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Finite slope Triple Product p-adic L-functions over totally real fields

We construct p-adic L-functions associated with triples of finite slope p-adic families of quaternionic automorphic eigenforms over totally real fields on Shimura curves. These results generalize a previous construction, joint work with D.Barrera, performed in the ordinary setting.


Introduction
The theory of p-adic L-functions has grown tremendously during the last decades due to its important arithmetic applications. In particular, it has been essential for recent developments towards the Birch and Swinnerton-Dyer conjecture and its generalizations. The seminal work of Kato provided deep results on the Birch and Swinnerton-Dyer conjecture in rank 0 for twists of elliptic curves over Q by Dirichlet characters. More recently, similar results with twists by certain Artin representations of dimension 2 and 4 have been obtained by Bertolini-Darmon-Rotger in [4] and Darmon-Rotger in [9] using analogous methods. In [16], [15], [17], [1] and [5] the development of such methods has been extended in different directions, as for example bounding certain Selmer groups and treating finite slope settings. In all these situations the p-adic L-function attached to a triple of p-adic families of modular forms has played a very important role. Such triple product p-adic L-functions were constructed in [11], [8], [12] in the ordinary case and in [1] for Coleman Families, namely, in the finite slope situation. This paper deals with the construction of triple product p-adic L-functions associated with families of quaternionic automorphic forms in Shimura curves over totally real fields in the finite slope situation. In a previous paper with D. Barrera [3], we constructed such p-adic L-functions in the ordinary setting. In fact, we were able to construct p-adic families of finite slope quaternionic automorphic forms over totally real fields, but we had to restrict ourselves to the ordinary setting for the construction of the corresponding triple product p-adic L-functions. The reason for such a restriction was the inability to iterate the Gauss-Manin connection in the finite slope situation.
In this article we manage to overcome these obstacles by following the strategy of Andreatta and Iovita in [1]. 1 Our setting is in many aspects analogous to the classical setting treated in [1], hence many of the techniques will be similar. Nevertheless, there are some novelties worth mentioning because they even provide improvements to the results of Andreatta-Iovita. In [1, §4] an expression for the χ-iteration of the Gauss-Manin connection is provided, for p-adic characters χ satisfying certain assumption [1,Assumption 4.7] but not for the universal character. This disadvantage reduces the domain of the triple product p-adic L-function. In this paper, we are able to get rid of such assumption and define the universal iteration of the Gauss-Manin connection.
Our strategy relies on defining the space of nearly overconvergent families using the theory of formal vector bundles also introduced in [1]. However, we define a new type of formal vector bundle, already used in [3] to define p-adic families of finite slope quaternionic automorphic forms ( §4.2). During the elaboration of this paper, Andreatta and Iovita reported me that in parallel they had reached the same new definition of formal vector bundle. This space of nearly overconvergent families admits a good definition of a Gauss-Manin connection (Theorem 5.5). We use the explicit description of the universal character given in §4.3.1 to define a formal power series of endomorphisms liable to be the universal iteration of the Gauss-Manin connection. The choice of this new type of formal vector bundle ensures the convergence of such power series, as verified using q-expansions on Serre-Tate coordinates. As mentioned before, these computations also apply in the classical setting, hence as a consequence one can define the classical triple product p-adic L-function in the finite slope situation without any restriction on the input weights ([1, Assumption 5.4]), improving the results known so far.
1.1. Main result. Let F be a totally real number field with integer ring O F and fix a real embedding τ 0 . Let p > 2 be a prime number and let p 0 the prime over p associated to τ 0 under a fixed embeddingQ ֒→ C p . During all this paper we will suppose that F p 0 = Q p .
