COMPLETED COHOMOLOGY OF SHIMURA CURVES AND A p-adic JACQUET-LANGLANDS CORRESPONDENCE

COMPLETED COHOMOLOGY OF SHIMURA CURVES AND A p-adic JACQUET-LANGLANDS CORRESPONDENCE JAMES NEWTON ABSTRACT. We study indefinite quaternion algebras over totally real fields F, and give an example of a cohomological construction of p-adic Jacquet-Langlands functoriality using completed cohomology. We also study the (tame) levels of p-adic automorphic forms on these quaternion algebras and give an analogue of Mazur s level lowering principle. 1. INTRODUCTION Our primary goal in this paper is to give some examples of p-adic interpolation of Jacquet-Langlands functoriality by studying the completed cohomology [11] of Shimura curves. The first example of a p-adic interpolation of Jacquet-Langlands functoriality appears in work of Chenevier [5], who produces a rigid analytic map between the eigencurve for a definite quaternion algebra over Q (as constructed in [1]) and the Coleman-Mazur eigencurve for GL 2 /Q [7]. In this paper we will be concerned with the Jacquet-Langlands transfer between automorphic representations of the multiplicative group of quaternion algebras over a totally real field F, which are split at one infinite place (so the attached Shimura varieties are one-dimensional). These quaternion algebras will always be split at places dividing p. For the purposes of this introduction we will suppose that F = Q. Let D be a quaternion algebra over Q, split at the infinite place and with discriminant N, and let D be a quaternion algebra over Q, split at the infinite place and with discriminant N ql for q, l distinct primes (coprime to Np). In this situation there is a rather geometric description of the Jacquet-Langlands transfer from automorphic representations of (D Q A Q ) to automorphic representations of (D Q A Q ), arising from the study of the reduction modulo q or l of integral models of Shimura curves attached to D and D (where we consider Shimura curves attached to D with Iwahori level at q and l). For the case N = 1 and F = Q this is one of the key results in [21], and it was extended to general totally real fields F by Rajaei [20]. Using the techniques from these papers, together with Emerton s completed cohomology [11], we can then describe a p-adic Jacquet-Langlands correspondence in a geometric way the result obtained is stronger than just the existence of a map between eigenvarieties, which is all that can be deduced from interpolatory techniques as in [5]. Our techniques also allow us to prove a p-adic analogue of Mazur s principle (as extended to the case of totally real fields by Jarvis [15]), generalising Theorem 6.2.4 of [12] to totally real fields. This should have applications to questions of local-global compatibility at l p in the p-adic setting. Having proved a level lowering result for completed cohomology, we can deduce a level lowering result for eigenvarieties. For F = Q these level lowering results have already been obtained by Paulin [19], using the results of [12] (in an indirect way, rather than by applying Emerton s construction of the eigencurve). A related approach to overconvergent Jacquet-Langlands functoriality is pursued in [14], using Stevens s cohomological construction of the eigencurve from overconvergent modular symbols. We briefly indicate the main results proven in what follows: Date: July 5, 2011. 1

2 JAMES NEWTON 1.1. Level lowering. We prove an analogue of Mazur s principle for p-adic automorphic forms (see Theorems 4.27 and 4.30 for precise statements). Roughly speaking, we show that if a two-dimensional p-adic representation of Gal(F /F ) (F a totally real field) is unramified at a finite place q p and occurs in the completed cohomology of a system of Shimura curves with tame level equal to U 0 (q) at the q-factor, then the system of Hecke eigenvalues attached to occurs in the completed cohomology of Shimura curves with tame level equal to GL 2 (O q ) at the q-factor. 1.2. A p-adic Jacquet-Langlands correspondence. We show the existence of a short exact sequence of admissible unitary Banach representations of GL 2 (F Q Q p ) (Theorem 4.35 in fact the exact sequence is defined integrally) which encodes a p-adic Jacquet- Langlands correspondence from a quaternion algebra D /F, split at one infinite place, with discriminant Nq 1 q 2, to a quaternion algebra D/F, split at the same infinite place and with discriminant N. Applying Emerton s eigenvariety construction to this exact sequence gives a map between eigenvarieties (Theorem 5.18), i.e. an overconvergent Jacquet-Langlands correspondence in the same spirit as the main theorem of [5]. As a corollary of these results we also obtain level raising results (Corollary 5.20). 2. PRELIMINARIES 2.1. Vanishing cycles. In this section we will follow the exposition of vanishing cycles given in the first chapter of [20] - we give some details to fix ideas and notation. Let O be a characteristic zero Henselian discrete valuation ring, residue field k of characteristic l, fraction field L. Let X S = Spec(O) be a proper, generically smooth curve with reduced special fibre, and define Σ to be the set of singular points on X k. Moreover assume that a neighbourhood of each point x Σ is étale locally isomorphic over S to Spec(O[t 1, t 2 ]/(t 1 t 2 a x )) with a x a non-zero element of O with valuation e x = v(a x ) > 0. Define morphisms i and j to be the inclusions: i : X k X, j : X L X. For F a constructible torsion sheaf on X, with torsion prime to l, the Grothendieck (or Leray) spectral sequence for the composition of the two functors Γ(X, ) and j gives an identification RΓ(X L, j F ) = RΓ(X, R(j) j F ). Applying the proper base change theorem we also have RΓ(X, R(j) j F ) RΓ(X k, i R(j) j F ). The natural adjunction id = R(j) j gives a morphism i F i R(j) j F of complexes on X k. We denote by RΦ(F ) the mapping cone of this morphism (the complex of vanishing cycles), and denote by RΨ(F ) the complex i R(j) j F (the complex of nearby cycles). There is a distinguished triangle i F RΨ(F ) RΦ(F ), whence a long exact sequence of cohomology H i (X k, i F ) H i (X k, RΨ(F )) H i (X k, RΦ(F )) As on page 36 of [20] there is a specialisation exact sequence, with (1) denoting a Tate twist of the Gal(L/L) action: (1) 0 H 1 (X k, i F )(1) H 1 (X L, j F )(1) γ H 2 (X k, i F )(1) β x Σ R1 Φ(F ) x(1) sp(1) H 2 (X L, j F )(1) 0. We can extend the above results to lisse étale Z p -sheaves (p l), by taking inverse limits of the exact sequences for the sheaves F /p i F. Since our groups satisfy the Mittag-Leffler

