THE Λ-ADIC SHIMURA-SHINTANI-WALDSPURGER CORRESPONDENCE

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THE Λ-ADIC SHIMURA-SHINTANI-WALDSPURGER CORRESPONDENCE

Matteo Longo, Marc-Hubert Nicole

To cite this version:

Matteo Longo, Marc-Hubert Nicole. THE

RESPONDENCE. Documenta Mathematica, Universität Bielefeld, 2013, 18, pp.1-21. �hal-01265178�

THE Λ-ADIC SHIMURA-SHINTANI-WALDSPURGER CORRESPONDENCE

MATTEO LONGO, MARC-HUBERT NICOLE

Abstract. We generalize the Λ-adic Shintani lifting for GL2(Q) to indefinite quaternion algebras overQ.

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 ofp-adic variants of Langlands’s functo- riality have been articulated in various special cases. We prove the existence of the Shimura-Shintani-Waldspurger lift forp-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 GL2(Q). This Λ-adic lifting can be seen as a formal power series with coefficients in a finite extension of the Iwasawa algebra Λ :=Zp[[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 algebraB, 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 givenp-adic family of automorphic forms on the quaternion algebra B. The Λ-adic 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 numberpand a positive integerN such thatp-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 7→ x^k−2. Let f_∞ be an Hida family of tame level N passing through a formf₀of level Γ₀(N p) and weightk₀. There is a neighborhoodU ofk₀ in Hom(Z^×p,Z^×p) such that, for anyk∈Z^≥2∩U, the weightk specialization off_∞

Date: January 12, 2013.

2000Mathematics Subject Classification. Primary 11F37, 11F30, 11F85.

1

gives rise to an elementfk ∈Sk(Γ0(N p)). Fix a factorizationN =M DwithD >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 M p contained in the indefinite rational quaternion algebraBof discriminantD. One can show that these modular forms can bep-adically interpolated, up to scaling, in a neighborhood ofk₀. More precisely, let O be the ring of integers of a finite extension F of Qp and let D denote theO-module ofO-valued measures on Z²p which are supported on the set of primitive elements inZ²p. Let Γ₀be the group of norm-one elements in an Eichler order R0 ⊆B containing R. There is a canonical action of Γ0 on D(see [9, §2.4]

for its description). Denote by Fk the extension of F generated by the Fourier coefficients offk. Then there is an element Φ∈H¹(Γ0,D) and maps

ρk:H¹(Γ0,D)−→H¹(Γ, Fk)

such thatρ(k)(Φ) =φk, the cohomology class associated tof_k^JL, withkin a neigh- borhood of k0 (for this we need a suitable normalization of the cohomology class associated tof_k^JL, which we do not touch for simplicity in this introduction). We view Φ as a quaternionic family of modular forms. To eachφk we may apply the Shimura-Shintani-Waldspurger lifting ([19]) and obtain a modular formhkof weigh k+ 1/2, level 4N pand 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 thep-adic family to weightk; JL stands for the Jacquet-Langlands correspondence;

SSW stands for the Shimura-Shintani-Waldspurger lift; and the dotted arrows are constructed in this paper:

f_∞ //

_

Λ−adic JL

f__k

JL

Φ_ ^ρk //

Λ−adic SSW

φ__k

SSW

Θ //hk

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

Theorem 1.1. There exists ap-adic neighborhoodU₀ofk₀inHom(Z^×p,Z^×p),p-adic periods Ωk fork∈U0∩Z^≥2 and a formal expansion

Θ =X

ξ≥1

aξq^ξ

with coefficients aξ in the ring of Cp-valued functions on U0, such that for all k∈U0∩Z^≥2 we have

Θ(k) = Ω_k·h_k. Further, Ωk₀ 6= 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 correspon- dence. For the quaternionic Shimura-Shintani-Waldspurger correspondence of in- terest 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 in- teger, 2k an even non-negative integer and χ a Dirichlet character modulo 4M such thatχ(−1) = 1. Recall that the space of half-integral weight modular forms S_k+1/2(4M, χ) consists of holomorphic cuspidal functionshon the upper-half place Hsuch that

h(γ(z)) =j^1/2(γ, z)^2k+1χ(d)h(z), for all γ = ^{a b}_{c d}

∈ Γ0(4M), where j^1/2(γ, z) is the standard square root of the usual automorphy factorj(γ, z) (cf. [15, 2.3]).

To any quaternionic integral weight modular form we may associate a half- integral 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 thatD >1 is a product of an even number of distinct primes. Fix a Dirichlet characterψmoduloM and a positive even integer 2k. Suppose that

ψ(−1) = (−1)^k. Define the Dirichlet characterχmodulo 4N by

χ(x) :=ψ(x) −1

x k

.

Let B be an indefinite quaternion algebra over Q of discriminant D. Fix a maximal orderOB ofB. For every prime `|M, choose an isomorphism

i<sub>`</sub>:B⊗<sub>Q</sub>Q`'M2(Q`)

such thati`(OB⊗<sub>Z</sub>Z`) =M2(Z`). LetR⊆ OB be the Eichler order ofBof levelM defined by requiring thati`(R⊗_ZZ`) is the suborder ofM2(Z`) of upper triangular matrices modulo`for all`|M. Let Γ denote the subgroup of the groupR^×₁ of norm 1 elements inR^× consisting of thoseγ such thati`(γ)≡ ¹_{0 1}^∗

mod`for all`|M. We denote byS2k(Γ) theC-vector space of weight 2kmodular forms on Γ, and by S2k(Γ, ψ²) the subspace of S2k(Γ) consisting of forms having character ψ² under the action ofR^×₁. Fix a Hecke eigenform

f ∈S2k(Γ, ψ²) as in [19, Section 3].

