Admissible p-adic Measures Attached to

Triple Products of Elliptic Cusp Forms

To dear John Coates for his sixtieth birthday

Siegfried Böcherer^{∗}, A. A. Panchishkin^{†}

Received: August 26, 2005 Revised: February 16, 2006

Abstract. We use the Siegel-Eisenstein distributions of degree three, and their higher twists with Dirichlet characters, in order to construct admissible p-adic measures attached to the triple products of elliptic cusp forms. We use an integral representation of Garrett’s type for triple products of three cusp eigenforms. For a prime p and for three primitive cusp eigenforms f_{1}, f_{2}, f_{3} of equal weights k_{1} = k_{2} = k_{3} = k, we study the critical values of Garrett’s triple product L(f_{1} ⊗ f_{2} ⊗ f_{3}, s, χ) twisted with Dirichlet characters χ. The result is stated in framework of a general program by John Coates, see [Co], [Co-PeRi].

2000 Mathematics Subject Classification: 11F60, 11S80 Keywords and Phrases: Siegel-Eisenstein series, triple products, admissible measures

0 Introduction

The purpose of this paper is to give a construction of p-adic admissible measures (in the sense of Amice-Vélu) attached to Garrett’s triple L-function attached to three primitive cusp eigenforms of equal weight k, where p is a prime. For this purpose we use the theory of p-adic integration with values in spaces of nearly-holomorphic modular forms (in the sense of Shimura, see [ShiAr]) over a normed 𝒪-algebra A where 𝒪 is the ring of integers in a finite extension K of ℚ_{p}. Often we simply assume that A = ℂ_{p}.

∗Supported by a French-German cooperation program by J.-P. Demailly (Institut Fourier, Grenoble)

†Supported by DFG - Forschergruppe ”Arithmetik Heidelberg-Mannheim”

Documenta Mathematica · Extra Volume Coates (2006) 77–132

0.1 Generalities on triple products

Consider three primitive cusp eigenforms

f_{j}(z) = ∑_{n=1}^{∞} a_{n,j}e(nz) ∈ 𝒮_{k_{j}}(N_{j}, ψ_{j}), (j = 1, 2, 3)  (0.1)

of weights k_{1}, k_{2}, k_{3}, of conductors N_{1}, N_{2}, N_{3}, and of nebentypus characters ψ_{j} mod N_{j} (j = 1, 2, 3), and let χ denote a Dirichlet character. The triple product twisted with Dirichlet characters χ is defined as the following complex L-function (an Euler product of degree eight):

L^{S}(f_{1} ⊗ f_{2} ⊗ f_{3}, s, χ) = ∏_{p∉S} L((f_{1} ⊗ f_{2} ⊗ f_{3})_{p}, χ(p)p^{−s}), where (0.2)
L((f_{1} ⊗ f_{2} ⊗ f_{3})_{p}, X)^{−1} =  (0.3)

det (1_{8} − X ({α_{p,1}^{(1)} \atop 0} {0 \atop α_{p,1}^{(2)}} ) ⊗ ({α_{p,2}^{(1)} \atop 0} {0 \atop α_{p,2}^{(2)}} ) ⊗ ({α_{p,3}^{(1)} \atop 0} {0 \atop α_{p,3}^{(2)}} ))

= ∏_{η}(1 − α_{p,1}^{(η(1))}α_{p,2}^{(η(2))}α_{p,3}^{(η(3))}X), η : {1, 2, 3} → {1, 2}, and

1 − a_{p,j}X − ψ_{j}(p)p^{k_{j}−1}X^{2} = (1 − α_{p,j}^{(1)}(p)X)(1 − α_{p,j}^{(2)}(p)X), j = 1, 2, 3, are the Hecke p–polynomials of forms f_{j} and the product is extended over all primes p ∉ S, and S = Supp(N_{1}N_{2}N_{3}) denotes the set of all prime divisors of the product N_{1}N_{2}N_{3}. We always assume that

k_{1} ≥ k_{2} ≥ k_{3},  (0.4)

including the case of equal weights k_{1} = k_{2} = k_{3} = k. We use the corresponding normalized motivic L function (see [De79], [Co], [Co-PeRi]), which in the case of “balanced” weights (i.e. k_{1} ≤ k_{2} + k_{3} − 2) has the form:

Λ^{S}(f_{1} ⊗ f_{2} ⊗ f_{3}, s, χ) =  (0.5)
Γ_{ℂ}(s)Γ_{ℂ}(s − k_{3} + 1)Γ_{ℂ}(s − k_{2} + 1)Γ_{ℂ}(s − k_{1} + 1)L(f_{1} ⊗ f_{2} ⊗ f_{3}, s, χ),

where Γ_{ℂ}(s) = 2(2π)^{−s}Γ(s). The motivic Gamma-factor

Γ_{ℂ}(s)Γ_{ℂ}(s − k_{3} + 1)Γ_{ℂ}(s − k_{2} + 1)Γ_{ℂ}(s − k_{1} + 1)

determines the critical values s = k_{1}, ⋯, k_{2} + k_{3} − 2 and a (conjectural) functional equation of the form: s ↦ k_{1} + k_{2} + k_{3} − 2 − s. Throughout the paper we fix an embedding

i_{p} : \overline{ℚ} ↪ ℂ_{p}, and define  (0.6)

λ(p) = α_{p,1}^{(1)}α_{p,2}^{(1)}α_{p,3}^{(1)}, where we assume that |i_{p}(α_{p,j}^{(1)})| ≤ |i_{p}(α_{p,j}^{(2)})|, j = 1, 2, 3. (0.7)

Admissible p-adic Measures … 79

0.2 Statement of main results

For a fixed positive integer N ∈ ℕ consider the profinite group

Y = Y_{N,p} = lim_{\overleftarrow{v}} Y_{v}, where Y_{v} = (ℤ/Np^{v}ℤ)^{×}.

There is a natural projection y_{p} : Y → ℤ_{p}^{×}. Let us fix a normed 𝒪-algebra A where 𝒪 is the ring of integers in a finite extension K of ℚ_{p}.

Definition 0.1 (a) For h ∈ ℕ, h ≥ 1 let 𝒫^{h}(Y, A) denote the A-module of locally polynomial functions of degree < h of the variable y_{p} : Y → ℤ_{p}^{×} ↪ A^{×}; in particular,

𝒫^{1}(Y, A) = 𝒞^{loc−const}(Y, A)

(the A-submodule of locally constant functions). We adopt the notation Φ(𝒰) := Φ(χ_{𝒰} ) for the characteristic function χ_{𝒰} of an open subset 𝒰 ⊂ Y. Let also denote 𝒞^{loc−an}(Y, A) the A-module of locally analytic functions and 𝒞(Y, A) the A-module of continuous functions so that

𝒫^{1}(Y, A) ⊂ 𝒫^{h}(Y, A) ⊂ 𝒞^{loc−an}(Y, A) ⊂ 𝒞(Y, A).

(b) For a given positive integer h we define an h-admissible measure on Y with values in an A-module M as a homomorphism of A-modules:

Φ̃ : 𝒫^{h}(Y, A) → M,

such that for all a ∈ Y and for v → ∞

|∫_{a+(Np^{v})}(y_{p} − a_{p})^{j}dΦ̃|_{p,M} = o(p^{−v(j−h)}) for all j = 0, 1, ⋯, h − 1,

where a_{p} = y_{p}(a).

We adopt the notation (a)_{v} = a + (Np^{v}) for both an element of Y_{v} and the corresponding open compact subset of Y.

U_{p}–Operator and Method of Canonical projection.

In Section 2.2, we construct an h-admissible measure Φ̃^{λ} : 𝒫^{h}(Y, A) → ℳ(A) out of a sequence of distributions

Φ_{r} : 𝒫^{1}(Y, A) → ℳ(A)

with values in an A-module M = ℳ(A) of nearly-holomorphic triple modular forms over A (for all r ∈ ℕ with r ≤ h − 1), where λ ∈ A^{×} is a fixed nonzero eigenvalue of triple Atkin’s operator U_{T} = U_{T,p}, acting on ℳ(A), and h = [2ord_{p}λ(p)] + 1. In our case ℳ(A) ⊂ A[[q_{1}, q_{2}, q_{3}]][R_{1}, R_{2}, R_{3}], and such modular forms are formal series

g = ∑_{n_{1},n_{2},n_{3}=0}^{∞} a(n_{1}, n_{2}, n_{3}; R_{1}, R_{2}, R_{3})q_{1}^{n_{1}}q_{2}^{n_{2}}q_{3}^{n_{3}} ∈ A[[q_{1}, q_{2}, q_{3}]][R_{1}, R_{2}, R_{3}]

such that for A = ℂ, for all z_{j} = x_{j} + iy_{j} ∈ ℍ and for R_{j} = (4πy_{j})^{−1} the series converges to a 𝒞^{∞}-modular form on ℍ^{3} of a given weight (k, k, k) and character (ψ_{1}, ψ_{2}, ψ_{3}), j = 1, 2, 3. The usual action of U = U_{p} on elliptic modular forms of one variable extends to triple Atkin’s operator U_{T} = U_{T,p} = (U_{p})^{⊗3} acting on triple modular forms by

U_{T} (g) = ∑_{n_{1},n_{2},n_{3}=0}^{∞} a(pn_{1}, pn_{2}, pn_{3}; pR_{1}, pR_{2}, pR_{3})q_{1}^{n_{1}}q_{2}^{n_{2}}q_{3}^{n_{3}}.  (0.8)

We consider the canonical projection operator π_{λ} : ℳ(A) → ℳ(A)^{λ} onto the maximal A-submodule ℳ(A)^{λ} over which the operator U_{T} − λI is nilpotent, and such that Ker π_{λ} = ⋂_{n≥1} Im (U_{T} − λI)^{n}. We define an A-linear map

Φ̃^{λ} : 𝒫^{h}(Y, A) → ℳ(A)

on local monomials y_{p}^{j} by

∫_{(a)_{v}} y_{p}^{j} dΦ̃^{λ} = π_{λ}(Φ_{j}((a)_{v}))

where Φ_{j} : 𝒫^{1}(Y, A) → ℳ(A) is a sequence of ℳ(A)-valued distributions on Y (for j = 0, 1, …, h − 1). Recall that for a primitive cusp eigenform f_{j} = ∑_{n=1}^{∞} a_{n}(f)q^{n} of conductor C = C_{f_{j}}, the function f_{j,0} = ∑_{n=1}^{∞} a_{n}(f_{j,0})q^{n} ∈ \overline{ℚ}[[q]] is defined as an eigenfunction of U = U_{p} with the eigenvalue α_{p,j}^{(1)} ∈ \overline{ℚ} (U(f_{0}) = αf_{0}) which satisfies the identity

f_{j,0} = f_{j} − α_{p,j}^{(2)}f_{j}|V_{p} = f_{j} − α_{p,j}^{(2)}p^{−k/2}f_{j}| ({p \atop 0} {0 \atop 1})  (0.9)

∑_{n=1}^{∞} a_{n}(f_{j,0})n^{−s} = ∑_{{n=1 \atop p∤n}}^{∞} a_{n}(f_{j})n^{−s}(1 − α_{p,j}^{(1)}p^{−s})^{−1}.

For any fixed n_{0} = n·p^{m} with p ∤ n we have a_{n_{0}}(f_{j,0}) = a_{n}(f_{j})·(α_{p,j}^{(1)})^{m} ∈ \overline{ℚ} and a_{n}(f_{j}) are eigenvalues of Hecke operators T_{n}. Therefore, U_{T} (f_{1,0}⊗f_{2,0}⊗f_{3,0}) = λ(f_{1,0} ⊗ f_{2,0} ⊗ f_{3,0}). Moreover,

f_{j}^{0} = f_{j,0}^{ρ}|_{k} ({0 \atop Np} {−1 \atop 0}) , where f_{j,0}^{ρ} = ∑_{n=1}^{∞} \overline{a(n, f_{0})}q^{n}.  (0.10)

Consider the triple product defined by (0.2) as an Euler product of degree eight: 𝒟(f_{1} ⊗ f_{2} ⊗ f_{3}, s, χ) = L^{(N)}(f_{1} ⊗ f_{2} ⊗ f_{3}, s, χ), attached to three cusp eigenforms f_{j}(z) = ∑_{n=1}^{∞} a_{n,j}e(nz) ∈ 𝒮_{k_{j}}(N_{j}, ψ_{j}), (j = 1, 2, 3) of weight k, of conductors N_{1}, N_{2}, N_{3}, and of nebentypus characters ψ_{j} mod N_{j} (j = 1, 2, 3), where χ mod Np^{v} is an arbitrary Dirichlet character, and the notation L^{(N)} means that the local factors at primes dividing N = LCM(N_{1}, N_{2}, N_{2}) are removed from an Euler product. Before giving the precise statements of our results on p-adic triple L-functions, we describe in more detail critical values of the L function 𝒟(f_{1} ⊗ f_{2} ⊗ f_{3}, s, χ). Let us introduce the following normalized L-function

𝒟^{⋆}(f_{1}^{ρ} ⊗ f_{2}^{ρ} ⊗ f_{3}^{ρ}, s + 2k − 2, ψ_{1}ψ_{2}χ) =  (0.11)
Γ_{ℂ}(s + 2k − 2)Γ_{ℂ}(s + k − 1)^{3}L^{(N)}(f_{1}^{ρ} ⊗ f_{2}^{ρ} ⊗ f_{3}^{ρ}, s + 2k − 2, ψ_{1}ψ_{2}χ),

where Γ_{ℂ}(s) = 2(2π)^{−s}Γ(s), and Γ_{ℂ}(s + 2k − 2)Γ_{ℂ}(s + k − 1)^{3} is the motivic Gamma-factor (compare with (0.5), and see [Co], [Co-PeRi], [Pa94]). For an arbitrary Dirichlet character χ mod Np^{v} consider the following Dirichlet characters:

χ_{1} mod Np^{v} = χ, χ_{2} mod Np^{v} = ψ_{2}ψ̄_{3}χ,  (0.12)
χ_{3} mod Np^{v} = ψ_{1}ψ̄_{3}χ, ψ = χ^{2}ψ_{1}ψ_{2}\overline{ψ}_{3};

later on we impose the condition that the conductors of the corresponding primitive characters χ_{0,1}, χ_{0,2}, χ_{0,3} are Np-complete (i.e., have the same prime divisors as those of Np).

Theorem A (Algebraic properties of the triple product) Assume that k ≥ 2. Then for all pairs (χ, r) such that the corresonding Dirichlet characters χ_{j} are Np-complete, and 0 ≤ r ≤ k − 2, we have that

\frac{𝒟^{⋆}(f_{1}^{ρ} ⊗ f_{2}^{ρ} ⊗ f_{3}^{ρ}, 2k − 2 − r, ψ_{1}ψ_{2}χ)}{⟨f_{1}^{ρ} ⊗ f_{2}^{ρ} ⊗ f_{3}^{ρ}, f_{1}^{ρ} ⊗ f_{2}^{ρ} ⊗ f_{3}^{ρ}⟩_{T}} ∈ \overline{ℚ}

where

⟨f_{1}^{ρ} ⊗ f_{2}^{ρ} ⊗ f_{3}^{ρ}, f_{1}^{ρ} ⊗ f_{2}^{ρ} ⊗ f_{3}^{ρ}⟩_{T} := ⟨f_{1}^{ρ}, f_{1}^{ρ}⟩_{N}⟨f_{2}^{ρ}, f_{2}^{ρ}⟩_{N}⟨f_{3}^{ρ}, f_{3}^{ρ}⟩_{N}
= ⟨f_{1}, f_{1}⟩_{N}⟨f_{2}, f_{2}⟩_{N}⟨f_{3}, f_{3}⟩_{N}.

For the p-adic construction, let ℂ_{p} = ˆ \overline{ℚ}_{p} denote the completion of an algebraic closure of the field ℚ_{p} of p-adic numbers. Fix a positive integer N, a Dirichlet character ψ mod N and consider the commutative profinite group Y = Y_{N,p} = lim_{m^{⟵}}(ℤ/Np^{m}ℤ)^{∗} and its group X_{N,p} = Hom_{cont}(Y, ℂ_{p}^{×}) of (contin-

uous) p-adic characters (this is a ℂ_{p}-analytic Lie group). The group X_{N,p} is isomorphic to a finite union of discs U = {z ∈ ℂ_{p} | |z|_{p} < 1}. A p-adic L-function L_{(p)} : X_{N,p} → ℂ_{p} is a certain meromorphic function on X_{N,p}. Such a function often come from a p-adic measure μ_{(p)} on Y (bounded or admissible in the sense of Amice-Vélu, see [Am-V]). In this case we write for all x ∈ X_{N,p}

L_{(p)}(x) = ∫_{Y_{N,p}} x(y)dμ_{(p)}(y)

In order to establish p-adic properties, let us use the product (0.7) λ = λ(p) = α_{p,1}^{(1)}α_{p,2}^{(1)}α_{p,3}^{(1)}, where we assume that |i_{p}(α_{p,j}^{(1)})| ≤ |i_{p}(α_{p,j}^{(2)})|, j = 1, 2, 3.

Theorem B (on admissible measures attached to the triple product). Under the assumptions as above there exist a ℂ_{p}-valued measure μ̃_{f_{1}⊗f_{2}⊗f_{3}}^{λ} on Y_{N,p}, and a ℂ_{p}-analytic function

𝒟_{(p)}(x, f_{1} ⊗ f_{2} ⊗ f_{3}) : X_{p} → ℂ_{p},

given for all x ∈ X_{N,p} by the integral

𝒟_{(p)}(x, f_{1} ⊗ f_{2} ⊗ f_{3}) = ∫_{Y_{N,p}} x(y)dμ̃_{f_{1}⊗f_{2}⊗f_{3}}^{λ}(y),

and having the following properties:

(i) for all pairs (r, χ) such that χ mod C_{χ} is a primitive Dirichlet character modulo C_{χ}, χ ∈ X_{N,p}^{tors}, assuming that all three corresonding Dirichlet characters χ_{j} given by (0.12) have Np-complete conductor (j = 1, 2, 3), and r ∈ ℤ is an integer with 0 ≤ r ≤ k − 2, the following equality holds:

𝒟_{(p)}(χx_{p}^{r}, f_{1} ⊗ f_{2} ⊗ f_{3}) =  (0.13)

i_{p}(\frac{(ψ_{1}ψ_{2})(2)C_{χ}^{4(2k−2−r)}}{G(χ_{1})G(χ_{2})G(χ_{3})G(ψ_{1}ψ_{2}χ_{1})λ(p)^{2v}}

\frac{𝒟^{⋆}(f_{1}^{ρ} ⊗ f_{2}^{ρ} ⊗ f_{3}^{ρ}, 2k − 2 − r, ψ_{1}ψ_{2}χ)}{⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, f_{1,0} ⊗ f_{2,0} ⊗ f_{3,0}⟩_{T,Np}} )

where v = ord_{p}(C_{χ}), χ_{1} mod Np^{v} = χ, χ_{2} mod Np^{v} = ψ_{2}ψ̄_{3}χ, χ_{3} mod Np^{v} = ψ_{1}ψ̄_{3}χ, G(χ) denotes the Gauß sum of a primitive Dirichlet character χ_{0} attached to χ (modulo the conductor of χ_{0}). (ii) if ord_{p}λ(p) = 0 then the holomorphic function in (i) is a bounded ℂ_{p}- analytic function; (iii) in the general case (but assuming that λ(p) ≠ 0) the holomorphic function in (i) belongs to the type o(log(x_{p}^{h})) with h = [2ord_{p}λ(p)] + 1 and it can be represented as the Mellin transform of the h-admissible ℂ_{p}-valued measure μ̃_{f_{1}⊗f_{2}⊗f_{3}}^{λ} (in the sense of Amice-Vélu) on Y (iv) if h ≤ k − 2 then the function 𝒟_{(p)} is uniquely determined by the above conditions (i).

Remark 0.2 It was checked by B.Gorsse and G.Robert that

⟨f_{1}^{0,ρ} ⊗ f_{2}^{0,ρ} ⊗ f_{3}^{0,ρ}, f_{1,0}^{ρ} ⊗ f_{2,0}^{ρ} ⊗ f_{3,0}^{ρ}⟩_{T,Np} = β · ⟨f_{1}, f_{1}⟩_{N}⟨f_{2}, f_{2}⟩_{N}⟨f_{3}, f_{3}⟩_{N}

for some β ∈ \overline{ℚ}^{∗} (see [Go-Ro]).

0.3 Scheme of the Proof

We construct \overline{ℚ}-valued distributions denoted by μ_{f_{1}⊗f_{2}⊗f_{3},r} on the profinite group Y_{N,p}, and attached to the special values at s = 2k−2−r with 0 ≤ r ≤ k−2 of the triple product L(f_{1}^{ρ}⊗f_{2}^{ρ}⊗f_{3}^{ρ}, s, ψ_{1}ψ_{2}χ) twisted with a Dirichlet character ψ_{1}ψ_{2}χ mod Np^{v}. We use an integral representation of this special value in terms of a 𝒞^{∞}-Siegel-Eisenstein series F_{χ,r} of degree 3 and of weight k (to be specified later), where 0 ≤ r ≤ k − 2. Such a series F_{χ,r} depends on the character χ, but its precise nebentypus character is ψ = χ^{2}ψ_{1}ψ_{2}\overline{ψ}_{3}, and it is defined by F_{χ,r} = G^{⋆}(𝒵, −r; k, (Np^{v})^{2}, ψ), where 𝒵 denotes a variable in the Siegel upper half space ℍ_{3}, and the normalized series G^{⋆}(𝒵, s; k, (Np^{v})^{2}, ψ) is given by (A.12). This series depends on s = −r, and for the critical values at integral points s ∈ ℤ such that 2 − k ≤ s ≤ 0, it represents a (nearly-) holomorphic Siegel modular form in the sense of Shimura [ShiAr]. Our construction consists of the following steps:

1) We consider the profinite ring A_{N,p} = lim_{\overleftarrow{v}}(ℤ/Np^{v}ℤ). Starting from any

sequence F_{r} of nearly-holomorphic Siegel modular forms we construct first a sequence Ψ_{F_{r}} of modular distributions on the additive profinite group

S = S_{N,p} := \{ε = ( \begin{matrix}0 & ε_{12} & ε_{13} \\ ε_{12} & 0 & ε_{23} \\ ε_{13} & ε_{23} & 0 \end{matrix} ) |ε_{12}, ε_{13}, ε_{23} ∈ A_{N,p}\};

such distributions take values in 𝒞^{∞}-(nearly-holomorphic) modular forms on the Siegel half plane ℍ_{3}. This construction, given in Section 1, generalizes the higher twist of F_{r}, already utilized in the work [Boe-Schm], in a simpler situation.

