = K(k), M 1 = D Ω and M 2 = D k, f is evaluation at k and φ is specialization at k. We obtain an S-equivariant chain map which we will denote by σ k : C i (D Ω ) C i (D k ). We can describe σ k alternatively in terms of the ξ i s of Definition 4 as the map ξ 1 i,k φri ξ i,ω, in the obvious notation. Remark 11. We compare elements in D k1 and D k2 by embedding them both into K(k 1, k 2 ) K D. Lemma 12. Let ẑ D 0. Then for any k Ω, (1 ẑ)(k) D 0 k and (1 ẑ)(k) = ẑ. Proof. By our conventions, both sides are equal to 1 1 ẑ O k O D 0. Let D be an R S-module. For each i 0, the chains C i (D) will be given the obvious structure as R-Banach module via the isomorphism ξ i : C i (D) D ri (Definition 4). Then C i (D 0 ) = C i (D) 0. Suppose D Ω is a ϖ-adic family of S- modules over Ω of type D. Then ξ i induces an isometry of Banach A Ω -modules C i (D Ω ) A Ω ˆ D ri via the canonical isomorphism D ri Ω A Ω ˆ D ri. We will also denote these isometries by ξ i, and we will use them in the proof of Lemma 25. We will denote the boundary maps by Ω in C (D Ω ) and k in C (D k ) for any k Ω. Lemma 13. Let D Ω be a ϖ-adic family of S-modules over Ω of type D and let i 0. (a) If s is given by formula (1) then s acts on the right of C i (D Ω ) giving a morphism of A Ω -Banach modules of operator norm 1. (b) The boundary map Ω : C i (D Ω ) C i 1 (D Ω ) is a map of A Ω -Banach modules of operator norm 1 and commutes with s. (c) Statements (a)-(b) are true if we replace Ω with k and A Ω with K(k), for any k Ω. Proof. Follows easily from Lemmas 3 and 5. 4 Families that are constant modulo ϖ Definition 14. We say that the ϖ-adic family D Ω of S-modules over Ω of type D is constant modulo ϖ if there exists an S-module structure on D such that D Ω is isomorphic as A Ω S module to A Ω F D, where S acts on the latter via the right factor and A Ω via the left factor. This is equivalent to saying that for any k 1, k 2 Ω, d D 0 and s S, then d k1 s d k2 s (mod ϖ). That is, (1 d) s(k 1 ) (1 d) s(k 2 ) ϖ(o k1,k 2 O D 0 ). In this sense, the action of S on any x D k is independent of k and will be denoted simply as x s. Also the homology groups H (D k ) as H-modules are independent of k. 7

8 For the rest of this section, fix k 0 Ω(K). There is no natural isomorphism C i (D k ) K(k) ˆ K C i (D k0 ), so it doesn t make sense to compare elements in these two Banach spaces. However, modulo ϖ we have the following lemma. Lemma 15. Let D Ω be a ϖ-adic family of S-modules over Ω of type D that is constant modulo ϖ. Let i 0. (1) Define η k : C i (D k ) O k /ϖ F C i (D k0 ) by η k ( β f β Γ d β ) = β f β Γ d β for any f β F i and d β D. Then η k is an isomorphism of O k /ϖ S Banach modules. (2) Define η Ω : C i (D Ω ) A Ω F C i (D k0 ) by η Ω ( β f β Γ (a β F d β )) = β a β F (f β Γ d β ) for any f β F i, d β D, and a β A Ω. Then η Ω is an isomorphism of A Ω S Banach modules. (3) η k σ k = σ k η Ω, where as above σ k : C (D Ω ) C (D k ) denotes specialization at k. Proof. The only thing that may not be clear is that η k and η Ω are well-defined. Consider (1). In the source of η k, for any γ Γ, we have f Γ d = f γ Γ d k γ. We must show that in the target f Γ d = f γ Γ d k γ. By the hypothesis of constancy modulo ϖ, we have that d k γ = d k0 γ, so that f γ Γ d k γ = f γ Γ d k0 γ = f Γ d in the target. To establish (2), note that for each k, ((a F d) γ)(k) = a(k) F d k γ, and argue as in (1). Denote reduction modulo ϖ by ρ. Definition 16. Let ˆx C i (D 0 k 0 ). A proper lift of ˆx is an element x C i (D 0 Ω ) such that (1) σ k0 (x ) = ˆx; (2) η k (ρ(σ k (x ))) = ρ(ˆx) for all k Ω. Lemma 17. Let D Ω be a ϖ-adic family of S-modules over Ω of type D that is constant modulo ϖ. Then proper lifts always exist. Proof. Let ˆx C i (D 0 k 0 ). We identify D and D k0 as K-Banach spaces. Let ξ Ω : C i (D Ω ) D ri Ω A ri Ω ˆ D k 0 and ξ k : C i (D k ) D ri k K(k) ˆ D ri k 0 be the maps ξ i of Banach modules defined in Definition 4. Let 1 Ω denote the constant function 1 on Ω. Take x = ξ 1 i,ω (1 Ω ξ k0 ˆx) C i (D 0 Ω ). We claim that x is a proper lift of ˆx. (1) We have σ k0 (x ) = σ k0 ξ 1 Ω (1 Ω ξ k0 ˆx) = ξ 1 k 0 σ k0 (1 Ω ξ k0 ˆx) which by Lemma 12 equals ξ 1 k 0 (ξ k0 ˆx) = ˆx. 8