Let B be a quaternion algebra over F that splits at any prime over p, and it is ramified at any real place but τ 0 . In [3], we construct p-adic families of automorphic forms on (B ⊗A) × as global sections of certain p-adic sheaves on certain unitary Shimura curves, extending the previous work of [6]. Such families are parametriced by the (d + 1)-dimensional Iwasawa algebra ]. Let µ 1 , µ 2 , µ 3 be three finite slope p-adic families of automorphic eigenforms for B and we denote by Λ 1 , Λ 2 and Λ 3 the rings over which they are defined, respectively. Let x, y, z be a triple of classical points corresponding to ((k 1 , ν 1 ), (k 2 , ν 2 ), (k 3 , ν 3 )), where µ i ∈ N and k i ∈ N [F :Q] . We denote by π x the automorphic representation of (B ⊗ A F ) × generated by the automorphic form obtained from the specialization of µ 1 at x, and Π x the corresponding cuspidal automorphic representation of GL 2 (A F ). We also denote by α p x and β p x the roots of the Hecke polynomial at p. In the same way we obtain π y , Π y , α p y , β p y , π z , Π z , α p z , β p z . We have (see Lemma 6.7 and Theorem 6.9 for more details): Theorem 1.1. There exists L p (µ 1 , µ 2 , µ 3 ) ∈ Λ 1⊗ Λ 2⊗ Frac(Λ 3 ) such that for each classical point (x, y, z) corresponding to a triple ((k 1 , ν 1 ), (k 2 , ν 2 ), (k 3 , ν 3 )) we have: which naturally decompose as O = p O p . We will denote by p 0 the prime corresponding to ι p and τ 0 . We suppose the following hypothesis: We also fix a quaternion algebra over F denoted by B such that: (i) split at τ 0 and at each p | p, We choose from now on λ ∈ Q such that λ < 0 and p split in Q( √ λ). Let E := F ( √ λ) and denote z → z the not-trivial automorphism of E/F . We denote D := B ⊗ F E which is a quaternion algebra over E and we consider the involution on D defined by l = b ⊗ z → l :=b ⊗ z whereb is the canonical involution of B. We fix δ ∈ D × such that δ = −δ and define a new involution on D by l → l * := δ −1 lδ. We denote by V to the underlying Q-vector space of D endowed with the natural left action of D. We have a symplectic bilinear form on V : Let G ′ be the reductive group over Q such that for each Q-algebra R we have: We introduce a way to cut certain modules endowed with an action of A D . We denote by ψ : Q( √ λ) −→ D the natural embedding given by z −→ 1 ⊗ z and fix an extension R/O E such that A ⊗ O F R = M 2 (R). For any R-module M endowed with a linear action of A D , we define Each M ± is equipped with an action of A ⊗ Z R = M 2 (R) ⊗ Z O F and we put M ±,1 := ( 1 0 0 0 )M ± and M ±,2 := ( 0 0 0 1 )M ± . Note that both are isomorphic R ⊗ Z O F -modules through the matrix ( 0 1 1 0 ). Moreover, by construction we have: and the inclusion is an equality if disc(Q( √ λ)) ∈ R × .

Unitary modular forms
We introduce unitary Shimura curves associated with G ′ and their moduli interpretation. We define the sheaves which give rise to modular forms. See [3, §4] for further details.
The order O D is maximal (locally) at every prime not dividing disc(B)disc(D) −1 . We denote by Let n be an integral ideal of F prime to disc(B) and consider the open compact subgroups of We define the unitary Shimura curve over C of level K ′ 1 (n) as: Let O B,m ⊆ O B be an Eichler order with a well chosen level m | n so that the conditions of [3, the Riemann surface X(C) has a model (also denoted X) defined over L. This curve solves the following moduli problem: if R is a L-algebra then X(R) corresponds to the set of the isomorphism classes of tuples (A, ι, θ, α) where: (i) A is an abelian scheme over R of relative dimension 4d. (iii) θ is a O m -invariant homogeneous polarization of A such that the Rosati involution sends [3,Remark 4.5] to provide a α p modulo K ′ 1 (n) p amounts to giving a point P ∈ A[nO F ⊗ Z p ] −,1 that generates a subgroup isomorphic to (O F ⊗ Z p )/(nO F ⊗ Z p ). We have an analogous description in case of Γ 0 (n)-structures.