Let V denote the Q-subspace of B consisting of elements with trace equal to zero. For anyv ∈V, which we view as a trace zero matrix inM2(R) (after fixing an isomorphismi_∞:B⊗R'M2(R)), set

G_v :={γ∈SL₂(R)|γ⁻¹vγ=v}

and put Γv:=Gv∩Γ. One can show that there exists an isomorphismω:R^{× ∼}→Gv

defined by ω(s) = β^{−1 s}_0s−1⁰

β, for some β ∈ SL2(R). Let tv be the order of Γv∩{±1}and letγvbe an element of Γvwhich generates Γv{±1}/{±1}. Changing γv to γ_v⁻¹ if necessary, we may assume γv = ω(t) with t > 0. Define V^∗ to be the Q-subspace of V consisting of elements with strictly negative norm. For any α= ^a_{c −a}^b

∈V^∗ andz∈ H, define the quadratic form Q_α(z) :=cz²−2az−b.

Fixτ∈ Hand set

P(f, α,Γ) :=− 2(−nr(α))^1/2/tα

Z γ_α(τ)

τ

Qα(z)^k−1f(z)dz

where nr :B→Qis the norm map. By [19, Lemma 2.1], the integral is independent on the choiceτ, 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 anyα∈ C.

DefineO⁰_Bto be the maximal order inBsuch thatO_B⁰ ⊗_ZZ`' O<sub>B</sub>⊗<sub>Z</sub>Z`for all

`-M andO<sub>B</sub><sup>0</sup> ⊗<sub>Z</sub>Z` is equal to the local order ofB⊗_QQ`consisting of elementsγ such thati`(γ) = _cM^{a b/M}_d

witha, b, c, d∈Z`, for all`|M. Givenα∈ O_B⁰ , we can find an integerbαsuch that

(1) i`(α)≡

∗ bα/M

∗ ∗

modi`(R⊗<sub>Z</sub>Z`), ∀`|M.

Define a locally constant function ηψ onV byηψ(α) =ψ(bα) if α∈ O⁰_B∩V and η(α) = 0 otherwise, with ψ(a) = 0 if (a, M) 6= 1 (for the definition of locally constant functions onV in this context, we refer to [19, p. 611]).

For anyC ∈R(Γ), fix αC ∈ C. For any integerξ≥1, define aξ(˜h) := 2µ(Γ\H)−1

· X

C∈R(Γ),q(C)=ξ

ηψ(αC)ξ^−1/2P(f,C,Γ).

Then, by [19, Theorem 3.1], h˜:=X

ξ≥1

aξ(˜h)q^ξ ∈S_k+1/2(4N, χ) is called the Shimura-Shintani-Waldspurger lifting off.

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]. IfG is a subgroup ofB^× andSa subsemigroup ofB^× such that (G, S) is an Hecke pair, we letH(G, S) denote the Hecke algebra corresponding to (G, S), whose elements are written asT(s) =GsG=`

iGs_i fors, s_i∈S (finite disjoint union). For any s∈ S, lets^∗ := norm(s)s⁻¹ and denote by S^∗ the set of elements of the form s^∗ for s∈S. For any Z[S^∗]-moduleM we let T(s) act on H¹(G, M) at the level of cochainsc∈Z¹(G, M) by the formula (c|T(s))(γ) =P

is^∗_ic(t_i(γ)), wheret_i(γ) are

defined by the equations Gsiγ=Gsj and siγ=ti(γ)sj. In the following, we will consider the case ofG= Γ and

S={s∈B^×|i`(s) is congruent to ¹₀^∗_∗

mod` for all`|M}.

For any field L and any integer n ≥0, let Vn(L) denote the L-dual of the L- vector spacePn(L) of homogeneous polynomials in 2 variables of degreen. We let M2(L) act from the right onP(x, y) asP|γ(x, y) :=P(γ(x, y)), where forγ= ^{a b}_{c d} we put

γ(x, y) := (ax+yb, cx+dy).

This also equipsVn(L) with a left action by γ·ϕ(P) := ϕ(P|γ). To simplify the notation, we will writeP(z) forP(z,1).

LetF denote the finite extension ofQgenerated by the eigenvalues of the Hecke action onf. For any fieldKcontainingF, set

Wf(K) :=H¹ Γ, Vk−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`(ω_∞) ≡ ^{1 0}₀₋₁

mod M for all primes `|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 generatorφ^±_f of W^±f(F).

To explicitly describeφ^±_f, let us introduce some more notation. Define f|ω∞(z) := (Cz+D)^−k/2f(ω∞(¯z))

where i∞(ω∞) = ^{A B}_{C D}

. Thenf|ω∞ ∈ S2k(Γ) as well. If the eigenvalues of the Hecke action on f are real, then we may assume, after multiplyingf by a scalar, that f|ω∞ =f (see [19, p. 627] or [10, Lemma 4.15]). In general, letI(f) denote the class inH¹(Γ, V_k−2(C)) represented by the cocycle

γ7−→

"

P 7→Iγ(f)(P) :=

Z γ(τ) 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/tα

·Iγ_αC(f) QαC(z)^k−1 .