9 (2) We have η k (ρ(σ k (x ))) = η k (σ k (x )) = σ k (η Ω (x )) by Lemma 15. Write ˆx = f j Γ d j where {f j } is the free basis of F i chosen in Section 2. Then we have σ k (η Ω (x )) = σ k (η Ω ( f j Γ (1 Ω d j ))) = σ k ( 1 Ω (f j Γ d j )) = fj Γ d j = ρ(ˆx). The following lemma gives us a general way to find ON bases for chain spaces. Lemma 18. Let D Ω be a ϖ-adic family of S-modules over Ω of type D that is constant modulo ϖ. Let {ẑ m } be a set of elements in C i (D 0 k 0 ). Let z m be the reduction modulo ϖ of ẑ m in C i (D k0 ). For each m, choose a proper lift z m of ẑ m. Assume that {z m } is an F-basis of C i (D k0 ). Then {z m} is an ON-basis of C i (D Ω ). Proof. By Lemma 9 it is enough to show that {ρ(z m)} is a free A Ω -basis of C i (D Ω ). By Lemma 15 it suffices to see this after applying η Ω. But η Ω (ρ(z m)) = 1 Ω z m A Ω F C i (D k0 ), as may be seen by specializing at each k, using Lemma 15 (3), and (2) in the definition of proper lift. The result is now clear. 5 Hypotheses on the U p operator For the rest of this paper we assume that D Ω is a ϖ-adic family of S-modules over Ω of type D and constant modulo ϖ. Let F be the residue field of K a, which is an algebraic closure of F. Let M be an H F-module and let α : H F be a ring homomorphism. We denote by M α the generalized α-eigenspace of M. Fix α for the rest of this paper. We make two hypotheses, an ordinarity assumption and a complete continuity assumption, to give us an analogue of the U p -operator. Hypothesis 19. There exists π S and a p-adic unit λ O such that u := λ 1 ΓπΓ H satisfies α(u) = 1. We fix a π, λ and u as in this hypothesis. Definition 20. Let π on C(D Ω ) be given by formula (1) (for s = π) and set υ = λ 1 π. Thus, for any j, we obtain υ acting on C j (D Ω ) and stabilizing C j (D 0 Ω ). And for any k Ω, we obtain k υ acting on C j (D k ) and stabilizing C j (D 0 k ). Recall from [11] the definition of complete continuity: Let M and N be Banach modules over the Banach algebra A and suppose L : M N is a continuous A-linear map. Then we say that L is completely continuous if L = lim j L j 9

10 for a sequence of continuous A-linear maps L j : M N such that for each j, L j (M) is contained in a finitely generated A-submodule of N. We also record Lemma 1.4 from [11]: Suppose {e i } is an ON-basis of M and {d j } is an ON-basis of N and L(e i ) = j n ij d j. Then L is completely continuous if and only if lim sup n ij = 0. j i It is easy to see that the dual of a completely continuous map is completely continuous. Hypothesis 21. For each j 0, υ acts completely continuously on C j (D Ω ). From now on we assume Hypotheses 2, 19 and 21. Lemma 22. Let j 0 and k Ω. (1) C j (D Ω ) υ is finitely generated over A Ω. (2) C i ( D k ) υ is finitely generated over F(k). (3) dim F(k) H j ( D k ) u is finite. (4) dim F(k) H j ( D k ) α is finite. Proof. Assertions (1)-(3) follow easily from complete continuity. Since α(u) = 1, H j ( D k ) α H j ( D k ) u, and (4) holds. Definition 23. Set d i (k) = dim F(k) H i 1 ( D k ) α. The integer d i (k) will have impact on the dimension of the deformation space of a Hecke eigenclass in H i (D 0 k ) whose reduction modulo ϖ has Hecke eigenvalues α. In the applications it is convenient to know the following: Lemma 24. If π acts completely continuously on D Ω, then Hypothesis 21 holds. Proof. The lemma follows easily from Formula (1) in Section 2. Lemma 25. Let j 0. Supppose {t a a = 1,..., n } is a finite subset of H and {c a a = 1,..., n } is a subset of O Ka. Let τ = ( t 1 c 1 )( t 2 c 2 ) ( t n c n ) where each t a is the lift of t a to the chain level given by Formula 1. Let e(υτ) = lim m (υτ)m!. Then the limit exists and e(υτ) acts on C j (D Ω ) as an A Ω -linear idempotent (necessarily of norm 1, unless e(υτ) = 0). Proof. This follows from Lemma 45 in Section 11 applied to ξ 1 i (υτ) ξ i (where ξ i is the isometry defined in Definition 4). 10