2) Next we consider the (real analytic) Siegel-Eisenstein series F_{χ,r} as a formal (nearly-holomorphic) Fourier series, whose coefficients admit explicit polynomial expressions (see Section 1 and Appendix A), and we use the fact that they may be written in terms of p-adic integrals of χ over Y (see [PaSE] and [PaIAS]). A crucial point of our construction is the higher twist in Section 1. We define the higher twist of the series F_{χ,r} by the characters (0.12) as the following formal nearly-holomorphic Fourier expansion:

F_{χ,r}^{χ̄_{1},χ̄_{2},χ̄_{3}} = ∑_{𝒯} χ̄_{1}(t_{12})χ̄_{2}(t_{13})χ̄_{3}(t_{23})Q(R, 𝒯; k − 2r, r)a_{χ,r}(𝒯)q^{𝒯} . (0.14)

The series (0.14) can be naturally interpreted as an integral of the Dirichlet character χ on the group Y with respect to a modular distribution Ψ_{r}:

F_{χ,r}^{χ̄_{1},χ̄_{2},χ̄_{3}} = ∫_{Y} χ(y)dΨ_{r}(y) =: Ψ_{r}(χ).  (0.15)

These modular distributions take values in the ring of formal Fourier expansion whose coefficients are polynomials in R = (4πIm (𝒵))^{−1} over the field \overline{ℚ} (which is imbedded into ℂ_{p} via (0.6). The distributions Ψ_{r} are uniformly bounded (coefficient-by-coefficient).

3) If we consider the diagonal embedding

diag : ℍ × ℍ × ℍ → ℍ_{3},

then the restriction produces a sequence Φ_{r} = 2^{r} diag^{∗} Ψ_{r} of distributions on Y with values in the tensor product ℳ_{k,r}(\overline{ℚ})⊗ℳ_{k,r}(\overline{ℚ})⊗ℳ_{k,r}(\overline{ℚ}) of three spaces of elliptic nearly-holomorphic modular forms on the Poincaré upper half plane ℍ (the normalizing factor 2^{r} is neeeded in order to prove certain congruences between Φ_{r} in Section 3). The important property of these distributions, established in Section 1, is that the nebentypus character of the triple modular form Φ_{r}(χ) is fixed and is equal to (ψ_{1}, ψ_{2}, ψ_{3}), see Proposition 1.5. Using this property, and applying the canonical projector π_{λ} of Section 2 to Φ_{r}(χ), we prove in Section 3 that the sequence of modular distributions Φ_{r} on Y produces a p-adic admissible measure Φ̃^{λ} (in the sense of Amice-Vélu, [Am-V]) with values in a finite dimensional subspace

ℳ^{λ}(ℂ_{p}) ⊂ ℳ(ℂ_{p}), ℳ(ℂ_{p}) = ℳ_{k,r}(ℂ_{p}) ⊗ ℳ_{k,r}(ℂ_{p}) ⊗ ℳ_{k,r}(ℂ_{p})

of the ℂ_{p}-vector space ℳ(ℂ_{p}) = ⋃_{v≥0} ℳ_{k,r}(Np^{v}, ψ_{1}, ψ_{2}, ψ_{3}; ℂ_{p}) of formal nearly-holomorphic triple modular forms of levels Np^{v} and the fixed nebentypus characters (ψ_{1}, ψ_{2}, ψ_{3}). We use congruences between triple modular forms Φ_{r}(χ) ∈ ℳ(\overline{ℚ}) (they have cyclotomic formal Fourier coefficients), and a general admissibility criterion (see Theorem 2.4). Proof of the Main Congruence is contained in Section 3.

4) Application of a \overline{ℚ}-valued linear form of type

ℒ : h ⟼ \begin{matrix}⟨ & f̃_{1} ⊗ f̃_{2} ⊗ f̃_{3}, h & ⟩ \\ ⟨f̃_{1}, f̃_{1}⟩ & ⟨f̃_{2}, f̃_{2}⟩ & ⟨f̃_{3}, f̃_{3}⟩ \end{matrix}

for h ∈ ℳ_{k,r}(\overline{ℚ}) ⊗ ℳ_{k,r}(\overline{ℚ}) ⊗ ℳ_{k,r}(\overline{ℚ}), produces a sequence of \overline{ℚ}-valued distributions given by μ_{r}^{λ}(χ) = ℒ(π_{λ}(Φ_{r})(χ)), λ ∈ \overline{ℚ}^{×}. More precisely, we consider three auxilliary modular forms

f̃_{j}(z) = ∑_{n=1}^{∞} ã_{n,j}e(nz) ∈ S_{k}(Γ_{0}(N_{j}p^{ν_{j}}), ψ_{j}) (1 ≤ j ≤ 3, ν_{j} ≥ 1), (0.16)

with the same eigenvalues as those of (0.1), for all Hecke operators T_{q}, with q prime to Np. In our construction we use as f̃_{j} certain “easy transforms” of primitive cusp forms in (0.1). In particular, we choose as f̃_{j} eigenfunctions f̃_{j} = f_{j}^{0} of the adjoint Atkin’s operator U_{p}^{∗}, in this case we denote by f_{j,0} the corresponding eigenfunctions of U_{p}. The \overline{ℚ}-linear form ℒ produces a ℂ_{p}-valued admissible measure μ̃^{λ} = ℓ(Φ̃^{λ}) starting from the modular p-adic admissible measure Φ̃^{λ} of stage 3), where ℓ : ℳ(ℂ_{p}) → ℂ_{p} denotes a ℂ_{p}-linear form, interpolating ℒ. See Section 4 for the construction of μ̃^{λ}.

5) We show in Section 5 that for any suitable Dirichlet character χ mod Np^{v} the integral

μ_{r}^{λ}(χ) = ℒ(π_{λ}(Φ_{r}(χ)))

coincides (up to a normalisation) with the special L-value

𝒟^{∗}(f_{1}^{ρ} ⊗f_{2}^{ρ} ⊗f_{3}^{ρ}, 2k −2−r, ψ_{1}ψ_{2}χ) (under the above assumptions on χ and r).

We use a general integral representation of Section B. The basic idea how a Dirichlet character χ is incorporated in the integral representation [Ga87, BoeSP] is somewhat similar to the one used in [Boe-Schm], but (surprisingly) more complicated to carry out. Note however that the existence of a ℂ_{p}-valued admissible measure μ̃^{λ} = ℓ(Φ̃^{λ}) established at stage 4), does not depend on this technical computation, and details will appear elsewhere.

Remark 0.3 Similar techniques can be applied in the case of three arbitrary “balanced” weights (0.4) k_{1} ≥ k_{2} ≥ k_{3}, i.e. when k_{1} ≤ k_{2}+k_{3}−2, using various differential operators acting on modular forms (the Maaß-Shimura differential operators (see [ShiAr], [Or]), and Ibukiyama’s differential operators (see [Ibu], [BSY]). More precisely, one applies these operators to a twisted Eisenstein series. In this case the critical values of the L function 𝒟(f_{1} ⊗ f_{2} ⊗ f_{3}, s, χ) correspond to s = k_{1}, ⋯, k_{2} + k_{3} − 2. The equality of weights in the present paper is made to avoid (for lack of space) the calculus of differential operators.

0.4 Conclusion: some advantages of our p-adic method

The whole construction works in various situations and it can be split into several independent steps:

1) Construction of modular distributions Φ_{r} (on a profinite or even adelic space Y of type Y = 𝔸_{K}^{∗}/K^{∗} for a number field K) with values in an infinite dimensional modular tower ℳ(𝒜) over complex numbers (or in an 𝒜-module of infinite rank over some p-adic algebra 𝒜).

2) Application of a canonical projector of type π_{λ} onto a finite dimensional subspace ℳ^{λ}(𝒜) of ℳ(𝒜) (or over a locally free 𝒜-module of finite rank over some 𝒜) in the form: π_{λ}(g) = (U^{λ})^{−v}π_{λ,1}(U^{v}(g)) ∈ ℳ^{λ}(Np, 𝒜) as in (2.3) of Section 2 (this method works only for λ ∈ 𝒜^{×}, and gives the λ-characteristic projector of g ∈ ℳ(Np^{v}, 𝒜) (independently of a sufficiently large v)).

3) One proves the admissibility criterium of Theorem 2.4 saying that the sequence π_{λ}(Φ_{r}) of distributions with values in ℳ^{λ}(𝒜) determines an hadmissible measure Φ̃^{λ} with values in this finite dimensional space for a suitable h (determined by the slope ord_{p}(λ)).

4) Application of a linear form ℓ of type g ↦ ⟨f^{0}, π_{λ}(g)⟩/⟨f, f⟩ to the modular distributions Φ_{r} produces a sequence of 𝒜-valued distributions μ_{r}^{λ} = ℓ(π_{λ}(Φ_{r})), and an 𝒜-valued admissible measure. The growth condition can be verified starting from congruences between modular forms Φ_{j}(χ), generalizing our Main Congruence of Section 3.

5) One shows that certain integrals μ_{j}^{λ}(χ) of the constructed distributions μ_{j}^{λ} coincide with normalized L-values; however, computing these integrals is not needed for the construction of p-adic admissible measures μ̃^{λ} (which is already done at stage 4)).

6) Under some assumptions, one can show a result on uniqueness for the constructed h-admissibles measures: they are determined by the integrals μ_{j}^{λ}(χ) over almost all Dirichlet characters and sufficiently many j = 0, 1, ⋯, h − 1 (this stage is not necessary, but it is nice to have uniqueness of the construction), see [JoH05].

7) If we are lucky, we can prove a functional equation for the constructed measure μ̃^{λ} (using the uniqueness in 6)), and using a functional equation for the L-values (over complex numbers), computed at stage 5), for almost all Dirichlet characters (again, this stage is not necessary, but it is nice to have a functional equation).

This strategy is applicable in various cases (described above), cf. [PaJTNB], [Puy], [Go02]. An interesting discussion in the Bourbaki talk [Colm03] of P.Colmez indicates the use of this method for constructing Euler systems.

Acknowledgement

Both authors are very grateful to the Institut Fourier (UJF, Grenoble-1) and to the Mathematical Institutes in Heidelberg (Forschergruppe ”Arithmetik Heidelberg-Mannheim”), and Oberwolfach for the permanent excellent working environment. A part of these results was exposed by the second author in a talk “Une nouvelle méthode pour construire des fonctions L p-adiques associées aux formes modulaires” at the Séminaire de théorie des nombres de Jussieu (Chevaleret). It is a great pleasure for us to thank P.Colmez, C.-G.Schmidt and R.Schulze-Pillot for valuable discussions. Our special thanks go to John Coates, Helmut Klingen and Yury I. Manin, for providing us with fruitful motivations and advice.

Contents

0 Introduction  77
1 Modular distributions attached to …  87

2 Computation of the canonical projection  96
3 Main Congruence for the higher …  101
4 A trilinear form on the …  107
5 Computation of p-adic integrals and L-values  112
A Nearly-holomorphic Siegel-Eisenstein series  116
B An integral representation for the triple product  122

1 Modular distributions attached to the higher twist of Eisenstein series

1.1 Higher twists of the Siegel-Eisenstein series

In this Section we study a 𝒞^{∞}-Siegel-Eisenstein series F_{χ,r} of degree 3 and of weight k, where 0 ≤ r ≤ k − 2. As in the Introduction, consider the Dirichlet characters (0.12) χ_{1} mod Np^{v} = χ, χ_{2} mod Np^{v} = ψ_{2}ψ̄_{3}χ, χ_{3} mod Np^{v} = ψ_{1}ψ̄_{3}χ. The series F_{χ,r} = G^{⋆}(𝒵, −r; k, (Np^{v})^{2}, ψ), depends on the character χ, but its precise nebentypus character is ψ = χ^{2}ψ_{1}ψ_{2}\overline{ψ}_{3}. Here 𝒵 denotes a variable in the Siegel upper half space ℍ_{3}, and the normalized series G^{⋆}(𝒵, s; k, (Np^{v})^{2}, ψ) is given by (A.12). This series depends on s = −r, and for the critical values at integral points s ∈ ℤ such that 2 − k ≤ s ≤ 0, it represents a (nearly-) holomorphic function in the sense of Shimura [ShiAr] viewed as formal (nearlyholomorphic) Fourier series, whose coefficients admit explicit polynomial expressions in terms of simple p-adic integrals for p ∤ det(𝒯):

F_{χ,r} = ∑_{𝒯∈B_{3}} det(𝒯)^{k−2r−κ}Q(R, 𝒯; k − 2r, r)a_{χ,r}(𝒯)q^{𝒯} ,

where B_{3} = {𝒯 = (𝒯_{ij}) ∈ M_{3}(ℝ) | 𝒯 = ^{t}𝒯, 𝒯 ≥ 0, 𝒯_{ij}, 2𝒯_{ii} ∈ ℤ} , and q^{𝒯} = exp(2πitr(𝒯𝒵)), R = (4πIm(𝒵))^{−1}. More precisely, for any 𝒯 with p ∤ det(𝒯) there exists a bounded measure ℱ_{𝒯} on Y with values in \overline{ℚ} such that

a_{χ,r}(𝒯) = ∫_{Y} y_{p}^{r}χ(y)dℱ_{𝒯} = ∏_{ℓ| det(2𝒯)} M_{ℓ}(𝒯, ψ(ℓ)ℓ^{−k+2r}),  (1.1)

where ψ = χ^{2}ψ_{1}ψ_{2}\overline{ψ}_{3} (see (A.17), Theorem A.2 in Appendix A, also in [PaSE], [PaIAS]). Here we use arithmetical nearly-holomorphic Siegel modular forms (see [ShiAr] and Appendix A.2 for more details) viewed as formal power series g = ∑_{𝒯∈B_{m}} a(𝒯, R_{i,j})q^{𝒯} ∈ \overline{ℚ}[[q^{B_{m}}]][R_{i,j}] such that for all 𝒵 ∈ ℍ_{m} the series

converges to a 𝒞^{∞}-Siegel modular form of a given weight k and character ψ. As in the introduction, (0.14), we define the higher twist of the series F_{χ,r} by the characters (0.12) as the following formal nearly-holomorphic Fourier expansion:

F_{χ,r}^{χ̄_{1},χ̄_{2},χ̄_{3}} = ∑_{𝒯} χ̄_{1}(t_{12})χ̄_{2}(t_{13})χ̄_{3}(t_{23})Q(R, 𝒯; k − 2r, r)a_{χ,r}(𝒯)q^{𝒯} = Ψ_{r}(χ).

We construct in this section a sequence of distributions Φ_{r} on Y using the restriction to the diagonal

Φ_{r}(χ) : = 2^{r} diag^{∗} Ψ_{r}(χ) = 2^{r}F_{χ,r}^{χ̄_{1},χ̄_{2},χ̄_{3}} ∘ diag  (1.2)

= 2^{r} ∑_{t_{1},t_{2},t_{3}≥0} ∑_{{𝒯:t_{11}=t_{1}, \atop t_{22}=t_{2},t_{23}=t_{3}}} χ̄_{1}(t_{12})χ̄_{2}(t_{13})χ̄_{3}(t_{23}) det(𝒯)^{k−2r−κ}×

× Q(diag(R_{1}, R_{2}, R_{3}), 𝒯; k − 2r, r)a_{χ,r}(𝒯)q_{1}^{t_{1}}q_{2}^{t_{2}}q_{3}^{t_{3}},

where χ̄_{1}(t_{12})χ̄_{2}(t_{13})χ̄_{3}(t_{23}) = χ̄(t_{12}t_{13}t_{23})ψ̄_{2}ψ_{3}(t_{13})ψ̄_{1}ψ_{3}(t_{23}),

taking values in the tensor product of three spaces of nearly-holomorphic elliptic modular forms on the Poincaré upper half plane ℍ (recall that the normalizing factor 2^{r} is neeeded in order to prove congruences between Φ_{r} in Section 3). We show in Proposition 1.5 that the (diagonal) nebentypus character of F_{χ,r}^{χ̄_{1},χ̄_{2},χ̄_{3}} is (ψ_{1}, ψ_{2}, ψ_{3}), thus it does not depend on χ.

1.2 The higher twist as a distribution

Let us fix a Dirichlet character χ mod Np^{v} as above with v ≥ 1, and an arbitrary 𝒞^{∞}-modular function

F ∈ ℳ_{k}^{(3)}(Γ_{0}(Np^{v}), ψ)^{∞},

with a Dirichlet character ψ mod Np^{v} which depends on χ mod Np^{v}, for example, the series F_{χ,r} with the nebentypus character ψ = χ^{2}ψ_{1}ψ_{2}\overline{ψ}_{3}. Then the higher twist of F with χ_{1}, χ_{2}, χ_{3} was initially defined by the formula

F̃ = ∑_{ε_{12},ε_{13},ε_{23} mod Np^{v}} χ_{1}(ε_{12})χ_{2}(ε_{13})χ_{3}(ε_{23})F |_{k}t_{ε,Np^{v}}  (1.3)

where we use the translation t_{ε,Np^{v}} = ( \begin{matrix}1_{3} & \frac{1}{Np^{v}} ε \\ 0_{3} & 1_{3} \end{matrix} ) on ℍ_{3} with ε =

(\begin{matrix}0 & ε_{12} & ε_{13} \\ ε_{12} & 0 & ε_{23} \\ ε_{13} & ε_{23} & 0 \end{matrix}) . The idea of the construction. We wish to interpret the series

(1.3) in terms of a distribution on a profinite group, using the following model example: consider the profinite ring A_{N,p} = lim_{\overleftarrow{v}}(ℤ/Np^{v}ℤ), and a compact

open subset α + (Np^{v}) ⊂ A_{N,p} with α an integer mod Np^{v}, and N is prime to p. For any formal series f = ∑_{n≥1} a_{n}q^{n} ∈ ℂ[[q]] and for any open subset α + (Np^{v}) ⊂ A_{N,p} consider the following partial series:

μ_{f}(α + (Np^{v})) = ∑_{{n≥1 \atop n≡α mod Np^{v}}} a_{n}q^{n} ∈ ℂ[[q]].

If q = exp(2πiz) with z ∈ ℍ, it follows from the orthogonality relations that

μ_{f}(α + (Np^{v})) = (Np^{v})^{−1} ∑_{β mod Np^{v}} exp(−2πiαβ/Np^{v})f (z + \frac{β}{Np^{v}} ) ,

and that for any Dirichlet character χ mod Np^{v} one has

∫_{A_{N,p}} χ(α)dμ_{f}(α) = ∑_{n≥1} χ(n)a_{n}q^{n} = f(χ) ∈ ℂ[[q]].

(the series f twisted by the character χ). In the same fashion, consider the additive profinite group

S = S_{N,p} := \{ε = ( \begin{matrix}0 & ε_{12} & ε_{13} \\ ε_{12} & 0 & ε_{23} \\ ε_{13} & ε_{23} & 0 \end{matrix} ) |ε_{12}, ε_{13}, ε_{23} ∈ A_{N,p}\};

equipped with the scalar product ⟨·, ·⟩ : S_{N,p} × S_{N,p} ⟶ A_{N,p}:

⟨ε^{(1)}, ε^{(2)}⟩ = tr(ε^{(1)}ε^{(2)}) = 2ε_{12}^{(1)}ε_{12}^{(2)} + 2ε_{13}^{(1)}ε_{13}^{(2)} + 2ε_{23}^{(1)}ε_{23}^{(2)}, where

ε^{(1)} = ( \begin{matrix}0 & ε_{12}^{(1)} & ε_{13}^{(1)} \\ ε_{12}^{(1)} & 0 & ε_{23}^{(1)} \\ ε_{13}^{(1)} & ε_{23}^{(1)} & 0 \end{matrix} ) , ε^{(2)} = ( \begin{matrix}0 & ε_{12}^{(2)} & ε_{13}^{(2)} \\ ε_{12}^{(2)} & 0 & ε_{23}^{(2)} \\ ε_{13}^{(2)} & ε_{23}^{(2)} & 0 \end{matrix} ) .

Proposition 1.1 Suppose that the function F is invariant with respect to any integer translation of type t_{ε,1} : F |t_{ε,1} = F. Then

1) The action F |t_{ε,Np^{v}} depends only on the class of ε ∈ S/Np^{v}S, and the additive character e_{ε}_{(0)} : ε ↦ exp(⟨ε, ε^{(0)}⟩/Np^{v}) on S is trivial iff ε^{(0)} ∈ Np^{v}S.

2) The formula

Ψ_{F} (ε^{(0)} + (Np^{v})) = (Np^{v})^{−3} ∑_{ε∈S mod Np^{v}S} exp(−2πi⟨ε, ε^{(0)}⟩/Np^{v})F |t_{ε,Np^{v}} (1.4)

= (Np^{v})^{−3} ∑_{ε∈S mod Np^{v}S} e(−⟨ε, ε^{(0)}⟩/Np^{v})F |t_{ε,Np^{v}}

defines a distribution with values in 𝒞^{∞}-functions on ℍ_{3}, where e(α/Np^{v}) := exp(2πiα/Np^{v}) is well-defined for all α ∈ A_{N}.

Proof: 1) Follows directly from the invariance: F |t_{ε,1} = F.

2) It suffices to check the finite-additivity condition:

Ψ_{F} (ε^{(0)} + (Np^{v})) = ∑_{ε^{(1)}∈S mod p} Ψ_{F} (ε^{(0)} + Np^{v}ε^{(1)} + (Np^{v+1})),  (1.5)

i.e.,

(Np^{v})^{−3} ∑_{ε∈S/Np^{v}S} e(−⟨ε, ε^{(0)}⟩/Np^{v})F |t_{ε,Np^{v}}  (1.6)

= (Np^{v+1})^{−3}×

∑_{ε^{(1)}∈S/pS} ∑_{ε^{(2)}∈S/Np^{v+1}S} e(−⟨ε^{(2)}, (ε^{(0)} + Np^{v}ε^{(1)})⟩/Np^{v+1})F |t_{ε^{(2)},Np^{v+1}}. (1.7)

For all ε^{(2)} the sum on the right on ε^{(1)} ∈ S/pS in (1.6) becomes

(Np^{v+1})^{−3} ∑_{ε^{(1)}∈S/pS} e(−⟨ε^{(2)}, (ε^{(0)} + Np^{v}ε^{(1)})⟩/Np^{v+1})F |t_{ε^{(2)},Np^{v+1}} (1.8)

= (Np^{v+1})^{−3}e(−\frac{⟨ε^{(2)}, ε^{(0)}⟩}{Np^{v+1}})F |t_{ε^{(2)},Np^{v+1}} ∑_{ε^{(1)}∈S/pS} e(−\frac{⟨ε^{(2)}, Np^{v}ε^{(1)}⟩}{Np^{v+1}})

= (Np^{v+1})^{−3}e(−⟨ε^{(2)}, ε^{(0)}⟩/Np^{v+1})F |t_{ε^{(2)},Np^{v+1}} ∑_{ε^{(1)}∈S/pS} e (−⟨ε^{(2)}, ε^{(1)}⟩) .