COMPLETED COHOMOLOGY OF SHIMURA CURVES 3 condition we preserve exactness. In particular we obtain the specialisation exact sequence (1) for F. From now on F will denote a Z p -sheaf, and we define so there is a short exact sequence X(F ) := ker(γ) = im(β) x Σ R 1 Φ(F ) x (1), 0 H 1 (X k, i F )(1) H 1 (X L, j F )(1) X(F ) 0. As in section 1.3 of [20] there is also a cospecialisation sequence (2) 0 H 0 ( X k, RΨ(F )) H 0 ( X γ k, F ) x Σ H1 x(x k, RΨ(F )) β H 1 (X k, RΨ(F )) H 1 (X k, F ) 0, where we write X k for the normalisation of X k, and (in the above, and from now on) we also denote by F the appropriate pullbacks of F and RΨ(F ) (e.g. i F ). The normalisation map is denoted r : X k X k. We can now define ˇX(F ) := im(β ), and apply Corollary 1 of [20] to get the following: Proposition 2.2. We have the following diagram made up of two short exact sequences: 0 ˇX(F ) 0 H 1 (X k, F ) H 1 (X L, F ) X(F )( 1) 0 H 1 (X k, r r F ) 0 Note that the map ˇX(F ) H 1 (X k, F ) is the one induced by the fact that ˇX(F ) H 1 (X k, RΨ(F )) is in the image of the injective map H 1 (X k, F ) H 1 (X k, RΨ(F )). Finally, as in (1.13) of [20] there is an injective map coming from the monodromy pairing. λ : X(F ) ˇX(F ) 3. SHIMURA CURVES AND THEIR BAD REDUCTION 3.1. Notation. Let F/Q be a totally real number field of degree d. We denote the infinite places of F by τ 1,..., τ d. We will be comparing the arithmetic of two quaternion algebras over F. The first quaternion algebra will be denoted by D, and we assume that it is split at τ 1 and at all places in F dividing p, and non-split at τ i for i 1 together with a finite set S of finite places. If d = 1 then we furthermore assume that D is split at some finite

4 JAMES NEWTON place (so we avoid working with non-compact modular curves). Fix two finite places q 1 and q 2 of F, which do not divide p and where D is split, and denote by D a quaternion algebra over F which is split at τ 1 and non-split at τ i for i 1 together with each finite place in S {q 1, q 2 }. We fix maximal orders O D and O D of D and D respectively, and an isomorphism D F A (q1q2) F = D F A (q1q2) F compatible with the choice of maximal orders in D and D. We write q to denote one of the places q 1, q 2, and denote the completion F q of F by L, with ring of integers O q and residue field k q. Denote F O q by O (q). We have the absolute Galois group G q = Gal(L/L) with its inertia subgroup I q. Let G and G denote the reductive groups over Q arising from the unit groups of D and D respectively. Note that G and G are both forms of Res F/Q (GL 2 /F ). For U a compact open subgroup of G(A f ) and V a compact open subgroup of G (A f ) we have complex (disconnected) Shimura curves M(U)(C) = G(Q)\G(A f ) (C R)/U M (U)(C) = G (Q)\G (A f ) (C R)/V, where G(Q) and G (Q) act on C R via the τ 1 factor of G(R) and G (R) respectively. Both these curves have canonical models over F, which we denote by M(U) and M (U). We follow the conventions of [3] to define this canonical model see [2] for a discussion of two different conventions used when defining canonical models of Shimura curves. 3.2. Integral models and coefficient sheaves. Let U G(A f ) be a compact open subgroup (we assume U is a product of local factors over the places of F ), with U q = U 0 (q) (the matrices in GL 2 (O q ) which have upper triangular image in GL 2 (k q )), and U unramified at all the finite places v where D is non-split (i.e. U v = O D,v ). Let Σ(U) denote the set of finite places where U is ramified. In Theorem 8.9 of [15], Jarvis constructs an integral model for M(U) over O q which we will denote by M q (U). In fact this integral model only exists when U is sufficiently small (a criterion for this is given in [15, Lemma 12.1]), but this will not cause us any difficulties (see the remark at the end of this subsection). Section 10 of [15] shows that M q (U) satisfies the conditions imposed on X in section 2.1 above. We also have integral models for places dividing the discriminant of our quaternion algebra. Let V G (A f ) be a compact open subgroup with V q = O D,q. We let M q(v ) denote the integral model for M (V ) over O q coming from Theorem 5.3 of [24] (this is denoted C in [20]). This arises from the q-adic uniformisation of M (V ) by a formal O q - scheme, and gives integral models for varying V which are compatible under the natural projections, and again satisfy the conditions necessary to apply section 2.1. We now want to construct sheaves on M q (U) and M q(v ) corresponding to the possible (cohomological) weights of Hilbert modular forms. Let k = (k 1,... k d ) be a d-tuple of integers (indexed by the embeddings τ i of F into R if the reader prefers), all 2 and of the same parity. The d-tuple of integers v is then characterised by having non-negative entries, at least one of which is zero, with k + 2v = (w,..., w) a parallel vector. We will now define p-adic lisse étale sheaves F k on M q (U), as in [4, 16, 22]. Denote by Z the centre of the algebraic group G. Note that Z = Res F/Q (G m,f ). We let Z s denote the maximal subtorus of Z that is split over R but which has no subtorus split over Q. It is straightforward to see that Z s is the subtorus given by the kernel of the norm map N F/Q : Res F/Q (G m,f ) G m,q. In the notation of [16, Ch. III] the algebraic group G c is the quotient of G by Z s. Take F C a number field of finite degree, Galois over Q and containing F such that F splits both D and D. Since F /Q is normal, F contains all the Galois conjugates of F and we

COMPLETED COHOMOLOGY OF SHIMURA CURVES 5 can identify the embeddings {τ i : F C} with the embedding {τ i : F F }, via the inclusion F C. Since D F,τi F = M 2 (F ) for each i, we obtain representations ξ i of D = G(Q) on the space W i = F 2. Let ν : G(Q) F denote the reduced norm. Then ξ (k) = d i=1(τ i ν) vi Sym ki 2 (ξ i ) is a representation of G(Q) acting on W k = d i=1 Symki 2 (W i ). In fact ξ (k) arises from an algebraic representation of G on the Q-vector space underlying W k (or rather the associated Q-vector space scheme). For z Z(Q) = F the action of ξ (k) (z) is multiplication by N F/Q (z) w 2, so ξ (k) factors through G c. The representation ξ (k) gives rise to a represpentation of G(Q p ) on W k Q Q p = W k F (F Q Q p ), so taking a prime p in F above p and projecting to the relevant factor of F Q Q p we obtain an F p-linear representation W k,p of G(Q p ). Now we may define a Q p -sheaf on M(U)(C): F k,qp,c = G(Q)\(G(A f ) (C R) W k,p /U, where G(Q) acts on W k,p by the representation described above (via its embedding into G(Q p )), and U acts only on the G(A f ) factor, by right multiplication. Note that this definition only works because ξ (k) factors through G c. Choose an O F p lattice Wk,p 0 in W k,p. After projection to G(Q p ), U acts on W k,p, and if U is sufficiently small it stabilises Wk,p 0. Pick a normal open subgroup U 1 U acting trivially on Wk,p 0 /pn. We now define a Z/p n Z-sheaf on M(U)(C) by F k,z/p n Z,C = (M(U 1 )(C) (W 0 k,p/p n ))/(U/U 1 ). Now on the integral model M q (U) we can define an étale sheaf of Z/p n Z-modules by F k,z/pn Z = (M q (U 1 ) (W 0 k,p/p n ))/(U/U 1 ). By section 2.1 of [4] this construction is independent of the choice of lattice W 0 k,p. Taking inverse limits of these Z/p n Z-sheaves we get a Z p - (in fact O F p -) lisse étale sheaf on M q (U) which will be denoted by F k. The same construction applies to M q(v ), and we also denote the resulting étale sheaves on M q(v ) by F k. Note that we chose F so that it split both D and D - this allows F p to serve as a field of coefficients for both M q(u) and M q(v ). From now on we work with a coefficient field E. This can be any finite extension of F p. Denote the ring of integers in E by O p, and fix a uniformiser ϖ. The sheaf F k on the curves M q (U) and M q(v ) will now denote the sheaves constructed above with coefficients extended to O p. Remark. Note that in the above discussion we assumed that the level subgroup U was small enough. In the applications below we will be taking direct limits of cohomology spaces as the level at p, U p, varies over all compact open subgroups of G(Q p ), so we can always work with a sufficiently small level subgroup by passing far enough through the direct limit. 3.3. Hecke algebras. We will briefly recall the definition of Hecke operators, as in [8, 20]. Let U and U be two (sufficiently small) compact open subgroups of G(A f ) as in the previous section, and suppose we have g G(A f ) satisfying g 1 U g U. Then there is a (finite) étale map g : M(U ) M(U), corresponding to right multiplication by g on G(A f ). Now for v a finite place of F, with D unramified at v, let η v denote ( the element ) of 1 0 G(A f ) which is equal to the identity at places away from v and equal to at the 0 ϖ v place v, where ϖ v is a fixed uniformiser of O F,v. We also use ϖ v to denote ( the element ) ϖv 0 of G(A f ) which is equal to the identity at places away from v and equal to at 0 ϖ v

