Admissible measures for standard L–functions and nearly holomorphic Siegel modular forms

A. A. Panchishkin^{∗}

Institut Fourier, B.P.74, 38402 St.–Martin d’Hères, FRANCE; e-mail: panchish@mozart.ujf-grenoble.fr, FAX: 33 (0) 4 76 51 44 78

§0. Introduction

0.1. Let p be a prime number (we often assume p ≥ 5).

The purpose of this paper is to give a new conceptual construction of admissible measures (in the sense of Amice-Vélu) attached to a standard L-function of a Siegel cusp eigenform. For this purpose we use the theory of p-adic integration in spaces of nearly holomorphic Siegel modular forms (in the sense of Shimura, see [Shi2000]) over an 𝒪algebra A where 𝒪 is the ring of integers in a finite extension K of ℚ_{p}. Often we simply assume that A = ℂ_{p}.

We view such modular forms as certain formal expansions. In a previous paper [PaAD] we studied the action of certain differential operators on Siegel-Eisenstein distributions and on theta distributions. The important property of these arithmetical differential operators is their commutation with Hecke operators (under an appropriate normalization). Their action produces natural families of distributions with values in spaces of nearly holomorphic Siegel modular forms.

Denote by ℳ = ℳ(A) an A-module (or simply a ℂ_{p}-vector space) of nearly holomorphic Siegel modular forms. In the present paper we apply to these distributions the canonical projector π_{α} : ℳ → ℳ^{α} onto the primary (characteristic) A-submodule associated to a non-zero eigenvalue α ∈ A^{×} of the Frobenius operator U = Π^{+}(p) = U_{p} (the Atkin-Lehner operator in the Siegel modular case; ℳ^{α} is the maximal A-submodule on which U − αI is nilpotent). This operator acts similar to the trace operator lowering the level of modular forms. On the other hand, U is invertible on ℳ^{α} if α ∈ A^{×} so that one can glue its action on forms of various levels. In this way one obtains the desired distributions with values in a finite dimensional vector space starting from naturally defined distributions with values in spaces of modular forms (like Siegel-Eisenstein distributions, theta distributions etc.). In order to obtain from them numerically valued distributions interpolating critical values attached to standard L-functions of Siegel modular forms one applies a suitable linear form coming from the Petersson scalar product (using the Andrianov identity of Rankin type for the standard L-function).

∗ Supported by The Max-Planck Institute of Mathematics during a visit of the author at MPIM in 2002.

In our previous works [PaAdm], [PaLNM] we constructed the non–Archimedean interpolation of these special values fixing a non zero Satake p–parameter α_{0}(p) of the cusp eigenform f. It follows that the normalized critical values D(s, f, χ) of the standard Lfunction can be explicitely rewritten in terms of certain ℂ_{p}–valued integrals of admissible p–adic measures (over a profinite group of the type ℤ_{p}^{×} of p–adic units) provided the character χ is non–trivial. More explicit description of these special values was given by M.Courtieu in his PhD thesis (Institut Fourier, 2000) using detailed study of the action of the Shimura differential operator on Siegel modular forms.

In the present paper we give a conceptual explanation of these p–adic properties satisfyied by the special values of the standard L-function D(s, f, χ), where f is a Siegel cusp form of an even degree m and of weight k > 2m+2, χ is a varying Dirichlet character. We show that these admissible measures can be lifted to arithmetical nearly holomorphic Siegel modular forms studied by G.Shimura [Shi2000]. This lifting is given by a universal sequence Φ_{s}^{±}(χ) of distributions with values in arithmetical nearly holomorphic Siegel modular forms (for critical pairs (s, χ), see Proposition 5.4). It would be interesting to extend these lifting results to Siegel cusp eigenforms of odd degree, using the method of Böcherer-Schmidt [BöSch].

