A p-adic Maass–Shimura operator on Mumford curves

We study a p-adic Maass–Shimura operator in the context of Mumford curves defined by [15]. We prove that this operator arises from a splitting of the Hodge filtration, thus answering a question in [15]. We also study the relation of this operator with generalized Heegner cycles, in the spirit of [1, 4, 19, 28].


Introduction
The main purpose of this paper is to study in the context of Mumford curves a p-adic variant of the Maass-Shimura operator, and relate it to generalized Heegner cycles.
The real analytic Maass-Shimura operator is defined by the formula where z is a variable in the complex upper half plane H, f is a real analytic modular form of weight k, and z →z denotes the complex conjugation; here δ k ( f ) is a real analytic modular form of weight k + 2. The relevance of this operator arises in studying algebraicity properties of Eisenstein series and L-functions: see Shimura [42], Hida [20,Chapter 10]. One of the  [42] is the following. Let δ r k = δ k+2(r −1) • δ k+2(r −2) • · · · • δ k for any r ≥ 1, and let K be an imaginary quadratic field. Then there exists K ∈ C × such that for every modular form f of weight k with algebraic Fourier coefficients, and for every CM point z ∈ K ∩ H, we have Katz described in [25] the Maass-Shimura operator in more abstract terms by means of the Gauss-Manin connection (see also [27]). Let N ≥ 1 be an integer, X 1 (N ) the modular curve of level 1 (N ) over Q, and let π : E → X 1 (N ) be the universal elliptic curve. Consider the relative de Rham cohomology sheaf on X 1 (N ), and define L r = Sym r (L 1 ). Let ω = π * 1 E/X 1 (N ) . The sheaf ω is invertible and we have the Hodge filtration Considering the associated real analytic sheaves, which we denote by a superscript ran , the Hodge exact sequence of real analytic sheaves associated with (3) admits indeed a splitting whereω ran is obtained from ω by applying the complex conjugation. The Maass-Shimura operator ∞,r : (ω ran ) ⊗r −→ (ω ran ) ⊗(r +2) can then be defined combining the splitting (4) with the Gauss-Manin connection and the Kodaira-Spencer map. For details, the reader is referred to [26,Sect. 1.8] and [4,Sect. 1.2]; for the case of Siegel modular forms, see [17,Sect. 4] while for the case of Shimura curves see [19,Sect. 3], [37,Sect. 2]. As hinted from the above discussion, Katz description of the Maass-Shimura operator rests on the fact that the real analytic Hodge sequence (3) splits. In [26,Sect. 1.11], Katz introduces a p-adic analogue of this splitting. Suppose that p N is a prime number, and let X ord 1 (N ) denote the ordinary locus of the modular curve, viewed as a rigid analytic scheme over Q p . Let F rig be the rigid analytic sheaf associated with a sheaf F on X 1 (N ). Then L rig 1 splits over X ord 1 (N ) as the direct sum has the property that the Frobenius endomorphism acts invertibly on this sheaf). This allows to define a differential operator p,r , which can be seen as a p-adic analogue of the Maass-Shimura operator; this operator can be described in terms of Atkin-Serre derivative. At CM points the splittings ∞ and p coincide, and therefore one deduces rationality results for the values of p,r at CM points from (2). For details, see [4,Proposition 1.12]. The p-adic Maass-Shimura operator is then used in [4,26] to construct p-adic L-functions and study their properties.
We now fix an integer N , a prime p N , and a quadratic imaginary field in which p is inert. In this context, Kritz introduced in [28] for modular forms of level N a new p-adic Maass-Shimura operator by using perfectoid techniques, and defined p-adic L-functions by means of this operator, thus removing the crucial assumption that p is split in K , but keeping the assumption that p is a prime of good reduction for the modular curve. Moreover, Andreatta-Iovita [1] introduced still another p-adic Maass-Shimura operator in [1], and obtained results analogous to those in [4], thus extending their work to the non-split case.
On the other hand, Franc in his thesis [15] proposed still another p-adic Maass-Shimura operator for primes p which are inert in K , in the following context. Let N ≥ 1 be an integer, K /Q a quadratic imaginary field, p N a prime number, p ≥ 5, which is inert in K , and let N p = N + · N − p be a factorization of N p into coprime integers such that N + is divisible only by primes which are split in K , and N − p is a square-free product of an even number of prime factors which are inert in K . Let B be the indefinite quaternion algebra of discriminant N − p, R an Eichler order of B of level N + , and X the Shimura curve attached to (B, R). The rigid analytic curve X rig over Q p is then a Mumford curve, namely X (C p ) is isomorphic to the rigid analytic quotient of the p-adic upper half plane H p (Q p ) = C p − Q p by an arithmetic subgroup of SL 2 (Q p ). Franc defines in this context a p-adic Maass-Shimura operator δ p,k by mimicking the definition (1), formally replacing the variable z ∈ H with the p-adic variable z ∈ H p (Q unr p ) =Q unr p − Q p , and replacing the complex conjugation with the Frobenius map (hereQ unr p is the completion of the maximal unramified extension of Q p ). Following the arguments of [42], Franc proves a statement analogous to (2) (see [15,Theorem 5.1.5]).
In [15,Sect. 6.1.3], Franc asks for a construction of his p-adic Maass-Shimura operator by means of a (non-rigid analytic) splitting p of the Hodge filtration, similar to what happens over X 1 (N ) (in the real analytic case, [42]) and X ord 1 (N ) (in the p-adic rigid analytic case, [26]). The first result of this paper is to provide such a splitting p , and define the associated p-adic Maass-Shimura operator. In particular, we show that our splitting p coincides at CM points with the Hodge splitting ∞ , and therefore, as in [26], we reprove the main results of [15] by the comparison of the two Shimura-Mass operators. We also derive a relation between our p-adic Maass-Shimura operator and generalized Heegner cycles in the context of Mumford curves, which can be viewed as an analogue of [4,Proposition 3.24]. In the remaining part of the introduction we describe more precisely the results of this paper.
Instead of the curve X attached to the Eichler order R, we follow [19] and consider a covering C → X where C is a geometrically connected curve defined over Q corresponding to a V 1 (N + )-level structure. The curve C is the solution of a moduli problem, and we have a universal false elliptic curve π : A → C (see Sect. 2.2). Following [18,19,37], we define a quaternionic projector e, acting on the relative de Rham cohomology of π : A → C, and define the sheaf and the line bundle We have a corresponding Hodge filtration The rigid analytic curve C rig associated with C admits a p-adic uniformization for a suitable subgroup p ⊆ SL 2 (Q p ). Modular forms on C rig are then p -invariant sections of H p , and therefore, to define a p-adic Maass-Shimura operator on C one is naturally led to consider the analogous problem for the covering H p of C rig . Let C 0 denote the C p -vector space of continuous (for the standard p-adic topology on both spaces) C p -valued functions on H p (Q unr p ), and let A denote theQ unr p -vector space of rigid analytic global sections of H p (Q unr p ). We have a map ofQ unr p -vector spaces r : A → C 0 and, following [15], we denote A * the image of the morphism of A-algebras A[X , Y ] → C 0 defined by sending X to the function z → 1/(z − σ (z)) and Y to the function z → σ (z), where σ :Q unr p →Q unr p is the Frobenius automorphism (note that the function z → z − σ (z) is invertible on H p (Q unr p )). DenoteĤ p the formal Z p -scheme whose generic fiber is H p , let H unr p be its base change toQ unr p and let G →Ĥ unr p be the universal special formal module with quaternionic multiplication. Denote , where e G :Ĥ unr p → G is the zero-section, and let Lie G ∨ be the Lie algebra of the Cartier dual G ∨ of G. Then ω G and Lie G ∨ are locally free OĤ unr p -modules, dual to each other and we have the Hodge-Tate exact sequence of OĤ unr Theorem 1. 1 The injection (5) of A * -algebras admits a canonical splitting p : This result corresponds to Theorem 4.5 below. The main tool which is used to prove Theorem 1.1 is Drinfel'd interpretation ofĤ unr p as moduli space of special formal modules with quaternionic multiplication; following [43], we call these objects SFD-modules. We study the relative de Rham cohomology of the universal SFD-module G →Ĥ unr p by means of techniques from [14,23,43]. The upshot of our analysis is an explicit description of the Gauss-Manin connection and the Kodaira-Spencer isomorphism for G →Ĥ unr p , once we apply to the relevant sheaves the projector e. This detailed study is contained in Sect. 4, which we believe is of independent interest and is the technical part of the paper. It should be noticed that related results on the splitting of the Hodge filtration have been obtained, even in greater generality, in [22,40]. Our result is in a way more explicit, but the price to pay is that it only works over the unramified upper half plane.
Using the splitting of the Hodge filtration in Theorem 1.1, we may then attach to p a p-adic Maass-Shimura operator where for each integer t ≥ 1 we put w * G,t = (w * G ) ⊗t . Using Teitelbaum's description of G, one can find a basis {dτ, dτ * } of w * G such that p (dτ * ) = 0. For each integer r ≥ 1, one may define the r -th iterate r p,k : w * G,k → w * G,k+2r and compare it with the r -th iterate δ r p,k : A * → A * of Franc p-adic Maass-Shimura operator defined in [15]. We prove that δ r p,k arises from the p-adic Maass-Shimura operator r p,k . More precisely, we have the following result (see (35) below). Corollary 1.2 There exists t p ∈ C × p , such that for each f ∈ A * and integers k ≥ 1 and r ≥ 1 we have