11 We are going to create this idempotent when τ is chosen to project onto the α-eigenspace, as follows: Lemma 26. Let J be a finite set of nonnegative integers and k Ω. There exists a finite extension K /K and T H K such that for all j J, (1) T induces an idempotent in End F(k) (H j ( D k ) u) and (2) T projects H j ( D k ) u onto H j ( D k ) α. Proof. Since J H j ( D k ) u is a finite dimensional F(k)-vector space, it can be decomposed over the algebraic closure F into generalized eigenspaces for the action of H. (Remember we assume H is commutative, and u H.) For each homomorphism β : H F such that β α and with the property that [ J H j ( D k ) u] β 0, choose a single double coset T β H such that α(t β ) β(t β ). Let e β be the cardinality of F(k)[β(T β )] and let δ be a p-power dim F(k) J H j ( D k ) u. Fix a lifting of β to ˆβ : H O Ka. Then we may take T = β α(t β ˆβ(T β )) e βδ. Fix k Ω and choose T β s as in the proof of Lemma 26. We lift each T β to a chain map T β as in equation (1). Set τ = ( T β ˆβ(T β )) e βδ β α where the product is taken in any (fixed) order. We also enlarge K if necessary, so that we may assume that k Ω(K) and all ˆβ(T β ) K. Definition 27. With the choices from the preceding paragraph, set e = e(υτ), an A Ω -linear idempotent on J H j (D Ω ). Lemma 28. J H j ( D k ) e = J H j ( D k ) α. Proof. This follows from the definition of e, the fact that u acts invertibly on J H j ( D k ) α, and Lemma ON bases of type (k, e, j) We continue the notation and choices of the preceding section. In particular, k Ω(K) and e is the idempotent defined in Definition 27. For any j 0, denote by ρ : C j (D 0 Ω ) C j(d Ω ) the canonical reduction map modulo ϖ. First we prove three lemmas about submodules of C j (D Ω ). Lemma 29. Let j 0 and M a closed A Ω -submodule of C j (D Ω ). Then (1) M 0 = M C j (D 0 Ω ). (2) ρ M 0 factors through M and induces a natural isomorphism M = ρ(m 0 ). 11

12 Proof. Since M inherits its norm from C j (D Ω ), we have (1), and (2) follows from (1). Lemma 30. Let j J. Any A Ω -submodule of C j (D Ω ) e is closed in C j (D Ω ) and in C j (D Ω ) e, and it is finitely generated over A Ω. Proof. By Theorem 7, page 210 of [8], it suffices to prove that C j (D Ω ) e itself is closed in C j (D Ω ), and has a finite ON-basis over A Ω. Because e is a continuous projector, C j (D Ω ) e, which is the kernel of 1 e, is closed in C j (D Ω ). Let {x µ µ I 1 } be an F-basis of C j (D k ) e. By Hypothesis 21, this is a finite basis. Let {x µ µ I 2 } be an F-basis of C j (D k ) (1 e). Together, {x µ µ I 1 I 2 } is an F-basis of C j (D k ). For µ I 1, lift x µ to some ˆx µ C j (D 0 k ) k e and choose a proper lift x µ C j (D 0 Ω ) e of ˆx µ. (This can be done by choosing any proper lift and then acting on it by e.) For µ I 2, lift x µ to some ˆx µ C j (D 0 k ) k (1 e) and choose a proper lift x µ C j (D 0 Ω ) (1 e) of ˆx µ. Now use Lemma 18. It follows that {x µ µ I 1 I 2 } is an ON basis of C j (D Ω ) and that {x µ µ I 1 } is an ON basis of C j (D Ω ) e. Lemma 31. Let j J. Then C j (D Ω ) = C j (D Ω ) e C j (D Ω ) (1 e). Any ONbasis of C j (D Ω ) e can be extended to an ON-basis of C j (D Ω ). The elements of the basis that span C j (D Ω ) (1 e) may be taken to be proper lifts in C j (D 0 Ω ) (1 e) of elements in C j (D 0 k ) (1 e). Proof. The first statement is clear. If {x µ µ I 1 } is an ON basis of C j (D Ω ) e, then {ρ(x µ ) µ I 1 } is a free A Ω -basis of A Ω F C j (D k ) e by Lemma 9. Since C j (D k ) = C j (D k ) e C j (D k ) (1 e), we can extend this free basis to one of all of C j (D k ) where the members of the basis that span C j (D k ) (1 e) are, say, {x µ µ I 2 }. The rest of the proof follows as in Lemma 30. We now define a special kind of ON basis for the Banach space of chains C j (D Ω ). In any chain complex, we denote the group of boundaries by B and the group of cycles by Z. Note that by Hypothesis 21, B j (D k ) e is a finite-dimensional F-vector space. The following lemma is obvious because ρ, e and all commute with each other. Lemma 32. {ρ( kˆv) ˆv C j+1 (D 0 k ) e} = B j(d k ) e Z j (D k ) e. Definition 33. Let j J and assume that j + 1 J. We fix k Ω(K), chain maps υ and τ as above, and denote by e the idempotent e(υτ). An ON basis of type (k, e, j) is a triple (B, X, Y ) where X and Y are finite sets, X C j (D 0 k ) e, Y C j+1(d 0 k ) e and B is an ON-basis of C j(d Ω ) such that: B = {b p, w q, c r } C j (D 0 Ω) where (1) For each q, there exists ˆx q X such that w q is a proper lift of ˆx q in C j (D 0 Ω ) e; 12