0.2. Arithmetical nearly holomorphic Siegel modular forms admit two different descriptions:

• as formal power series g = ∑_{ξ∈B_{m}} a(ξ, R_{i,j})q^{ξ} ∈ ℚ̄[[q^{B_{m}}]][R_{i,j}] such that for all R = (R_{i,j})_{i,j=1,···,m} = (4πIm(z))^{−1} = (2πi(z̄− z))^{−1} and z ∈ ℍ_{m} in the Siegel upper half plane of degree m the series converges to a 𝒞^{∞}-Siegel modular form of a given weight k and character ψ;

• as certain 𝒞^{∞}-Siegel modular forms g taking values in ℚ̄ at all CM-points (up to a factor independnt of a concrete form).

Throughout the paper we fix embeddings

i_{∞} : ℚ̄ → ℂ, i_{p} : ℚ̄ → ℂ_{p},

and we shall often view algebraic numbers (via these embeddings) as both complex and p–adic numbers, where ℂ_{p} = ˆ \overline{ℚ}_{p} is the Tate field (the completion of a fixed algebraic closure ℚ̄_{p} of ℚ_{p}), which is endowed with a unique norm | · |_{p} such that | p |_{p}= p^{−1}.

0.3. For a fixed positive integer N ∈ ℕ 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}^{×}.

Definition 0.4. a) For h ∈ ℕ, h ≥ 1 let C^{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,

C^{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 continous functions so that

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

b) For a given positive integer h an h-admissible measure on Y with values in M is an A-module homomorphism

Φ̃ : C^{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. 0.5. We explain in Section 3, how to construct an h-admissible measure Φ̃^{α} : C^{h}(Y, A) → ℳ(A) out of a sequence of distributions

Φ_{j} : C^{1}(Y, A) → ℳ

with values in an A-module M = ℳ(A) of nearly holomorphic modular forms over A (for all j ∈ ℕ with j ≤ h−1, where A is an 𝒪-algebra, and α ∈ A^{×} is a fixed non-zero eigenvalue of the Frobenius operator U = U(p) = Π^{+}(p) acting on

g = ∑_{ξ∈B_{m},\underline{n}≥0} a(ξ, \underline{n})R\underline{n}q^{ξ} ∈ A[[q^{B_{m}}]][R_{i,j}],

by

g|U(p) = ∑_{ξ∈B_{m},\underline{n}≥0} a(pξ, \underline{n})(pR)\underline{n}q^{ξ}

(over the complex numbers R = (4π)^{−1}Im(z)^{−1} and this notation corresponds to

f|U(p)(z) = ∑_{^{t}u=u∈M_{m}(ℤ) mod p} f|_{k} ( \begin{matrix}1_{m} & u \\ 0_{m} & p1_{m} \end{matrix} ) =

p^{−κm} ∑_{^{t}u=u∈M_{m}(ℤ) mod p} f((z + u)/p),

Im ((z + u)/p) = Im(z/p) = Im(z)/p = (4π)^{−1}(pR)^{−1}.

(we use the Petersson-Andrianov notation for the action of matrices). Then we consider (for an α ∈ A^{×}) the canonical projection operator π_{α} : ℳ(A) → ℳ(A)^{α}. We define an A-linear map

Φ̃^{α} : C^{h}(Y, A) → ℳ

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

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

where Φ_{j} : C^{1}(Y, A) → ℳ(A) are certain ℳ(A)-valued distributions on Y for all j = 0, 1, …, h − 1.

0.6. The standard zeta function of a Siegel cusp eigenform. In order to describe the main result of the paper let

f = ∑_{ξ>0} a(ξ)e_{m}(ξz)

be a Siegel cusp form of the even degree m of weight k on the congruence subgroup

Γ_{0}(C) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Sp_{m}(ℤ) | c ≡ 0 mod C\}

with a Dirichlet character ψ mod C. Here z belongs to the Siegel upper half plane

ℍ_{m} = {z ∈ GL_{m}(ℂ)|z =^{t} z, Im(z) > 0}

and we adopt the standard notation e_{m}(z) = exp(2πitr(z)).

Suppose that f is an eigenfunction of the global Hecke algebra

ℒ^{(m)}(C) = ⊗_{q |̸C}ℒ_{(q)}^{(m)}(C)

with the eigenvalue given by a homomorphism λ_{f} : ℒ^{(m)}(C) → ℂ, i.e.

f | X = λ_{f}(X)f for all X ∈ ℒ_{(q)}^{(m)}(C).