6 JAMES NEWTON the place v. Then the Hecke operator T v (or U v if U v is not maximal compact) is defined by means of the correspondence on M(U) coming from the maps 1, ηv whilst S v is defined by the correspondence : M(U η v Uηv 1 ) M(U), 1, ϖv : M(U) M(U). We can make the same definitions for the curve M (V ). Definition 3.4. The Hecke algebra T k (U) is defined to be the O p -algebra of endomorphisms of H 1 (M(U) F, F k ) E generated by Hecke operators T v, S v for finite places v pq 1 q 2 such that U and D are both unramified at v. The Hecke algebra T k (V ) is defined to be the O p -algebra of endomorphisms of H 1 (M (V ) F, F k ) E generated by Hecke operators T v, S v for finite places v p such that V and D are both unramified at v. The Hecke algebras T k (U) and T k (V ) are semilocal, and their maximal ideals correspond to mod p Galois representations arising from Hecke eigenforms occuring in H 1 (M(U) F, F k ) E and H 1 (M (V ) F, F k ) E respectively. Given a maximal ideal m of T k (U) we denote the Hecke algebra s localisation at m by T k (U), where : Gal(F /F ) GL 2 (F p ) is the mod p Galois representation attached to m, and more generally for any T k (U)-module M, denote its localisation at m by M. We apply the same notational convention to T k (V ) and modules for this Hecke algebra. The Hecke algebra T k (U) acts faithfully on the O p -torsion free quotient H 1 (M(U) F, F k ) tf of H 1 (M(U) F, F k ). Moreover, since the action of the Hecke operators on H 1 (M(U) F, F k ) tors is Eisenstein (see [8, Lemma 4]), if as above is irreducible then H 1 (M(U) F, F k ) tf can be naturally viewed as an O p-direct summand of H 1 (M(U) F, F k ), and we denote it by H 1 (M(U) F, F k ). The same consideration applies to any subquotient M of H 1 (M(U) F, F k ), and for the Hecke algebra T k (V ). Note that if k = (2,..., 2) then H 1 (M(U) F, F k ) = H 1 (M(U) F, O p ) is already O p -torsion free. 3.5. Mazur s principle. In the next two subsections we summarise the results of [15] and [20]. First we fix a (sufficiently small) compact open subgroup U G(A f ) such that U q = GL 2 (O q ). For brevity we denote the compact open subgroup U U 0 (q) by U(q). We apply the theory of Section 2.1 to the curve M q (U(q)) and the Z p -sheaf given by F k, for some weight vector k, to obtain O p -modules X q (U(q), F k ) and ˇX q (U(q), F k ). We have short exact sequences (3) 0 H 1 (M q(u(q)) k q, F k ) H 1 (M q(u(q)) L, F k ) X q(u(q), F k )( 1) 0, (4) 0 ˇX q(u(q), F k ) H 1 (M q(u(q)) k q, F k ) H 1 (M q(u(q)) k q, r r F k ) 0. The above short exact sequences are equivariant with respect to the G q action, as well as the action of T k (U(q)) and the Hecke operator U q. The sequence 4 underlies the proof of Mazur s principle (see [15] and Theorem 4.27 below). 3.6. Ribet-Rajaei exact sequence. We also apply the theory of Section 2.1 to the curve M q(v ) and the sheaf F k, where V G (A f ) is a (sufficiently small) compact open subgroup which is unramified at q. We obtain O p -modules Y q (V, F k ) and ˇY q (V, F k ), where Y and ˇY correspond to X and ˇX respectively. As in equation 3.2 of [20] these modules satisfy the (G q and T k (V ) equivariant) short exact sequences 0 H 1 (M q(v ) k q, F k ) H 1 (M q(v ) L, F k ) Y q (V, F k )( 1) 0, 0 ˇY q (V, F k ) H 1 (M q(v ) k q, F k ) H 1 (M q(v ) k q, r r F k ) 0.

COMPLETED COHOMOLOGY OF SHIMURA CURVES 7 It is remarked in section 3 of [20] that in fact H 1 (M q(v ) k q, r r F k ) = 0, since the irreducible components of the special fibre of M q(v ) are rational curves, so we have ˇY q (V, F k ) = H 1 (M q(v ) k q, F k ). We can now relate our constructions for the two quaternion algebras D and D. This requires us to now distinguish between the two places q 1 and q 2. We make some further assumptions on the levels U and V. First we assume that U q1 = O D,q1 = GL2 (O q1 ) and U q2 = O D,q2 = GL2 (O q2 ), and similarly that V is also maximal at both q 1 and q 2. We furthermore assume that the factors of U away from q 1 and q 2, U q1q2, match with V q1q2 under the fixed isomorphism D F A (q1q2) F,f = D F A (q1q2) F,f. We will use the notation U(q 1 ), U(q 2 ), U(q 1 q 2 ) to denote U U 0 (q 1 ), U U 0 (q 2 ), U U 0 (q 1 ) U 0 (q 2 ) respectively. We fix a Galois representation : Gal(F /F ) GL 2 (F p ), satisfying two assumptions. Firstly, we assume that arises from a Hecke eigenform in H 1 (M (V ) F, F k ). Secondly, we assume that is irreducible. Equivalently, the corresponding maximal ideal m in T k (V ) is not Eisenstein. There is a surjection T k (U(q 1 q 2 )) T k(v ) arising from the classical Jacquet Langlands correspondence, identifying T k (V ) as the q 1 q 2 -new quotient of T k (U(q 1 q 2 )). Hence there is a (non-eisenstein) maximal ideal of T k (U(q 1 q 2 )) associated to (in fact this can be established directly from the results contained on pg. 55 of [20], but for expository purposes it is easier to rely on the existence of the Jacquet Langlands correspondence). We can therefore regard all T k (V ) -modules as T k (U(q 1 q 2 )) -modules. Now [20, Theorem 3] implies the following: Theorem 3.7. There are T k (U(q 1 q 2 )) -equivariant short exact sequences 0 Y q1 (V, F k ) X q2 (U(q 1 q 2 ), F k ) i X q2 (U(q 2 ), F k ) 2 0 ˇX q2 (U(q 2 ), F k ) 2 0, i ˇX q2 (U(q 1 q 2 ), F k ) ˇY q1 (V, F k ) 0. In the theorem, the maps labelled i and i are the natural level raising map and its adjoint. To be more precise i = (i 1, i ηq ) where i g is the map coming from the composition of the natural map H 1 (M(U) F, F k ) H 1 (M(U(q)) F, gf k ) with the map coming from the sheaf morphism gf k F k as in [8, pg. 449]. The map i = (i 1, i η q ) where i g is the map coming from the composition of the map arising from the sheaf morphism F k gf k (again see [8, pg. 449]) with the map H 1 (M(U(q)) F, gf k ) H 1 (M(U) F, F k ) arising from the trace map g gf k F k. We think of the short exact sequences in the above Theorem as a geometric realisation of the Jacquet Langlands correspondence between automorphic forms for G and for G. The proof of the above theorem relies on relating the spaces involved to the arithmetic of a third quaternion algebra D (which associated reductive algebraic group G/Q) which is non-split at all the infinite places of F and at the places in S {q 2 }. However, we will just focus on the groups G and G, and view G as playing an auxiliary role.

