The Λ-Adic

Shimura-Shintani-Waldspurger Correspondence

Matteo Longo, Marc-Hubert Nicole

Received: October 25, 2012

Communicated by Takeshi Saito

Abstract. We generalize the Λ-adic Shintani lifting for GL_{2}(ℚ) to indefinite quaternion algebras over ℚ.

2010 Mathematics Subject Classification: Primary 11F37, 11F30, 11F85

1. Introduction

Langlands’s principle of functoriality predicts the existence of a staggering wealth of transfers (or lifts) between automorphic forms for different reductive groups. In recent years, attempts at the formulation of p-adic variants of Langlands’s functoriality have been articulated in various special cases. We prove the existence of the Shimura-Shintani-Waldspurger lift for p-adic families. More precisely, Stevens, building on the work of Hida and Greenberg-Stevens, showed in [21] the existence of a Λ-adic variant of the classical Shintani lifting of [20] for GL_{2}(ℚ). This Λ-adic lifting can be seen as a formal power series with coefficients in a finite extension of the Iwasawa algebra Λ := ℤ_{p}[[X]] equipped with specialization maps interpolating classical Shintani lifts of classical modular forms appearing in a given Hida family. Shimura in [19], resp. Waldspurger in [22] generalized the classical Shimura-Shintani correspondence to quaternion algebras over ℚ, resp. over any number field. In the p-adic realm, Hida ([7]) constructed a Λ-adic Shimura lifting, while Ramsey ([17]) (resp. Park [12]) extended the Shimura (resp. Shintani) lifting to the overconvergent setting. In this paper, motivated by ulterior applications to Shimura curves over ℚ, we generalize Stevens’s result to any non-split rational indefinite quaternion algebra B, building on work of Shimura [19] and combining this with a result of Longo-Vigni [9]. Our main result, for which the reader is referred to Theorem 3.8 below, states the existence of a formal power series and specialization maps interpolating Shimura-Shintani-Waldspurger lifts of classical forms in a given p-adic family of automorphic forms on the quaternion algebra B. The Λadic variant of Waldspurger’s result appears computationally challenging (see remark in [15, Intro.]), but it seems within reach for real quadratic fields (cf. [13]). As an example of our main result, we consider the case of families with trivial character. Fix a prime number p and a positive integer N such that p ∤ N. Embed the set ℤ^{≥2} of integers greater or equal to 2 in Hom(ℤ_{p}^{×}, ℤ_{p}^{×}) by sending k ∈ ℤ^{≥2} to the character x ↦ x^{k−2}. Let f_{∞} be an Hida family of tame level N passing through a form f_{0} of level Γ_{0}(Np) and weight k_{0}. There is a neighborhood U of k_{0} in Hom(ℤ_{p}^{×}, ℤ_{p}^{×}) such that, for any k ∈ ℤ^{≥2} ∩ U, the weight k specialization of f_{∞} gives rise to an element f_{k} ∈ S_{k}(Γ_{0}(Np)). Fix a factorization N = MD with D > 1 a square-free product of an even number of primes and (M, D) = 1 (we assume that such a factorization exists). Applying the Jacquet-Langlands correspondence we get for any k ∈ ℤ^{≥2} ∩ U a modular form f_{k}^{JL} on Γ, which is the group of norm-one elements in an Eichler order R of level Mp contained in the indefinite rational quaternion algebra B of discriminant D. One can show that these modular forms can be p-adically interpolated, up to scaling, in a neighborhood of k_{0}. More precisely, let O be the ring of integers of a finite extension F of ℚ_{p} and let 𝔻 denote the O-module of O-valued measures on ℤ_{p}^{2} which are supported on the set of primitive elements in ℤ_{p}^{2}. Let Γ_{0} be the group of norm-one elements in an Eichler order R_{0} ⊆ B containing R. There is a canonical action of Γ_{0} on 𝔻 (see [9, §2.4] for its description). Denote by F_{k} the extension of F generated by the Fourier coefficients of f_{k}. Then there is an element Φ ∈ H^{1}(Γ_{0}, 𝔻) and maps ρ_{k} : H^{1}(Γ_{0}, 𝔻) ⟶ H^{1}(Γ, F_{k}) such that ρ(k)(Φ) = ϕ_{k}, the cohomology class associated to f_{k}^{JL}, with k in a neighborhood of k_{0} (for this we need a suitable normalization of the cohomology class associated to f_{k}^{JL}, which we do not touch for simplicity in this introduction). We view Φ as a quaternionic family of modular forms. To each ϕ_{k} we may apply the Shimura-Shintani-Waldspurger lifting ([19]) and obtain a modular form h_{k} of weigh k + 1/2, level 4Np and trivial character. We show that this collection of forms can be p-adically interpolated. For clarity’s sake, we present the liftings and their Λ-adic variants in a diagram, in which the horizontal maps are specialization maps of the p-adic family to weight k; JL stands for the Jacquet-Langlands correspondence; SSW stands for the Shimura-Shintani-Waldspurger lift; and the dotted arrows are constructed in this paper:

[figure: f_{∞}; Λ−adic JL; f_{k}; JL; Φ; ρ_{k}; Λ−adic SSW; ϕ_{k}; SSW; Θ; h_{k}]

More precisely, as a particular case of our main result, Theorem 3.8, we get the following

Theorem 1.1. There exists a p-adic neighborhood U_{0} of k_{0} in Hom(ℤ_{p}^{×}, ℤ_{p}^{×}), p-adic periods Ω_{k} for k ∈ U_{0} ∩ ℤ^{≥2} and a formal expansion

Θ = ∑_{ξ≥1} a_{ξ}q^{ξ}

with coefficients a_{ξ} in the ring of ℂ_{p}-valued functions on U_{0}, such that for all k ∈ U_{0} ∩ ℤ^{≥2} we have

Θ(k) = Ω_{k} · h_{k}.

Further, Ω_{k_{0}} ≠ 0.

2. Shintani integrals and Fourier coefficients of half-integral weight modular forms

We express the Fourier coefficients of half-integral weight modular forms in terms of period integrals, thus allowing a cohomological interpretation which is key to the production of the Λ-adic version of the Shimura-Shintani-Waldspurger correspondence. For the quaternionic Shimura-Shintani-Waldspurger correspondence of interest to us (see [15], [22]), the period integrals expressing the values of the Fourier coefficients have been computed generally by Prasanna in [16].

2.1. The Shimura-Shintani-Waldspurger lifting. Let 4M be a positive integer, 2k an even non-negative integer and χ a Dirichlet character modulo 4M such that χ(−1) = 1. Recall that the space of half-integral weight modular forms S_{k+1/2}(4M, χ) consists of holomorphic cuspidal functions h on the upperhalf place ℌ such that

h(γ(z)) = j^{1/2}(γ, z)^{2k+1}χ(d)h(z),

for all γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(4M), where j^{1/2}(γ, z) is the standard square root of the usual automorphy factor j(γ, z) (cf. [15, 2.3]). To any quaternionic integral weight modular form we may associate a halfintegral weight modular form following Shimura’s work [19], as we will describe below. Fix an odd square free integer N and a factorization N = M · D into coprime integers such that D > 1 is a product of an even number of distinct primes. Fix a Dirichlet character ψ modulo M and a positive even integer 2k. Suppose that

ψ(−1) = (−1)^{k}.

Define the Dirichlet character χ modulo 4N by

χ(x) := ψ(x) (\frac{−1}{x})^{k} .

Let B be an indefinite quaternion algebra over ℚ of discriminant D. Fix a maximal order O_{B} of B. For every prime ℓ|M, choose an isomorphism

i_{ℓ} : B ⊗_{ℚ} ℚ_{ℓ} ≃ 𝕄_{2}(ℚ_{ℓ})

such that i_{ℓ}(O_{B} ⊗_{ℤ} ℤ_{ℓ}) = 𝕄_{2}(ℤ_{ℓ}). Let R ⊆ O_{B} be the Eichler order of B of level M defined by requiring that i_{ℓ}(R ⊗_{ℤ} ℤ_{ℓ}) is the suborder of 𝕄_{2}(ℤ_{ℓ}) of upper triangular matrices modulo ℓ for all ℓ|M. Let Γ denote the subgroup of the group R_{1}^{×} of norm 1 elements in R^{×} consisting of those γ such that i_{ℓ}(γ) ≡ ( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ) mod ℓ for all ℓ|M. We denote by S_{2k}(Γ) the ℂ-vector space of weight 2k modular forms on Γ, and by S_{2k}(Γ, ψ^{2}) the subspace of S_{2k}(Γ) consisting of forms having character ψ^{2} under the action of R_{1}^{×}. Fix a Hecke eigenform

f ∈ S_{2k}(Γ, ψ^{2})

as in [19, Section 3]. Let V denote the ℚ-subspace of B consisting of elements with trace equal to zero. For any v ∈ V, which we view as a trace zero matrix in 𝕄_{2}(ℝ) (after fixing an isomorphism i_{∞} : B ⊗ ℝ ≃ 𝕄_{2}(ℝ)), set

G_{v} := {γ ∈ SL_{2}(ℝ)| γ^{−1}vγ = v}

and put Γ_{v} := G_{v} ∩ Γ. One can show that there exists an isomorphism

ω : ℝ^{×} ⟶^{∼} G_{v}

defined by ω(s) = β^{−1}( \begin{matrix}s & 0 \\ 0 & s^{−1} \end{matrix} )β, for some β ∈ SL_{2}(ℝ). Let 𝔱_{v} be the order of Γ_{v} ∩ {±1} and let γ_{v} be an element of Γ_{v} which generates Γ_{v} {±1} / {±1}. Changing γ_{v} to γ_{v}^{−1} if necessary, we may assume γ_{v} = ω(t) with t > 0. Define V^{∗} to be the ℚ-subspace of V consisting of elements with strictly negative norm. For any α = ( \begin{matrix}a & b \\ c & −a \end{matrix} ) ∈ V^{∗} and z ∈ H, define the quadratic form

Q_{α}(z) := cz^{2} − 2az − b.