Let

α_{0}(q), α_{1}(q), ..., α_{m}(q)

be the (m + 1)–tuple of Satake q–parameters of λ_{f}, which uniquely determine λ_{f}. It is known [An79] that the relation

α_{0}^{2}(q)α_{1}(q)...α_{m}(q) = ψ(q)^{m}q^{m(k−(m+1)/2)}

holds. Recall that the standard zeta function of f with a Dirichlet character χ mod N is defined as the Euler product

D(s, f, χ) = ∏_{q |̸C} D^{(q)}(s, f, χ),  (0.1)

with

\begin{aligned}D^{(q)} & (s, f, χ)^{−1} = \\ & (1 − \frac{χ(q)ψ(q}{q^{s}} ) ∏_{i=1}^{m} (1 − \frac{χ(q)ψ(q)α_{i}(q)}{q^{s}}) (1 − \frac{χ(q)ψ(q)α_{i}^{−1}(q)}{q^{s}}) , \end{aligned}

(the product is absolutely convergent for Re(s) > 1 + m). Recall the three types of the normalized zeta function: for κ = (m + 1)/2, s ∈ C

\begin{matrix}D^{∗}(s, f, χ) = \\ (2π)^{−m(s+k−κ)}Γ((s + δ)/2) ∏_{j=1}^{m} Γ(s + k − j)D(s, f, χ), \end{matrix}  (0.2)

\begin{matrix}D^{−}(s, f, χ) = Γ((s + δ)/2)^{−1}D^{∗}(s, f, χ) = \\ (2π)^{−m(s+k−κ)} ∏_{j=1}^{m} Γ(s + k − j)D(s, f, χ), \end{matrix}  (0.3)

\begin{matrix}D^{+}(s, f, χ) = \frac{2 i^{δ}Γ(s) cos(π(s − δ)/2)}{(2πi)^{s}}D^{−}(s, f, χ) = \\ \frac{i^{δ}π^{(1−2s)/2}}{Γ((1 + δ − s)/2)}D^{∗}(s, f, χ), \end{matrix}  (0.4)

where δ = 0 or 1 according as ψχ(−1) = (−1)^{δ} with f ∈ S_{k}^{m}(C, ψ) be an eigenfunction of the global Hecke algebra ℒ^{m}(C) = ⊗_{q |̸C}ℒ_{q}^{m}(C). Define

h = [2ord_{p}α_{0}(p)] + 1,

Assume that a(ξ_{0}) ≠ 0 for some ξ_{0} > 0 with det(2ξ_{0}) = 1.

Moreover, we make the essential assumption (*) that

a_{0}(ξ_{0}) ≠ 0  (∗)

for a Fourier coefficient a_{0}(ξ_{0}) of the U-eigenform f_{0}(z) = ∑_{ξ>0} a_{0}(ξ)e_{m}(ξz) defined in Section 4 and in §1, (1.47), chapter 2 of [PaLNM]. The importance of the above non– vanishing condition was pointed out to the author by S. Böcherer. It turns out that it actually may not be satisfied by certain Eisenstein series (note that we deal here with cusp forms). However it is not known in general, whether there exist cusp forms violating the non–vanishing condition.

Main Theorem 0.7. Let f ∈ S_{m}^{k}(N, ψ) be a Siegel cusp eigenform of even degree m and of weight k > 2m + 2 satisfying the above nonvanishing condition (*). Define h = [2ord_{p}(α_{0}(p))] + 1 then for any positive integer c > 1, (c, Np) = 1 there exist two ℂ_{p}- valued h-admissible measures D^{c±} on Y such that:

(i) For all pairs (s, χ) with s ∈ ℤ, 0 ≤ s ≤ k − m − 2 and a primitive non-trivial Dirichlet character χ ∈ X_{S}^{tors} of conductor C_{χ} = p^{v} one has