3.2. Modular sheaves. We introduce the sheaves which give rise to the modular forms for G ′ .
Let L 0 /F be an extension such that B ⊗ F L 0 = M 2 (L 0 ), write L = L 0 ( √ λ) ⊃ E, and denote by X L the base change to L of the unitary Shimura curve X. Using the universal abelian variety π : A → X L we define the following coherent sheaves on X L : The sheaf H is endowed with a Gauss-Manin connection ▽ : H → H ⊗ Ω 1 X L . The natural O Dequivariant exact sequence: A/X L ) +,1 and (Ω 1 A/X L ) +,2 are isomorphic then condition (ii)(2) of the moduli problem of X imply that ω τ 0 is locally free of rank 1, while ω τ is of rank 2 for τ = τ 0 . Moreover, ω − is locally free of rank 1. Thus, ω τ = H τ for each τ = τ 0 , and we have the exact sequence we introduce the modular sheaves over X L considered in this paper: If τ = τ 0 , also the polarization provides an isomorphism: Katz Modular forms. Let R 0 be a L-algebra and let f ∈ H 0 (X/R 0 , ω k ). If R is a R 0algebra, (A, ι, θ, α) is a tuple corresponding to a point of X(Spec(R)) and w = (f 0 , (f τ , e τ ) τ =τ 0 ) is a R-basis of ω A = Ω 1 , then there exists f (A, ι, θ, α, w) ∈ τ =τ 0 Sym kτ R 2 ∨ such that Thus a section f ∈ H 0 (X/R 0 , ω k ) is characterized as a rule that assigns to any R 0 -algebra R and (A, ι, θ, α, w) over R a linear form satisfying: (A1) The element f (A, ι, θ, α, w) depends only on the R-isomorphism class of (A, ι, θ, α).
where (t, g) acts on w in a natural way (see [3, §4.4] for further details).

Overconvergent families of modular forms
In this section we construct families of modular sheaves that p-adically interpolate the modular sheaves introduced in 3.2. Again, for further details see [3, §5].

4.1.
Integral models, Hasse invariants and canonical subgroups. Let n be an integral ideal of F prime to p and disc(B). Write K ′ 1 n, p =p 0 p := K ′ 1 (n) ∩ K ′ 0 p =p 0 p , and denote by X the unitary Shimura curve of level K ′ 1 n, p =p 0 p similarly as in §3.1. By [7, §5.3] X admits a canonical model X int over O p 0 , representing the analogue moduli problem described in §3.1 but exchanging an E-algebra by an O p 0 -algebra R. Namely, it classifies quadruples (A, ι, θ, α p 0 ) over R, where α p 0 is a class of O D -linear symplectic similitudes outside p 0 . By [7, §5.4], X int has good reduction. Let π : A → X int be the universal abelian variety. Since we have added Γ 0 (p)-structure for all p = p 0 , A is endowed with a subgroup C p ⊂ A[p] −,1 isomorphic to O p /p by Remark 3.1.
Let X be denote the formal scheme over Spf(O p 0 ) obtained as the completion of X int along its special fiber which is denoted byX int .
There is a dichotomy in X int which says that any point in the generic fiberX int is ordinary or supersingular (with respect to G 0 ), and there are finitely many supersingular points inX int . From We denote by Hdg the locally principal ideal of OX int described as follows: for each U = Spec(R) ⊂X int if ω 0 | U = Rw and Ha | U = Hw ⊗(p−1) then Hdg | U = HR ⊆ R. Let Hdg the inverse image of Hdg in O X , which is also a locally principal ideal. Note that Ha p n extends canonically to a section of H 0 (X, ω  We introduce the corresponding neighborhoods of the ordinary locus. For each integer r ≥ 1 we denote by X r the formal scheme over X which represents the functor that classifies for each p-adically completeÔ-algebra R: here the brackets means the equivalence class given by There exists a canonical subgroup C n of G 0 [p n 0 ] for n ≤ r over X r . This is unique and satisfy the compatibility If we write Ω 0 ⊆ ω 0 for the subsheaf generated by the lifts of the image of the Hodge-Tate map, by [2, Proposition A.3] we have that Ω 0 = δω 0 . Thus, we obtain a morphism where I n := p n Hdg − p n p−1 .
By the moduli interpretation, the p-divisible group p =p 0 G p → X r isétale isomorphic to Assume that r ≥ n. We denote by X r,n → X r the formal scheme obtained by adding to the moduli interpretation a point of order p n for each p = p 0 whose multiples generate Remark 3.1). It is clear that the extension X r,n → X r isétale and its Galois group Moreover, X r,n has also good reduction (see [7, §5.4]).