Fix τ ∈ H and set

P (f, α, Γ) := −(2(−nr(α))^{1/2}/𝔱_{α}) ∫_{τ}^{γ_{α}(τ)} Q_{α}(z)^{k−1}f(z)dz

where nr : B → ℚ is the norm map. By [19, Lemma 2.1], the integral is independent on the choice τ, which justifies the notation.

Remark 2.1. The definition of P (f, α, Γ) given in [19, (2.5)] looks different: the above expression can be derived as in [19, page 629] by means of [19, (2.20) and (2.22)].

Let R(Γ) denote the set of equivalence classes of V^{∗} under the action of Γ by conjugation. By [19, (2.6)], P (f, α, Γ) only depends on the conjugacy class of α, and thus, for C ∈ R(Γ), we may define P (f, C, Γ) := P (f, α, Γ) for any choice of α ∈ C. Also, q(C) := −nr(α) for any α ∈ C. Define O′_{B} to be the maximal order in B such that O′_{B} ⊗_{ℤ} ℤ_{ℓ} ≃ O_{B} ⊗_{ℤ} ℤ_{ℓ} for all ℓ ∤ M and O′_{B} ⊗_{ℤ} ℤ_{ℓ} is equal to the local order of B ⊗_{ℚ} ℚ_{ℓ} consisting of elements γ such that i_{ℓ}(γ) = ( \begin{matrix}a & b/M \\ cM & d \end{matrix} ) with a, b, c, d ∈ ℤ_{ℓ}, for all ℓ|M. Given α ∈ O′_{B}, we can find an integer b_{α} such that

(1)  i_{ℓ}(α) ≡ ( \begin{matrix}∗ & b_{α}/M \\ ∗ & ∗ \end{matrix} ) mod i_{ℓ}(R ⊗_{ℤ} ℤ_{ℓ}), ∀ℓ|M.

Define a locally constant function η_{ψ} on V by η_{ψ}(α) = ψ(b_{α}) if α ∈ O′_{B} ∩V and η(α) = 0 otherwise, with ψ(a) = 0 if (a, M) ≠ 1 (for the definition of locally constant functions on V in this context, we refer to [19, p. 611]). For any C ∈ R(Γ), fix α_{C} ∈ C. For any integer ξ ≥ 1, define

a_{ξ}(h̃) := (2μ(Γ\ℌ))^{−1} · ∑_{C∈R(Γ),q(C)=ξ} η_{ψ}(α_{C})ξ^{−1/2}P (f, C, Γ).

Then, by [19, Theorem 3.1],

h̃ := ∑_{ξ≥1} a_{ξ}(h̃)q^{ξ} ∈ S_{k+1/2}(4N, χ)

is called the Shimura-Shintani-Waldspurger lifting of f.

2.2. Cohomological interpretation. We introduce necessary notation to define the action of the Hecke action on cohomology groups; for details, see [9, §2.1]. If G is a subgroup of B^{×} and S a subsemigroup of B^{×} such that (G, S) is an Hecke pair, we let H(G, S) denote the Hecke algebra corresponding to (G, S), whose elements are written as T (s) = GsG = ∐_{i} Gs_{i} for s, s_{i} ∈ S (finite disjoint union). For any s ∈ S, let s^{∗} := norm(s)s^{−1} and denote by S^{∗} the set of elements of the form s^{∗} for s ∈ S. For any ℤ[S^{∗}]-module M we let T (s) act on H^{1}(G, M) at the level of cochains c ∈ Z^{1}(G, M) by the formula (c|T (s))(γ) = ∑_{i} s_{i}^{∗}c(t_{i}(γ)), where t_{i}(γ) are defined by the equations Gs_{i}γ = Gs_{j} and s_{i}γ = t_{i}(γ)s_{j}. In the following, we will consider the case of G = Γ and

S = \{s ∈ B^{×}|i_{ℓ}(s) is congruent to ( \begin{matrix}1 & ∗ \\ 0 & ∗ \end{matrix} ) mod ℓ for all ℓ|M\}.

For any field L and any integer n ≥ 0, let V_{n}(L) denote the L-dual of the L-vector space P_{n}(L) of homogeneous polynomials in 2 variables of degree n. We let 𝕄_{2}(L) act from the right on P (x, y) as P |γ(x, y) := P (γ(x, y)), where for γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) we put

γ(x, y) := (ax + yb, cx + dy).

This also equips V_{n}(L) with a left action by γ · φ(P) := φ(P |γ). To simplify the notation, we will write P (z) for P (z, 1). Let F denote the finite extension of ℚ generated by the eigenvalues of the Hecke action on f. For any field K containing F, set

𝕎_{f}(K) := H^{1}(Γ, V_{k−2}(K))^{f}

where the superscript f denotes the subspace on which the Hecke algebra acts via the character associated with f. Also, for any sign ±, let 𝕎_{f}^{±}(K) denote the ±-eigenspace for the action of the archimedean involution ι. Remember that ι is defined by choosing an element ω_{∞} of norm −1 in R^{×} such that such that i_{ℓ}(ω_{∞}) ≡ ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) mod M for all primes ℓ|M and then setting ι := T (w_{∞}) (see [9, §2.1]). Then 𝕎_{f}^{±}(K) is one dimensional (see, e.g., [9, Proposition 2.2]); fix a generator ϕ_{f}^{±} of 𝕎_{f}^{±}(F). To explicitly describe ϕ_{f}^{±}, let us introduce some more notation. Define

f|ω_{∞}(z) := (Cz + D)^{−k/2}\overline{f(ω_{∞}(z̄))}

where i_{∞}(ω_{∞}) = ( \begin{matrix}A & B \\ C & D \end{matrix} ). Then f|ω_{∞} ∈ S_{2k}(Γ) as well. If the eigenvalues of the Hecke action on f are real, then we may assume, after multiplying f by a scalar, that f|ω_{∞} = f (see [19, p. 627] or [10, Lemma 4.15]). In general, let I(f) denote the class in H^{1}(Γ, V_{k−2}(ℂ)) represented by the cocycle

γ ⟼ [P ↦ I_{γ}(f)(P) := ∫_{τ}^{γ(τ)} f(z)P (z)dz]

for any τ ∈ H (the corresponding class is independent on the choice of τ). With this notation,

P (f, α, Γ) = −(2(−nr(α))^{1/2}/𝔱_{α}) · I_{γ_{α_{C}}} (f)(Q_{α_{C}}(z)^{k−1}).

Denote by I^{±}(f) := (1/2) · I(f) ± (1/2) · I(f)|ω_{∞}, the projection of I(f) to the eigenspaces for the action of ω_{∞}. Then I(f) = I^{+}(f)+I^{−}(f) and I_{f}^{±} = Ω_{f}^{±}·ϕ_{f}^{±}, for some Ω_{f}^{±} ∈ ℂ^{×}. Given α ∈ V^{∗} of norm −ξ, put α′ := ω_{∞}^{−1}αω_{∞}. By [19, 4.19], we have

η(α)ξ^{−1/2}P (f, α, Γ) + η(α′)ξ^{−1/2}P (f, α′, Γ) = −η(α) · 𝔱_{α}^{−1} · I_{γ_{α}}^{+}(Q_{α_{C}}(z)^{k−1}).

We then have

a_{ξ}(h̃) = ∑_{C∈R_{2}(Γ),q(C)=ξ} \frac{−η_{ψ}(α_{C})}{2μ(Γ\H) · 𝔱_{α_{C}}} · I_{γ_{α_{C}}}^{+} (Q_{α_{C}}(z)^{k−1}).

We close this section by choosing a suitable multiple of h which will be the object of the next section. Given Q_{α}(z) = cz^{2} − 2az − b as above, with α in V^{∗}, define Q̃_{α}(z) := M · Q_{α}(z). Then, clearly, I^{±}(f)(Q̃_{α_{C}}(z)^{k−1}) is equal to M^{k−1}I^{±}(f)(Q_{α_{C}}(z)^{k−1}). We thus normalize the Fourier coefficients by setting (2)

a_{ξ}(h) := −\frac{a_{ξ}(h̃) · M^{k−1} · 2μ(Γ\H)}{Ω_{f}^{+}} = ∑_{C∈R(Γ),q(C)=ξ} \frac{η_{ψ}(α_{C})}{𝔱_{α_{C}}} · ϕ_{f}^{+}(Q̃_{α_{C}}(z)^{k−1}).

