A p-adic Maass–Shimura operator on Mumford curves

Matteo Longo

doi:10.1007/s40316-022-00193-x

A p-adic Maass–Shimura operator on Mumford curves

Longo, Matteo 2023-04-01 00:00:00 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]. Résumé Nous étudions un opérateur de Maass–Shimura p-adique pour les courbes de Mumford déﬁni par [15]. Nous montrons que cet opérateur peut être déﬁni en terme d’une scission de la ﬁltration de Hodge, répondant à une question posée dans [15]. Nous étudions aussi la relation de cet opérateur avec les cycles de Heegner généralisés, comme dans [1, 4, 19, 28]. Keywords p-adic uniformisation · Shimura curves · Maass–Shimura operators Mathematics Subject Classiﬁcation 11F03 · 14F40 · 11R52 1 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 1 ∂ k δ ( f )(z) = + f (z) (1) 2πi ∂z z −¯z 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 ( 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 To Bernadette Perrin-Riou on the occasion of her 65th birthday. B Matteo Longo mlongo@math.unipd.it Dipartimento di Matematica Tullio Levi-Civita, Università degli Studi di Padova, Via Trieste 63, 35121 Padua, Italy 123 M. Longo main results in [42] is the following. Let δ = δ ◦ δ ◦ ··· ◦ δ k+2(r −1) k+2(r −2) k for any r ≥ 1, and let K be an imaginary quadratic ﬁeld. Then there exists ∈ C such that for every modular form f of weight k with algebraic Fourier coefﬁcients, and for every CM point z ∈ K ∩ H,wehave δ ( f )(z) ∈ Q. (2) (k+2r) Katz described in [25] the Maass–Shimura operator in more abstract terms by means of the Gauss–Manin connection (see also [27]). Let N ≥ 1beaninteger, X (N) the modular curve of level (N) over Q,and letπ : E → X (N) be the universal elliptic curve. Consider 1 1 the relative de Rham cohomology sheaf 1 1 L = R π 0 → O → 1 ∗ E E/X (N) r 1 on X (N), and deﬁne L = Sym (L ).Let ω = π . The sheaf ω is invertible 1 r 1 ∗ E/X (N) and we have the Hodge ﬁltration −1 0 → ω → L → ω → 0. (3) ran Considering the associated real analytic sheaves, which we denote by a superscript ,the Hodge exact sequence of real analytic sheaves associated with (3) admits indeed a splitting ran ran ran : L −→ ω ⊕¯ω , (4) where ω ¯ is obtained from ω by applying the complex conjugation. The Maass–Shimura ran operator ran ⊗r ran ⊗(r +2) : (ω ) −→ (ω ) ∞,r 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 ord X (N) denote the ordinary locus of the modular curve, viewed as a rigid analytic scheme rig rig over Q .Let F be the rigid analytic sheaf associated with a sheaf F on X (N).Then L p 1 ord splits over X (N) as the direct sum rig rig Frob : L ω ⊕ L 1 1 Frob (where L has the property that the Frobenius endomorphism acts invertibly on this sheaf). This allows to deﬁne a differential operator , which can be seen as a p-adic analogue of the p,r Maass–Shimura operator; this operator can be described in terms of Atkin-Serre derivative. At CM points the splittings and coincide, and therefore one deduces rationality ∞ p results for the values of at CM points from (2). For details, see [4,Proposition 1.12]. p,r The p-adic Maass–Shimura operator is then used in [4, 26] to construct p-adic L-functions and study their properties. 123 Shimura–Mass operator We nowﬁxaninteger 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 anew 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 ≥ 1beaninteger, K/Q a quadratic imaginary ﬁeld, p N a prime number, p ≥ 5, which is inert in K,and let + − + Np = N · N p be a factorization of Np 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 rig analytic curve X over Q is then a Mumford curve, namely X(C ) is isomorphic to the p p rigid analytic quotient of the p-adic upper half plane H (Q ) = C − Q by an arithmetic p p p p subgroup of SL (Q ). Franc deﬁnes in this context a p-adic Maass–Shimura operatorδ by 2 p p,k mimicking the deﬁnition (1), formally replacing the variable z ∈ H with the p-adic variable unr unr ˆ ˆ z ∈ H (Q ) = Q − Q , and replacing the complex conjugation with the Frobenius p p p p unr map (here Q is the completion of the maximal unramiﬁed extension of Q ). 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 of the Hodge ﬁltration, similar to what happens ord over X (N) (in the real analytic case, [42]) and X (N) (in the p-adic rigid analytic case, [26]). The ﬁrst result of this paper is to provide such a splitting , and deﬁne the associated p-adic Maass–Shimura operator. In particular, we show that our splitting 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 (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 L = e · H (A/C) dR and the line bundle ω = e · π . A/C We have a corresponding Hodge ﬁltration −1 0 −→ ω −→ L −→ ω −→ 0. rig The rigid analytic curve C associated with C admits a p-adic uniformization rig C (C ) \H (C ) p p p p 123 M. Longo rig for a suitable subgroup ⊆ SL (Q ). Modular forms on C are then -invariant sections p 2 p p of H , and therefore, to deﬁne a p-adic Maass–Shimura operator on C one is naturally led rig to consider the analogous problem for the covering H of C . Let C denote the C -vector space of continuous (for the standard p-adic topology on unr unr ˆ ˆ both spaces) C -valued functions on H (Q ),and let A denote the Q -vector space of p p p p unr unr 0 ˆ ˆ rigid analytic global sections of H (Q ).Wehaveamapof Q -vector spaces r : A → C p p ∗ 0 and, following [15], we denote A the image of the morphism of A-algebras A[X, Y]→ C deﬁned by sending X to the function z → 1/(z − σ(z)) and Y to the function z → σ(z), unr unr ˆ ˆ whereσ : Q → Q is the Frobenius automorphism (note that the function z → z −σ(z) p p unr ˆ ˆ is invertible on H (Q )). Denote H the formal Z -scheme whose generic ﬁber is H ,let p p p p unr unr unr ˆ ˆ ˆ H be its base change to Q and let G → H be the universal special formal module p p p ∗ 1 unr with quaternionic multiplication. Denote ω = e ( ),where e : H → G is the G G unr G/H zero-section, and let Lie ∨ be the Lie algebra of the Cartier dual G of G.Then ω and Lie are locally free O -modules, dual to each other and we have the Hodge-Tate exact G unr sequence of O -modules ˆ unr 1 unr 0 −→ ω −→ H (G/H ) −→ Lie ∨ −→ 0. G dR p 0 1 0 0 ∗ unr 0 unr ∗ Set L = e ·H (G/H ) andω = e ·ω .Deﬁne = H (H , L ), = ⊗ A , G G A p p G dR G G G G 0 unr 0 ∗ ∗ ∗ w = H (H ,ω ), w = w ⊗ A . We have then an injective map of A -algebras G G A p G G ∗ ∗ w −→ . (5) G G ∗ ∗ ∗ Theorem 1.1 The injection (5) of A -algebras admits a canonical splitting : → w . G G This result corresponds to Theorem 4.5 below. The main tool which is used to prove unr Theorem 1.1 is Drinfel’d interpretation of H as moduli space of special formal modules with quaternionic multiplication; following [43], we call these objects SFD-modules.We unr study the relative de Rham cohomology of the universal SFD-module G → H by means of techniques from [14, 23, 43]. The upshot of our analysis is an explicit description of the unr Gauss–Manin connection and the Kodaira–Spencer isomorphism for G → H , 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 a p-adic Maass–Shimura operator ∗ ∗ : w −→ w p,k G,k G,k+2 ∗ ∗ ⊗t where for each integer t ≥ 1 we put w = (w ) . Using Teitelbaum’s description of G,t G ∗ ∗ ∗ G, one can ﬁnd a basis {dτ, dτ } of w such that (dτ ) = 0. For each integer r ≥ 1, r ∗ ∗ one may deﬁne the r-th iterate : w → w and compare it with the r-th iterate p,k G,k G,k+2r r ∗ ∗ r δ : A → A of Franc p-adic Maass–Shimura operator deﬁned in [15]. We prove thatδ p,k p,k arises from the p-adic Maass–Shimura operator . More precisely, we have the following p,k result (see (35)below). 123 Shimura–Mass operator × ∗ Corollary 1.2 There exists t ∈ C , such that for each f ∈ A and integers k ≥ 1 and r ≥ 1 we have δ ( f )(z) p,k r ⊗k ⊗(k+2r) f (z)dτ = dτ . p,k Remark 1.3 The p-adic number t arises as a period comparing two pairings on the coho- mology 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 embedding Q → Q ; we say that a p-adic number x ∈ Q is algebraic if it p p rig belongs to the image of this embedding, in which case we simply write x ∈ Q.Let S ( ) be the C -vector space of rigid analytic quaternionic modular forms of weight k and level rig ;elementsof S ( ) are functions from H (C ) = C − Q to C which transform p p p p p p p rig under the action of by the automorphic factor of weight k. We say that f ∈ S ( ) is p p algebraic if it corresponds, via the Cerednik–Drinfel’d theorem, to an algebraic modular form of weight k on C which is deﬁned over Q (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). rig Corollary 1.4 Let f ∈ S ( ) be an algebraic modular form of weight k and level .Then p p unr for every CM point z ∈ K ∩ H (Q ),wehave δ ( f )(z) p,k ∈ Q. 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 intro- duced 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 with CM by O . For any isogeny ϕ : A → A of false elliptic curves, we construct a 0 K 0 cycle ϒ in the Chow group CH (A × A ) of the Chow motive A × A ,where m = n/2 ϕ 0 0 with n = k − 2. Brooks introduces in [19] a projector in the ring of correspondences of m m X = A × A , which deﬁnes the motive D = (X ,). The generalised Heegner cycle m ϕ is the image of ϒ in CH (D) via this projector. We construct a p-adic Abel-Jacobi map rig m m 1 AJ : CH (D) −→ S () ⊗ Sym eH (A ) p 0 k dR ∨ 0 where denotes linear dual (see Sect. 7.1 for details). We prove that the sheaf L = 1 unr eH (G/H ) is equipped with two canonical sections ω and η , such that ω is can can can dR p 0 0 a generator of the invertible sheaf ω .Let ω ∈ w be the -invariant differential form G G rig associated with an algebraic modular form f ∈ S (),and let F denote its Coleman primitive satisfying ∇(F ) = ω ,where ∇ is the Gauss–Manin connection. Denote , the f f n 1 Poincaré pairing on Sym eH (A ),where A is the ﬁber of A at z. Deﬁne the function z z dR H(z) =F (z),ω (z). can 123 M. Longo Theorem 1.5 Let ϕ : A → A be an isogeny of degree d prime to N p, and z the point of 0 ϕ A C whose ﬁber is A. Then for each integer j = n/2,..., n we have n− j δ (H)(z ) A n! p,k j n− j = · AJ ( )(ω ⊗ ω η ). p ϕ f can can n− j j t j!· d p ϕ 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. 2 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 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 and Q → Q for each prime number . 2.1 Quaternion algebras Let B/Q be the indeﬁnite quaternion algebra of discriminant N p.Weneed to ﬁx aconve- nient basis for the Q-algebra B, called Hashimoto model. Denote M = Q( p ) the splitting ﬁeld of the quadratic polynomial X − p ,where p if an auxiliary prime number ﬁxed as 0 0 in [37,Sect. 1.1] and [19,Sect. 2.1], such that: − − (1) for all primes we have(p , pN ) =−1 if and only if | pN ,where(a, b) denotes the Hilbert symbol, √ √ (2) all primes | N are split in the real quadratic ﬁeld M = Q( p ),where p is a 0 0 square root of p in Q. 2 − The choice of p ﬁxes a Q-basis of B as in [18,Sect. 2] given by {1, i, j, k} with i =−pN , j = p , k = ij =−ji,and 1 the unit of B; of course, if x ∈ Q we will often just write x for x · 1.Let R be the maximal order of B which contains the Z-span of this basis. max Since M splits B,wehaveisomorphisms ι : B ⊗ M M (M) and for place v N p M Q 2 of M we have ι (R ⊗ O ) = M (O ),where O is the ring of algebraic integers M max Z M,v 2 M,v M of M,and O is the v-adic completion of O [19,Lemma 2.1]. For each prime number M,v M | N , choose a place v of M above with M = Q . We thus obtain an isomorphism ι : B ⊗ Q M (Q ) such that ι (R ⊗ Z ) = M (Z ).Deﬁne V (N ) to be Q 2 p max Z 2 1 ∗∗ + the subgroup of R consisting of elements x such that ι (x) ≡ (mod N ).This max subgroup is contained in the standard Eichler order R ⊆ R of level N . Let ﬁnally be max ∗∗ × + the subgroup of elements x ∈ R having norm one and such that i (x) ≡ (mod N ) for all | N . 123 Shimura–Mass operator 2.2 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 : R → End (A).An isogeny of false elliptic curves A max S is an isogeny which commutes with the action of R .A full level N -structure on A is an max + + isomorphism of group schemes α : A[N ] (R ⊗ (Z/N Z)) , where for any group A max Z S G we denote G the constant group scheme G over S.A level structure of V (N )-type is S 1 + + an equivalence class of full level N -structures under the (right) action of V (N ). The moduli problem which associates to any Z[1/Np]-scheme S the set of isomorphism classes of false elliptic curves equipped with a V (N )-level structure is representable by a smooth proper scheme C deﬁned over Spec(Z[1/Np]) [19,Theorem 2.2]. Let π : A C be the universal false elliptic curve. For any Z[1/Np]-algebra R,let π : A C be the base R R R change of π to R.Wehave \H C (C),where H is the complex upper half plane and acts on it by Moebius transformations. 2.3 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 • • • • ( , d ),orsimply understanding the differentials d , the complex of sheaves X/S X/S X/S X/S of relative differential forms for the morphismφ. For a sheaf F of O -modules over a scheme ∨ ⊗k X, we denote F its O -linear dual. If F is invertible, for an integer k we let F denote the usual tensor product operation. Fix a ﬁeld F of characteristic zero. The relative de Rham cohomology bundle for the morphism A → C is deﬁned by F F q • H (A /C ) = R π . F F F ∗ dR A /C F F We ﬁrst recall the construction of the Gauss–Manin connection. We have a canonical short exact sequence of locally free sheaves ∗ 1 1 1 0 −→ π −→ −→ −→0(6) F C /F A /C A /F F F F F (the exactness is because π is smooth). This exact sequence induces maps A /F •−i ∗ i • ⊗ π −→ A /F A F C /F A /F F F F i • i • for each integer i, deﬁning a ﬁltration F = Im(ψ ) on with associated A /F A /F A /F F F F graded objects i • •−i ∗ i gr = ⊗ π . A /F F C /F F A /F A F F F p,q p,q Let E denote the spectral sequence associated with this ﬁltration. The E terms are then i 1 given by p,q p q E ⊗ H (A /C ) O F F 1 C /F C dR F F 123 M. Longo [27,(7)]. The Gauss–Manin connection i 1 i ∇: H (A /C ) −→ ⊗ H (A /C ) F F O F F dR C /F C dR 0,i 0,i 1,i is then deﬁned as the differential d : E → E in this spectral sequence. 1 1 1 We now recall various descriptions of the Kodaira–Spencer map. It is deﬁned to be the boundary map 1 1 ∗ 1 KS : π −→ R π π A /C F ∗ F ∗ F F A /C F C /F F F F 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 : A → C denote the dual abelian F F variety. By a result of Buzzard ([9,Sect. 1], see also [19,page 4183]), it is known that the abelian surface A is equipped with a canonical principal polarization ι : A A over F A F F F C , which we use to identify A and A in the following without explicit mentioning it; we F F recall that this polarization is characterized by the fact that the associated Rosati involution in End (A ) restricted to the image of R via the map R → End (A ) coincides C F max max C F F F † † −1 with the involution x → x of R ,deﬁnedby x = i bi (as usual, if x = a +bi +cj+dk, max then x¯ = a − bi − cj − dk). Using the principal polarization and the isomorphism between 1 1 ∨ R π ( ) and the tangent bundle of A , the Hodge exact sequence can be written as F ∗ A /F F 1 1 1 0 −→ π −→ H (A /C ) −→ π −→0(7) F ∗ F F F ∗ A /C dR A /C F F F F (cf. [37,(2.2)], [19,Sect. 2.6]). The Kodaira–Spencer map can be deﬁned using the Gauss– Manin connection as the composition (7) (7) 1 1 i 1 KS : π −→ H (A /C ) −→ H (A /C ) ⊗ − A /C F ∗ F F F F O F F A /C dR dR C C /F F F F (7) ∨ 1 1 − π ⊗ F ∗ O A /C C /F F F F F 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 -modules, denoted again with the same symbol, ⊗2 1 1 KS : π −→ . A /C F ∗ F F A /C C /F F F 2.4 Idempotents and line bundles Let 1 1 √ e = 1 ⊗ 1 + j ⊗ p ∈ R = R ⊗ O [1/(2 p )] 0 M max Z M 0 2 p be the idempotent in [37,(1.10)], [19,Sect. 2.1], where O is the ring of integers of M. 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 embedding Q → C)or F ⊆ Q (and ¯ ¯ then we require that F contains the image of M via the ﬁxed embedding Q → Q ). 123 Shimura–Mass operator Since M is contained in F,wehaveanactionof R on the sheaves π and M F ∗ A /C F F H (A /C ), and we may therefore deﬁne the invertible sheaf of O -modules F F C dR F ω = e · π (8) F ∗ A /C F F and the sheaf of O -modules L = e · H (A /C ). (9) F F F dR Using that e is ﬁxed by the Rosati involution, the Hodge exact sequence (7) becomes −1 0 −→ ω −→ L −→ ω −→ 0 (see [19,Sect. 2.6]). For any integer n ≥ 1, deﬁne L = Sym (L ). (10) F,n F The Gauss–Manin connection is compatible with the quaternionic action [37,Proposition 2.2]. Therefore, restricting to L and using the Leibniz rule (see for example [19,Sect. 3.2]), the F,1 Gauss–Manin connection deﬁnes a connection ∇ : L −→ L ⊗ . n F,n F,n C /F ⊗2 By [37,Theorem 2.5], restricting the Kodaira–Spencer map to ω gives an isomorphism ⊗2 1 KS : ω −→ . F C /F We may then deﬁne a map ∇ : L → L by the composition n F,n F,n+2 −1 id⊗KS n F 1 ⊗2 ∇ : L L ⊗ L ⊗ ω L ⊗ L L n F,n F,n O F,n O F,n O F,2 F,n+2 C C /F C F C L L L (11) where the last map is the product map in the symmetric algebras. 2.5 Algebraic modular forms As in the previous section, let F e a ﬁeld of characteristic 0. For any F-algebra R,wedeﬁne the R-algebra alg ⊗k S , R = H (C ,ω ) ( ) k R of algebraic modular forms of weight k and level V (N ) over R. One can show [19,Sect. alg 3.1] that the R-algebra S (, R) can be alternatively described in modular terms. Let R be an R algebra. A test triple over R is a triplet (A , t ,ω ) consisting of a false elliptic curve A /R ,a V (N )-level structure t and a global section ω of ω . An isomorphism of test A /R triples (A , t ,ω ) and (A , t ,ω ) is an isomorphism of false elliptic curves φ : A → A such that φ(t ) = t and φ (ω ) = ω .A test pair over R is a pair (A , t ) obtained from a test triplet by forgetting the datum of the global section. Then one can identify global sections ⊗k of ω with: (1) A rule F which assigns, to each R-algebra R and each isomorphism class of test triplets (A , t ,ω ) over R ,anelement F(A , t ,ω ) ∈ R , subject to the base change axiom (for all maps of R-algebras φ : R → R ,wehave F(A , t ,φ (ω )) = F(A ,φ(t ),ω ), where A is the base change of A via φ)and the weight k condition (for all λ ∈ (R ) , −k we have F(A , t ,λω ) = λ F(A , t ,ω ))[19,Deﬁnition 3.2]. 123 M. Longo (2) A rule F which assigns to each R-algebra R and each isomorphism class of test pairs ⊗k (A , t ) over R , a translation invariant section F(A , t ) ∈ ω subject to the base A /R change axiom (for all maps of R-algebrasφ : R → R , we have the relation F(A , t ) = φ (F(A ,φ(t )),where A is the base change of A via φ)[19,Deﬁnition 3.3]. Let us make the relations between these deﬁnitions more explicit [19,page 4193]. Given a 0 ⊗k global section f ∈ H (C ,ω ), we get a function as in (2) above associating to each test pair (A , t ) over R the point x ∈ C (R ), and taking the value of f at x ;if F (A ,t ) R (A ,t ) is as in (2), we get a function G on test triples (A , t ,ω ) over R as in (1) by the formula ⊗k F(A , t ) = G(A , t ,ω )ω where ω ∈ ω is the choice of any translation invariant A /R global section. 3 Special values of L-series In this section we review the work of Brooks [19] expressing special values of certain L- functions of modular forms in terms of CM-values of the Maass–Shimura operator applied to the modular form in question. 