8 JAMES NEWTON 4. COMPLETED COHOMOLOGY OF SHIMURA CURVES 4.1. Completed cohomology. We now consider compact open subgroups U p of G(A (p) f ), with factor at v equal to O D v at all places v of F where D is non-split. Then we have the following definitions, as in [11, 12]: Definition 4.2. We write H n D (U p, F k /ϖ s ) for the O p module lim Up H n (M(U p U p ) F, F k /ϖ s ), which has a smooth action of G(Q p ) = v p GL 2(F v ) and a continuous action of G F. We then define a ϖ-adically complete O p -module H n (U p, F k ) := lim s H n (U p, F k /ϖ s ), which has a continuous action of G(Q p ) and G F. Lemma 4.3. The O p -module H n D (U p, F k ) is a ϖ-adically admissible G(Q p ) representation over O p (in the sense of [13, Definition 2.4.7]). Proof. This follows from [11, Theorem 2.2.11], taking care to remember the O p -integral structure. Write H n (U p, F k ) E for the admissible continuous Banach G(Q p ) representation H n (U p, F k ) Op E. Lemma 4.4. There is a canonical isomorphism H n (U p, O p ) = lim s H n (U p, O p )/ϖ s H n (U p, O p ), compatible with G F and G(Q p ) actions. Proof. Since M(U p U p ) F is proper of dimension 1, this follows from Corollary 2.2.27 of [11] Remark. Note that in the notation of [11] the above lemma says that H n (U p ) = Ĥn (U p ). The following lemma allows us to restrict to considering completed cohomology with trivial coefficients. Lemma 4.5. There is a canonical G(Q p ), G F equivariant isomorphism H n (U p, F k ) = H n (U p, O p ) Op W 0 k,p. Proof. This follows from Theorem 2.2.17 of [11]. Proposition 4.6. Let F be the closure of F in A f, F. There is an isomorphism between H 0 (U p, O p ) and the space of continuous functions C (F \A f, F / det(u p ), O p ), with GL 2 (Q p ) action on the right hand side induced by g acting on A f, F by multiplication by det(g). Proof. This follows from the fact that the product of the sign map on (C R) and the reduced norm map on G(A f ) induces a map G(Q)\G(A f ) (C R)/U F \A F,f {±}/ det(u) = F \A F,f / det(u) with connected fibres. Proposition 4.7. The space H 2 (U p, O p ) is equal to 0.

COMPLETED COHOMOLOGY OF SHIMURA CURVES 9 Proof. Exactly as for Proposition 4.3.6 of [11] If v is a prime in F such that U p and D are both unramified at v then there are Hecke operators T v, S v acting on the spaces H 1 (U p, O p /ϖ s ), defined as in Section 3.3. By continuity this extends to an action of the Hecke operators on the space H 1 D (U p, O p ), so we can make the following definition: Definition 4.8. The Hecke algebra T(U p ) is defined to be lim Up T (2,...,2) (U p U p ). We endow it with the projective limit topology given by taking the ϖ-adic topology on each finitely generated O p -algebra T (2,...,2) (U p U p ). The topological O p -algebra T(U p ) acts faithfully on H 1 D (U p, O p ), and is topologically generated by Hecke operators T v, S v for primes v p such that U p and D are both unramified at v. Note that the isomorphisms of lemmas 4.4 and 4.5 are equivariant with respect to the Hecke action, so we will often just work with the completed cohomology spaces defined with trivial coefficients. Definition 4.9. An ideal I of T(U p ) is Eisenstein if the map λ : T (U p ) T (U p )/I satisfies λ(t q ) = ɛ 1 (q) + ɛ 2 (q) and λ(qs q ) = ɛ 1 (q)ɛ 2 (q) for all q / {p} Σ, for some characters ɛ 1, ɛ 2 : Z pσ T (U p )/I. The Hecke algebra T(U p ) is semilocal and Noetherian, with maximal ideals corresponding to mod p Galois representations arising from Hecke eigenforms in (an extension of scalars of) HD 1 (U p, O p /ϖ). Given a maximal ideal m of T(U p ) we denote the Hecke algebra s localisation at m by T(U p ), where : Gal(F /F ) GL 2 (F p ) is the mod p Galois representation attached to m, and more generally for any T(U p )-module M, denote its localisation at m by M. We now fix such a, and assume that it is irreducible. There is a deformation,u m of p to T(U p ) (taking a projective limit, over compact open subgroups U p of G(Q p ), of the deformations to T (2,...,2) (U p U p ) ). For a closed point P Spec(T(U p ) [ 1 p ]) we denote by (P) the attached Galois representation, defined over the characteristic 0 field k(p). Definition 4.10. Given a T(U p ) -module M and a closed point P of Spec(T(U p ) [ 1 p ]), we denote by M[P] the k(p)-vector space {m M Op k(p) : T m = 0 for all T P}. Suppose we have a homomorphism λ : T(U p ) E, where E /E is a finite field extension. We say that the system of Hecke eigenvalues λ occurs in a T(U p ) -module M if M Op E contains a non-zero element where T(U p ) acts via the character λ. In other words, λ occurs in M if and only if M[ker(λ)] 0. We say that λ is Eisenstein whenever ker λ is. We have the following result, generalising [12, Proposition 5.5.3]. Denote by X the O p -module Hom T(U p ) [G F ]( m,u p, H 1 D (U p, O p ) ). Proposition 4.11. The natural evaluation map is an isomorphism. ev : m,u p T(U p ) X H 1 D(U p, O p ) Proof. First let P Spec(T(U p ) [ 1 p ]) be the prime ideal of the Hecke algebra corresponding to a classical point coming from a Hecke eigenform in Then the image of the map H 1 (M(U p U p ), O p ) Op k(p). (P) k(p) X[P] H 1 D(U p, O p ) Op k(p)