So

(3)  h := ∑_{ξ≥1} a_{ξ}(h)q^{ξ}

belongs to S_{k+1/2}(4N, χ) and is a non-zero multiple of h̃.

Documenta Mathematica 18 (2013) 1–21

3. The Λ-adic Shimura-Shintani-Waldspurger correspondence

At the heart of Stevens’s proof lies the control theorem of Greenberg-Stevens, which has been worked out in the quaternionic setting by Longo–Vigni [9]. Recall that N ≥ 1 is a square free integer and fix a decomposition N = M · D where D is a square free product of an even number of primes and M is coprime to D. Let p ∤ N be a prime number and fix an embedding ℚ̄ ↪ ℚ̄_{p}.

3.1. The Hida Hecke algebra. Fix an ordinary p-stabilized newform

(4)  f_{0} ∈ S_{k_{0}}(Γ_{1}(Mp^{r_{0}}) ∩ Γ_{0}(D), ϵ_{0})

of level Γ_{1}(Mp^{r_{0}}) ∩ Γ_{0}(D), Dirichlet character ϵ_{0} and weight k_{0}, and write O for the ring of integers of the field generated over ℚ_{p} by the Fourier coefficients of f_{0}. Let Λ (respectively, O[[ℤ_{p}^{×}]]) denote the Iwasawa algebra of W := 1 + pℤ_{p} (respectively, ℤ_{p}^{×}) with coefficients in O. We denote group-like elements in Λ and O[[ℤ_{p}^{×}]] as [t]. Let 𝔥_{∞}^{ord} denote the p-ordinary Hida Hecke algebra with coefficients in O of tame level Γ_{1}(N). Denote by L := Frac(Λ) the fraction field of Λ. Let R denote the integral closure of Λ in the primitive component K of 𝔥_{∞}^{ord} ⊗_{Λ} L corresponding to f_{0}. It is well known that the Λ-algebra R is finitely generated as Λ-module. Denote by X the O-module Hom_{O-alg}^{cont}(R, ℚ̄_{p}) of continuous homomorphisms of O-algebras. Let X^{arith} the set of arithmetic homomorphisms in X, defined in [9, §2.2] by requiring that the composition

W ⟶Λ ⟶^{κ} ℚ̄_{p}

has the form γ ↦ ψ_{κ}(γ)γ^{n_{κ}} with n_{κ} = k_{κ} − 2 for an integer k_{κ} ≥ 2 (called the weight of κ) and a finite order character ψ_{κ} : W → ℚ̄_{p} (called the wild character of κ). Denote by r_{κ} the smallest among the positive integers t such that 1 + p^{t}ℤ_{p} ⊆ ker(ψ_{κ}). For any κ ∈ X^{arith}, let P_{κ} denote the kernel of κ and R_{P_{κ}} the localization of R at κ. The field F_{κ} := R_{P_{κ}}/P_{κ}R_{P_{κ}} is a finite extension of Frac(O). Further, by duality, κ corresponds to a normalized eigenform

f_{κ} ∈ S_{k_{κ}}(Γ_{0}(Np^{r_{κ}}), ϵ_{κ})

for a Dirichlet character ϵ_{κ} : (ℤ/Np^{r_{κ}}ℤ)^{×} → ℚ̄_{p}. More precisely, if we write ψ_{R} for the character of R, defined as in [6, Terminology p. 555], and we let ω denote the Teichmüller character, we have ϵ_{κ} := ψ_{κ} · ψ_{R} · ω^{−n_{κ}} (see [6, Cor. 1.6]). We call (ϵ_{κ}, k_{κ}) the signature of κ. We let κ_{0} denote the arithmetic character associated with f_{0}, so f_{0} = f_{κ_{0}}, k_{0} = k_{κ_{0}}, ϵ_{0} = ϵ_{κ_{0}}, and r_{0} = r_{κ_{0}}. The eigenvalues of f_{κ} under the action of the Hecke operators T_{n} (n ≥ 1 an integer) belong to F_{κ}. Actually, one can show that f_{κ} is a p-stabilized newform on Γ_{1}(Mp^{r_{κ}}) ∩ Γ_{0}(D). Let Λ_{N} denote the Iwasawa algebra of ℤ_{p}^{×} × (ℤ/Nℤ)^{×} with coefficients in O. To simplify the notation, define Δ := (ℤ/Npℤ)^{×}. We have a canonical isomorphism of rings Λ_{N} ≃ Λ[Δ], which makes Λ_{N} a Λ-algebra, finitely generated as Λ-module. Define the tensor product of Λ-algebras

R_{N} := R ⊗_{Λ} Λ_{N},

which is again a Λ-algebra (resp. Λ_{N}-algebra) finitely generated as a Λ-module, (resp. as a Λ_{N}-module). One easily checks that there is a canonical isomorphism of Λ-algebras

R_{N} ≃ R[Δ]

(where Λ acts on R); this is also an isomorphism of Λ_{N}-algebras, when we let Λ_{N} ≃ Λ[Δ] act on R[Δ] in the obvious way. We can extend any κ ∈ X^{arith} to a continuous O-algebra morphism

κ_{N} : R_{N} ⟶ ℚ̄_{p}

setting

κ_{N} (∑_{i=1}^{n} r_{i} · δ_{i}) := ∑_{i=1}^{n} κ(r_{i}) · ψ_{R}(δ_{i})

for r_{i} ∈ R and δ_{i} ∈ Δ. Therefore, κ_{N} restricted to ℤ_{p}^{×} is the character t ↦ ϵ_{κ}(t)t^{n_{κ}}. If we denote by X_{N} the O-module of continuous O-algebra homomorphisms from R_{N} to ℚ̄_{p}, the above correspondence sets up an injective map X^{arith} ↪ X_{N}. Let X_{N}^{arith} denote the image of X^{arith} under this map. For κ_{N} ∈ X_{N}^{arith}, we define the signature of κ_{N} to be that of the corresponding κ.

3.2. The control theorem in the quaternionic setting. Recall that B/ℚ is a quaternion algebra of discriminant D. Fix an auxiliary real quadratic field F such that all primes dividing D are inert in F and all primes dividing Mp are split in F, and an isomorphism i_{F} : B ⊗_{ℚ} F ≃ 𝕄_{2}(F). Let O_{B} denote the maximal order of B obtained by taking the intersection of B with 𝕄_{2}(O_{F}), where O_{F} is the ring of integers of F. More precisely, define

O_{B} := ι^{−1}(i_{F}^{−1}(i_{F} (B ⊗ 1) ∩ 𝕄_{2}(O_{F})))

where ι : B ↪ B ⊗_{ℚ} F is the inclusion defined by b ↦ b ⊗ 1. This is a maximal order in B because i_{F} (B ⊗ 1) ∩ 𝕄_{2}(O_{F}) is a maximal order in i_{F} (B ⊗ 1). In particular, i_{F} and our fixed embedding of ℚ̄ into ℚ̄_{p} induce an isomorphism

i_{p} : B ⊗_{ℚ} ℚ_{p} ≃ 𝕄_{2}(ℚ_{p})

such that i_{p}(O_{B} ⊗_{ℤ} ℤ_{p}) = 𝕄_{2}(ℤ_{p}). For any prime ℓ|M, also choose an embedding ℚ̄ ↪ ℚ̄_{ℓ} which, composed with i_{F}, yields isomorphisms

i_{ℓ} : B ⊗_{ℚ} ℚ_{ℓ} ≃ 𝕄_{2}(ℚ_{ℓ})

such that i_{p}(O_{B} ⊗_{ℤ} ℤ_{ℓ}) = 𝕄_{2}(ℤ_{ℓ}). Define an Eichler order R ⊆ O_{B} of level M by requiring that for all primes ℓ|M the image of R ⊗_{ℤ} ℤ_{ℓ} via i_{ℓ} consists of upper triangular matrices modulo ℓ. For any r ≥ 0, let Γ_{r} denote the subgroup of the group R_{1}^{×} of norm-one elements in R consisting of those γ such that i_{ℓ}(γ) = ( \begin{matrix}a & b \\ c & d \end{matrix} ) with c ≡ 0 mod Mp^{r} and a ≡ d ≡ 1 mod Mp^{r}, for all primes ℓ|Mp. To conclude this list of notation and definitions, fix an embedding F ↪ ℝ and let

i_{∞} : B ⊗_{ℚ} ℝ ≃ 𝕄_{2}(ℝ)

be the induced isomorphism. Let 𝕐 := ℤ_{p}^{2} and denote by 𝕏 the set of primitive vectors in 𝕐. Let 𝔻 denote the O-module of O-valued measures on 𝕐 which are supported on 𝕏. Note that 𝕄_{2}(ℤ_{p}) acts on 𝕐 by left multiplication; this induces an action of 𝕄_{2}(ℤ_{p}) on the O-module of O-valued measures on 𝕐, which induces an action on 𝔻. The group R^{×} acts on 𝔻 via i_{p}. In particular, we may define the group:

𝕎 := H^{1}(Γ_{0}, 𝔻).