3.1 The real analytic Maass–Shimura operator an an We denote (X, O ) (X , O ) the analytiﬁcation functor which takes a scheme of ﬁnite type over C to its associated complex analytic space. For each sheaf F of O -modules on an X,wealsodenote F the analytiﬁcation of F, and for each morphism ϕ : F → G of an an an O -modules, we let ϕ : F → G the corresponding morphism of analytic sheaves. If ran (X, O ) is an analytic space, we denote O the ring of real analytic functions on X;thisis ran ran a sheaf of O -modules, and for any sheaf F of O -modules, we let F = F ⊗ O ; X X O an ran an ran when F = F , we simplify the notation by writing F instead of (F ) . an Since C is proper and smooth over C, the analytiﬁcation functor F F induces an equivalence of categories between the category of coherent sheaves C and the category an of analytic coherent sheaves of O -modules. Also, the analytic sheaf obtained from the sheaf of algebraic de Rham cohomology H (A /C ) coincides with the derived functor C C dR 1 an 1 an R π in the category of analytic sheaves over C [41,Theorem 1]. an an C∗ C A /C C C Hodge theory gives a splitting ran 1 ran 1 H (A /C ) −→ π C C C∗ dR A /C C C of the corresponding Hodge exact sequence of real analytic sheaves obtained from (7). Since ran 1 1 ran this splitting is the identity on the image of π in H (A /C ) ,itgives C∗ C C A /C dR C C ran ran rise to a map : L → ω (cf. [37,Proposition 2.8]). We may then consider the induced C,1 C ran ⊗n ran maps : L → (ω ) for any integer n ≥ 1. Further, the map ∇ gives rise to a ∞,n n C,n C ran ran ran map ∇ : L → L of real analytic sheaves. The composition C,n C,n+2 ran n ∞,n ⊗n ran ran ran ⊗n+2 ran : (ω ) L L (ω ) ∞,n C C,n C,n+2 C is the real-analytic Shimura–Maas operator. The effect of on modular forms is described in [19,Proposition 3.4] and ∞,n + × + [37,Proposition 2.9]. Denote = (N ) the subgroup of B ∩ V (N ) consisting of 1 1 123 Shimura–Mass operator elements of norm equal to 1. Fix an isomorphism B ⊗ R M (R) and denote the Q 2 ∞ image of in GL (R).Let S ( ) denote the C-vector space of holomorphic modular forms 2 k ∞ of weight k and level consisting of those holomorphic functions on H ,the complex ∞ ∞ upper half plane, such that f (γ(z)) = j(γ, z) f (z) for all γ ∈ ;here acts on H by ∞ ∞ ∞ fractional linear transformations via the map B → B ⊗ R M (R).Wehave(cf.[19,Sect. Q 2 2.7]) 0 an ⊗k an S ( ) H C ,(ω ) . k ∞ ran Deﬁne the space S ( ) of real analytic modular forms of level and weight k to be the ∞ ∞ C-vector space of real analytic functions f : H → C such that f (γ(z)) = j(γ, z) f (z) for all γ ∈ . One then has ran 0 an ⊗k ran S ( ) H C ,(ω ) . (12) k C ran The operator gives then rise to a map δ : S ( ) → S ( ) and we have ∞,k ∞,k k ∞ ∞ k+2 1 d k δ ( f )(z) = + f (z). ∞,k 2πi dz z +¯z 3.2 CM points and triples Fix an embedding Q → C. For any embedding ϕ : K → B there exists a unique τ ∈ H such that ι (ϕ(K ))(τ) = τ . The additive map K → C deﬁned by α → j(ι (ϕ(α)),τ) ∞ ∞ gives an embedding K → C; we say that ϕ is normalized if α → j(ι (ϕ(α)),τ) is the identity (with respect to our ﬁxed embedding Q → C). We say that τ ∈ H is a CM point if there exists an embedding ϕ : K → B which has τ as ﬁxed point as above, and that a CM point τ is normalized if ϕ is normalized. Finally, we say that a CM point τ ∈ H is a Heegner point if ϕ(O ) ⊆ R [19,Sect. 2.4 and page 4188]. Fix a CM point τ corresponding to an embedding ϕ : K → B.Let a be an integral ideal of O , and deﬁne the R -ideal a = R · ϕ(a). This ideal is principal, generated by K max B max an element α = α ∈ B. Right multiplication by α gives an isogeny A → A , whose −1 a τ α τ kernel is A [a].Let be the subgroup of R consisting of elements of norm equal to 1. τ max max The image of ατ by the canonical projection map ρ : H → \H does not depend on max max the choice of the representative α, and therefore one may write A for the corresponding aτ abelian surface. Shimura’s reciprocity law states thatρ (τ) is deﬁned over the Hilbert class max −1 (a ,H/K) −1 ﬁeld H of K,and thatρ (τ) = ρ (aτ),where (a , H/K) denotes the Artin max max symbol. Fix a primitive N -root of unity ζ . Fix a normalized Heegner point τ , and ﬁx a point + + P ∈ A [N ] of exact order N such that e · P = P.Let (A , P ) denote the point on C(F) τ τ τ τ + + + corresponding to the level structure μ + × μ + Z/N Z × Z/N Z → A [N ] which N N + + takes (1, 0) ∈ Z/N Z × Z/N Z to P .A CM triple is an isomorphism class of triples (A , P ,ω ) with (A , P ) as above and a non-vanishing section ω in e · . τ τ τ τ τ τ A /F There is an action of Cl(O ) on the set of CM triples, given by a(A , P ,π (ω)) = (A /A [a],π(P ),ω) τ τ τ τ τ where π : A A /A [a] is the canonical projection. τ τ τ 123 M. Longo 3.3 Special value formulas Fix a CM triple (A, P,ω) = (A , P ,ω ) with ω deﬁned over H, the Hilbert class ﬁeld of τ τ τ K ; recall that A is also deﬁned over H. The complex structure J on M (R) deﬁnes a differential formω = J (2πidz ),and let τ 2 C 1 ∈ C be deﬁned by ω = ·ω ; clearly, different choices of ω correspond to changing ∞ ∞ C by a multiple in H. + − We now let f be a modular form of weight k,level (N ) ∩ (N ), and character 1 0 JL ε ,and let f be the modular form on the Shimura curve C associated with f by the f C JL Jacquet-Langlands correspondence. We can normalise the choice of f so that the ratio JL JL f , f / f , f belongs to K [19,Sect. 2.7 and page 4232]. (2) Let be the set of Hecke characters χ of K of inﬁnite type ( , ) with ≥ k and 1 2 1 (2) ≤ 0. We say that χ ∈ is central critical if + = k, so that the inﬁnite type of χ 2 1 2 (2) (2) is (k + j, − j) for some integer j ≥ 0. Denote the subset of consisting of central cc critical characters. ran ran For each positive integer j,let δ : S ( ) → D ( ) denote the j-th iterate of ∞ ∞ ∞,k k k+2 j the Shimura–Mass operator deﬁned by δ = δ ◦ ··· ◦ δ ◦ δ . ∞,k+2( j −1) ∞,k+2 ∞,k ∞,k (2) −1 For any Hecke character, one may consider the L-function L( f,χ , s),and forχ ∈ cc −1 central critical deﬁne the algebraic part L ( f,χ ) of its special value at s = 0as in alg (2) −1 [19,Proposition 8.7]. By [19,Proposition 8.7], if χ ∈ then L ( f,χ ) ∈ Q,and we cc alg have ⎛ ⎞ −1 −1 JL ⎝ ⎠ L ( f,χ ) = χ (a) · δ ( f )(a(A, t,ω)) alg j ∞,k a∈Cl(O ) − j where χ = χ · nr and nr is the norm map on ideals of O . In this formula we view the j K JL real analytic modular δ ( f ) as a function on test triplets, as in [19,Proposition 8.5] via ∞,k (12) (see also the discussion in [4,page 1094] in the GL case). For each ideal class a in Cl(O ),let α be the corresponding element in B, as in Sect. K a 3.2. Then using the dictionary between real analytic forms as functions on H or functions on test triples, and recalling that A = A for a normalized Heegner point τ,wehave ⎛ ⎞ k+2 j j −1 −1 JL ⎝ ⎠ L ( f,χ ) = · χ (a) · δ ( f )(α ·τ) . alg ∞ a j ∞,k a∈Cl(O ) JL In this formula we view δ ( f ) as a function on H. ∞,k 4 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. 123 Shimura–Mass operator Let H denote Drinfel’d p-adic upper half plane; this is a Z -formal scheme, and we p p denote H its generic ﬁber, which is a Q -rigid space [3,Chapitre I]. p p 4.1 Drinfel’d Theorem Denote D the unique division quaternion algebra over Q ,and let O be its maximal order. p D The unramiﬁed quadratic extension Q of Q 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 unr σ denote the absolute Frobenius automorphism of Gal(Q /Q ).Ifﬁxanelement ∈ O p D such that = p and x = σ(x) for x ∈ Q ,then D = Q [ ]. We will denote x →¯x 2 2 p p the restriction of σ to Gal(Q 2/Q ). For any Z -algebra B,a formal O -module over B is a commutative 2-dimensional p D formal group G over B equipped with an embedding ι : O → End(G). A formal O - G D D module is said to be special if for each geometric point P of Spec(B/pB), the representation of O / O on the tangent space Lie(G ) of G = G × k is the sum of two distinct D D P P P characters of O / O ,where k is the residue ﬁeld of P;see [43,Deﬁnition 1] for more D D P details on this deﬁnition. By an SFD-module over B, we mean a special formal O -module over B.If G is a SFD-module over a Z -algebra B in which p is nilpotent, we denote M(G) the (covariant) Cartier-Dieudonné module of G [3,Chapitre II, Sect. 1]; we also denote F and V (or simply F and V when there is no confusion) the Frobenius and Verschiebung endomorphisms of M(G).If B is Z 2-algebra, where Z 2 is the valuation ring of Q 2,and p p p G a formal O -module, then we may deﬁne Lie (G) ={m ∈ Lie(G) : ι (a) = am, a ∈ Z 2 }, Lie (G) ={m ∈ Lie(G) : ι (a) =¯am, a ∈ Z } 0 1 and, since G is special, both Lie (G) and Lie (G) are free B-modules of rank 1, (recall that x¯ = σ(x),so x →¯x is the non-trivial automorphism of Gal(Q 2/Q )). Moreover, M(G) is 0 1 also equipped with a graduation M(G) = M (G) ⊕ M (G) where M (G) ={m ∈ M(G) : ι (a) = am, a ∈ Z 2 }, M (G) ={m ∈ M(G) : ι (a) =¯am, a ∈ Z 2 }. Fix a SFD-module ! = G × G over F ,where G is the reduction modulo p of a Lubin- unr ˆ ˆ Tate formal group E of height 2 over Z , the completion of the valuation ring of the maximal unr unramifed extension Z of Z ;so E is the formal group of a supersingular elliptic curve unr E over Z (see [43,Deﬁnition 9 and Remark 27]). The Dieudonné module M(!) of ! is unr 0 1 0 0 the Z [F, V]-module with V-basis g and g , satisfying the relations F(g ) = V(g ) and 1 1 0 1 1 F(g ) = V(g ). The quaternionic order O acts via the rules (g ) = V(g ), (g ) = 0 0 0 1 1 0 V(g ) and a(g ) = ag , a(g ) =¯ag for a ∈ Z .By[43,Corollary 30],η (!) is generated 0 1 1 1 0 over Z by [g , 0] and [V(g ), 0],and η (!) is generated over Z by [g , 0] and [V(g ), 0]. p p Let Nilp denote the category of Z -algebras in which p is nilpotent. Denote SFD the functor on Nilp which associates to each B ∈ Nilp the set SFD(B) of isomorphism classes of triples (ψ, G,ρ) where (1) ψ : F → B/pB is an homomorphism, (2) G is a SFD-module over B of height 4, (3) ρ : ψ ! → G = G ⊗ B/pB is a quasi-isogeny of height 0, called rigidiﬁcation. ∗ B/pB B 123 M. Longo See [43,page 663] or [3,Chapitre II (8.3)] for more details on the deﬁnition of the functor SFD. Drinfel’d shows in [13] that the functor SFD is represented by the Z -formal scheme unr unr ˆ ˆ ˆ H = H ⊗ Z p Z p p p unr unr (see [43,Theorem 28], [3,Chapitre II (8.4)]). Note that H , considered as Z -formal p p unr scheme, represents the restriction SFD of SFD to the category Nilp of Z -algebras in unr which p is nilpotent (cf. [3,Chapitre II, Sect. 8]). Unless otherwise stated, we will see H unr as a Z -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 -formal scheme H as the solution of a moduli prob- p p lem. For B ∈ Nilp, a compatible data on S = Spf(B) consists of a quadruplet (η, T, u,ρ) where 0 1 (1) η = η ⊕ η is a sheaf of ﬂat Z/2Z-graded Z [ ]-modules on S, 0 1 i (2) T = T ⊕ T is a Z/2Z-graded sheaf of O [ ]-modules with T invertible, (3) u : η → T is a homogeneous degree zero map such that u ⊗ 1 : η ⊗ O → T is Z S surjective, (4) ρ : (Q ) → η ⊗ Q is a Q -linear isomorphism, S 0 Z p p p p which satisfy natural compatibilities, denoted(C1),(C2),(C3) in [43,page 652], to which we refer for details. The ﬁrst step in Drinfel’d work is to show that the Z -formal sheme H repre- p p sents 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 → H of H , 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 unr ˆ ˆ (η , T , u ,ρ ) which corresponds to an S = Spf(B)-valued point on H ⊗ Z .If X X X X p Z p p X = (ψ, G,ρ) ∈ SFD(B) is given as above, the quadruplet (η , T , u ,ρ ) can be explic- X X X X itly constructed as follows: • T = T(G) = M(G)/VM(G) the tangent space to G at the origin, equipped with its graduation deﬁned previously; • Deﬁne N(G) = M(G) × M(G)/ ∼ where (V(x), 0) ∼ (0, (x)); we denote [x, y] the class in N represented by the pair (x, y).Let λ : N(G) → M(G) be the map deﬁned by λ([x, y]) = (x) − V(y).There is amap N(G) → M(G)/VM(G) induced by the projection onto the ﬁrst component. Further, one easily shows that there exists a unique map L : M(G) → N(G) satisfying the relation λ ◦ L = F.Let φ : N(G) → N(G) be φ=Id deﬁned by φ([x, y]) = L(x) +[y, 0].Then η = η(G) = N . The graduation of 0 1 M(G) deﬁnes a graduation η(G) = η (G) ⊕ η (G). • u = u(G) is induced by the projection map N(G) → M(G)/VM(G). • Fix an isomorphism η (!) Z ⊕ Z . The quasi-isogeny ρ induces a map p p 0 0 ρ = ρ(G) : Q ⊕ Q η (ψ (!)) ⊗ Q → η (!) ⊗ Q . X p p ∗ Z p Z p p p 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 unr unr ˆ ˆ z : SFD(Z ) −→ H (Q ) (13) 0 p p p unr such that the map X = (ψ, G,ρ) → (z (X),ψ) gives a bijection between SFD(Z ) and unr unr unr unr unr ˆ ˆ ˆ ˆ ˆ ˆ (H ⊗ Z )(Z ), which we identify with the set H (Q ) × Hom(Z , Z ). We call p Z p p p p p p p 123 Shimura–Mass operator the map X → z (X) a rigid analytic parameter on SFD.Ifwelet pro-Nilp the category of projective limits of objects in Nilp, and we still denote SFD the restriction of SFD to pro-Nilp, unr this implies that the map X = (ψ, G,ρ) → z (X) gives a bijection between SFD(Z ) and unr unr ˆ ˆ H (Q ).By[43,Thm. 45], for each z ∈ H (Q ), there exists triple X = (ψ, G,ρ) in p p p p unr SFD(Z ) such that z (X) = z. 4.2 Filtered Frobenius modules Let E be an unramiﬁed ﬁeld extension of Q .A Frobenius module E over E is a pair E = (V,φ) consisting of a ﬁnite dimensional E-vector space V with aσ -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 φ- modules these objects. If G is a p-divisible formal group over F , one can deﬁne its ﬁrst crystalline coho- mology 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 self- contained expositions). In the following we will denote H (G) the global sections of the cris unr crystalline Dieudonné functor (deﬁned as in [2,Théorème 4.2.8.1]) tensored over Z with unr 1 Q . By construction, H (G) is then a Frobenius module. Moreover, the canonical iso- p cris 1 1 morphism between H (G) and the ﬁrst de Rham cohomology group H (G) of G equips cris dR H (G) with a canonical ﬁltration (arising from the Hodge ﬁltration in the de Rham coho- cris mology), making H (G) a ﬁltered Frobenius module; see [38]. dR Let G be a SFD-module over F . Then the Frobenius module H (G) is a four- cris unr dimensional Q -vector space, equipped with its σ -linear Frobenius φ (G).Itisalso cris equipped with a D-module structure j : D → End H (G) which commutes G ˆ unr cris with φ (G),and a Q -algebra embedding i : M (Q ) → End H (G) induced cris p G 2 p ˆ unr cris by the isomorphism End (G) M (Q ), which commutes with the D-action. Deﬁne O 2 p −1 φ (G) = j ( ) φ (G) and put G cris cris 1 φ (G)=Id cris V (G) = H (G) . cris cris Denote φ = j ( ) the restriction of j ( ) to V (G). Moreover, denote V (G) G |V (G) G cris cris cris i ∨ i (η (G) ⊗ Q ) = Hom (η (G) ⊗ Q , Q ) Z p Q Z p p p p p the Q -linear dual of η (G) ⊗ Q . p Z p The following lemma is crucial in what follows, and identiﬁes V (G) with (η (G) ⊗ cris Z Q ) , from which one deduces a complete description of the ﬁltered Frobenius module H (G). It appears in a slightly different version in the proof of [23,Lemma 5.10]. Since we cris 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 (G) η(G) ⊗ Q of Q -vector cris Z p p 1 unr spaces. Moreover, H (G) = V (G) ⊗ Q , where the right hand side is equipped with cris Q cris p p unr the structure of Q -vector space given by x ·(v ⊗α) = v ⊗(σ(x)α) forv ∈ V (G),x,α ∈ cris unr Q . Finally, under this isomorphism the Frobenius φ (G) corresponds to φ ⊗ σ . cris V (G) p cris 123 M. Longo Proof Recall that the Frobenius module H (G) is canonically isomorphic to the con- cris unr travariant Dieudonné module of G with p inverted, and with Q -action twisted by the unr Frobenius automorphism σ of Q , equipped with the canonical Frobenius of the con- travariant Dieudonné module (see [2,4.2.14]). More precisely, denote unr D(G) = Hom M(G)[1/p], Q ˆ unr unr the Q -linear dual of the covariant Dieudonné module M(G) of G with p inverted, and let σ unr D(G) = D(G) ⊗ Q , ˆ unr Q ,σ unr where the tensor product is taken with respect to the Frobenius endomorphismσ of Q .Then unr 1 σ as Q -vector spaces, we have H (G) D(G) . Under this isomorphism the Frobenius p cris φ (G) is given by the map ϕ → σ ◦ ϕ ◦ V for ϕ ∈ D(G). cris G −1 Now, by [3,Lemme (5.12)], we have an isomorphism of σ -isocrystals i −1 i unr −1 M (G)[1/p], V η (G) ⊗ Q ,σ (14) G Z p p −1 i unr for each index i = 0, 1 (where the action of σ on η (G) ⊗ Q is on the second factor p p i unr −1 only). We may therefore compute V (G) in terms of the isocrystal η (G) ⊗ Q ,σ . cris Z p p As above, deﬁne i i unr D (G) = Hom (M (G)[1/p], Q ) unr ˆ p unr i σ i unr ˆ ˆ (Q -linear dual) and let D (G) denote the base change D (G) ⊗ Q via σ.Since ˆ unr p p Q ,σ 0 1 M(G) = M (G) ⊕ M (G),wehave σ 0 σ 1 σ D(G) = D (G) ⊕ D (G) , σ i σ and we may write any element ϕ ∈ D(G) as a pair (ϕ ,ϕ ) with ϕ ∈ D (G) , i = 0 1 i 0, 1. By deﬁnition, an element ϕ = (ϕ ,ϕ ) ∈ D(G) belongs to V (G) if and only if 0 1 cris −1 −1 i ϕ (V (m )) is equal to σ (ϕ (m )) for all m ∈ M (G)[1/p],and forall i = 0, 1. i G i i i i unr i unr unr ˆ ˆ ˆ Using (14), identify ϕ with a Q -linear homomorphism ϕ : η (G) ⊗ Q → Q , i i Z p p p p denoted with a slight abuse of notation with the same symbol; then the above equation describing V (G) becomes cris −1 −1 ϕ (n ⊗ σ (x)) = σ (ϕ (n ⊗ x)) i i i unr unr ˆ ˆ for all n ∈ η (G) and all x ∈ Q .Since ϕ is Q -linear, p p −1 −1 −1 σ (ϕ (n ⊗ x)) = σ (x)σ (ϕ (n ⊗ 1)), i i −1 −1 −1 i and we deduce the equality ϕ (n ⊗ σ (x)) = σ (x)σ (ϕ (n ⊗ 1)) for all n ∈ η (G). i i −1 Taking x = 1 we see that ϕ (n ⊗ 1) = σ (ϕ (n ⊗ 1)) and we conclude that ϕ (n ⊗ 1) ∈ Q i i i p i unr for all n ∈ η (G).So ϕ is the Q -linear extension of a Q -linear homomorphism i p η (G) ⊗ Q → Q . Z p p 0 1 Since η(G) = η (G) ⊕ η (G), we then conclude that V (G) η(G) ⊗ Q as Q - cris Z p p vector spaces (here denotes the Q -dual). If n ,..., n is a Q -basis of η(G) ⊗ Q , p 1 4 p Z p then dn ,..., dn deﬁned by dn (n ) = δ (as usual, δ = 1if i = j and 0 otherwise) is 1 4 i j i, j i, j 123 Shimura–Mass operator ∨ unr unr ∨ ˆ ˆ abasis of (η(G) ⊗ Q ) and, by Q -linear extension, also of (η(G) ⊗ Q ) .Ifwe Z p Z p p p p unr unr ∨ ˆ ˆ now base change the Q -vector space (η(G) ⊗ Q ) via σ , we see that dn ,..., dn Z 1 4 p p p unr is still a Q -basis, and we have (x · dn )(n ) = σ(x)δ i j i, j unr 1 σ for all x ∈ Q . Using the above description of H (G) in terms of D(G) ,and the p cris unr description of V (G) in terms of η(G), we have an isomorphism of Q -vector spaces, cris 1 unr H (G) V (G) ⊗ Q , cris Q cris p p unr where the upper index σ on the right hand side means that the structure of Q -vector −1 −1 space is twisted by σ as explained above. Moreover, the σ -linear isomorphism V −1 −1 unr of M(G)[1/p] corresponds to the σ -linear isomorphism σ of η(G) ⊗ Q (act- p p ing on the second component only), and therefore the isomorphism ϕ → σ ◦ ϕ ◦ V ∨ ∨ unr of M(G)[1/p] (where denotes the Q -linear dual) corresponds to the isomorphism ∨ unr ⊗ σ of (η(G) ⊗ Q ) ⊗ Q given by dn ⊗ x → (dn ◦ ) ⊗ σ(x) where Z p Q i i p p p (dn ◦ )(n) = n ( n), which corresponds to φ ⊗ σ by deﬁnition of φ . i i V (G) V (G) cris cris 4.3 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]. unr Let E ⊆ Q be an unramiﬁed extension of Q , with valuation ring O .If (X, O ) is a p E X rig rig p-adic O -formal scheme, we denote (X , O ) the associated E-rigid analytic space (or rig its generic ﬁber), and if F is a sheaf of O -modules, we denote F its associated sheaf of rig O -modules [8,Sect. 7.4], [5,Sect. 1]. The notion of convergent isocrystal on a p-adic, formally smooth O -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 ) of X a coherent O ⊗ E-module E satisfying a natural cocycle condition for morphism T O T 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 : 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 .By[39,1.20, 2.81] is a canonical rig integrable connection ∇ : E → E ⊗ . Accordingly with [23,Deﬁnition 3.3], a X X rig rig ﬁltered convergent F-isocrystal on X is a F-isocrystal (E,φ ) such that E is equipped rig rig with an exhaustive and separated decreasing ﬁltration F E of coherent O -submodules X X rig rig rig i i −1 1 such that ∇ (F E ) is contained in F E ⊗ for all i; in these deﬁnitions, F rig rig X X X O X is the absolute Frobenius of the reduced closed subscheme of the closed subscheme of X deﬁned by the ideal pO . We denote E(O ) 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 ) O ⊗ E; it is equipped with canonical Frobenius and ﬁltration (see loc. T T O cit. for details). We also denote E(V) = V ⊗ E(O ) the ﬁltered convergent F-isocrystal Q unr p ˆ attached to a Q -rational, ﬁnite dimensional representation ρ : GL × GL → GL(V) of p 2 2 123 M. Longo the algebraic group GL × GL ;see [23,pages 345–346] for the deﬁnition of Frobenius and 2 2 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,ρ ) be the universal triple, arising from the representability of the functor SFD by G G unr unr ˆ ˆ H ; denote λ : G → H be universal map. Then p p E(G) = R λ (O ) ∗ ˆ unr G/Q unr has a structure of convergent F-isocrystal of H , which interpolates crystalline cohomology unr sheaves (see [39,Theorem (3.1), Theorem (3.7)]; see also [6, 7]). The coherent O -module rig E(G) is canonically isomorphic to the relative rigid de Rham cohomology sheaf unr rig 1 rig unr 1 • H (G /H ) = R λ . ∗ rig unr dR p G /H rig The canonical integrable connection ∇ coincides with the Gauss–Manin connection unr rig 1 rig unr 1 1 rig unr ∇ : H (G /H ) −→ ⊗ H (G /H ) unr G dR p unr ˆ unr dR p H /Q p p p whose construction in this context follows [27]; see [23,Example 3.4(c)], [39,Theorem (3.10)]. 