• ∫_{Y} χx_{p}^{s} dD^{c+} = \frac{G_{C_{χ}}(1_{m}, χ)C_{χ}^{m(2s+2k−m)/4}}{(iC_{χ})^{m^{2}/2}α_{0}(C_{χ}^{2})} \frac{C_{χ̄ψ}^{s+1}}{G(χ̄ψ)}(1 − (χ̄ψ)^{2}(c)c^{−2s−2})·

· ∏_{q|N} \frac{(1 − (χψ̄)_{0}(q)q^{s})}{(1 − (χ̄ψ)_{0}(q)q^{−s−1})} \frac{D^{+}(s + 1, f, χ̄)}{⟨f, f⟩_{N}}.

• ∫_{Y} χx_{p}^{s} dD^{c−} = \frac{G_{C_{χ}}(1_{m}, χ̄)C_{χ}^{m(2−2s+2k−m)/4}}{(iC_{χ})^{m^{2}/2}α_{0}(C_{χ}^{2})} (1 − (χ̄ψ̄)^{2}(c)c^{−2s−2})·

· \frac{D^{−}(−s, f, χ)}{⟨f, f⟩_{N}}.

These integrals are algebraic numbers ℚ̄ viewed as elements of the Tate field ℂ_{p} (via the fixed embedding i_{p} : ℚ̄ → ℂ_{p}).

(ii) If ord_{p}(α_{0}(p)) = 0 (i.e. f is p-ordinary), then the measures D^{c±} are bounded ℂ_{p}-analytic functions.

Content of the paper: in Section 1 we recall some properties and definitions concerning nearly holomorphic Siegel modular forms. Then in Section 2 we describe algebraic differential operators and their action on Siegel-Eisenstein distributions. Then we give in Section 3 a general result on admissible measures with values in nearly holomorphic Siegel modular forms. In Section 4 we describe the definition of the standard zeta functions D(s, f, χ), where f is a Siegel cusp form of the degree m and of the weight k, and we give their critical values. Then in Section 5 we construct certain algebraic linear forms on arithmetical nearly holomorphic Siegel modular forms. In Section 6 we prove the Main theorem using congruences between nearly holomorphic modular forms. We give a more detailed formulation of our main results in Theorem 6.9 containing also analytic properties of the p-adic Mellin transforms of D^{c±}, and their uniqueness property. §1. Nearly holomorphic Siegel modular forms. §2. Algebraic differential operators and Siegel-Eisenstein distributions. §3. A general result on admissible measures with values in nearly holomorphic Siegel

modular forms. §4. The standard L function of a Siegel cusp eigenform and its critical values. §5. Algebraic linear forms on arithmetical nearly holomorphic Siegel modular forms. §6. Congruences between nearly holomorphic modular forms and proof of the Main the-

orem.

Acknowledgement. A part of these results was exposed in a lecture “A new way of constructing p-adic L functions” on July 4, 2001 in the Oberseminar of the Mathematical Institute in Heidelberg; it develops some techniques of [PaIAS] found during a visit of the author to the Institute for Advanced Study in Princeton in 1999-2000. The new proof of the main result was obtained during a visit of the author at MPIM in 2002. Let these institutions be thanked for their kind hospitality.

In Bonn I understood how to use the Frobenius operator in order to prove congruences between nearly holomorphic Siegel modular forms, and then I prepared the manuscript consulting the PhD Thesis of M.Courtieu (where only the holomorphic forms where used).

It is a great pleasure for me to thank S.Böcherer, Don Zagier, V.Gritsenko, and G.Robert for valuable discussions and observations.

I am especially grateful to Prof. Yu. I. Manin, who invited me to Bonn in 2002 and who provided me since 1973 with all essential motivations for my work.

§1. Nearly holomorphic Siegel modular forms

1.1. We use an algebraic version of nearly holomorphic Siegel modular forms over an 𝒪algebra A where 𝒪 is the ring of integers in a finite extension K of ℚ_{p}. We fix embeddings i_{∞}: ℚ̄ ↪ ℂ, i_{p}: ℚ̄ ↪ ℂ_{p} (ℂ_{p} = ˆ ℚ̄_{p} is the Tate field). We assume that A is a closed subring of ℂ_{p} containing i_{p}(ℚ_{p}).