It remains to notice that

∑_{ε^{(1)}∈S/pS} e(−⟨ε^{(2)}, ε^{(1)}⟩/p) = \begin{cases}p^{3}, & if ε^{(2)} = pε^{(3)}, ε^{(3)} ∈ S \\ 0, & otherwise, \end{cases}  (1.9)

because ε^{(1)} ↦ e(−⟨ε^{(2)}, ε^{(1)}⟩/p) is a non trivial character of S/pS iff ε^{(2)} ∈ pS. The right hand side of (1.6) becomes

(Np^{v+1})^{−3} ∑_{ε^{(1)}∈S/pS} ∑_{ε^{(2)}∈S/Np^{v+1}S} e(−⟨ε^{(2)}, (ε^{(0)} + Np^{v}ε^{(1)})⟩/Np^{v+1})F |t_{ε^{(2)},Np^{v+1}} (1.10)

= (Np^{v+1})^{−3}p^{3} ∑_{ε^{(3)}∈S/Np^{v}S} e(−⟨ε^{(3)}, ε^{(0)}⟩/Np^{v})F |t_{ε^{(3)},Np^{v}}.

Remark 1.2 The Fourier expansions of the nearly-holomorphic Siegel modular form

F_{ε,v} :=

Ψ_{F} (ε + (Np^{v})) = (Np^{v})^{−3} ∑_{ε′∈S mod Np^{v}S} exp(−2πi⟨ε′, ε⟩/Np^{v})F |t_{ε′,Np^{v}}.

is given as the following partial Fourier series

F_{ε,v}(𝒵) = ∑_{{𝒯,t_{12}≡ε_{12} mod Np^{v} \atop t_{13}≡ε_{13},t_{23}≡ε_{23} mod Np^{v}}} a(𝒯, R)q^{𝒯} ,  (1.11)

where F is a nearly-holomorphic Siegel modular form, which is a periodic func-

tion on ℍ_{3}: F = ∑_{𝒯}a(𝒯, R)q^{𝒯} , and 𝒯 = (\begin{matrix}t_{11} & t_{12} & t_{13} \\ t_{12} & t_{22} & t_{23} \\ t_{13} & t_{23} & t_{33} \end{matrix}) runs over half

integral symmetric non negative matrices.

Indeed,

F |t_{ε′,Np^{v}} = ∑_{𝒯}a(𝒯, R)q^{𝒯} |t_{ε′,Np^{v}} = ∑_{𝒯}exp(2πitr(ε′𝒯)/Np^{v})a(𝒯, R)q^{𝒯} ,

hence

F_{ε,v} = (Np^{v})^{−3} ∑_{ε′∈S mod Np^{v}S} exp(−2πi⟨ε′, ε⟩/Np^{v}) ∑_{𝒯}exp(2πitr(ε′𝒯)/Np^{v})a(𝒯, R)q^{𝒯} .

It suffices to notice that

tr(ε′𝒯) = tr ((\begin{matrix}0 & ε′_{12} & ε′_{13} \\ ε′_{12} & 0 & ε′_{23} \\ ε′_{13} & ε′_{23} & 0 \end{matrix}) (\begin{matrix}t_{11} & t_{12} & t_{13} \\ t_{12} & t_{22} & t_{23} \\ t_{13} & t_{23} & t_{33} \end{matrix}))= 2(ε′_{12}t_{12}+ε′_{13}t_{13}+ε′_{23}t_{23}).

Let us consider now three Dirichlet characters χ_{1}, χ_{2}, χ_{3} mod Np^{v}, and let us compute the corresponding integrals against the constructed modular distribution (1.4) of the locally constant function ε ↦ χ_{1}(ε_{12})χ_{2}(ε_{13})χ_{3}(ε_{23}) on the profinite additive group

S = S_{N} := \{ε = ( \begin{matrix}0 & ε_{12} & ε_{13} \\ ε_{12} & 0 & ε_{23} \\ ε_{13} & ε_{23} & 0 \end{matrix} ) |ε_{12}, ε_{13}, ε_{23} ∈ A_{N} \} .

Proposition 1.3 Let F be a function invariant with respect to any translation of type t_{ε,1} : F |t_{ε,1} = F. Let us write F_{χ̄_{1},χ̄_{2},χ̄_{3}} = ∫_{S} χ̄_{1}(ε_{12})χ̄_{2}(ε_{13})χ̄_{3}(ε_{23})dΨ_{F} (ε). Then

F_{χ̄_{1},χ̄_{2},χ̄_{3}} =  (1.12)

(Np^{v})^{−3} ∑_{ε∈S/Np^{v}S} G_{Np^{v}}(χ̄_{1}, −ε_{12})G_{Np^{v}}(χ̄_{2}, −ε_{13})G_{Np^{v}}(χ̄_{3}, −ε_{23})F |t_{ε,Np^{v}}.

Here G_{Np^{v}}(χ, ε) := ∑_{α′} e(εα′/Np^{v})χ(α′) denotes the Gauß sum (of a non nec-

essarily primitive Dirichlet character χ).

Remarks 1.4 1) The advantage of the expression (1.12) in compare with (1.3) is that it does not depend on a choice of v. 2)It follows from (1.11, that the Fourier expansion of the series (1.12) is given by

F_{χ̄_{1},χ̄_{2},χ̄_{3}} = ∑_{𝒯} χ̄_{1}(t_{12})χ̄_{2}(t_{13})χ̄_{3}(t_{23})a(𝒯, R)q^{𝒯} .  (1.13)

Proof is similar to that of Proposition 1.1, and it follows from the definitions.

1.3 The level of the higher twist

Let us consider the symplectic inclusion:

i : SL_{2}(ℤ) × SL_{2}(ℤ) × SL_{2}(ℤ) → Sp_{3}(ℤ)  (1.14)

({a_{1} \atop c_{1}} {b_{1} \atop d_{1}} ) , ({a_{2} \atop c_{2}} {b_{2} \atop d_{2}} ) , ({a_{3} \atop c_{3}} {b_{3} \atop d_{3}} ) ↦ ( \begin{aligned}a_{1} \\ & a_{2} \\ & & a_{3} \end{aligned} ; \begin{aligned}c_{1} \\ & c_{2} \\ & & c_{3} \end{aligned} \begin{aligned}b_{1} \\ & b_{2} \\ & & b_{3} \end{aligned} ; \begin{aligned}d_{1} \\ & d_{2} \\ & & d_{3} \end{aligned} )

We study the behaviour of the modular form F_{χ̄_{1},χ̄_{2},χ̄_{3}} with respect to the subgroup

i(Γ_{0}(N^{2}p^{2v})^{3}) ⊂ Γ_{0}^{(3)}(N^{2}p^{2v}),

where (χ_{1} ⊗ χ_{2} ⊗ χ_{3})(ε) = χ_{1}(ε_{12})χ_{2}(ε_{13})χ_{3}(ε_{23}). We will have to study two different types of twist; we can treat them simultaneously if we consider a function

ϕ : ℤ/Nℤ ⟼ ℂ

which is “φ-spherical” i.e.

ϕ(gXh) = φ(g)φ(h)ϕ(X)

for all g, h ∈ (ℤ/Nℤ)^{×}, X ∈ ℤ/Nℤ, where φ is a Dirichlet character mod N. Let us use Proposition 1.12 and the spherical function

ϕ : (ε_{12}, ε_{13}, ε_{23}) ↦ G_{Np^{v}}(χ̄_{1}, −ε_{12})G_{Np^{v}}(χ̄_{2}, −ε_{13})G_{Np^{v}}(χ̄_{3}, −ε_{23}),

with respect to three variables (ε_{12}, ε_{13}, ε_{23}), and the Dirichlet characters

(ε_{12}, ε_{13}, ε_{23}) ↦ χ_{1}(ε_{12})χ_{2}(ε_{13})χ_{3}(ε_{23}).

Proposition 1.5 Consider a (nearly-holomorphic) Siegel modular form F for the group Γ_{0}^{(3)}(Np^{v}) and the Dirichlet character ψ = χ^{2}ψ_{1}ψ_{2}\overline{ψ}_{3}). Then for all M = i (({a_{1} \atop c_{1}} {b_{1} \atop d_{1}} ) , ({a_{2} \atop c_{2}} {b_{2} \atop d_{2}} ) , ({a_{3} \atop c_{3}} {b_{3} \atop d_{3}} )) ∈ Γ_{0}^{(3)}(N^{2}p^{2v}) one has:

1) F̃|M = \underbrace{ψχ̄_{1}χ̄_{2}(d_{1})}_{ψ_{1}} \underbrace{ψχ̄_{1}χ̄_{3}(d_{2})}_{ψ_{2}} \underbrace{ψχ̄_{2}χ̄_{3}(d_{3})}_{ψ_{3}} F̃, where F̃ is defined by (1.3),

2) F_{χ̄_{1},χ̄_{2},χ̄_{3}}|M = \underbrace{ψχ̄_{1}χ̄_{2}(d_{1})}_{ψ_{1}} \underbrace{ψχ̄_{1}χ̄_{3}(d_{2})}_{ψ_{2}} \underbrace{ψχ̄_{2}χ̄_{3}(d_{3})}_{ψ_{3}} F_{χ̄_{1},χ̄_{2},χ̄_{3}}, where F_{χ̄_{1},χ̄_{2},χ̄_{3}}

is defined by (1.12).

Proof. We study modular forms on ℍ_{3}. Let us consider a more general situation and write N instead of Np^{v}. We use the (somewhat unconventional) congruence subgroup (with N | M):

Γ_{1}^{(3)}(M, N) := \{γ = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Γ_{0}^{(3)}(M) | D ≡ diag(D_{1}, D_{2}, D_{3}) mod N\} .

Here the D_{i} denote integers along the diagonal of D. It is easy to see that this defines a subgroup of Sp(3, ℤ) and that a similar congruence also holds for A. The appropriate space of modular forms, denoted by ℳ_{k}^{(3)}(M, N; χ; ψ_{1}, ψ_{2}, ψ_{3}), with Dirichlet characters ψ_{j} mod N and a Dirichlet character χ mod M is then the set of holomorphic functions on ℍ_{3} satisfying

f |_{k} γ = χ(detD) (∏_{j=1}^{3} ψ_{j}(D_{j})) f

for all γ = ( \begin{matrix}A & B \\ C & D \end{matrix} ) ∈ Γ_{1}^{(3)}(M, N). For any α ∈ ℝ and any 1 ≤ i < j ≤ 3

we define a symmetric matrix of size 3 by

S_{ij}^{(3)}(α) := ( \begin{matrix}& α \\ α \end{matrix} )

(the number α sits in the (i, j)th and (j, i)positions). Then, for a function F ∈ ℳ_{k}^{(3)}(M, N; χ; ψ_{1}, ψ_{2}, ψ_{3}) we define a new function F_{ij}^{ϕ} on ℍ_{3} by

F_{ij}^{ϕ}(𝒵) = ∑_{α mod N} ϕ(α) · F (𝒵 + S_{ij}^{(3)}(\frac{α}{N}))

Proposition 1.6 Assume that N^{2} | M, χ is a character mod \frac{M}{N}, and F ∈ ℳ_{k}^{(3)}(M, N; χ; ψ_{1}, ψ_{2}, ψ_{3}). Then

F_{ij}^{ϕ} ∈ ℳ_{k}^{(3)}(M, N; χ; ψ′_{1}, ψ′_{2}, ψ′_{3})

with

ψ′_{r} = \{ \begin{matrix}ψ_{r} & if & r ∉ \{i, j\} \\ ψ_{r} · \overline{φ} & if & r ∈ \{i, j\} \end{matrix}

Remarks 1.7 1) We mention here two basic types of φ-spherical functions ϕ : ℤ/Nℤ: Type I: “Dirichlet character” ϕ(X) := φ(X) Type II: “Gauß sum” ϕ(X) = G(\overline{φ}, −X) where G(φ, X) denotes a Gauß sum (a version of such spherical functions of matrix argument was studied in [Boe-Schm]):

G(φ, X) := ∑_{α mod N} φ(α) exp(2πi\frac{1}{N} αX)

2) Our basic example is as follows: let φ_{1}, φ_{2}, φ_{3} be three Dirichlet characters mod N and let ϕ_{i} be φ_{i}-spherical functions on ℤ/Nℤ. Furthermore let F ∈ ℳ_{k}^{(3)}(Γ_{0}(M), χ) with N^{2} | M and χ a Dirichlet character mod \frac{M}{N}. Then

h(z_{1}, z_{2}, z_{3}) := ∑_{α,β,γ mod N} ϕ_{1}(α)ϕ_{2}(β)ϕ_{3}(γ)F (( \begin{matrix}z_{1} & \frac{α}{N} & \frac{β}{N} \\ \frac{α}{N} & z_{2} & \frac{γ}{N} \\ \frac{β}{N} & \frac{γ}{N} & z_{3} \end{matrix} ))

is an element of

ℳ_{k}(Γ_{0}(M), χφ̄_{1}φ̄_{2}) ⊗ ℳ_{k}(Γ_{0}(M), χφ̄_{1}φ̄_{3}) ⊗ ℳ_{k}(Γ_{0}(M), χφ̄_{2}φ̄_{3})

(note that the definition of h depends on N)

3) Other important cases are treated in [Boe-Schm] it can also (by iteration) be applied to cases of block matrices of different size which e.g. occur in the work [Boe-Ha] on the L-function for GSp(2) × GL(2).

Proof. We first try to find X ∈ Sym_{3}(\frac{1}{N} ℤ) such that

( \begin{matrix}1_{3} & S(\frac{α}{N}) \\ 0_{3} & 1_{3} \end{matrix} ) ( \begin{matrix}𝒜 & ℬ \\ 𝒞 & 𝒟 \end{matrix} ) ( \begin{matrix}1_{3} & −X \\ 0_{3} & 1_{3} \end{matrix} )

= ( \begin{matrix}𝒜 + S(\frac{α}{N})𝒞 & −𝒜X + ℬ − S(\frac{α}{N})𝒞X + S(\frac{α}{N})𝒟 \\ 𝒞 & −𝒞X + 𝒟 \end{matrix} )

is in Γ_{0}^{(3)}(M) (for the moment we only assume here that ( \begin{matrix}𝒜 & ℬ \\ 𝒞 & 𝒟 \end{matrix} ) is integral.

The conditions N^{2} | M and the congruences mod M and N will then be forced to hold). The first (evident) condition is that 𝒞 ≡ 0 mod M. It is easy to see that the two numbers on the diagonal

−𝒞X + 𝒟  and  𝒜 + S(\frac{α}{N})𝒞

are integers, if 𝒞 is congruent to 0 modulo N.

The remaining condition is that

−𝒜X + ℬ − S(\frac{α}{N})𝒞X + S(\frac{α}{N})𝒟

is integral, which is satisfied if 𝒞 ≡ 0 mod N^{2} and −𝒜 · X + S(\frac{α}{N})𝒟 is integral. Therefore we should choose any X satisfying

(NX) ≡ \overline{𝒜}S(α)𝒟 mod N

where \overline{𝒜} is a (multiplicative) inverse of the matrix 𝒜 mod N. Now we use the fact that 𝒜 ≡ diag(A_{1}, A_{2}, A_{3}) mod N and 𝒟 ≡ diag(D_{1}, D_{2}, D_{3}) mod N are matrices which are diagonal modulo N, we may therefore choose the integral symmetric matrix NX to be modulo N equal to

NX := S_{ij}^{(3)} (\overline{A_{i}} · α · D_{j}) ⇒ X = X(α) = S_{ij}^{(3)} (\frac{\overline{A_{i}} · α · D_{j}}{N} ) .

By the above,

F_{ij}^{ϕ} |_{k} ( \begin{matrix}𝒜 & ℬ \\ 𝒞 & 𝒟 \end{matrix} ) = ∑_{α mod N} ϕ(α)F |_{k} ( \begin{matrix}1 & S(\frac{α}{N}) \\ 0 & 1 \end{matrix} ) ( \begin{matrix}𝒜 & ℬ \\ 𝒞 & 𝒟 \end{matrix} )

= ∑_{α mod N} ϕ(α)F |_{k} ( \begin{matrix}𝒜̃ & ℬ̃ \\ 𝒞̃ & 𝒟̃ \end{matrix} ) ( \begin{matrix}1 & X(α) \\ 0 & 1 \end{matrix} )

where ( \begin{matrix}𝒜̃ & ℬ̃ \\ 𝒞̃ & 𝒟̃ \end{matrix} ) ∈ Γ_{1}^{(3)}(M, N) with

𝒜̃ ≡ 𝒜 mod \frac{M}{N} and 𝒟̃ ≡ 𝒟 mod \frac{M}{N}

(in particular, these congruences hold mod N). Therefore

F_{ij}^{ϕ} |_{k} (\begin{matrix}𝒜 & ℬ \\ 𝒞 & 𝒟 \end{matrix}) = χ(det(𝒟))ψ_{1}(D_{1}) … ψ_{n}(D_{n}) ∑_{α mod N} ϕ(α)F |_{k} (\begin{matrix}1_{3} & X(α) \\ 0_{3} & 1_{3} \end{matrix}) .

Instead of summing over α we may as well sum over β := D_{i} · α · D_{j} mod N. Then we obtain

χ(det(𝒟))ψ_{1}(D_{1}) … ψ_{n}(D_{n})\overline{φ}(D_{i})\overline{φ}(D_{j}) ∑_{β mod N} ϕ(β)F |_{k} ( \begin{matrix}1_{3} & S_{ij}^{(3)}(\frac{β}{N}) \\ 0_{3} & 1_{3} \end{matrix} )

= χ(det(𝒟)ψ_{1}(D_{1}) … ψ_{n}(D_{n})\overline{φ}(D_{i})\overline{φ}(D_{j})F_{ij}^{ϕ}.

Notice that the properties of Propositions 1.6 hold for the iterated twists, and Propositions 1.5 follows from Propositions 1.6 by three iterated twists with N equal to Np^{v}.

2 Computation of the canonical projection

2.1 A general construction: the canonical λ-characteristic projection

We explain now a general method which associates a p-adic measure μ_{λ,Φ} on a profinite group Y, to a sequence of distributions Φ_{r} on Y with values in a suitable (infinite dimensional) vector space ℳ of modular forms, and to a nonzero eigenvalue λ of the Atkin operator U = U_{p} acting on ℳ. We consider holomorphic (or nearly-holomorphic) modular forms in a space of the type

ℳ = ℳ_{k}(ψ, \overline{ℚ}) = ⋃_{v≥0} ℳ_{k}(Np^{v}, ψ, \overline{ℚ}) ⊂ ℳ(ℂ_{p}) = ⋃_{v≥0} ℳ_{k}(Np^{v}, ψ, ℂ_{p}),

with finite dimensional vector spaces ℳ_{k}(Np^{v}, ψ, \overline{ℚ}) at each fixed level, endowed with a natural \overline{ℚ}-rational structure (for example, given by the Fourier coefficients). The parameters here are triples k = (k_{1}, k_{2}, k_{3}), ψ = (ψ_{1}, ψ_{2}, ψ_{3}) of weights and characters. The important property of our construction is that does not use passage to a p-adic limit. We put

ℳ_{k}(Np^{v}, ψ, A) = ℳ_{k}(Np^{v}, ψ, \overline{ℚ}) ⊗\overline{ℚ} A.

for any \overline{ℚ}-algebra A.

Definition 2.1 Let A = ℂ_{p} , A = \overline{ℚ}, or A = ℂ, and ℳ = ℳ(A).

(a) For a λ ∈ A let us define ℳ^{(λ)} = Ker (U − λI) the subspace of eigenvectors with eigenvalue λ).

(b) Let us define the λ-characteristic subspace of U on ℳ by

ℳ^{λ} = ⋃_{n≥1} Ker (U − λI)^{n}

(c) Let us define for any v ≥ 0

ℳ^{λ}(Np^{v}) = ℳ^{λ} ∩ ℳ(Np^{v}), ℳ^{(λ)}(Np^{v}) = ℳ^{(λ)} ∩ ℳ(Np^{v}).

Proposition 2.2 Let ψ mod N be a fixed Dirichlet character, then U^{v}(ℳ(Np^{v+1}, ψ)) ⊂ ℳ(Np, ψ).

Proof follows from a known formula of J.-P. Serre: for g ∈ ℳ_{k}(Np^{v+1}, ψ),

g|_{k}U^{v} = p^{3v(k/2−1)}g|_{k}W_{Np^{v+1}}Tr_{Np}^{Np^{v+1}}W_{Np},  (2.1)

where W_{N} : ℳ(N, ψ) → ℳ(N, ψ̄) is the involution (over ℂ) of level N (see [Se73] for the elliptic modular case, which extends to the triple modular case).

Proposition 2.3 Let A = ℂ_{p} or A = \overline{ℚ}, ℳ = ℳ(A), λ ∈ A^{∗}, and let U^{λ} be the restriction of U on ℳ^{λ}, then

(a) (U^{λ})^{v} : ℳ^{λ}(Np^{v+1}) →^{∼} ℳ^{λ}(Np) is an A-linear invertible operator, where U^{λ} = U|_{ℳ^{λ}(Np^{v+1})} .

(b) The vector subspace ℳ^{λ}(Np^{v+1}) = ℳ^{λ}(Np) does not depend on v.