10 JAMES NEWTON induced by ev contains H 1 (M(U p U p ), O p )[P]. From this, it is straightforward to to see that the image of the map ev E : E Op m,u p T(U p ) X H 1 D(U p, O p ),E is dense, since it will contain the dense subspace H 1 D (U p, O p ),E. Now X is a ϖ-adically admissible G(Q p )-representation over T(U p ), hence by [12, Proposition 3.1.3] the image of ev E is closed, therefore ev E is surjective. We now note that the evaluation map k(p) Hom GF (, H 1 D(U p, O p /ϖ) ) H 1 D(U p, O p /ϖ) is injective, by the irreducibility of. We conclude the proof of the proposition by applying the following lemma (see [12, Lemma 3.1.6]). Lemma 4.12. Let π 1 and π 2 be two ϖ-adically admissible G(Q p ) representations over a complete local Noetherian O p -algebra A (with maximal ideal m), which are torsion free as O p -modules. Suppose f : π 1 π 2 is a continuous A[G(Q p )]-linear morphism, satisfying (1) The induced map π 1 Op E π 2 Op E is a surjection (2) The induced map π 1 /ϖ[m] π 2 /ϖ[m] is an injection. Then f is an isomorphism. Proof. First we show that f is an injection. Denote the kernel of f by K. It is a ϖ-adically admissible G(Q p ) representation over A, so in particular K/ϖ = i 1 K/ϖ[m i ]. Our second assumption on f then implies that K/ϖ = 0, which implies K = 0, since K is ϖ adically separated. We now know that f is an injection, and out first assumption on f implies that it has O p -torsion cokernel C. Since π 1 is O p -torsion free there is an exact sequence 0 C[ϖ] π 1 /ϖ π 2 /ϖ. Since π 1 /ϖ = i 1 π 1/ϖ[m i ] we have C[ϖ] = i 1 C[ϖ][mi ], but the second assumption on f implies that C[ϖ][m] = 0. Therefore C[ϖ] = 0, and since C is O p -torsion we have C = 0. Therefore f is an isomorphism. 4.13. Completed vanishing cycles. We now assume that our tame level U p is unramified at the prime q, and let U p (q) denote the tame level U p U 0 (q). Fix a non-eisenstein maximal ideal m of T(U p (q)). We are going to ϖ-adically complete the constructions of subsection 3.5. Definition 4.14. For F equal to either F k or r r F k we define O p -modules with smooth G(Q p ) actions H 1 red(u p (q), F ) = lim Up H 1 (M q (U p U p (q)) k q, F ), X q (U p (q), F k ) = lim Up X q (U p U p (q), F k ), and ˇX q (U p (q), F k ) = lim Up ˇXq (U p U p (q), F k ). Proposition 4.15. Equation 3 induces a short exact sequence, equivariant with respect to Hecke, G(Q p ) and G q actions: 0 H 1 red (U p (q), F k ) H 1 (U p (q), F k ) X q (U p (q), F k )( 1) 0. Similarly, Equation 4 induces 0 ˇX q (U p (q), F k ) H 1 red (U p (q), F k ) H 1 red (U p (q), r r F k ) 0.

COMPLETED COHOMOLOGY OF SHIMURA CURVES 11 Proof. This just follows from taking the direct limit of the relevant exact sequences. Definition 4.16. We define O p -modules with continuous G(Q p ) actions (again F equals either F k or r r F k ) H 1 red(u p (q), F ) = lim s H 1 red(u p (q), F )/ϖ s H 1 red(u p (q), F ), and X q (U p (q), F k ) = lim s X q (U p (q), F k )/ϖ s X q (U p (q), F k ) ˇX q (U p (q), F k ) = lim s ˇXq (U p (q), F k )/ϖ s ˇXq (U p (q), F k ). Now ϖ-adically completing the short exact sequences of Proposition 4.15, we obtain Proposition 4.17. We have short exact sequences of ϖ-adically admissible G(Q p )- representations over O p, equivariant with respect to Hecke, G(Q p ) and G q actions: 0 H 1 red (U p (q), F k ) m H 1 D (U p (q), F k ) m X q (U p (q), F k ) m ( 1) 0, 0 ˇXq (U p (q), F k ) m H 1 red (U p (q), F k ) m H 1 red (U p (q), r r F k ) m 0. Proof. The existence of a short exact sequence of O p -modules follows from the fact that X q (U p U p (q), F k ) m and Hred 1 (U p (q), r r F k ) m is always O p -torsion free. The module X q (U p U p (q), F k ) is torsion free even without localising at m (see the remark following [20, Corollary 1]), whilst for Hred 1 (U p (q), r r F k ) m this follows as in [8, Lemma 4]. The fact that we obtain a short exact sequence of ϖ-adically admissible O p [G(Q p )]-modules follows from Proposition 2.4.4 of [13]. It is now straightforward to deduce the analogue of Lemma 4.5: Lemma 4.18. There are canonical G(Q p ), G q and Hecke equivariant isomorphisms H 1 red(u p (q), F k ) m = H1 red (U p (q), O p ) m Op W 0 k,p, ˇX q (U p (q), F k ) m = ˇXq (U p (q), O p ) m Op W 0 k,p, X q (U p (q), F k ) m = Xq (U p (q), O p ) m Op W 0 k,p. 4.19. Mazur s principle. We can now proceed as in Theorem 6.2.4 of [12], following Jarvis s approach to Mazur s principle over totally real fields [15]. First we fix an absolutely irreducible mod p Galois representation, coming from a maximal ideal of T(U p (q)). We have the following proposition Proposition 4.20. The injection H 1 red(u p (q), O p ) H 1 D(U p (q), O p ) induces an isomorphism H 1 red(u p (q), O p ) = ( H1 D (U p (q), O p ) ) Iq. Proof. This follows from Proposition 4 and (2.2) in [20]. There is a Hecke operator U q acting on H 1 (U p (q), O p ), which induces an action on the spaces ˇX q (U p (q), O p ) and X q (U p (q), O p ). This extends by continuity to give an action of U q on ˇXq (U p (q), O p ) and X q (U p (q), O p ). Lemma 4.21. The Frobenius element of Gal(k q /k q ) acts on ˇXq (U p (q), O p ) by (Nq)U q and on X q (U p (q), O p ) by U q. Proof. Section 6 of [4] tells us that Frobenius acts as required on the dense subspaces ˇX q (U p (q), O p ) and X q (U p (q), O p ), so by continuity we are done.