13 (2) for each p, there exists ŷ p Y such that b p = Ω (y p) for some proper lift y p of ŷ p in C j+1 (D 0 Ω ) e; (3) the set {ρ( k ŷ) ŷ Y } is an F-basis of B j (D k ) e, and is an F-basis of Z j (D k ) e. {ρ(ˆx), ρ( k ŷ) ˆx X, ŷ Y } Lemma 34. An ON basis {b p, w q, c r } of type (k, e, j) exists for C j (D Ω ). The index q takes on d j+1 (k) values (Definition 23). Proof. Let d = d j+1 (k). We know that C j (D k ) e is finite and that Z j (D k ) e/b j (D k ) e = H j (D k ) e = H j (D k ) α has F-dimension d (see Lemma 28.) Therefore, in view of Lemma 32, we can choose a finite set Y = {ŷ p } C j+1 (D 0 k ) e such that {ρ( k ŷ) ŷ Y } is an F-basis of B j (D k ) e, and a finite set X = {ˆx q } C j (D 0 k ) e of cardinality d such that {ρ(ˆx), ρ( k ŷ) ˆx X, ŷ Y } is an F-basis of Z j (D k ) e. Let w q be a proper lift of ˆx q in C j (D 0 Ω ) e and let b p = Ω (yp) where yp is a proper lift of ŷ p in C j+1 (D 0 Ω ) e. Next choose ĉ r C j (D 0 k ) such that {ρ(ŷ p) ρ(ˆx q ), ρ(ĉ r )} is an F-basis of C j (D k ). Let c r be a proper lift of ĉ r. Using Lemma 18 we see that {b p, w q, c r } is an ON basis of C j (D 0 Ω ). By construction it is of type (k, e, j). 13

14 7 Main theorem Theorem 35. Assume Hypotheses 2, 19 and 21. Let k 0 Ω(K). Fix i and set J = {i 1, i, i + 1}. Let υ, τ, and e be as in Definition 33 for k = k 0. Let (B, X, Y ) = {b p, w q, c r } be an ON basis of C i 1 (D Ω ) of type (k 0, e, i 1). For each ŷ p Y, set β p = yp, so that b p = Ω (β p ). Consider a chain z C i (D 0 k 0 ). Let z C i (D k0 ) be the reduction modulo ϖ of z. Assume that z is a cycle and let ζ denote its homology class. Assume further that ζ H i (D k0 ) α {0}. Choose a proper lift z of z. Write Ω (z ) e = d i(k 0) f p b p + g q w q + h r c r (2) p r q=1 for some functions f p, g q, h r A Ω. Set Z = (z ) e p f p β p and let V = V (z) be the zero locus of the ideal generated by the g q in A Ω. Then (1) If k V, Z(k) is a cycle in Z i (D 0 k ) and the homology class of the reduction modulo ϖ of Z(k) is congruent to ζ modulo ϖ and in particular is nonzero. (2) If k V, then for any r, h r (k) = 0. Proof. Recall that for any k Ω, ϖ k is a uniformizer in O k. Write a O k /ϖ for the reduction of a modulo ϖ (not modulo ϖ k ) for a O k. (1) First note that in fact f p, g q, h r A 0 Ω, because the left hand side of Equation (2) is integral and B is an ON basis. Since the family D Ω is constant modulo ϖ and z is a cycle, we obtain upon reducing (2) modulo ϖ and specializing at any k Ω, 0 = p f p (k)b p (k) + g q (k)w q (k) + r h r (k)c r (k). By the freeness of {b p (k), w q (k), c r (k)} over O k /ϖ, we conclude that 0 f p (k) g q (k) h r (k) (mod ϖ) for all k, p, q, r. In particular, for all k, Z(k) z k e (mod ϖ). Therefore, the homology class of Z(k) (mod ϖ) is congruent to ζ e = ζ. Set d = d i (k 0 ). By equation (2), Therefore for any k Ω, k (Z(k)) = Ω Z = d d q=1 g q w q + r q=1 g q (k)w q (k) + r 14 h r c r. h r (k)c r (k).

15 Fix a k such that g q (k) = 0 for all q. Then k (Z(k)) = r h r (k)c r (k). Now suppose that r h r(k)c r (k) 0. Let m be the largest integer such that h r (k) is divisible by ϖk m for all r. Since k and k e both commute with multiplication by constants in K(k), and because Z e = Z, we have that ( ) Z(k) k ϖk m = h r (k) ϖ m c r (k) r k is fixed under k e. Also, for some r, ϖ m+1 k does not divide h r (k). Now reduce both sides modulo ϖ k. Let x denote the reduction of x modulo ϖ k. Because k 2 = 0, the left hand side reduces to a cycle in Z i 1(D 0 k /ϖ kd 0 k ) e O k /ϖ k O Z i 1 (D k0 ) e. Hence it is in the O k /ϖ k -span of { b p (k), w q (k)}. But the right hand side reduces modulo ϖ k to something nonzero in the O k /ϖ k -span of { c r (k)}. This contradicts the freeness of { b p (k), w q (k), c r (k)} over O k /ϖ k. Hence k (Z(k)) = r h r(k)c r (k) = 0. (2) What we have just seen implies: if k V then h r (k) = 0 for all r, since the c r are part of an ON basis. Remark 36. We do not know whether V depends on the various choices made, such as the choice of z modulo boundaries, the choice of proper lift z, the idempotent e, the resolution F, or the choice of ON-basis of type (k 0, e, i 1). There is no apparent reason that V should not depend on all these choices. Probably, V will be just one piece through k 0 of the projection of an eigenvariety to weight space. Note that V may be empty, in general. But we have the following: Corollary 37. Assume that A Ω is a Tate algebra and let d = d i (k 0 ). Given k 0 Ω(K) and η k0 H i (D 0 k 0 ) such that the reduction modulo ϖ of η k0 is an α-eigenclass. Then there exists a Zariski-closed subspace V of Ω of dimension at least dim Ω d such that for each k V, there exists η k H i (D 0 k ) such that η k H i (D 0 k /ϖ kd 0 k ) α {0}. The family {η k } is analytic in the sense that there exists Z A V C i (D k0 ) such that for each k V, Z(k) is a cycle and η k is its homology class. Moreover, if η k0 k0 e = η k0, then the homology class of Z(k 0 ) is η k0. Proof. Choose for z in Theorem 35 a cycle that represents η k0. Then k0 z = 0. From Equation (2) specialized at k 0, we obtain 0 = k0 z k0 e = p f p (k 0 )b p (k 0 ) + d q=1 g q (k 0 )w q (k 0 ) + r h r (k 0 )c r (k 0 ). 15