(c) Let π_{λ,v+1} : ℳ(Np^{v+1}) → ℳ^{λ}(Np^{v+1}) be the projector on the λcharacteristic subspace of U with the kernel Ker (π_{λ,v}) = ⋂_{n≥1} Im (U − λI)^{n} =

⨁_{β≠λ} ℳ^{β}(N_{0}p^{v})), then the following diagram is commutative

\begin{matrix}ℳ(Np^{v+1}) & ⟶_{π_{λ,v+1}} & ℳ^{λ}(Np^{v+1}) \\ U^{v}↓ & & ↓≀ U^{v} \\ ℳ(Np) & ⟶_{π_{λ,1}} & ℳ^{λ}(Np) \end{matrix}  (2.2)

Let us use the notation

π_{λ}(g) = (U^{λ})^{−v}π_{λ,1}(U^{v}(g)) ∈ ℳ^{λ}(Γ_{0}(Np), ψ, ℂ)  (2.3)

for the canonical λ-characteristic projection of g ∈ ℳ(Γ_{0}(Np^{v+1}), ψ, ℂ). Proof of (a). The linear operator (U^{λ})^{v} acts on the A-linear vector space ℳ^{λ}(Np^{v+1}) of finite dimension, and its determinant is in A^{∗}, hence the Alinear operator (U^{λ})^{v} is invertible. Proof of (b). We have the obvious inclusion of vector spaces: ℳ^{λ}(Np) ⊂ ℳ^{λ}(Np^{v+1}). On the other hand the A-vector spaces ℳ^{λ}(Np^{v+1}) and ℳ^{λ}(Np) are isomorphic by (a), hence they coincide:

ℳ^{λ}(Np) ⊂ ℳ^{λ}(Np^{v+1}) = U^{v}(ℳ^{λ}(Np^{v+1})) ⊂ ℳ^{λ}(Np).

Proof of (c). Following the theory of reduction of endomorphisms in finite dimensional vector spaces over a field K, the canonical projector π_{λ,v} onto the λ-characteristic subspace ⋃_{n≥1} Ker (U − λI)^{n} with the kernel ⋂_{n≥1} Im (U − λI)^{n} can be expressed, on one hand, as a polynomial of U over K, hence π_{λ,v} commutes with U. On the other hand, the restriction of π_{λ,v+1} on ℳ(Np) coincides with π_{λ,1} : ℳ(Np) → ℳ^{λ}(Np), because its image is

⋃_{n≥1} Ker (U − λI)^{n} ∩ ℳ(Np) = ⋃_{n≥1} Ker (U|_{ℳ(Np)} − λI)^{n},

and its kernel is

⋂_{n≥1} Im (U − λI)^{n} ∩ ℳ(Np) = ⋂_{n≥1} Im (U|_{ℳ(Np)} − λI)^{n}.

2.2 A general result on admissible measures with values in modular forms (a criterion for admissibility)

Consider the profinite group Y = lim_{\overleftarrow{v}} Y_{v} where Y_{v} = (ℤ/Np^{v}ℤ)^{×}. There is a

natural projection y_{p} : Y → ℤ_{p}^{×}. Let A be a normed ring over ℤ_{p}, and M be a normed A-module with the norm | · |_{p,M}. Let us recall Definition 0.1, c): for a given positive integer h an h-admissible measure on Y with values in M is an A-module homomorphism

Φ̃ : 𝒫^{h}(Y, A) → M

such that for fixed a ∈ Y and for v → ∞

|∫_{a+(Np^{v})}(y_{p} − a_{p})^{h′}dΦ̃|_{p,M} = o(p^{−v(h′−h)}) for all h′ = 0, 1, …, h − 1,

where a_{p} = y_{p}(a), 𝒫^{h}(Y, A) denotes the A-module of locally polynomial functions of degree < h of the variable y_{p} : Y → ℤ_{p}^{×} ↪ A^{×}. We adopt the notation (a)_{v} = a+(Np^{v}) for both an element of Y_{v} and the corresponding open compact subset of Y. We wish now to construct an h-admissible measure Φ̃^{λ} : 𝒫^{h}(Y, A) → ℳ(A) out of a sequence of distributions

Φ_{r}^{λ} : 𝒫^{1}(Y, A) → ℳ(A)

with values in an A-module M = ℳ(A) of modular forms over A as in Section 2.1). For this purpose we recall first Proposition 2.3, (c). Suppose that λ ∈ A^{×} is an invertible element of the algebra A. Recall that the λ-characteristic projection operator

π_{λ,v} : ℳ(Np^{v}; A) → ℳ(Np^{v}; A)^{λ} ⊂ ℳ(Np^{v}; A) (v ≥ 1)

is determined by the kernel ⋂_{n≥1} Im(U − λI)^{n}; this projector is given as a poly-

nomial of U over A whose degree is bounded by the rank of ℳ(Np^{v}; A). Using Proposition 2.3(c), the sequence of projectors π_{λ,v} can be glued to the canonical projection operator

π_{λ} : ℳ(A) → ℳ(A)^{λ} ⊂ ℳ(A)  (2.4)

given for all g ∈ ℳ(A) by

π_{λ}(g) = g^{λ} = U^{−v} [π_{λ,1}U^{v}(g)]

(g^{λ} is well defined if v is sufficiently large so that g ∈ ℳ(Np^{v+1})).

Next we construct an admissible measure

Φ̃^{λ} : 𝒫^{h}(Y, A) → ℳ(Np; A)

such that

∫_{(a)_{v}} y_{p}^{r} dΦ̃^{λ} = Φ_{r}^{λ}((a)_{v}) = π_{λ}(Φ_{r}((a)_{v}))

where Φ_{r} : 𝒫^{1}(Y, A) → ℳ(A) are ℳ(A)-valued distributions on Y for r = 0, 1, …, h − 1 , and Φ_{r}^{λ}((a)_{v}) are their λ-characteristic projections given by

Φ_{r}^{λ}((a)_{v}) = U^{−v′} [π_{λ,1}U^{v′}Φ_{r}((a)_{v})]

for any sufficiently large v′. Note first of all that the definition

∫_{(a)_{v}} y_{p}^{r} dΦ̃^{λ} = Φ_{r}^{λ}((a)_{v}) = U^{−κv} [π_{λ,1}U^{κv}Φ_{r}((a)_{v})] .

of the linear form Φ̃^{λ} : 𝒫^{h}(Y, A) → ℳ(A) is independent on the choice of the level: for any sufficiently large v′, we have by Proposition 2.3 the following comutative diagram

ℳ(Np^{v′+1}; A) ⟶^{π_{λ,v′+1}} ℳ(Np^{v′+1}; A)^{λ}
U^{v′} ↓  ↓≀ U^{v′}
ℳ(Np; A) ⟶^{π_{λ,1}}  ℳ(Np; A)^{λ}

in which the right vertical arrow is an A-isomorphism by Proposition 2.3 (b), and the A-linear endomorphism U commutes with the characteristic projectors π_{λ,v′+1}, π_{λ,1}. Hence the following sequence stabilizes: for some v′_{0} and for all v′ ≥ v′_{0} we have that

U^{−v′} [π_{λ,1}U^{v′}Φ_{r}((a)_{v})] = U^{−v′_{0}} [π_{λ,1}U^{v′_{0}}Φ_{r}((a)_{v})] .

Theorem 2.4 Let λ ∈ A be an element whose absolute value is a positive constant with 0 < |λ|_{p} < 1. Suppose that there exists a positive integer κ such that for any (a)_{v} ⊂ Y the following two conditions are satisfied:

Φ_{r}((a)_{v}) ∈ ℳ(N′p^{κv}), with N′ independent of v,  (level)

|U^{κv}(∑_{r′=0}^{r} \binom{r}{r′}(−y_{p}^{0})^{r−r′}Φ_{r′}((a)_{v})))|_{p} ≤ Cp^{−vr}  (growth)

for all r = 0, 1, …, h − 1 with h = [κord_{p}(λ)] + 1. Then there exists an h-admissible measure Φ̃^{λ} : 𝒫^{h}(Y, A) → ℳ such that for all ((a)_{v}) ⊂ Y and for all r = 0, 1, …, h − 1 one has

∫_{(a)_{v}} y_{p}^{r} dΦ̃^{λ} = Φ_{r}^{λ}((a)_{v})

where

Φ_{r}^{λ}((a)_{v}) = π_{λ}(Φ_{r}((a)_{v})) := U^{−κv} [π_{λ,1}U^{κv}Φ_{r}((a)_{v})]

is the canonical projection of π_{λ} of the modular form Φ_{r}((a)_{v}) (note that U^{κv}Φ_{r}((a)_{v}) ∈ ℳ(Np^{κv}; A)^{λ} = ℳ(Np; A)^{λ} because of the inclusion U^{κv−1}(ℳ(Np^{κv}; A)) ⊂ ℳ(Np; A) for all v ≥ 1, see Proposition 2.3 (a))

Proof. We need to check the h-growth condition of Definition 0.1, c) for the linear form

Φ̃^{λ} : 𝒫^{h}(Y, A) → ℳ(A)^{λ}

(given by the condition of Theorem 2.4). This growth condition says that for all a ∈ Y and for v → ∞

|∫_{(a)_{v}}(y_{p} − y_{p}^{0})^{r} dΦ̃^{λ}|_{p,ℳ} = o(p^{−v(r−h)})

for all r = 0, 1, …, h − 1, where h = [κord_{p}(λ)] + 1 and y_{p}^{0} = y_{p}(a). Let us develop the definition of Φ̃^{λ} using the binomial formula:

∫_{(a)_{v}}(y_{p} − y_{p}^{0})^{r} dΦ̃^{λ} = ∑_{r′=0}^{r} \binom{r}{r′}(−y_{p}^{0})^{r−r′}Φ_{r′}^{λ}((a)_{v}) = λ^{−vκ}·

λ^{vκ} · U^{−vκ} [π_{λ,1}U^{κv}(∑_{r′=0}^{r} \binom{r}{r′}(−y_{p}^{0})^{r−r′}Φ_{r′}((a)_{v}))] .  (2.5)

First we notice that all the operators

λ^{vκ} · U^{−vκ} = (λ^{−1}U)^{−vκ} = (I + λ^{−1}Z)^{−vκ} = ∑_{j=0}^{n−1} \binom{−vκ}{j}(λ^{−1}Z)^{j}

are uniformly bounded for v → ∞ by a positive constant C_{1} (where U = λI + Z and Z^{n} = 0 where n is the rank of ℳ(Np; A)). Note that the binomial

coefficients \binom{−vκ}{j} are all ℤ_{p}-integral.

On the other hand by the condition (growth) of the theorem (for the distributions Φ_{r}) we have the following inequality:

|U^{κv}(∑_{r′=0}^{r} \binom{r}{r′}(−y_{p}^{0})^{r−r′}Φ_{r′}((a)_{v}))|_{p,ℳ} ≤ Cp^{−vr}

for all r = 0, 1, …, κh − 1. If we apply to this estimate the previous bounded operators we get

|∫_{(a)_{v}}(y_{p} − y_{p}^{0})^{r} dΦ̃^{λ}|_{p,ℳ} ≤ C · C_{1}|λ^{−vκ}|_{p} · p^{−vr} = o(p^{−v(r−h)}) because of the estimate

|λ^{−vκ}|_{p} = (p^{ord_{p}(λ)})^{vκ} = o(p^{vh}), and κord_{p}(λ) < h = [κord_{p}(λ)] + 1. (2.6)

We apply Theorem 2.4 in Section 5.1 in order to obtain a p-adic measure in the form μ_{λ,Φ} = ℓ(π_{λ}(Φ)). Here λ is a non-zero eigenvalue of Atkin’s operator U = U_{p} acting on ℳ, ℓ : ℳ^{λ}(Np; A) → A is an A-linear form, applied to the projection π_{λ} : ℳ → ℳ^{λ} ⊂ ℳ^{λ}(Np; A) of a modular distribution Φ, where A = ℂ_{p}.

3 Main Congruence for the higher twists of the Siegel-Eisenstein series

The purpose of this section is to show that the admissibility criterion of Theorem 2.4 with h^{∗} = 2 is satisfied by a sequence of modular distributions (1.2), constructed in Section 1.

3.1 Construction of a sequence of modular distributions

As in the Introduction, consider the series F_{χ,r} = G^{⋆}(𝒵, −r; k, (Np^{v})^{2}, ψ), given by (A.12), viewed as formal (nearly-holomorphic) Fourier series, whose coefficients admit explicit polynomial expressions. The only property that we use in this section is the fact that they can be written in terms of simple p-adic integrals:

F_{χ,r} = ∑_{𝒯}det(𝒯)^{k−2r−κ}Q(R, 𝒯; k − 2r, r)a_{χ,r}(𝒯)q^{𝒯} ,

[PaSE], [PaIAS] and (1.1)). Here we use a universal polynomial, described in [CourPa], Theorem 3.14 as follows:

Q(R, 𝒯) = Q(R, 𝒯; k − 2r, r)  (3.1)

= ∑_{t=0}^{r} \binom{r}{t} det(𝒯)^{r−t} ∑_{|L|≤mt−t} R_{L}(κ − k + r)Q_{L}(R, 𝒯),

Q_{L}(R, 𝒯) = tr (^{t}ρ_{m−l_{1}}(R)ρ_{l_{1}}^{⋆}(𝒯)) · . . . · tr (^{t}ρ_{m−l_{t}}(R)ρ_{l_{t}}^{⋆}(𝒯))),

where we use the natural representation ρ_{r} : GL_{m}(ℂ) ⟶ GL(∧^{r}ℂ^{m}) (0 ≤ r ≤ m) of the group GL_{m}(ℂ) on the vector space Λ^{r}ℂ^{m}. Thus ρ_{r}(z) is a matrix of size \binom{m}{r} × \binom{m}{r} composed of the subdeterminants of z of degree r. Put ρ_{r}^{⋆}(z) = det(z)ρ_{m−r}(^{t}z)^{−1}. Then the representations ρ_{r} and ρ_{r}^{⋆} turn out to be polynomial representations so that for each z ∈ M_{m}(ℂ) the linear operators ρ_{r}(z), ρ_{r}^{⋆}(z) are well defined. In (3.1), L runs over all the multi-indices 0 ≤ l_{1} ≤ ⋯ ≤ l_{t} ≤ m, such that |L| = l_{1} + ⋯ + l_{t} ≤ mt − t. The coefficients R_{L}(β) ∈ ℤ[1/2][β] in (3.1) are polynomials in β of degree (mt − |L|) and with coefficients in the ring ℤ[1/2].

3.2 Utilizing the admissibility criterion

Recall an important property of the sequence of distributions Φ_{r} defined by (1.2), Section 1: the nebentypus character of Φ_{r}(χ) is (ψ_{1}, ψ_{2}, ψ_{3}), so that it does not depend on χ. Now let us prove that the sequence of distributions Φ_{r} on Y produces a certain admissible measure Φ̃ with values in a finite dimensional ℂ_{p}-vector subspace

ℳ^{λ} ⊂ ℳ, ℳ = ℳ_{k,r}(ℂ_{p}) ⊗ ℳ_{k,r}(ℂ_{p}) ⊗ ℳ_{k,r}(ℂ_{p}),

(of nearly-holomorphic triple modular forms over ℂ_{p}) using a general admissibility criterion (see Theorem 2.4).

3.3 Sufficient conditions for admissibility of measures with values in nearly-holomorphic modular forms

In order to construct the admissible measures of Theorem B we use the admissible measures μ̃^{λ}(f_{1} ⊗ f_{2} ⊗ f_{3}, y) constructed in Section 5 out of the modular distributions Φ_{r} in the form

μ̃^{λ}(f_{1} ⊗ f_{2} ⊗ f_{3})(χy_{p}^{r}) = ℓ(π_{λ}(Φ_{r})(χ)).

The growth condition for μ̃^{λ} follows then from a growth condition for Φ_{r}:

sup_{a∈Y} |∫_{a+(Np^{v})}(y_{p} − a_{p})^{r}dΦ̃^{λ}|_{p} = o (|Np^{v}|_{p}^{r−2ord_{p}λ}) ,  (3.2)

where

Φ̃^{λ}(χy_{p}^{r}) = π_{λ}(Φ_{r}(χ)).

Let us use a general result giving a sufficient condition for the admissibility of measures with values in nearly-holomorphic Siegel modular forms (given in Theorem 2.4) with κ = 2, h = [2ord_{p}λ] + 1. Then we need to check that the nearly-holomorphic triple modular forms Φ_{r}(χ) are of level N^{2}χ^{2v}, nebentypus (ψ_{1}, ψ_{2}, ψ_{3}), and satisfy the congruences

|U_{T}^{2v}(∑_{r′=0}^{r} \binom{r}{r′}(−a_{p}^{0})^{r−r′}Φ_{r′}((a)_{v})))|_{p} ≤ Cp^{−vr}  (3.3)

and for all r = 0, 1, ⋯, k − 2.

3.4 Special Fourier coefficients of the higher twist of the Siegel-Eisenstein distributions

Let us use the Fourier expansions (1.13) for Ψ_{r}(χ). These formulas directly imply the Fourier expansion of Φ_{r}(χ)|U_{p}^{2v} as follows

Φ_{r}(χ)|U_{p}^{2v} = ∑_{t_{1},t_{2}t_{3}≥0} a(p^{2v}t_{1}, p^{2v}t_{2}, p^{2v}t_{3}; p^{2v}R_{1}, p^{2v}R_{2}, p^{2v}R_{3}, r)q_{1}^{t_{1}}q_{2}^{t_{2}}q_{3}^{t_{3}} (3.4)

Admissible p-adic Measures … 103

with

a(p^{2v}t_{1}, p^{2v}t_{2}, p^{2v}t_{3}; p^{2v}R_{1}, p^{2v}R_{2}, p^{2v}R_{3}, r)

= ∑_{𝒯:diag(𝒯)=(p^{2v}t_{1},p^{2v}t_{2},p^{2v}t_{3})} χ̄(t_{12}t_{13}t_{23})ψ̄_{2}ψ_{3}(t_{13})ψ̄_{1}ψ_{3}(t_{23})×

× det(𝒯)^{k−2r−κ}Q(p^{2v} diag(R_{1}, R_{2}, R_{3}), 𝒯; k − 2r, r)2^{r}a_{χ,r}(𝒯)

= ∑_{𝒯:diag(𝒯)=(p^{2v}t_{1},p^{2v}t_{2},p^{2v}t_{3})} v_{χ,r}(𝒯, diag(R_{1}, R_{2}, R_{3})),

where

v_{χ,r}(𝒯, diag(R_{1}, R_{2}, R_{3})) = χ̄(t_{12}t_{13}t_{23})ψ̄_{2}ψ_{3}(t_{13})ψ̄_{1}ψ_{3}(t_{23})×  (3.5)
× det(𝒯)^{k−2r−κ}Q(p^{2v} diag(R_{1}, R_{2}, R_{3}), 𝒯; k − 2r, r)2^{r}a_{χ,r}(𝒯)
= χ^{(p)}(2)χ̄^{(p)}(𝒯)χ^{∘}(t_{12}t_{13}t_{23})ψ̄_{2}ψ_{3}(t_{13})ψ̄_{1}ψ_{3}(t_{23})×
× det(𝒯)^{k−2r−κ}Q(p^{2v} diag(R_{1}, R_{2}, R_{3}), 𝒯; k − 2r, r)2^{r}a_{χ,r}(𝒯).

Let us notice that, for any 𝒯 with diag(𝒯) = (p^{2v}t_{1}, p^{2v}t_{2}, p^{2v}t_{3}) one has

det(𝒯) ≡ 2t_{12}t_{13}t_{23} mod p^{2v},

χ^{(p)}(2t_{12}t_{13}t_{23}) = χ^{(p)}(det(𝒯)) = χ(det(𝒯)\overline{χ^{∘}(det(𝒯)},
2^{r}a_{χ,r}(𝒯) = ∫_{Y} y_{p}^{r}χ(y)dℱ_{𝒯} ,
with χ = χ^{(p)}χ^{∘}, χ^{(p)} mod p^{v}, χ^{∘} mod N, and p ∤ N,

for a bounded measure ℱ_{𝒯} on Y with values in \overline{ℚ}. It follows that

v_{χ,r}(𝒯, diag(R_{1}, R_{2}, R_{3}))
= χ^{(p)}(2)χ̄(det(𝒯)) det(𝒯)^{−r}\overline{χ^{∘}(det(𝒯)}ψ̄_{2}ψ_{3}(t_{13})ψ̄_{1}ψ_{3}(t_{23})·  (3.6)
· det(𝒯)^{k−r−κ}Q(p^{2v} diag(R_{1}, R_{2}, R_{3}), 𝒯; k − 2r, r)2^{r}a_{χ,r}(𝒯)  (3.7)

= det(𝒯)^{k−r−κ}Q(p^{2v} diag(R_{1}, R_{2}, R_{3}), 𝒯; k − 2r, r)\overline{χ^{∘}}(2) ∫_{Y} χy_{p}^{r}dℱ_{𝒯;χ^{∘},ψ_{1},ψ_{2},ψ_{3}} ,

Documenta Mathematica · Extra Volume Coates (2006) 77–132

where ℱ_{𝒯;χ^{∘},ψ_{1},ψ_{2},ψ_{3}} denotes the bounded measure defined by the equality:

∫_{Y} χy_{p}^{r}dℱ_{𝒯;χ^{∘},ψ_{1},ψ_{2},ψ_{3}}  (3.8)

= χ^{(p)}(2)χ^{∘}(2)2^{r}χ̄(det(𝒯)) det(𝒯)^{−r}\overline{χ^{∘}(det(𝒯)}ψ̄_{2}ψ_{3}(t_{13})ψ̄_{1}ψ_{3}(t_{23})a_{χ,r}(𝒯).