12 JAMES NEWTON Lemma 4.22. There is a natural isomorphism H 1 red(u p (q), r r O p ) = H 1 D(U p, O p ) 2. Proof. This follows from Lemma 16.1 in [15]. Lemma 4.23. The monodromy pairing map λ : X q (U p (q), O p ) ˇX q (U p (q), O p ) induces a G q and T(U p (q))-equivariant isomorphism X q (U p (q), O p ) = ˇXq (U p (q), O p ). Proof. This follows from Proposition 5 in [20]. Remark. Note that the isomorphism in the above lemma is not U q equivariant. Before giving an analogue of Mazur s principle, we will make explicit the connection between the modules X q (U p (q), O p ) and a space of newforms. There are natural level raising maps i : H1 D (U p, O p ) 2 H D 1 (U p (q), O p ), and their adjoints i : H D 1 (U p (q), O p ) H D 1 (U p, O p ) 2, defined by taking the limit of maps at finite level defined as in the paragraph after Theorem 3.7. Definition 4.24. The space of q-newforms H 1 D(U p (q), O p ) q-new is defined to be the kernel of the map i : H 1 D(U p (q), O p ) H 1 D(U p, O p ) 2. Composing the injection H red 1 (U p (q), O p ) H D 1 (U p (q), O p ) with i gives a map, which we also denote by i, from H red 1 (U p (q), O p ) to H D 1 (U p, O p ) 2. Recall that we can also define a map ω between these spaces, by composing the natural map H 1 red(u p (q), O p ) H 1 red(u p (q), r r O p ) with the isomorphism of Lemma 4.22. However these maps are not equal, as remarked in [21, pg. 447]. Fortunately, they have the same kernel, as we now show (essentially by following the proof of [21, Theorem 3.11]). Proposition 4.25. The kernel of the map i : H 1 red(u p (q), O p ) H 1 D(U p, O p ) 2 is equal to the image of ˇXq (U p (q), O p ) in H 1 red (U p (q), O p ). Proof. Since all our spaces are O p -torsion free, it suffices to show that the kernel of the map i : H 1 (M q (U p U p (q)) k q, O p ) H 1 (M q (U p U p ) k q, O p ) 2 is equal to the image of ˇXq (U p U p (q), O p ) in H 1 (M q (U p U p (q)) k q, O p ) for every U p. In fact this just needs to be checked after tensoring everything with E. For brevity we write U for U p U p and U(q) for U p U p (q). We complete the proof by relating our constructions to the Jacobians of Shimura curves. For simplicity we assume that our Shimura curves have just one connected component, but the argument is easily modified to work in the disconnected case by taking products of Jacobians of connected components. The degeneracy maps 1, ηq induce by Picard functoriality a map : Jac(M(U)/F ) Jac(M(U)/F ) Jac(M(U(q))/F ) with finite kernel, giving the map i on cohomology. The dual map (induced by Albanese functoriality from the degeneracy maps) gives the map i on cohomology. Let J 0 denote the connected component of 0 in the special fibre at q of the Néron model of Jac(M(U(q))/F ). The map ω arises from a map Ω : J 0 Jac(M q (U) k q ) Jac(M q (U) k q ), which identifies Jac(M q (U) k q ) Jac(M q (U) k q ) as the maximal Abelian variety quotient of J 0

COMPLETED COHOMOLOGY OF SHIMURA CURVES 13 (which is a semi-abelian variety). The composition of Ω with the map induced by on the special fibre gives an auto-isogeny A of Jac(M q (U) k q ) Jac(M q (U) k q ). The quasi-auto-isogeny (deg ) 1 A gives an automorphism of H 1 (M q (U p U p ) k q, E) 2. Denote the inverse of this automorphism by γ. Then the construction of the automorphism γ implies that there is a commutative diagram H 1 (M q (U p U p (q)) k q, E) ω H 1 (M q (U p U p ) k q, E) 2 γ H 1 (M q (U p U p (q)) k q, E) i H 1 (M q (U p U p ) k q, E) 2, which identifies the kernel of i with the kernel of ω. Proposition 4.26. A system of Hecke eigenvalues λ : T(U p (q)) E occurs in if and only if it occurs in X q (U p (q), O p ). H 1 D(U p (q), O p ) q-new Proof. Proposition 4.17 and Lemma 4.22, together with Definition 4.24 and Proposition 4.25 imply that we have the following commutative diagram of ϖ-adically admissible G(Q p ) representations over O p : 0 ˇXq (U p (q), O p ) H red 1 (U p ω (q), O p ) H1 D (U p, O p ) 2 0 α β γ 0 H 1 D (U p (q), O p ) q-new H 1 D (U p (q), O p ) i H 1 D (U p, O p ) 2, where the two rows are exact and the maps α, β and γ are injections. Note that both the existence of the map γ and its injectivity follow from Proposition 4.25. Moreover coker(β) = X q (U p (q), O p ) ( 1) = ˇXq (U p (q), O p ) ( 1), where the second isomorphism comes from the monodromy pairing (see Lemma 4.23). Applying the snake lemma we see that coker(α) coker(β), so (applying Lemma 4.23 once more to the source of the map α) we have an exact sequence 0 X q (U p (q), O p ) α H1 D (U p (q), O p ) q-new X q (U p (q), O p ) ( 1). From this exact sequence it is easy to deduce that the systems of Hecke eigenvalues occurring in H 1 D (U p (q), O p ) q-new and X q (U p (q), O p ) are the same. Theorem 4.27. Let : Gal(F /F ) GL 2 (Q p ) be a Galois representation, with irreducible reduction : Gal(F /F ) GL 2 (F p ). Suppose the following two conditions are verified: (1) is unramified at the prime q (2) (Frob q ) is not a scalar (3) There is a system of Hecke eigenvalues λ : T(U p ) E attached to (i.e. = (ker(λ))), with λ occurring in H 1 D (U p (q), O p ). Then λ occurs in H 1 D (U p, O p ). Proof. We take as in the statement of the theorem. The third assumption, combined with Proposition 4.11 implies that there is a prime ideal P = ker(λ) of T(U p (q)), and an embedding H 1 D(U p (q), O p )[P]. Our aim is to show that H 1 D (U p, O p )[P] 0.

14 JAMES NEWTON Let T(U p (q)) denote the finite ring extension of T(U p (q)) obtained by adjoining the operator U q. Since is irreducible there is a prime P of T(U p (q)) such that there is an embedding H 1 D(U p (q), O p )[P ]. Recall that by Propositions 4.17 and 4.20 and Lemma 4.22 there is a G q and T(U p (q))- equivariant short exact sequence 0 ˇXq (U p (q), O p ) H 1 D (U p (q), O p ) Iq ω H 1 D (U p, O p ) 2 0. Suppose for a contradiction that H D 1 (U p, O p ) [P] = 0 Let V denote the G q -representation obtained from by restriction. Since is unramified at q, there is an embedding V H D 1 (U p (q), O p ) Iq [P ]. The above short exact sequence then implies that the embedding ˇX q (U p (q), O p ) [P] H 1 D(U p (q), O p ) Iq [P] is an isomorphism, and hence the embedding is also an isomorphism. ˇX q (U p (q), O p ) [P ] H 1 D(U p (q), O p ) Iq [P ] This implies that there is an embedding V ˇXq (U p (q), O p ) [P ], so by Lemma 4.21 (Frob q ) is a scalar (Nq)α where α is the U q eigenvalue coming from P, contradicting our second assumption. 4.28. Mazur s principle and the Jacquet functor. After taking locally analytic vectors and applying Emerton s locally analytic Jacquet functor (see [9]) we can prove a variant of Theorem 4.27. We let B denote the Borel subgroup of G(Q p ) consisting of upper triangular matrices, and let T denote the maximal torus contained in B. We then have a locally analytic Jacquet functor J B as defined in [9], which can be applied to the space of locally analytic vectors (V ) an of an admissible continuous Banach E[G(Q p )]-representation V. The following lemma follows from Proposition 4.11, but we give a more elementary separate proof. Lemma 4.29. Let λ : T(U p (q)) E be a system of Hecke eigenvalues such that λ occurs in J B ( H 1 D (U p (q), E) an ). Suppose the attached Galois representation is absolutely irreducible.then there is a non-zero Gal(F /F )-equivariant map (necessarily an embedding, since is irreducible) J B ( H 1 D(U p (q), E) an ). Proof. To abbreviate notation we let M denote J B ( H D 1 (U p (q), E) an ). Since M is an essentially admissible T representation, the system of Hecke eigenvalues λ occurs in the χ-isotypic subspace M χ for some continuous character χ of T. So (again by essential admissibility) M χ,λ is a non-zero finite dimensional Gal(F /F )-representation over E, and therefore by the Eichler-Shimura relations (as in section 10.3 of [3]) and the irreducibility of the desired embedding exists. Theorem 4.30. Suppose that the system of Hecke eigenvalues λ : T(U p (q)) E occurs in J B ( H 1 D (U p (q), E) an ), that the attached Galois representation : Gal(F /F ) GL 2 (E) is unramified at q, with (F rob q ) not a scalar, and that the reduction : Gal(F /F ) GL 2 (F p ) is irreducible. Then λ occurs in J B ( H 1 D (U p, E) an ).