1 rig unr The Hodge ﬁltration on the de Rham cohomology H (G /H ) makes then E(G) a ﬁltered dR convergent F-isocrystal. To describe this ﬁltration more explicitly, denote 1 1 unr H (G) = H (G/H ) dR dR p the dual of the Lie algebra of the universal vectorial extension of G, equipped with its structure of convergent F-isocrystal ([35,Chapter IV, §2], [36,Sect. 1,9,11)]). By [2,Sect. 3.3], we have an isomorphism of convergent F-isocrystals E(G) H (G) dR such that the Hodge ﬁltration on the de Rham cohomology groups coincides with the Hodge- Tate ﬁltration rig 1,rig rig 0 −→ ω −→ H (G) −→ Lie −→ 0. (15) G dR 1,rig ∗ 1 unr Here ω = e ( ),where e : H → G is the zero-section, H (G) = G p G ˆ unr dR G/H rig 1 ∨ H (G) ,and Lie is the Lie algebra of the Cartier dual G of G. A result of Falt- dR ˆ unr ings [14,Sect. 5] (see also [23,Lemma 5.10]) shows that, as ﬁltered convergent F-isocrystals, we have H (G) V (!) ⊗ E(O ) E(M ), (16) cris unr 2 Q ˆ dR p H where ρ : GL × GL → GL(M ) is the representation deﬁned by ρ (A)(B) = AB and 2 2 2 1 ab d −b ¯ ¯ ρ (A)B = B A (here if A = ,then A = ). cd −ca 123 Shimura–Mass operator 4.4 The Kodaira–Spencer map unr The Kodaira–Spencer map forλ : G → H , where as above G is the universal SFD-module, is the composition rig (15) (15) rig rig 1,rig G 1,rig rig 1 1 KS : ω −→ H (G) −→ H (G) ⊗ −→ Lie ⊗ O unr ∨ O unr G G dR dR unr unr unr unr H ˆ G H ˆ p H /Q p H /Q p p p p in which the ﬁrst and the last map come from the Hodge-Tate exact sequence (15). Recalling the duality between ω and Lie , we therefore obtain, as in the algebraic case, a symmetric rig rig rig unr map of O -modules, again denoted KS : ω ⊗ ω → (tensor product H ∨ p G G G unr ˆ unr H /Q p p over O unr ). By ﬁxing a formal polarization ι : G G of G [3,Chapitre III, Lemma 4.4], H G ∨ unr we obtain isomorphism ω ω of O -modules, and the Kodaira–Spencer map takes G H the form rig rig ⊗2 1 KS : (ω ) −→ G G unr ˆ unr H /Q p p unr where the tensor product is again over O . We now describe the Kodaira–Spencer map more explicitly, mimicking, in the complex case, [17, 37]. Denote unr H (G) = H (G/H ) 1,dR 1,dR the universal vectorial extension of G, which is equipped with a structure of ﬁltered convergent rig rig F-isocrystal as before; see [14,Sect. 5]. Put H (G) = H (G) . By deﬁnition, the 1,dR 1,dR ˆ unr universal vectorial extensions of G and G are dual to each other. We therefore obtain a O rig-bilinear skew-symmetric map 1,rig rig H (G) × H (G) −→ O unr. dR 1,dR 1,rig rig The principal polarization ι : G G identiﬁes canonically H (G) and H (G),and dR 1,dR we therefore obtain a pairing rig 1,rig 1,rig unr , : H (G) × H (G) −→ O dR dR dR p rig rig 1,rig satisfying dx, y =x, d y for all x, y sections in H (G) and all d ∈ D (because G G dR ι (dy) = d ι (y)) which we call rigid polarization pairing. We may therefore construct a G G map 1,rig 1,rig rig ρ : H (G) −→ H (G) −→ (ω ) dR dR G rig where the ﬁrst map takes a section s to the map deﬁned for a section t by t →s, t and dR rig 1,rig the second map is induced by duality from the inclusionω → H (G). Fix now a section G dR 0 ∨ s ∈ H (U,( ) ) over some afﬁnoid U. Then we may compose the maps to get unr unr H /Q p p rig G 1⊗s rig 1,rig 1,rig 0 0 0 1 ρ : H U,ω −→H U, H (G) −→ H U, H (G) ⊗ −→ unr unr G dR dR ˆ H /Q p p 1⊗s ρ 1,rig rig 0 0 ∨ −→ H U, H (G) −→ H U,(ω ) . dR G (17) 123 M. Longo The association s → ρ deﬁnes then a map of sheaves rig rig rig ∨ 1 ∨ ∨ (KS ) : ( ) −→ Hom ω ,(ω ) . O unr G unr ˆ unr H G G H /Q p p p By construction, the dual of this map is the Kodaira–Spencer map, under the canonical rig rig rig ∨ ∨ ⊗2 identiﬁcation between Hom(ω ,(ω ) ) and (ω ) . G G G 4.5 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 and V be constant sheaves of one-dimensional Q -vector spaces on the Q -rigid 0 1 p p univ analytic space H with basis t and t respectively. Deﬁne two invertible sheaves T and p 0 1 univ univ T on H by T = O ⊗ V for i = 0, 1, where O is the structural sheaf of rigid p H i H p p 1 i univ univ univ univ analytic functions on H .Deﬁne T = T ⊕T .For i = 0, 1, letη be the constant 0 1 i sheaf of two-dimensional Q -vector spaces on H with basis e and e .One ﬁxes p p i,0 i,1 univ i η = η (!) ⊗ Q (18) Z p univ univ univ univ univ as in [43,page 664]. Deﬁne u : η → T by u (e ) = zt and u (e ) = t , 0,0 0 0,1 0 0 0 0 0 0 univ univ univ univ univ and u : η → T by u (e ) = (p/z)t and u (e ) = t ,where z denotes 1,0 1 1,1 1 1 1 1 1 1 univ univ univ the standard coordinate function on H .Deﬁne η = η ⊕ η and similarly deﬁne 0 1 univ univ univ univ 2 univ u = u ⊕ u . We write ρ : (Q ) → η for the isomorphism determined 0 1 H 0 ab by the choice of the basis {e , e }.For γ = ∈ M (Q ) and i = 0, 1, deﬁne 0,0 0,1 2 p cd (γ) univ endomorphisms φ in End (T ) by i p i (γ) φ f (z) ⊗ t = (cz + d) f (γ(z)) ⊗ t , ( ) 0 0 (19) (γ) φ ( f (z) ⊗ t ) = (a + b/z) f (γ(z)) ⊗ t 1 1 univ for any f ∈ O (U), and any afﬁnoid U ⊆ H . Deﬁne an action of SL (Q ) on η H p 2 p univ for i = 0,1tomake u equivariant with respect to these actions; in other words, for x x x ab/ p x ab ∗ 0,0 ab 0,0 ∗ 1,0 1,0 γ = in GL (Q ), put γ = and γ = .Let Z [ ] 2 p p cd x cd x x pc d x 0,1 0,1 1,1 1,1 univ univ act on T by t = (p/z)t and t = zt .Welet Z [ ] act on η in such a way that 0 1 1 0 p univ u commutes with this action. We call the quadruplet univ univ univ univ univ D = (η , T , u ,ρ ) the universal rigid data. univ Passing to the associated normed sheaves [43,Deﬁnition 6], we obtain from D a quadru- univ univ univ univ univ ˆ ˆ ˆ ˆ plet D = (η ˆ , T , u ˆ ,ρˆ ) on H , corresponding to a H -valued point, which p p is universal in the following sense: for each B ∈ Nilp and each : S = Spec(B) → H corresponding to a quadruplet (η, T, u,ρ),wehave −1 univ ∗ univ −1 univ −1 univ (η, T, u,ρ) = ( η ˆ , T , u ˆ , ρˆ ). (20) See [43,Corollary 18 and Theorem 19] for more precise and complete statements. We call univ univ ˆ ˆ D the universal formal data, and we denote the quadruplet on the RHS of (20)by D to simplify the notation. 123 Shimura–Mass operator unr univ The universal SFD-module G over H can be recovered from a universal rigid data D . unr ˆ ˆ Pulling back via the projection π : H → H , we obtain a quadruplet ˆ p unr unr unr unr unr −1 univ ∗ univ −1 univ −1 univ ˆ ˆ ˆ D = (η ˆ , T , u ˆ ,ρˆ ) = (π η ˆ ,π T ,π u ˆ ,π ρˆ ) ˆ ˆ ˆ ˆ H p H H p p p unr on H . Comparing (20) with the universal property satisﬁed by G, we see that the quadru- unr plet (η , T , u ,ρ ) associated to G coincides with the quadruplet D . In particular, the G G G G rig rig rig rig unr rig associated quadruplet (η , T , u ,ρ ) on the rigid Q -rigid analytic space G is the G G G G p quadruplet −1 −1 −1 unr unr unr unr unr univ ∗ univ univ univ D = (η , T , u ,ρ ) = (π η ,π T ,π u ,π ρ ) unr unr unr unr H H H H p p p p univ unr obtained from the quadruplet D ,where π unr : H → H is the canonical projection. H p unr ∨ unr unr ∨ unr Let (T ) denote the O -dual of T , and, as above, denote (η ⊗ Q ) the H Z p p p unr unr unr unr Q -linear dual of η ⊗ Q . From the surjective map u : η ⊗ O unr T p Z p Z H p p univ induced by u we obtain an injective map unr ∨ unr ∨ unr τ : (T ) −→ (η ⊗ Q ) ⊗ O p H Z Q p p p Proposition 4.2 We have canonical isomorphisms rig unr ∨ (T ) −→ ω , 1,rig unr ∨ unr (η ⊗ Q ) ⊗ O −→ H (G). Z p Q H p p p dR under which the map τ corresponds to the ﬁrst map in (15). Proof The ﬁrst statement follows from the canonical isomorphism between T = Lie and G G unr T , while the second follows from Proposition 4.1 combined with (16). For the statement about τ , note that for each SFD-module G over F ,the map u corresponds under the p G unr unr ˆ ˆ identiﬁcation between η(G) ⊗ Q and M(G) ⊗ Q to the canonical projection Z unr p p ˆ p M(G)/V M(G) → T ,where T is the tangent space of G at the origin. G G G 4.6 The action of the idempotent e Fixanisomorphism Q ( p ) Q 2. By means of this isomorphism, and the ﬁxed embed- p 0 √ √ ding Q → D, we may identify elements a + b p in Q ( p ) (where a, b ∈ Q ) with 0 p 0 p elements of D in what follows without explicitly mentioning it. 0 0 Lemma 4.3 e · η(!) ⊗ Z 2 = η (!) ⊗ Z 2 and e · T(!) ⊗ Z 2 = T (!) ⊗ Z p Z p Z p Z p p p p Z . Proof The action of O on η(!) is induced by duality from the action on M(!),soany 0 1 element a ∈ Z → O acts onη (!) by multiplication by a and onη (!) by multiplication by a ¯. On the other hand, the action of 1 ⊗a onη(!) ⊗ Q 2 is given by multiplication by a. p p An immediate calculation shows then that the action of e is just the projectionη(!) → η (!). The argument for T(!) is similar. −1 −1 unr univ unr ∗ univ unr univ Write η = π unrη , T = π unr T , u = π unr u . 0 H 0 0 H 0 0 H 0 p p p 123 M. Longo unr unr unr unr Proposition 4.4 e · η = η and e · T = T . 0 0 Proof This is clear from Lemma 4.3 and (18). unr unr For i = 0, 1, the sheaf T is a free O -module of rank 1, so it is invertible; denote 0 p unr ∨ unr ∨ ∨ unr unr (T ) its O -dual. W thus get a map du : (T ) → (η ⊗ Q ) ⊗ O (where H 0 0 Z p Q H p p 0 p 0 p the RHS denotes the Q -dual). We set up the following notation: rig • ω = e · ω ; G G 1,rig • L = e · H (G). G dR 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: du unr ∨ ∨ unr 0 (T ) (η ⊗ Q ) ⊗ O (21) 0 Z p Q H p p 0 p 0 0 0 ω L G G 4.7 Differential calculus on the p-adic upper half plane We now set up the following notation. Recall that the map u takes x e + x e to 0 0,0 0,0 0,1 0,1 (zx + x ) ⊗ t ; dualizing, du can be described in coordinates by the map which takes 0,0 0,1 0 0 unr ∨ unr the canonical generator t of the O -module (T ) (satisfying the relation dt (t ) = 1) 0 H 0 0 p 0 to the map x e + x e → zx + x . If we denote de the dual basis of e 0,0 0,0 0,1 0,1 0,0 0,1 0,i 0,i (satisfying the condition de (e ) = δ ), we may write this map as zde + de .To 0,i 0, j i, j 0,0 0,1 simplify the notation, we put from now on τ = t , dτ = dt , x = e , y = e , dx = de 0 0 0,0 0,1 0,0 and dy = de , so that the above map reads simply as 0,1 dτ = zdx + dy. 0 0 unr Let C = C (H (Q ), C ) denote the C -vector space of continuous (for the stan- p p p unr dard p-adic topology on both spaces) C -valued functions on H (Q ). Denote A = p p 0 unr unr H (H , O unr) the Q -vector space of global sections of O unr . Each f ∈ A is, in H H p p p p unr unr particular, continuous on H for the standard p-adic topology of Q − Q , and therefore p p unr 0 ∗ restriction induces a map of Q -vector spaces r : A → C . Denote A the image of the mor- phism of A-algebras A[X, Y]→ C deﬁned by sending X to the function z → 1/(z −σ(z)) and Y to the function z → σ(z) (note that the function z → z − σ(z) is invertible on unr unr H (Q )). To simplify the notation, we put from now on p p z = σ(z). Set up the following notation (here n ≥ 1isaninteger) ⊗n 0 unr 0 • = H (H , L ) and = , G G,n G G ⊗n ∗ ∗ ∗ ∗ • = ⊗ A and = , G A G G,n G,n ⊗n 0 unr 0 • w = H (H ,ω ) and w = w , G G,n p G G ⊗n ∗ ∗ ∗ ∗ • w = w ⊗ A and w = w . G A G G,n G,n 123 Shimura–Mass operator unr d The Q -algebra A is equipped with the standard derivation on power series. The dz 1 1 0 unr A-module = H (H , ) is then one dimensional and generated by dz satisfying unr A p H d d ∂ ∗ ∗ dz = 1. We extend the differential operator to a differential operator : A → A dz dz ∂z unr ∂ ∗ ∂ 1 −1 by Q -linearity using the product formula and setting (z ) = 0and = . p ∗ 2 ∂z ∂z z−z (z−z ) ∂ ∂ ∂ ∗ ∗ ∗ Similarly, we deﬁne a differential operator : A → A setting (z) = 0, (z ) = 1 ∗ ∗ ∗ ∂z ∂z ∂z ∂ 1 1 1 ∗ and = .Deﬁne to be the A -subalgebra of the algebra of derivations ∗ ∗ ∗ ∗ 2 ∂z z−z (z−z ) ∗ ∂ ∂ ∗ ∂ generated by dz and dz satisfying the usual rules dz = 1, dz = 0, dz = 0, ∗ ∗ ∂z ∂z ∂z ∗ ∂ dz = 1. ∂z 4.8 Splitting of the rigid analytic Hodge filtration Recall the notation ﬁxed before for the differential form dτ = zdx + dy.Deﬁne ∗ ∗ dτ = z dx + dy. ∗ ∗ unr Then dτ belongs to w . Taking global sections, restricting to Q , and extending linearly ∗ ∗ with A we obtain a short exact sequence of A -algebras ∗ ∗ 0 −→ w −→ . (22) G G ∗ ∗ Theorem 4.5 The exact sequence (22) admits a canonical splitting : → w . G G Proof We have ∗ ∗ ∗ dτ − dτ zdτ − z dτ dx = , dy = . ∗ ∗ z − z z − z We may therefore write any differential form ω = f (z)dx + g(z)dy with f , g ∈ A as f (z) − g(z)z zg(z) − f (z) ω = dτ + dτ . ∗ ∗ z − z z − z f (z)−g(z)z One then deﬁnes the sough-for splitting sending ω → dτ . z−z 4.9 The p-adic Maass–Shimura operator rig Taking global section, the Gauss–Manin connection gives rise to a map ∇ : → rig 1 ∗ ∗ ∗ 1 ⊗ .Weextend ∇ to a map ∇ : → ⊗ as follows. First deﬁne G ∗ A G G G G A 1,0 1,0 ∗ ∗ 1 ∇ : → ⊗ to be the derivation satisfying the rules ∇ (dτ) = dx ⊗ dz, G G G A G 1,0 1,0 1,0 0,1 ∗ ∗ ∗ ∇ (dτ ) = 0, ∇ (z) = 1, ∇ (z ) = 0. Deﬁne similarly the derivation ∇ : → G G G G G 0,1 0,1 0,1 0,1 ∗ 1 ∗ ∗ ∗ ⊗ by the rules ∇ (dτ) = 0, ∇ (dτ ) = dx ⊗ dz , ∇ (z) = 0, ∇ (z ¯) = dz . G A G G G G We ﬁnally deﬁne 1,0 0,1 ∗ ∗ ∗ 1 ∇ =∇ +∇ : −→ ⊗ ∗. G G G A G G ⊗2 1 Taking global sections, the Kodaira–Spencer map gives rise to a map KS : w → , G A which we extend A -linearly to a map ∗ ∗ ⊗2 1 KS : (w ) −→ ∗. G G A 123 M. Longo Note that dτ − dτ rig ∇ (dτ) =∇ (dτ) = ⊗ dz. (23) G G z − z ∗ ∗ and, since ∇ (z ) = 0, we have ∗ ∗ ∇ (dτ ) = 0. (24) In particular, if f (z) ⊗ dτ ∈ w we have ∂ dτ − dτ rig ∇ ( f (z) ⊗ dτ) = f (z) ⊗ dτ + f (z) ⊗ ⊗ dz. (25) ∂z z − z rig Taking global sections, we can form the pairing , : ⊗ → A. Extending G A G linearly by A , we obtain a new pairing ∗ ∗ ∗ ∗ , : ⊗ ∗ −→ A . G G G Using the description of the Kodaira–Spencer map in Sect. 4.4,wesee that ∗ ∗ −dτ, dτ rig rig ∗ ∗ G ∗ dτ, ∇ (dτ) =dτ, ∇ (dτ) = dz =−dx, dy dz G G G G G z − z where for the second equality we use (23), while the last equality follows easily from the equation ∗ ∗ ∗ ∗ zdx + dy, z dx + dy = (z − z )dx, dy . G G Therefore rig rig KS (dτ ⊗ dτ) =−dx, dy dz. G G rig rig So, to compute KS (dτ ⊗ dτ) = KS (dτ ⊗ dτ) we are reduced to compute dx, dy . G G G For this, we switch to de Rham homology and follow the computations in [19, 37]. To begin with, let W denote the order O viewed as free left O -module of rank 1; then D D W R ⊗ Z .By[3,Ch. III, Lemma 1.9], the collection of bilinear skew-symmetric max Z p mapsψ : W ×W → Z which satisfyψ(dx, y) = ψ(x, d y) (for all x, y ∈ W and d ∈ O ) p D is a free Z -module of rank 1, and every generator ψ of this Z -module is a perfect duality p 0 p on W ; the pairing tr(iy x) ψ (x, y) = is such a generator, which we ﬁx once and for all (recall the notation introduced in Sects. 2.1 and 2.3 for i and d ). 1 unr Recall that H (!) is a free D⊗ Q -module of rank 1 (cf. [23,page 354]); the structure cris p p unr ∨ of D ⊗ Q -module is induced from the D-module structure of (η(!) ⊗ Q ) via the Q Z p p p p ∨ 1 unr isomorphisms (η(!) ⊗ Q ) V (!) and H (!) V (!) ⊗ Q in Lemma Z p cris cris Q p p p cris 4.1.WehavethenfromLemma 4.1 then canonical isomorphisms of convergent F-isocrystals: 1 1 H (G/H ) H (!) ⊗ E(O ) p unr unr dR cris ˆ ˆ Q H p p unr D ⊗ Q ⊗ E(O ) unr Q ˆ unr ˆ p p H (26) D ⊗ Q 2 ⊗ E(O ) Q Q unr p p 2 ˆ M (Q 2) ⊗ E(O ). unr 2 Q ˆ p 2 p H 123 Shimura–Mass operator unr Under the isomorphism (26), the Q -linear extension of ψ deﬁnes a pairing 1 1 unr ψ : H (!) × H (!) −→ Q cris cris p rig rig still denoted with the same symbol. If we denote , the restriction of , to H (!),it G G cris follows from the unicity of ψ up to constant that there exists an element t ∈ C such that 0 p rig , = · ψ . (27) G,W Moreover, under the isomorphism (26), the element dτ = zdx + dy of H (G) corresponds dR 10 01 00 00 to the element e ⊗ z + e ⊗ 1, where e = , e = , e = , e = 1 2 1 2 3 4 00 00 10 01 is the standard basis of M (Q ). We therefore obtain the sought-for recipe to compute the Kodaira–Spencer image of dτ ⊗ dτ in terms of ψ : rig KS (dτ ⊗ dτ) = · ψ (e ⊗ z, e ⊗ 1). (28) 0 1 2 Remark 4.6 The number t 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 at the denominator in (27). We now make more explicit the equations (23)and (28) using Hashimoto basis {1, i, j, k} ﬁxed in Sect. 2.1.Asin[37,Proposition 2.3] and [18,(2)], deﬁne = 1, = (1 + j)/2, 1 2 = (i + ij)/2, = (apN j + ij)/p and use these elements to deﬁne a symplectic basis 3 4 0 p −1 of W with respect to the pairing ψ as in [18,(5)] by η = − , η =−aD − , 0 1 3 4 2 1 4 η = , η = (note that ψ we consider above is equal to the pairing (x, y) → tr(xiy ) 3 1 4 2 0 ∨ ∨ ∨ ∨ ∨ ∨ in [18,(3)]). Denote η ,η ,η ,η the dual basis of W ,and let η be the column vector 1 2 3 4 ∨ ∨ ∨ ∨ ∨ 1 with entries η ,η ,η ,η . The elements η give rise to elements of H (G), denoted with 1 2 3 4 dR rig rig the same symbol, which are horizontal with respect to ∇ , namely ∇ (η ) = 0. Write G G dτ = (z) · η . A simple calculation shows that α −1 1 + + + (z) = (α az + 1), (α az + 1), z, α z . (29) √ √ 2 p p 2 0 0 rig d (z) Sinceη are horizontal sections of ∇ ,using (29) to calculate shows that (23) becomes i G dz − + + α α a −α az 1 rig + ∨ ∇ (dτ) = , , 1, α · η ⊗ dz. (30) √ √ 2 p p 2 0 0 The recipe (17) to compute the Kodaira–Spencer map combined with (27)and (30)gives then 1 d (z) 1 0 I rig 2 KS (dτ ⊗ dτ) = (z) dz = dz. (31) −I 0 t dz 2 t p p rig In particular, (31) shows that KS is an isomorphism, and therefore the same is true for KS . This allows us to deﬁne the operator ∗ ∗ −1 ∇ (KS ) G,n G ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∇ : ⊗ ⊗ w ⊗ A ∗ A A G,n G,n G,n A G,n G,2 G,n G,2 G,n+2 ∗ ∗ where ∇ is obtained from ∇ using the Leibniz rule (see [19,Sect. 3.2]). The splitting in G,n G ∗ ∗ ∗ Theorem 4.5 induces a morphism of A -modules : → w . p,n G,n G,n 123 M. Longo Deﬁnition 4.7 The composition G,n p,n+2 ∗ ∗ ∗ ∗ : w w p,n G,n G,n G,n+2 G,n+2 is the p-adic Maass–Shimura operator. We also need to consider iterates of this operator. Deﬁne j,∗ ∗ ∗ ∗ ˜ ˜ ˜ ˜ ∇ = ∇ ◦ ··· ◦ ∇ ◦ ∇ . n+2( j −1) n+2 n Deﬁne then j,∗ ∇ p,n+2 j j n ∗ ∗ ∗ ∗ : w w (32) p,n G,n G,n G,n+2 j G,n+2 j ∗ ∗ ∗ where the morphism of A -modules : → w is induced as above p,n+2 j G,n+2 j G,n+2 j by the splitting in Theorem 4.5.Wecall the j-th iterate of the p-adic Maass–Shimura p,n operator. 4.10 Relation with Franc Maass–Shimura operator The work accomplished so far allows us to explicitly describe , thus relating it to Franc p,k Shimura–Masss operator. We ﬁrst introduce some differential operators, similar in shape to the Maass–Shimura operator in the real analytic setting. For each integer k ≥ 0, we may then unr deﬁne the Q -linear function ∂ k ∗ ∗ δ = + : A −→ A . (33) p,k ∂z z − z ∗ ∗ For each integer j ≥ 0 weget amap δ : A → A deﬁned by p,k δ = δ ◦ ··· ◦ δ ◦ δ . p,k+2( j −1) p,k+2 p,k p,k We callδ the Maass–Shimura operator, andδ its j-th iteration. Applying (31) to compute p,k p,k the inverse of the Kodaira–Spencer map to (25), we obtain 1 ∂ dτ − dσ(τ) ∗ ⊗k ⊗k+2 ⊗k+1 ∇ f (z) ⊗ dτ = · f (z) ⊗ dτ + kf (z) ⊗ ⊗ dτ . t ∂z z −σ(z) Applying the splitting of the Hodge ﬁltration which annihilate dτ , we ﬁnally obtain p,k+2 1 ∂ kf (z) 1 ⊗k ⊗k+2 ⊗k+2 f (z) ⊗ dτ = · f (z) + ⊗ dτ = · δ ( f ) ⊗ dτ . p,k p,k t ∂z z −σ(z) t p p (34) Iterating (34) we obtain j j ⊗k ⊗k+2 j f (z) ⊗ dτ = δ ( f (z)) ⊗ dτ . (35) p,k p,k This proves Corollary 1.2 stated in the Introduction. 