Denote by I^±(f) := (1/2)·I(f)±(1/2)·I(f)|ω∞, the projection ofI(f) to the eigenspaces for the action ofω∞. ThenI(f) =I⁺(f) +I⁻(f) and I_f^±= Ω^±_f ·φ^±_f, for some Ω^±_f ∈C^×.

Givenα∈V^∗ of norm−ξ, putα⁰:=ω⁻¹_∞αω∞. By [19, 4.19], we have η(α)ξ^−1/2P(f, α,Γ) +η(α⁰)ξ^−1/2P(f, α⁰,Γ) =−η(α)·tα−1·I_γ⁺_α QαC(z)^k−1

. We then have

aξ(˜h) = X

C∈R2(Γ),q(C)=ξ

−ηψ(α_C) 2µ(Γ\H)·tα_C

·I_γ⁺_α

C QαC(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) = cz²−2az−b as above, with α in

V^∗, define ˜Qα(z) := M ·Qα(z). Then, clearly, I^±(f)( ˜QαC(z)^k−1) is equal to M^k−1I^±(f)(QαC(z)^k−1). We thus normalize the Fourier coefficients by setting (2) aξ(h) :=−aξ(˜h)·M^k−1·2µ(Γ\H)

Ω⁺_f = X

C∈R(Γ),q(C)=ξ

ηψ(αC) tα_C

·φ⁺_f Q˜αC(z)^k−1 .

So

(3) h:=X

ξ≥1

aξ(h)q^ξ

belongs toS_k+1/2(4N, χ) and is a non-zero multiple of ˜h.

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 thatN ≥1 is a square free integer and fix a decompositionN =M·D where D is a square free product of an even number of primes and M is coprime toD. Letp-N be a prime number and fix an embedding ¯Q,→Q¯p.

3.1. The Hida Hecke algebra. Fix an ordinaryp-stabilized newform (4) f0∈Sk₀ Γ1(M p^r0)∩Γ0(D), 0

of level Γ1(M p^r0)∩Γ0(D), Dirichlet character0 and weight k0, and write O for the ring of integers of the field generated overQp by the Fourier coefficients off0. Let Λ (respectively,O[[Z^×p]]) denote the Iwasawa algebra ofW := 1+pZp(respec- tively,Z^×p) with coefficients in O. We denote group-like elements in Λ andO[[Z^×p]]

as [t]. Let h^ord_∞ denote the p-ordinary Hida Hecke algebra with coefficients inO of tame level Γ1(N). Denote by L := Frac(Λ) the fraction field of Λ. Let Rdenote the integral closure of Λ in the primitive componentK ofh^ord_∞ ⊗ΛLcorresponding tof₀. It is well known that the Λ-algebra Ris finitely generated as Λ-module.

Denote by X theO-module Hom^cont_O-alg(R,Q¯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 Q¯p

has the formγ7→ψκ(γ)γ^nκwithnκ=kκ−2 for an integerkκ≥2 (called the weight of κ) and a finite order character ψκ : W → Q¯p (called the wild character of κ).

Denote byrκthe smallest among the positive integerstsuch that 1+p^tZp⊆ker(ψκ).

For any κ ∈ X^arith, let Pκ denote the kernel of κand RP_κ the localization of R at κ. The field Fκ := RP_κ/PκRP_κ is a finite extension of Frac(O). Further, by duality,κcorresponds to a normalized eigenform

fκ∈Sk_κ Γ0(N p^rκ), κ

for a Dirichlet character κ : (Z/N p^rκZ)^× → Q¯p. More precisely, if we write ψ_R for the character ofR, defined as in [6, Terminology p. 555], and we letω denote the Teichm¨uller character, we haveκ:=ψκ·ψR·ω^−nκ (see [6, Cor. 1.6]). We call (κ, kκ) the signature of κ. We letκ0 denote the arithmetic character associated withf0, sof0=fκ0,k0=kκ0,0=κ0, andr0=rκ0. The eigenvalues offκ under the action of the Hecke operators T_n (n ≥1 an integer) belong to F_κ. Actually, one can show thatf_κis a p-stabilized newform on Γ₁(M p^rκ)∩Γ₀(D).

Let ΛN denote the Iwasawa algebra ofZ^×p ×(Z/NZ)^× with coefficients inO. To simplify the notation, define ∆ := (Z/N pZ)^×. We have a canonical isomorphism of rings ΛN 'Λ[∆], which makes ΛN a Λ-algebra, finitely generated as Λ-module.

Define the tensor product of Λ-algebras

RN :=R ⊗ΛΛN,

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[∆]

(where Λ acts on R); this is also an isomorphism of ΛN-algebras, when we let Λ_N 'Λ[∆] act onR[∆] in the obvious way.