16 Since {b p (k 0 ), w q (k 0 ), c r (k 0 )} form an orthonormal basis, the coefficients in the sums must all vanish. In particular, g q (k 0 ) = 0 for all q, so that k 0 V. The Tate algebra A Ω is a catenary ring. Since k 0 V, V is not empty. Hence the dimension of V is at least dim Ω d by the Hauptidealsatz and Lemma 34. The analytic nature of Z(k) as a function of k is clear from its definition in Theorem 35. The last assertion of the corollary is clear. Remark 38. Assume that η k0 e = η k0 and that η k0 is a Hecke eigenclass. The homology class of the cycle Z(k) may not be a Hecke eigenclass. However, by Lemma 47, we can find a dense open affinoid subset U of a finite integral cover of V and an analytic family Z (k ) over U consisting of cycles whose homology classes are Hecke eigenclasses whose eigenvalues modulo ϖ k are given by α. If in addition a certain multiplicity one result holds for the homology over V, as specified in (3) of Lemma 47, we can find U such that there is a point k in U over k 0 and the eigenvalues of the homology class of Z (k ) are the same as those of η k0. Note that H i (D 0 k ) may have ϖ k-torsion, so that some or all of the homology classes η k of Corollary 37 may be ϖ k -power-torsion classes. The following theorem gives us a way to rule out this possibility under a certain condition. Theorem 39. Suppose H i+1 (D 0 k 0 /ϖd 0 k 0 ) α = 0. Then for each k V, the homology class of Z(k) in Corollary 37 is non-ϖ k -power-torsion. Proof. The short exact sequence of S-modules 0 D 0 k ϖ k D 0 k D 0 k/ϖ k D 0 k 0 gives us a long exact sequence of Hecke-modules, of which there is a segment as follows: H i+1 (D 0 k/ϖ k D 0 k) H i (D 0 k) ϖ k H i (D 0 k). Act on it by e. Since e is an idempotent, we obtain an exact sequence: 0 = H i+1 (D 0 k/ϖ k D 0 k) α = H i+1 (D 0 k/ϖ k D 0 k) k e H i (D 0 k) k e ϖ k H i (D 0 k) k e. The second equality follows from the fact (Universal Coefficient Theorem) that H i+1 (D 0 k /ϖ kd 0 k ) = O k/ϖ k O H i+1 (D k0 ) and Lemma 26. If Z(k) is a ϖ k -power-torsion class, let ϖk m be the lowest positive power of ϖ k such that ϖk m Z(k) = 0. But ϖm 1 k Z(k) H i (D 0 k ) k e and hence must vanish. Contradiction. 8 Arithmetic groups In this section we briefly sketch how the theory developed above can apply to the homology of arithmetic groups. 16

17 By a theorem of Borel and Serre, any arithmetic group Γ is of type V F L (see, for example, page 218 of [9]). Therefore, by Proposition 5.1, page 197 of [9], Γ is of type F P. Then by Proposition 4.5, page 195 of [9], the trivial Γ-module Z p admits a resolution by free, finitely generated Z[Γ]-modules. Thus, Hypothesis 2 is satisfied for Γ. For the family of Banach modules we sketch a construction due to Glenn Stevens. This construction may be found in detail in [6] for any reductive Q- group G split at p. Let K = Q p. Let T denote a maximal K-split torus of G p. Let X denote the K-rigid analytic space that parametrizes C p -valued characters of T (Z p ). If k X (K a ), we use the notation t k to denote the evaluation of k on t. We fix an open K-affinoid neighborhood Ω of k 0 with the properties (1) A Ω = O X (Ω) is a Tate algebra; (2) for any t T (Z p ), the function on Ω sending t to t k is in A Ω ; (3) t k t k0 (mod p) for all k Ω and t T (Z p ). Let I be an Iwahori subgroup of G(Z p ) containing T (Z p ). Let X denote the corresponding big cell. It is a p-adic manifold. There is a cone T + in T (Q p ) such that the semigroup T + I acts naturally on X on the right. We construct D := D k0 as a family of distributions on X, making T act through the character k 0. Then D Ω becomes a family of T + I-modules over Ω of type D which is constant modulo ϖ. For each k, T + I acts on D k through the character k. Let Γ G(Z) be a congruence subgroup and S G(Q) a subsemigroup containing Γ. Set Γ = Γ I and S = S T + I. We assume that (Γ, S) is a Hecke pair with commutative Hecke algebra. We choose an element π T + such that all the negative roots evaluated on π have positive p-adic ord. Suppose we have chosen α and λ so that Hypothesis 19 holds for π. Then Hypothesis 21 also holds for these choices. To compute the integer d in our example in section 10 we need the following comparison theorem [6]. If k is a dominant integral weight, let V k denote the finite dimensional irreducible right K[G(K)]-module with highest weight vector v k of weight k. We define V k to be the same as V k except that the action of T + is twisted in such a way that it acts trivially on v k. There is a surjective map φ k : D k V k of A Ω T + I-modules, which takes the Dirac distribution at 1 to v k. Let L k = φ k((d 0 k ) ), a lattice in V Theorem 40. Let e be the projector corresponding to u = λ 1 ΓπΓ from Lemma 25. For any dominant integral weight k Ω(K), the induced map on homology (φ k ) : H i (D 0 k) e H i (L k) e is an isomorphism. The same is true for the reductions modulo ϖ: (φ k ) : H i (D k ) e H i (L k/ϖl k) e. k. 17