123 Shimura–Mass operator 5The 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. 5.1 p-adic uniformization of Shimura curves In this subsection we review the Cerednik–Drinfel’d Theorem. Recall the subgroup V (N ) of the Eichler order R of level N of B deﬁned in Sect. 2.1.Let U be the open compact ˆ ˆ subgroup of the group R (where R is the proﬁnite completion of R) consisting of elements × × ab (x ) ∈ R such thatι (x ) = for elements a ∈ Z and b ∈ Z , for each prime number + (p) × | N .Let U denote the subgroup of R 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 − ( p) algebra over Q of discriminant N . Fix isomorphisms B B for all primes = p;so U × (p) ( p) ˆ ˆ deﬁnes a subgroup of (B ) , still denoted with the same symbol U (as above, B is the × (p) ˆ ˆ adele ring of B and (B ) is the subgroup of invertible elements of B whose -component belongs to the-component of U via the isomorphism B B , for all primes = p). Deﬁne × (p) ˜ ˜ ˜ = B ∩ U . We still denote the image of in GL (Q ) via a ﬁxed isomorphism p p p 2 p i : B ⊗ Q M (Q ), and we let denote the subgroup of consisting of elements p Q p 2 p p p whose determinant has even p-power order. Base changing from Z to the valuation ring Z 2 of Q 2 gives a Z 2-formal scheme p p p H , whose generic ﬁber H is the base change of the Q -rigid analytic space H to 2 2 p p p p Q 2. The group GL (Q ) acts on the Z -formal scheme H [3,Chapitre I, Sect. 6] and 2 p p p acts on Spf(Z ) via the inverse of the arithmetic Frobenius raised to the determinant map [3,Chapitre II, Sect. 9]. Therefore, the group GL (Q ) also acts on the Z 2-formal 2 p scheme H 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 2-rigid analytic spaces rig \H 2 C , (36) rig rig \G A . (37) The isomorphism (37) is equivariant with respect to the quaternionic actions on both sides. See also [11,Sect. 6.2], [31,Sect. 6], [29]. 5.2 Rigid analytic modular forms A rigid analytic function f : H (C ) → C is said to be a rigid analytic modular form of p p p weight k and level if f (γ z) = (cz + d) f (z) rig for all z ∈ H (C ) and γ ∈ ,where γ(z) = (az + b)/(cz + d). Denote S ( ) the p p p p C -vector space of rigid analytic modular forms of weight k and level .See [11,Sect. 5.2] p p for details. Given a Z[ ]-module M,wedenote M the submodule consisting of -invariant ele- p p unr ments of M. With notation as in Sect. 4.8, consider the Q -subspace w of w consisting 123 M. Longo of global sections which are invariant for the -action. In particular, w is a A -module. rig ⊗k Given f ∈ S ( ),deﬁne ω = f (z) ⊗ dτ in w = w ⊗ C . The correspon- p f G,C G unr p p ˆ k Q rig p dence f → ω sets up a C -linear isomorphism between S ( ) and w . f p p k G,C unr unr For any sheaf F on H , denote F the sheaf on \H deﬁned by taking -invariant p p p p sections. Also, recall the sheaves ω and L introduced in (8)and (9). It is easy to see unr unr ˆ ˆ Q Q p p rig rig 0 0 p p that (36)and (37) induce isomorphisms (ω ) ω and (L ) L of sheaves. G G ˆ unr ˆ unr Q Q p p rig Combining this with the isomorphism between S ( ) and w , we obtain a canonical k G,C isomorphism of C -vector spaces rig 0 ⊗k S ( ) H (C ,ω ). (38) p C k C rig rig If E is a subﬁeld of C containing Q 2, we denote S ( , E) the E-subspace of S ( ) p p p p k k 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 spaces rig alg S (, F) S ( , F). (39) k k 5.3 The p-adic Maass–Shimura operator Taking -invariants deﬁnes a map, for integers k ≥ 0and j ≥ 0 ∗ ∗ p p : (w ) −→ (w ) G,n G,n+2 j p,k where recall that was introduced in (32), and we understand that = , p,k p,k p,k An alternative way to introduce is the following. Recall the operator ∇ in (11) and, p,k for any integer j ≥ 0, deﬁne ∇ : L → L by the formula n unr unr ˆ ˆ Q ,n Q ,n+2 j p p ˜ ˜ ˜ ˜ ∇ = ∇ ◦ ··· ◦ ∇ ◦ ∇ . n n+2( j −1) n+2 n Considering the associated rigid analytic sheaves, and taking global sections, we obtain a j,rig p p map of A-modules ∇ : → . One may deﬁne the operator G,n G,n+2 j j,rig i p p G,n ∇ p,n+2 j j n p p p ∗ ∗ : w w . p,n G,n G,n G,n+2 j G,n+2 j G,n+2 j j j ∗ ∗ By (24), dτ is horizontal for ∇ , and therefore coincides with the restriction of p,n G p,k to w , thus justifying the abuse of notation. G,n 5.4 Comparison of Maass–Shimura operators at CM points Identify the set of Q 2-points in the rigid space H with the set of Q -algebra homomor- p p phisms Hom(Q , M (Q )) as follows: any ∈ Hom(Q , M (Q )) deﬁnes an action 2 2 2 p 2 p p p of Q on H (Q ) = Q − Q by fractional linear transformations, and the point 2 2 2 p p p p z z z ∈ H (Q 2) associated with is characterised by the property (a) = a ,for 1 1 all a ∈ Q . 123 Shimura–Mass operator Given a representation ρ = (ρ ,ρ ) : GL × GL → GL(V),the stalk E(V) of E(V) 1 2 2 2 at a point ∈ Hom(Q 2, M (Q )) can be described explicitly. One ﬁrst observes that the 2 p structure of ﬁltered convergent F-isocrystal of E(V) induces a structure of ﬁltered Frobe- nius module on the ﬁber E(V) . On the other hand, one attaches to such a pair (V,) a unr unr ﬁltered Frobenius module V whose underlying vector space is V = V ⊗ Q (see Q Q p p [23,Sect. 4.2] for details). Denote F V the ﬁltration of the ﬁltered Frobenius module V ; i • i i −1 this is then a ﬁltration on V unr which depends on .Let gr (F V ) = F V /F V be the graded pieces of the ﬁltration. If V is pure of weight n,wehaveanisomorphism i • gr (F V ) V (where V is be the subspace of V unr consisting of elements v satisfy- i j Q j n− j ing the property ρ((a))(v) = a σ(a) v for all a ∈ Q 2) as well as a decomposition i • V unr = gr (F V ).For ∈ Hom(Q , M (Q )), denote the morphism of Q 2 p i ∈Z p Q -algebras obtained by composition with the main involution of M (Q ); therefore, p 2 p ab d −b if (x) = then (x) = .If V is pure of weight n, then the graduate pieces cd −ca i • n−i • gr (F V ) and gr (F V ) are equal, for all i ∈ Z. In particular, for V = M we have ¯ 2 1 • 2 • gr F (M ) gr F (M ) . (40) 2 2 ¯ and therefore there is an exact sequence: 1 • 1 • 0 −→ gr F (M ) −→ (M ) −→ gr F (M ) −→ 0 2 2 2 ¯ and a canonical decomposition 1 • 1 • (M ) gr F (M ) gr F (M ) . (41) 2 2 2 ¯ unr 1 • One can choose generatorsω ,ω of the Q -vector space gr (F (M ) ) so thatω andω 1 2 2 1 2 unr 1 • are deﬁned over Q .Thenω andω are generators of the Q -vector space gr F (M ) , 1 2 2 ¯ p p where ω → ω for i = 1, 2 denotes the action of Gal(Q 2/Q ) on ω . It follows that the i i p i 1 • Hodge splitting coincides on quadratic points with the projection (M ) → gr (F (M ) ) 2 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 ⊗k a imaginary quadratic ﬁeld and f ∈ H (C ,ω ) 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 → C ∞ p and f : H → C for the holomorphic and the rigid analytic modular forms corresponding p p p 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 (K) be a Heegner point, and assume that P ∈ C (C) is represented by the point τ ∈ H modulo , while C ∞ ∞ ∞ ¯ ¯ P ∈ C (C ) is represented by the point τ ∈ H modulo . Fix embeddings Q → Q C p p p p p and Q → C which allows us to view algebraic numbers as complex and p-adic numbers. Theorem 5.1 For any positive integer j we have the equality j j ( f )(τ ) = ( f )(τ ). ∞ ∞ p p ∞,k p,k Proof We mimic a well known argument of Katz when p is split in K ([26,Theorems 2.4.5, 2.4.7]; see also [4,Proposition 1.12], [19,Theorem 3.5], [37,Proposition 2.12]). Let A be the false elliptic curve corresponding to the Heegner point P. The algebraic CM splitting of A coincides both with the Hodge splitting and the p-adic splitting, and therefore the values of and at CM points are the same. Since the construction of the Maass–Shimura ∞,n p,n rig ran operators is algebraic, we see that ∇ ( f ) coincides with ∇ ( f ), and the same still holds ∞ n p for the iterates of the Maass–Shimura operator, which also admit an algebraic construction. The result follows. 123 M. Longo 5.5 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 -subspace of the C -vector space C of continuous functions, which p p 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 ( j) ( j) N = N where for each integer j ≥ 0, we denote N the sub-A-algebra of N j ≥0 consisting of functions f which can be written in the form f (z) f (z) = (z −σ(z)) i =0 with f ∈ A. The Maass–Shimura operator δ restricts to an operator (denoted with the i p,k ( j) ( j +2) same symbol) δ : N → N which takes N to N . p,k Deﬁne now N ( ) = N to be the C -subalgebra of N consisting of functions which k p p are invariant under the weight k action of on N , i.e. those functions satisfying the trans- k unr formation property f (γ z) = (cz + d) f (z) for all z ∈ Q − Q and γ ∈ . Note that p p rig S ( ) ⊆ N ( ). We call N ( ) the C -vector space of nearly rigid analytic modular p k p k p p ( j) j ( j) forms of weight k and level . Deﬁne also N ( ) = N ( ) ∩ N . The operator δ p p k p k p,k introduced in Sect. 4.7 restricts to a mapδ : N ( ) → N ( ) [15,Lemma 4.3.8]. By k p k+2 j p p,k [15,Theorem 4.3.11], for each integer r ≥ 0 we have an isomorphism of C -vector spaces rig (r) S ( ) N ( ) p p k+2(r − j) p,k+2r j =0 k+2(r − j) k+2(r − j) j which maps (h ) to δ (h ). j j j =0 j =0 p,k δ ( f )(τ ) p,k Corollary 5.2 (Franc) Let τ ∈ H corresponds to a Heegner point. The values p p are algebraic for each integer j ≥ 0. Proof The result is clear from Theorem 5.1 since this is known for ( f )(τ ). ∞,k Remark 5.3 Equation (35) answers afﬁrmatively one of the questions left in [15,Sect. 6.1] whether if it was possible to describe the p-adic Maass–Shimura operator δ introduced p,k in [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. 6 The Coleman primitive 1,0 0,1 ∗ 1,0 0,1 ∗ Write ∇= ∇ , ∇ =∇ , ∇ =∇ and ,=, to simplify the notation. G,n G G G,n For any n and any j, whenever there is not possible confusion, we write = and p p,n 123 Shimura–Mass operator j j = for the p-adic Maass–Shimura operator, and = for the splitting of the p,n p p p,n Hodge ﬁltration. dτ We set up the notation ω = dτ and η = .Since dx, dy=−1, we have can can ∗ z −z j n− j ⊗ j ⊗n− j ω ,η = 1. We also write ω η = ω ⊗ η . can can can can can can The computation of the Gauss–Manin connection gives can ∇(ω ) = − + η ⊗ dz, can can z − z ω ⊗ dz η ⊗ dz can can ∇(η ) =− + . can ∗ 2 ∗ (z − z) z − z ⊗k Let f : H → C be a rigid modular form giving rise to a sectionω = f (z) ⊗dτ . Put p p f ⊗n n = k − 2. Using the Kodaira–Spencer map, we identify this withω = f (z)dz ⊗ dτ . Let F be the Coleman primitive of the differential form ω , satisfying the differential equation f f ∇(F ) = ω . f f Deﬁne for j = n/2,..., n an integer j n− j n−2 j G (z) =F (z),ω η ⊗ ω . (42) j f can can can j +1 Theorem 6.1 (G ) = j !ω . p j f Proof This result, which is proved by means of a simple and explicit computation, is the analogue of [4,Proposition 3.24] (and also of [19,Theorem 7.3]), but we provide a complete proof since our formalism is quite different from that in [4], where one can use the Tate curve and the q-expansion principle. As in loc. cit. we show that (G (z)) = ω and p 0 f (G (z)) = jG (z). p j j −1 We ﬁrst compute ∇(G (z)).Wehave: ∇(G (z)) n n =∇ F (z),η ⊗ ω can can n n n n n n = ∇(F (z)),η ⊗ ω +F (z), ∇ η ⊗ ω + F (z),η ⊗∇(ω ) f f f can can can can can can n n n n n n n = f (z)dz ⊗ ω ,η ⊗ ω +F (z), ∇ η ⊗ ω + F (z),η ⊗∇(ω ). f f can can can can can can can We now compute the last two pieces: n n F (z), ∇ η ⊗ ω can can n−1 n = F (z), nη ∇(η ) ⊗ ω f can can can −ω ⊗ dz η ⊗ dz can can n−1 n = F (z), nη + ⊗ ω can can ∗ 2 ∗ (z − z) z − z n−1 ∗ n nη ω ⊗ dz nη ⊗ dz can can n can n =− F (z), ⊗ ω + F (z), ⊗ ω f f can can ∗ 2 ∗ (z − z) z − z n ∗ n nω ⊗ dz nω ⊗ dz n−1 can n can =− F (z),η ω ⊗ + F (z),η ⊗ f can f can can ∗ 2 ∗ (z − z) z − z 123 M. Longo and n n F (z),η ⊗∇(ω ) can can n n = F (z),η ⊗∇(ω ) can can n n−1 = F (z),η ⊗ nω ∇(ω ) f can can can can n n−1 = F (z),η ⊗ nω − + η ⊗ dz f can can can z − z nω ⊗ dz n can n n−1 =− F (z),η ⊗ + F (z),η ⊗ nω η ⊗ dz. f f can can can can z − z Therefore the sum of these two pieces gives: n ∗ nω ⊗ dz can n−1 n n−1 − F (z),η ω ⊗ + F (z),η ⊗ nω η ⊗ dz. f can f can can can can ∗ 2 (z − z ) Recall now that (η ) = 0and (dz ) = 0. Therefore, using the Kodaira–Spencer map can to replace dz with ω , and applying we have can n n (G (z)) = ω ω ,η = ω . p 0 f f can can We now compute ∇(G (z)) for j ≥ 1. The Gauss–Manin connection j n− j n−2 j ∇(G (z)) =∇ F (z),ω η ⊗ ω j f can can can is the sum of three terms j n− j n−2 j ∇(F (z)),ω η ⊗ ω can can can j n− j n−2 j j n− j n−2 j +F (z), ∇(ω η )⊗ ω +F (z),ω η ⊗∇(ω ) (43) f f can can can can can can which we calculate separately as before. First, since j > 0, we have j n− j n−2 j n j n− j ∇(F (z)),ω η ⊗ ω = f (z)dz ⊗ ω ,ω η = 0. can can can can can can Next, a simple computation shows that dz dz j n− j j n− j j −1 n− j +1 j +1 n− j −1 ∇(ω η ) = (n − 2 j)ω η ⊗ + jω η ⊗ dz − (n − j)ω η ⊗ can can can can can can can can ∗ ∗ 2 z − z (z − z) and therefore the second summand in (43)is dz j n− j n−2 j j n− j n−2 j F (z), ∇(ω η )⊗ ω =(n − 2 j) F (z),ω η ⊗ ω ⊗ f f can can can can can can z − z j −1 n− j +1 n−2 j − j F (z),ω η ⊗ ω ⊗ dz+ can can can dz j +1 n− j −1 n−2 j − (n − j) F (z),ω η ⊗ ω ⊗ can can can ∗ 2 (z − z ) 123 Shimura–Mass operator Finally, the third summand is j n− j n−2 j F (z),ω η ⊗∇(ω ) can can can j n− j n−2 j −1 = (n − 2 j)F (z),ω η ⊗ ω ∇(ω ) f can can can can can j n− j n−2 j −1 = (n − 2 j)F (z),ω η ⊗ ω − + η ⊗ dz f can can can can z − z dz j n− j n−2 j =−(n − 2 j)F (z),ω η ⊗ ω ⊗ + can can can z − z j n− j n−2 j −1 + (n − 2 j)F (z),ω η ⊗ ω η ⊗ dz. f can can can can Summing up the pieces in (43), using the Kodaira–Spencer map to replace dz with ω ,and can applying the splitting of the Hodge ﬁltration which kills η and dz ,wehave can j −1 n−( j −1) n−2( j −1) (G (z)) = j F (z),ω η ⊗ ω = jG (z). p j f j −1 can can can The result follows. 7 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. 7.1 The generalised Kuga-Sato motive Fixaneveninteger k ≥ 2 and put n = k − 2, m = n/2. Let A 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 ; the action of O is required to commute with K K the quaternionic action, and this implies that A is isogenous to E × E for an elliptic curve E with CM by O .Fix aﬁeld F ⊃ H and consider the (2n + 1)-dimensional variety X K m over F given by m m X := A × A . Here and in the following we simplify the notation and simply write A, C and A for A , 0 F C and (A ) , unless we need to stress the ﬁeld of deﬁnition in which case we keep the F 0 F full notation. The variety X is equipped with a proper morphism π : X → C with 2n- m m m m dimensional ﬁbers. The ﬁbers above points of C are products of the form A × A . The de Rham cohomology of C attached to L , denoted H (C, L , ∇),isdeﬁnedtobe n n dR the 1-st hypercohomology of the complex 0 −→ L −→ L ⊗ −→ 0. n n As shown in [19,Corollary 6.3], one can deﬁne a projector (denoted P in loc. cit.)inthe m m ring of correspondences Corr (A , A ), such that ∗ n+1 H (A /F) ⊆ H (A /F), (44) A m m dR dR ∗ 1 H (A /F) H (C, L , ∇). (45) A m n dR dR 123 M. Longo m m On the other hand, by [19,Proposition 6.4], we can deﬁne a projector ∈ Corr(A , A ) 0 0 (which is deﬁned by means of ) such that ∗ m n 1 H (A /F) = Sym eH (A /F). (46) A 0 0 dR 0 dR The projectors and are commuting idempotents when viewed in the ring Corr (X , A A C m X ).Wedeﬁne = and denote D the motive (X ,).By[19,Proposition 6.5] and m A A m (44), (45), (46) we see that 1 n 1 m H (C, L , ∇) ⊗ Sym eH (A /F), if i = 2n + 1, i dR dR 0 H (X /F) = (47) dR 0, if i = 2n + 1. Let v be the place of F above p induced by the inclusion F ⊆ Q → C ,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 rig n+1 n 1 AJ : CH (D)(F ) −→ S ( , F ) ⊗ Sym eH (A/F ) . p v p v F v v dR 0 k Here (·) denotes the F -linear dual. For details, see [23,page 362] and [30,Sect. 4.2]; see also [32]and [33]. 7.2 Generalized Heegner cycles Let ϕ : A → A be an isogeny (deﬁned over K ) of false elliptic curves, of degree d prime 0 ϕ to Np.Let P be the point on C corresponding to A with level structure given by composing ϕ with the level structure of A .Weassociatetoany pair (ϕ, A) a codimension n + 1cycle ϒ on X by deﬁning ϕ m m m ϒ := ( ) ⊂ (A × A ) ϕ ϕ 0 where ={(ϕ(x), x) : x ∈ A }⊂ A × A is the graph of ϕ.Wethenset ϕ 0 0 := ϒ . ϕ ϕ m m The cycle of D is supported on the ﬁber above P and has codimension n +1in A × A , ϕ A n+1 thus ∈ CH (D).By(47), the cycle is homologous to zero. See [30] for details. ϕ ϕ We now compute the image of under the Abel-Jacobi map. The de Rham cohomology group H (A/F) of a false elliptic curve A deﬁned over a ﬁeld F containing the Hilbert class dR ﬁeld H of K is equipped with the Poincaré pairing , , which we simply denote H (A/F) dR , . Recall the canonical differentials ω , η introduced in Sect. 6; taking the ﬁber at A can can A , the universal differential ω deﬁnes a differential ω in H (A /F), and we choose 0 can A 0 0 dR η so that ω ,η = 1and {ω ,η } is a F-basis of eH (A /F); this is possible A A A A A A 0 0 0 0 0 0 0 dR because the Hodge exact sequence 1 1 1 0 −→ −→ H (A /F) −→ H (A , O ) −→ 0 0 0 A A /F dR 0 n 1 splits, since A has CM. This yields a basis for Sym eH (A /F) given by the elements 0 0 dR j n− j ω ⊗ η for j an integer such that 0 ≤ j ≤ n. A A 0 0 + − Let f be a newform of level (N ) ∩ (N p) and weight k deﬁned over F.Let v 1 0 ¯ ¯ be the prime of F determined by the embedding F ⊆ Q → Q ; we assume that F /Q p v p is unramiﬁed and contains Q 2. The Jacquet-Langlands correspondence associates to f 123 Shimura–Mass operator alg JL an algebraic modular form f in S (, F ), and therefore, using (39), a rigid analytic rig JL modular form f ∈ S ( , F ).Let ω be differential associated with the rigid analytic p v f rig k JL modular form f . Recall now the deﬁnition of the function G in (42), and for integers rig j = n/2,..., n deﬁne the function j n− j H (z) =F (z),ω (z)η (z). j f can can n−2 j So we have G (z) = H (z) ⊗ ω . j j can Theorem 7.1 For each j = n/2,..., n we have j n− j d · H (z ) = AJ( )(ω ⊗ ω η ). j A ϕ f A A 0 0 Proof The universal differentialω deﬁnes a differential formω in H (A/F), and choose can A dR as above η in H (A/F) in such a way that ω ,η = 1and {ω ,η } is a basis of A A A A A A dR H (A/F). It follows from [30,Theorem 5.5] that dR j n− j j n− j AJ( )(ω ⊗ ω η ) =ϕ (F (z )),ω η . ϕ f f A A A A A A 0 0 0 0 0 ∗ ∗ ∗ ∗ Since ϕ (ω ) = ω and ϕ (ω ),ϕ (η ) = d ,wehave ϕ (η ) = d η ,sowe A A A A A ϕ A ϕ A 0 0 deduce j n− j j n− j ∗ j −n ∗ ∗ ϕ (F (z )),ω η = d ·ϕ (F (z )),ϕ (ω η f A A f A A A A 0 ϕ A A 0 0 0 j n− j = d ·F (z ),ω η f A A ϕ A A and the last expression is equal to d · H (z ). ϕ j A Corollary 7.2 For each j = n/2,..., n we have n− j δ (H )(z ) n! p n A j n− j = AJ( )(ω ∧ ω η ). ϕ f A A n− j j 0 0 t j!· d p ϕ Proof From the proof of Theorem 6.1 we see that (G (z)) = jG (z), and therefore we p j j −1 n− j n− j n− j have j ! (G (z)) = n!G (z).Sousing (35) we conclude j !δ (H (z)) = n!t H (z). p n j p n p j The result follows from Theorem 7.1. Acknowledgements The author thanks the referee for carefully reading the manuscript and for giving valuable suggestions which helped improving the quality of the paper. The author is supported by PRIN 2017, INdAM– GNSAGA. 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Abstract