3.5 Main Congruence for the Fourier expansions

Let us use the orthogonality relations for Dirichlet characters in order to prove the admissibility of the distributions given by the sequence π_{λ}(Φ_{r}(χ)) using the Fourier expansions (3.4). According to the admissibility criterion of Theorem 2.4 we need to check the following Main Congruence:

| ∑_{r′=0}^{r} \binom{r}{r′}(−a_{p}^{0})^{r−r′} \frac{1}{φ(Np^{v})} ∑_{χ mod Np^{v}} χ^{−1}(a)v_{χ,r′}(𝒯, p^{2v} diag(R_{1}, R_{2}, R_{3}))|_{p}
≤ Cp^{−vr},  (3.9)

where we use the notation (3.6) for v_{χ,r′}(𝒯, diag(R_{1}, R_{2}, R_{3})), implying that the coefficients

i_{p}(v_{χ,r′}(𝒯, diag(R_{1}, R_{2}, R_{3})))

in (3.5) are given as sums of the following expressions:

B_{r}(χ, 𝒯) = \overline{χ^{∘}}(2) det(𝒯)^{k−r−κ} ∫_{Y} χy_{p}^{r}dℱ_{𝒯;χ^{∘},ψ_{1},ψ_{2},ψ_{3}} ·  (3.10)

· ∑_{t=0}^{r} \binom{r}{t} det(𝒯)^{r−t} ∑_{|L|≤mt−t} R_{L}(κ − k + r)Q_{L}(p^{2v} diag(R_{1}, R_{2}, R_{3}), 𝒯),

where ℱ_{𝒯;χ^{∘},ψ_{1},ψ_{2},ψ_{3}} denotes the bounded measure defined by (3.8). Using the expressions (3.10), the main congruence (3.9) is reduced to proving the congruence for the numbers B_{r}(χ, 𝒯): there exists a non-zero integer C_{k} such that

C_{k} · ∑_{r′=0}^{r} \binom{r}{r′}(−a_{p}^{0})^{r−r′} \frac{1}{φ(Np^{v})} ∑_{χ mod Np^{v}} χ^{−1}(a)B_{r′}(χ, 𝒯) ≡ 0 mod p^{vr} (3.11)

⟺ C_{k} · A ≡ 0 mod Np^{v}^{r}, where we use the notation

A =A_{r}(𝒯; χ^{∘}, ψ_{1}, ψ_{2}, ψ_{3}) = ∑_{r′=0}^{r} \binom{r}{r′}(−a_{p}^{0})^{r−r′} \frac{1}{φ(Np^{v})} ∑_{χ mod Np^{v}} χ^{−1}(a)· (3.12)

· \overline{χ^{∘}}(2) det(𝒯)^{k−r′−κ} ∫_{Y} χy_{p}^{r′}dℱ_{𝒯;χ^{∘},ψ_{1},ψ_{2},ψ_{3}} ∑_{t=0}^{r′} \binom{r′}{t} det(𝒯)^{r′−t}

∑_{|L|≤mt−t} R_{L}(κ − k + r′)Q_{L}(p^{2v} diag(R_{1}, R_{2}, R_{3}), 𝒯).

Note that R_{L}(κ − k + r′) is a polynomial of degree mt − |L| = 3t − |L| in κ − k + r′ (see (3.1)), hence in r′, and \binom{r′}{t} is a polynomial of degree t in r′. One can therefore write

\binom{r′}{t}R_{L}(κ − k + r) = ∑_{n=0}^{4t−|L|} μ_{n} \frac{(r′ + n + 1)!}{(r′ + 1)!}.

Here the coefficients μ_{n} are fixed rational numbers (independent of r′). Using the orthogonality relations for Dirichlet characters mod Np^{v}, we see that the sum over r′ in (3.12), denoted by C = C_{r}(t, L, 𝒯; χ^{∘}, ψ_{1}, ψ_{2}, ψ_{3}), takes the form

C_{r}(t, L, 𝒯; χ^{∘}, ψ_{1}, ψ_{2}, ψ_{3}) = \overline{χ^{∘}}(2) det(𝒯)^{k−t−κ}

∫_{y≡a mod p^{v}} ∑_{n=0}^{4t−|L|} μ_{n} \underbrace{∑_{r′=0}^{r} \binom{r}{r′}(−a)^{r−r′} \frac{(r′ + n + 1)!}{(r′ + 1)!}y^{r′}}_{y^{−n} \frac{∂^{n}}{∂y^{n}} (y^{n+1}(y − a)^{r})} dℱ_{𝒯;χ^{∘},ψ_{1},ψ_{2},ψ_{3}} (y)

Note that we write χ = χ^{∘}χ^{(p)}, fix χ^{∘}, and sum over all characters χ^{(p)} mod p^{v}. We have therefore (y − a)^{r} ≡ 0 mod (p^{v})^{r} in the integration domain y ≡ a mod p^{v}, implying the congruence

c_{k}C_{r}(t, L, 𝒯; χ^{∘}, ψ_{1}, ψ_{2}, ψ_{3}) ≡ 0 (mod (p^{v})^{r−n}) ≡ 0 (mod (p^{v})^{r−4t+|L|}),

(3.13)

where c_{k} ∈ ℚ^{∗} is a nonzero constant coming from the denominators of the fixed rational numbers μ_{n}, and of the bounded distributions ℱ_{𝒯;χ^{∘},ψ_{1},ψ_{2},ψ_{3}} .

3.6 Proof of the Main Congruence

Now the expression (3.12) transforms to

A_{r}(𝒯) = ∑_{t=0}^{r} ∑_{|L|≤2t} det(𝒯)^{t} · C(t, L, 𝒯) det(𝒯)^{k−2r−κ}Q_{L}(p^{2v} diag(R_{1}, R_{2}, R_{3}), 𝒯),

(3.14) where Q_{L}(p^{2v} diag(R_{1}, R_{2}, R_{3}), 𝒯) is a homogeneuos polynomial of degree 3t − |L| in the variables R_{ij} implying the congruence

Q_{L}(p^{2v} diag(R_{1}, R_{2}, R_{3}), 𝒯) ≡ 0 (mod (p^{2v})^{(3t−|L|)}).  (3.15)

On the other hand we know from the description (3.1) of the polynomial

Q(R, 𝒯) = Q(R, 𝒯; k − 2r, r) = ∑_{t=0}^{r} \binom{r}{t} det(𝒯)^{r−t} ∑_{|L|≤2t} R_{L}(κ − k + r)Q_{L}(R, 𝒯),

Q_{L}(R, 𝒯) = tr (^{t}ρ_{3−l_{1}}(R)ρ_{l_{1}}^{⋆}(𝒯)) · . . . · tr (^{t}ρ_{3−l_{t}}(R)ρ_{l_{t}}^{⋆}(𝒯)) ,

that 2t − |L| ≥ 0 so we obtain the desired congruence as follows

\begin{cases}c_{k}C_{r}(t, L, 𝒯) ≡ 0 (mod (p^{v})^{r−4t+|L|}) \\ Q_{L}(p^{2v} diag(R_{1}, R_{2}, R_{3}), 𝒯) ≡ 0 (mod (p^{2v})^{(3t−|L|)}) \end{cases}  (3.16)
⇒ c_{k}A_{r}(𝒯) ≡ 0 (mod p^{vr}),

since v(r − 4t + |L|) + 2v(3t − |L|) = vr + 2vt − v|L| ≥ vr, proving (3.9).

3.7 Construction of admissible measures with values in nearlyholomorphic modular forms

We wish now to construct an h-admissible measure Φ̃^{λ} : 𝒫^{h}(Y, A) → ℳ_{T} (A) on Y out of the following sequence of the higher twists of Siegel-Eisenstein distributions given by the equality (1.2):

Φ_{r} := 2^{r} diag^{∗} Ψ_{r} = 2^{r}F_{χ,r}^{χ̄_{1},χ̄_{2},χ̄_{3}}, Φ_{r} : 𝒫^{1}(Y, A) → ℳ_{T} (A)

(they take values in the A-module

M = ℳ_{T} (ψ_{1}, ψ_{2}, ψ_{3}; A) ⊂ ℳ_{k,r}(ψ_{1}; A) ⊗ ℳ_{k,r}(ψ_{2}; A) ⊗ ℳ_{k,r}(ψ_{3}; A)

of triple modular forms over A = ℂ_{p} or A = \overline{ℚ}).

Theorem 3.1 Let λ ∈ A be an element whose absolute value is a positive constant with 0 < |λ|_{p} < 1, and define h = [2ord_{p}(λ)] + 1. Then the sequence (1.2) satisfies for any (a)_{v} ⊂ Y the following two conditions:

Φ_{r}((a)_{v}) ∈ ℳ(N′p^{2v}), with N′ independent of v,  (level)
|U_{T}^{2v}(∑_{r′=0}^{r} \binom{r}{r′}(−y_{p}^{0})^{r−r′}Φ_{r′}((a)_{v})))|_{p} ≤ Cp^{−vr}  (growth)

for all r = 0, 1, …, h − 1. Moreover, there exists an h-admissible measure Φ̃^{λ} : 𝒫^{h}(Y, A) → ℳ_{T} such that for all ((a)_{v}) ⊂ Y and for all r = 0, 1, …, h − 1 one has

∫_{(a)_{v}} y_{p}^{r} dΦ̃^{λ} = Φ_{r}^{λ}((a)_{v})

where

Φ_{r}^{λ}((a)_{v}) = π_{λ,T} (Φ_{r}((a)_{v})) := U_{T}^{−2v} [π_{λ,1}U_{T}^{2v}Φ_{r}((a)_{v})]

is the canonical projection of π_{λ} of the triple modular form Φ_{r}((a)_{v}) (note that U_{T}^{2v}Φ_{r}((a)_{v}) ∈ ℳ_{T} (Np^{2v}; A)^{λ} = ℳ_{T} (Np; A)^{λ} because of the inclusion U_{T}^{2v−1}(ℳ_{T} (Np^{2v}; A)) ⊂ ℳ_{T} (Np; A) for all v ≥ 1, see Proposition 2.3 (a)).

Proof. We use Theorem 2.4 with κ = 2, and we to check the h-growth condition for the A-linear map

Φ̃^{λ} : 𝒫^{h}(Y, A) → ℳ_{T} (A)

defined in Theorem 3.1. We have to check that for any ((a)_{v}) ∈ Y the following two conditions are satisfied: for all r = 0, 1, …, h − 1,

Φ_{r}((a)_{v}) ∈ ℳ(N^{2}p^{2v}),  (level)

|U_{T}^{2v}(∑_{r′=0}^{r} \binom{r}{r′}(−y_{p}^{0})^{r−r′}Φ_{r′}((a)_{v}))|_{p} ≤ Cp^{−vr}.  (growth)

The (level) condition is implied by the definition (1.2)

Φ_{r}(χ) =2^{r} diag^{∗} F_{χ,r}^{χ̄_{1},χ̄_{2},χ̄_{3}},

and Proposition 1.5. The (growth) is deduced from the Main Congruence (3.9) (proved in Section 3.6) for the Fourier coefficients of the functions (1.2).

4 A trilinear form on the characteristic subspace of the Uoperator

4.1 The adjoint operator U^{∗}

Let f = ∑_{n=1}^{∞} a_{n}q^{n} denote a primitive cusp eigenform of conductor dividing Np, with coefficients i_{p}(a_{n}) in a finite extension K of ℚ_{p} and of Dirichlet character ψ modulo N. Let α ∈ K be a root of the Hecke polynomial x^{2} − a_{p}(f)x + ψ(p)p^{k−1} as above, and let α′ denote the other root. Recall that the function f_{0} = ∑_{n=1}^{∞} a_{n}(f_{0})q^{n} ∈ \overline{ℚ}[[q]] is defined by (0.9) as an eigenfunction of U = U_{p} with the eigenvalue α ∈ \overline{ℚ}. In the following proposition, let U^{∗} denote the operator adjoint to

U = U_{p} : ℳ_{r,k}(Γ_{1}(Np), ℂ) → ℳ_{r,k}(Γ_{1}(Np), ℂ)

in the complex vector space ℳ_{r,k}(Γ_{1}(Np), ℂ) with respect to the Petersson inner product.

Proposition 4.1 (a) The following operator identity holds: U^{∗} = W_{Np}^{−1}UW_{Np} (in the complex vector space ℳ_{r,k}(Γ_{1}(Np), ℂ)).

(b)There are the following identities in ℳ_{r,k}(Γ_{1}(Np), ℂ):

f^{0}|U^{∗} = \overline{α}f^{0} and T_{l}(f^{0}) = a_{l}(f)f^{0}

for all “good primes” l ∤ Np.

(c) The linear form g ↦ ⟨f^{0}, g⟩_{Np} on ℳ_{r,k}(Γ_{1}(Np), ℂ) vanishes on the complex vector subspace Kerπ_{α,1} = Im(U −αI)^{n_{1}} where n_{1} = dim ℳ_{r,k}(Γ_{1}(Np), ℂ), and we use the same notation as above

π_{α,1} : ℳ_{r,k}(Γ_{1}(Np), ℂ) → ℳ_{r,k}^{α}(Γ_{1}(Np), ℂ)

for the complex characteristic projection onto the α-primary subspace of the operator U (acting on the finite-dimensional complex vector space ℳ_{r,k}(Γ_{1}(Np), ℂ)) hence

⟨f^{0}, g⟩_{Np} = ⟨f^{0}, π_{α,1}(g)⟩_{Np}

(d) If g ∈ ℳ(Np^{v+1}; \overline{ℚ}) and α ≠ 0, then we have the equality

⟨f^{0}, π_{α}(g)⟩_{Np} = α^{−v}⟨f^{0}, U^{v}g⟩_{Np}

where

π_{α}(g) = g^{α} = U^{−v}[π_{α,1}U^{v}g] ∈ ℳ^{α}(Np)

is the α-part of g.

(e) The linear form

ℒ_{f,α} : ℳ(Np^{v}; ℂ) → ℂ, g ↦ \frac{⟨f^{0}, α^{−v}U^{v}(g)⟩_{Np}}{⟨f^{0}, f_{0}⟩_{Np}}

is defined over \overline{ℚ}:

ℒ_{f,α} : ℳ(Np^{v}; \overline{ℚ}) → \overline{ℚ}

and there exists a unique ℂ_{p}-linear form ℓ_{f,α} on ℳ(Np^{v}; ℂ_{p}) = ℳ(Np^{v}; \overline{ℚ})⊗_{i_{p}} ℂ_{p} such that ℓ_{f,α}(g) = i_{p}(ℒ_{f,α}(g)) for all g ∈ i_{p}(ℳ(Np^{v}; \overline{ℚ})).

Proof (a) See [Miy], Theorem 4.5.5 (see also [Ran90]).

(b) Let us use directly the statement a):

f^{0}|U^{∗} = f_{0}^{ρ}|W_{Np}W_{Np}^{−1}UW_{Np} = ᾱf^{ρ}|W_{Np} = ᾱf^{0}.

(c) If g ∈ Kerπ_{α,1} = Im(U − αI)^{n_{1}} then g = (U − αI)^{n_{1}}g_{1} and

⟨f^{0}, (U − αI)^{n_{1}}g_{1}⟩_{Np} = ⟨(U^{∗} − ᾱI)f^{0}, (U − αI)^{n_{1}−1}g_{1}⟩_{Np} = 0

hence ⟨f^{0}, g⟩_{Np} = 0; moreover

⟨f^{0}, g⟩_{Np} = ⟨f^{0}, π_{α,1}(g) + (g − π_{α,1}(g))⟩_{Np} = ⟨f^{0}, π_{α,1}(g)⟩_{Np}.

(d) Let us use the definitions and write the following product:

α^{v}⟨f^{0}, π_{α}g⟩_{Np} = ⟨U^{∗}^{v}(f^{0}), U^{−v}[π_{α,1}U^{v}g]⟩_{Np}
= ⟨f^{0}, π_{α,1}(U^{v}g)⟩_{Np} = ⟨f^{0}, U^{v}g⟩_{Np}

by (c) as U^{v}g ∈ ℳ(Np).

(e) Note that ℒ_{f,α}(f_{0}) = 1, f_{0} ∈ ℳ(Np; \overline{ℚ}). Consider the complex vector space

Kerℒ_{f,α} = ⟨f^{0}⟩^{⊥} = {g ∈ ℳ(Np^{v}; ℂ) | ⟨f^{0}, g⟩_{Np^{v}} = 0}.

It admits a \overline{ℚ}-rational basis (as it is stable under all “good” Hecke operators T_{l} (l ∤ Np):

⟨f^{0}, g⟩_{Np^{v}} = 0 ⇒ ⟨f^{0}, T_{l}g⟩_{Np^{v}} = ⟨T_{l}^{∗}f^{0}, g⟩_{Np^{v}} = 0

and diagonalizing the action of T_{l} (over \overline{ℚ}) we get such a basis establishing e). We obtain then the ℂ_{p}-linear form ℓ_{f,α} on ℳ(Np^{v}; ℂ_{p}) = ℳ(Np^{v}; \overline{ℚ}) ⊗_{i_{p}} ℂ_{p} such that ℓ_{f,α}(g) = i_{p}(ℒ_{f,α}(g)) by extending scalars from \overline{ℚ} to ℂ_{p} via the imbedding i_{p}. Note that we use here only the α-part ℳ(Np^{v}; A)^{α} because the constructed linear form ℓ_{f,α} passes through the π_{α} (for A = ℂ_{p} , A = \overline{ℚ}, or A = ℂ). Moreover, f_{0} can be included to a basis {f_{0}, g_{i}}_{i=2,··· ,n} of ℳ(Np^{v}; A)^{α}, where g_{i} are eigenfunctions of all Hecke operators T_{l} for primes l ∤ Np; they are algebraically orthogonal to f_{0} (in the sense of the algebraic Petersson product studied by Hida [Hi90]) so that projection to the f_{0} part of this basis gives such an A-linear form.

4.2 The triple U-operator

In the following proposition, we consider the triple U-operator

U_{T} = U_{1,p} ⊗ U_{2,p} ⊗ U_{3,p} : ℳ_{T} (Γ_{1}(Np), ℂ) → ℳ_{T} (Γ_{1}(Np), ℂ), where (4.1)

ℳ_{T} (Γ_{1}(Np), ℂ) = ℳ_{k_{1}}(Γ_{1}(Np), ℂ) ⊗ ℳ_{k_{2}}(Γ_{1}(Np), ℂ) ⊗ ℳ_{k_{3}}(Γ_{1}(Np), ℂ),

acting on the complex vector space ℳ_{T} (Γ_{1}(Np), ℂ) endowed with the triple Petersson inner product ⟨·, ·⟩ defined by

⟨g_{1} ⊗ g_{2} ⊗ g_{3}, h_{1} ⊗ h_{2} ⊗ h_{3}⟩_{T} = ⟨g_{1}, h_{1}⟩_{Np}⟨g_{2}, h_{2}⟩_{Np}⟨g_{3}, h_{3}⟩_{Np}.

Let

U_{T}^{∗} = U_{1,p}^{∗} ⊗ U_{2,p}^{∗} ⊗ U_{3,p}^{∗}

denote the adjoint operator on ℳ_{T} (Γ_{1}(Np), ℂ) for the triple Petersson inner product. Recall the notation (0.9) and (0.10):

f_{j,0} = f_{j} − α_{p,j}^{(2)}f_{j}|V_{p} = f_{j} − α_{p,j}^{(2)}p^{−k/2}f_{j}| ({p \atop 0} {0 \atop 1})

f_{j,0}^{ρ} = ∑_{n=1}^{∞} \overline{a(n, f_{0})}q^{n}, f_{j}^{0} = f_{j,0}^{ρ}|_{k} W_{Np} = f_{j,0}^{ρ}|_{k} ({0 \atop Np} {−1 \atop 0}) .

Proposition 4.2 (a) The following operator identity holds:

U_{T}^{∗} = W_{Np}^{−1}U_{p,1}W_{Np} ⊗ W_{Np}^{−1}U_{p,2}W_{Np} ⊗ W_{Np}^{−1}U_{p,3}W_{Np}

(in the complex vector space ℳ_{T} (Γ_{1}(Np), ℂ)).

(b) There are the following identities in ℳ_{T} (Γ_{1}(Np), ℂ):

U_{T}^{∗}(f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}) = \overline{λ}(f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}).

(c)The linear form on ℳ_{T} (Γ_{1}(Np), ℂ) defined by

g_{1}⊗g_{2}⊗g_{3}⊗ ↦ ⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, g_{1} ⊗ g_{2} ⊗ g_{3}⟩_{T} = ⟨f_{1}^{0}, g_{1}⟩_{Np}⟨f_{2}^{0}, g_{2}⟩_{Np}⟨f_{3}^{0}, g_{3}⟩_{Np}

vanishes on the complex vector subspace Ker π_{λ,T,1} = Im(U_{T} − λI)^{n_{T}} where we write n_{T} = dim ℳ_{T} (Γ_{1}(Np), ℂ), and we use the notation

π_{λ,T,1} : ℳ_{T} (Γ_{1}(Np), ℂ) → ℳ_{T}^{λ} (Γ_{1}(Np), ℂ)

for the complex characteristic projection onto the λ-primary subspace of the operator U_{T} acting on the finite-dimensional complex vector space ℳ_{T} (Γ_{1}(Np), ℂ). Moreover, the following equality holds

⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, g_{1} ⊗ g_{2} ⊗ g_{3}⟩_{T} = ⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, π_{λ,T,1}(g_{1} ⊗ g_{2} ⊗ g_{3})⟩_{T} .

(d) If g ∈ ℳ_{T} (Np^{v+1}; \overline{ℚ}) and λ ≠ 0, then we have the equality

⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, π_{λ,T} (g)⟩_{T,Np} = λ^{−v}⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, U_{T}^{v}g⟩_{T,Np}

where

π_{λ,T} (g) = g^{λ} = U_{T}^{−v}[π_{λ,T,1}U_{T}^{v}g] ∈ ℳ_{T}^{λ} (Np)

is the λ-part of g.

(e) The linear form

ℒ_{T,λ} : ℳ_{T} (Np^{v}; ℂ) → ℂ, g ↦ \frac{⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, λ^{−v}U_{T}^{v}g⟩_{T,Np}}{⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, f_{1,0} ⊗ f_{2,0} ⊗ f_{3,0}⟩_{T,Np}}

is defined over \overline{ℚ}:

ℒ_{T,λ} : ℳ_{T} (Np^{v}; \overline{ℚ}) → \overline{ℚ}

and there exists a unique ℂ_{p}-linear form ℓ_{T,λ} on ℳ_{T} (Np^{v}; ℂ_{p}) = ℳ_{T} (Np^{v}; \overline{ℚ})⊗_{i_{p}} ℂ_{p} such that ℓ_{T,λ}(g) = i_{p}(ℒ_{f,α}(g)) for all g ∈ i_{p}(ℳ_{T} (Np^{v}; \overline{ℚ})).

Remark 4.3 We may view the trilinear form

(g_{1}, g_{2}, g_{3}) ↦ ℓ_{T,λ}(g_{1} ⊗ g_{2} ⊗ g_{3})

as a p-adic version of the triple Petersson product following Hida [Hi90].

Proof of Proposition 4.2, a), b) follows directly from that of Proposition 4.1. In order to prove c) we need to show that the linear form on ℳ_{T} (Γ_{1}(Np), ℂ) defined by

g_{1} ⊗ g_{2} ⊗ g_{3}⊗ ↦ ⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, g_{1} ⊗ g_{2} ⊗ g_{3}⟩_{T,Np}
= ⟨f_{1}^{0}, g_{1}⟩_{Np}⟨f_{2}^{0}, g_{2}⟩_{Np}⟨f_{3}^{0}, g_{3}⟩_{Np}

vanishes on the complex vector subspace

Ker π_{λ,T,1} = Im (U_{T} − λI)^{n_{T}} = (Ker (U_{T}^{∗} − \overline{λ}I)^{n_{T}})^{⊥}.