18 9 Cohomology In this section we indicate what changes are necessary to state and prove the analogues for cohomology of the preceding theory. We keep Hypothesis 2 and can use the same resolution F. If M is a right Z p [Γ]-module we denote the cochains by C (M) = Hom Zp[Γ](F, M). Denote the boundary maps by δ. Formula (1) is modified as follows. Let s S. Leave the definitions of = s 1 Γs, the Z p [ ]-resolution F and the homotopy equivalence τ : F F of Γ -modules as they are in Section 2. Write Γ \Γ = t (Γ )γ t. We now define s on c C (M) by the formula (c s)(f) = t c(τ(f (γ t) 1 )) sγ t (3) for any f F. The analogue of Lemma 5 is easily proven. The definitions of a ϖ-adic family of S-modules over Ω of type D and of its being constant modulo ϖ stay the same. The analogues for cochains of the definition of proper lift and the rest of the results in Section 4 are easily given. Hypothesis 19 stays the same and Hypothesis 21 stays the same except for cochains in place of chains. Instead of Definition 23 we have Definition 41. Set d i (k) = dim F(k) H i+1 (D k ) α. The analogues of Lemmas 25 and 26 for cochains and cohomology are clear. Now we come to the analogue of Theorem 35 for cohomology. We define modules and ON bases of type (k, e, j) for cochains C j (D Ω ) in the obvious way, imitating their definitions in the case of homology, but replacing the boundary operator with the coboundary operator δ and replacing degree j 1 with j + 1. It is then easy to prove the analagous lemmas about these objects and their consequences. For ease of reference we will here state the main theorem and corollary in the case of cohomology: Theorem 42. Let k 0 Ω(K). Assume Hypotheses 2, 19 and the cohomology version of Hypothesis 21, with corresponding values for i and α. Let e be the cohomological version of the idempotent of Definition 27. Let B(X, Y ) = {b p, w q, c r } be an ON basis of C i+1 (D Ω ) of type (k 0, e, i + 1). For each y p Y, set β p = yp, so that b p = δ Ω (β p ). Consider a cochain z C i (D 0 k 0 ). Let z C i (D k0 ) be the reduction modulo ϖ of z. Assume that z is a cocycle and let ζ denote the cohomology class of z. Assume further that ζ H i (D k0 ) α {0}. Choose a proper lift z of z. Write δ Ω (z ) e = d i (k0) f p b p + g q w q + h r c r (4) p r for some functions f p, g q, h r A Ω. q=1 18

19 Set Z = (z ) e p f p β p and let V = V (z) be the zero locus of the ideal generated by the g q in A Ω. Then (1) If k V, Z(k) is a cocycle in Z i (D 0 k ) and the cohomology class of the reduction modulo ϖ of Z(k) is congruent to ζ modulo ϖ and in particular is nonzero. (2) If k V, then for any r, h r (k) = 0. Proof. The proof is the same as for Theorem 35, changing chains to cochains, boundary maps to coboundary maps, cycles to cocycles, and homology to cohomology, and switching i 1 with i + 1 as needed. Corollary 43. Assume A Ω is a Tate algebra. Suppose k 0 Ω(K) and η k0 H i (D 0 k 0 ) is such that the reduction modulo ϖ of η k0 is an α-eigenclass. Let d = d i (k 0). Then there exists a Zariski-closed subspace V of Ω of dimension at least dim Ω d such that for each k V, there exists η k H i (D 0 k ) such that η k H i (D 0 k /ϖ kd 0 k ) α {0}. The family {η k } is analytic in the sense that there exists Z A V C i (D k0 ) such that for each k V, Z(k) is a cocycle and η k is its cohomology class. As in the case of homology, we can replace Z(k) with an analytically varying family of Hecke eigenclasses. The criterion for Z(k) to be non-ϖ k -torsion now reads: Theorem 44. Suppose H i 1 (D 0 k 0 /ϖd 0 k 0 ) α = 0. Then the cohomology class of Z(k) in Corollary 43 is non-ϖ k -power-torsion. The analogue of Theorem 40 also holds for cohomology 10 Remarks on Computations The deformation space V in Theorem 35 is effectively computable up to any desired degree of accuracy. We can construct an ON basis of type (k 0, e, i 1) in such a way that there is a partition of the r-indices into R 1 and R 2, with R 1 finite with the following properties: (1) b p, w q and the c r, r R 1 form a finite ON basis for C i 1 (D 0 Ω ) e. (2) the c r, r R 2 form an ON basis for C i 1 (D 0 Ω ) (1 e). (Cf. Lemma 31.) For any n, the computation modulo ϖ n of f p, g q and the h r, r R 1 in Equation (2) up to any desired degree of accuracy in the k-variable is a finite computation. We do not need to compute the h r, r R 2. In particular we can compute the g q modulo ϖ n and thus their common zero set V modulo ϖ n up to any degree of accuracy. To get a resolution F, we can use an explicit finite cell complex for a classifying space of a normal torsion-free subgroup of Γ. 19