Now we trivialize the subgroup D n : Let X r,n be the adic generic fiber of X r,n . By [2, Corollaire A.2], the group scheme D n → X r,n is alsoétale isomorphic to p −n O p 0 /O p 0 . We denote by IG r,n the adic space over X r,n that trivialize D n . Then the map IG r,n → X r,n is a finiteétale with Galois We denote by IG r,n the normalization IG r,n in X r,n which is finite over X r,n and it is also endowed with an action of (O p 0 /p n O p 0 ) × . These constructions are captured by the following tower of formal schemes: Let us consider the morphism The following result is completely analogous to [1, Lemma 3.3]: IGr,n/Ô has kernel and cokernel annihilated by a power of δ and, in particular, by a power of p, depending on n.

Formal vector bundles.
We briefly recall in this subsection constructions performed in [1, §2] and [3, §6]. Let S be a formal scheme, I its (invertible) ideal of definition and E a locally free O S -module of rank n. We writeS the scheme with structural sheaf O S /I and put E the corresponding O S -module. We fix marked sections s 1 , · · · , s m of E, namely, the sections where Q is a locally free OS-module of rank n − m. Let S − Sch be the category of the formal S-schemes. There exists a formal scheme V(E) over S called the formal vector bundle attached to E which represents the functor, denoted by the same [1] is the construction of the so called formal vector bundles with marked sections which is the hereρ is the reduction of ρ modulo I.
Given the fixed decomposition E = Q⊕ s i i , let us consider now the sub-functor V Q (E, s 1 , · · · , s m ) that associates to any formal S-scheme t : T → S the subset of sections ρ ∈ V 0 (E, s 1 , · · · , s m )(T ) whose reductionρ modulo I also satisfiesρ(t * (m)) = 0 for every m ∈ Q.
Remark 4.5. Notice that this construction is also functorial with respect to (E, Q, s 1 , · · · , s m ).
Indeed, given a morphism ϕ : E ′ → E of locally free O S -modules of finite rank and marked sections we have the morphisms making the following diagram commutative The filtration is characterized by its local description: Over U = Spf(R) ⊂ S, an open affine subscheme such that F, E are free over U , we have that .
consists of the polynomials of degree ≤ n in the variables Moreover, if we assume that we are in the situation of the previous section with a locally free O S -module and a direct summand F ⊂ E, we consider the filtrations Fil • f 0, * O V 0 (E,s 1 ,··· ,s d ) . By [1, Lemma 2.9] the connection ▽ 0 satisfies Griffith's transversality with respect to the filtration , namely for every integer n we have Furthermore the induced map We fix a decomposition: we consider the algebras: We also consider the following formal schemes W 0 = Spf(Λ 0 F ), W n := Spf(Λ n ) and W 0 n := Spf(Λ 0 n ). By construction we have W = n W n and W 0 = n W 0 n . Moreover, we have the following explicit description: We denote by k : The following Lemma can be found in [3, Lemma 6.4], but it also can be deduced from the computations below.
Lemma 4.7. Let R be a p-adically complete Λ 0 n -algebra. Then k 0 n extends locally analytically to a

Recall that hypothesis 2.1 imply a decomposition of rings
put O τ 0 ,0 := 1 + pO τ 0 . Analogously as above we introduce: Thus, we have decompositions Λ 0 n = Λ 0 τ 0 ,n⊗ Λ τ 0 ,0 n and Λ n = Λ τ 0 ,n⊗ Λ τ 0 n . We denote by k 0 τ 0 ,n : In this part we will describe the universal character k 0 τ 0 ,n . By the above lemma k 0 τ 0 ,n is analytic when restricted to 1 + p n Z p . Moreover, it is given by where v : Z → N is the p-adic valuation. This implies that where the last equality comes from the following equality in This implies that i≥0 Thus, we obtain that the locally analytic expression of k 0 τ 0 ,n is given by The following examples help us to visualize that the above expression can be specialized at both analytic character and locally constant characters that factor through (1 + p Z p )/(1 + p n+1 Z p ).