It suffices to notice that

f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0} ∈ Ker (U_{T}^{∗} − \overline{λ}I) ⊂ Ker (U_{T}^{∗} − \overline{λ}I)^{n_{T}} ,

because of the equality

U_{T}^{∗}(f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}) = U_{1,p}^{∗}(f_{1}^{0}) ⊗ U_{2,p}^{∗}(f_{2}^{0}) ⊗ U_{3,p}^{∗}(f_{3}^{0}) = \overline{λ}(f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}).

More precisely, if g ∈ Ker π_{λ,T,1} = Im(U_{T} − λI)^{n_{T}} then g = (U_{T} − λI)^{n_{T}} g_{1} and

⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, (U_{T} − λI)^{n}g_{1}⟩_{T,Np}
= ⟨(U_{T}^{∗} − \overline{λ}I)(f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, (U_{T} − λI)^{n−1}g_{1})⟩_{T,Np} = 0

hence ⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, g⟩_{T,Np} = 0. Moreover, the following equality holds

⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, g_{1} ⊗ g_{2} ⊗ g_{3}⟩_{T} = ⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, π_{λ,T,1}(g_{1} ⊗ g_{2} ⊗ g_{3})⟩_{T} ,

by the definition of the projection π_{λ,T,1}:

g_{1} ⊗ g_{2} ⊗ g_{3} − π_{λ,T,1}(g_{1} ⊗ g_{2} ⊗ g_{3}) ∈ Ker π_{λ,T,1}.

d) Let us use the definitions and write the following product:

λ^{v}⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, π_{λ,T} g⟩_{T,Np} = ⟨U_{T}^{∗}^{v}(f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}), U_{T}^{−v}[π_{λ,T,1}U_{T}^{v}g]⟩_{T,Np} =

⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, π_{λ,T,1}(U_{T}^{v}g)⟩_{T,Np} = ⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, U_{T}^{v}g⟩_{T,Np}

by c) as U_{T}^{v}g ∈ ℳ_{T} (Np).

e) Note that ℒ_{T,λ}(f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}) = 1, f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0} ∈ ℳ_{T} (Np; \overline{ℚ}). Consider the complex vector space

Ker ℒ_{T,λ} = ⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}⟩^{⊥} = {g ∈ ℳ_{T} (Np^{v}; ℂ) | ⟨f^{0}, g⟩_{T,Np^{v}} = 0}.

It admits a \overline{ℚ}-rational basis (as in Proposition 4.1) establishing e). We obtain then the ℂ_{p}-linear form ℓ_{T,λ} on ℳ_{T} (Np^{v}; ℂ_{p}) = ℳ_{T} (Np^{v}; \overline{ℚ})⊗_{i_{p}} ℂ_{p} such that ℓ_{T,λ}(g) = i_{p}(ℒ_{T,λ}(g)) by extending scalars from \overline{ℚ} to ℂ_{p} via the imbedding i_{p}.

5 Computation of p-adic integrals and L-values

5.1 Construction of p-adic measures

Let ℳ = ℳ_{T} (A) = ⋃_{v≥0} ℳ_{k,r}(Np^{v}, ψ_{1}; A) ⊗_{A} ℳ_{k,r}(Np^{v}, ψ_{2}; A) ⊗_{A} ℳ_{k,r}(Np^{v}, ψ_{3}; A) be the A-module of nearly-holomorphic triple modular forms with formal Fourier coefficients in A, where A = ℂ_{p}. Let us define an A-valued measure

μ̃^{λ}(y; f_{1} ⊗ f_{2} ⊗ f_{3}) : 𝒞^{loc−an}(Y, A) → A

by applying the trilinear form ℓ_{T,λ} : ℳ(Np^{v}; A) → A of Proposition 4.2

μ̃^{λ}(y; f_{1} ⊗ f_{2} ⊗ f_{3}) = ℓ_{T,λ}(Φ̃^{λ})  (5.1)

to the h-admissible measure Φ̃^{λ} of Theorem 2.4 on Y with values in ℳ(A)^{λ} ⊂ ℳ(Np; A). That h-admissible measure was defined as an A-linear map Φ̃^{λ} : 𝒫^{h}(Y, A) → ℳ(A)^{λ} satisfying for any (a)_{ν} ⊂ Y and for all r = 0, 1, …, h − 1 the following equality:

∫_{(a)_{ν}} y_{p}^{r} dΦ̃^{λ} = π_{λ}(Φ_{r}((a)_{ν})) ∈ ℳ(Np),

where h = [2ord_{p}λ(p)] + 1, hence

∫_{(a)_{ν}} y_{p}^{r} dμ̃^{λ}(y; f_{1} ⊗ f_{2} ⊗ f_{3}) = ℓ_{T,λ} (∫_{(a)_{ν}} y_{p}^{r} dΦ̃^{λ}(y)) .  (5.2)

5.2 Evaluation of the integral

∫_{Y} χ(y) y_{p}^{r} dμ̃^{λ}(y; f_{1} ⊗ f_{2} ⊗ f_{3})  (5.3)

for r ∈ ℕ, 0 ≤ r ≤ k − 2. The result is given in terms of Garrett’s triple L function 𝒟^{⋆}(f_{1}^{ρ} ⊗ f_{2}^{ρ} ⊗ f_{3}^{ρ}, 2k − 2 − r, ψ_{1}ψ_{2}χ). Let us use the action of the

involution W_{N_{j}} = (\begin{matrix}0 & −1 \\ N_{j} & 0 \end{matrix}) of the exact level N_{j} of f_{j}:

f_{j}|_{k}W_{N_{j}} = (\begin{matrix}0 & −1 \\ N_{j} & 0 \end{matrix}) = γ_{j} · f_{j}^{ρ}, f_{j}^{ρ}|_{k}W_{N_{j}} = (\begin{matrix}0 & −1 \\ N_{j} & 0 \end{matrix}) = γ̄_{j} · f_{j},

where f_{j}^{ρ}(z) = ∑_{n=1}^{∞} ā_{n,j}e(nz) ∈ 𝒮_{k}(N_{j}, ψ̄_{j}),  (5.4)
(j = 1, 2, 3) and γ_{j} is the corresponding root number.  (5.5)

Recall the notation (0.9) and (0.10):

f_{j,0} = f_{j} − α_{p,j}^{(2)}f_{j}|V_{p} = f_{j} − α_{p,j}^{(2)}p^{−k/2}f_{j}| ({p \atop 0} {0 \atop 1})

f_{j,0}^{ρ} = ∑_{n=1}^{∞} \overline{a(n, f_{0})}q^{n}, f_{j}^{0} = f_{j,0}^{ρ}|_{k} W_{Np} = f_{j,0}^{ρ}|_{k} ({0 \atop Np} {−1 \atop 0}) .

Proposition 5.1 Under the notations and assumptions as in Theorem B.2, the value of the integral (5.3) is given for 0 ≤ r ≤ k − 2 by the image under i_{p} of the following algebraic number

T · λ^{−2v}𝔏_{Np}(−r)\frac{𝒟^{⋆}(f_{1}^{ρ} ⊗ f_{2}^{ρ} ⊗ f_{3}^{ρ}, 2k − 2 − r, ψ_{1}ψ_{2}χ)}{⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, f_{1,0} ⊗ f_{2,0} ⊗ f_{3,0}⊗⟩_{T,N^{2}p^{2v}}} ,

where

T = 2^{−r} \frac{((Np)^{3}/N_{1}N_{2}N_{3})^{k/2}γ̄_{1}γ̄_{2}γ̄_{3}(χ_{1}χ_{2}χ_{3})(2)p^{3·v(k−2)}}{N_{1,1}N_{1,2}N_{1,3}G(χ_{1,0})G(χ_{2,0})G(χ_{3,0})} ×

× (Np^{2v})^{k−2r} \frac{N^{2}p^{2v}φ(N^{2}p^{2v})φ(Np^{v})}{[Γ_{0}(N^{2}p^{2v}) : Γ(N^{2}p^{2v})]^{3}} .

γ_{j} is the corresponding root number, given by (5.4), and the factor 𝔏_{Np}(−r), given by (5.13).

Remark. In particular, Propostion 5.1 implies Theorem A, using a computation by B.Gorsse and G.Robert (see [Go-Ro]) that for some β ∈ \overline{ℚ}^{∗}

⟨f_{1}^{0,ρ} ⊗ f_{2}^{0,ρ} ⊗ f_{3}^{0,ρ}, f_{1,0}^{ρ} ⊗ f_{2,0}^{ρ} ⊗ f_{3,0}^{ρ}⟩_{T,Np} = β · ⟨f_{1}, f_{1}⟩_{N}⟨f_{2}, f_{2}⟩_{N}⟨f_{3}, f_{3}⟩_{N}.

5.3 Evaluation of the trilinear form

In order to compute the p-adic integral, the next step of the proof uses computations similar to those in [Hi85], §4 and §7. More precisely let us write the integral in the form

∫_{Y} χ(y) y_{p}^{r} dμ̃_{λ}(y; f_{1} ⊗ f_{2} ⊗ f_{3}) = ∑_{a∈Y_{v}} χ(a) ∫_{(a)_{v}} y_{p}^{r} dℓ_{T,λ}(Φ̃^{λ})(y)) =

= ℓ_{T,λ} ( ∑_{a∈Y_{v}} χ(a) ∫_{(a)_{v}} y_{p}^{r} dΦ̃^{λ}(y)) = ℓ_{T,λ} ( ∑_{a∈Y_{v}} χ(a)Φ_{r}^{λ}((a)_{v})) ,  (5.6)

where (a)_{v} = (a + (Np^{v})) ⊂ Y, and by definition (5.1)

μ̃^{λ}(y; f_{1} ⊗ f_{2} ⊗ f_{3}) = ℓ_{T,λ}(Φ̃^{λ})(y),  (5.7)
∫_{(a)_{v}} y_{p}^{r} d(Φ̃^{λ}) = Φ_{r}^{λ}((a)_{v}) ∈ ℳ_{T}^{λ} (Np)  (5.8)

for r = 0, 1, …, h−1. Moreover Φ_{r}((a)_{v}) is a triple modular form given by (1.2) of level N^{2}p^{2v} as a value of a higher twist of a Siegel-Eisenstein distributions, hence

Φ_{r}^{λ}(χ) = U_{T}^{−2v} [π_{λ,T,1}U_{T}^{2v}(2^{r}F_{χ,r}^{χ̄_{1},χ̄_{2},χ̄_{3}} ∘ diag)] .  (5.9)

Taking into account the equalities (5.9), the integral (5.6) transforms to the following

∫_{Y} χ(y) y_{p}^{r} dμ̃^{λ}(y; f_{1} ⊗ f_{2} ⊗ f_{3}) = ℓ_{T,λ} ( ∑_{a∈Y_{v}} χ(a)Φ_{r}^{λ}((a)_{v}))  (5.10)
= ℓ_{T,λ}(U_{T}^{−2v}[π_{λ,T,1}U_{T}^{2v}(2^{r}F_{χ,r}^{χ̄_{1},χ̄_{2},χ̄_{3}} ∘ diag)])

Notice that then it follows that the sum in the right hand side of the equality (5.10) can be expressed through the functions (1.2):

∫_{Y} χ(y) y_{p}^{r} dμ̃^{λ}(y; f_{1} ⊗ f_{2} ⊗ f_{3})(y)
ℓ_{T,λ} (U_{T}^{−2v}[π_{λ,T,1}U_{T}^{2v}(2^{r}F_{χ,r}^{χ̄_{1},χ̄_{2},χ̄_{3}} ∘ diag)])  (5.11)

where we use the functions (1.2). The function

g = Φ_{r}(χ) = 2^{r}F_{χ,r}^{χ̄_{1},χ̄_{2},χ̄_{3}} ∘ diag

is computed in (B.5), Appendix B as follows:

ℰ(z_{1}, z_{2}, z_{3}; −r, k, Np^{v}, ψ, χ_{1}, χ_{2}, χ_{3})
= N_{1,1}N_{1,2}N_{1,3}(χ̄_{1}χ̄_{2}χ̄_{3})(2)G(χ_{0,1})G(χ_{0,2})G(χ_{0,3})2^{−r}Φ_{r}(χ),

thus it is a nearly-holomorphic triple modular form in in the ℚ^{ab}-module

M(ℚ^{ab}) = ℳ_{T} (N^{2}p^{2v}, ψ_{1} ⊗ ψ_{2} ⊗ ψ_{3}; ℚ^{ab})

⊂ ℳ_{k,r}(N^{2}p^{2v}, ψ_{1}; ℚ^{ab}) ⊗ ℳ_{k,r}(N^{2}p^{2v}, ψ_{2}; ℚ^{ab}) ⊗ ℳ_{k,r}(N^{2}p^{2v}, ψ_{3}; ℚ^{ab}).

Then by the general formula of Proposition 4.2 e) we have:

ℒ_{T,λ} : ℳ_{T} (N^{2}p^{2v}; ℂ) → ℂ, g ↦ \frac{⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, λ^{−2v}U_{T}^{2v}g⟩_{T,N^{2}p}}{⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, f_{1,0} ⊗ f_{2,0} ⊗ f_{3,0}⟩_{T,N^{2}p}} , (5.12)

ℓ_{T,λ} (U_{T}^{−2v}[π_{λ,T,1}U_{T}^{2v}(g)]) = i_{p} (\frac{⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, λ^{−2v}U_{T}^{2v}(g)⟩_{T,N^{2}p}}{⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, f_{1,0} ⊗ f_{2,0} ⊗ f_{3,0}⟩_{N^{2}p}} )

= i_{p} (λ^{−2v}p^{3·2v(k−1)} · \frac{⟨V^{2v}(f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}), g⟩_{T,N^{2}p^{2v+1}}}{⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, f_{1,0} ⊗ f_{2,0} ⊗ f_{3,0}⟩_{T,N^{2}p}} ) .

Documenta Mathematica · Extra Volume Coates (2006) 77–132

The scalar products in 5.12 can be computed using Theorem B.2, but we omit here the details. This implies Proposition 5.1 using the integral representation

of Theorem B.2 for modular forms f̃_{j,2v}(z) = ∑_{n=1}^{∞} a_{j,n,2v}e(nz) as above:

𝒟^{⋆}(f_{1}^{ρ} ⊗ f_{2}^{ρ} ⊗ f_{3}^{ρ}, 2k − 2 − r, ψ_{1}ψ_{2}χ_{1})  (5.13)

(Np^{2v})^{k−2r} \frac{N^{2}p^{2v}φ(N^{2}p^{2v})φ(Np^{v})}{[Γ_{0}(N^{2}p^{2v}) : Γ(N^{2}p^{2v})]^{3}} × 𝔏_{Np}(−r)

= ⟨f̃_{1,2v} ⊗ f̃_{2,2v} ⊗ f̃_{3,2v}, ℰ(z_{1}, z_{2}, z_{3}; −r, k, N^{2}p^{2v}, ψ, χ_{1}, χ_{2}, χ_{3})⟩_{T,N^{2}p^{2v}},

where

𝔏_{Np}(s)=𝔏_{Np}(s; f̃_{1,2v} ⊗ f̃_{2,2v} ⊗ f̃_{3,2v}):= ∑_{n|N^{∞}}G_{N}(\overline{ψ_{1}ψ_{2}χ_{1}}, 2n)\frac{a_{n,1,2v}a_{n,2,2v}a_{n,3,2v}}{n^{2s+2k−2}} .

5.4 Proof of Theorem B

Let us use Propostion 5.1 and (5.13):

2^{−r} ∫_{Y} χ(y) y_{p}^{r} dμ̃^{λ}(y; f_{1} ⊗ f_{2} ⊗ f_{3})(y) = 2^{−r}ℓ_{T,λ} (U_{T}^{−2v}[π_{λ,T,1}U_{T}^{2v}(g)]) (5.14)

= \frac{((Np)^{3}/N_{1}N_{2}N_{3})^{k/2}γ̄_{1}γ̄_{2}γ̄_{3}(χ_{1}χ_{2}χ_{3})(2)p^{3·v(k−2)}}{λ^{2v}N_{1,1}N_{2,1}N_{3,1}G(χ_{1,0})G(χ_{2,0})G(χ_{3,0})} ×

× (Np^{2v})^{k−2r} \frac{N^{2}p^{2v}φ(N^{2}p^{2v})φ(Np^{v})}{[Γ_{0}(N^{2}p^{2v}) : Γ(N^{2}p^{2v})]^{3}} 𝔏_{Np}(−r)×

× \frac{𝒟^{⋆}(f_{1}^{ρ} ⊗ f_{2}^{ρ} ⊗ f_{3}^{ρ}, 2k − 2 − r, ψ_{1}ψ_{2}χ_{1})}{⟨f_{1}^{0} ⊗ f_{2}^{0} ⊗ f_{3}^{0}, f_{1,0} ⊗ f_{2,0} ⊗ f_{3,0}⟩_{T,N^{2}p}}

Let us show that under the assumptions as above there exist an admissible ℂ_{p}-valued measure μ̃_{f_{1}⊗f_{2}⊗f_{3}}^{λ} on Y_{N,p}, and a ℂ_{p}-analytic function

𝒟_{(p)}(x, f_{1} ⊗ f_{2} ⊗ f_{3}) : X_{p} → ℂ_{p},

given for all x ∈ X_{N,p} by the integral

𝒟_{(p)}(x, f_{1} ⊗ f_{2} ⊗ f_{3}) = ∫_{Y_{N,p}} x(y)dμ̃_{f_{1}⊗f_{2}⊗f_{3}}^{λ}(y),

and having the following properties: for all pairs (r, χ) such that for χ ∈ X_{p}^{tors} the corresponding Dirichlet characters χ_{j} are Np-complete, and r ∈ ℤ with 0 ≤ r ≤ k − 2, the following equality holds:

𝒟_{(p)}(χx_{p}^{r}, f_{1} ⊗ f_{2} ⊗ f_{3}) =  (5.15)

i_{p}(\frac{(ψ_{1}ψ_{2})(2)C_{χ}^{4(2k−3−r)}}{G(χ_{1})G(χ_{2})G(χ_{3})G(ψ_{1}ψ_{2}χ_{1})λ(p)^{2v}}

\frac{𝒟^{⋆}(f_{1}^{ρ} ⊗ f_{2}^{ρ} ⊗ f_{3}^{ρ}, 2k − 2 − r, ψ_{1}ψ_{2}χ)}{⟨f_{1}^{ρ} ⊗ f_{2}^{ρ} ⊗ f_{3}^{ρ}, f_{1}^{ρ} ⊗ f_{2}^{ρ} ⊗ f_{3}^{ρ}⟩_{T}})

where v = ord_{p}(C_{χ}), χ_{1} mod Np^{v} = χ, χ_{2} mod Np^{v} = ψ_{2}ψ̄_{3}χ, χ_{3} mod Np^{v} = ψ_{1}ψ̄_{3}χ, G(χ) denotes the Gauß sum of a primitive Dirichlet character χ_{0} attached to χ (modulo the conductor of χ_{0}). Indeed, we may write

𝒟_{(p)}(x, f_{1} ⊗ f_{2} ⊗ f_{3}) = C · x(2) ∫_{Y} x(y)dμ̃^{λ}(y; f_{1} ⊗ f_{2} ⊗ f_{3})

with an appropriate constant, given by the RHS of (5.14), where v = ord_{p}(C_{χ}). Moreover, it follows from the properties of the constructed measure

μ̃_{f_{1}⊗f_{2}⊗f_{3}}^{λ}(y) := C · μ̃_{λ}(2^{−1}y; f_{1} ⊗ f_{2} ⊗ f_{3})

that

(ii) if ord_{p}λ(p) = 0 then the holomorphic functions in (i), (ii) are bounded ℂ_{p}-analytic functions: it suffices to use the equality (2.5) with r = 0 in order to show that in this case the measure Φ̃^{λ} is bounded because of |λ(p)|_{p} = 1);

(iii) in the general case (but assuming that λ(p) ≠ 0) the holomorphic functions in (i) belong to the type o(log(x_{p}^{h})) with h = [2ord_{p}λ(p)]+1 and they can be represented as the Mellin transform of the h-admissible measure μ̃_{f_{1}⊗f_{2}⊗f_{3}}^{λ} (in the sense of Amice-Vélu);

(iv) if h = [2ord_{p}λ] + 1 ≤ k − 2 then the function 𝒟_{(p)} is uniquely determined by the above conditions (i).

A Nearly-holomorphic Siegel-Eisenstein series

A.1 Fourier expansions of Siegel-Eisenstein series

In this section χ denotes a Dirichlet character modulo an arbitrary integer N (not to be confused with N in the Introduction). We recall some standard facts about the Fourier expansions of the Siegel-Eisenstein series defined by:

E(𝒵, s; k, χ, N) = E(𝒵, s)  (A.1)

= det(y)^{s} ∑_{γ∈P ∩Γ\Γ} χ(det(d_{γ}))j(γ, 𝒵)^{−k}|j(γ, 𝒵)|^{−2s},

for k + 2Re(s) > m + 1, s ∈ ℂ, k ∈ ℤ, and by analytic continuation over s for other values of s ∈ ℂ (see [Sh83]). It is assumed in the identity (A.1) that N > 1, χ is a Dirichlet character mod N (not necessarily primitive, e.g. trivial modulo N > 1), and

γ = ({a_{γ} \atop c_{γ}} {b_{γ} \atop d_{γ}} ) ∈ Γ = Γ_{0}^{m}(N) ⊂ Γ^{m} = Sp(m, ℤ).

Recall an explicit computation of the Fourier expansion of the series

E^{⋆}(𝒵, s) = E^{⋆}(𝒵, s; k, χ, N) := E(−𝒵^{−1}, s) det(𝒵)^{−k},  (A.2)

obtained from (A.1) by applying the involution

J_{m} = ({0_{m} \atop 1_{m}} {−1_{m} \atop 0_{m}} ) .

Note that for k > m + 1 and N = 1 both series coincide and were studied by Siegel:

E(𝒵) = E_{k}^{m}(𝒵) = E(𝒵, 0) = E^{⋆}(𝒵, 0).