Then 𝔻 has a canonical structure of O[[ℤ_{p}^{×}]]-module, as well as 𝔥_{∞}^{ord}-action, as described in [9, §2.4]. In particular, let us recall that, for any [t] ∈ O[[ℤ_{p}^{×}]], we have

∫_{𝕏} φ(x, y)d([t] · ν) = ∫_{𝕏} φ(tx, ty)dν,

for any locally constant function φ on 𝕏. For any κ ∈ X^{arith} and any sign ± ∈ {−, +}, set

𝕎_{κ}^{±} := 𝕎_{f_{κ}^{JL}}^{±}(F_{κ}) = H^{1}(Γ_{r_{κ}}, V_{n_{κ}}(F_{κ}))^{f_{κ},±}

where f_{κ}^{JL} is any Jacquet-Langlands lift of f_{κ} to Γ_{r_{κ}}; recall that the superscript f_{κ} denotes the subspace on which the Hecke algebra acts via the character associated with f_{κ}, and the superscript ± denotes the ±-eigenspace for the action of the archimedean involution ι. Also, recall that 𝕎_{κ}^{±} is one dimensional and fix a generator ϕ_{κ}^{±} of it. We may define specialization maps

ρ_{κ} : 𝔻 ⟶ V_{n_{κ}}(F_{κ})

by the formula

(5)  ρ_{κ}(ν)(P) := ∫_{ℤ_{p}×ℤ_{p}^{×}} ϵ_{κ}(y)P (x, y)dν

which induces (see [9, §2.5]) a map:

ρ_{κ} : 𝕎^{ord} ⟶ 𝕎_{κ}^{ord}.

Here 𝕎^{ord} and 𝕎_{κ}^{ord} denote the ordinary submodules of 𝕎 and 𝕎_{κ}, respectively, defined as in [3, Definition 2.2] (see also [9, §3.5]). We also let 𝕎_{R} := 𝕎 ⊗_{Λ} R, and extend the above map ρ_{κ} to a map

ρ_{κ} : 𝕎_{R}^{ord} ⟶ 𝕎_{κ}^{ord}

by setting ρ_{κ}(x ⊗ r) := ρ_{κ}(x) · κ(r).

Theorem 3.1. There exists a p-adic neighborhood U_{0} of κ_{0} in X, elements Φ^{±} in 𝕎_{R}^{ord} and choices of p-adic periods Ω_{κ}^{±} ∈ F_{κ} for κ ∈ U_{0} ∩ X^{arith} such that, for all κ ∈ U_{0} ∩ X^{arith}, we have

ρ_{κ}(Φ^{±}) = Ω_{κ}^{±} · ϕ_{κ}^{±}

and Ω_{κ_{0}}^{±} ≠ 0.

Proof. This is an easy consequence of [9, Theorem 2.18] and follows along the lines of the proof of [21, Theorem 5.5], cf. [10, Proposition 3.2]. □

We now normalize our choices as follows. With U_{0} as above, define

U_{0}^{arith} := U_{0} ∩ X^{arith}.

Fix κ ∈ U_{0}^{arith} and an embedding ℚ̄_{p} ↪ ℂ. Let f_{κ}^{JL} denote a modular form on Γ_{r_{κ}} corresponding to f_{κ} by the Jacquet-Langlands correspondence, which is well defined up to elements in ℂ^{×}. View ϕ_{κ}^{±} as an element in H^{1}(Γ_{r_{κ}}, V_{n}(ℂ))^{±}. Choose a representative Φ_{γ}^{±} of Φ^{±}, by which we mean that if Φ^{±} = ∑_{i} Φ_{i}^{±} ⊗r_{i}, then we choose a representative Φ_{i,γ}^{±} for all i. Also, we will write ρ_{κ}(Φ)(P) as

∫_{ℤ_{p}×ℤ_{p}^{×}} ϵ_{κ}(y)P (x, y)dΦ_{γ}^{±} := ∑_{i} κ(r_{i}) · ∫_{ℤ_{p}×ℤ_{p}^{×}} ϵ_{κ}(y)P (x, y)dΦ_{i,γ}^{±}.

With this notation, we see that the two cohomology classes

γ ⟼ ∫_{ℤ_{p}×ℤ_{p}^{×}} ϵ_{κ}(y)P (x, y)dΦ_{γ}^{±}(x, y)

and

γ ⟼ Ω_{κ}^{±} · ∫_{τ}^{γ(τ)} P (z, 1)f_{κ}^{JL,±}(z)dz

are cohomologous in H^{1}(Γ_{r_{κ}}, V_{n_{κ}}(ℂ)), for any choice of τ ∈ H.

3.3. Metaplectic Hida Hecke algebras. Let σ : Λ_{N} → Λ_{N} be the ring homomorphism associated to the group homomorphism t ↦ t^{2} on ℤ_{p}^{×}×(ℤ/Nℤ)^{×}, and denote by the same symbol its restriction to Λ and O[[ℤ_{p}^{×}]]. We let Λ_{σ}, O[[ℤ_{p}^{×}]]_{σ} and Λ_{N,σ} denote, respectively, Λ, O[[ℤ_{p}^{×}]] and Λ_{N} viewed as algebras over themselves via σ. The ordinary metaplectic p-adic Hida Hecke algebra we will consider is the Λ-algebra

R̃ := R ⊗_{Λ} Λ_{σ}.

Define as above

X̃ := Hom_{O-alg}^{cont}(R̃, ℚ̄_{p})

and let the set X̃^{arith} of arithmetic points in X̃ to consist of those κ̃ such that the composition

\xymatrix{W \ar@{^{(}->}[r] & Λ \ar@{^{(}->}[r]^{λ↦1⊗λ} & R̃ \ar@{^{(}->}[r]^{κ̃} & ℚ̄_{p}}

has the form γ ↦ ψκ̃(γ)γ^{nκ̃} with nκ̃ := kκ̃ − 2 for an integer kκ̃ ≥ 2 (called the weight of κ̃) and a finite order character ψκ̃ : W → ℚ̄ (called the wild character of κ̃). Let rκ̃ the smallest among the positive integers t such that 1 + p^{t}ℤ_{p} ⊆ ker(ψκ̃). We have a map p : X̃ → X induced by pull-back from the canonical map R → R̃. The map p restricts to arithmetic points.

As above, define the Λ-algebra (or Λ_{N}-algebra)

(6)  R̃_{N} := R ⊗_{Λ} Λ_{N,σ}

via λ ↦ 1 ⊗ λ. We easily see that R̃_{N} ≃ R̃[Δ] as Λ_{N}-algebras, where we enhance R̃[Δ] with the following structure of Λ_{N} ≃ Λ[Δ]-algebra: for ∑_{i} λ_{i}·δ_{i} ∈ Λ[Δ] (with λ_{i} ∈ Λ and δ_{i} ∈ Δ) and ∑ r_{j} · δ′_{j} ∈ R̃[Δ] (with r_{j} = ∑_{h} r_{j,h} ⊗ λ_{j,h} ∈ R̃, r_{j,h} ∈ R, λ_{j,h} ∈ Λ_{σ}, and δ′_{j} ∈ Δ), we set

( ∑_{i} λ_{i} · δ_{i}) · ( ∑_{j} r_{j} · δ′_{j}) := ∑_{i,j,h} (r_{j,h} ⊗ (λ_{i}λ_{j,h})) · (δ_{i}δ′_{j}).

As above, extend κ̃ ∈ X̃^{arith} to a continuous O-algebra morphism

κ̃_{N} : R̃_{N} ⟶ ℚ̄_{p}

by setting

κ̃_{N} (∑_{i=1}^{n} x_{i} · δ_{i}) := ∑_{i=1}^{n} κ̃(x_{i}) · ψ_{R}(δ_{i})

for x_{i} ∈ R̃ and δ_{i} ∈ Δ, where ψ_{R} is the character of R. If we denote by X̃_{N} the O-module of continuous O-linear homomorphisms from R̃_{N} to ℚ̄_{p}, the above correspondence sets up an injective map X̃^{arith} ↪ X̃_{N} and we let X̃_{N}^{arith} denote the image of X̃^{arith}. Put ϵκ̃ := ψκ̃ · ψ_{R} · ω^{−nκ̃}, which we view as a Dirichlet character of (ℤ/Np^{rκ̃}ℤ)^{×}, and call the pair (ϵκ̃, kκ̃) the signature of κ̃_{N}, where κ̃ is the arithmetic point corresponding to κ̃_{N}. We also have a map p_{N} : X̃_{N} → X_{N} induced from the map R_{N} → R̃_{N} taking r ↦ r ⊗ 1 by pull-back. The map p_{N} also restricts to arithmetic points. The maps p and p_{N} make the following diagram commute:

\xymatrix{X̃^{arith} \ar@{^{(}->}[r] \ar[d]^{p} & X̃_{N}^{arith} \ar[d]^{p_{N}} \\ X^{arith} \ar@{^{(}->}[r] & X_{N}^{arith}}

where the projections take a signature (ϵ, k) to (ϵ^{2}, 2k).