COMPLETED COHOMOLOGY OF SHIMURA CURVES 15 Proof. We take as in the statement of the theorem. Let m be the maximal ideal of T(U p (q)) attached to, and let P be the prime ideal of T(U p (q)) attached to, so applying Lemma 4.29 gives an embedding J B ( H D(U 1 p (q), E) an )[P]. Our aim is to show that J B ( H D 1 (U p, E) an)[p] 0. Let T(U p (q)) denote the finite ring extension of T(U p (q)) obtained by adjoining the operator U q. Since is irreducible there is a prime P of T(U p (q)) such that there is an embedding J B ( H 1 D(U p (q), E) an )[P ]. By Propositions 4.17 and 4.20 and Lemma 4.22 there is a short exact sequence 0 ˇXq (U p (q), E) H D 1 (U p (q), E) Iq H D 1 (U p, E) 2 0. Taking locally analytic vectors (which is exact) and applying the Jacquet functor (which is left exact), we get an exact sequence 0 J B( ˇXq(U p (q), E) an ) J B(( H D(U 1 p (q), E) Iq )an ) ω J B( H 1 D(U p, E) an ) 2. Note that J B (( H D 1 (U p (q), E) Iq )an ) is equal to J B ( H D 1 (U p (q), E) an, since under )Iq the isomorphism of Proposition 4.11 they both correspond to ( m,u p (q) )Iq T(U p (q)) J B (X an E ). Let V denote the G q -representation obtained from by restriction. Since is unramified at q, there is an embedding V J B (( H D 1 (U p (q), E) Iq )an )[P ]. Suppose for a contradiction that J B ( H D 1 (U p, E) an )[P] = 0. The above exact sequence then implies that the embedding J B ( ˇXq (U p (q), E) an )[P] J B (( H D(U 1 p (q), E) Iq )an )[P] is an isomorphism, and hence the embedding J B ( ˇXq (U p (q), E) an )[P ] J B (( H D(U 1 p (q), E) Iq )an )[P ] is also an isomorphism. This implies that there is an embedding V J B ( ˇXq (U p (q), E) an)[p ], so by Lemma 4.22 (Frob q ) is a scalar (Nq)α corresponding to the (Nq)U q eigenvalue in P. We then obtain a contradiction exactly as in the proof of Theorem 4.27. 4.31. A p-adic Jacquet Langlands correspondence. We now wish to relate the arithmetic of p-adic automorphic forms for G and G. The definitions and results of subection 4.1 apply to G. We denote the associated completed cohomology spaces of tame level V p by H 1 D (V p, F k ), and its Hecke algebra by T (V p ). The integral models M q and the vanishing cycle formalism also allows us to define some more topological O p -modules: Definition 4.32. We define O p -modules with smooth G(Q p ) actions Y q (V p, F k ) = lim Vp Y q (V p V p, F k ), and ˇY q (V p, F k ) = lim Vp ˇYq (V p V p, F k ). We define O p -modules with continuous G(Q p ) actions Ỹ q (V p, F k ) = lim s Y (V p, F k )/ϖ s Y (V p, F k )

16 JAMES NEWTON and ˇY q (V p, F k ) = lim s ˇYq (V p, F k )/ϖ s ˇYq (V p, F k ). Fixing a non-eisenstein maximal ideal m of T (V p ), then in exactly the same way as we obtained Proposition 4.17 we get: Proposition 4.33. We have a short exact sequence of ϖ-adically admissible G(Q p ) representations over O p, equivariant with respect to T (V p ) m, G(Q p ) and G q actions: 0 ˇY q (V p, F k ) m H 1 D (V p, F k ) m Ỹq(V p, F k ) m ( 1) 0. Fix U p G(A (p) f ) such that U p is unramified at the places q 1, q 2, and matches with V p at all other places. We now fix an irreducible mod p Galois representation, arising from a (necessarily non-eisenstein) maximal ideal m of T (V p ). There is a corresponding maximal ideal m 0 of T (2,...,2) (V p V p ) for V p a small enough compact open subgroup of G (Q p ), and this pulls back by the map T (2,...,2) (U p (q 1 q 2 )U p ) T (2,...,2) (V p V p ) to a maximal ideal m 0 of T (2,...,2) (U p (q 1 q 2 )U p ) for U p the compact open subgroup of G(Q p ) which is identified with V p under the isomorphism G(Q p ) = G (Q p ). Hence there is a map T(U p (q 1 q 2 )) T (V p ). We can give a refinement of Proposition 4.11 in this situation. Firstly we note that the Galois representation,v m : G p F GL 2 (T (V p ) Op E) has an unramified sub-g q representation V 0,E (which is a free T (V p ) Op E-module of rank one), with unramified quotient V 1,E = V0,E ( 1) (also a free T (V p ) Op E-module of rank one). This follows from the fact that each representation : G F GL 2 (T (2,...,2) (V p V p ) Op E) has a unique unramified subrepresentation of the right kind. Denote by Y the O p -module Hom T (V p ) [G F ]( m,v p, H D 1 (V p, O p ) ). By proposition 4.11 we have an isomorphism ev E : E Op m,v p T (V p ) Y = H 1 D (V p, O p ). Proposition 4.34. The isomorphism ev E identifies the image of the embedding ˇY q (V p, O p ),E H 1 D (V p, O p ),E with V 0,E T (V p ) Y. It follows that the quotient Ỹq(V p, O p ),E ( 1) is identified with V 1,E T (V p ) Y. Proof. We proceed as in the proof of proposition 4.11. First let P Spec(T (V p ) Op E) be the prime ideal of the Hecke algebra corresponding to a classical point coming from a Hecke eigenform in H 1 (M (V p V p ), O p ) Op k(p). Taking V p invariants, extending scalars to k(p) and taking the P torsion parts, the isomorphism ev E induces an isomorphism (P) k(p) Hom k(p)[gf ]((P), H 1 (M (V p V p), O p),e[p]) = H 1 (M (V p V p), O p),e[p], under which ˇY q(v p V p, O p),e[p] is identified with V 0,E(P) k(p) Hom k(p)[gf ]((P), H 1 (M (V p V p), O p),e[p]). A density argument now completes the proof. Passing to the direct limit over compact open subgroups U p of G(Q p ) and ϖ-adically completing the short exact sequences of Theorem 3.7, we obtain Theorem 4.35. We have the following short exact sequences of ϖ-adically admissible G(Q p )-representations over O p, equivariant with respect to T(U p (q 1 q 2 )) and G(Q p ) = G (Q p ) actions: 0 Ỹq 1 (V p, O p ) X q2 (U p (q 1 q 2 ), O p ) i X q2 (U p (q 2 ), O p ) 2 0, 0 ˇXq2 (U p (q 2 ), O p ) 2 ˇXq2 (U p (q 1 q 2 ), O p ) ˇY q1 (V p, O p ) 0.