Remark 1.3
The p-adic number t p arises as a period comparing two pairings on the cohomology of SFD-modules, and can be understood as a p-adic analogue of the complex period 2πi. See (27) for details. It is independent of f , k and r .
Fix an embeddingQ →Q p ; we say that a p-adic number x ∈Q p is algebraic if it belongs to the image of this embedding, in which case we simply write x ∈Q. Let S rig k ( p ) be the C p -vector space of rigid analytic quaternionic modular forms of weight k and level under the action of p by the automorphic factor of weight k. We say that f ∈ S rig k ( p ) is algebraic if it corresponds, via the Cerednik-Drinfel'd theorem, to an algebraic modular form of weight k on C which is defined overQ (see Sect. 2.5 for the notion of algebraic modular forms, and Sect. 5.2 for the comparison between rigid analytic and algebraic modular forms).

Corollary 1.4 Let f ∈ S
rig k ( p ) be an algebraic modular form of weight k and level p . Then As remarked above, Corollary 1.4 is the main result of Franc thesis [15], which he proves via an explicit approach following Shimura. Instead, we derive this result in Theorem 5.1 from a comparison between the values at CM points of our p-adic Maass-Shimura operator and the real analytic Maass-Shimura operator.
We explain now the connection with generalized Heegner cycles. These cycles were introduced in [4] with the aim of studying certain anticyclotomic p-adic L-functions. Generalized Heegner cycles have been also studied in the context of Shimura curves with good reduction at p in [19], and in the context of Mumford curves in [30,34]. Fix a false elliptic curve A 0 with CM by O K . For any isogeny ϕ : A 0 → A of false elliptic curves, we construct a cycle ϒ ϕ in the Chow group CH m (A × A 0 ) of the Chow motive A × A 0 , where m = n/2 with n = k − 2. Brooks introduces in [19] a projector in the ring of correspondences of X m = A m × A m 0 , which defines the motive D = (X m , ). The generalised Heegner cycle ϕ is the image of ϒ ϕ in CH m (D) via this projector. We construct a p-adic Abel-Jacobi map where ∨ denotes linear dual (see Sect. 7.1 for details). We prove that the sheaf L 0 G = eH 1 dR (G/Ĥ unr p ) is equipped with two canonical sections ω can and η can , such that ω can is a generator of the invertible sheaf ω 0 G . Let ω f ∈ w 0 G be the -invariant differential form associated with an algebraic modular form f ∈ S Theorem 1.5 relates the Maass-Shimura operator with generalised Heegner cycles, and corresponds to Corollary 7.2.
We finally make a remark on p-adic L-functions. It would be interesting to use our p-adic Maass-Shimura operator to construct p-adic L-functions interpolating special values of the complex L-function of f twisted by Hecke characters as in [1,4,19,28]. We would like to come back to this problem in a future work.

Algebraic de Rham cohomology of Shimura curves
Throughout this section, let k ≥ 2 be an even integer and N ≥ 1 an integer. Fix an imaginary quadratic field K /Q of discriminant D K prime to N and factor N = N + · N − by requiring that all primes dividing N + (respectively N − ) split in K (respectively, are inert in K ). Assume that N − is a square-free of an odd number of primes, and let p N be a prime number which is inert in K (thus N − p is a square-free of an even number of primes). Fix also embeddings Q → C andQ →Q for each prime number .