Exemple 4.9. Let k ∈ W 0 τ 0 ,n (C p ) be the point corresponding to the character β → β k , for some k ∈ Z. It clearly satisfies the characterization of Equation (9). It corresponds to the maximal ideal (T − exp(kp) + 1). We check that Exemple 4.10. Let χ ∈ W 0 τ 0 ,n (C p ) be the point corresponding to the character χ(exp(αp)) = ξ α n , for a fix (n − 1)-th root of unity ξ n . It also satisfies the characterization of Equation (9) and corresponds to the maximal ideal (T − ξ n + 1). We check that where Ω 0 is the O IGr,n -submodule of ω 0 generated by all the lifts of dlog 0 (D n ). By construction, there exist on IG r,n a universal canonical generator P 0,n of D n , and universal points P p,n of order p n in G p [p n ]. We put: where Ω 0 = p =p 0 ω p , and we denote Ω the associated O IGr,n -module. Now we produce marked sections in Ω as follows. Let p | p and we consider two cases: • if p = p 0 we denote by s 0 ∈ Ω the image dlog 0 (P 0,n ) using the isomorphism (11).
• if p = p 0 we firstly consider the decomposition ω p = ⊕ τ ∈Σp ω p,τ over IG r,n and the dlog map: Hence the image of P p,n through dlog p provides a set of sections {s p,τ } τ ∈Σp of ω p,τ /I n ω p,τ ⊆ ω p /I n ω p .
By For any X r -scheme T we have: Let s 0 := p =p 0 {s p,τ } τ ∈Σp then from (12) we have: As we are interested in locally analytic distributions (rather than functions) we perform the following construction. Let t p,τ ∈ω p,τ be any section such that ϕ τ (s p,τ ∧ t p,τ ) = 1 and Q s 0 ⊂ Ω 0 be the direct summand generated by the sections in s 0 . We put t 0 := p =p 0 {t p,τ } τ ∈Σp and The sections t p,τ are well defined modulo Q s 0 which is fine because remark 4.6. We have The morphism g n is endowed with an action of (O/p n O) × , then both V Q s 0 (Ω 0 , t 0 )/X r and Since r ≥ n by Lemma 4.7 (with λ = Hdg p n p−1 ) the character k 0 n extends to a locally analytic character Definition 4.11. We consider the following sheaves over X r × W n : , The formal overconvergent modular sheaf over X r × W n is defined as  • f (tx, ty) = k(t) · f (x, y) for each t ∈ O τ 0 × and (x, y) ∈ O τ 0 × × O τ 0 ; • the function y → f (1, y) is analytic on the disks y 0 + p n O τ 0 where y 0 below to a system of is the module of analytic functions and naturally contains Sym k (R 2 ). We obtain a natural projection We have a natural action of the subgroup K 0 (p) τ 0 ⊂ GL 2 (O τ 0 ) of upper triangular matrices modulo p on C k n (O τ 0 , R) and D k n (O τ 0 , R) given by: where g ∈ K 0 (p) τ 0 , f ∈ C k n (O τ 0 , R) and µ ∈ D k n (O τ 0 , R). Since y → f (1, y) is analytic on the disks y 0 + p n O τ 0 this action extends to an action of K 0 (p) τ 0 (1 + p n M 2 (R ⊗ Z p O τ 0 )).

Nearly overconvergent modular forms
Let us consider the sheaf R 1 π * Ω • A/X , where A → X is the universal abelian variety. By the moduli interpretation of X Notice that H 0 is endowed with: Fixing integers r ≥ n as above and having the morphisms IG r,n → X r,n → X r → X, we can base-change the triple (H 0 , ▽, Fil • ) over IG r,n and denote it the same way.
Write H ♯ 0 := Ω 0 + δ p H 0 . Since δ is a locally free O IGr,n -module of rank 1, H ♯ 0 is a O IGr,n -module of rank 2 with Hodge filtration (15) 0 This implies that s 0 is also a marked section of H ♯ 0 . Similarly as in [1, Lemma 6.1] one can prove that the construction of the exact sequence (15) is functorial. But Dα ∈ p 2n δ −p n −1 = p n δ −1 I n , thus HdgDαs = δ p−1 Dαs ≡ 0 modulo I n .
In order to prove the claim, we will work locally. Let ρ : Spf(R) → IG r,n be a morphism of formal schemes without p-torsion such that ρ * ω 0 , and ρ * H 0 are free R-modules of rank 1 and 2, respectively. We choose basis ρ * ω 0 = Rf , ρ * H 0 = Rf + Re. Recall that the Kodaira-Spencer isomorphism KS is obtained from restricting ▽ to ω 0 and composing with the morphism ǫ of (14).