3.4. The Λ-adic correspondence. In this part, we combine the explicit integral formula of Shimura and the fact that the toric integrals can be p-adically interpolated to show the existence of a Λ-adic Shimura-Shintani-Waldspurger correspondence with the expected interpolation property. This follows very closely [21, §6]. Let κ̃_{N} ∈ X̃_{N}^{arith} of signature (ϵκ̃, kκ̃). Let L_{r} denote the order of 𝕄_{2}(F) consisting of matrices ( \begin{matrix}a & b/Mp^{r} \\ Mp^{r}c & d \end{matrix} ) with a, b, c, d ∈ O_{F}. Define

O_{B,r} := ι^{−1}(i_{F}^{−1}(i_{F} (B ⊗ 1) ∩ L_{r}))

Then O_{B,r} is the maximal order introduced in §2.1 (and denoted O′_{B} there) defined in terms of the maximal order O_{B} and the integer Mp^{r}. Also,

S := O_{B} ∩ O_{B,r}

is an Eichler order of B of level Mp containing the fixed Eichler order R of level M. With α ∈ V^{∗} ∩ O_{B,1}, we have

(7)  i_{F} (α) = ( \begin{matrix}a & b/(Mp) \\ c & −a \end{matrix} )

in 𝕄_{2}(F) with a, b, c ∈ O_{F} and we can consider the quadratic forms

Q_{α}(x, y) := cx^{2} − 2axy − (b/(Mp))y^{2},

and

(8)  Q̃_{α}(x, y) := Mp · Q_{α}(x, y) = Mpcx^{2} − 2Mpaxy − by^{2}.

Then Q̃_{α}(x, y) has coefficients in O_{F} and, composing with F ↪ ℝ and letting x = z, y = 1, we recover Q_{α}(z) and Q̃_{α}(z) of §2.1 (defined by means of the isomorphism i_{∞}). Since each prime ℓ|Mp is split in F, the elements a, b, c can be viewed as elements in ℤ_{ℓ} via our fixed embedding ℚ̄ ↪ ℚ̄_{ℓ}, for any prime ℓ|Mp (we will continue writing a, b, c for these elements, with a slight abuse of notation). So, letting b_{α} ∈ ℤ such that i_{ℓ}(α) ≡ ( \begin{matrix}∗ & b_{α}/(Mp) \\ ∗ & ∗ \end{matrix} ) modulo i_{ℓ}(S⊗_{ℤ}ℤ_{ℓ}), for all ℓ|Mp, we have b ≡ b_{α} modulo Mpℤ_{ℓ} as elements in ℤ_{ℓ}, for all ℓ|Mp, and thus we get

(9)  η_{ϵκ̃}(α) = ϵκ̃(b_{α}) = ϵκ̃(b)

for b as in (7). For any ν ∈ 𝔻, we may define an O-valued measure j_{α}(ν) on ℤ_{p}^{×} by the formula:

∫_{ℤ_{p}^{×}} f(t)dj_{α}(ν)(t) := ∫_{ℤ_{p}×ℤ_{p}^{×}} f(Q̃_{α}(x, y))dν(x, y).

for any continuous function f : ℤ_{p}^{×} → ℂ_{p}. Recall that the group of O-valued measures on ℤ_{p}^{×} is isomorphic to the Iwasawa algebra O[[ℤ_{p}^{×}]], and thus we may view j_{α}(ν) as an element in O[[ℤ_{p}^{×}]] (see, for example, [1, §3.2]). In particular, for any group-like element [λ] ∈ O[[ℤ_{p}^{×}]] we have:

∫_{ℤ_{p}^{×}} f(t)d([λ] · j_{α}(ν))(t) = ∫_{ℤ_{p}^{×}} (∫_{ℤ_{p}^{×}} f(ts)d[λ](s)) dj_{α}(ν)(t) = ∫_{ℤ_{p}^{×}} f(λt)dj_{α}(ν)(t).

On the other hand,

∫_{ℤ_{p}×ℤ_{p}^{×}} f(Q̃_{α}(x, y))d(λ · ν) = ∫_{ℤ_{p}×ℤ_{p}^{×}} f(Q̃_{α}(λx, λy))dν = ∫_{ℤ_{p}×ℤ_{p}^{×}} f(λ^{2}Q̃_{α}(x, y))dν

and we conclude that j_{α}(λ · ν) = [λ^{2}] · j_{α}(ν). In other words, j_{α} is a O[[ℤ_{p}^{×}]]- linear map

j_{α} : 𝔻 ⟶ O[[ℤ_{p}^{×}]]_{σ}.

Before going ahead, let us introduce some notation. Let χ be a Dirichlet character modulo Mp^{r}, for a positive integer r, which we decompose accordingly with the isomorphism (ℤ/Np^{r}ℤ)^{×} ≃ (ℤ/Nℤ)^{×} × (ℤ/p^{r}ℤ)^{×} into the product χ = χ_{N} · χ_{p} with χ_{N} : (ℤ/Nℤ)^{×} → ℂ^{×} and χ_{p} : (ℤ/p^{r}ℤ)^{×} → ℂ^{×}. Thus, we will write χ(x) = χ_{N}(x_{N}) · χ_{p}(x_{p}), where x_{N} and x_{p} are the projections of x ∈ (ℤ/Np^{r}ℤ)^{×} to (ℤ/Nℤ)^{×} and (ℤ/p^{r}ℤ)^{×}, respectively. To simplify the notation, we will suppress the N and p from the notation for x_{N} and x_{p}, thus simply writing x for any of the two. Using the isomorphism (ℤ/Nℤ)^{×} ≃ (ℤ/Mℤ)^{×} × (ℤ/Dℤ)^{×}, decompose χ_{N} as χ_{N} = χ_{M} · χ_{D} with χ_{M} and χ_{D} characters on (ℤ/Mℤ)^{×} and (ℤ/Dℤ)^{×}, respectively. In the following, we only need the case when χ_{D} = 1. Using the above notation, we may define a O[[ℤ_{p}^{×}]]-linear map J_{α} : 𝔻 → O[[ℤ_{p}^{×}]] by

J_{α}(ν) = ϵ_{κ̃,M}(b) · ϵ_{κ̃,p}(−1) · j_{α}(ν)

with b as in (7). Set 𝔻_{N} := 𝔻 ⊗_{O[[ℤ_{p}^{×}]]} Λ_{N}, where the map O[[ℤ_{p}^{×}]] → Λ_{N} is induced from the map ℤ_{p}^{×} → ℤ_{p}^{×} × (ℤ/Nℤ)^{×} on group-like elements given by x ↦ x ⊗ 1. Then J_{α} can be extended to a Λ_{N}-linear map J_{α} : 𝔻_{N} → Λ_{N,σ}. Setting 𝔻_{R_{N}} := R_{N} ⊗_{Λ_{N}} 𝔻_{N} and extending by R_{N}-linearity over Λ_{N} we finally obtain a R_{N}-linear map, again denoted by the same symbol,

J_{α} : 𝔻_{R_{N}} ⟶ R̃_{N}.

For ν ∈ 𝔻_{N} and r ∈ R_{N} we thus have

J_{α}(r ⊗ ν) = ϵ_{κ̃,M}(b) · ϵ_{κ̃,p}(−1) · r ⊗ j_{α}(ν).

For the next result, for any arithmetic point κ_{N} ∈ X_{N}^{arith} coming from κ ∈ X^{arith}, extend ρ_{κ} in (5) by R_{N}-linearity over O[[ℤ_{p}^{×}]], to get a map

ρ_{κ_{N}} : 𝔻_{R_{N}} ⟶ V_{n_{κ}}

defined by ρ_{κ_{N}} (r ⊗ ν) := ρ_{κ}(ν) · κ_{N}(r), for ν ∈ 𝔻 and r ∈ R_{N}. To simplify the notation, set

(10)  ⟨ν, α⟩_{κ_{N}} := ρ_{κ_{N}} (ν)(Q̃_{α}^{nκ̃/2}).

The following is essentially [21, Lemma (6.1)].

Lemma 3.2. Let κ̃_{N} ∈ X̃_{N}^{arith} with signature (ϵκ̃, kκ̃) and define κ_{N} := p_{N}(κ̃_{N}). Then for any ν ∈ 𝔻_{R_{N}} we have:

κ̃_{N}(J_{α}(ν)) = η_{ϵκ̃}(α) · ⟨ν, α⟩_{κ_{N}} .

Proof. For ν ∈ 𝔻_{N} and r ∈ R_{N} we have

κ̃_{N}(J_{α}(r ⊗ ν)) = κ̃_{N}(ϵ_{κ̃,M}(b) · ϵ_{κ̃,p}(−1) · r ⊗ j_{α}(ν))
= ϵ_{κ̃,M}(b) · ϵ_{κ̃,p}(−1) · κ̃_{N}(r ⊗ 1) · κ̃_{N}(1 ⊗ j_{α}(ν))
= ϵ_{κ̃,M}(b) · ϵ_{κ̃,p}(−1) · κ_{N}(r) · ∫_{ℤ_{p}^{×}} κ̃_{N}(t)dj_{α}(ν)

and thus, noticing that κ̃_{N} restricted to ℤ_{p}^{×} is κ̃_{N}(t) = ϵ_{κ̃,p}(t)t^{nκ̃}, we have

κ̃_{N}(J_{α}(r⊗ν)) = ϵ_{κ̃,M}(b)·ϵ_{κ̃,p}(−1)·κ_{N}(r) ∫_{ℤ_{p}×ℤ_{p}^{×}} ϵ_{κ̃,p}(Q̃_{α}(x, y))Q̃_{α}(x, y)^{nκ̃/2}dν.