We can extend anyκ∈ X^arith to a continuousO-algebra morphism κN :RN −→Q¯p

setting

κN n

X

i=1

ri·δi

! :=

n

X

i=1

κ(ri)·ψ_R(δi)

forri∈ Randδi∈∆. Therefore,κN restricted toZ^×p is the charactert7→κ(t)t^nκ. If we denote byXN theO-module of continuousO-algebra homomorphisms from RN to ¯Qp, the above correspondence sets up an injective mapX^arith,→ XN. Let X_N^arith denote the image of X^arith under this map. ForκN ∈ X_N^arith, we define the signature ofκN to be that of the correspondingκ.

3.2. The control theorem in the quaternionic setting. Recall thatB/Qis a quaternion algebra of discriminantD. Fix an auxiliary real quadratic field F such that all primes dividing D are inert in F and all primes dividing M pare split in F, and an isomorphismiF :B⊗_QF 'M2(F). LetOB denote the maximal order ofB obtained by taking the intersection ofB withM2(OF), whereOF is the ring of integers ofF. More precisely, define

O_B :=ι⁻¹ i⁻¹_F i_F(B⊗1)∩M2(O_F)

whereι:B ,→B⊗_QFis the inclusion defined byb7→b⊗1. This is a maximal order inB becauseiF(B⊗1)∩M2(OF) is a maximal order iniF(B⊗1). In particular, i_F and our fixed embedding of ¯Qinto ¯Qp induce an isomorphism

ip:B⊗_QQp'M2(Qp)

such thati_p(O_B⊗_ZZp) =M2(Zp). For any prime`|M, also choose an embedding Q¯ ,→ ¯

Q` which, composed withi<sub>F</sub>, yields isomorphisms i`:B⊗_QQ`'M2(Q`)

such that ip(OB ⊗_ZZ`) =M2(Z`). Define an Eichler order R ⊆ OB of level M by requiring that for all primes`|M the image ofR⊗<sub>Z</sub>Z` viai` consists of upper triangular matrices modulo `. For any r ≥ 0, let Γr denote the subgroup of the groupR^×₁ of norm-one elements inRconsisting of thoseγsuch thati`(γ) = <sup>a b</sup><sub>c d</sub> withc≡0 modM p<sup>r</sup> anda≡d≡1 modM p<sup>r</sup>, for all primes `|M p. To conclude this list of notation and definitions, fix an embeddingF ,→Rand let

i_∞:B⊗_QR'M2(R) be the induced isomorphism.

Let Y:=Z²p and denote byX the set of primitive vectors inY. Let D denote the O-module of O-valued measures on Y which are supported on X. Note that M2(Zp) acts onYby left multiplication; this induces an action of M2(Zp) on the O-module of O-valued measures onY, which induces an action on D. The group R^× acts onDviai_p. In particular, we may define the group:

W:=H¹(Γ₀,D).

Then Dhas a canonical structure ofO[[Z^×p]]-module, as well as h^ord_∞ -action, as de- scribed in [9,§2.4]. In particular, let us recall that, for any [t]∈ O[[Z^×p]], we have

Z

X

ϕ(x, y)d [t]·ν

= Z

X

ϕ(tx, ty)dν, for any locally constant functionϕonX.

For anyκ∈ X^arith and any sign± ∈ {−,+}, set W^±κ :=W^±_fJL

κ (Fκ) =H¹ Γr_κ, Vn_κ(Fκ)^fκ,±

where f_κ^JL is any Jacquet-Langlands lift offκ to Γr_κ; recall that the superscript fκ denotes the subspace on which the Hecke algebra acts via the character asso- ciated 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 generatorφ^±_κ of it.

We may define specialization maps

ρ_κ:D−→V_nκ(F_κ) by the formula

(5) ρκ(ν)(P) :=

Z

Zp×Z^×p

κ(y)P(x, y)dν which induces (see [9,§2.5]) a map:

ρ_κ:W^ord−→W^ordκ .

Here W^ord and W^ordκ denote the ordinary submodules ofW andWκ, respectively, defined as in [3, Definition 2.2] (see also [9,§3.5]). We also letWR:=W⊗ΛR, and extend the above mapρ_κto a map

ρκ:W^ordR −→W^ordκ

by settingρκ(x⊗r) :=ρκ(x)·κ(r).

Theorem 3.1. There exists a p-adic neighborhoodU0of κ₀ inX, elementsΦ^± in W^ordR and choices of p-adic periods Ω^±_κ ∈F_κ forκ∈ U0∩ X^arith such that, for all κ∈ U0∩ X^arith, we have

ρκ(Φ^±) = Ω^±_κ ·φ^±_κ andΩ^±_κ0 6= 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. WithU0 as above, define U₀^arith:=U₀∩ X^arith.

Fix κ ∈ U₀^arith and an embedding ¯Qp ,→ C. Let f_κ^JL denote a modular form on Γ_rκ corresponding to f_κ by the Jacquet-Langlands correspondence, which is well

defined up to elements inC^×. Viewφ^±_κ as an element inH¹(Γr_κ, Vn(C))^±. Choose a representative Φ^±_γ of Φ^±, by which we mean that if Φ^± =P

iΦ^±_i ⊗ri, then we choose a representative Φ^±_i,γ for alli. Also, we will writeρκ(Φ)(P) as

Z

Zp×Z^×p

κ(y)P(x, y)dΦ^±_γ :=X

i

κ(ri)· Z

Zp×Z^×p

κ(y)P(x, y)dΦ^±_i,γ. With this notation, we see that the two cohomology classes

γ7−→

Z

Zp×Z^×p

κ(y)P(x, y)dΦ^±_γ(x, y) and

γ7−→Ω^±_κ · Z γ(τ)

τ

P(z,1)f_κ^JL,±(z)dz are cohomologous inH¹(Γr_κ, Vn_κ(C)), for any choice ofτ∈ H.