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]. Résumé Nous étudions un opérateur de Maass–Shimura p-adique pour les courbes de Mumford déﬁni par [15]. Nous montrons que cet opérateur peut être déﬁni en terme d’une scission de la ﬁltration de Hodge, répondant à une question posée dans [15]. Nous étudions aussi la relation de cet opérateur avec les cycles de Heegner généralisés, comme dans [1, 4, 19, 28]. Keywords p-adic uniformisation · Shimura curves · Maass–Shimura operators Mathematics Subject Classiﬁcation 11F03 · 14F40 · 11R52 1 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 1 ∂ k δ ( f )(z) = + f (z) (1) 2πi ∂z z −¯z 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 ( 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 To Bernadette Perrin-Riou on the occasion of her 65th birthday. B Matteo Longo mlongo@math.unipd.it Dipartimento di Matematica Tullio Levi-Civita, Università degli Studi di Padova, Via Trieste 63, 35121 Padua, Italy 123 M. Longo main results in [42] is the following. Let δ = δ ◦ δ ◦ ··· ◦ δ k+2(r −1) k+2(r −2) k for any r ≥ 1, and let K be an imaginary quadratic ﬁeld. Then there exists ∈ C such that for every modular form f of weight k with algebraic Fourier coefﬁcients, and for every CM point z ∈ K ∩ H,wehave δ ( f )(z) ∈ Q. (2) (k+2r) Katz described in [25] the Maass–Shimura operator in more abstract terms by means of the Gauss–Manin connection (see also [27]). Let N ≥ 1beaninteger, X (N) the modular curve of level (N) over Q,and letπ : E → X (N) be the universal elliptic curve. Consider 1 1 the relative de Rham cohomology sheaf 1 1 L = R π 0 → O → 1 ∗ E E/X (N) r 1 on X (N), and deﬁne L = Sym (L ).Let ω = π . The sheaf ω is invertible 1 r 1 ∗ E/X (N) and we have the Hodge ﬁltration −1 0 → ω → L → ω → 0. (3) ran Considering the associated real analytic sheaves, which we denote by a superscript ,the Hodge exact sequence of real analytic sheaves associated with (3) admits indeed a splitting ran ran ran : L −→ ω ⊕¯ω , (4) where ω ¯ is obtained from ω by applying the complex conjugation. The Maass–Shimura ran operator ran ⊗r ran ⊗(r +2) : (ω ) −→ (ω ) ∞,r 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 ord X (N) denote the ordinary locus of the modular curve, viewed as a rigid analytic scheme rig rig over Q .Let F be the rigid analytic sheaf associated with a sheaf F on X (N).Then L p 1 ord splits over X (N) as the direct sum rig rig Frob : L ω ⊕ L 1 1 Frob (where L has the property that the Frobenius endomorphism acts invertibly on this sheaf). This allows to deﬁne a differential operator , which can be seen as a p-adic analogue of the p,r Maass–Shimura operator; this operator can be described in terms of Atkin-Serre derivative. At CM points the splittings and coincide, and therefore one deduces rationality ∞ p results for the values of at CM points from (2). For details, see [4,Proposition 1.12]. p,r The p-adic Maass–Shimura operator is then used in [4, 26] to construct p-adic L-functions and study their properties. 123 Shimura–Mass operator We nowﬁxaninteger 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 anew 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 ≥ 1beaninteger, K/Q a quadratic imaginary ﬁeld, p N a prime number, p ≥ 5, which is inert in K,and let + − + Np = N · N p be a factorization of Np 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 rig analytic curve X over Q is then a Mumford curve, namely X(C ) is isomorphic to the p p rigid analytic quotient of the p-adic upper half plane H (Q ) = C − Q by an arithmetic p p p p subgroup of SL (Q ). Franc deﬁnes in this context a p-adic Maass–Shimura operatorδ by 2 p p,k mimicking the deﬁnition (1), formally replacing the variable z ∈ H with the p-adic variable unr unr ˆ ˆ z ∈ H (Q ) = Q − Q , and replacing the complex conjugation with the Frobenius p p p p unr map (here Q is the completion of the maximal unramiﬁed extension of Q ). 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 of the Hodge ﬁltration, similar to what happens ord over X (N) (in the real analytic case, [42]) and X (N) (in the p-adic rigid analytic case, [26]). The ﬁrst result of this paper is to provide such a splitting , and deﬁne the associated p-adic Maass–Shimura operator. In particular, we show that our splitting 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 (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 L = e · H (A/C) dR and the line bundle ω = e · π . A/C We have a corresponding Hodge ﬁltration −1 0 −→ ω −→ L −→ ω −→ 0. rig The rigid analytic curve C associated with C admits a p-adic uniformization rig C (C ) \H (C ) p p p p 123 M. Longo rig for a suitable subgroup ⊆ SL (Q ). Modular forms on C are then -invariant sections p 2 p p of H , and therefore, to deﬁne a p-adic Maass–Shimura operator on C one is naturally led rig to consider the analogous problem for the covering H of C . Let C denote the C -vector space of continuous (for the standard p-adic topology on unr unr ˆ ˆ both spaces) C -valued functions on H (Q ),and let A denote the Q -vector space of p p p p unr unr 0 ˆ ˆ rigid analytic global sections of H (Q ).Wehaveamapof Q -vector spaces r : A → C p p ∗ 0 and, following [15], we denote A the image of the morphism of A-algebras A[X, Y]→ C deﬁned by sending X to the function z → 1/(z − σ(z)) and Y to the function z → σ(z), unr unr ˆ ˆ whereσ : Q → Q is the Frobenius automorphism (note that the function z → z −σ(z) p p unr ˆ ˆ is invertible on H (Q )). Denote H the formal Z -scheme whose generic ﬁber is H ,let p p p p unr unr unr ˆ ˆ ˆ H be its base change to Q and let G → H be the universal special formal module p p p ∗ 1 unr with quaternionic multiplication. Denote ω = e ( ),where e : H → G is the G G unr G/H zero-section, and let Lie ∨ be the Lie algebra of the Cartier dual G of G.Then ω and Lie are locally free O -modules, dual to each other and we have the Hodge-Tate exact G unr sequence of O -modules ˆ unr 1 unr 0 −→ ω −→ H (G/H ) −→ Lie ∨ −→ 0. G dR p 0 1 0 0 ∗ unr 0 unr ∗ Set L = e ·H (G/H ) andω = e ·ω .Deﬁne = H (H , L ), = ⊗ A , G G A p p G dR G G G G 0 unr 0 ∗ ∗ ∗ w = H (H ,ω ), w = w ⊗ A . We have then an injective map of A -algebras G G A p G G ∗ ∗ w −→ . (5) G G ∗ ∗ ∗ Theorem 1.1 The injection (5) of A -algebras admits a canonical splitting : → w . G G This result corresponds to Theorem 4.5 below. The main tool which is used to prove unr Theorem 1.1 is Drinfel’d interpretation of H as moduli space of special formal modules with quaternionic multiplication; following [43], we call these objects SFD-modules.We unr study the relative de Rham cohomology of the universal SFD-module G → H by means of techniques from [14, 23, 43]. The upshot of our analysis is an explicit description of the unr Gauss–Manin connection and the Kodaira–Spencer isomorphism for G → H , 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 a p-adic Maass–Shimura operator ∗ ∗ : w −→ w p,k G,k G,k+2 ∗ ∗ ⊗t where for each integer t ≥ 1 we put w = (w ) . Using Teitelbaum’s description of G,t G ∗ ∗ ∗ G, one can ﬁnd a basis {dτ, dτ } of w such that (dτ ) = 0. For each integer r ≥ 1, r ∗ ∗ one may deﬁne the r-th iterate : w → w and compare it with the r-th iterate p,k G,k G,k+2r r ∗ ∗ r δ : A → A of Franc p-adic Maass–Shimura operator deﬁned in [15]. We prove thatδ p,k p,k arises from the p-adic Maass–Shimura operator . More precisely, we have the following p,k result (see (35)below). 123 Shimura–Mass operator × ∗ Corollary 1.2 There exists t ∈ C , such that for each f ∈ A and integers k ≥ 1 and r ≥ 1 we have δ ( f )(z) p,k r ⊗k ⊗(k+2r) f (z)dτ = dτ . p,k Remark 1.3 The p-adic number t arises as a period comparing two pairings on the coho- mology 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 embedding Q → Q ; we say that a p-adic number x ∈ Q is algebraic if it p p rig belongs to the image of this embedding, in which case we simply write x ∈ Q.Let S ( ) be the C -vector space of rigid analytic quaternionic modular forms of weight k and level rig ;elementsof S ( ) are functions from H (C ) = C − Q to C which transform p p p p p p p rig under the action of by the automorphic factor of weight k. We say that f ∈ S ( ) is p p algebraic if it corresponds, via the Cerednik–Drinfel’d theorem, to an algebraic modular form of weight k on C which is deﬁned over Q (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). rig Corollary 1.4 Let f ∈ S ( ) be an algebraic modular form of weight k and level .Then p p unr for every CM point z ∈ K ∩ H (Q ),wehave δ ( f )(z) p,k ∈ Q. 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 intro- duced 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 with CM by O . For any isogeny ϕ : A → A of false elliptic curves, we construct a 0 K 0 cycle ϒ in the Chow group CH (A × A ) of the Chow motive A × A ,where m = n/2 ϕ 0 0 with n = k − 2. Brooks introduces in [19] a projector in the ring of correspondences of m m X = A × A , which deﬁnes the motive D = (X ,). The generalised Heegner cycle m ϕ is the image of ϒ in CH (D) via this projector. We construct a p-adic Abel-Jacobi map rig m m 1 AJ : CH (D) −→ S () ⊗ Sym eH (A ) p 0 k dR ∨ 0 where denotes linear dual (see Sect. 7.1 for details). We prove that the sheaf L = 1 unr eH (G/H ) is equipped with two canonical sections ω and η , such that ω is can can can dR p 0 0 a generator of the invertible sheaf ω .Let ω ∈ w be the -invariant differential form G G rig associated with an algebraic modular form f ∈ S (),and let F denote its Coleman primitive satisfying ∇(F ) = ω ,where ∇ is the Gauss–Manin connection. Denote , the f f n 1 Poincaré pairing on Sym eH (A ),where A is the ﬁber of A at z. Deﬁne the function z z dR H(z) =F (z),ω (z). can 123 M. Longo Theorem 1.5 Let ϕ : A → A be an isogeny of degree d prime to N p, and z the point of 0 ϕ A C whose ﬁber is A. Then for each integer j = n/2,..., n we have n− j δ (H)(z ) A n! p,k j n− j = · AJ ( )(ω ⊗ ω η ). p ϕ f can can n− j j t j!· d p ϕ 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. 2 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 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 and Q → Q for each prime number . 2.1 Quaternion algebras Let B/Q be the indeﬁnite quaternion algebra of discriminant N p.Weneed to ﬁx aconve- nient basis for the Q-algebra B, called Hashimoto model. Denote M = Q( p ) the splitting ﬁeld of the quadratic polynomial X − p ,where p if an auxiliary prime number ﬁxed as 0 0 in [37,Sect. 1.1] and [19,Sect. 2.1], such that: − − (1) for all primes we have(p , pN ) =−1 if and only if | pN ,where(a, b) denotes the Hilbert symbol, √ √ (2) all primes | N are split in the real quadratic ﬁeld M = Q( p ),where p is a 0 0 square root of p in Q. 2 − The choice of p ﬁxes a Q-basis of B as in [18,Sect. 2] given by {1, i, j, k} with i =−pN , j = p , k = ij =−ji,and 1 the unit of B; of course, if x ∈ Q we will often just write x for x · 1.Let R be the maximal order of B which contains the Z-span of this basis. max Since M splits B,wehaveisomorphisms ι : B ⊗ M M (M) and for place v N p M Q 2 of M we have ι (R ⊗ O ) = M (O ),where O is the ring of algebraic integers M max Z M,v 2 M,v M of M,and O is the v-adic completion of O [19,Lemma 2.1]. For each prime number M,v M | N , choose a place v of M above with M = Q . We thus obtain an isomorphism ι : B ⊗ Q M (Q ) such that ι (R ⊗ Z ) = M (Z ).Deﬁne V (N ) to be Q 2 p max Z 2 1 ∗∗ + the subgroup of R consisting of elements x such that ι (x) ≡ (mod N ).This max subgroup is contained in the standard Eichler order R ⊆ R of level N . Let ﬁnally be max ∗∗ × + the subgroup of elements x ∈ R having norm one and such that i (x) ≡ (mod N ) for all | N . 123 Shimura–Mass operator 2.2 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 : R → End (A).An isogeny of false elliptic curves A max S is an isogeny which commutes with the action of R .A full level N -structure on A is an max + + isomorphism of group schemes α : A[N ] (R ⊗ (Z/N Z)) , where for any group A max Z S G we denote G the constant group scheme G over S.A level structure of V (N )-type is S 1 + + an equivalence class of full level N -structures under the (right) action of V (N ). The moduli problem which associates to any Z[1/Np]-scheme S the set of isomorphism classes of false elliptic curves equipped with a V (N )-level structure is representable by a smooth proper scheme C deﬁned over Spec(Z[1/Np]) [19,Theorem 2.2]. Let π : A C be the universal false elliptic curve. For any Z[1/Np]-algebra R,let π : A C be the base R R R change of π to R.Wehave \H C (C),where H is the complex upper half plane and acts on it by Moebius transformations. 2.3 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 • • • • ( , d ),orsimply understanding the differentials d , the complex of sheaves X/S X/S X/S X/S of relative differential forms for the morphismφ. For a sheaf F of O -modules over a scheme ∨ ⊗k X, we denote F its O -linear dual. If F is invertible, for an integer k we let F denote the usual tensor product operation. Fix a ﬁeld F of characteristic zero. The relative de Rham cohomology bundle for the morphism A → C is deﬁned by F F q • H (A /C ) = R π . F F F ∗ dR A /C F F We ﬁrst recall the construction of the Gauss–Manin connection. We have a canonical short exact sequence of locally free sheaves ∗ 1 1 1 0 −→ π −→ −→ −→0(6) F C /F A /C A /F F F F F (the exactness is because π is smooth). This exact sequence induces maps A /F •−i ∗ i • ⊗ π −→ A /F A F C /F A /F F F F i • i • for each integer i, deﬁning a ﬁltration F = Im(ψ ) on with associated A /F A /F A /F F F F graded objects i • •−i ∗ i gr = ⊗ π . A /F F C /F F A /F A F F F p,q p,q Let E denote the spectral sequence associated with this ﬁltration. The E terms are then i 1 given by p,q p q E ⊗ H (A /C ) O F F 1 C /F C dR F F 123 M. Longo [27,(7)]. The Gauss–Manin connection i 1 i ∇: H (A /C ) −→ ⊗ H (A /C ) F F O F F dR C /F C dR 0,i 0,i 1,i is then deﬁned as the differential d : E → E in this spectral sequence. 1 1 1 We now recall various descriptions of the Kodaira–Spencer map. It is deﬁned to be the boundary map 1 1 ∗ 1 KS : π −→ R π π A /C F ∗ F ∗ F F A /C F C /F F F F 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 : A → C denote the dual abelian F F variety. By a result of Buzzard ([9,Sect. 1], see also [19,page 4183]), it is known that the abelian surface A is equipped with a canonical principal polarization ι : A A over F A F F F C , which we use to identify A and A in the following without explicit mentioning it; we F F recall that this polarization is characterized by the fact that the associated Rosati involution in End (A ) restricted to the image of R via the map R → End (A ) coincides C F max max C F F F † † −1 with the involution x → x of R ,deﬁnedby x = i bi (as usual, if x = a +bi +cj+dk, max then x¯ = a − bi − cj − dk). Using the principal polarization and the isomorphism between 1 1 ∨ R π ( ) and the tangent bundle of A , the Hodge exact sequence can be written as F ∗ A /F F 1 1 1 0 −→ π −→ H (A /C ) −→ π −→0(7) F ∗ F F F ∗ A /C dR A /C F F F F (cf. [37,(2.2)], [19,Sect. 2.6]). The Kodaira–Spencer map can be deﬁned using the Gauss– Manin connection as the composition (7) (7) 1 1 i 1 KS : π −→ H (A /C ) −→ H (A /C ) ⊗ − A /C F ∗ F F F F O F F A /C dR dR C C /F F F F (7) ∨ 1 1 − π ⊗ F ∗ O A /C C /F F F F F 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 -modules, denoted again with the same symbol, ⊗2 1 1 KS : π −→ . A /C F ∗ F F A /C C /F F F 2.4 Idempotents and line bundles Let 1 1 √ e = 1 ⊗ 1 + j ⊗ p ∈ R = R ⊗ O [1/(2 p )] 0 M max Z M 0 2 p be the idempotent in [37,(1.10)], [19,Sect. 2.1], where O is the ring of integers of M. 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 embedding Q → C)or F ⊆ Q (and ¯ ¯ then we require that F contains the image of M via the ﬁxed embedding Q → Q ). 123 Shimura–Mass operator Since M is contained in F,wehaveanactionof R on the sheaves π and M F ∗ A /C F F H (A /C ), and we may therefore deﬁne the invertible sheaf of O -modules F F C dR F ω = e · π (8) F ∗ A /C F F and the sheaf of O -modules L = e · H (A /C ). (9) F F F dR Using that e is ﬁxed by the Rosati involution, the Hodge exact sequence (7) becomes −1 0 −→ ω −→ L −→ ω −→ 0 (see [19,Sect. 2.6]). For any integer n ≥ 1, deﬁne L = Sym (L ). (10) F,n F The Gauss–Manin connection is compatible with the quaternionic action [37,Proposition 2.2]. Therefore, restricting to L and using the Leibniz rule (see for example [19,Sect. 3.2]), the F,1 Gauss–Manin connection deﬁnes a connection ∇ : L −→ L ⊗ . n F,n F,n C /F ⊗2 By [37,Theorem 2.5], restricting the Kodaira–Spencer map to ω gives an isomorphism ⊗2 1 KS : ω −→ . F C /F We may then deﬁne a map ∇ : L → L by the composition n F,n F,n+2 −1 id⊗KS n F 1 ⊗2 ∇ : L L ⊗ L ⊗ ω L ⊗ L L n F,n F,n O F,n O F,n O F,2 F,n+2 C C /F C F C L L L (11) where the last map is the product map in the symmetric algebras. 2.5 Algebraic modular forms As in the previous section, let F e a ﬁeld of characteristic 0. For any F-algebra R,wedeﬁne the R-algebra alg ⊗k S , R = H (C ,ω ) ( ) k R of algebraic modular forms of weight k and level V (N ) over R. One can show [19,Sect. alg 3.1] that the R-algebra S (, R) can be alternatively described in modular terms. Let R be an R algebra. A test triple over R is a triplet (A , t ,ω ) consisting of a false elliptic curve A /R ,a V (N )-level structure t and a global section ω of ω . An isomorphism of test A /R triples (A , t ,ω ) and (A , t ,ω ) is an isomorphism of false elliptic curves φ : A → A such that φ(t ) = t and φ (ω ) = ω .A test pair over R is a pair (A , t ) obtained from a test triplet by forgetting the datum of the global section. Then one can identify global sections ⊗k of ω with: (1) A rule F which assigns, to each R-algebra R and each isomorphism class of test triplets (A , t ,ω ) over R ,anelement F(A , t ,ω ) ∈ R , subject to the base change axiom (for all maps of R-algebras φ : R → R ,wehave F(A , t ,φ (ω )) = F(A ,φ(t ),ω ), where A is the base change of A via φ)and the weight k condition (for all λ ∈ (R ) , −k we have F(A , t ,λω ) = λ F(A , t ,ω ))[19,Deﬁnition 3.2]. 123 M. Longo (2) A rule F which assigns to each R-algebra R and each isomorphism class of test pairs ⊗k (A , t ) over R , a translation invariant section F(A , t ) ∈ ω subject to the base A /R change axiom (for all maps of R-algebrasφ : R → R , we have the relation F(A , t ) = φ (F(A ,φ(t )),where A is the base change of A via φ)[19,Deﬁnition 3.3]. Let us make the relations between these deﬁnitions more explicit [19,page 4193]. Given a 0 ⊗k global section f ∈ H (C ,ω ), we get a function as in (2) above associating to each test pair (A , t ) over R the point x ∈ C (R ), and taking the value of f at x ;if F (A ,t ) R (A ,t ) is as in (2), we get a function G on test triples (A , t ,ω ) over R as in (1) by the formula ⊗k F(A , t ) = G(A , t ,ω )ω where ω ∈ ω is the choice of any translation invariant A /R global section. 3 Special values of L-series In this section we review the work of Brooks [19] expressing special values of certain L- functions of modular forms in terms of CM-values of the Maass–Shimura operator applied to the modular form in question. 