COMPLETED COHOMOLOGY OF SHIMURA CURVES 17 Proof. The proof is as for Proposition 4.17. We just need to check that the short exact sequences of Theorem 3.7 are compatible as U p and V p vary, but this is clear from the proof of [20, Theorem 3], since the descriptions of the dual graph of the special fibres of M q2 (U p (q 1 q 2 )U p ) and M q 1 (V p V p ) are compatible as U p and V p vary. We regard Theorem 4.35 (especially the first exact sequence therein) as a geometric realisation of a p-adic Jacquet Langlands correspondence. The following propositions explain why this is reasonable. We let X q2 (U p (q 1 q 2 ), O p ) q1-new denote the kernel of the map i : X q2 (U p (q 1 q 2 ), O p ) X q2 (U p (q 2 ), O p ) 2. Proposition 4.36. A system of Hecke eigenvalues λ : T(U p (q 1 q 2 )) E occurs in H D 1 (U p (q 1 q 2 ), O p ) q1q2-new if and only if it occurs in X q2 (U p (q 1 q 2 ), O p ) q1-new. Proof. We have the following commutative diagram: 0 0 X q2 (U p (q 1 q 2 ), O p ) q1-new X q2 (U p (q 1 q 2 ), O p ) X q2 (U p (q 2 ), O p ) 2 α β γ H D 1 (U p (q 1 q 2 ), O p ) q1q2-new H D 1 (U p (q 1 q 2 ), O p ) q2-new ( H 1 D (U p (q 2 ), O p ) q2-new 0 where the two columns are exact, and the maps α, β and γ are injections. Note that we are applying Lemma 4.23. By the snake lemma we have an exact sequence 0 coker(α) cokerβ cokerγ, and the proof of Proposition 4.26 implies that there are injections cokerβ X q2 (U p (q 1 q 2 ), O p ) ( 1), cokerγ X q2 (U p (q 2 ), O p ) 2 ( 1). This shows that there is a Hecke-equivariant injection Therefore, we have an exact sequence coker(α) X q2 (U p (q 1 q 2 ), O p ) q1-new ( 1). 0 X q2 (U p (q 1 q 2 ), O p ) q1-new α ) 2 H 1 D (U p (q 1 q 2 ), O p ) q1q2-new X q2 (U p (q 1 q 2 ), O p ) q1-new ( 1). From this last exact sequence it is easy to deduce that the systems of Hecke eigenvalues occurring in H D 1 (U p (q 1 q 2 ), O p ) q1q2-new and X q2 (U p (q 1 q 2 ), O p ) q1-new are the same. Proposition 4.37. A system of Hecke eigenvalues λ : T (V p ) E occurs in H 1 D (V p, O p ) if and only if it occurs in Ỹq 1 (V p, O p ). Proof. This follows from Proposition 4.34. Combined with Theorem 4.35 this has as a consequence

18 JAMES NEWTON Corollary 4.38. The first exact sequence of Theorem 4.35 induces a bijection between the systems of eigenvalues for T (V p ) appearing in H 1 D (V p, O p ) and the systems of eigenvalues for T(U p (q 1 q 2 )) appearing in H 1 D (U p (q 1 q 2 ), O p ) q1q2-new. 5. EIGENVARIETIES AND AN OVERCONVERGENT JACQUET LANGLANDS CORRESPONDENCE In this section we will apply Emerton s construction of eigenvarieties to deduce an overconvergent Jacquet Langlands correspondence from Theorem 4.35. 5.1. Locally algebraic vectors. Fix a maximal ideal m of T(U p (q)) which is not Eisenstein. Recall that given an admissible G(Q p )-representation V over E, Schneider and Teitelbaum [23] define an exact functor by passing to the locally Q p -analytic vectors V an, and the G(Q p )-action on V an differentiates to give an action of the Lie algebra g of G(Q p ). Proposition 5.2. There are natural isomorphisms (for F = F k or r r F k ) (1) H 0 (g, H 1 D (U p (q), F k ) an m,e ) = H 1 D (U p (q), F k ) m,e, (2) H 1 (g, H 1 D (U p (q), F k ) an m,e ) = 0, (3) H 0 (g, H 1 red (U p (q), F ) an m,e ) = H 1 red (U p (q), F ) m,e, (4) H 1 (g, H 1 red (U p (q), F ) an m,e ) = 0, (5) H 0 (g, X q (U p (q), F k ) an m,e ) = X q (U p (q), F k ) m,e, (6) H 0 (g, ˇXq (U p (q), F k ) an m,e ) = ˇX q (U p (q), F k ) m,e, Proof. The first and second natural isomorphisms comes from the spectral sequence (for completed étale cohomology) discussed in [11, Proposition 2.4.1], since localising at m kills H i D (U p (q), F k ) and H i D (U p (q), F k ) for i = 0, 2. The same argument works for the third and fourth isomorphisms, since as noted in the proof of [11, Proposition 2.4.1] the spectral sequence of [11, Corollary 2.2.18] can be obtained using the Hochschild-Serre spectral sequence for étale cohomology, which applies equally well to the cohomology of the special fibre of M q (U p U p (q)). We now turn to the fifth isomorphism. By the first short exact sequence of Proposition 4.17 and the exactness of localising at m, inverting ϖ and then passing to locally analytic vectors, there is a short exact sequence 0 H 1 red (U p (q), F k ) an m,e H 1 D (U p (q), F k ) an m,e X q (U p (q), F k )( 1) an m,e 0. Taking the long exact sequence of g-cohomology then gives an exact sequence 0 H 0 (g, H 1 red (U p (q), F k ) an m,e ) H 0 (g, H 1 D (U p (q), F k ) an m,e ) H 0 (g, X q (U p (q), F k )( 1) an m,e ) H 1 (g, H 1 red (U p (q), F k ) an m,e ), so applying the third, first and fourth part of the proposition to the first, second and fourth terms in this sequence respectively we get a short exact sequence 0 H 1 red (U p (q), F k ) m,e H 1 D (U p (q), F k ) m,e H 0 (g, X q (U p (q), F k )( 1) an m,e ) 0, so identifying this with the short exact sequence of Proposition 2.3.2 gives the desired isomorphism H 0 (g, X q (U p (q), F k ) an m,e) = X q (U p (q), F k ) m,e.