3.3. Metaplectic Hida Hecke algebras. Letσ : ΛN →ΛN be the ring homo- morphism associated to the group homomorphismt7→t² on Z^×p ×(Z/NZ)^×, and denote by the same symbol its restriction to Λ andO[[Z^×p]]. We let Λσ,O[[Z^×p]]σand ΛN,σ denote, respectively, Λ, O[[Z^×p]] and ΛN viewed as algebras over themselves viaσ. The ordinary metaplecticp-adic Hida Hecke algebra we will consider is the Λ-algebra

Re:=R ⊗ΛΛσ. Define as above

Xe:= Hom^cont_O-alg(R,e ¯ Q^p)

and let the setXe^arith of arithmetic points inXe to consist of those ˜κsuch that the composition

W  //Λ  ^λ7→1⊗λ //

Re  ^κ˜ //Q¯p

has the form γ 7→ ψκ˜(γ)γ^nκ˜ with n˜κ := k˜κ−2 for an integer k˜κ ≥ 2 (called the weight of ˜κ) and a finite order characterψ˜κ:W →Q¯ (called the wild character of

˜

κ). Letrκ˜ the smallest among the positive integerstsuch that 1 +p^tZp⊆ker(ψκ˜).

We have a mapp:X → Xe induced by pull-back from the canonical mapR →R.e The mapprestricts to arithmetic points.

As above, define the Λ-algebra (or ΛN-algebra)

(6) ReN :=R ⊗ΛΛN,σ

viaλ7→1⊗λ.

We easily see that

ReN 'R[∆]e

as Λ_N-algebras, where we enhanceR[∆] with the following structure of Λe _N 'Λ[∆]- algebra: forP

iλi·δi∈Λ[∆] (withλi∈Λ andδi∈∆) andPrj·δ_j⁰ ∈R[∆] (withe rj=P

hrj,h⊗λj,h∈R,e rj,h∈ R,λj,h∈Λσ, andδ⁰_j∈∆), we set X

i

λi·δi

· X

j

rj·δ⁰_j :=X

i,j,h

rj,h⊗(λiλj,h)

·(δiδ⁰_j).

As above, extend ˜κ∈Xe^arithto a continuousO-algebra morphism ˜κN :ReN →Q¯p

by setting

˜ κ_N

n

X

i=1

x_i·δ_i

! :=

n

X

i=1

˜

κ(x_i)·ψ_R(δ_i)

for xi ∈ Re and δi ∈ ∆, where ψ_R is the character of R. If we denote by XeN

theO-module of continuousO-linear homomorphisms fromReN to ¯Qp, the above correspondence sets up an injective mapXe^arith,→Xe_N and we letXe_N^arith denote the image ofXe^arith. Put_κ˜:=ψ_˜κ·ψ_R·ω^−n˜κ, which we view as a Dirichlet character of (Z/N p^rκ˜Z)^×, and call the pair (_κ˜, k_κ˜) the signature of ˜κ_N, where ˜κis the arithmetic point corresponding to ˜κ_N.

We also have a map p_N :Xe_N → X_N induced from the map R_N →Re_N taking r7→r⊗1 by pull-back. The mapp_N also restricts to arithmetic points. The maps pandp_N make the following diagram commute:

Xe^arith  //

p

Xe_N^arith

p_N

X^arith  //X_N^arith where the projections take a signature (, k) to (²,2k).

3.4. The Λ-adic correspondence. In this part, we combine the explicit inte- gral 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 cor- respondence with the expected interpolation property. This follows very closely [21,

§6].

Let ˜κN ∈Xe_N^arith of signature (κ˜, k˜κ). LetLrdenote the order ofM2(F) consist- ing of matrices _{M p}^arc^{b/M p}d ^r

witha, b, c, d∈ OF. Define OB,r:=ι⁻¹ i⁻¹_F iF(B⊗1)∩Lr

ThenO_B,ris the maximal order introduced in§2.1 (and denotedO⁰_Bthere) defined in terms of the maximal orderO_B and the integerM p^r. Also,S:=O_B∩ O_B,r is an Eichler order ofB of levelM pcontaining the fixed Eichler orderRof level M. Withα∈V^∗∩ OB,1, we have

(7) iF(α) =

a b/(M p)

c −a

inM2(F) witha, b, c∈ OF and we can consider the quadratic forms Q_α(x, y) :=cx²−2axy− b/(M p)

y², and

(8) Q˜α(x, y) :=M p·Qα(x, y) =M pcx²−2M paxy−by².

Then ˜Qα(x, y) has coefficients inOF and, composing withF ,→Rand lettingx=z, y = 1, we recoverQα(z) and ˜Qα(z) of§2.1 (defined by means of the isomorphism i∞). Since each prime `|M p is split in F, the elements a, b, c can be viewed as elements in Z` via our fixed embedding ¯Q ,→ Q¯`, for any prime `|M p (we will continue writing a, b, c for these elements, with a slight abuse of notation). So,

lettingbα∈Zsuch that i`(α)≡ ^∗_∗^{bα/(M p)}_∗

moduloi`(S⊗<sub>Z</sub>Z`), for all`|M p, we haveb≡bα moduloM pZ` as elements inZ`, for all`|M p, and thus we get (9) η_˜κ(α) =˜κ(bα) =˜κ(b)

forbas in (7).