20 Consider the case of GL(3, Q). Fix a positive integer N. Let S 0 (N) denote the subgroup of GL(3, Z) consisting of matrices whose first row is congruent to (, 0, 0) modulo N, and with positive determinant. Set Γ 0 (N) = S 0 (N) SL(3, Z). Fix a prime p not dividing N. Let I denote the Iwahori subgroup of GL(3, Z p ) consisting of matrices that become upper triangular modulo p. Set S = S 0 (N) I and Γ = Γ 0 (N) I. We work with the Hecke pair (Γ, S). If l is a prime, let D l,i denote the diagonal matrix with 3 i 1 s and then i l s down the diagonal, for 1 i 3. Let T l,i denote the Hecke operator corresponding to the double coset ΓD l,i Γ. We choose N = 61, p = 5, and k 0 = the trivial character. In [2] we computed the cohomology of Γ 0 (61) with trivial coefficients, together with the Hecke eigenvalues for l 29. Choose a square root w of 3 in (Q 5 ) a and set K = Q 5 [w]. Then these computations show that up to scalar multiples, there is a unique cuspidal Hecke eigenclass in H 2 (Γ 0 (61), K) with T 2,1 acting by w. On this class, T 5,1 acts by 2w and T 5,2 acts by 2w. This class fulfills all of our hypotheses and Corollary 43 applies to it. Moreover, using Theorem of [1] and Theorem 44 we can show that all the eigenclasses constructed are non-ϖ k -torsion. In this case, Theorem 40 plus unpublished computations of David Pollack show that d = 1. By the main results of [3], if we look at the subset V 0 of V consisting of k V such that k is trivial on all diag (1, 1, x) T (Z p ), then at most finitely many of the weights in V 0 can be dominant integral. Therefore, in this example, the dimension of V is exactly 2. Existing computer programs are not adequate for the necessary computations to determine approximations to V. I hope to report in the future on computations of V in collaboration with David Pollack. 11 Lemmas Lemma 45. Let A be a reduced K-affinoid Banach algebra, and D an ON-able Banach K-module. Let D A = A ˆ D. Let f 0 : D D and f 1 : D A D A be completely continuous maps of norm 1, linear over K and A respectively. Let f = f 0 + ϖf 1 : D A D A. Then lim m f m! exists and is an A-linear idempotent. The rate of convergence is uniform over Sp(A). Proof. For any K-Banach algebra R and ON-able Banach R-module N, let L R (N, N) denote the Banach space of bounded R-linear maps from N to N with the operator norm. Given an ON basis of N, elements of L R (N, N) can be represented by matrices, which act on the right of row vectors. The corresponding norm of a matrix is the sup of the norms of its entries. Although the ON basis might not be countable, we will employ the usual matrix and block matrix notations. Let {d t } be an ON basis for D, so that {1 d t } is an ON basis for D A. Let M be the matrix of f with respect to this basis. We view elements of A as functions of k Sp(A), so that we can write M(k). 20

21 We want to show that M m! has a limit as m and that this limit is an idempotent. This will follow if we can show (*) for any n > 0 and for any k Sp(A), there exists a finite extension K /K(k) and matrices M n (k) and E k representing linear maps of norm 1 in L K (K K D, K K D) such that M n (k) M(k) mod ϖ n, E 2 k = E k, and M n (k) m! E k uniformly in k. Indeed, assume (*). Then for any n > 0, there exists µ such that for any k Sp(A), s, t µ M n (k) s! M n (k) t! p n. We also have for any u > 0 and any k Sp(A) that M(k) u! M n (k) u! p n. Therefore, for any n > 0, there exists µ such that for any k Sp(A), s, t µ M(k) s! M(k) t! = M(k) s! M n (k) s! +M n (k) s! M n (k) t! +M n (k) t! M(k) t! p n. Hence {M m! } is a Cauchy sequence in the uniform (operator) norm, so it converges to some E. For each k and n, M n (k) m! E k, so that E E k modulo ϖ n, whence E 2 = E modulo ϖ n for every n. Hence E 2 = E. To prove (*), fix n > 0. Using the hypotheses of complete continuity, note that M is congruent modulo ϖ n to a matrix [ ] B 0 Z 0 where the block B is a square matrix of finite size (say r r) with entries in A 0. By hypothesis, there exists B 0 M r (O K ) and B 1 M r (A 0 ) such that B = B 0 + ϖb 1. Thus for any k Sp(A), B(k) = B 0 + ϖb 1 (k) M r (O k ). Consider the rigid analytic space X whose K a points are M r (O Ka ), i.e the closed ball of radius 1 about the 0-matrix in M r (K a ). We have the polynomials on X given by d(x) = det(x) and δ(x) = the discriminant of the characteristic polynomial of x. The sup norm of dδ is 1. To see this, choose t > 0 such that p t 1 > r and let x = diag (ζ 1,..., ζ r ), where the ζ i are pairwise distinct (p t 1)-th roots of unity. Then d(x) = δ(x) = 1. Temporarily fix k. It follows that the sup norm of dδ(b 0 + ϖb 1 (k) + x) is 1 and hence its Gauss norm is also 1. Let q be the degree of the polynomial dδ. Then the Gauss norm (which equals the sup norm) of dδ(b 0 + ϖb 1 (k) + ϖ n x) is p nq. So U = {x X dδ(b 0 + ϖb 1 (k) + ϖ n x) p nq } is a non-empty open set in X. Note that since d(y) and δ(y) are integral for any y X, dδ(u) p nq implies d(u) p nq and δ(u) p nq. Hence for any k, we can choose C(k) X (not necessarily analytically in k) and set G(k) = B 0 +ϖb 1 (k)+ϖ n C(k) in such a way that, det(g(k)) p nq and δ(g(k)) p nq. We claim: (**) There exists an idempotent F k X such that G(k) m! F k uniformly in k. 21