Recall some notations and definitions concerning Siegel modular forms.

Let A be a commutative ring with identity, then M_{r,s}(A) denote the set of all (r × s)- matrices with coefficients in A. For z ∈ M_{r}(ℂ) put e_{r}(z) = e(tr(z)) with e(u) = exp(2iπu) for u ∈ ℂ. We denote by ^{t}z ∈ M_{r,s}(A) the matrix, which is transpose to z ∈ M_{r,s}(A), and write ξ[η] for ^{t}ηξη. For an invertible matrix ξ we put ξ^{⋆} = ^{t}ξ^{−1}. If ξ is a hermitian matrix then we write ξ ≥ 0 or ξ > 0 according as ξ is non negative or positive definite.

Let ℍ_{m} denote the Siegel upper half plane on the degree m,

ℍ_{m} = {z ∈ M_{m}(ℂ) | ^{t}z = z = x + iy, y > 0},

so that ℍ_{m} is a complex analytic variety whose dimension is denoted by ⟨m⟩ = m(m+1)/2.

Let the symbol A_{m} denote the lattice of all half integral symetric matrices in the vector space V = V_{m} = {y ∈ M_{m}(ℝ) | ^{t}y = y}. This lattice is dual to the latice L = M_{m}(ℤ) ∩ V with respect to the pairing given by (u, v) ⟼ e_{m}(uv). Let G = GSp_{2m} be the algebraic subgroup of GL_{2m} defined by

G_{A} = {γ ∈ GL_{2m}(A) | ^{t}γJ_{m}γ = ν(γ)J_{m}, ν(γ) ∈ A^{×}},

for any commutative ring A, where

J_{2m} = ( \begin{matrix}0_{m} & −1_{m} \\ 1_{m} & 0_{m} \end{matrix} ) .

The elements of G_{A} are characterized by the conditions

b^{t}a − a^{t}b = d^{t}c − c^{t}d = 0_{m}, d^{t}a − c^{t}b = 1_{m}.

The multiplier ν defines a homomorphism ν : G_{A} → A^{×} so that ν(γ)^{2m} = det(γ)^{2} and ker(ν) is denoted by Sp_{m}(A). We also put

G_{∞} = G_{ℝ}, G_{∞}^{+} = {γ ∈ G_{∞} | ν(γ) > 0}, G_{ℚ}^{+} = G_{∞}^{+} ∩ G_{ℚ}.

The group G_{∞}^{+} acts transitively on the upper half plane ℍ_{m} by the rule

z ⟼ γ(z) = (az + b)(cz + d)^{−1} (γ = ({a \atop c} {b \atop d}) ∈ G_{∞}^{+}, z ∈ ℍ_{m})

so that the scalar matrices acts trivially, and ℍ_{m} can be identified with a homogeneous space of the group Sp_{m}(ℝ). Let K_{m} denote the stabilizer of the point i1_{m} ∈ ℍ_{m} in the group Sp_{m}(ℝ),

K_{m} = {γ ∈ Sp_{m}(ℝ) | γ(i1_{m}) = i1_{m}},

then there is a bijection Sp_{m}(ℝ)/K_{m} ≃ ℍ_{m} and K_{m} = Sp_{m}(ℝ) ∩ SO_{2m}. The group K_{m} is a maximal compact subgroup of the Lie group Sp_{m}(ℝ) and it can be identified with the group U(m) of all unitary m × m-matrices via the map γ = ({a \atop c} {b \atop d}) ⟼ a + ib.