For anyν∈D, we may define anO-valued measurejα(ν) onZ^×p by the formula:

Z

Z^×p

f(t)djα(ν)(t) :=

Z

Zp×Z^×p

f Q˜α(x, y)

dν(x, y).

for any continuous function f : Z^×p → Cp. 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 inO[[Z^×p]] (see, for example, [1, §3.2]). In particular, for any group-like element [λ]∈ O[[Z^×p]] we have:

Z

Z^×p

f(t)d [λ]·jα(ν) (t) =

Z

Z^×p

Z

Z^×p

f(ts)d[λ](s)

!

djα(ν)(t) = Z

Z^×p

f(λt)djα(ν)(t).

On the other hand, Z

Zp×Z^×p

f Q˜α(x, y)

d(λ·ν) = Z

Zp×Z^×p

f Q˜α(λx, λy) dν =

Z

Zp×Z^×p

f λ²Q˜α(x, y) dν

and we conclude thatj_α(λ·ν) = [λ²]·j_α(ν). In other words,j_α is aO[[Z^×p]]-linear map

j_α:D−→ O[[Z^×p]]

σ.

Before going ahead, let us introduce some notation. Letχ be a Dirichlet char- acter moduloM p^r, for a positive integerr, which we decompose accordingly with the isomorphism (Z/N p^rZ)^×'(Z/NZ)^××(Z/p^rZ)^×into the productχ=χN·χp

with χ_N : (Z/NZ)^× → C^× and χ_p : (Z/p^rZ)^× → C^×. Thus, we will write χ(x) =χ_N(x_N)·χ_p(x_p), wherex_N andx_pare the projections ofx∈(Z/N p^rZ)^× to (Z/NZ)^× and (Z/p^rZ)^×, respectively. To simplify the notation, we will suppress the N and p from the notation for x_N and x_p, thus simply writing xfor 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 whenχD= 1.

Using the above notation, we may define aO[[Z^×p]]-linear mapJα:D→ O[[Z^×p]]

by

Jα(ν) =κ,M˜ (b)·˜κ,p(−1)·jα(ν) with b as in (7). Set DN := D⊗_O[[

Z^×p]] ΛN, where the map O[[Z^×p]] → ΛN is induced from the map Z^×p → Z^×p ×(Z/NZ)^× on group-like elements given by x7→x⊗1. ThenJαcan be extended to a ΛN-linear mapJα:DN →ΛN,σ. Setting DRN :=RN ⊗Λ_N DN and extending by RN-linearity over ΛN we finally obtain a RN-linear map, again denoted by the same symbol,

Jα:DRN −→ReN. Forν∈DN andr∈ R_N we thus have

J_α(r⊗ν) =_κ,M˜ (b)·_˜κ,p(−1)·r⊗j_α(ν).

For the next result, for any arithmetic pointκN ∈ X_N^arithcoming fromκ∈ X^arith, extendρκ in (5) byRN-linearity overO[[Z^×p]], to get a map

ρκ_N :DRN −→Vn_κ

defined by ρκ_N(r⊗ν) := ρκ(ν)·κN(r), for ν ∈ Dand r ∈ RN. To simplify the notation, set

(10) hν, αiκ_N :=ρκ_N(ν)( ˜Qⁿ_α^˜κ/2).

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

Lemma 3.2. Let κ˜N ∈ Xe_N^arith with signature (˜κ, kκ˜) and define κN :=pN(˜κN).

Then for anyν ∈DR_N we have:

˜

κN Jα(ν)

=η_κ˜(α)· hν, αiκ_N. Proof. Forν∈DN andr∈ RN we have

˜

κ_N J_α(r⊗ν)

= ˜κ_{N κ,M˜} (b)·_˜κ,p(−1)·r⊗j_α(ν)

=κ,M˜ (b)·κ,p˜ (−1)·κ˜N(r⊗1)·κ˜N 1⊗jα(ν)

=_κ,M˜ (b)·_κ,p˜ (−1)·κ_N(r)· Z

Z^×p

˜

κ_N(t)dj_α(ν) and thus, noticing that ˜κN restricted toZ^×p is ˜κN(t) =˜κ,p(t)t^n˜κ, we have

˜

κN Jα(r⊗ν)

=˜κ,M(b)·κ,p˜ (−1)·κN(r) Z

Zp×Z^×p

˜κ,p( ˜Qα(x, y)) ˜Qα(x, y)^nκ˜/2dν.

Recalling (8), and viewinga, b, cas elements inZp, we have, for (x, y)∈Zp×Z^×p, κ,p˜ Q˜α(x, y)

=κ,p˜ (−by²) =˜κ,p(−b)κ,p˜ (y²) =˜κ,p(−b)²_˜κ,p(y) =˜κ,p(−b)κ,p(y).