3.1 The real analytic Maass–Shimura operator an an We denote (X, O ) (X , O ) the analytiﬁcation functor which takes a scheme of ﬁnite type over C to its associated complex analytic space. For each sheaf F of O -modules on an X,wealsodenote F the analytiﬁcation of F, and for each morphism ϕ : F → G of an an an O -modules, we let ϕ : F → G the corresponding morphism of analytic sheaves. If ran (X, O ) is an analytic space, we denote O the ring of real analytic functions on X;thisis ran ran a sheaf of O -modules, and for any sheaf F of O -modules, we let F = F ⊗ O ; X X O an ran an ran when F = F , we simplify the notation by writing F instead of (F ) . an Since C is proper and smooth over C, the analytiﬁcation functor F F induces an equivalence of categories between the category of coherent sheaves C and the category an of analytic coherent sheaves of O -modules. Also, the analytic sheaf obtained from the sheaf of algebraic de Rham cohomology H (A /C ) coincides with the derived functor C C dR 1 an 1 an R π in the category of analytic sheaves over C [41,Theorem 1]. an an C∗ C A /C C C Hodge theory gives a splitting ran 1 ran 1 H (A /C ) −→ π C C C∗ dR A /C C C of the corresponding Hodge exact sequence of real analytic sheaves obtained from (7). Since ran 1 1 ran this splitting is the identity on the image of π in H (A /C ) ,itgives C∗ C C A /C dR C C ran ran rise to a map : L → ω (cf. [37,Proposition 2.8]). We may then consider the induced C,1 C ran ⊗n ran maps : L → (ω ) for any integer n ≥ 1. Further, the map ∇ gives rise to a ∞,n n C,n C ran ran ran map ∇ : L → L of real analytic sheaves. The composition C,n C,n+2 ran n ∞,n ⊗n ran ran ran ⊗n+2 ran : (ω ) L L (ω ) ∞,n C C,n C,n+2 C is the real-analytic Shimura–Maas operator. The effect of on modular forms is described in [19,Proposition 3.4] and ∞,n + × + [37,Proposition 2.9]. Denote = (N ) the subgroup of B ∩ V (N ) consisting of 1 1 123 Shimura–Mass operator elements of norm equal to 1. Fix an isomorphism B ⊗ R M (R) and denote the Q 2 ∞ image of in GL (R).Let S ( ) denote the C-vector space of holomorphic modular forms 2 k ∞ of weight k and level consisting of those holomorphic functions on H ,the complex ∞ ∞ upper half plane, such that f (γ(z)) = j(γ, z) f (z) for all γ ∈ ;here acts on H by ∞ ∞ ∞ fractional linear transformations via the map B → B ⊗ R M (R).Wehave(cf.[19,Sect. Q 2 2.7]) 0 an ⊗k an S ( ) H C ,(ω ) . k ∞ ran Deﬁne the space S ( ) of real analytic modular forms of level and weight k to be the ∞ ∞ C-vector space of real analytic functions f : H → C such that f (γ(z)) = j(γ, z) f (z) for all γ ∈ . One then has ran 0 an ⊗k ran S ( ) H C ,(ω ) . (12) k C ran The operator gives then rise to a map δ : S ( ) → S ( ) and we have ∞,k ∞,k k ∞ ∞ k+2 1 d k δ ( f )(z) = + f (z). ∞,k 2πi dz z +¯z 3.2 CM points and triples Fix an embedding Q → C. For any embedding ϕ : K → B there exists a unique τ ∈ H such that ι (ϕ(K ))(τ) = τ . The additive map K → C deﬁned by α → j(ι (ϕ(α)),τ) ∞ ∞ gives an embedding K → C; we say that ϕ is normalized if α → j(ι (ϕ(α)),τ) is the identity (with respect to our ﬁxed embedding Q → C). We say that τ ∈ H is a CM point if there exists an embedding ϕ : K → B which has τ as ﬁxed point as above, and that a CM point τ is normalized if ϕ is normalized. Finally, we say that a CM point τ ∈ H is a Heegner point if ϕ(O ) ⊆ R [19,Sect. 2.4 and page 4188]. Fix a CM point τ corresponding to an embedding ϕ : K → B.Let a be an integral ideal of O , and deﬁne the R -ideal a = R · ϕ(a). This ideal is principal, generated by K max B max an element α = α ∈ B. Right multiplication by α gives an isogeny A → A , whose −1 a τ α τ kernel is A [a].Let be the subgroup of R consisting of elements of norm equal to 1. τ max max The image of ατ by the canonical projection map ρ : H → \H does not depend on max max the choice of the representative α, and therefore one may write A for the corresponding aτ abelian surface. Shimura’s reciprocity law states thatρ (τ) is deﬁned over the Hilbert class max −1 (a ,H/K) −1 ﬁeld H of K,and thatρ (τ) = ρ (aτ),where (a , H/K) denotes the Artin max max symbol. Fix a primitive N -root of unity ζ . Fix a normalized Heegner point τ , and ﬁx a point + + P ∈ A [N ] of exact order N such that e · P = P.Let (A , P ) denote the point on C(F) τ τ τ τ + + + corresponding to the level structure μ + × μ + Z/N Z × Z/N Z → A [N ] which N N + + takes (1, 0) ∈ Z/N Z × Z/N Z to P .A CM triple is an isomorphism class of triples (A , P ,ω ) with (A , P ) as above and a non-vanishing section ω in e · . τ τ τ τ τ τ A /F There is an action of Cl(O ) on the set of CM triples, given by a(A , P ,π (ω)) = (A /A [a],π(P ),ω) τ τ τ τ τ where π : A A /A [a] is the canonical projection. τ τ τ 123 M. Longo 3.3 Special value formulas Fix a CM triple (A, P,ω) = (A , P ,ω ) with ω deﬁned over H, the Hilbert class ﬁeld of τ τ τ K ; recall that A is also deﬁned over H. The complex structure J on M (R) deﬁnes a differential formω = J (2πidz ),and let τ 2 C 1 ∈ C be deﬁned by ω = ·ω ; clearly, different choices of ω correspond to changing ∞ ∞ C by a multiple in H. + − We now let f be a modular form of weight k,level (N ) ∩ (N ), and character 1 0 JL ε ,and let f be the modular form on the Shimura curve C associated with f by the f C JL Jacquet-Langlands correspondence. We can normalise the choice of f so that the ratio JL JL f , f / f , f belongs to K [19,Sect. 2.7 and page 4232]. (2) Let be the set of Hecke characters χ of K of inﬁnite type ( , ) with ≥ k and 1 2 1 (2) ≤ 0. We say that χ ∈ is central critical if + = k, so that the inﬁnite type of χ 2 1 2 (2) (2) is (k + j, − j) for some integer j ≥ 0. Denote the subset of consisting of central cc critical characters. ran ran For each positive integer j,let δ : S ( ) → D ( ) denote the j-th iterate of ∞ ∞ ∞,k k k+2 j the Shimura–Mass operator deﬁned by δ = δ ◦ ··· ◦ δ ◦ δ . ∞,k+2( j −1) ∞,k+2 ∞,k ∞,k (2) −1 For any Hecke character, one may consider the L-function L( f,χ , s),and forχ ∈ cc −1 central critical deﬁne the algebraic part L ( f,χ ) of its special value at s = 0as in alg (2) −1 [19,Proposition 8.7]. By [19,Proposition 8.7], if χ ∈ then L ( f,χ ) ∈ Q,and we cc alg have ⎛ ⎞ −1 −1 JL ⎝ ⎠ L ( f,χ ) = χ (a) · δ ( f )(a(A, t,ω)) alg j ∞,k a∈Cl(O ) − j where χ = χ · nr and nr is the norm map on ideals of O . In this formula we view the j K JL real analytic modular δ ( f ) as a function on test triplets, as in [19,Proposition 8.5] via ∞,k (12) (see also the discussion in [4,page 1094] in the GL case). For each ideal class a in Cl(O ),let α be the corresponding element in B, as in Sect. K a 3.2. Then using the dictionary between real analytic forms as functions on H or functions on test triples, and recalling that A = A for a normalized Heegner point τ,wehave ⎛ ⎞ k+2 j j −1 −1 JL ⎝ ⎠ L ( f,χ ) = · χ (a) · δ ( f )(α ·τ) . alg ∞ a j ∞,k a∈Cl(O ) JL In this formula we view δ ( f ) as a function on H. ∞,k 4 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. 123 Shimura–Mass operator Let H denote Drinfel’d p-adic upper half plane; this is a Z -formal scheme, and we p p denote H its generic ﬁber, which is a Q -rigid space [3,Chapitre I]. p p 4.1 Drinfel’d Theorem Denote D the unique division quaternion algebra over Q ,and let O be its maximal order. p D The unramiﬁed quadratic extension Q of Q 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 unr σ denote the absolute Frobenius automorphism of Gal(Q /Q ).Ifﬁxanelement ∈ O p D such that = p and x = σ(x) for x ∈ Q ,then D = Q [ ]. We will denote x →¯x 2 2 p p the restriction of σ to Gal(Q 2/Q ). For any Z -algebra B,a formal O -module over B is a commutative 2-dimensional p D formal group G over B equipped with an embedding ι : O → End(G). A formal O - G D D module is said to be special if for each geometric point P of Spec(B/pB), the representation of O / O on the tangent space Lie(G ) of G = G × k is the sum of two distinct D D P P P characters of O / O ,where k is the residue ﬁeld of P;see [43,Deﬁnition 1] for more D D P details on this deﬁnition. By an SFD-module over B, we mean a special formal O -module over B.If G is a SFD-module over a Z -algebra B in which p is nilpotent, we denote M(G) the (covariant) Cartier-Dieudonné module of G [3,Chapitre II, Sect. 1]; we also denote F and V (or simply F and V when there is no confusion) the Frobenius and Verschiebung endomorphisms of M(G).If B is Z 2-algebra, where Z 2 is the valuation ring of Q 2,and p p p G a formal O -module, then we may deﬁne Lie (G) ={m ∈ Lie(G) : ι (a) = am, a ∈ Z 2 }, Lie (G) ={m ∈ Lie(G) : ι (a) =¯am, a ∈ Z } 0 1 and, since G is special, both Lie (G) and Lie (G) are free B-modules of rank 1, (recall that x¯ = σ(x),so x →¯x is the non-trivial automorphism of Gal(Q 2/Q )). Moreover, M(G) is 0 1 also equipped with a graduation M(G) = M (G) ⊕ M (G) where M (G) ={m ∈ M(G) : ι (a) = am, a ∈ Z 2 }, M (G) ={m ∈ M(G) : ι (a) =¯am, a ∈ Z 2 }. Fix a SFD-module ! = G × G over F ,where G is the reduction modulo p of a Lubin- unr ˆ ˆ Tate formal group E of height 2 over Z , the completion of the valuation ring of the maximal unr unramifed extension Z of Z ;so E is the formal group of a supersingular elliptic curve unr E over Z (see [43,Deﬁnition 9 and Remark 27]). The Dieudonné module M(!) of ! is unr 0 1 0 0 the Z [F, V]-module with V-basis g and g , satisfying the relations F(g ) = V(g ) and 1 1 0 1 1 F(g ) = V(g ). The quaternionic order O acts via the rules (g ) = V(g ), (g ) = 0 0 0 1 1 0 V(g ) and a(g ) = ag , a(g ) =¯ag for a ∈ Z .By[43,Corollary 30],η (!) is generated 0 1 1 1 0 over Z by [g , 0] and [V(g ), 0],and η (!) is generated over Z by [g , 0] and [V(g ), 0]. p p Let Nilp denote the category of Z -algebras in which p is nilpotent. Denote SFD the functor on Nilp which associates to each B ∈ Nilp the set SFD(B) of isomorphism classes of triples (ψ, G,ρ) where (1) ψ : F → B/pB is an homomorphism, (2) G is a SFD-module over B of height 4, (3) ρ : ψ ! → G = G ⊗ B/pB is a quasi-isogeny of height 0, called rigidiﬁcation. ∗ B/pB B 123 M. Longo See [43,page 663] or [3,Chapitre II (8.3)] for more details on the deﬁnition of the functor SFD. Drinfel’d shows in [13] that the functor SFD is represented by the Z -formal scheme unr unr ˆ ˆ ˆ H = H ⊗ Z p Z p p p unr unr (see [43,Theorem 28], [3,Chapitre II (8.4)]). Note that H , considered as Z -formal p p unr scheme, represents the restriction SFD of SFD to the category Nilp of Z -algebras in unr which p is nilpotent (cf. [3,Chapitre II, Sect. 8]). Unless otherwise stated, we will see H unr as a Z -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 -formal scheme H as the solution of a moduli prob- p p lem. For B ∈ Nilp, a compatible data on S = Spf(B) consists of a quadruplet (η, T, u,ρ) where 0 1 (1) η = η ⊕ η is a sheaf of ﬂat Z/2Z-graded Z [ ]-modules on S, 0 1 i (2) T = T ⊕ T is a Z/2Z-graded sheaf of O [ ]-modules with T invertible, (3) u : η → T is a homogeneous degree zero map such that u ⊗ 1 : η ⊗ O → T is Z S surjective, (4) ρ : (Q ) → η ⊗ Q is a Q -linear isomorphism, S 0 Z p p p p which satisfy natural compatibilities, denoted(C1),(C2),(C3) in [43,page 652], to which we refer for details. The ﬁrst step in Drinfel’d work is to show that the Z -formal sheme H repre- p p sents 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 → H of H , 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 unr ˆ ˆ (η , T , u ,ρ ) which corresponds to an S = Spf(B)-valued point on H ⊗ Z .If X X X X p Z p p X = (ψ, G,ρ) ∈ SFD(B) is given as above, the quadruplet (η , T , u ,ρ ) can be explic- X X X X itly constructed as follows: • T = T(G) = M(G)/VM(G) the tangent space to G at the origin, equipped with its graduation deﬁned previously; • Deﬁne N(G) = M(G) × M(G)/ ∼ where (V(x), 0) ∼ (0, (x)); we denote [x, y] the class in N represented by the pair (x, y).Let λ : N(G) → M(G) be the map deﬁned by λ([x, y]) = (x) − V(y).There is amap N(G) → M(G)/VM(G) induced by the projection onto the ﬁrst component. Further, one easily shows that there exists a unique map L : M(G) → N(G) satisfying the relation λ ◦ L = F.Let φ : N(G) → N(G) be φ=Id deﬁned by φ([x, y]) = L(x) +[y, 0].Then η = η(G) = N . The graduation of 0 1 M(G) deﬁnes a graduation η(G) = η (G) ⊕ η (G). • u = u(G) is induced by the projection map N(G) → M(G)/VM(G). • Fix an isomorphism η (!) Z ⊕ Z . The quasi-isogeny ρ induces a map p p 0 0 ρ = ρ(G) : Q ⊕ Q η (ψ (!)) ⊗ Q → η (!) ⊗ Q . X p p ∗ Z p Z p p p 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 unr unr ˆ ˆ z : SFD(Z ) −→ H (Q ) (13) 0 p p p unr such that the map X = (ψ, G,ρ) → (z (X),ψ) gives a bijection between SFD(Z ) and unr unr unr unr unr ˆ ˆ ˆ ˆ ˆ ˆ (H ⊗ Z )(Z ), which we identify with the set H (Q ) × Hom(Z , Z ). We call p Z p p p p p p p 123 Shimura–Mass operator the map X → z (X) a rigid analytic parameter on SFD.Ifwelet pro-Nilp the category of projective limits of objects in Nilp, and we still denote SFD the restriction of SFD to pro-Nilp, unr this implies that the map X = (ψ, G,ρ) → z (X) gives a bijection between SFD(Z ) and unr unr ˆ ˆ H (Q ).By[43,Thm. 45], for each z ∈ H (Q ), there exists triple X = (ψ, G,ρ) in p p p p unr SFD(Z ) such that z (X) = z. 4.2 Filtered Frobenius modules Let E be an unramiﬁed ﬁeld extension of Q .A Frobenius module E over E is a pair E = (V,φ) consisting of a ﬁnite dimensional E-vector space V with aσ -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 φ- modules these objects. If G is a p-divisible formal group over F , one can deﬁne its ﬁrst crystalline coho- mology 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 self- contained expositions). In the following we will denote H (G) the global sections of the cris unr crystalline Dieudonné functor (deﬁned as in [2,Théorème 4.2.8.1]) tensored over Z with unr 1 Q . By construction, H (G) is then a Frobenius module. Moreover, the canonical iso- p cris 1 1 morphism between H (G) and the ﬁrst de Rham cohomology group H (G) of G equips cris dR H (G) with a canonical ﬁltration (arising from the Hodge ﬁltration in the de Rham coho- cris mology), making H (G) a ﬁltered Frobenius module; see [38]. dR Let G be a SFD-module over F . Then the Frobenius module H (G) is a four- cris unr dimensional Q -vector space, equipped with its σ -linear Frobenius φ (G).Itisalso cris equipped with a D-module structure j : D → End H (G) which commutes G ˆ unr cris with φ (G),and a Q -algebra embedding i : M (Q ) → End H (G) induced cris p G 2 p ˆ unr cris by the isomorphism End (G) M (Q ), which commutes with the D-action. Deﬁne O 2 p −1 φ (G) = j ( ) φ (G) and put G cris cris 1 φ (G)=Id cris V (G) = H (G) . cris cris Denote φ = j ( ) the restriction of j ( ) to V (G). Moreover, denote V (G) G |V (G) G cris cris cris i ∨ i (η (G) ⊗ Q ) = Hom (η (G) ⊗ Q , Q ) Z p Q Z p p p p p the Q -linear dual of η (G) ⊗ Q . p Z p The following lemma is crucial in what follows, and identiﬁes V (G) with (η (G) ⊗ cris Z Q ) , from which one deduces a complete description of the ﬁltered Frobenius module H (G). It appears in a slightly different version in the proof of [23,Lemma 5.10]. Since we cris 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 (G) η(G) ⊗ Q of Q -vector cris Z p p 1 unr spaces. Moreover, H (G) = V (G) ⊗ Q , where the right hand side is equipped with cris Q cris p p unr the structure of Q -vector space given by x ·(v ⊗α) = v ⊗(σ(x)α) forv ∈ V (G),x,α ∈ cris unr Q . Finally, under this isomorphism the Frobenius φ (G) corresponds to φ ⊗ σ . cris V (G) p cris 123 M. Longo Proof Recall that the Frobenius module H (G) is canonically isomorphic to the con- cris unr travariant Dieudonné module of G with p inverted, and with Q -action twisted by the unr Frobenius automorphism σ of Q , equipped with the canonical Frobenius of the con- travariant Dieudonné module (see [2,4.2.14]). More precisely, denote unr D(G) = Hom M(G)[1/p], Q ˆ unr unr the Q -linear dual of the covariant Dieudonné module M(G) of G with p inverted, and let σ unr D(G) = D(G) ⊗ Q , ˆ unr Q ,σ unr where the tensor product is taken with respect to the Frobenius endomorphismσ of Q .Then unr 1 σ as Q -vector spaces, we have H (G) D(G) . Under this isomorphism the Frobenius p cris φ (G) is given by the map ϕ → σ ◦ ϕ ◦ V for ϕ ∈ D(G). cris G −1 Now, by [3,Lemme (5.12)], we have an isomorphism of σ -isocrystals i −1 i unr −1 M (G)[1/p], V η (G) ⊗ Q ,σ (14) G Z p p −1 i unr for each index i = 0, 1 (where the action of σ on η (G) ⊗ Q is on the second factor p p i unr −1 only). We may therefore compute V (G) in terms of the isocrystal η (G) ⊗ Q ,σ . cris Z p p As above, deﬁne i i unr D (G) = Hom (M (G)[1/p], Q ) unr ˆ p unr i σ i unr ˆ ˆ (Q -linear dual) and let D (G) denote the base change D (G) ⊗ Q via σ.Since ˆ unr p p Q ,σ 0 1 M(G) = M (G) ⊕ M (G),wehave σ 0 σ 1 σ D(G) = D (G) ⊕ D (G) , σ i σ and we may write any element ϕ ∈ D(G) as a pair (ϕ ,ϕ ) with ϕ ∈ D (G) , i = 0 1 i 0, 1. By deﬁnition, an element ϕ = (ϕ ,ϕ ) ∈ D(G) belongs to V (G) if and only if 0 1 cris −1 −1 i ϕ (V (m )) is equal to σ (ϕ (m )) for all m ∈ M (G)[1/p],and forall i = 0, 1. i G i i i i unr i unr unr ˆ ˆ ˆ Using (14), identify ϕ with a Q -linear homomorphism ϕ : η (G) ⊗ Q → Q , i i Z p p p p denoted with a slight abuse of notation with the same symbol; then the above equation describing V (G) becomes cris −1 −1 ϕ (n ⊗ σ (x)) = σ (ϕ (n ⊗ x)) i i i unr unr ˆ ˆ for all n ∈ η (G) and all x ∈ Q .Since ϕ is Q -linear, p p −1 −1 −1 σ (ϕ (n ⊗ x)) = σ (x)σ (ϕ (n ⊗ 1)), i i −1 −1 −1 i and we deduce the equality ϕ (n ⊗ σ (x)) = σ (x)σ (ϕ (n ⊗ 1)) for all n ∈ η (G). i i −1 Taking x = 1 we see that ϕ (n ⊗ 1) = σ (ϕ (n ⊗ 1)) and we conclude that ϕ (n ⊗ 1) ∈ Q i i i p i unr for all n ∈ η (G).So ϕ is the Q -linear extension of a Q -linear homomorphism i p η (G) ⊗ Q → Q . Z p p 0 1 Since η(G) = η (G) ⊕ η (G), we then conclude that V (G) η(G) ⊗ Q as Q - cris Z p p vector spaces (here denotes the Q -dual). If n ,..., n is a Q -basis of η(G) ⊗ Q , p 1 4 p Z p then dn ,..., dn deﬁned by dn (n ) = δ (as usual, δ = 1if i = j and 0 otherwise) is 1 4 i j i, j i, j 123 Shimura–Mass operator ∨ unr unr ∨ ˆ ˆ abasis of (η(G) ⊗ Q ) and, by Q -linear extension, also of (η(G) ⊗ Q ) .Ifwe Z p Z p p p p unr unr ∨ ˆ ˆ now base change the Q -vector space (η(G) ⊗ Q ) via σ , we see that dn ,..., dn Z 1 4 p p p unr is still a Q -basis, and we have (x · dn )(n ) = σ(x)δ i j i, j unr 1 σ for all x ∈ Q . Using the above description of H (G) in terms of D(G) ,and the p cris unr description of V (G) in terms of η(G), we have an isomorphism of Q -vector spaces, cris 1 unr H (G) V (G) ⊗ Q , cris Q cris p p unr where the upper index σ on the right hand side means that the structure of Q -vector −1 −1 space is twisted by σ as explained above. Moreover, the σ -linear isomorphism V −1 −1 unr of M(G)[1/p] corresponds to the σ -linear isomorphism σ of η(G) ⊗ Q (act- p p ing on the second component only), and therefore the isomorphism ϕ → σ ◦ ϕ ◦ V ∨ ∨ unr of M(G)[1/p] (where denotes the Q -linear dual) corresponds to the isomorphism ∨ unr ⊗ σ of (η(G) ⊗ Q ) ⊗ Q given by dn ⊗ x → (dn ◦ ) ⊗ σ(x) where Z p Q i i p p p (dn ◦ )(n) = n ( n), which corresponds to φ ⊗ σ by deﬁnition of φ . i i V (G) V (G) cris cris 4.3 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]. unr Let E ⊆ Q be an unramiﬁed extension of Q , with valuation ring O .If (X, O ) is a p E X rig rig p-adic O -formal scheme, we denote (X , O ) the associated E-rigid analytic space (or rig its generic ﬁber), and if F is a sheaf of O -modules, we denote F its associated sheaf of rig O -modules [8,Sect. 7.4], [5,Sect. 1]. The notion of convergent isocrystal on a p-adic, formally smooth O -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 ) of X a coherent O ⊗ E-module E satisfying a natural cocycle condition for morphism T O T 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 : 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 .By[39,1.20, 2.81] is a canonical rig integrable connection ∇ : E → E ⊗ . Accordingly with [23,Deﬁnition 3.3], a X X rig rig ﬁltered convergent F-isocrystal on X is a F-isocrystal (E,φ ) such that E is equipped rig rig with an exhaustive and separated decreasing ﬁltration F E of coherent O -submodules X X rig rig rig i i −1 1 such that ∇ (F E ) is contained in F E ⊗ for all i; in these deﬁnitions, F rig rig X X X O X is the absolute Frobenius of the reduced closed subscheme of the closed subscheme of X deﬁned by the ideal pO . We denote E(O ) 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 ) O ⊗ E; it is equipped with canonical Frobenius and ﬁltration (see loc. T T O cit. for details). We also denote E(V) = V ⊗ E(O ) the ﬁltered convergent F-isocrystal Q unr p ˆ attached to a Q -rational, ﬁnite dimensional representation ρ : GL × GL → GL(V) of p 2 2 123 M. Longo the algebraic group GL × GL ;see [23,pages 345–346] for the deﬁnition of Frobenius and 2 2 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,ρ ) be the universal triple, arising from the representability of the functor SFD by G G unr unr ˆ ˆ H ; denote λ : G → H be universal map. Then p p E(G) = R λ (O ) ∗ ˆ unr G/Q unr has a structure of convergent F-isocrystal of H , which interpolates crystalline cohomology unr sheaves (see [39,Theorem (3.1), Theorem (3.7)]; see also [6, 7]). The coherent O -module rig E(G) is canonically isomorphic to the relative rigid de Rham cohomology sheaf unr rig 1 rig unr 1 • H (G /H ) = R λ . ∗ rig unr dR p G /H rig The canonical integrable connection ∇ coincides with the Gauss–Manin connection unr rig 1 rig unr 1 1 rig unr ∇ : H (G /H ) −→ ⊗ H (G /H ) unr G dR p unr ˆ unr dR p H /Q p p p whose construction in this context follows [27]; see [23,Example 3.4(c)], [39,Theorem (3.10)]. 