Quaternion algebras
Let B/Q be the indefinite quaternion algebra of discriminant N − p. We need to fix a convenient basis for the Q-algebra B, called Hashimoto model. We thus obtain an isomorphism ι : to be the subgroup of R max consisting of elements x such that ι (x) ≡ * * 0 1 (mod N + ). This subgroup is contained in the standard Eichler order R ⊆ R max of level N + . Let finally be the subgroup of elements x ∈ R × having norm one and such that i (x) ≡ * * 0 1 (mod N + ) for all | N + .

Moduli problem
A false elliptic curve A over a scheme S is a an abelian scheme A → S of relative dimension 2 equipped with an embedding ι A : R max → End S (A). An isogeny of false elliptic curves is an isogeny which commutes with the action of R max . A full level N + -structure on A is an isomorphism of group schemes α A : A[N + ] (R max ⊗ Z (Z/N + Z)) S , where for any group G we denote G S the constant group scheme G over S. A level structure of V 1 (N + )-type is an equivalence class of full level N + -structures under the (right) action of V 1 (N + ).
The moduli problem which associates to any Z[1/N p]-scheme S the set of isomorphism classes of false elliptic curves equipped with a V 1 (N + )-level structure is representable by a smooth proper scheme C defined over Spec(Z[1/N p]) [19,Theorem 2.2]. Let π : A C be the universal false elliptic curve. For any Z[1/N p]-algebra R, let π R : A R C R be the base change of π to R. We have \H C C (C), where H is the complex upper half plane and acts on it by Moebius transformations.

Algebraic de Rham cohomology
We review some preliminaries on the algebraic de Rham cohomology of Shimura curves, including the Gauss-Manin connection and the Kodaira-Spencer map, referring for details to [27], and especially to [37, Sect. 2.1] for the case under consideration of Shimura curves.
We first recall some general notation. For any morphism of schemes φ : X → S, denote ( • X /S , d • X /S ), or simply • X /S understanding the differentials d • X /S , the complex of sheaves of relative differential forms for the morphism φ. For a sheaf F of O X -modules over a scheme X , we denote F ∨ its O X -linear dual. If F is invertible, for an integer k we let F ⊗k denote the usual tensor product operation.
Fix a field F of characteristic zero. The relative de Rham cohomology bundle for the We first recall the construction of the Gauss-Manin connection. We have a canonical short exact sequence of locally free sheaves (the exactness is because π F is smooth). This exact sequence induces maps denote the spectral sequence associated with this filtration. The E p,q 1 terms are then given by [27, (7)]. The Gauss-Manin connection is then defined as the differential d 0,i 1 : E 0,i 1 → E 1,i 1 in this spectral sequence. We now recall various descriptions of the Kodaira-Spencer map. It is defined to be the boundary map in the long exact sequence of derived functors obtained from (6). It can also be reconstructed from the Gauss-Manin connection as follows. Let π ∨ F : A ∨ F → C F denote the dual abelian variety. By a result of Buzzard ([9, Sect. 1], see also [19, page 4183]), it is known that the abelian surface A F is equipped with a canonical principal polarization ι A F : A F A ∨ F over C F , which we use to identify A ∨ F and A F in the following without explicit mentioning it; we recall that this polarization is characterized by the fact that the associated Rosati involution . Using the principal polarization and the isomorphism between R 1 π F * ( 1 A F /F ) and the tangent bundle of A ∨ F , the Hodge exact sequence can be written as . The Kodaira-Spencer map can be defined using the Gauss-Manin connection as the composition in which the first and the last map come from the Hodge exact sequence (7). Therefore the Kodaira-Spencer map can also be seen as a map of O C F -modules, denoted again with the same symbol,

Idempotents and line bundles
Let be the idempotent in [ Suppose we have an embedding M → F, allowing us to identify M with a subfield of F; in the cases we are interested in, either F ⊆Q (and then we require that F contains M), or F = C (and then we view M → C via the fixed embeddingQ → C) or F ⊆Q p (and then we require that F contains the image of M via the fixed embeddingQ →Q p ).
Since M is contained in F, we have an action of R M on the sheaves π F * Using that e is fixed by the Rosati involution, the Hodge exact sequence (7) becomes [19,Sect. 2.6]). For any integer n ≥ 1, define The Gauss-Manin connection is compatible with the quaternionic action [37, Proposition 2.2]. Therefore, restricting to L F,1 and using the Leibniz rule (see for example [19,Sect. 3.2]), the Gauss-Manin connection defines a connection By [37, Theorem 2.5], restricting the Kodaira-Spencer map to ω ⊗2 F gives an isomorphism We may then define a map∇ n : L F,n → L F,n+2 by the compositioñ (11) where the last map is the product map in the symmetric algebras.

Algebraic modular forms
As in the previous section, let F e a field of characteristic 0. For any F-algebra R, we define the R-algebra obtained from a test triplet by forgetting the datum of the global section. Then one can identify global sections of ω ⊗k R with: (1) A rule F which assigns, to each R-algebra R and each isomorphism class of test triplets (A , t , ω ) over R , an element F(A , t , ω ) ∈ R , subject to the base change axiom (for all maps of R-algebras φ : where A is the base change of A via φ) and the weight k condition (for all λ ∈ (R ) × , (2) A rule F which assigns to each R-algebra R and each isomorphism class of test pairs Let us make the relations between these definitions more explicit [19, page 4193]. Given a global section f ∈ H 0 (C R , ω ⊗k R ), we get a function as in (2) above associating to each test

Special values of L-series
In this section we review the work of Brooks [19] expressing special values of certain Lfunctions of modular forms in terms of CM-values of the Maass-Shimura operator applied to the modular form in question.