Thus, since_˜κ(−1)²= 1, we get:

˜

κN Jα(r⊗ν)

=κN(r)·˜κ,M(b)·κ,p˜ (b)·ρκ(ν)( ˜Qⁿ_α^κ˜/2) =η_κ(α)· hν, αiκ_N

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

Define

WRN :=W⊗_O[[

Z^×p]]R_N,

the structure of O[[Z^×p]]-module of RN being that induced by the composition of the two mapsO[[Z^×p]]→ΛN → RN described above. There is a canonical map

ϑ:WRN −→H¹(Γ₀,DRN)

described as follows: if ν_γ is a representative of an element ν in W and r∈ RN, thenϑ(ν⊗r) is represented by the cocycle ν_γ⊗r.

Forν∈WRN represented byν_γ andξ≥1 an integer, define θξ(ν) := X

C∈R(Γ1),q(C)=ξ

JαC(νγ_αC) tα_C

.

Definition 3.3. Forν ∈WR_N, the formal Fourier expansion Θ(ν) :=X

ξ≥1

θξ(ν)q^ξ

in RN[[q]] is called the Λ-adic Shimura-Shintani-Waldspurger lift of ν. For any

˜

κ∈Xe^arith, the formal power series expansion Θ(ν)(˜κN) :=X

ξ≥1

˜

κN θξ(ν) q^ξ

is called the ˜κ-specialization of Θ(ν).

There is a natural map

WR−→WRN

takingν⊗rto itself (use thatRhas a canonical map toRN ' R[∆], as described above). So, for any choice of sign, Φ^±∈WR will be viewed as an element inWR_N. From now on we will use the following notation. Fix ˜κ₀ ∈ Xe^arith and put κ₀ := p(˜κ₀)∈ X^arith. Recall the neighborhood U0 of κ₀ in Theorem 3.1. Define Ue₀:=p⁻¹(U₀) and

Ue₀^arith:=Ue0∩Xe^arith.

For each ˜κ∈Ue₀^arith put κ=p(˜κ)∈ U₀^arith. Recall that if (_˜κ, k_˜κ) is the signature of ˜κ, then (_κ, k_κ) := (²_˜κ,2k_κ˜) is that of κ₀. For anyκ:=p(˜κ) as above, we may consider the modular form

f_κ^JL ∈Sk_κ(Γr_κ, κ) and its Shimura-Shintani-Waldspurger lift

h_κ=X

ξ

a_ξ(h_κ)q^ξ∈S_kκ+1/2(4N p^rκ, χ_κ), whereχ_κ(x) :=_κ˜(x) −1

x ^kκ

,

normalized as in (2) and (3). For our fixedκ₀, recall the elements Φ := Φ⁺ chosen as in Theorem 3.1 and defineφκ:=φ⁺_κ and Ωκ:= Ω⁺_κ forκ∈ U₀^arith.

Proposition 3.4. For allκ˜∈Ue₀^arith such that r_κ= 1we have

˜

κ_N θ_ξ(Φ)

= Ω_κ·a_ξ(h_κ) and Θ(Φ)(˜κ_N) = Ω_κ·h_κ. Proof. By Lemma 3.2 we have

˜

κ_N θ_ξ(Φ)

= X

C∈R(Γ1),q(C)=ξ

η_κ˜(α_C) tαC

ρ_κN(Φ)( ˜Qⁿ_α^κ˜_C^/2).

Using Theorem 3.1, we get

˜

κN θξ(Φ)

= X

C∈R(Γ1),q(C)=ξ

η_κ˜(αC)·Ωκ

tαC

φκ( ˜Q^k_α^κ_C⁻¹).

Now (2) shows the statement on ˜κN(θξ(Φ)), while that on Θ(Φ)(˜κN) is a formal

consequence of the previous one.

Corollary 3.5. Let ap denote the image of the Hecke operator Tp in R. Then Θ(Φ)|T_p²=ap·Θ(Φ).

Proof. For anyκ∈ X^arith, letap(κ) := κ(Tp), which is a p-adic unit by the ordi- narity assumption. For all ˜κ∈Ue₀^arith withrκ= 1, we have

Θ(Φ)(˜κ_N)|T_p²= Ω_κ·h_κ|T_p²=a_p(κ)·Ω_κ·h_κ=a_p(κ)·Θ(Φ)(˜κ_N).

Consequently,

˜

κN θ_ξp2(Φ)

=ap(κ)·˜κN θξ(Φ)

for all ˜κ such that r_κ = 1. Since this subset is dense in Xe_N, we conclude that θ_ξp2(Φ) =a_p·θ_ξ(Φ) and so Θ(Φ)|T_p²=a_p·Θ(Φ).

For any integern≥1 and any quadratic formQwith coefficients inF, write [Q]n

for the class ofQmodulo the action ofiF(Γn). DefineFn,ξto be the subset of theF- vector space of quadratic forms with coefficients inF consisting of quadratic forms Q˜αsuch thatα∈V^∗∩ OB,n and−nr(α) =ξ. WritingδQ˜α for the discriminant of Qα, the above set can be equivalently described as

Fn,ξ :={Q˜α|α∈V^∗∩ OB,n, δQ˜α=N pⁿξ}.

Define F_n,ξ/Γ_n to be the set {[ ˜Q_α]_n|Q˜_α ∈ F_n,ξ} of equivalence classes of F_n,ξ under the action ofi_F(Γ_n). A simple computation shows thatQ_g−1αg=Q_α|g for allα∈V^∗ and allg∈Γn, and thus we find

Fn,ξ/Γn ={[ ˜QC_α]n| C ∈R(Γn), δQ˜_α =N pⁿξ}.