1 rig unr The Hodge ﬁltration on the de Rham cohomology H (G /H ) makes then E(G) a ﬁltered dR convergent F-isocrystal. To describe this ﬁltration more explicitly, denote 1 1 unr H (G) = H (G/H ) dR dR p the dual of the Lie algebra of the universal vectorial extension of G, equipped with its structure of convergent F-isocrystal ([35,Chapter IV, §2], [36,Sect. 1,9,11)]). By [2,Sect. 3.3], we have an isomorphism of convergent F-isocrystals E(G) H (G) dR such that the Hodge ﬁltration on the de Rham cohomology groups coincides with the Hodge- Tate ﬁltration rig 1,rig rig 0 −→ ω −→ H (G) −→ Lie −→ 0. (15) G dR 1,rig ∗ 1 unr Here ω = e ( ),where e : H → G is the zero-section, H (G) = G p G ˆ unr dR G/H rig 1 ∨ H (G) ,and Lie is the Lie algebra of the Cartier dual G of G. A result of Falt- dR ˆ unr ings [14,Sect. 5] (see also [23,Lemma 5.10]) shows that, as ﬁltered convergent F-isocrystals, we have H (G) V (!) ⊗ E(O ) E(M ), (16) cris unr 2 Q ˆ dR p H where ρ : GL × GL → GL(M ) is the representation deﬁned by ρ (A)(B) = AB and 2 2 2 1 ab d −b ¯ ¯ ρ (A)B = B A (here if A = ,then A = ). cd −ca 123 Shimura–Mass operator 4.4 The Kodaira–Spencer map unr The Kodaira–Spencer map forλ : G → H , where as above G is the universal SFD-module, is the composition rig (15) (15) rig rig 1,rig G 1,rig rig 1 1 KS : ω −→ H (G) −→ H (G) ⊗ −→ Lie ⊗ O unr ∨ O unr G G dR dR unr unr unr unr H ˆ G H ˆ p H /Q p H /Q p p p p in which the ﬁrst and the last map come from the Hodge-Tate exact sequence (15). Recalling the duality between ω and Lie , we therefore obtain, as in the algebraic case, a symmetric rig rig rig unr map of O -modules, again denoted KS : ω ⊗ ω → (tensor product H ∨ p G G G unr ˆ unr H /Q p p over O unr ). By ﬁxing a formal polarization ι : G G of G [3,Chapitre III, Lemma 4.4], H G ∨ unr we obtain isomorphism ω ω of O -modules, and the Kodaira–Spencer map takes G H the form rig rig ⊗2 1 KS : (ω ) −→ G G unr ˆ unr H /Q p p unr where the tensor product is again over O . We now describe the Kodaira–Spencer map more explicitly, mimicking, in the complex case, [17, 37]. Denote unr H (G) = H (G/H ) 1,dR 1,dR the universal vectorial extension of G, which is equipped with a structure of ﬁltered convergent rig rig F-isocrystal as before; see [14,Sect. 5]. Put H (G) = H (G) . By deﬁnition, the 1,dR 1,dR ˆ unr universal vectorial extensions of G and G are dual to each other. We therefore obtain a O rig-bilinear skew-symmetric map 1,rig rig H (G) × H (G) −→ O unr. dR 1,dR 1,rig rig The principal polarization ι : G G identiﬁes canonically H (G) and H (G),and dR 1,dR we therefore obtain a pairing rig 1,rig 1,rig unr , : H (G) × H (G) −→ O dR dR dR p rig rig 1,rig satisfying dx, y =x, d y for all x, y sections in H (G) and all d ∈ D (because G G dR ι (dy) = d ι (y)) which we call rigid polarization pairing. We may therefore construct a G G map 1,rig 1,rig rig ρ : H (G) −→ H (G) −→ (ω ) dR dR G rig where the ﬁrst map takes a section s to the map deﬁned for a section t by t →s, t and dR rig 1,rig the second map is induced by duality from the inclusionω → H (G). Fix now a section G dR 0 ∨ s ∈ H (U,( ) ) over some afﬁnoid U. Then we may compose the maps to get unr unr H /Q p p rig G 1⊗s rig 1,rig 1,rig 0 0 0 1 ρ : H U,ω −→H U, H (G) −→ H U, H (G) ⊗ −→ unr unr G dR dR ˆ H /Q p p 1⊗s ρ 1,rig rig 0 0 ∨ −→ H U, H (G) −→ H U,(ω ) . dR G (17) 123 M. Longo The association s → ρ deﬁnes then a map of sheaves rig rig rig ∨ 1 ∨ ∨ (KS ) : ( ) −→ Hom ω ,(ω ) . O unr G unr ˆ unr H G G H /Q p p p By construction, the dual of this map is the Kodaira–Spencer map, under the canonical rig rig rig ∨ ∨ ⊗2 identiﬁcation between Hom(ω ,(ω ) ) and (ω ) . G G G 4.5 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 and V be constant sheaves of one-dimensional Q -vector spaces on the Q -rigid 0 1 p p univ analytic space H with basis t and t respectively. Deﬁne two invertible sheaves T and p 0 1 univ univ T on H by T = O ⊗ V for i = 0, 1, where O is the structural sheaf of rigid p H i H p p 1 i univ univ univ univ analytic functions on H .Deﬁne T = T ⊕T .For i = 0, 1, letη be the constant 0 1 i sheaf of two-dimensional Q -vector spaces on H with basis e and e .One ﬁxes p p i,0 i,1 univ i η = η (!) ⊗ Q (18) Z p univ univ univ univ univ as in [43,page 664]. Deﬁne u : η → T by u (e ) = zt and u (e ) = t , 0,0 0 0,1 0 0 0 0 0 0 univ univ univ univ univ and u : η → T by u (e ) = (p/z)t and u (e ) = t ,where z denotes 1,0 1 1,1 1 1 1 1 1 1 univ univ univ the standard coordinate function on H .Deﬁne η = η ⊕ η and similarly deﬁne 0 1 univ univ univ univ 2 univ u = u ⊕ u . We write ρ : (Q ) → η for the isomorphism determined 0 1 H 0 ab by the choice of the basis {e , e }.For γ = ∈ M (Q ) and i = 0, 1, deﬁne 0,0 0,1 2 p cd (γ) univ endomorphisms φ in End (T ) by i p i (γ) φ f (z) ⊗ t = (cz + d) f (γ(z)) ⊗ t , ( ) 0 0 (19) (γ) φ ( f (z) ⊗ t ) = (a + b/z) f (γ(z)) ⊗ t 1 1 univ for any f ∈ O (U), and any afﬁnoid U ⊆ H . Deﬁne an action of SL (Q ) on η H p 2 p univ for i = 0,1tomake u equivariant with respect to these actions; in other words, for x x x ab/ p x ab ∗ 0,0 ab 0,0 ∗ 1,0 1,0 γ = in GL (Q ), put γ = and γ = .Let Z [ ] 2 p p cd x cd x x pc d x 0,1 0,1 1,1 1,1 univ univ act on T by t = (p/z)t and t = zt .Welet Z [ ] act on η in such a way that 0 1 1 0 p univ u commutes with this action. We call the quadruplet univ univ univ univ univ D = (η , T , u ,ρ ) the universal rigid data. univ Passing to the associated normed sheaves [43,Deﬁnition 6], we obtain from D a quadru- univ univ univ univ univ ˆ ˆ ˆ ˆ plet D = (η ˆ , T , u ˆ ,ρˆ ) on H , corresponding to a H -valued point, which p p is universal in the following sense: for each B ∈ Nilp and each : S = Spec(B) → H corresponding to a quadruplet (η, T, u,ρ),wehave −1 univ ∗ univ −1 univ −1 univ (η, T, u,ρ) = ( η ˆ , T , u ˆ , ρˆ ). (20) See [43,Corollary 18 and Theorem 19] for more precise and complete statements. We call univ univ ˆ ˆ D the universal formal data, and we denote the quadruplet on the RHS of (20)by D to simplify the notation. 123 Shimura–Mass operator unr univ The universal SFD-module G over H can be recovered from a universal rigid data D . unr ˆ ˆ Pulling back via the projection π : H → H , we obtain a quadruplet ˆ p unr unr unr unr unr −1 univ ∗ univ −1 univ −1 univ ˆ ˆ ˆ D = (η ˆ , T , u ˆ ,ρˆ ) = (π η ˆ ,π T ,π u ˆ ,π ρˆ ) ˆ ˆ ˆ ˆ H p H H p p p unr on H . Comparing (20) with the universal property satisﬁed by G, we see that the quadru- unr plet (η , T , u ,ρ ) associated to G coincides with the quadruplet D . In particular, the G G G G rig rig rig rig unr rig associated quadruplet (η , T , u ,ρ ) on the rigid Q -rigid analytic space G is the G G G G p quadruplet −1 −1 −1 unr unr unr unr unr univ ∗ univ univ univ D = (η , T , u ,ρ ) = (π η ,π T ,π u ,π ρ ) unr unr unr unr H H H H p p p p univ unr obtained from the quadruplet D ,where π unr : H → H is the canonical projection. H p unr ∨ unr unr ∨ unr Let (T ) denote the O -dual of T , and, as above, denote (η ⊗ Q ) the H Z p p p unr unr unr unr Q -linear dual of η ⊗ Q . From the surjective map u : η ⊗ O unr T p Z p Z H p p univ induced by u we obtain an injective map unr ∨ unr ∨ unr τ : (T ) −→ (η ⊗ Q ) ⊗ O p H Z Q p p p Proposition 4.2 We have canonical isomorphisms rig unr ∨ (T ) −→ ω , 1,rig unr ∨ unr (η ⊗ Q ) ⊗ O −→ H (G). Z p Q H p p p dR under which the map τ corresponds to the ﬁrst map in (15). Proof The ﬁrst statement follows from the canonical isomorphism between T = Lie and G G unr T , while the second follows from Proposition 4.1 combined with (16). For the statement about τ , note that for each SFD-module G over F ,the map u corresponds under the p G unr unr ˆ ˆ identiﬁcation between η(G) ⊗ Q and M(G) ⊗ Q to the canonical projection Z unr p p ˆ p M(G)/V M(G) → T ,where T is the tangent space of G at the origin. G G G 4.6 The action of the idempotent e Fixanisomorphism Q ( p ) Q 2. By means of this isomorphism, and the ﬁxed embed- p 0 √ √ ding Q → D, we may identify elements a + b p in Q ( p ) (where a, b ∈ Q ) with 0 p 0 p elements of D in what follows without explicitly mentioning it. 0 0 Lemma 4.3 e · η(!) ⊗ Z 2 = η (!) ⊗ Z 2 and e · T(!) ⊗ Z 2 = T (!) ⊗ Z p Z p Z p Z p p p p Z . Proof The action of O on η(!) is induced by duality from the action on M(!),soany 0 1 element a ∈ Z → O acts onη (!) by multiplication by a and onη (!) by multiplication by a ¯. On the other hand, the action of 1 ⊗a onη(!) ⊗ Q 2 is given by multiplication by a. p p An immediate calculation shows then that the action of e is just the projectionη(!) → η (!). The argument for T(!) is similar. −1 −1 unr univ unr ∗ univ unr univ Write η = π unrη , T = π unr T , u = π unr u . 0 H 0 0 H 0 0 H 0 p p p 123 M. Longo unr unr unr unr Proposition 4.4 e · η = η and e · T = T . 0 0 Proof This is clear from Lemma 4.3 and (18). unr unr For i = 0, 1, the sheaf T is a free O -module of rank 1, so it is invertible; denote 0 p unr ∨ unr ∨ ∨ unr unr (T ) its O -dual. W thus get a map du : (T ) → (η ⊗ Q ) ⊗ O (where H 0 0 Z p Q H p p 0 p 0 p the RHS denotes the Q -dual). We set up the following notation: rig • ω = e · ω ; G G 1,rig • L = e · H (G). G dR 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: du unr ∨ ∨ unr 0 (T ) (η ⊗ Q ) ⊗ O (21) 0 Z p Q H p p 0 p 0 0 0 ω L G G 4.7 Differential calculus on the p-adic upper half plane We now set up the following notation. Recall that the map u takes x e + x e to 0 0,0 0,0 0,1 0,1 (zx + x ) ⊗ t ; dualizing, du can be described in coordinates by the map which takes 0,0 0,1 0 0 unr ∨ unr the canonical generator t of the O -module (T ) (satisfying the relation dt (t ) = 1) 0 H 0 0 p 0 to the map x e + x e → zx + x . If we denote de the dual basis of e 0,0 0,0 0,1 0,1 0,0 0,1 0,i 0,i (satisfying the condition de (e ) = δ ), we may write this map as zde + de .To 0,i 0, j i, j 0,0 0,1 simplify the notation, we put from now on τ = t , dτ = dt , x = e , y = e , dx = de 0 0 0,0 0,1 0,0 and dy = de , so that the above map reads simply as 0,1 dτ = zdx + dy. 0 0 unr Let C = C (H (Q ), C ) denote the C -vector space of continuous (for the stan- p p p unr dard p-adic topology on both spaces) C -valued functions on H (Q ). Denote A = p p 0 unr unr H (H , O unr) the Q -vector space of global sections of O unr . Each f ∈ A is, in H H p p p p unr unr particular, continuous on H for the standard p-adic topology of Q − Q , and therefore p p unr 0 ∗ restriction induces a map of Q -vector spaces r : A → C . Denote A the image of the mor- phism of A-algebras A[X, Y]→ C deﬁned by sending X to the function z → 1/(z −σ(z)) and Y to the function z → σ(z) (note that the function z → z − σ(z) is invertible on unr unr H (Q )). To simplify the notation, we put from now on p p z = σ(z). Set up the following notation (here n ≥ 1isaninteger) ⊗n 0 unr 0 • = H (H , L ) and = , G G,n G G ⊗n ∗ ∗ ∗ ∗ • = ⊗ A and = , G A G G,n G,n ⊗n 0 unr 0 • w = H (H ,ω ) and w = w , G G,n p G G ⊗n ∗ ∗ ∗ ∗ • w = w ⊗ A and w = w . G A G G,n G,n 123 Shimura–Mass operator unr d The Q -algebra A is equipped with the standard derivation on power series. The dz 1 1 0 unr A-module = H (H , ) is then one dimensional and generated by dz satisfying unr A p H d d ∂ ∗ ∗ dz = 1. We extend the differential operator to a differential operator : A → A dz dz ∂z unr ∂ ∗ ∂ 1 −1 by Q -linearity using the product formula and setting (z ) = 0and = . p ∗ 2 ∂z ∂z z−z (z−z ) ∂ ∂ ∂ ∗ ∗ ∗ Similarly, we deﬁne a differential operator : A → A setting (z) = 0, (z ) = 1 ∗ ∗ ∗ ∂z ∂z ∂z ∂ 1 1 1 ∗ and = .Deﬁne to be the A -subalgebra of the algebra of derivations ∗ ∗ ∗ ∗ 2 ∂z z−z (z−z ) ∗ ∂ ∂ ∗ ∂ generated by dz and dz satisfying the usual rules dz = 1, dz = 0, dz = 0, ∗ ∗ ∂z ∂z ∂z ∗ ∂ dz = 1. ∂z 4.8 Splitting of the rigid analytic Hodge filtration Recall the notation ﬁxed before for the differential form dτ = zdx + dy.Deﬁne ∗ ∗ dτ = z dx + dy. ∗ ∗ unr Then dτ belongs to w . Taking global sections, restricting to Q , and extending linearly ∗ ∗ with A we obtain a short exact sequence of A -algebras ∗ ∗ 0 −→ w −→ . (22) G G ∗ ∗ Theorem 4.5 The exact sequence (22) admits a canonical splitting : → w . G G Proof We have ∗ ∗ ∗ dτ − dτ zdτ − z dτ dx = , dy = . ∗ ∗ z − z z − z We may therefore write any differential form ω = f (z)dx + g(z)dy with f , g ∈ A as f (z) − g(z)z zg(z) − f (z) ω = dτ + dτ . ∗ ∗ z − z z − z f (z)−g(z)z One then deﬁnes the sough-for splitting sending ω → dτ . z−z 4.9 The p-adic Maass–Shimura operator rig Taking global section, the Gauss–Manin connection gives rise to a map ∇ : → rig 1 ∗ ∗ ∗ 1 ⊗ .Weextend ∇ to a map ∇ : → ⊗ as follows. First deﬁne G ∗ A G G G G A 1,0 1,0 ∗ ∗ 1 ∇ : → ⊗ to be the derivation satisfying the rules ∇ (dτ) = dx ⊗ dz, G G G A G 1,0 1,0 1,0 0,1 ∗ ∗ ∗ ∇ (dτ ) = 0, ∇ (z) = 1, ∇ (z ) = 0. Deﬁne similarly the derivation ∇ : → G G G G G 0,1 0,1 0,1 0,1 ∗ 1 ∗ ∗ ∗ ⊗ by the rules ∇ (dτ) = 0, ∇ (dτ ) = dx ⊗ dz , ∇ (z) = 0, ∇ (z ¯) = dz . G A G G G G We ﬁnally deﬁne 1,0 0,1 ∗ ∗ ∗ 1 ∇ =∇ +∇ : −→ ⊗ ∗. G G G A G G ⊗2 1 Taking global sections, the Kodaira–Spencer map gives rise to a map KS : w → , G A which we extend A -linearly to a map ∗ ∗ ⊗2 1 KS : (w ) −→ ∗. G G A 123 M. Longo Note that dτ − dτ rig ∇ (dτ) =∇ (dτ) = ⊗ dz. (23) G G z − z ∗ ∗ and, since ∇ (z ) = 0, we have ∗ ∗ ∇ (dτ ) = 0. (24) In particular, if f (z) ⊗ dτ ∈ w we have ∂ dτ − dτ rig ∇ ( f (z) ⊗ dτ) = f (z) ⊗ dτ + f (z) ⊗ ⊗ dz. (25) ∂z z − z rig Taking global sections, we can form the pairing , : ⊗ → A. Extending G A G linearly by A , we obtain a new pairing ∗ ∗ ∗ ∗ , : ⊗ ∗ −→ A . G G G Using the description of the Kodaira–Spencer map in Sect. 4.4,wesee that ∗ ∗ −dτ, dτ rig rig ∗ ∗ G ∗ dτ, ∇ (dτ) =dτ, ∇ (dτ) = dz =−dx, dy dz G G G G G z − z where for the second equality we use (23), while the last equality follows easily from the equation ∗ ∗ ∗ ∗ zdx + dy, z dx + dy = (z − z )dx, dy . G G Therefore rig rig KS (dτ ⊗ dτ) =−dx, dy dz. G G rig rig So, to compute KS (dτ ⊗ dτ) = KS (dτ ⊗ dτ) we are reduced to compute dx, dy . G G G For this, we switch to de Rham homology and follow the computations in [19, 37]. To begin with, let W denote the order O viewed as free left O -module of rank 1; then D D W R ⊗ Z .By[3,Ch. III, Lemma 1.9], the collection of bilinear skew-symmetric max Z p mapsψ : W ×W → Z which satisfyψ(dx, y) = ψ(x, d y) (for all x, y ∈ W and d ∈ O ) p D is a free Z -module of rank 1, and every generator ψ of this Z -module is a perfect duality p 0 p on W ; the pairing tr(iy x) ψ (x, y) = is such a generator, which we ﬁx once and for all (recall the notation introduced in Sects. 2.1 and 2.3 for i and d ). 1 unr Recall that H (!) is a free D⊗ Q -module of rank 1 (cf. [23,page 354]); the structure cris p p unr ∨ of D ⊗ Q -module is induced from the D-module structure of (η(!) ⊗ Q ) via the Q Z p p p p ∨ 1 unr isomorphisms (η(!) ⊗ Q ) V (!) and H (!) V (!) ⊗ Q in Lemma Z p cris cris Q p p p cris 4.1.WehavethenfromLemma 4.1 then canonical isomorphisms of convergent F-isocrystals: 1 1 H (G/H ) H (!) ⊗ E(O ) p unr unr dR cris ˆ ˆ Q H p p unr D ⊗ Q ⊗ E(O ) unr Q ˆ unr ˆ p p H (26) D ⊗ Q 2 ⊗ E(O ) Q Q unr p p 2 ˆ M (Q 2) ⊗ E(O ). unr 2 Q ˆ p 2 p H 123 Shimura–Mass operator unr Under the isomorphism (26), the Q -linear extension of ψ deﬁnes a pairing 1 1 unr ψ : H (!) × H (!) −→ Q cris cris p rig rig still denoted with the same symbol. If we denote , the restriction of , to H (!),it G G cris follows from the unicity of ψ up to constant that there exists an element t ∈ C such that 0 p rig , = · ψ . (27) G,W Moreover, under the isomorphism (26), the element dτ = zdx + dy of H (G) corresponds dR 10 01 00 00 to the element e ⊗ z + e ⊗ 1, where e = , e = , e = , e = 1 2 1 2 3 4 00 00 10 01 is the standard basis of M (Q ). We therefore obtain the sought-for recipe to compute the Kodaira–Spencer image of dτ ⊗ dτ in terms of ψ : rig KS (dτ ⊗ dτ) = · ψ (e ⊗ z, e ⊗ 1). (28) 0 1 2 Remark 4.6 The number t 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 at the denominator in (27). We now make more explicit the equations (23)and (28) using Hashimoto basis {1, i, j, k} ﬁxed in Sect. 2.1.Asin[37,Proposition 2.3] and [18,(2)], deﬁne = 1, = (1 + j)/2, 1 2 = (i + ij)/2, = (apN j + ij)/p and use these elements to deﬁne a symplectic basis 3 4 0 p −1 of W with respect to the pairing ψ as in [18,(5)] by η = − , η =−aD − , 0 1 3 4 2 1 4 η = , η = (note that ψ we consider above is equal to the pairing (x, y) → tr(xiy ) 3 1 4 2 0 ∨ ∨ ∨ ∨ ∨ ∨ in [18,(3)]). Denote η ,η ,η ,η the dual basis of W ,and let η be the column vector 1 2 3 4 ∨ ∨ ∨ ∨ ∨ 1 with entries η ,η ,η ,η . The elements η give rise to elements of H (G), denoted with 1 2 3 4 dR rig rig the same symbol, which are horizontal with respect to ∇ , namely ∇ (η ) = 0. Write G G dτ = (z) · η . A simple calculation shows that α −1 1 + + + (z) = (α az + 1), (α az + 1), z, α z . (29) √ √ 2 p p 2 0 0 rig d (z) Sinceη are horizontal sections of ∇ ,using (29) to calculate shows that (23) becomes i G dz − + + α α a −α az 1 rig + ∨ ∇ (dτ) = , , 1, α · η ⊗ dz. (30) √ √ 2 p p 2 0 0 The recipe (17) to compute the Kodaira–Spencer map combined with (27)and (30)gives then 1 d (z) 1 0 I rig 2 KS (dτ ⊗ dτ) = (z) dz = dz. (31) −I 0 t dz 2 t p p rig In particular, (31) shows that KS is an isomorphism, and therefore the same is true for KS . This allows us to deﬁne the operator ∗ ∗ −1 ∇ (KS ) G,n G ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∇ : ⊗ ⊗ w ⊗ A ∗ A A G,n G,n G,n A G,n G,2 G,n G,2 G,n+2 ∗ ∗ where ∇ is obtained from ∇ using the Leibniz rule (see [19,Sect. 3.2]). The splitting in G,n G ∗ ∗ ∗ Theorem 4.5 induces a morphism of A -modules : → w . p,n G,n G,n 123 M. Longo Deﬁnition 4.7 The composition G,n p,n+2 ∗ ∗ ∗ ∗ : w w p,n G,n G,n G,n+2 G,n+2 is the p-adic Maass–Shimura operator. We also need to consider iterates of this operator. Deﬁne j,∗ ∗ ∗ ∗ ˜ ˜ ˜ ˜ ∇ = ∇ ◦ ··· ◦ ∇ ◦ ∇ . n+2( j −1) n+2 n Deﬁne then j,∗ ∇ p,n+2 j j n ∗ ∗ ∗ ∗ : w w (32) p,n G,n G,n G,n+2 j G,n+2 j ∗ ∗ ∗ where the morphism of A -modules : → w is induced as above p,n+2 j G,n+2 j G,n+2 j by the splitting in Theorem 4.5.Wecall the j-th iterate of the p-adic Maass–Shimura p,n operator. 4.10 Relation with Franc Maass–Shimura operator The work accomplished so far allows us to explicitly describe , thus relating it to Franc p,k Shimura–Masss operator. We ﬁrst introduce some differential operators, similar in shape to the Maass–Shimura operator in the real analytic setting. For each integer k ≥ 0, we may then unr deﬁne the Q -linear function ∂ k ∗ ∗ δ = + : A −→ A . (33) p,k ∂z z − z ∗ ∗ For each integer j ≥ 0 weget amap δ : A → A deﬁned by p,k δ = δ ◦ ··· ◦ δ ◦ δ . p,k+2( j −1) p,k+2 p,k p,k We callδ the Maass–Shimura operator, andδ its j-th iteration. Applying (31) to compute p,k p,k the inverse of the Kodaira–Spencer map to (25), we obtain 1 ∂ dτ − dσ(τ) ∗ ⊗k ⊗k+2 ⊗k+1 ∇ f (z) ⊗ dτ = · f (z) ⊗ dτ + kf (z) ⊗ ⊗ dτ . t ∂z z −σ(z) Applying the splitting of the Hodge ﬁltration which annihilate dτ , we ﬁnally obtain p,k+2 1 ∂ kf (z) 1 ⊗k ⊗k+2 ⊗k+2 f (z) ⊗ dτ = · f (z) + ⊗ dτ = · δ ( f ) ⊗ dτ . p,k p,k t ∂z z −σ(z) t p p (34) Iterating (34) we obtain j j ⊗k ⊗k+2 j f (z) ⊗ dτ = δ ( f (z)) ⊗ dτ . (35) p,k p,k This proves Corollary 1.2 stated in the Introduction. 