The real analytic Maass-Shimura operator
We denote (X , O X ) (X an , O an X ) the analytification functor which takes a scheme of finite type over C to its associated complex analytic space. For each sheaf F of O X -modules on X , we also denote F an the analytification of F , and for each morphism ϕ : F → G of O X -modules, we let ϕ an : F an → G an the corresponding morphism of analytic sheaves. If (X , O X ) is an analytic space, we denote O ran X the ring of real analytic functions on X ; this is a sheaf of O X -modules, and for any sheaf F of O X -modules, we let F ran = F ⊗ O X O ran X ; when F = F an , we simplify the notation by writing F ran instead of (F an ) ran .
Since C C is proper and smooth over C, the analytification functor F F an induces an equivalence of categories between the category of coherent sheaves C C and the category of analytic coherent sheaves of O an C C -modules. Also, the analytic sheaf obtained from the sheaf of algebraic de Rham cohomology H i dR (A C /C C ) coincides with the derived functor in the category of analytic sheaves over C an C [41, Theorem 1]. Hodge theory gives a splitting of the corresponding Hodge exact sequence of real analytic sheaves obtained from (7). Since this splitting is the identity on the image of π C * . We may then consider the induced maps ∞,n : L ran C,n → (ω ⊗n C ) ran for any integer n ≥ 1. Further, the map∇ n gives rise to a map∇ ran n : L ran C,n → L ran C,n+2 of real analytic sheaves. The composition is the real-analytic Shimura-Maas operator. The effect of ∞,n on modular forms is described in [19,Proposition 3.4] and [37, Proposition 2.9]. Denote = 1 (N + ) the subgroup of B × ∩ V 1 (N + ) consisting of elements of norm equal to 1. Fix an isomorphism B ⊗ Q R M 2 (R) and denote ∞ the image of in GL 2 (R). Let S k ( ∞ ) denote the C-vector space of holomorphic modular forms of weight k and level ∞ consisting of those holomorphic functions on H ∞ , the complex upper half plane, such that Define the space S ran k ( ∞ ) of real analytic modular forms of level ∞ and weight k to be the C-vector space of real analytic functions f : The operator ∞,k gives then rise to a map δ ∞,k :

CM points and triples
Fix an embeddingQ → C. For any embedding ϕ : Fix a CM point τ corresponding to an embedding ϕ : K → B. Let a be an integral ideal of O K , and define the R max -ideal a B = R max · ϕ(a). This ideal is principal, generated by an element α = α a ∈ B. Right multiplication by α gives an isogeny A τ → A α −1 τ , whose kernel is A τ [a]. Let max be the subgroup of R × max consisting of elements of norm equal to 1. The image of ατ by the canonical projection map ρ max : H → max \H does not depend on the choice of the representative α, and therefore one may write A a τ for the corresponding abelian surface. Shimura's reciprocity law states that ρ max (τ ) is defined over the Hilbert class field H of K , and that ρ max (τ ) ( Fix a primitive N + -root of unity ζ . Fix a normalized Heegner point τ , and fix a point A CM triple is an isomorphism class of triples (A τ , P τ , ω τ ) with (A τ , P τ ) as above and a non-vanishing section ω τ in e · A τ /F .
There is an action of Cl(O K ) on the set of CM triples, given by is the canonical projection.

Special value formulas
Fix a CM triple (A, P, ω) = (A τ , P τ , ω τ ) with ω defined over H , the Hilbert class field of K ; recall that A is also defined over H . The complex structure J τ on M 2 (R) defines a differential form ω C = J * τ (2πidz 1 ), and let ∞ ∈ C be defined by ω = ∞ · ω C ; clearly, different choices of ω correspond to changing ∞ by a multiple in H .
We now let f be a modular form of weight k, level 1 (N + ) ∩ 0 (N − ), and character ε f , and let f JL be the modular form on the Shimura curve C C associated with f by the Jacquet-Langlands correspondence. We can normalise the choice of f JL so that the ratio Let (2) be the set of Hecke characters χ of K of infinite type ( 1 , 2 ) with 1 ≥ k and cc the subset of (2) consisting of central critical characters.
For each positive integer j, let δ For any Hecke character, one may consider the L-function L( f , χ −1 , s), and for χ ∈ (2) cc central critical define the algebraic part L alg ( f , χ −1 ) of its special value at s = 0 as in [19,Proposition 8.7]. By [19,Proposition 8.7 where χ j = χ · nr − j and nr is the norm map on ideals of O K . In this formula we view the real analytic modular δ ∞,k ( f JL ) as a function on test triplets, as in [19,Proposition 8.5] via (12) (see also the discussion in [4, page 1094] in the GL 2 case).
For each ideal class a in Cl(O K ), let α a be the corresponding element in B, as in Sect. 3.2. Then using the dictionary between real analytic forms as functions on H or functions on test triples, and recalling that A = A τ for a normalized Heegner point τ , we have In this formula we view δ j ∞,k ( f JL ) as a function on H.

The Maass-Shimura operator on the p-adic upper half plane
In this section we define a p-adic Maass-Shimura operator in the context of Drinfel'd upper half plane. These results will be used in the next section to define a p-adic Maass-Shimura operator on Shimura curves, whose values at CM points will be compared with their complex analogue. As in the complex case, we will see that this operator plays a special role in defining p-adic L-functions.
LetĤ p denote Drinfel'd p-adic upper half plane; this is a Z p -formal scheme, and we denote H p its generic fiber, which is a Q p -rigid space [3, Chapitre I].

Drinfel'd Theorem
Denote D the unique division quaternion algebra over Q p , and let O D be its maximal order. The unramified quadratic extension Q p 2 of Q p can be embedded in D, and in the following we will see it as a maximal commutative subfield of D without explicitly mentioning it. Let σ denote the absolute Frobenius automorphism of Gal(Q unr p /Q p ). If fix an element ∈ O D such that 2 = p and x = σ (x) for x ∈ Q p 2 , then D = Q p 2 [ ]. We will denote x →x the restriction of σ to Gal(Q p 2 /Q p ).  . Unless otherwise stated, we will seeĤ unr p as â Z unr p -formal scheme.
For later use, we review some of the steps involved in the proof of Drinfel'd Theorem. The crucial step is the interpretation of the Z p -formal schemeĤ p as the solution of a moduli problem. For B ∈ Nilp, a compatible data on S = Spf(B) consists of a quadruplet (η, T , u, ρ) where [43, page 652], to which we refer for details. The first step in Drinfel'd work is to show that the Z p -formal shemeĤ p represents the functor which associates to each B ∈ Nilp the set of admissible quadruplets over B.
To each compatible data D = (η, T , u, ρ) on S one associates a S-valued point : S →Ĥ p ofĤ p , as explained in [43, pages 652-655]. The second step to prove the representability of SFD is to associate with any B ∈ Nilp and X = (ψ, G, ρ) ∈ SFD(B) a quadruplet (η X , T X , u X , ρ X ) which corresponds to an S = Spf(B)-valued point onĤ p⊗Z pẐ unr p . If X = (ψ, G, ρ) ∈ SFD(B) is given as above, the quadruplet (η X , T X , u X , ρ X ) can be explicitly constructed as follows: We finally discuss rigid analytic parameters [43]. With an abuse of notation, let SFD be the functor from the category pro-Nilp of projective limits of objects in Nilp associated with SFD. In [43, Def. 10], Teitelbaum introduces a function such that the map X = (ψ, G, ρ) → (z 0 (X ), ψ) gives a bijection between SFD(Ẑ unr p ) and (Ĥ p⊗Z pẐ unr p )(Ẑ unr p ), which we identify with the set H p (Q unr p ) × Hom(Ẑ unr p ,Ẑ unr p ). We call the map X → z 0 (X ) a rigid analytic parameter on SFD. If we let pro-Nilp the category of projective limits of objects in Nilp, and we still denote SFD the restriction of SFD to pro-Nilp, this implies that the map X = (ψ, G, ρ) → z 0 (X ) gives a bijection between SFD(Ẑ unr p ) and H p (Q unr p ). By [43,Thm. 45], for each z ∈ H p (Q unr p ), there exists triple X = (ψ, G, ρ) in SFD(Ẑ unr p ) such that z 0 (X ) = z.