We also note that, in the notation of §2.1, if f has weight character ψ, defined moduloN pⁿ, and level Γ_n, the Fourier coefficientsa_ξ(h) of the Shimura-Shintani- Waldspurger lifthoff are given by

(11) a_ξ(h) = X

[Q]∈F_n,ξ/Γ_n

ψ(Q) tQ

φ⁺_f Q(z)^k−1

and, ifQ= ˜Qα, we putψ(Q) :=ηψ(bα) andtQ :=tα. Also, if we let Fn/Γn :=a

ξ

Fn,ξ/Γn

we can write

(12) h= X

[Q]∈F_n/Γ_n

ψ(Q) tQ

φ⁺_f Q(z)^k−1

q^{δQ/(N pn)}.

Fix now an integer m≥1 and let n∈ {1, m}. For any t ∈(Z/pⁿZ)^× and any integer ξ ≥1, define Fn,ξ,t to be the subset ofFn,ξ consisting of forms such that N pⁿbα≡t modN p^m. Also, define Fn,ξ,t/Γn to be the set of equivalence classes ofFn,ξ,tunder the action ofiF(Γn). Ifα∈V^∗∩ OB,m andiF(α) = ^a_c−a^b

, then (13) Q˜α(x, y) =N pⁿcx²−2N pⁿaxy−N pⁿby²

from which we see that there is an inclusion F_m,ξ,t ⊆ F_1,ξpm−1,t. If ˜Q_α and ˜Q_α⁰ belong to Fm,ξ,t, and α⁰ = gαg⁻¹ for some g ∈ Γm, then, since Γm ⊆ Γ1, we see that ˜Q_α and ˜Q_α⁰ represent the same class in F_1,ξpm−1,t/Γ₁. This shows that [ ˜Q_α]_m7→[ ˜Q_α]₁ gives a well-defined map

πm,ξ,t:Fm,ξ,t/Γm−→ F_1,ξp^m−1_,t/Γ1. Lemma 3.6. The map π_m,ξ,t is bijective.

Proof. We first show the injectivity. For this, suppose ˜Qαand ˜Qα⁰are inFm,ξ,tand [ ˜Qα]1= [ ˜Qα⁰]1. So there existsg= ^{α β}_{γ δ}

iniF(Γ1) such that such that ˜Qα= ˜Qα⁰|g.

If ˜Qα=cx²−2axy−by², and easy computation shows that ˜Qα⁰ =c⁰x²−2a⁰xy−b⁰y² with

c⁰=cα²−2aαγ−bγ² a⁰=−cαβ+aβγ+aαδ+bγδ

b⁰=−cβ²+ 2aβδ+bδ².

The first condition shows that γ ≡0 modN p^m. We haveb ≡b⁰ ≡t modN p^m, soδ²≡1 modN p^m. Sinceδ≡1 modN p, we see thatδ≡1 modN p^mtoo.

We now show the surjectivity. For this, fix [ ˜QαC]1in the target ofπ, and choose a representative

Q˜αC=cx²−2axy−by²

(recallM p^mξ|δQ˜αC,N p|c,N p|a, andb∈ O_F^×, the last condition due toηψ(α_C)6= 0).

By the Strong Approximation Theorem, we can find ˜g∈Γ₁such that i`(˜g)≡

1 0 ab⁻¹ 1

modN p^m

for all`|N p. Takeg :=iF(˜g), and putα:=g⁻¹α_Cg. An easy computation, using the expressions fora⁰, b⁰, c⁰ in terms ofa, b, c and g = ^{α β}_{γ δ}

as above, shows that α∈V^∗∩ OB,m,ηψ(α) =tandδQ˜α =N p^mξ, and it follows that ˜Qα∈ Fm,ξ,t. Now

π [ ˜Q_α]_m

= [ ˜Q_α]₁= [ ˜Q_g−1αCg]₁= [ ˜Q_αC]₁

where the last equality follows becauseg∈Γ₁.

Proposition 3.7. For allκ˜∈Ue₀^arith we have

Θ(Φ)(˜κ_N)|T_p^rκ−1= Ω_κ·h_κ.

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

Θ(Φ)(˜κN) =X

ξ≥1





X

C∈R(Γ1),q(C)=ξ

η_κ˜(α_C)·Ωκ

tα_C

φκ( ˜Q^k_α^κ_C⁻¹)



q^ξ

which, by (11) and (12) above we may rewrite as Θ(Φ)(˜κN) = X

[Q]∈F1/Γ₁

˜κ(Q)·Ωκ

tQ

φκ(Q^kκ−1)q^{δQ/(N p)} By definition of the action ofTp on power series, we have

Θ(Φ)(˜κN)|T_p^rκ−1= X

[Q]∈F1/Γ1, p^rκ|δQ

κ˜(Q)·Ωκ

t_Q φκ(Q^kκ−1)q^{δQ/(N prκ)}. SettingFn,t/Γn :=`

ξ≥1Fn,t,ξ/Γn forn∈ {1, rκ}, Lemma 3.6 shows that F_1,t^∗ :={[Q]∈ F1,t/Γ_1,t such thatp^rκ|δQ}is equal toFrκ,t.