Recalling (8), and viewing a, b, c as elements in ℤ_{p}, we have, for (x, y) ∈ ℤ_{p}×ℤ_{p}^{×},

ϵ_{κ̃,p}(Q̃_{α}(x, y)) = ϵ_{κ̃,p}(−by^{2}) = ϵ_{κ̃,p}(−b)ϵ_{κ̃,p}(y^{2}) = ϵ_{κ̃,p}(−b)ϵ_{κ̃,p}^{2}(y) = ϵ_{κ̃,p}(−b)ϵ_{κ,p}(y).

Thus, since ϵκ̃(−1)^{2} = 1, we get:

κ̃_{N}(J_{α}(r ⊗ ν)) = κ_{N}(r) · ϵ_{κ̃,M}(b) · ϵ_{κ̃,p}(b) · ρ_{κ}(ν)(Q̃_{α}^{nκ̃/2}) = η_{ϵ_{κ}}(α) · ⟨ν, α⟩_{κ_{N}}

where for the last equality use (9) and (10). □

Define

𝕎_{R_{N}} := 𝕎 ⊗_{O[[ℤ_{p}^{×}]]} R_{N},

the structure of O[[ℤ_{p}^{×}]]-module of R_{N} being that induced by the composition of the two maps O[[ℤ_{p}^{×}]] → Λ_{N} → R_{N} described above. There is a canonical map

ϑ : 𝕎_{R_{N}} ⟶ H^{1}(Γ_{0}, 𝔻_{R_{N}})

described as follows: if ν_{γ} is a representative of an element ν in 𝕎 and r ∈ R_{N}, then ϑ(ν ⊗ r) is represented by the cocycle ν_{γ} ⊗ r. For ν ∈ 𝕎_{R_{N}} represented by ν_{γ} and ξ ≥ 1 an integer, define

θ_{ξ}(ν) := ∑_{C∈R(Γ_{1}),q(C)=ξ} \frac{J_{α_{C}}(ν_{γ_{α_{C}}})}{𝔱_{α_{C}}}.

Definition 3.3. For ν ∈ 𝕎_{R_{N}}, the formal Fourier expansion

Θ(ν) := ∑_{ξ≥1} θ_{ξ}(ν)q^{ξ}

in R_{N}[[q]] is called the Λ-adic Shimura-Shintani-Waldspurger lift of ν. For any κ̃ ∈ X̃^{arith}, the formal power series expansion

Θ(ν)(κ̃_{N}) := ∑_{ξ≥1} κ̃_{N}(θ_{ξ}(ν))q^{ξ}

is called the κ̃-specialization of Θ(ν).

There is a natural map

𝕎_{R} ⟶ 𝕎_{R_{N}}

taking ν ⊗ r to itself (use that R has a canonical map to R_{N} ≃ R[Δ], as described above). So, for any choice of sign, Φ^{±} ∈ 𝕎_{R} will be viewed as an element in 𝕎_{R_{N}}. ¿From now on we will use the following notation. Fix κ̃_{0} ∈ X̃^{arith} and put κ_{0} := p(κ̃_{0}) ∈ X^{arith}. Recall the neighborhood U_{0} of κ_{0} in Theorem 3.1. Define Ũ_{0} := p^{−1}(U_{0}) and

Ũ_{0}^{arith} := Ũ_{0} ∩ X̃^{arith}.

For each κ̃ ∈ Ũ_{0}^{arith} put κ = p(κ̃) ∈ U_{0}^{arith}. Recall that if (ϵκ̃, kκ̃) is the signature of κ̃, then (ϵ_{κ}, k_{κ}) := (ϵ2 ; κ̃, 2kκ̃) is that of κ_{0}. For any κ := p(κ̃) as above, we may consider the modular form

f_{κ}^{JL} ∈ S_{k_{κ}}(Γ_{r_{κ}}, ϵ_{κ})

and its Shimura-Shintani-Waldspurger lift

h_{κ} = ∑_{ξ} a_{ξ}(h_{κ})q^{ξ} ∈ S_{k_{κ}+1/2}(4Np^{r_{κ}}, χ_{κ}), where χ_{κ}(x) := ϵκ̃(x) (\frac{−1}{x})^{k_{κ}} ,

normalized as in (2) and (3). For our fixed κ_{0}, recall the elements Φ := Φ^{+} chosen as in Theorem 3.1 and define ϕ_{κ} := ϕ_{κ}^{+} and Ω_{κ} := Ω_{κ}^{+} for κ ∈ U_{0}^{arith}.

Proposition 3.4. For all κ̃ ∈ Ũ_{0}^{arith} such that r_{κ} = 1 we have

κ̃_{N}(θ_{ξ}(Φ)) = Ω_{κ} · a_{ξ}(h_{κ}) and Θ(Φ)(κ̃_{N}) = Ω_{κ} · h_{κ}.

Proof. By Lemma 3.2 we have

κ̃_{N}(θ_{ξ}(Φ)) = ∑_{C∈R(Γ_{1}),q(C)=ξ} \frac{η_{ϵκ̃}(α_{C})}{𝔱_{α_{C}}}ρ_{κ_{N}} (Φ)(Q̃_{α_{C}}^{nκ̃/2}).

Using Theorem 3.1, we get

κ̃_{N}(θ_{ξ}(Φ)) = ∑_{C∈R(Γ_{1}),q(C)=ξ} \frac{η_{ϵκ̃}(α_{C}) · Ω_{κ}}{𝔱_{α_{C}}} ϕ_{κ}(Q̃_{α_{C}}^{k_{κ}−1}).

Now (2) shows the statement on κ̃_{N}(θ_{ξ}(Φ)), while that on Θ(Φ)(κ̃_{N}) is a formal consequence of the previous one. □

Corollary 3.5. Let a_{p} denote the image of the Hecke operator T_{p} in R. Then Θ(Φ)|T_{p}^{2} = a_{p} · Θ(Φ).

Proof. For any κ ∈ X^{arith}, let a_{p}(κ) := κ(T_{p}), which is a p-adic unit by the ordinarity assumption. For all κ̃ ∈ Ũ_{0}^{arith} with r_{κ} = 1, we have

Θ(Φ)(κ̃_{N})|T_{p}^{2} = Ω_{κ} · h_{κ}|T_{p}^{2} = a_{p}(κ) · Ω_{κ} · h_{κ} = a_{p}(κ) · Θ(Φ)(κ̃_{N}).

Consequently,

κ̃_{N}(θ_{ξp^{2}}(Φ)) = a_{p}(κ) · κ̃_{N}(θ_{ξ}(Φ))

for all κ̃ such that r_{κ} = 1. Since this subset is dense in X̃_{N}, we conclude that θ_{ξp^{2}}(Φ) = a_{p} · θ_{ξ}(Φ) and so Θ(Φ)|T_{p}^{2} = a_{p} · Θ(Φ). □

For any integer n ≥ 1 and any quadratic form Q with coefficients in F, write [Q]_{n} for the class of Q modulo the action of i_{F} (Γ_{n}). Define F_{n,ξ} to be the subset of the F -vector space of quadratic forms with coefficients in F consisting of quadratic forms Q̃_{α} such that α ∈ V^{∗} ∩ O_{B,n} and −nr(α) = ξ. Writing δQ̃_{α} for the discriminant of Q_{α}, the above set can be equivalently described as

F_{n,ξ} := {Q̃_{α}| α ∈ V^{∗} ∩ O_{B,n}, δQ̃_{α} = Np^{n}ξ}.

Define F_{n,ξ}/Γ_{n} to be the set {[Q̃_{α}]_{n}| Q̃_{α} ∈ F_{n,ξ}} of equivalence classes of F_{n,ξ} under the action of i_{F} (Γ_{n}). A simple computation shows that Q_{g^{−1}αg} = Q_{α}|g for all α ∈ V^{∗} and all g ∈ Γ_{n}, and thus we find

F_{n,ξ}/Γ_{n} = {[Q̃_{C_{α}}]_{n}| C ∈ R(Γ_{n}), δQ̃_{α} = Np^{n}ξ}.

We also note that, in the notation of §2.1, if f has weight character ψ, defined modulo Np^{n}, and level Γ_{n}, the Fourier coefficients a_{ξ}(h) of the Shimura-Shintani-Waldspurger lift h of f are given by

(11)  a_{ξ}(h) = ∑_{[Q]∈F_{n,ξ}/Γ_{n}} \frac{ψ(Q)}{𝔱_{Q}}ϕ_{f}^{+}(Q(z)^{k−1})

and, if Q = Q̃_{α}, we put ψ(Q) := η_{ψ}(b_{α}) and 𝔱_{Q} := 𝔱_{α}. Also, if we let

F_{n}/Γ_{n} := ∐_{ξ} F_{n,ξ}/Γ_{n}

we can write

(12)  h = ∑_{[Q]∈F_{n}/Γ_{n}} \frac{ψ(Q)}{𝔱_{Q}}ϕ_{f}^{+}(Q(z)^{k−1})q^{δ_{Q}/(Np^{n})}.