22 Assume the claim. For each k, set M n (k) = [ ] G(k) 0. Z(k) 0 For a given k, let K denote the finite extension of K(k) generated by the entries of G(k). Then M n (K) represents an element of L K (K K D, K K D) such that M n (k) M(k) mod ϖ n. Moreover, [ ] M n (k) m! G(k) = m! 0 Z(k)G(k) 1 G(k) m!. 0 By (**), G(k) m! tends to F k uniformly in k, and hence Z(k)G(k) 1 G(k) m! tends to Z(k)G(k) 1 F k uniformly in k, because the norm of the denominators in Z(k)G(k) 1 are bounded uniformly away from zero by det(g(k)) p nr. Therefore M n (k) m! E k uniformly in k where [ ] F E k = k 0 Z(k)G(k) 1. F k 0 Since F k is an idempotent, so is E k. It remains to prove ( ). We have G(k) = B 0 + ϖh(k) for some H(k) M r (O Ka ). Let p 0 (X) be the characteristic polynomial of B 0 and p(x) the characteristic polynomial of G(k). Then p(x) p 0 (X) modulo ϖ. Factor and p(x) = r (X α i (k)) i=1 p 0 (X) = r (X β i ) where the α i (k) and β i are all integral. Then for j = 1,..., r, i=1 0 = p(α j (k)) p 0 (α j (k)) = r (α j (k) β i ) where the congruence is modulo ϖ. Hence r i=1 α j(k) β i p 1. It follows that for every j and k there exists i(j, k) with the property that (α j (k) β i(j,k) ) ϖ 1/r O Ka, where we choose and fix an r-th root of ϖ. Diagonalize G(k) over K a : i=1 G(k) = P (k) 1 D(k)P (k) where D(k) = diag (α 1 (k),..., α r (k)) is diagonal and P (k) is invertible. We normalize P (k) so that its columns, which are eigenvectors of G(k), are each 22

23 integral with at least one entry equal to 1. It follows that P (k) 1 is (det(p (k)) 1 times an integral matrix. Let J 0 (k) = diag (β i(1,k),..., β i(r,k) ) and J 1 (k) = ϖ (1/r) (D(k) J 0 (k)), two diagonal and integral matrices. Then D(k) = J 0 (k) + ϖ 1/r J 1 (k) and G(k) = P (k) 1 (J 0 (k) + ϖ 1/r J 1 (k))p (k). For each k, J 0 (k) is one of r r possible r r matrices with entries in a fixed finite extension of K. So by Lemma 1, page 201 of Hida [15], lim m J 0 (k) m! exists and is an idempotent. The rate of convergence is uniform in k. Use of the binomial formula then shows that lim m (J 0 (k) + ϖ 1/r J 1 (k)) m! exists and is the same idempotent with rate of convergence uniform in k. Suppose we have (***) (det(p (k)) is bounded away from 0, independently of k. Then lim m G(k) m! exists and is an idempotent with rate of convergence uniform in k and the proof is finished. The following sublemma immediately implies (***). Sublemma 46. Let δ(x) denote the discriminant of the characteristic polynomial of a matrix x. Fix constants ɛ, η > 0. Suppose G M r (O Ka ) such that δ(g) η 2. Let λ 1,..., λ r be the eigenvalues of G. They are distinct and integral. Let P M r (O Ka ) be a matrix whose columns are v 1,..., v r, where for each i, Gv i = λ i v i and at least one entry of v i equals 1. Then det(p ) η r(r 1). Proof. Let us say that a vector in Ka r is normalized if it is in OK r a and at least one of its entries equals 1. The norm on OK r a is the sup of the norm of the components of a vector. Note that i<j (λ i λ j ) 2 = δ(p ) η 2 implies that λ i λ j η for all i j, since the λ s are integral. Set φ = η r(r 1) and suppose det(p ) < φ. Let µ 1,..., µ r be the eigenvalues of P. Then r i=1 µ i < φ and hence for some j, µ j < φ 1/r. If c is a corresponding normalized eigenvector, then P c = µ j c. Write c 1,..., c r for the components of c. Without loss of generality we may assume that c 1 = 1. Therefore we have v 1 + c 2 v c r v r = y 1 O r K a and y 1 < φ 1/r. Multiply both sides of this equation by G, also multiply both sides by λ r and subtract: (λ 1 λ r )v 1 + (λ 2 λ r )c 2 v (λ r 1 λ r )c r 1 v r 1 = Gy 1 λ r y 1. Dividing through by λ 1 λ r gives v 1 + c 2v c r 1v r 1 = y 2 O r K a and y 2 < η 1 φ 1/r 23