The detailed study of the series E^{⋆}(𝒵, s; k, χ, N) was made by G. Shimura [Sh83] and P. Feit ([Fei86], §10).

On the other hand, it is convenient to use the following notation. Let ϕ be a Dirichlet character mod Q > 1 and consider the Eisenstein series of degree m ≥ 1

F_{α,β}(𝒵, Q, ϕ) := det(y)^{β} ∑_{c,d} ϕ(det c) det(c𝒵 + d)^{−α,−β}  (A.3)
= det(y)^{β} ∑_{c,d} ϕ(det c) det(c𝒵 + d)^{−α} det(c\overline{𝒵} + d)^{−β}
= det(y)^{β} ∑_{c,d} ϕ(det c) det(c𝒵 + d)^{β−α}| det(c𝒵 + d)|^{−2β} (A.4)

where (c, d) runs over all “non-associated coprime symmetric pairs” with det(c) coprime to Q. A more conceptual description would be to sum over 𝒯^{m}(Q)_{∞}\𝒯^{m}(Q), where

𝒯^{m}(Q) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sp(m, ℤ)|A ≡ 0 mod Q\} = ( \begin{matrix}0_{m} & −1_{m} \\ 1_{m} & 0_{m} \end{matrix} ) Γ_{0}^{m}(Q)

𝒯^{m}(Q)_{∞} = (( \begin{matrix}0_{m} & −1_{m} \\ 1_{m} & 0_{m} \end{matrix} ) Γ_{0}^{m}(Q) ( \begin{matrix}0_{m} & −1_{m} \\ 1_{m} & 0_{m} \end{matrix} )^{−1})_{∞}

\{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sp(m, ℤ)|c = 0, b ≡ 0 mod Q\} ⊂ Γ^{m,0}(Q) ⊂ Sp(m, ℤ),

where Γ^{m,0}(Q) = ( \begin{matrix}0_{m} & −1_{m} \\ 1_{m} & 0_{m} \end{matrix} ) Γ_{0}^{m}(Q) ( \begin{matrix}0_{m} & −1_{m} \\ 1_{m} & 0_{m} \end{matrix} )^{−1} ⊂ Sp(m, ℤ) is the

stabilizer of ℳ = ( \begin{matrix}0_{m} & −1_{m} \\ 1_{m} & 0_{m} \end{matrix} ) Γ_{0}^{m}(Q), and more generally, for any set ℳ ⊂ Sp(m, ℤ) of symplectic matrices we denote by ℳ_{∞} the set of those matrices γ = ({a \atop c} {b \atop d}) ∈ Sp(m, ℤ) satisfying the conditions c = 0 and γℳ ⊂ ℳ.

Action of σ ∈ Sp(m, ℤ) on the Eisenstein series Note that for any σ ∈ Sp(m, ℤ) one has

E(𝒵, s; k, χ, N)|_{k}σ = ∑_{γ∈Γ_{0}^{m}(N)_{∞}\Γ_{0}^{m}(N)} ϕ(det d_{γ})(1|_{k}γσ)(𝒵)(Im (γσ(𝒵))^{s}

= det(y)^{s} ∑_{γ∈Γ_{0}^{m}(N)_{∞}\Γ_{0}^{m}(N)} ϕ(det d_{γ})j(γσ, 𝒵)^{−k}|j(γσ, 𝒵)|^{−2s}

= det(y)^{s} ∑_{γ̃∈(Γ_{0}^{m}(N))_{∞}\Γ_{0}^{m}(N)σ} ϕ(det d_{σ^{−1}γ̃})j(γ̃, 𝒵)^{−k}|j(γ̃, 𝒵)|^{−2s},

by writing γ̃ = σγ, σ^{−1}γ̃ = γ: P γ_{1} = P γ_{2} ⟺ P γ̃_{1} = P γ̃_{2}.

In particular, for σ = J_{m} = ( \begin{matrix}0_{m} & −1_{m} \\ 1_{m} & 0_{m} \end{matrix} ) one has ({a \atop c} {b \atop d}) J_{m} = ({b \atop d} {−a \atop −c}) ∈

Γ_{0}^{m}(N)J_{m}, hence

E(𝒵, s; k, χ, N)| ( \begin{matrix}0_{m} & −1_{m} \\ 1_{m} & 0_{m} \end{matrix} ) = E^{⋆}(𝒵, s; k, χ, N)

= det(y)^{s} ∑_{({b \atop d} {−a \atop −c})∈(Γ_{0}^{m}(N))_{∞}\Γ_{0}^{m}(N)σ} χ(det d) det(d𝒵 − c)^{−k}| det(d𝒵 − c)|^{−2s}.

Notice that J_{m}(N)Γ_{0}^{m}(N) = Γ_{0}^{m}(N)J_{m}(N), where J_{m}(N) =

( \begin{matrix}0_{m} & −1_{m} \\ N · 1_{m} & 0_{m} \end{matrix} ), and

J_{m}(N) ({a \atop c} {b \atop d}) = ({−c \atop Na} {−d \atop Nb}) = ({a_{1} \atop c_{1}} {b_{1} \atop d_{1}} ) J_{m}(N) = ({Nb_{1} \atop Nd_{1}} {−a_{1} \atop −c_{1}} ) .

Therefore (Nd_{1}, −c_{1}) = (Na, Nb), and (a, b) runs over all “non-associated coprime symmetric pairs” with det(a) coprime to N. We may therefore write (Nd_{1}, −c_{1}) = (Na, Nb), and

E^{⋆}(N𝒵, s; k, χ, N)  (A.5)

= det(Ny)^{s} ∑_{( {b_{1} \atop d_{1}} {−a_{1} \atop −c_{1}} )∈(Γ_{0}^{m}(N))_{∞}\Γ_{0}^{m}(N)σ} χ(det d_{1}) det(d_{1}N𝒵 − c_{1})^{−k}| det(d_{1}N𝒵 − c_{1})|^{−2s}

= N^{−m(k+s)} det(y)^{s} ∑_{a,b} χ(det a) det(a𝒵 + b)^{−k−s,−s}  (A.6)
= N^{−m(k+s)}F_{k+s,s}(𝒵, N, χ)  (A.7)

A.2 Arithmetical variables of nearly-holomorphic Siegel modular forms and differential operators

Consider a commutative ring A, the formal variables q = (q_{i,j})_{i,j=1,...,m}, R = (R_{i,j})_{i,j=1,...,m}, and the ring of formal arithmetical Fourier series

A[[q^{B_{m}}]][R_{i,j}] = {f = ∑_{𝒯∈B_{m}} a(𝒯, R)q^{𝒯} | a(𝒯, R) ∈ A[R_{i,j}]}  (A.8)

using the semi-group

B_{m} = {𝒯 = (𝒯_{ij}) ∈ M_{m}(ℝ) | 𝒯 = ^{t}𝒯, 𝒯 ≥ 0, 𝒯_{ij}, 2𝒯_{ii} ∈ ℤ}

and the symbols

q^{𝒯} = ∏_{i=1}^{m} q_{ii}^{𝒯_{ii}} ∏_{i<j} q_{ij}^{2𝒯_{ij}} ⊂ A[[q_{11}, …, q_{mm}]][q_{ij}, q_{ij}^{−1}]_{i,j=1,··· ,m}

(over the complex numbers this notation corresponds to q^{𝒯} = exp(2πitr(𝒯𝒵)), R = (4πIm(𝒵))^{−1}). The formal Fourier expansion of a nearly-holomorphic Siegel modular form f with coefficients in A is an element of A[[q^{B_{m}}]][R_{i,j}]. Let

ℳ_{k}^{m}(N, ψ) ⊂ ℳ̃_{k}^{m}(N, ψ) ⊂ ℳ_{k}^{m}(N, ψ)^{∞}

denote the complex vector spaces of holomorphic, nearly-holomorphic, and 𝒞^{∞}- Siegel modular forms of weight k and character ψ for Γ_{0}^{m}(N), see [ShiAr],

[CourPa] so that ℳ_{k}^{m}(N, ψ) ⊂ ℂ[[q^{B_{m}}]], ℳ̃_{k}^{m}(N, ψ) ⊂ ℂ[[q^{B_{m}}]][R_{i,j}], and ℳ_{k}^{m}(N, ψ)^{∞} ⊂ 𝒞^{∞}(ℍ_{m}).

A.3 Formal Fourier expansions of nearly-holomorphic Siegel-Eisenstein series

In the Siegel modular case Γ^{m} = Sp_{2m}(ℤ) ⊃ Γ_{0}^{m}(N) the series

E(𝒵, s; k, χ, N) = E(𝒵, s)  (A.9)

= det(y)^{s} ∑_{γ∈P ∩Γ\Γ} χ(det(d_{γ}))j(γ, 𝒵)^{−k}|j(γ, 𝒵)|^{−2s} ∈ ℳ_{k}^{∞}(Γ_{0}(N), χ̄)

is absolutely convergent for k + 2Re(s) > m + 1, but can be continued to all s ∈ ℂ. However, for N > 1, the Fourier expansion is known only for the involuted series E(·, s)|W (N), where W (N) = ({0_{m} \atop N·1_{m}} {−1_{m} \atop 0_{m}} ), and for some critical values s ∈ ℤ (for N = 1 both series coincide). Here 𝒵 ∈ ℍ_{m} is in the Siegel upper half-space:

ℍ_{m} = \{𝒵 = ^{t}𝒵 ∈ M_{m}(ℂ)|Im 𝒵 > 0\} , and P = \{(\begin{matrix}a & b \\ 0 & c \end{matrix}) ∈ Sp_{2m}(ℝ)\}

is the Siegel parabolic subgroup.

Example A.1 (Involuted Siegel-Eisenstein series) Let χ be a Dirichlet character modulo N. Recall that by (A.5)

E^{⋆}(N𝒵, s; k, χ, N) = N^{−m(k+s)}F_{k+s,s}(𝒵, N, χ)  (A.10)

= N^{−m(k+s)} det(y)^{s} ∑_{a,b} χ(det a) det(a𝒵 + b)^{−k−s,−s}, where

E^{⋆}(N𝒵, s) = E(−(N𝒵)^{−1}, s) det(N𝒵)^{−k} = N^{−km/2}E|W (N),  (A.11)
G^{⋆}(𝒵, s) = G^{∗}(𝒵, s; k, χ, N) = N^{m(k+s)}E^{∗}(N𝒵, s)·  (A.12)

· Γ̃(k, s)L_{N}(k + 2s, χ) ( ∏_{i=1}^{[m/2]} L_{N}(2k + 4s − 2i, χ^{2}))

κ = (m + 1)/2, and for m odd the Γ-factor has the form:

Γ̃(k, s) = i^{mk}2^{−m(k+1)}π^{−m(s+k)}Γ_{m}(k + s),

where Γ_{m}(s) = π^{m(m−1)/4} ∏_{j=0}^{m−1} Γ(s − (j/2))).

In order to describe the formal Fourier expansions explicitly let us consider the Maass differential operator Δ_{m}, acting on 𝒞^{∞}-functions over V ⊗ ℂ of degree m, which is defined by the equality:

Δ_{m} = det(∂_{ij}),  ∂_{ij} = 2^{−1}(1 + δ_{ij})∂/∂_{ij}.  (A.13)

For an integer n ≥ 0 and a complex number β consider the polynomial

R_{m}(𝒵; n, β) = (−1)^{mn}e^{tr(𝒵)} det(𝒵)^{n+β}Δ_{m}^{n} [e^{−tr(𝒵)} det(𝒵)^{−β}] ,  (A.14)

with 𝒵 ∈ V ⊗ ℂ, where the exponentiation is well defined by

det(y)^{β} = exp (β log[det(y)]) ,

for det(y) > 0, y ∈ Y ⊗ ℂ. According to definition (A.14) the degree of the polynomial R_{m}(𝒵; n, β) is equal to mn and the term of the highest degree coincides with det(𝒵)^{n}. We have also that for β ∈ ℚ the polynomial R_{m}(𝒵; n, β) has rational coefficients.

Theorem A.2 Let m be an odd integer such that 2k > m, and N > 1 be an integer, then: For an integer s such that s = −r ≤ 0, 0 ≤ r ≤ k − κ, there is the following Fourier expansion

G^{⋆}(𝒵, −r) = G^{⋆}(𝒵, −r; k, χ, N) = ∑_{A_{m}∋𝒯≥0} b^{⋆}(𝒯, y, −r)q^{𝒯} = ∑_{A_{m}∋𝒯≥0} a(𝒯, R)q^{𝒯} ,

(A.15)

where for s > (m + 2 − 2k)/4 in (A.15) the only non-zero terms occur for positive definite 𝒯 > 0, and for all s = −r with 0 ≤ r ≤ k − κ, and for all 𝒯 > 0, 𝒯 ∈ A_{m}, where

b^{⋆}(𝒯, y, −r) = a(𝒯, R) = W^{⋆}(y, 𝒯, −r)M(𝒯, χ, k − 2r),  (A.16)
W^{⋆}(y, 𝒯, −r) = 2^{−mκ} det(𝒯)^{k−2r−κ}Q(R, 𝒯; k − 2r, r).

Here a(𝒯, R) = a(𝒯, R; r, N, χ) is a homogeneous polynomial with rational coefficients in the variables R_{ij} and 𝒯_{ij}, and

M(𝒯, k − 2r, χ) = ∏_{ℓ| det(2𝒯)} M_{ℓ}(𝒯, χ(ℓ)ℓ^{−k+2r})  (A.17)

is a finite Euler product, in which M_{ℓ}(𝒯, x) ∈ ℤ[x]; we use the notation q^{𝒯} = exp(2πitr(𝒯𝒵)), R = (4πIm(𝒵))^{−1} as above, and polynomials Q(R, 𝒯; k − 2r, r) are given by (3.1).

Proof: see [Sh83], [Fei86], Theorem 2.14 and formulas (2.137) in [CourPa]. The use of definitions gives

W^{⋆}(y, 𝒯, −r) =2^{−mκ} det(𝒯)^{k−2r−κ} det(4πy)^{−r}R_{m}(4π𝒯y; r, κ − k + r)

where R_{m}(y; n, β) is defined by (A.14). Moreover, let us use the polynomials (3.1):

Q(R, 𝒯; k − 2r, r) det(𝒯)^{−r} = det(4π𝒯y)^{−r}R_{m}(4π𝒯y; r, κ − k + r),

it follows

W^{⋆}(y, 𝒯, −r) =2^{−mκ} det(𝒯)^{k−2r−κ} det(4πy)^{−r}R_{m}(4π𝒯y; r, κ − k + r)
= 2^{−mκ} det(𝒯)^{k−2r−κ}Q(R, 𝒯; k − 2r, r).

B An integral representation for the triple product

B.1 Summary of analytic results

In this section we use the following data :

• Three equal weights k = k_{1} = k_{2} = k_{3}

• Three Dirichlet characters modN_{j} with ψ_{j}(−1) = (−1)^{k}

• Three cusp forms f̃_{j}(z) = ∑_{n=1}^{∞} ã_{n,j}e(nz) ∈ 𝒮_{k}(Ñ_{j}, ψ_{j}), (j = 1, 2, 3) with N_{j}|Ñ_{j}, assumed to be eigenforms for all Hecke operators T_{q}, with q prime to N. In our construction we use as f̃_{j} some “easy transforms” of primitive cusp forms f_{j} ∈ 𝒮_{k}(N_{j}, ψ_{j}) in the Introduction, so that they have the same eigenvalues for all Hecke operators T_{q}, for q prime to N. For example, f̃_{j} could be chosen as eigenfunctions f̃_{j} = f_{j}^{0} of the conjugate Atkin’s operator U_{p}^{∗} given by (0.10), in this case we denote by f_{j,0} the corresponding eigenfunctions of U_{p}.

• Assume that Ñ|Np^{v}, where Ñ := LCM{Ñ_{1}, Ñ_{2}, Ñ_{3}}

• Consider a non necessary primitive Dirichlet character χ mod Np^{v}, and the Dirichlet characters as in (0.12).

Using the notation z_{j} = x_{j} + iy_{j} ∈ ℍ, one associates to this data the following function

ℰ(z_{1}, z_{2}, z_{3}) = ℰ(z_{1}, z_{2}, z_{3}; s, k, ψ, χ_{1}, χ_{2}, χ_{3}) :=  (B.1)

i^{3k}2^{−3(k+1)−2s−2k+2}π^{3(s+k)+2}Γ(2s + 2k − 1)Γ(s + k − 1)×

× L^{(Np)}(k + 2s, ψ)L^{(Np)}(4s + 2k − 2, ψ^{2}) ∑_{ε_{12},ε_{13},ε_{23} mod Np^{v}} χ_{1}(ε_{12})χ_{2}(ε_{13})χ_{3}(ε_{23})

F_{k+s,s}(⋆, N^{2}p^{2v}, ψ)| (\begin{matrix}1 & 0 & 0 & 0 & \frac{ε_{12}}{Np^{v}} & \frac{ε_{13}}{Np^{v}} \\ & 1 & 0 & \frac{ε_{12}}{Np^{v}} & 0 & \frac{ε_{23}}{Np^{v}} \\ & & 1 & \frac{ε_{13}}{Np^{v}} & \frac{ε_{23}}{Np^{v}} & 0 \\ & & & 1 & 0 & 0 \\ & & & & 1 & 0 \\ & & & & & 1 \end{matrix} ) (z_{1}, z_{2}, z_{3})y_{1}^{s}y_{2}^{s}y_{3}^{s}.

Note that the product of the normalizing Gamma-factor and of the two Dirichlet L-functions come from the definitions (A.11) and (A.10) of the Siegel-Eisenstein series.

B.2 Fourier expansion of the Eisenstein series (B.1)

Consider again the Dirichlet characters (0.12), and the corresponding function (B.1) of level Np^{v}. We wish to express the series (B.1), evaluated at s = −r, through the series (1.2) in the case of Np-complete conductors.

Proposition B.1 For F (𝒵) = ∑_{𝒯}a(𝒯, R)q^{𝒯} one has F^{ϕ}(𝒵) =

∑_{𝒯}g_{t}(ϕ, 𝒯)a(𝒯, R)q^{𝒯} , where ε  =  (\begin{matrix}0 & ε_{12} & ε_{13} \\ ε_{12} & 0 & ε_{23} \\ ε_{13} & ε_{23} & 0 \end{matrix}) , ϕ(ε)  =

χ_{1}(ε_{12})χ_{2}(ε_{13})χ_{3}(ε_{23}), 𝒯 denotes the (half integral) block matrix and

g_{t}(ϕ, 𝒯) = ∑_{ε∈S_{N,p}/Np^{v}S_{N,p}} ϕ(ε) exp(2πitr(\frac{1}{Np^{v}} 𝒯ε)), where ϕ(ε) = χ_{1}(ε_{12})χ_{2}(ε_{13})χ_{3}(ε_{23}).

Proof. Indeed,

F |t_{ε,Np^{v}} = ∑_{𝒯}a(𝒯, R)q^{𝒯} |t_{ε,Np^{v}} = ∑_{𝒯}exp(2πitr(ε𝒯)/Np^{v})a(𝒯, R)q^{𝒯} , and it

suffices to notice again that

tr(ε𝒯) = tr ( ; (\begin{matrix}0 & ε_{12} & ε_{13} \\ ε_{12} & 0 & ε_{23} \\ ε_{13} & ε_{23} & 0 \end{matrix}) ; (\begin{matrix}t_{11} & t_{12} & t_{13} \\ t_{12} & t_{22} & t_{23} \\ t_{13} & t_{23} & t_{33} \end{matrix}) ; ) = 2(ε_{12}t_{12} + ε_{13}t_{13} + ε_{23}t_{23}).

Using this formula for F = G^{⋆}(𝒵, s; k − 2r, (Np^{v})^{2}, ψ) at s = −r (see (A.3)), gives:

ℰ(z_{1}, z_{2}, z_{3}; −r, k, ψ, χ_{1}, χ_{2}, χ_{3}) =  (B.2)

∑_{ε∈S/Np^{v}S} χ_{1}(ε_{12})χ_{2}(ε_{13})χ_{3}(ε_{23})G^{⋆}(𝒵, −r; k − 2r, (Np^{v})^{2}, ψ)|t_{ε,Np^{v}}(z_{1}, z_{2}, z_{3})

= (∑_{𝒯} ∑_{ε∈S/Np^{v}S} χ_{1}(ε_{12})χ_{2}(ε_{13})χ_{3}(ε_{23}) exp(2πitr(ε𝒯)/Np^{v})a(𝒯, R)q^{𝒯} )∘ diag

then the sum over ε ∈ S/Np^{v}S transforms simply to the product

G_{Np^{v}}(χ_{1}, 2t_{12})G_{Np^{v}}(χ_{2}, 2t_{13})G_{Np^{v}}(χ_{3}, 2t_{23}),

which is easily evaluated by the general formula for a generalized Gauss sum G_{N}(χ, c) = ∑_{b mod N} χ(b)e(bcN^{−1}). This last sum admits the following known expression in terms of the usual Gauss sums (see for example [PaTV], Section 2, (2.20)): let χ_{0} denote the primitive Dirichlet character modulo N_{0} associated with χ, N_{1} = NN_{0}^{−1}, then

G_{N}(χ, c) = G(χ_{0})N_{1} ∑_{d|N_{1}} μ(d)χ_{0}(d)d^{−1}δ(\frac{c}{N_{1}d^{−1}} )χ̄_{0}(\frac{c}{N_{1}d^{−1}} ).