Fix now an integer m ≥ 1 and let n ∈ {1, m}. For any t ∈ (ℤ/p^{n}ℤ)^{×} and any integer ξ ≥ 1, define F_{n,ξ,t} to be the subset of F_{n,ξ} consisting of forms such that Np^{n}b_{α} ≡ t mod Np^{m}. Also, define F_{n,ξ,t}/Γ_{n} to be the set of equivalence classes of F_{n,ξ,t} under the action of i_{F} (Γ_{n}). If α ∈ V^{∗} ∩ O_{B,m} and

i_{F} (α) = ( \begin{matrix}a & b \\ c & −a \end{matrix} ) ,

then

(13)  Q̃_{α}(x, y) = Np^{n}cx^{2} − 2Np^{n}axy − Np^{n}by^{2}

from which we see that there is an inclusion F_{m,ξ,t} ⊆ F_{1,ξp^{m−1},t}. If Q̃_{α} and Q̃_{α′} belong to F_{m,ξ,t}, and α′ = gαg^{−1} for some g ∈ Γ_{m}, then, since Γ_{m} ⊆ Γ_{1}, we see that Q̃_{α} and Q̃_{α′} represent the same class in F_{1,ξp^{m−1},t}/Γ_{1}. This shows that [Q̃_{α}]_{m} ↦ [Q̃_{α}]_{1} gives a well-defined map

π_{m,ξ,t} : F_{m,ξ,t}/Γ_{m} ⟶ F_{1,ξp^{m−1},t}/Γ_{1}.

Lemma 3.6. The map π_{m,ξ,t} is bijective.

Proof. We first show the injectivity. For this, suppose Q̃_{α} and Q̃_{α′} are in F_{m,ξ,t} and [Q̃_{α}]_{1} = [Q̃_{α′}]_{1}. So there exists g = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) in i_{F} (Γ_{1}) such that such that Q̃_{α} = Q̃_{α′}|g. If Q̃_{α} = cx^{2} − 2axy − by^{2}, and easy computation shows that Q̃_{α′} = c′x^{2} − 2a′xy − b′y^{2} with

c′ = cα^{2} − 2aαγ − bγ^{2}
a′ = −cαβ + aβγ + aαδ + bγδ
b′ = −cβ^{2} + 2aβδ + bδ^{2}.

The first condition shows that γ ≡ 0 mod Np^{m}. We have b ≡ b′ ≡ t mod Np^{m}, so δ^{2} ≡ 1 mod Np^{m}. Since δ ≡ 1 mod Np, we see that δ ≡ 1 mod Np^{m} too.

We now show the surjectivity. For this, fix [Q̃_{α_{C}}]_{1} in the target of π, and choose a representative

Q̃_{α_{C}} = cx^{2} − 2axy − by^{2}

(recall Mp^{m}ξ|δQ̃_{α_{C}}, Np|c, Np|a, and b ∈ O_{F}^{×}, the last condition due to η_{ψ}(α_{C}) ≠ 0). By the Strong Approximation Theorem, we can find g̃ ∈ Γ_{1} such that

i_{ℓ}(g̃) ≡ ( \begin{matrix}1 & 0 \\ ab^{−1} & 1 \end{matrix} ) mod Np^{m}

for all ℓ|Np. Take g := i_{F} (g̃), and put α := g^{−1}α_{C}g. An easy computation, using the expressions for a′, b′, c′ in terms of a, b, c and g = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) as above, shows that α ∈ V^{∗} ∩ O_{B,m}, η_{ψ}(α) = t and δQ̃_{α} = Np^{m}ξ, and it follows that Q̃_{α} ∈ F_{m,ξ,t}. Now

π([Q̃_{α}]_{m}) = [Q̃_{α}]_{1} = [Q̃_{g^{−1}α_{C}g}]_{1} = [Q̃_{α_{C}}]_{1}

where the last equality follows because g ∈ Γ_{1}. □

Proposition 3.7. For all κ̃ ∈ Ũ_{0}^{arith} we have

Θ(Φ)(κ̃_{N})|T_{p}^{r_{κ}−1} = Ω_{κ} · h_{κ}.

Proof. For r_{κ} = 1, this is Proposition 3.4 above, so we may assume r_{κ} ≥ 2. As in the proof of Proposition 3.4, combining Lemma 3.2 and Theorem 3.1 we get

Θ(Φ)(κ̃_{N}) = ∑_{ξ≥1} ( ∑_{C∈R(Γ_{1}),q(C)=ξ} \frac{η_{ϵκ̃}(α_{C}) · Ω_{κ}}{𝔱_{α_{C}}} ϕ_{κ}(Q̃_{α_{C}}^{k_{κ}−1})) q^{ξ}

which, by (11) and (12) above we may rewrite as

Θ(Φ)(κ̃_{N}) = ∑_{[Q]∈F_{1}/Γ_{1}} \frac{ϵκ̃(Q) · Ω_{κ}}{𝔱_{Q}} ϕ_{κ}(Q^{k_{κ}−1})q^{δ_{Q}/(Np)}

By definition of the action of T_{p} on power series, we have

Θ(Φ)(κ̃_{N})|T_{p}^{r_{κ}−1} = ∑_{[Q]∈F_{1}/Γ_{1}, p^{rκ} |δ_{Q}} \frac{ϵκ̃(Q) · Ω_{κ}}{𝔱_{Q}} ϕ_{κ}(Q^{k_{κ}−1})q^{δ_{Q}/(Np^{rκ})}.

Setting F_{n,t}/Γ_{n} := ∐_{ξ≥1} F_{n,t,ξ}/Γ_{n} for n ∈ {1, r_{κ}}, Lemma 3.6 shows that

F_{1,t}^{∗} := {[Q] ∈ F_{1,t}/Γ_{1,t} such that p^{r_{κ}}|δ_{Q}} is equal to F_{r_{κ},t}.

Therefore, splitting the above sum over t ∈ (ℤ/Np^{r_{κ}}ℤ)^{×}, we get

Θ(Φ)(κ̃_{N})|T_{p}^{r_{κ}−1} = ∑_{t∈(ℤ/p^{rκ−1}ℤ)^{×}} ∑_{[Q]∈F_{1,t}^{∗}} \frac{ϵκ̃(Q) · Ω_{κ}}{𝔱_{Q}} ϕ_{κ}(Q^{k_{κ}−1})q^{δ_{Q}/(Np^{rκ})}

= ∑_{t∈(ℤ/p^{rκ−1}ℤ)^{×}} ∑_{[Q]∈F_{m,t}/Γ_{m}} \frac{ϵκ̃(Q) · Ω_{κ}}{𝔱_{Q}} ϕ_{κ}(Q^{k_{κ}−1})q^{δ_{Q}/(Np^{rκ})}

= ∑_{[Q]∈F_{m}/Γ_{m}} \frac{ϵκ̃(Q) · Ω_{κ}}{𝔱_{Q}} ϕ_{κ}(Q^{k_{κ}−1})q^{δ_{Q}/(Np^{rκ})}.

Comparing this expression with (12) gives the result. □

We are now ready to state the analogue of [21, Thm. 3.3], which is our main result. For the reader’s convenience, we briefly recall the notation appearing below. We denote by X the points of the ordinary Hida Hecke algebra, and by X^{arith} its arithmetic points. For κ_{0} ∈ X^{arith}, we denote by U_{0} the padic neighborhood of κ_{0} appearing in the statement of Theorem 3.1 and put U_{0}^{arith} := U_{0} ∩ X^{arith}. We also denote by Φ = Φ^{+} ∈ 𝕎_{R}^{ord} the cohomology class appearing in Theorem 3.1. The points X̃ of the metaplectic Hida Hecke algebra defined in §3.3 are equipped with a canonical map p : X̃^{arith} → X^{arith} on arithmetic points. Let Ũ_{0}^{arith} := Ũ_{0} ∩ X̃^{arith}. For each κ̃ ∈ Ũ_{0}^{arith}, put κ = p(κ̃) ∈ U_{0}^{arith}. Recall that if (ϵκ̃, kκ̃) is the signature of κ̃, then the signature of κ is (ϵ_{κ}, k_{κ}) := (ϵ2 ; κ̃, 2kκ̃). For any κ := p(κ̃) as above, we may consider the modular form

f_{κ}^{JL} ∈ S_{k_{κ}}(Γ_{r_{κ}}, ϵ_{κ})

and its Shimura-Shintani-Waldspurger lift

h_{κ} = ∑_{ξ} a_{ξ}(h_{κ})q^{ξ} ∈ S_{k_{κ}+1/2}(4Np^{r_{κ}}, χ_{κ}), where χ_{κ}(x) := ϵκ̃(x) (\frac{−1}{x})^{k_{κ}} ,

normalized as in (2) and (3). Finally, for κ̃ ∈ X̃^{arith}, we denote by κ̃_{N} its extension to the metaplectic Hecke algebra R̃_{N} defined in §3.3.

Theorem 3.8. Let κ_{0} ∈ X^{arith}. Then there exists a choice of p-adic periods Ω_{κ} for κ ∈ U_{0} such that the Λ-adic Shimura-Shintani-Waldspurger lift of Φ

Θ(Φ) := ∑_{ξ≥1} θ_{ξ}(Φ)q^{ξ}

in R_{N}[[q]] has the following properties:

(1) Ω_{κ_{0}} ≠ 0.