Filtered Frobenius modules
Let E be an unramified field extension of Q p . A Frobenius module E over E is a pair E = (V , φ) consisting of a finite dimensional E-vector space V with a σ -linear isomorphism φ, called Frobenius [44, Chapter VI, §1]; we also call φ-modules these objects. A filtered Frobenius module is a Frobenius module (V , φ) equipped with an exhaustive and separate filtration F • V ; we also call filtered φmodules these objects. If G is a p-divisible formal group overF p , one can define its first crystalline cohomology cohomology group as in [6,16], [2, Définition 2.5.7], in terms of the crystalline Dieudonné functor (among many other references, see for example [10,12,21] for selfcontained expositions). In the following we will denote H 1 cris (G) the global sections of the crystalline Dieudonné functor (defined as in [ [38]. Let G be a SFD-module overF p . Then the Frobenius module H 1 cris (G) is a four-dimensionalQ unr p -vector space, equipped with its σ -linear Frobenius φ cris (G). It is also equipped with a D-module structure j G : D → EndQ unr Denote φ V cris (G) = j G ( ) |V cris (G) the restriction of j G ( ) to V cris (G). Moreover, denote The following lemma is crucial in what follows, and identifies V cris (G) with (η i (G) ⊗ Z p Q p ) ∨ , from which one deduces a complete description of the filtered Frobenius module H 1 cris (G). It appears in a slightly different version in the proof of [23, Lemma 5.10]. Since we did not find an reference for this fact in the form we need it, we add a complete proof.

Lemma 4.1 There is a canonical isomorphism V cris
for each index i = 0, 1 (where the action of σ −1 on η i (G) ⊗ Z pQ unr p is on the second factor only). We may therefore compute V cris (G) in terms of the isocrystal η i (G) ⊗ Z pQ unr p , σ −1 . As above, define  [1/ p], and for all i = 0, 1. Using (14), identify ϕ i with aQ unr p -linear homomorphism ϕ i : η i (G) ⊗ Z pQ unr p →Q unr p , denoted with a slight abuse of notation with the same symbol; then the above equation describing V cris (G) becomes for all n ∈ η i (G) and all x ∈Q unr p . Since ϕ i isQ unr p -linear, and we deduce the equality ϕ i (n ⊗ σ −1 (x)) = σ −1 (x)σ −1 (ϕ i (n ⊗ 1)) for all n ∈ η i (G). Taking x = 1 we see that ϕ i (n ⊗ 1) = σ −1 (ϕ i (n ⊗ 1)) and we conclude that ϕ i (n ⊗ 1) ∈ Q p for all n ∈ η i (G). So ϕ i is theQ unr p -linear extension of a Q p -linear homomorphism Since η(G) = η 0 (G) ⊕ η 1 (G), we then conclude that V cris (G) η(G) ⊗ Z p Q p ∨ as Q pvector spaces (here ∨ denotes the Q p -dual). If n 1 , . . . , n 4 is a Q p -basis of η(G) ⊗ Z p Q p , then dn 1 , . . . , dn 4 defined by dn i (n j ) = δ i, j (as usual, δ i, j = 1 if i = j and 0 otherwise) is a basis of (η(G) ⊗ Z p Q p ) ∨ and, byQ unr p -linear extension, also of (η(G) ⊗ Z pQ unr p ) ∨ . If we now base change theQ unr p -vector space (η(G) ⊗ Z pQ unr p ) ∨ via σ , we see that dn 1 , . . . , dn 4 is still aQ unr p -basis, and we have for all x ∈Q unr p . Using the above description of H 1 cris (G) in terms of D(G) σ , and the description of V cris (G) in terms of η(G), we have an isomorphism ofQ unr p -vector spaces, where the upper index σ on the right hand side means that the structure ofQ unr p -vector space is twisted by σ as explained above. Moreover, the σ −1 -linear isomorphism V G

Filtered convergent F-isocrystals
To describe the relative de Rham cohomology of the p-adic upper half plane, we first need some preliminaries on the notion of filtered convergent F-isocrystals introduced in [23]. Let E ⊆ Q unr p be an unramified extension of Q p , with valuation ring O E . If (X , O X ) is a p-adic O E -formal scheme, we denote (X rig , O rig X ) the associated E-rigid analytic space (or its generic fiber), and if F is a sheaf of O X -modules, we denote F rig its associated sheaf of The notion of convergent isocrystal on a p-adic, formally smooth O E -formal scheme X is introduced in [23, Definition 3.1], and we refer to loc. cit. for details; we only recall that a convergent isocrystal on X is a rule E which assigns to each enlargement (T , z T ) of X a coherent O T ⊗ O E E-module E T satisfying a natural cocycle condition for morphism of enlargements. Also recall from [23, Definition 3.2] that a convergent F-isocrystal on X is a convergent isocrystal E on X equipped with an isomorphism of convergent isocrystals φ E : F * E E; here F is the absolute Frobenius of the reduced closed subscheme of the closed subscheme of X defined by the ideal pO X . By [39, 1.20, 2.81] is a canonical integrable connection ∇ X : X rig for all i; in these definitions, F is the absolute Frobenius of the reduced closed subscheme of the closed subscheme of X defined by the ideal pO X .
We denote E(O X ) the identity object of the additive tensor category of convergent filtered F-isocrystals on X , introduced in [23, Example 3.4(a)] and defined on enlargements by the rule (T , z T ) it is equipped with canonical Frobenius and filtration (see loc. cit. for details). We also denote E(V ) = V ⊗ Q p E(OĤ unr p ) the filtered convergent F-isocrystal attached to a Q p -rational, finite dimensional representation ρ : GL 2 × GL 2 → GL(V ) of the algebraic group GL 2 × GL 2 ; see [23, pages 345-346] for the definition of Frobenius and filtration.
The following more articulated example of filtered convergent F-isocrystal arises from relative de Rham cohomology of the universal SFD-module. We follow closely [14,23]. Let (λ G , G, ρ G ) be the universal triple, arising from the representability of the functor SFD bŷ H unr p ; denote λ : G →Ĥ unr p be universal map. Then has a structure of convergent F-isocrystal ofĤ unr p , which interpolates crystalline cohomology sheaves (see [39,Theorem (3.1), Theorem (3.7)]; see also [6,7] such that the Hodge filtration on the de Rham cohomology groups coincides with the Hodge- Here where ρ : GL 2 × GL 2 → GL(M 2 ) is the representation defined by ρ 1 (A)