Thus, if the derivation D is dual to Θ = KS(f, e) ∈ Ω 1 X/ Zp , we have that ▽(D)(f ) = af + e, for some a ∈ R.
The above lemma provides all the ingredients to construct the formal vector bundle V Q (H ♯ 0 , s 0 ) → X r . Similarly as in [3, §6.3], we define an action of Z × p (1 + I n G a ) on V Q (H ♯ 0 , s 0 ) → X r as follows: For any X r -scheme T and the action of Definition 5.2. We consider the following sheaf over X r × W n : corresponding to sections such that t * s = k 0 τ 0 ,n (t)s, for all t ∈ Z × p (1 + I n G a ). The formal nearly overconvergent modular sheaf over X r × W n is defined as is called a family of locally analytic nearly overconvergent modular forms.
The inclusion Ω 0 ⊂ H ♯ 0 provides a filtration of locally free sheaves with marked sections (Ω 0 , s 0 ) ֒→ . Analogously as in [1,Theorem 3.11.], the action of Recall that, by (10), the universal character k 0 τ 0 ,n is given by Theorem 5.5. The above connection descends to an integrable connection for some c n ∈ N depending on n, such that the induced O X -linear map on the n graded piece is an isomorphism times w n − h.
Proof. Let us work locally. Let ρ : Spf(R) → IG r,n be a morphism of formal schemes without p-torsion such that ρ * ω 0 , and ρ * H 0 are free R-modules of rank 1 and 2, respectively. We choose basis ρ * ω 0 = Rf , ρ * H 0 = Rf + Re and assume that δ = δR. Moreover, we assume that δf ≡ s 0 Since δf is in the image of dlog modulo I n , we have δf = s + s 0 , where s ∈ dlog(T p G 0 ) and s 0 ∈ δ −an I n Ω 0 , for some a n ∈ N. Since ▽(d)δ −an I n Ω 0 ⊆ δ −an−1 I n Ω 0 , we deduce that ▽(d) 2 (δf ) ∈ δ −bn I n , for some b n ∈ N. Applying the Leibniz rule, we compute that Since da ∈ p n δ −1 I n , this implies that, for some c n ∈ N, where A, C ∈ I n and B, D ∈ R.
Fix a generator α ∈ I n . As explained in [3, §6.4], the sections ρ * O V Q (H ♯ 0 ,s 0 ) (Spf(R)) are given by where X corresponds to δf and Y corresponds to δ p e. This implies that the connection on is given by ▽(δ cn d)(X) = AX + BY , ▽(δ cn d)(Y ) = CX + DY . We observe that this connection Thus, by Lemma 4.3, after possibly enlarging c n we obtain In [3, §6.4] we have also the local description of ρ * W k 0 τ 0 ,n 0 : It is given by being P n the analytic extension of k 0 τ 0 ,n around 1 + I n R Z . We compute using Leibniz rule: From this we deduce the first assertion.
Moreover, the above computation shows that the induced morphism in graded peaces is given by multiplication by (w n − m)αB, and recall by (16) that B = δ cn+1−p . Hence, once we invert p (and therefore δ) we obtain the second assertion.
Together with the connection on Ω k f [1/p] provided by the universal derivation on IG r,n , the above ▽ kτ 0 ,n induces a connection on W kτ 0 ,n := W Similarly as in [2,Proposition 3.3], the morphism p 2 : X r+1 → X r is finite flat of degree p hence there is a well defined trace map with respect to O Xr → p 2, * (O X r+1 ). We define the U p 0 -operator Lemma 5.6. The morphism induced by U p 0 on H 0 (X, W kτ 0 ,n /Fil m W kτ 0 ,n ) is nilpotent modulo denotes the integral part.
Proof. Analogously to [1, Proposition 3.24] we will prove that the morphism U induces a morphism on the mth graded part of the filtration that is zero modulo p [m/p] , this automatically implies the claim.
By [1, Lemma 6.4] the map induced on de Rham cohomology λ : and identifies (6) we deduce that the induced morphism on graded parts is zero modulo p m δ (1−p 2 )m . By construction pHdg −p r+2 is a well defined section of X r+1 , since r ≥ 1, hence the result follows.
Remark 5.7. We note that the above definition of U p 0 fits with the one given in [3, §7.3] acting on H 0 (X, Ω kn ). Moreover, in [3, §7] one can also find the definition of the Hecke operators V p 0 and U p , for p = p 0 , acting on H 0 (X, Ω kn ).