Writing χ_{0,j} for the primitive Dirichlet character modulo N_{0,j} associated with χ_{j} mod Np^{v}, and using the notation Np^{v} = N_{0,j}N_{1,j}, gives

G_{Np^{v}}(χ_{1}, 2t_{12})

= G(χ_{0,1})N_{1,1} ∑_{d_{1}|N_{1,1}} μ(d_{1})χ_{0,1}(d_{1})d_{1}^{−1}δ(\frac{2t_{12}}{N_{1,1}d_{1}^{−1}} )χ̄_{0,1}(\frac{2t_{12}}{N_{1,1}d_{1}^{−1}} )

G_{Np^{v}}(χ_{2}, 2t_{13})

= G(χ_{0,2})N_{1,2} ∑_{d_{2}|N_{1,2}} μ(d_{2})χ_{0,2}(d_{2})d_{2}^{−1}δ(\frac{2t_{12}}{N_{1,2}d_{2}^{−1}} )χ̄_{0,2}(\frac{2t_{13}}{N_{1,2}d_{2}^{−1}} )

G_{Np^{v}}(χ_{3}, 2t_{23})

= G(χ_{0,3})N_{1,3} ∑_{d_{3}|N_{1,3}} μ(d_{3})χ_{0,3}(d_{3})d_{3}^{−1}δ(\frac{2t_{23}}{N_{1,3}d_{3}^{−1}} )χ̄_{0,3}(\frac{2t_{23}}{N_{1,3}d_{3}^{−1}} )

Let us take the product of these expressions using the notation

2t′_{12} = \frac{2t_{12}}{N_{1,1}/d_{1}} (modN_{0,1}d_{1}),

2t′_{13} = \frac{2t_{13}}{N_{1,2}/d_{2}} (modN_{0,2}d_{2}),

2t′_{23} = \frac{2t_{23}}{N_{1,3}/d_{3}} (modN_{0,3}d_{3})

It follows

G_{Np^{v}}(χ_{1}, 2t_{12})G_{Np^{v}}(χ_{2}, 2t_{13})G_{Np^{v}}(χ_{3}, 2t_{23})

= N_{1,1}N_{1,2}N_{1,3} ∑_{d_{1}|N_{1,1} ; {d_{2}|N_{1,2} \atop d_{3}|N_{1,3}}} μ(d_{1})μ(d_{2})μ(d_{3})χ_{0,1}(d_{1})χ_{0,2}(d_{2})χ_{0,3}(d_{3})(d_{1}d_{2}d_{3})^{−1}

G(χ_{0,1})G(χ_{0,2})G(χ_{0,3})χ̄_{0,1}(2t′_{12})χ̄_{0,2}(2t′_{13})χ̄_{0,3}(2t′_{23}).

The formula (B.3) transforms to

ℰ(z_{1}, z_{2}, z_{3}; −r, k, ψ, χ_{1}, χ_{2}, χ_{3})  (B.3)

= (∑_{𝒯}G_{Np^{v}}(χ_{1}, 2t_{12})G_{Np^{v}}(χ_{2}, 2t_{13})G_{Np^{v}}(χ_{3}, 2t_{23})a(𝒯, R)q^{𝒯} ) ∘ diag

= N_{1,1}N_{1,2}N_{1,3} ∑_{d_{1}|N_{1,1} ; {d_{2}|N_{1,2} \atop d_{3}|N_{1,3}}} μ(d_{1})μ(d_{2})μ(d_{3})χ_{0,1}(d_{1})χ_{0,2}(d_{2})χ_{0,3}(d_{3})(d_{1}d_{2}d_{3})^{−1}

G(χ_{0,1})G(χ_{0,2})G(χ_{0,3}) ∑_{{𝒯:t_{12}=d_{1}t′_{12}, \atop t_{13}=d_{2}t′_{13},t_{23}=d_{3}t′_{23},}} χ̄_{0,1}(2t′_{12})χ̄_{0,2}(2t′_{13})χ̄_{0,3}(2t′_{23})a(𝒯, R)q_{1}^{t_{11}}q_{2}^{t_{22}}q_{3}^{t_{33}}.

Later on we impose the condition that the conductors of χ_{0,1}, χ_{0,2}, χ_{0,3} are complete (i.e. have the same prime divisors as those of Np), when χ_{0,j}(d_{j}) = 0 unless all d_{j} = 1, when χ_{0,j}(d_{j}) = 1. In this complete case χ_{0,j}(n) = χ_{j}(n) for all n ∈ ℤ, hence the equality (B.3) simplifies to the following:

ℰ(z_{1}, z_{2}, z_{3}; −r, k, ψ, χ_{1}, χ_{2}, χ_{3})  (B.4)

= (∑_{𝒯}G_{Np^{v}}(χ_{1}, 2t_{12})G_{Np^{v}}(χ_{2}, 2t_{13})G_{Np^{v}}(χ_{3}, 2t_{23})a(𝒯, R)q^{𝒯} ) ∘ diag

= N_{1,1}N_{1,2}N_{1,3}G(χ_{0,1})G(χ_{0,2})G(χ_{0,3})

(∑_{𝒯} χ̄_{1}(2t_{12})χ̄_{2}(2t_{13})χ̄_{3}(2t_{23})a(𝒯, R)q^{𝒯} ) ∘ diag

= N_{1,1}N_{1,2}N_{1,3}(χ̄_{1}χ̄_{2}χ̄_{3})(2)G(χ_{0,1})G(χ_{0,2})G(χ_{0,3})

(∑_{𝒯}a(𝒯, R)χ̄_{1}(t_{12})χ̄_{2}(t_{13})χ̄_{3}(t_{23})q^{𝒯} ) ∘ diag .

Thus we have expressed the series (B.1) through the series (1.2) in the case of Np-complete conductors:

ℰ(z_{1}, z_{2}, z_{3}; −r, k, Np^{v}, ψ, χ_{1}, χ_{2}, χ_{3})  (B.5)
= N_{1,1}N_{1,2}N_{1,3}(χ̄_{1}χ̄_{2}χ̄_{3})(2)G(χ_{0,1})G(χ_{0,2})G(χ_{0,3})F_{χ,r}^{χ̄_{1},χ̄_{2},χ̄_{3}} ∘ diag

= N_{1,1}N_{1,2}N_{1,3}(χ̄_{1}χ̄_{2}χ̄_{3})(2)G(χ_{0,1})G(χ_{0,2})G(χ_{0,3})2^{−r}Φ_{r}(χ).

B.3 The integral representation

Consider three auxilliary modular forms as in (0.16):

f̃_{j}(z) = ∑_{n=1}^{∞} ã_{n,j}e(nz) ∈ S_{k}(Γ_{0}(N_{j}p^{ν_{j}}), ψ_{j}) (1 ≤ i ≤ 3)

with the same eigenvalues, as those of (0.1), for all Hecke operators T_{q}, with q prime to Np.

Theorem B.2 Under the assumptions and notations as in section B.1, the following integral representation holds:

∫ ∫ ∫_{(Γ_{0}(N^{2}p^{2v})\ℍ)^{3}} \overline{f̃_{1}(z_{1})f̃_{2}(z_{2})f̃_{3}(z_{3})}ℰ(z_{1}, z_{2}, z_{3}; s, k, N^{2}p^{2v}, ψ, χ_{1}, χ_{2}, χ_{3})) ×

∏_{j} y_{j}^{k}(\frac{dx_{j}dy_{j}}{y_{j}^{2}} )

= i^{−3k+3}(2π)^{−4s}Γ(s + 2k − 2)Γ(s + k − 1)^{3}

(Np^{v})^{k+2s} \frac{N^{2}p^{2v}φ(N^{2}p^{2v})φ(Np^{v})}{[Γ_{0}(N^{2}p^{2v}) : Γ(N^{2}p^{2v})]^{3}} × 𝔏_{Np}(s)

L^{(Np)}(f_{1}^{ρ} ⊗ f_{2}^{ρ} ⊗ f_{3}^{ρ}, s + 2k − 2, ψ_{1}ψ_{2}χ_{1}),

where

(2π)^{−4s}Γ(s + 2k − 2)Γ(s + k − 1)^{3} = 2^{−4}Γ_{ℂ}(s + 2k − 2)Γ_{ℂ}(s + k − 1)^{3}, Γ_{ℂ}(s) = 2(2π)^{−s}Γ(s)

is the motivic Gamma-factor,

𝔏_{Np}(s) = 𝔏_{Np}(s; f̃_{1} ⊗ f̃_{2} ⊗ f̃_{3}) := ∑_{n|(Np)^{∞}} G_{Np^{v}}(\overline{ψ_{1}ψ_{2}χ_{1}}, 2n)\frac{ã_{n,1}ã_{n,2}ã_{n,3}}{n^{2s+2k−2}} . (B.6)

Remark. In the special case when the character ψ_{1}ψ_{2}χ has Np-complete con-

ductor, or if it is primitive mod Np^{v}, and f̃_{1}, f̃_{2}, f̃_{3} are primitive normalized cusp eigenforms, one can show that 𝔏_{Np}(s) = (ψ_{1}ψ_{2}χ_{1})(2)G(\overline{ψ_{1}ψ_{2}χ_{1}}).

Theorem B.2 follows from a computation, similar to that in [BoeSP], Theorem 4.2, (triple product, no twisting character) and [Boe-Schm], Section 2 (standard L-function, with twisting character). Details will appear elsewhere.

Corollary B.3 Under the notations and assumptions, for all critical values s = 2k − 2 − r, r = 0, ⋯, k − 2 the following integral representation holds

(2π)^{4r}Γ(−r + 2k − 2)Γ(−r + k − 1)^{3}L^{(N)}(f_{1}^{ρ} ⊗ f_{2}^{ρ} ⊗ f_{3}^{ρ}, 2k − 2 − r, ψ_{1}ψ_{2}χ_{1})

(Np^{v})^{k−2r} \frac{N^{2}p^{2v}φ(N^{2}p^{2v})φ(Np^{v})}{[Γ_{0}(N^{2}p^{2v}) : Γ(N^{2}p^{2v})]^{3}} × 𝔏_{Np}(s)

= ⟨f̃_{1} ⊗ f̃_{2} ⊗ f̃_{3}, ℰ(z_{1}, z_{2}, z_{3}; −r, k, k, N^{2}p^{2v}, ψ, χ_{1}, χ_{2}, χ_{3})⟩_{T,N^{2}p^{2v}}.

Documenta Mathematica · Extra Volume Coates (2006) 77–132

References

[Am-V] Amice, Y. and Vélu, J., Distributions p-adiques associées aux séries de Hecke, Journées Arithmétiques de Bordeaux (Conf. Univ. Bordeaux, 1974), Astérisque no. 24/25, Soc. Math. France, Paris 1975, 119 - 131.

[Boe1] Böcherer, S., Über die Funktionalgleichung automorpher L–Funktionen zur Siegelscher Modulgruppe, J. reine angew. Math. 362 (1985) 146–168.

[Boe2] Böcherer S., Über die Fourier–Jacobi Entwickelung Siegelscher Eisensteinreihen. I.II., Math. Z. 183 (1983) 21-46; 189 (1985) 81– 100.

[BHam] Böcherer, S., Ein Rationalitätssatz für formale Heckereihen zur Siegelschen Modulgruppe. Abh. Math. Sem. Univ. Hamburg 56, 35– 47 (1986)

[Boe-Ha] Boecherer, S., Heim, B. L-functions for GSp_{2} × Gl_{2} of mixed weights. Math. Z. 235, 11-51(2000).

[BSY] Böcherer, S., Satoh, T., and Yamazaki, T., On the pullback of a differential operator and its application to vector valued Eisenstem series, Comm. Math. Univ. S. Pauli, 41 (1992), 1-22.

[Boe-Schm] Böcherer, S., and Schmidt, C.-G., p-adic measures attached to Siegel modular forms, Ann. Inst. Fourier 50, N^{∘}5, 1375-1443 (2000).

[BoeSP] Böcherer, S. and Schulze-Pillot, R., On the central critical value of the triple product L-function. David, Sinnou (ed.), Number theory. Séminaire de Théorie des Nombres de Paris 1993–94. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 235, 1-46 (1996).

[Co] Coates, J. On p–adic L–functions. Sem. Bourbaki, 40eme annee, 1987-88, n^{∘} 701, Asterisque (1989) 177–178.

[Co-PeRi] Coates, J. and Perrin-Riou, B., On p-adic L-functions attached to motives over ℚ, Advanced Studies in Pure Math. 17, 23–54 (1989).

[Colm98] Colmez, P. Fonctions L p-adiques, Séminaire Bourbaki, exposé n^{∘} 851, novembre 1998.

[Colm03] Colmez, P. La conjecture de Birch et Swinnerton-Dyer p-adique. Séminaire Bourbaki, exposé n^{∘} 919, juin 2003.

[CourPa] Courtieu M., Panchishkin A.A., Non-Archimedean L-Functions and Arithmetical Siegel Modular Forms, Lecture Notes in Mathematics 1471, Springer-Verlag, 2004 (2nd augmented ed.)

[De69] Deligne P., Formes modulaires et reprèsentations l-adiques, Sem. Bourb. 1968/69 ,exp. no 335. Springer-Verlag, Lect. Notes in Math. N^{∘}179 (1971) 139 - 172.

[De79] Deligne P., Valeurs de fonctions L et périodes d’intégrales, Proc. Symp. Pure Math AMS 33 (part 2) (1979) 313–342.

[Fei86] Feit, P., Poles and Residues of Eisenstein Series for Symplectic and Unitary Groups, Memoir AMS 61 N^{∘}346 (1986), 89 p.

[Ga87] Garrett, P.B., Decomposition of Eisenstein series: Rankin triple products, Ann. of Math. 125 (1987), 209–235.

[GaHa] Garrett, P.B. and Harris, M., Special values of triple product L–functions, Amer. J. Math 115 (1993) 159–238.

[Go02] Gorsse, B., Carrés symétriques de formes modulaires et intégration p-adique. Mémoire de DEA de l’Institut Fourier, juin 2002.

[Go-Ro] Gorsse B., Robert G., Computing the Petersson product ⟨f^{0}, f_{0}⟩. Prépublication de l’Institut Fourier, N^{∘}654, (2004).

[Ha93] Harris, M., Hodge–de Rham structures and periods of automorphic forms, Proceedings of the Joint AMS Summer Conference on Motives, Seattle, July 20–August 2 1991, Seattle, Providence, R.I., 1993.

[Ha93] Harris, M., and Kudla, S., The central critical value of a triple product L – functions, Ann. Math. 133 (1991), 605–672.

[Hasse] Hasse, H., Vorlesungen über Zahlentheorie. Zweite neubearbeitete Auflage. Die Grundlehren der Mathematischen Wissenschaften, Band 59 Springer-Verlag, Berlin-New York 1964 xv+504 pp.

[Hi85] Hida, H., A p-adic measure attached to the zeta functions associated with two elliptic cusp forms. I, Invent. Math. 79 (1985) 159–195.

[Hi86] Hida, H., Galois representations into GL_{2}(ℤ_{p}[[X]]) attached to ordinary cusp forms, Invent. Math. 85 (1986) 545–613.

[Hi90] Hida, H., Le produit de Petersson et de Rankin p-adique, Séminaire de Théorie des Nombres, Paris 1988–1989, 87–102, Progr. Math., 91, Birkhäuser Boston, Boston, MA, 1990.

[Hi93] Hida, H., Elementary theory of L-functions and Eisenstein series, London Mathematical Society Student Texts. 26 Cambridge: Cambridge University Press. ix, 386 p. (1993).

[JoH05] Jory–Hugue, F., Unicité des h–mesures admissibles p-adiques données par des valeurs de fonctions L sur les caractères, Prépublication de l’Institut Fourier (Grenoble), N^{∘}676, 1-33, 2005.

[Iw] Iwasawa, K., Lectures on p–adic L–functions, Ann. of Math. Studies, N^{∘}74. Princeton University Press, 1972.

[Ibu] Ibukiyama, T. On differential operators on automorphic forms and invariant pluriharmonic polynomials, Comm. Math. Univ. S. Pauli 48 (1999), 103-118.

[Ka76] Katz, N.M., p-adic interpolation of real analytic Eisenstein series, Ann. of Math. 104 (1976) 459–571

[Ka78] Katz, N.M., p-adic L-functions for CM–fields, Invent. Math. 48 (1978) 199–297.

[KiK] Kitagawa,Koji, On standard p-adic L-functions of families of elliptic cusp forms, Mazur, Barry (ed.) et al.: p-adic monodromy and the Birch and Swinnerton-Dyer conjecture. A workshop held August 12-16, 1991 in Boston, MA, USA. Providence, R: American Mathematical Society. Contemp. Math. 165, 81-110, 1994

[Ku-Le] Kubota T. and Leopoldt H.-W. , Eine p-adische Theorie der Zetawerte, J. reine angew. Math. 214/215 (1964) 328–339.

[La] Lang S., Introduction to Modular Forms, Springer Verlag, 1976.

[Man73] Manin, Yu.I., Periods of cusp forms and p-adic Hecke series, Mat. Sbornik 92 (1973) 378–401(in Russian); Math. USSR, Sb. 21(1973), 371-393 (1975) (English translation).

[Man74] Manin, Yu.I., The values of p–adic Hecke series at integer points of the critical strip. Mat. Sbornik 93 (1974) 621 - 626 (in Russian).

[Man76] Manin, Yu.I., Non–Archimedean integration and p-adic L-functions of Jacquet – Langlands, Uspekhi Mat. Nauk 31 (1976) 5–54 (in Russian); Russ. Math. Surv. 31, No.1, 5-57 (1976) (English translation).

[Man-Pa] Manin, Yu.I. and Panchishkin, A.A. , Convolutions of Hecke series and their values at integral points, Mat. Sbornik, 104 (1977) 617–651 (in Russian); Math. USSR, Sb. 33, 539-571 (1977) (English translation).

[MTT] Mazur, B., Tate, J. and Teitelbaum, J., On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84, 1-48 (1986).

[Miy] Miyake, Toshitsune, Modular forms. Transl. from the Japanese by Yoshitaka Maeda. Berlin etc.: Springer-Verlag. viii, 335 p. (1989).

[Or] Orloff T., Special values and mixed weight triple products (with an Appendix by D.Blasius), Invent. Math. 90 (1987) 169–180.

[Pa94] Panchishkin, A.A., Admissible Non-Archimedean standard zeta functions of Siegel modular forms, Proceedings of the Joint AMS Summer Conference on Motives, Seattle, July 20–August 2 1991, Seattle, Providence, R.I., 1994, vol.2, 251 – 292.

[PaViet] Panchishkin, A. A. Non-Archimedean Mellin transform and p-adic L-Functions, Vietnam Journal of Mathematics, 1997, N3, 179–202.

[PaSE] Panchishkin, A.A., On the Siegel–Eisenstein measure. Israel Journal of Mathematics, Vol. 120, Part B(2000), 467–509.

[PaIAS] Panchishkin, A.A., On p-adic integration in spaces of modular forms and its applications, J. Math. Sci., New York 115, No.3, 2357-2377 (2003).

[PaMMJ] Panchishkin, A.A., A new method of constructing p-adic Lfunctions associated with modular forms, Moscow Mathematical Journal, 2 (2002), Number 2, 1-16.

[PaTV] Panchishkin, A.A., Two variable p-adic L functions attached to eigenfamilies of positive slope, Invent. Math. v. 154, N3 (2003), pp. 551 - 615.

[PaB1] Panchishkin, A.A., The Maass–Shimura differential operators and congruences between arithmetical Siegel modular forms, (accepted in Moscow Mathematical Journal in October 2005 (39 p.).

[PaB2] Panchishkin, A.A., Admissible measures for standard L-functions and nearly holomorphic Siegel modular forms , Preprint MPI 2002 - 42, 1-65.

[PaJTNB] Panchishkin A.A., Sur une condition suffisante pour l’existence des mesures p-adiques admissibles, Journal de Théorie des Nombres de Bordeaux, v. 15 (2003), pp. 1-24.

[PSRa] Piatetski–Shapiro, I.I., and Rallis, S., Rankin triple L– functions, Compositio Math. 64 (1987) 333–399.

[Puy] Puydt, J., Valeurs spéciales de fonctions L de formes modulaires adéliques, Thèse de Doctorat, Institut Fourier (Grenoble), 19 décembre 2003.

[Ran39] Rankin, R.A., Contribution to the theory of Ramanujan’s function τ(n) and similar arithmetical functions. I.II, Proc. Camb. Phil. Soc 35 (1939) 351–372.

[Ran52] Rankin, R.A., The scalar product of modular forms, Proc. London math. Soc. 2 (3) (1952) 198-217.

[Ran90] Rankin, R.A., The adjoint Hecke operator. Automorphic functions and their applications, Int. Conf., Khabarovsk/USSR 1988, 163-166 (1990).

[Sa] Satoh,T., Some remarks on triple L-functions. Math.Ann.276, 687-698(1987).

[Schm] Schmidt, C.–G., The p-adic L-functions attached to Rankin convolutions of modular forms, J. reine angew. Math. 368 (1986) 201–220.

[Scho] Scholl, A., Motives for modular forms, Inv. Math. 100 (1990), 419– 430.

[SchEu] Scholl, A.J., An introduction to Kato’s Euler systems, Scholl, A. J. (ed.) et al., Galois representations in arithmetic algebraic geometry. Proceedings of the symposium, Durham, UK, July 9-18, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 254, 379-460 (1998).

[Se73] Serre, J.–P., Formes modulaires et fonctions zêta p-adiques, Lect Notes in Math. 350 (1973) 191–268 (Springer Verlag).

[Shi71] Shimura G., Introduction to the Arithmetic Theory of Automorphic Functions, Princeton Univ. Press, 1971.

[Shi75] Shimura G., On the holomorphy of certain Dirichlet series, Proc. Lond. Math. Soc. 31 (1975) 79–98.

[Shi76] Shimura G., The special values of the zeta functions associated with cusp forms. Comm. Pure Appl. Math. 29 (1976), no. 6, 783–804.

[Shi77] Shimura G., On the periods of modular forms, Math. Annalen 229 (1977) 211–221.

[Shi82] Shimura, G., Confluent Hypergeometric Functions on Tube Domains, Math. Ann. 260 (1982), p. 269-302.

[Sh83] Shimura, G., On Eisenstein Series, Duke Math. J; 50 (1983), p. 417-476.

[ShiAr] Shimura, G., Arithmeticity in the theory of automorphic forms, Mathematical Surveys and Monographs. 82. Providence, RI: American Mathematical Society (AMS) . x, 302 p. (2000).

[V] Visik, M.M., Non-Archimedean measures connected with Dirichlet series, Math. USSR Sb. 28 (1976), 216-228 (1978).

[MV] Vishik, M.M. and Manin, Yu.I., p-adic Hecke series of imaginary quadratic fields, Math. USSR, Sbornik 24 (1974), 345-371 (1976).

[We56] Weil, A., On a certain type of characters of the idèle-class group of an algebraic number-field, Proceedings of the international symposium on algebraic number theory, Tokyo & Nikko, 1955, pp. 1–7, Science Council of Japan, Tokyo, 1956.

[Wi95] Wiles A., Modular elliptic curves and Fermat’s Last Theorem, Ann. Math., II. Ser. 141, No.3 (1995) 443–55.

Siegfried Böcherer Kunzenhof 4B, 79117 Freiburg, Germany boech@siegel.math.unimannheim.de A. A. Panchishkin Institut Fourier, B.P.74 38402 St.-Martin d’Hères France panchish@mozart.ujfgrenoble.fr