The Kodaira-Spencer map
The Kodaira-Spencer map for λ : G →Ĥ unr p , where as above G is the universal SFD-module, is the composition in which the first and the last map come from the Hodge-Tate exact sequence (15). Recalling the duality between ω G and Lie G , we therefore obtain, as in the algebraic case, a symmetric map of O H unr p -modules, again denoted KS p -modules, and the Kodaira-Spencer map takes the form where the tensor product is again over O H unr p . We now describe the Kodaira-Spencer map more explicitly, mimicking, in the complex case, [17,37]. Denote The association s → ρ s defines then a map of sheaves By construction, the dual of this map is the Kodaira-Spencer map, under the canonical identification between Hom(ω

Universal rigid data
The aim of this subsection is to use the results of [43] to describe the Hodge filtration (15). For this, we need to recall the universal rigid data introduced in [43]. Let V 0 and V 1 be constant sheaves of one-dimensional Q p -vector spaces on the Q p -rigid analytic space H p with basis t 0 and t 1 respectively. Define two invertible sheaves T univ 0 and T univ 1 on H p by T univ for any f ∈ O H p (U ), and any affinoid U ⊆ H p . Define an action of SL 2 (Q p ) on η univ i for i = 0, 1 to make u univ i equivariant with respect to these actions; in other words, for x 0,1 and γ * x 1,0 x 1,1 . Let Z p [ ] act on T univ by t 0 = ( p/z)t 1 and t 1 = zt 0 . We let Z p [ ] act on η univ in such a way that u univ commutes with this action. We call the quadruplet D univ = (η univ , T univ , u univ , ρ univ ) the universal rigid data.
See [43,Corollary 18 and Theorem 19] for more precise and complete statements. We call D univ the universal formal data, and we denote the quadruplet on the RHS of (20) byD univ to simplify the notation.
The universal SFD-module G overĤ unr p can be recovered from a universal rigid data D univ . Pulling back via the projection πĤ p :Ĥ unr p →Ĥ p , we obtain a quadruplet onĤ unr p . Comparing (20) with the universal property satisfied by G, we see that the quadru- plet (η G , T G , u G , ρ G ) associated to G coincides with the quadrupletD unr . In particular, the associated quadruplet (η obtained from the quadruplet D univ , where π H unr p : H unr p → H p is the canonical projection. Let (T unr ) ∨ denote the O H unr p -dual of T unr , and, as above, denote (η unr ⊗ Z p Q p ) ∨ the Q p -linear dual of η unr ⊗ Z p Q p . From the surjective map u unr : η unr ⊗ Z p O H unr p T unr induced by u univ we obtain an injective map

Proposition 4.2 We have canonical isomorphisms
under which the map τ corresponds to the first map in (15).

Proof
The first statement follows from the canonical isomorphism between T G = Lie G and T unr , while the second follows from Proposition 4.1 combined with (16). For the statement about τ , note that for each SFD-module G overF p , the map u G corresponds under the identification between η(G) ⊗ Z pQ

The action of the idempotent e
Fix an isomorphism Q p ( √ p 0 ) Q p 2 . By means of this isomorphism, and the fixed embed- elements of D in what follows without explicitly mentioning it.
Proof The action of O D on η( ) is induced by duality from the action on M( ), so any element a ∈ Z p 2 → O D acts on η 0 ( ) by multiplication by a and on η 1 ( ) by multiplication byā. On the other hand, the action of 1 ⊗ a on η( ) ⊗ Q p Q p 2 is given by multiplication by a. An immediate calculation shows then that the action of e is just the projection η( ) → η 0 ( ). The argument for T ( ) is similar.
Proof This is clear from Lemma 4.3 and (18).
where the RHS denotes the Q p -dual). We set up the following notation: . Applying the idempotent e and using Propositions 4.2 and 4.4 we then obtain a diagram with exact rows in which the vertical arrows are isomorphisms:

Differential calculus on the p-adic upper half plane
We now set up the following notation. Recall that the map u 0 takes x 0,0 e 0,0 + x 0,1 e 0,1 to (zx 0,0 + x 0,1 ) ⊗ t 0 ; dualizing, du 0 can be described in coordinates by the map which takes the canonical generator t 0 of the O H unr p -module (T unr 0 ) ∨ (satisfying the relation dt 0 (t 0 ) = 1) to the map x 0,0 e 0,0 + x 0,1 e 0,1 → zx 0,0 + x 0,1 . If we denote de 0,i the dual basis of e 0,i (satisfying the condition de 0,i (e 0, j ) = δ i, j ), we may write this map as zde 0,0 + de 0,1 . To simplify the notation, we put from now on τ = t 0 , dτ = dt 0 , x = e 0,0 , y = e 0,1 , dx = de 0,0 and dy = de 0,1 , so that the above map reads simply as Moreover, under the isomorphism (26) is the standard basis of M 2 (Q p 2 ). We therefore obtain the sought-for recipe to compute the Kodaira-Spencer image of dτ ⊗ dτ in terms of ψ 0 : Remark 4. 6 The number t p may be viewed as the p-adic analogue of the complex period 2πi, relating de Rham cohomology with homology [37, (2.7)], [19, p. 4197]. This explains why we prefer to keep t p at the denominator in (27).
We now make more explicit the equations (23) and (28)