De Rham cohomology with coefficients in W
Moreover, by the Kodaira-Spenser isomorphism we can identify Ω 1 Xr with Fil 2 W 2 in the adic space X r . Hence, the connection ▽ kτ 0 ,n can be reinterpreted as the morphism of de Rham complexes of sheaves on the adic space X r × W n W • kτ 0 ,n : W kτ 0 ,n Let us denote by H i dR (X r × W n , •) the i-th hypercohomology group of such de Rham complexes. The same identification as above allows us to consider the Gauss-Manin operator given by Griffith's transversality Notice that the sheaves ω k−m 0 ⊗ Sym m H 0 define a filtration Fil m of Sym k H 0 . Moreover, on graded pieces the connection ▽ m defines an isomorphism times k − m. This implies that , we obtain that ▽ k defines a morphism As shown in [3, §8.2], the classical computations on q-expansions also apply on this setting considering Serre-Tate coordinates. This implies that the exactly same proof of [1, Lemma 3.32] works showing that H 1 dR (X r × W n , Fil m (W • kτ 0 ,n )) lies in the exact sequence (17) where j ρ is the closed immersion X r ֒→ X r × W n defined by a point such that w n = i. Moreover, the torsion free part H 1 dR (X r × W n , Fil m (W • kτ 0 ,n )) tf lies in the exact sequence where the morphisms are equivariant for the action of U p 0 and the action of U p 0 on H 0 (X r × W n , j ρ, * (ω 0 ) −i ) is divided by p i+1 .
Let I m be the ideal m i=0 (w n − i) and Λ Im the localization of Λ τ 0 ,n at I m . The above exact sequence (17) provides an isomorphism Since Fil m W kn is coherent and U p 0 is compact, the usual discussion on slope decompositions applies to H 0 (X r × W n , Fil m W kn ). Hence, given a finite slope h ≥ 0 we have, locally on the weight space, a slope h decomposition. This implies that H i dR (X r × W n , Fil m (W • kτ 0 ,n )) admits a slope h decomposition. Due to Lemma 5.6, the operator U p 0 on H i dR (X r × W n , (W /Fil m (W)) • ) is divisible by p h+1 for large enough m, where (W /Fil m (W)) • is the quotient complex of W • kτ 0 ,n and Fil m (W • kτ 0 ,n ). By the long exact sequence of de Rham complexes this implies that The above isomorphism together with (17) provides a morphism (20) where m is an natural number depending on h.
The spectral theory developed in [2, Appendice B] provides a overconvergent projection e ≤h onto slope ≤ h subspace, this gives rise to the composition: where the left down arrow is the natural projection.

Proof.
A laborious but straightforward computation shows that This implies that .
Hence the result follows from the fact that the class of ▽ m 6.2. p-adic interpolation of the operator Θ m . We aim to construct the operator Θ kτ 0 ,n 3 for the universal character k τ 0 ,n 3 : Z × p −→ Λ n 3 . A strategy one can think of is to p-adically iterate the operators Θ m as m varies. Following this strategy, the problem that one immediately finds is that as m grows the ideal I m grows as well. Since we have to invert I m in order to define H † m , we lose control of the denominators.
One can think of the strategy of defining Θ kτ 0 ,n 3 in the space of p-adic modular forms, namely, sections on the ordinary locus. This is the strategy used in [3]. By equation (19), we have to restrict ourselves to the case of slope ≤ h in order to control the filtration where Θ kτ 0 ,n 3 lives.
Using the interpolation of the Serre operator introduced in [3], one could define Θ kτ 0 ,n 3 provided by the existence of a slope ≤ h projector e ≤h on the space of p-adic modular forms. But there is no such projector acting on the space of p-adic modular forms except for the ordinary case.
The strategy followed by Andreatta and Iovita in [1] relies on p-adically interpolate the operator ▽ m kτ 0 ,n acting on Since on the space of nearly overconvergent modular forms we have the projector e ≤h , we have the operator H †,≤h of the diagram (21), hence we can proceed with a well defined construction of Θ kτ 0 ,n 3 on the space of nearly overconvergent modular forms. In order to ensure the convergence of the series involved in the definition of ▽ s kτ 0 ,n , for a universal character s, one has to control de valuation of the corresponding terms, and this can be done by computing the Gauss-Manin connection on the ordinary locus. There we can use Serre-Tate coordinates and the corresponding q-expansions.