123 Shimura–Mass operator 5The 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. 5.1 p-adic uniformization of Shimura curves In this subsection we review the Cerednik–Drinfel’d Theorem. Recall the subgroup V (N ) of the Eichler order R of level N of B deﬁned in Sect. 2.1.Let U be the open compact ˆ ˆ subgroup of the group R (where R is the proﬁnite completion of R) consisting of elements × × ab (x ) ∈ R such thatι (x ) = for elements a ∈ Z and b ∈ Z , for each prime number + (p) × | N .Let U denote the subgroup of R 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 − ( p) algebra over Q of discriminant N . Fix isomorphisms B B for all primes = p;so U × (p) ( p) ˆ ˆ deﬁnes a subgroup of (B ) , still denoted with the same symbol U (as above, B is the × (p) ˆ ˆ adele ring of B and (B ) is the subgroup of invertible elements of B whose -component belongs to the-component of U via the isomorphism B B , for all primes = p). Deﬁne × (p) ˜ ˜ ˜ = B ∩ U . We still denote the image of in GL (Q ) via a ﬁxed isomorphism p p p 2 p i : B ⊗ Q M (Q ), and we let denote the subgroup of consisting of elements p Q p 2 p p p whose determinant has even p-power order. Base changing from Z to the valuation ring Z 2 of Q 2 gives a Z 2-formal scheme p p p H , whose generic ﬁber H is the base change of the Q -rigid analytic space H to 2 2 p p p p Q 2. The group GL (Q ) acts on the Z -formal scheme H [3,Chapitre I, Sect. 6] and 2 p p p acts on Spf(Z ) via the inverse of the arithmetic Frobenius raised to the determinant map [3,Chapitre II, Sect. 9]. Therefore, the group GL (Q ) also acts on the Z 2-formal 2 p scheme H 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 2-rigid analytic spaces rig \H 2 C , (36) rig rig \G A . (37) The isomorphism (37) is equivariant with respect to the quaternionic actions on both sides. See also [11,Sect. 6.2], [31,Sect. 6], [29]. 5.2 Rigid analytic modular forms A rigid analytic function f : H (C ) → C is said to be a rigid analytic modular form of p p p weight k and level if f (γ z) = (cz + d) f (z) rig for all z ∈ H (C ) and γ ∈ ,where γ(z) = (az + b)/(cz + d). Denote S ( ) the p p p p C -vector space of rigid analytic modular forms of weight k and level .See [11,Sect. 5.2] p p for details. Given a Z[ ]-module M,wedenote M the submodule consisting of -invariant ele- p p unr ments of M. With notation as in Sect. 4.8, consider the Q -subspace w of w consisting 123 M. Longo of global sections which are invariant for the -action. In particular, w is a A -module. rig ⊗k Given f ∈ S ( ),deﬁne ω = f (z) ⊗ dτ in w = w ⊗ C . The correspon- p f G,C G unr p p ˆ k Q rig p dence f → ω sets up a C -linear isomorphism between S ( ) and w . f p p k G,C unr unr For any sheaf F on H , denote F the sheaf on \H deﬁned by taking -invariant p p p p sections. Also, recall the sheaves ω and L introduced in (8)and (9). It is easy to see unr unr ˆ ˆ Q Q p p rig rig 0 0 p p that (36)and (37) induce isomorphisms (ω ) ω and (L ) L of sheaves. G G ˆ unr ˆ unr Q Q p p rig Combining this with the isomorphism between S ( ) and w , we obtain a canonical k G,C isomorphism of C -vector spaces rig 0 ⊗k S ( ) H (C ,ω ). (38) p C k C rig rig If E is a subﬁeld of C containing Q 2, we denote S ( , E) the E-subspace of S ( ) p p p p k k 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 spaces rig alg S (, F) S ( , F). (39) k k 5.3 The p-adic Maass–Shimura operator Taking -invariants deﬁnes a map, for integers k ≥ 0and j ≥ 0 ∗ ∗ p p : (w ) −→ (w ) G,n G,n+2 j p,k where recall that was introduced in (32), and we understand that = , p,k p,k p,k An alternative way to introduce is the following. Recall the operator ∇ in (11) and, p,k for any integer j ≥ 0, deﬁne ∇ : L → L by the formula n unr unr ˆ ˆ Q ,n Q ,n+2 j p p ˜ ˜ ˜ ˜ ∇ = ∇ ◦ ··· ◦ ∇ ◦ ∇ . n n+2( j −1) n+2 n Considering the associated rigid analytic sheaves, and taking global sections, we obtain a j,rig p p map of A-modules ∇ : → . One may deﬁne the operator G,n G,n+2 j j,rig i p p G,n ∇ p,n+2 j j n p p p ∗ ∗ : w w . p,n G,n G,n G,n+2 j G,n+2 j G,n+2 j j j ∗ ∗ By (24), dτ is horizontal for ∇ , and therefore coincides with the restriction of p,n G p,k to w , thus justifying the abuse of notation. G,n 5.4 Comparison of Maass–Shimura operators at CM points Identify the set of Q 2-points in the rigid space H with the set of Q -algebra homomor- p p phisms Hom(Q , M (Q )) as follows: any ∈ Hom(Q , M (Q )) deﬁnes an action 2 2 2 p 2 p p p of Q on H (Q ) = Q − Q by fractional linear transformations, and the point 2 2 2 p p p p z z z ∈ H (Q 2) associated with is characterised by the property (a) = a ,for 1 1 all a ∈ Q . 123 Shimura–Mass operator Given a representation ρ = (ρ ,ρ ) : GL × GL → GL(V),the stalk E(V) of E(V) 1 2 2 2 at a point ∈ Hom(Q 2, M (Q )) can be described explicitly. One ﬁrst observes that the 2 p structure of ﬁltered convergent F-isocrystal of E(V) induces a structure of ﬁltered Frobe- nius module on the ﬁber E(V) . On the other hand, one attaches to such a pair (V,) a unr unr ﬁltered Frobenius module V whose underlying vector space is V = V ⊗ Q (see Q Q p p [23,Sect. 4.2] for details). Denote F V the ﬁltration of the ﬁltered Frobenius module V ; i • i i −1 this is then a ﬁltration on V unr which depends on .Let gr (F V ) = F V /F V be the graded pieces of the ﬁltration. If V is pure of weight n,wehaveanisomorphism i • gr (F V ) V (where V is be the subspace of V unr consisting of elements v satisfy- i j Q j n− j ing the property ρ((a))(v) = a σ(a) v for all a ∈ Q 2) as well as a decomposition i • V unr = gr (F V ).For ∈ Hom(Q , M (Q )), denote the morphism of Q 2 p i ∈Z p Q -algebras obtained by composition with the main involution of M (Q ); therefore, p 2 p ab d −b if (x) = then (x) = .If V is pure of weight n, then the graduate pieces cd −ca i • n−i • gr (F V ) and gr (F V ) are equal, for all i ∈ Z. In particular, for V = M we have ¯ 2 1 • 2 • gr F (M ) gr F (M ) . (40) 2 2 ¯ and therefore there is an exact sequence: 1 • 1 • 0 −→ gr F (M ) −→ (M ) −→ gr F (M ) −→ 0 2 2 2 ¯ and a canonical decomposition 1 • 1 • (M ) gr F (M ) gr F (M ) . (41) 2 2 2 ¯ unr 1 • One can choose generatorsω ,ω of the Q -vector space gr (F (M ) ) so thatω andω 1 2 2 1 2 unr 1 • are deﬁned over Q .Thenω andω are generators of the Q -vector space gr F (M ) , 1 2 2 ¯ p p where ω → ω for i = 1, 2 denotes the action of Gal(Q 2/Q ) on ω . It follows that the i i p i 1 • Hodge splitting coincides on quadratic points with the projection (M ) → gr (F (M ) ) 2 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 ⊗k a imaginary quadratic ﬁeld and f ∈ H (C ,ω ) 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 → C ∞ p and f : H → C for the holomorphic and the rigid analytic modular forms corresponding p p p 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 (K) be a Heegner point, and assume that P ∈ C (C) is represented by the point τ ∈ H modulo , while C ∞ ∞ ∞ ¯ ¯ P ∈ C (C ) is represented by the point τ ∈ H modulo . Fix embeddings Q → Q C p p p p p and Q → C which allows us to view algebraic numbers as complex and p-adic numbers. Theorem 5.1 For any positive integer j we have the equality j j ( f )(τ ) = ( f )(τ ). ∞ ∞ p p ∞,k p,k Proof We mimic a well known argument of Katz when p is split in K ([26,Theorems 2.4.5, 2.4.7]; see also [4,Proposition 1.12], [19,Theorem 3.5], [37,Proposition 2.12]). Let A be the false elliptic curve corresponding to the Heegner point P. The algebraic CM splitting of A coincides both with the Hodge splitting and the p-adic splitting, and therefore the values of and at CM points are the same. Since the construction of the Maass–Shimura ∞,n p,n rig ran operators is algebraic, we see that ∇ ( f ) coincides with ∇ ( f ), and the same still holds ∞ n p for the iterates of the Maass–Shimura operator, which also admit an algebraic construction. The result follows. 123 M. Longo 5.5 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 -subspace of the C -vector space C of continuous functions, which p p 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 ( j) ( j) N = N where for each integer j ≥ 0, we denote N the sub-A-algebra of N j ≥0 consisting of functions f which can be written in the form f (z) f (z) = (z −σ(z)) i =0 with f ∈ A. The Maass–Shimura operator δ restricts to an operator (denoted with the i p,k ( j) ( j +2) same symbol) δ : N → N which takes N to N . p,k Deﬁne now N ( ) = N to be the C -subalgebra of N consisting of functions which k p p are invariant under the weight k action of on N , i.e. those functions satisfying the trans- k unr formation property f (γ z) = (cz + d) f (z) for all z ∈ Q − Q and γ ∈ . Note that p p rig S ( ) ⊆ N ( ). We call N ( ) the C -vector space of nearly rigid analytic modular p k p k p p ( j) j ( j) forms of weight k and level . Deﬁne also N ( ) = N ( ) ∩ N . The operator δ p p k p k p,k introduced in Sect. 4.7 restricts to a mapδ : N ( ) → N ( ) [15,Lemma 4.3.8]. By k p k+2 j p p,k [15,Theorem 4.3.11], for each integer r ≥ 0 we have an isomorphism of C -vector spaces rig (r) S ( ) N ( ) p p k+2(r − j) p,k+2r j =0 k+2(r − j) k+2(r − j) j which maps (h ) to δ (h ). j j j =0 j =0 p,k δ ( f )(τ ) p,k Corollary 5.2 (Franc) Let τ ∈ H corresponds to a Heegner point. The values p p are algebraic for each integer j ≥ 0. Proof The result is clear from Theorem 5.1 since this is known for ( f )(τ ). ∞,k Remark 5.3 Equation (35) answers afﬁrmatively one of the questions left in [15,Sect. 6.1] whether if it was possible to describe the p-adic Maass–Shimura operator δ introduced p,k in [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. 6 The Coleman primitive 1,0 0,1 ∗ 1,0 0,1 ∗ Write ∇= ∇ , ∇ =∇ , ∇ =∇ and ,=, to simplify the notation. G,n G G G,n For any n and any j, whenever there is not possible confusion, we write = and p p,n 123 Shimura–Mass operator j j = for the p-adic Maass–Shimura operator, and = for the splitting of the p,n p p p,n Hodge ﬁltration. dτ We set up the notation ω = dτ and η = .Since dx, dy=−1, we have can can ∗ z −z j n− j ⊗ j ⊗n− j ω ,η = 1. We also write ω η = ω ⊗ η . can can can can can can The computation of the Gauss–Manin connection gives can ∇(ω ) = − + η ⊗ dz, can can z − z ω ⊗ dz η ⊗ dz can can ∇(η ) =− + . can ∗ 2 ∗ (z − z) z − z ⊗k Let f : H → C be a rigid modular form giving rise to a sectionω = f (z) ⊗dτ . Put p p f ⊗n n = k − 2. Using the Kodaira–Spencer map, we identify this withω = f (z)dz ⊗ dτ . Let F be the Coleman primitive of the differential form ω , satisfying the differential equation f f ∇(F ) = ω . f f Deﬁne for j = n/2,..., n an integer j n− j n−2 j G (z) =F (z),ω η ⊗ ω . (42) j f can can can j +1 Theorem 6.1 (G ) = j !ω . p j f Proof This result, which is proved by means of a simple and explicit computation, is the analogue of [4,Proposition 3.24] (and also of [19,Theorem 7.3]), but we provide a complete proof since our formalism is quite different from that in [4], where one can use the Tate curve and the q-expansion principle. As in loc. cit. we show that (G (z)) = ω and p 0 f (G (z)) = jG (z). p j j −1 We ﬁrst compute ∇(G (z)).Wehave: ∇(G (z)) n n =∇ F (z),η ⊗ ω can can n n n n n n = ∇(F (z)),η ⊗ ω +F (z), ∇ η ⊗ ω + F (z),η ⊗∇(ω ) f f f can can can can can can n n n n n n n = f (z)dz ⊗ ω ,η ⊗ ω +F (z), ∇ η ⊗ ω + F (z),η ⊗∇(ω ). f f can can can can can can can We now compute the last two pieces: n n F (z), ∇ η ⊗ ω can can n−1 n = F (z), nη ∇(η ) ⊗ ω f can can can −ω ⊗ dz η ⊗ dz can can n−1 n = F (z), nη + ⊗ ω can can ∗ 2 ∗ (z − z) z − z n−1 ∗ n nη ω ⊗ dz nη ⊗ dz can can n can n =− F (z), ⊗ ω + F (z), ⊗ ω f f can can ∗ 2 ∗ (z − z) z − z n ∗ n nω ⊗ dz nω ⊗ dz n−1 can n can =− F (z),η ω ⊗ + F (z),η ⊗ f can f can can ∗ 2 ∗ (z − z) z − z 123 M. Longo and n n F (z),η ⊗∇(ω ) can can n n = F (z),η ⊗∇(ω ) can can n n−1 = F (z),η ⊗ nω ∇(ω ) f can can can can n n−1 = F (z),η ⊗ nω − + η ⊗ dz f can can can z − z nω ⊗ dz n can n n−1 =− F (z),η ⊗ + F (z),η ⊗ nω η ⊗ dz. f f can can can can z − z Therefore the sum of these two pieces gives: n ∗ nω ⊗ dz can n−1 n n−1 − F (z),η ω ⊗ + F (z),η ⊗ nω η ⊗ dz. f can f can can can can ∗ 2 (z − z ) Recall now that (η ) = 0and (dz ) = 0. Therefore, using the Kodaira–Spencer map can to replace dz with ω , and applying we have can n n (G (z)) = ω ω ,η = ω . p 0 f f can can We now compute ∇(G (z)) for j ≥ 1. The Gauss–Manin connection j n− j n−2 j ∇(G (z)) =∇ F (z),ω η ⊗ ω j f can can can is the sum of three terms j n− j n−2 j ∇(F (z)),ω η ⊗ ω can can can j n− j n−2 j j n− j n−2 j +F (z), ∇(ω η )⊗ ω +F (z),ω η ⊗∇(ω ) (43) f f can can can can can can which we calculate separately as before. First, since j > 0, we have j n− j n−2 j n j n− j ∇(F (z)),ω η ⊗ ω = f (z)dz ⊗ ω ,ω η = 0. can can can can can can Next, a simple computation shows that dz dz j n− j j n− j j −1 n− j +1 j +1 n− j −1 ∇(ω η ) = (n − 2 j)ω η ⊗ + jω η ⊗ dz − (n − j)ω η ⊗ can can can can can can can can ∗ ∗ 2 z − z (z − z) and therefore the second summand in (43)is dz j n− j n−2 j j n− j n−2 j F (z), ∇(ω η )⊗ ω =(n − 2 j) F (z),ω η ⊗ ω ⊗ f f can can can can can can z − z j −1 n− j +1 n−2 j − j F (z),ω η ⊗ ω ⊗ dz+ can can can dz j +1 n− j −1 n−2 j − (n − j) F (z),ω η ⊗ ω ⊗ can can can ∗ 2 (z − z ) 123 Shimura–Mass operator Finally, the third summand is j n− j n−2 j F (z),ω η ⊗∇(ω ) can can can j n− j n−2 j −1 = (n − 2 j)F (z),ω η ⊗ ω ∇(ω ) f can can can can can j n− j n−2 j −1 = (n − 2 j)F (z),ω η ⊗ ω − + η ⊗ dz f can can can can z − z dz j n− j n−2 j =−(n − 2 j)F (z),ω η ⊗ ω ⊗ + can can can z − z j n− j n−2 j −1 + (n − 2 j)F (z),ω η ⊗ ω η ⊗ dz. f can can can can Summing up the pieces in (43), using the Kodaira–Spencer map to replace dz with ω ,and can applying the splitting of the Hodge ﬁltration which kills η and dz ,wehave can j −1 n−( j −1) n−2( j −1) (G (z)) = j F (z),ω η ⊗ ω = jG (z). p j f j −1 can can can The result follows. 7 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. 7.1 The generalised Kuga-Sato motive Fixaneveninteger k ≥ 2 and put n = k − 2, m = n/2. Let A 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 ; the action of O is required to commute with K K the quaternionic action, and this implies that A is isogenous to E × E for an elliptic curve E with CM by O .Fix aﬁeld F ⊃ H and consider the (2n + 1)-dimensional variety X K m over F given by m m X := A × A . Here and in the following we simplify the notation and simply write A, C and A for A , 0 F C and (A ) , unless we need to stress the ﬁeld of deﬁnition in which case we keep the F 0 F full notation. The variety X is equipped with a proper morphism π : X → C with 2n- m m m m dimensional ﬁbers. The ﬁbers above points of C are products of the form A × A . The de Rham cohomology of C attached to L , denoted H (C, L , ∇),isdeﬁnedtobe n n dR the 1-st hypercohomology of the complex 0 −→ L −→ L ⊗ −→ 0. n n As shown in [19,Corollary 6.3], one can deﬁne a projector (denoted P in loc. cit.)inthe m m ring of correspondences Corr (A , A ), such that ∗ n+1 H (A /F) ⊆ H (A /F), (44) A m m dR dR ∗ 1 H (A /F) H (C, L , ∇). (45) A m n dR dR 123 M. Longo m m On the other hand, by [19,Proposition 6.4], we can deﬁne a projector ∈ Corr(A , A ) 0 0 (which is deﬁned by means of ) such that ∗ m n 1 H (A /F) = Sym eH (A /F). (46) A 0 0 dR 0 dR The projectors and are commuting idempotents when viewed in the ring Corr (X , A A C m X ).Wedeﬁne = and denote D the motive (X ,).By[19,Proposition 6.5] and m A A m (44), (45), (46) we see that 1 n 1 m H (C, L , ∇) ⊗ Sym eH (A /F), if i = 2n + 1, i dR dR 0 H (X /F) = (47) dR 0, if i = 2n + 1. Let v be the place of F above p induced by the inclusion F ⊆ Q → C ,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 rig n+1 n 1 AJ : CH (D)(F ) −→ S ( , F ) ⊗ Sym eH (A/F ) . p v p v F v v dR 0 k Here (·) denotes the F -linear dual. For details, see [23,page 362] and [30,Sect. 4.2]; see also [32]and [33]. 7.2 Generalized Heegner cycles Let ϕ : A → A be an isogeny (deﬁned over K ) of false elliptic curves, of degree d prime 0 ϕ to Np.Let P be the point on C corresponding to A with level structure given by composing ϕ with the level structure of A .Weassociatetoany pair (ϕ, A) a codimension n + 1cycle ϒ on X by deﬁning ϕ m m m ϒ := ( ) ⊂ (A × A ) ϕ ϕ 0 where ={(ϕ(x), x) : x ∈ A }⊂ A × A is the graph of ϕ.Wethenset ϕ 0 0 := ϒ . ϕ ϕ m m The cycle of D is supported on the ﬁber above P and has codimension n +1in A × A , ϕ A n+1 thus ∈ CH (D).By(47), the cycle is homologous to zero. See [30] for details. ϕ ϕ We now compute the image of under the Abel-Jacobi map. The de Rham cohomology group H (A/F) of a false elliptic curve A deﬁned over a ﬁeld F containing the Hilbert class dR ﬁeld H of K is equipped with the Poincaré pairing , , which we simply denote H (A/F) dR , . Recall the canonical differentials ω , η introduced in Sect. 6; taking the ﬁber at A can can A , the universal differential ω deﬁnes a differential ω in H (A /F), and we choose 0 can A 0 0 dR η so that ω ,η = 1and {ω ,η } is a F-basis of eH (A /F); this is possible A A A A A A 0 0 0 0 0 0 0 dR because the Hodge exact sequence 1 1 1 0 −→ −→ H (A /F) −→ H (A , O ) −→ 0 0 0 A A /F dR 0 n 1 splits, since A has CM. This yields a basis for Sym eH (A /F) given by the elements 0 0 dR j n− j ω ⊗ η for j an integer such that 0 ≤ j ≤ n. A A 0 0 + − Let f be a newform of level (N ) ∩ (N p) and weight k deﬁned over F.Let v 1 0 ¯ ¯ be the prime of F determined by the embedding F ⊆ Q → Q ; we assume that F /Q p v p is unramiﬁed and contains Q 2. The Jacquet-Langlands correspondence associates to f 123 Shimura–Mass operator alg JL an algebraic modular form f in S (, F ), and therefore, using (39), a rigid analytic rig JL modular form f ∈ S ( , F ).Let ω be differential associated with the rigid analytic p v f rig k JL modular form f . Recall now the deﬁnition of the function G in (42), and for integers rig j = n/2,..., n deﬁne the function j n− j H (z) =F (z),ω (z)η (z). j f can can n−2 j So we have G (z) = H (z) ⊗ ω . j j can Theorem 7.1 For each j = n/2,..., n we have j n− j d · H (z ) = AJ( )(ω ⊗ ω η ). j A ϕ f A A 0 0 Proof The universal differentialω deﬁnes a differential formω in H (A/F), and choose can A dR as above η in H (A/F) in such a way that ω ,η = 1and {ω ,η } is a basis of A A A A A A dR H (A/F). It follows from [30,Theorem 5.5] that dR j n− j j n− j AJ( )(ω ⊗ ω η ) =ϕ (F (z )),ω η . ϕ f f A A A A A A 0 0 0 0 0 ∗ ∗ ∗ ∗ Since ϕ (ω ) = ω and ϕ (ω ),ϕ (η ) = d ,wehave ϕ (η ) = d η ,sowe A A A A A ϕ A ϕ A 0 0 deduce j n− j j n− j ∗ j −n ∗ ∗ ϕ (F (z )),ω η = d ·ϕ (F (z )),ϕ (ω η f A A f A A A A 0 ϕ A A 0 0 0 j n− j = d ·F (z ),ω η f A A ϕ A A and the last expression is equal to d · H (z ). ϕ j A Corollary 7.2 For each j = n/2,..., n we have n− j δ (H )(z ) n! p n A j n− j = AJ( )(ω ∧ ω η ). ϕ f A A n− j j 0 0 t j!· d p ϕ Proof From the proof of Theorem 6.1 we see that (G (z)) = jG (z), and therefore we p j j −1 n− j n− j n− j have j ! (G (z)) = n!G (z).Sousing (35) we conclude j !δ (H (z)) = n!t H (z). p n j p n p j The result follows from Theorem 7.1. Acknowledgements The author thanks the referee for carefully reading the manuscript and for giving valuable suggestions which helped improving the quality of the paper. The author is supported by PRIN 2017, INdAM– GNSAGA. 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Published: Apr 1, 2023

Keywords: p-adic uniformisation; Shimura curves; Maass–Shimura operators; 11F03; 14F40; 11R52

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A p-adic Maass–Shimura operator on Mumford curves

A p-adic Maass–Shimura operator on Mumford curves

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A p-adic Maass–Shimura operator on Mumford curves

A p-adic Maass–Shimura operator on Mumford curves

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