The p-adic Maass-Shimura operator on Shimura curves
The aim of this section is to apply the results in Sect. 4 to define a p-adic Maass-Shimura operator on Shimura curves.

p-adic uniformization of Shimura curves
In this subsection we review the Cerednik-Drinfel'd Theorem. Recall the subgroup V 1 (N + ) of the Eichler order R of level N + of B defined in Sect. 2.1. Let U be the open compact subgroup of the groupR × (whereR is the profinite completion of R) consisting of elements (x ) ∈R × such that ι (x ) = a b 0 1 for elements a ∈ Z × and b ∈ Z , for each prime number | N + . Let U ( p) denote the subgroup ofR × consisting of elements whose -component belongs to the -component of U , for all primes = p. Let B/Q be the quaternion algebra obtained from B by interchanging the invariants at ∞ and p; so B is the definite quaternion algebra over Q of discriminant N − . Fix isomorphisms B B for all primes = p; so U ( p) defines a subgroup of (B × ) ( p) , still denoted with the same symbol U ( p) (as above,B is the adele ring of B and (B × ) ( p) is the subgroup of invertible elements ofB whose -component belongs to the -component of U via the isomorphism B B , for all primes = p). Definẽ . We still denote˜ p the image of˜ p in GL 2 (Q p ) via a fixed isomorphism i p : B ⊗ Q Q p M 2 (Q p ), and we let p denote the subgroup of˜ p consisting of elements whose determinant has even p-power order.
Base changing from Z p to the valuation ring Z p 2 of Q p 2 gives a Z p 2 -formal schemê H p 2 , whose generic fiber H p 2 is the base change of the Q p -rigid analytic space H p to Q p 2 . The group GL 2 (Q p ) acts on the Z p -formal schemeĤ p [3, Chapitre I, Sect. 6] and acts on Spf(Z p 2 ) via the inverse of the arithmetic Frobenius raised to the determinant map [3, Chapitre II, Sect. 9]. Therefore, the group GL 2 (Q p ) also acts on the Z p 2 -formal schemê H p 2 , its generic fiber, and on G. The Cerednik-Drinflel'd theorem [3,Sect. 3.5.3], [24,Theorem 4.3'] implies that there are isomorphisms of Q p 2 -rigid analytic spaces The isomorphism (37) is equivariant with respect to the quaternionic actions on both sides. See also [11,Sect. 6.2], [31,Sect. 6], [29].

Rigid analytic modular forms
A rigid analytic function f : H p (C p ) → C p is said to be a rigid analytic modular form of weight k and level p if for all z ∈ H p (C p ) and γ ∈ p , where γ (z) = (az + b)/(cz + d). Denote S rig k ( p ) the C p -vector space of rigid analytic modular forms of weight k and level p . See [11,Sect. 5 The correspondence f → ω f sets up a C p -linear isomorphism between S rig k ( p ) and w p G,C p . For any sheaf F on H unr p , denote F p the sheaf on p \H unr p defined by taking p -invariant sections. Also, recall the sheaves ωQ unr p and LQ unr p introduced in (8) and (9). It is easy to see that (36) and (37) Combining this with the isomorphism between S rig k ( p ) and w p G,C p , we obtain a canonical isomorphism of C p -vector spaces If E is a subfield of C p containing Q p 2 , we denote S rig k ( p , E) the E-subspace of S rig k ( p ) consisting of those rigid analytic functions which are defined over E. Using the Cerednik-Drinfel'd theorem (36) it is easy to see that (38) induces an isomorphism of E-rigid analytic

The p-adic Maass-Shimura operator
Taking p -invariants defines a map, for integers k ≥ 0 and j ≥ 0 Given a representation ρ = (ρ 1 , ρ 2 ) : GL 2 × GL 2 → GL(V ), the stalk E(V ) of E(V ) at a point ∈ Hom(Q p 2 , M 2 (Q p )) can be described explicitly. One first observes that the structure of filtered convergent F-isocrystal of E(V ) induces a structure of filtered Frobenius module on the fiber E(V ) . On the other hand, one attaches to such a pair (V , ) a filtered Frobenius module V whose underlying vector space is V Q unr p = V ⊗ Q p Q unr p (see [23,Sect. 4.2] for details). Denote F • V the filtration of the filtered Frobenius module V ; this is then a filtration on V Q unr p which depends on . Let gr i (F • V ) = F i V /F i−1 V be the graded pieces of the filtration. If V is pure of weight n, we have an isomorphism gr i (F • V ) V i (where V j is be the subspace of V Q unr p consisting of elements v satisfying the property ρ( (a))(v) = a j σ (a) n− j v for all a ∈ Q p 2 ) as well as a decomposition V Q unr p = i∈Z gr i (F • V ). For ∈ Hom(Q p 2 , M 2 (Q p )), denote¯ the morphism of Q p -algebras obtained by composition with the main involution of If V is pure of weight n, then the graduate pieces gr i (F • V ) and gr n−i (F • V¯ ) are equal, for all i ∈ Z. In particular, for V = M 2 we have and therefore there is an exact sequence: and a canonical decomposition One can choose generators ω 1 , ω 2 of theQ unr p -vector space gr 1 (F • (M 2 ) ) so that ω 1 and ω 2 are defined over Q p 2 . Then ω 1 and ω 2 are generators of theQ unr p -vector space gr 1 F • (M 2 )¯ , where ω i → ω i for i = 1, 2 denotes the action of Gal(Q p 2 /Q p ) on ω i . It follows that the Hodge splitting coincides on quadratic points with the projection (M 2 ) → gr 1 (F • (M 2 ) ) to the first factor in the decomposition (41).
We now apply the above results to the situation of the previous sections. Recall that K is a imaginary quadratic field and f ∈ H 0 (C Q , ω ⊗k Q ) is an algebraic modular form of weight k and level N + N − p with p N = N + N − and (N + , N − ) = 1. We write f ∞ : H p → C and f p : H p → C p for the holomorphic and the rigid analytic modular forms corresponding to f , respectively. Assume N − p is a product of an even number of distinct primes, each of them inert in K , and that all primes dividing N + are split in K . Let P ∈ C Q (K ) be a Heegner point, and assume that P ∈ C C (C) is represented by the point τ ∞ ∈ H ∞ modulo ∞ , while P ∈ C C p (C p ) is represented by the point τ p ∈ H p modulo p . Fix embeddingsQ →Q p andQ → C p which allows us to view algebraic numbers as complex and p-adic numbers. . Let A P be the false elliptic curve corresponding to the Heegner point P. The algebraic CM splitting of A P coincides both with the Hodge splitting and the p-adic splitting, and therefore the values of ∞,n and p,n at CM points are the same. Since the construction of the Maass-Shimura operators is algebraic, we see that ∇ ran n ( f ∞ ) coincides with ∇ rig n ( f p ), and the same still holds for the iterates of the Maass-Shimura operator, which also admit an algebraic construction. The result follows.