(2) For any κ̃ ∈ Ũ_{0}^{arith}, the κ̃-specialization of Θ(Φ)

Θ(ν)(κ̃_{N}) := ∑_{ξ≥1} κ̃(θ_{ξ}(Φ))q^{ξ} belongs to S_{k_{κ}+1/2}(4Np^{r_{κ}}, χ′_{κ}),

where χ′_{κ}(x) := χ_{κ}(x) · (\frac{p}{x})^{k_{κ}−1}, and satisfies

Θ(Φ)(κ̃_{N}) = Ω_{κ} · h_{κ}|T_{p}^{1−r_{κ}}.

Proof. The elements Ω_{κ} are those Ω_{κ}^{+} appearing in Theorem 3.1, which we used in Propositions 3.4 and 3.7 above, so (1) is clear. Applying T_{p}^{r_{κ}−1} to the formula of Proposition 3.7, using Corollary 3.5 and applying a_{p}(κ)^{1−r_{κ}} on both sides gives

Θ(Φ)(κ̃_{N}) = a_{p}(κ)^{1−r_{κ}}Ω_{κ} · h_{κ}|T_{p}^{r_{κ}−1}.

By [18, Prop. 1.9], each application of T_{p} has the effect of multiplying the character by (\frac{p}{·}), hence

h′_{κ} := h_{κ}|T_{p}^{r_{κ}−1} ∈ S_{k_{κ}+1/2}(4Np^{r_{κ}}, χ′_{κ}) with χ′_{κ} as in the statement. This gives the first part of (2), while the last formula follows immediately from Proposition 3.7. □

Remark 3.9. Theorem 1.1 is a direct consequence of Theorem 3.8, as we briefly show below. Recall the embedding ℤ^{≥2} ↪ Hom(ℤ_{p}^{×}, ℤ_{p}^{×}) which sends k ∈ ℤ^{≥2} to the character x ↦ x^{k−2}. Extending characters by O-linearity gives a map

ℤ^{≥2} ⟶X (Λ) := Hom_{O-alg}^{cont}(Λ, ℚ̄_{p}).

We denote by k^{(Λ)} the image of k ∈ ℤ^{≥2} in X (Λ) via this embedding. We also denote by ϖ : X → X (Λ) the finite-to-one map obtained by restriction of homomorphisms to Λ. Let k^{(R)} be a point in X of signature (k, 1) such that ϖ(k^{(R)}) = k^{(Λ)}. A well-known result by Hida (see [6, Cor. 1.4]) shows that R/Λ is unramified at k^{(Λ)}. As shown in [21, §1], this implies that there is a section s_{k^{(Λ)}} of ϖ which is defined on a neighborhood U_{k^{(Λ)}} of k^{(Λ)} in X (Λ) and sends k^{(Λ)} to k^{(R)}. Fix now k_{0} as in the statement of Theorem 1.1, corresponding to a cuspform f_{0} of weight k_{0} with trivial character. The form f_{0} corresponds to an arithmetic character k_{0}^{(R)} of signature (1, k_{0}) belonging to X. Let U′_{0} be the inverse image of U_{0} under the section s_{k_{0}^{(Λ)}}^{−1}, where U_{0} ⊆ X is the neighborhood of k_{0}^{(R)}

in Theorem 3.8. Extending scalars by O gives, as above, an injective continuos map Hom(ℤ_{p}^{×}, ℤ_{p}^{×}) ↪ X (Λ), and we let U_{0} be any neighborhood of the character x ↦ x^{k_{0}−2} which maps to U′_{0} and is contained in the residue class of k_{0} modulo p − 1. Composing this map with the section U′_{0} ↪ U_{0} gives a continuous injective map

ς : U_{0} ⟶ U′_{0} ⟶ U_{0}

which takes k_{0} to k_{0}^{(R)}, since by construction the image of k_{0} by the first map is k_{0}^{(Λ)}. We also note that, more generally, ς(k) = k^{(R)} because by construction ς(k) restricts to k^{(Λ)} and its signature is (1, k), since the character of ς(k) is trivial. To show the last assertion, recall that the character of ς(k) is ψ_{k} · ψ_{R} · ω^{−k}, and note that ψ_{k} is trivial because k^{(Λ)}(x) = x^{k−1}, and ψ_{R} · ω^{−k} is trivial because the same is true for k_{0} and k ≡ k_{0} modulo p−1. In other words, arithmetic points in ς(U_{0}) correspond to cuspforms with trivial character. This is the Hida family of forms with trivial character that we considered in the Introduction. We can now prove Theorem 1.1. For all k ∈ U_{0} ∩ ℤ^{≥2}, put Ω_{k} := Ω_{k^{(R)}} and define Θ := Θ(Φ) ∘ ς with Φ as in Theorem 3.8 for κ_{0} = k_{0}^{(R)}. Applying Theorem 3.8 to k_{0}^{(R)}, and restricting to ς(U_{0}), shows that U_{0}, Ω_{k} and Θ satisfy the conclusion of Theorem 1.1.

Remark 3.10. For κ̃ ∈ Ũ_{0}^{arith} of signature (ϵκ̃, kκ̃) with rκ̃ = 1 as in the above theorem, h_{κ} is trivial if (−1)^{kκ̃} ≠ ϵκ̃(−1). However, since ϕ_{κ_{0}} ≠ 0, it follows that h_{κ_{0}} is not trivial as long as the necessary condition (−1)^{k_{0}} = ϵ_{0}(−1) is verified.

Remark 3.11. This result can be used to produce a quaternionic Λ-adic version of the Saito-Kurokawa lifting, following closely the arguments in [8, Cor. 1].

References

[1] J. Coates, R. Sujatha, Cyclotomic fields and zeta values. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2006.

[2] H. Darmon, G. Tornaria, Stark-Heegner points and the Shimura correspondence. Compositio Math., 144 (2008) 1155-1175.

[3] R. Greenberg, G. Stevens, p-adic L-functions and p-adic periods of modular forms. Invent. Math. 111 (1993), no. 2, 407–447.

[4] Koblitz, N., Introduction to elliptic curves and modular forms. Graduate Texts in Mathematics, 97. Springer-Verlag, New York, 1984. viii+248 pp.

[5] W. Kohnen, Fourier coefficients of modular forms of half-integral weight. Math. Ann. 271 (1985), no. 2, 237–268.

[6] Hida H., Galois representations into GL_{2}(ℤ_{p}[[X]]) attached to ordinary cusp forms. Invent. Math. 85 (1986), no. 3, 545–613.

[7] Hida H., On Λ-adic forms of half-integral weight for SL(2)/ℚ. Number Theory (Paris 1992-3). Lond. Math. Soc. Lect. Note Ser.

[8] M. Longo, M.-H. Nicole, The Saito-Kurokawa lifting and Darmon points, to appear in Math. Ann. DOI 10.1007/s00208-012-0846-5

[9] M. Longo, S. Vigni, A note on control theorems for quaternionic Hida families of modular forms, Int. J. Number Theory, 2012 (to appear).

[10] M. Longo, S. Vigni, The rationality of quaternionic Darmon points over genus fields of real quadratic fields, preprint 2011.

[11] J. Nekovář, A. Plater, On the parity of ranks of Selmer groups. Asian J. Math. 4 (2000), no. 2, 437–497.

[12] J. Park, p-adic family of half-integral weight modular forms via overconvergent Shintani lifting Manuscripta Mathematica, Volume 131, 3-4, 2010, 355-384.

[13] A. Popa, Central values of Rankin L-series over real quadratic fields. Compos. Math. 142 (2006), no. 4, 811–866.

[14] K. Prasanna, Integrality of a ratio of Petersson norms and level-lowering congruences. Ann. of Math. (2) 163 (2006), no. 3, 901–967.

[15] K. Prasanna, Arithmetic properties of the Shimura-Shintani-Waldspurger correspondence. With an appendix by Brian Conrad. Invent. Math. 176 (2009), no. 3, 521–600.

[16] K. Prasanna, On the Fourier coefficients of modular forms of half-integral weight. Forum Math. 22 (2010), no. 1, 153–177.

[17] Ramsey, N., The overconvergent Shimura lifting, Int. Math. Res. Not., 2009, no. 2, p. 193-220.

[18] G. Shimura, On modular forms of half integral weight. Ann. of Math. (2) 97 (1973), 440–481.

[19] G. Shimura, The periods of certain automorphic forms of arithmetic type. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 605-632 (1982).

[20] T. Shintani, On construction of holomorphic cusp forms of half integral weight. Nagoya Math. J. 58 (1975), 83–126.

[21] G. Stevens, Λ-adic modular forms of half-integral weight and a Λ-adic Shintani lifting. Arithmetic geometry (Tempe, AZ, 1993), 129–151, Contemp. Math., 174, Amer. Math. Soc., Providence, RI, 1994.

[22] J.-L. Waldspurger, Correspondances de Shimura et quaternions. Forum Math. 3 (1991), no. 3, 219–307.

Matteo Longo Dipartimento di Matematica Università di Padova Via Trieste 63 35121 Padova Italy mlongo@math.unipd.it Marc-Hubert Nicole Institut de mathématiques de Luminy Université d’Aix-Marseille campus de Luminy, case 907 13288 Marseille cedex 9 France nicole@iml.univ-mrs.fr 22

Documenta Mathematica 18 (2013)