6.3. Twists by finite characters. This section is analogous to [1, §3.8]. As always r ≥ n and let G 0 be the universal formal group over IG 2n,r+n . Let C n ⊂ G 0 [p n ] be the canonical subgroup and let us consider It is clear that C ′ n is the canonical subgroup of G ′ 0 and the dual isogeny λ : G ′ 0 → G 0 maps C ′ n to C n . This construction defines a morphism t : IG 2n,r+n → IG n,r , since a trivialization of D 2n induces by multiplication by p n a trivialization of D ′ n = (C ′ n ) ∨ . If C ′′ n := ker λ, we have that G 0 [p n ] = C ′ n × C ′′ n as group schemes over IG 2n,r+n . The universal trivialization provides isomorphisms since the Weil pairing and the polarization identifies C ′ n ≃ (C ′′ n ) ∨ . A choice of a primitive p n -root of unity ξ provides an identification Z /p n Z ≃ Hom(Z /p n Z, µ p n ), j → (1 → ξ j ). Thus, we obtain a bijection Write ρ j ∈ Hom(C ′′ n , C ′ n ) for the morphism corresponding to j ∈ Z /p n Z, and let us consider t j : IG 2n,r+n → IG n,r for the morphism induced by making quotient by (ρ j ×Id)(C ′′ n ) ⊂ G ′ 0 [p n ], with the induced trivialization of (C n,j ) ∨ , where C n,j is the image of C ′ n in G j 0 := G 0 /(ρ j × Id)(C ′′ n ) which coincides with the canonical subgroup. Write also t j : IG 2n,r+n → IG n,r for the corresponding morphism taking normalizations. Over IG 2n,r+n we have isogenies on the universal formal groups respecting canonical groups and trivializations. By functoriality, this gives rise to the corresponding maps By [1, Lemma 6.4] the images coincide Imλ ♯ = Imλ ♯ j , hence it gives rise to a isomorphism f j : (H j 0 ) ♯ → H ♯ 0 preserving Hodge exact sequences and Gauss-Manin connections. Again by functoriality, this gives rise to a morphism over IG 2n,r+n preserving filfrations and Gauss-Manin connections. Thus, after extending scalars in order to have the root ξ, we obtain a morphism for any finite character χ : where g χ −1 := n χ(n) −1 ξ n is a Gauss sum.
The following result is completely analogous to [1, Lemma 3.31]: Lemma 6.2. The morphism θ χ provides a morphism 6.4. q-expansions on unitary Shimura curves. Over the ordinary locus X ord it is constructed in [3] an affine Spf(W )-formal scheme X(∞), where W is the ring of Witt vectors ofF p . X(∞) represents ordinary abelian varieties (A, ι, θ, α p 0 ) together with frames: Moreover, X(∞) is equipped with a classifying morphism c : X(∞)/W → G m /W . The local coordinates given by the morphism c are called Serre-Tate coordinates.
In the rest we use the following notation: If (x, y, z) ∈ W 1 × W 1 × W we denote by µ x , µ y and µ z the specializations of the families µ 1 at x, µ 2 at y, and µ 3 at z respectively. If z ∈ W is a classical weight then the specialization µ z is a classical automorphic form of weight k z , where k z is the specialization of k 3 at z. Recall the triple product of (29) defines a triple product The following result is analogous to [ Lemma 6.7. There exists L p (µ 1 , µ 2 , µ 3 ) ∈ R ′ such that for each classical point (x, y, z) ∈ W 1 × W 1 × W, we have: L p (µ 1 , µ 2 , µ 3 )(x, y, z) = µ * z , t(µ 1 , µ 2 ) (x,y,z) µ * z , µ z where ·, · is the Petersson inner product and µ * z defines the dual basis of µ z . Definition 6.8. Let µ 1 , µ 2 , µ 3 as above and assume that they are test vectors for three families of eigenvectors. The functions L p (µ 1 , µ 2 , µ 3 ) ∈ R ′ introduced in 6.7 is called the triple product p-adic L-function of µ 1 , µ 2 , µ 3 .