Nearly rigid analytic modular forms
In this subsection we make explicit the relation between the results of this paper and those of Franc's thesis [15]; it is independent from the rest of the paper. We first introduce a C p -subspace of the C p -vector space C 0 of continuous functions, which plays a role analogue to that of nearly holomorphic functions in the real analytic setting. For this part, we closely follow [15]. The assignment X → 1/(z − z * ) defines an injective homomorphism A[X ] → C [15,Proposition 4.3.3]. Define the A-algebra N of nearly rigid analytic functions to be the image of this map (cf. [15,Definition 4.3.5]). By definition, N is a sub-A-algebra of A * . The A-algebra N is equipped with a canonical graduation N = j≥0 N ( j) where for each integer j ≥ 0, we denote N ( j) the sub-A-algebra of N consisting of functions f which can be written in the form The Maass-Shimura operator δ p,k restricts to an operator (denoted with the same symbol) δ p,k : N → N which takes N ( j) to N ( j+2) .
Define now N k ( p ) = N p k to be the C p -subalgebra of N consisting of functions which are invariant under the weight k action of p on N , i.e. those functions satisfying the transformation property f (γ z) = (cz + d) k f (z) for all z ∈Q unr p − Q p and γ ∈ p . Note that S rig k ( p ) ⊆ N k ( p ). We call N k ( p ) the C p -vector space of nearly rigid analytic modular forms of weight k and level p . Define also N   [15] in a more conceptual way, similar to that in the complex case. Corollary 5.2 is the main result of [15], which was obtained via a completely different method, following more closely the complex analytic approach of Shimura. and F f (z), η n can ⊗ ∇(ω n can ) = F f (z), η n can ⊗ ∇(ω n can ) = F f (z), η n can ⊗ nω n−1 can ∇(ω can ) = F f (z), η n can ⊗ nω n−1 Finally, the third summand is Summing up the pieces in (43), using the Kodaira-Spencer map to replace dz with ω 2 can , and applying the splitting of the Hodge filtration which kills η can and dz * , we have The result follows.

Generalised Heegner cycles and p-adic Maass-Shimura operator
The aim of this section is to prove Theorem 1.5 stated in the Introduction, which relates generalised Heegner cycles and the p-adic Maass-Shimura operator.

The generalised Kuga-Sato motive
Fix an even integer k ≥ 2 and put n = k − 2, m = n/2. Let A 0 be a false elliptic curve with quaternionic multiplication and full level-M structure, defined over H (the Hilbert class field of K ) and with complex multiplication by O K ; the action of O K is required to commute with the quaternionic action, and this implies that A 0 is isogenous to E × E for an elliptic curve E with CM by O K . Fix a field F ⊃ H and consider the (2n + 1)-dimensional variety X m over F given by Here and in the following we simplify the notation and simply write A, C and A 0 for A F , C F and (A 0 ) F , unless we need to stress the field of definition in which case we keep the full notation. The variety X m is equipped with a proper morphism π : X m → C with 2ndimensional fibers. The fibers above points of C are products of the form A m × A m 0 . The de Rham cohomology of C attached to L n , denoted H 1 dR (C, L n , ∇), is defined to be the 1-st hypercohomology of the complex As shown in [19,Corollary 6.3], one can define a projector A (denoted P in loc. cit.) in the ring of correspondences Corr C (A m , A m ), such that A H * dR (A m /F) H 1 dR (C, L n , ∇).
On the other hand, by [19,Proposition 6.4], we can define a projector A ∈ Corr(A m 0 , A m 0 ) (which is defined by means of A ) such that The projectors A and A are commuting idempotents when viewed in the ring Corr C (X m , X m ). We define = A A 0 and denote D the motive (X m , ). By [19, Proposition 6.5] and (44), (45), (46) we see that Let v be the place of F above p induced by the inclusion F ⊆ Q → C p , which for simplicity we assume to be unramified over p. Using the explicit description (47) of the Hodge filtration, one can see that the p-adic Abel-Jacobi map for the nullhomologous (n + 1)-th Chow cycles of the motive D can be viewed as a map Here (·) ∨ denotes the F v -linear dual. For details, see [23, page 362] and [30,Sect. 4.2]; see also [32] and [33].

Generalized Heegner cycles
Let ϕ : A 0 → A be an isogeny (defined overK ) of false elliptic curves, of degree d ϕ prime to N p. Let P A be the point on C corresponding to A with level structure given by composing ϕ with the level structure of A 0 . We associate to any pair (ϕ, A) a codimension n + 1 cycle ϒ ϕ on X m by defining The cycle ϕ of D is supported on the fiber above P A and has codimension n +1 in A m × A m 0 , thus ϕ ∈ CH n+1 (D). By (47), the cycle ϕ is homologous to zero. See [30] for details.
We now compute the image of ϕ under the Abel-Jacobi map. The de Rham cohomology group H 1 dR (A/F) of a false elliptic curve A defined over a field F containing the Hilbert class field H of K is equipped with the Poincaré pairing , H 1 dR (A/F) , which we simply denote , A . Recall the canonical differentials ω can , η can introduced in Sect. 6; taking the fiber at A 0 , the universal differential ω can defines a differential ω A 0 in H 1 dR (A 0 /F), and we choose η A 0 so that ω A 0 , η A 0 A 0 = 1 and {ω A 0 , η A 0 } is a F-basis of eH 1 dR (A 0 /F); this is possible because the Hodge exact sequence splits, since A 0 has CM. This yields a basis for Sym n eH 1 dR (A 0 /F) given by the elements ω j A 0 ⊗ η n− j A 0 for j an integer such that 0 ≤ j ≤ n. Let f be a newform of level 1 (N + ) ∩ 0 (N − p) and weight k defined over F. Let v be the prime of F determined by the embedding F ⊆Q →Q p ; we assume that F v /Q p is unramified and contains Q p 2 . The Jacquet-Langlands correspondence associates to f