For a function f : ℍ_{m} ⟶ ℂ of the form

f = ∑_{ξ∈A_{m}} c(ξ)e_{m}(ξz)

and for an integer k we use the notation of Petersson for the action of γ ∈ GSp_{2m}on f:

(f|_{k}γ)(z) = det(γ)^{k−κ}det(cz + d)^{−k}f(γ(z)) (where κ = (m + 1)/2),

we put also for N ∈ ℕ:

f|U(N)(z) = ∑_{ξ∈A_{m}} c(Nξ)e_{m}(ξz)

= ∑_{^{t}u=u∈M_{m}(ℤ) mod N} f|_{k} ( \begin{matrix}1_{m} & u \\ 0_{m} & N1_{m} \end{matrix} ) ,

f|V (N)(z) = ∑_{ξ∈A_{m}} c(ξ)e_{m}(Nξz) = f(Nz)

= N^{m(κ−k)}f|_{k} ( \begin{matrix}N1_{m} & 0_{m} \\ 0_{m} & 1_{m} \end{matrix} ) ,

f|W (N)(z) = det(\sqrt{N}z)^{−k}f(−(Nz)^{−1})

= N^{m(κ−k/2)}f|_{k}W (N), W (N) = ( \begin{matrix}0_{m} & −1_{m} \\ N1_{m} & 0_{m} \end{matrix} ) .

There are the following relations for these operators:

(f|W (N))|W (N) = (−1)^{mk}f, f|W (N′N) = N^{mk/2}f|W (N′)V (N).

We adopt also the notations

f^{ρ} = ∑_{ξ∈A_{m}} \overline{c(ξ)}e_{m}(ξz), Y = \{y ∈ M_{m}(ℝ) | ^{t}y = y > 0\},

B_{m} = {h ∈ A_{m} | h ≥ 0}, C_{m} = {h ∈ A_{m} | h > 0}.

The bloc notation g = \binom{a b}{c d} ∈ Sp_{2m}(ℤ) defines the congruence subgroup

Γ_{0}^{m}(N) = \{g = \binom{a b}{c d} ∈ Sp_{2m}(ℤ)|c ≡ 0 mod N\}

of the Siegel modular group Γ^{m} = Sp_{2m}(ℤ), and its normal subgroup

Γ_{1}^{m}(N) = \{\binom{a b}{c d} ∈ Sp_{2m}(ℤ)|det(d) ≡ 1 mod N, c ≡ 0 mod N\} ⊂ Γ_{0}^{m}(N).

A classical Siegel modular form f ∈ ℳ_{k}(N, ψ) of weight k and character ψ for Γ_{0}^{m}(N) is a holomorphic function f: ℍ_{m} → ℂ such that for every γ = \binom{a b}{c d} ∈ Γ_{0}^{m}(N) one has

f(γ(z)) = ψ(det d)det(cz + d)^{k}f(z)

(for m > 1 the regularity at ∞ is automatically satisfied by Koecher).

The Fourier expansion of such f uses the semi-group B_{m}:

f(z) = ∑_{ξ∈B_{m}} a(ξ)q^{ξ} ∈ ℂ[[q^{B_{m}}]]

where the symbols

q^{ξ} = exp(2πitr(ξz)) = ∏_{i=1}^{m} q_{ii}^{ξ_{ii}} ∏_{i<j} q_{ij}^{2ξ_{ij}} ⊂ ℂ[[q_{11}, …, q_{mm}]][q_{ij}, q_{ij}^{−1}]_{i,j=1,···,m}

are used (with q_{ij} = exp(2π(\sqrt{−1}z_{i,j}))); they form a multiplicative semi-group so that one may consider f as a formal q-expansion and one can introduce Siegel modular forms over A as certain elements of A[[q^{B_{m}}]]. Let ℳ̃_{k}(N, ψ) denote the complex vector space of 𝒞^{∞}-Siegel modular forms of weight k and character ψ for Γ_{0}^{m}(N).

For a non negative integer r and variables q, R = (R_{i,j})_{i,j=1,...,m} consider the ring A[[q^{B_{m}}]][R_{i,j}] (over the complex numbers this notation corresponds to q^{ξ} = exp(2πitr(ξz)), R = (4πIm(z))^{−1}). Consider the following A-submodules

\begin{matrix}P_{r}(A) = & \{g = ∑_{ξ∈B_{m}} a(ξ, R_{i,j})q^{ξ} ∈ A[[q^{B_{m}}]][R_{i,j}]| \\ a(ξ, R_{i,j}) ∈ A[R_{i,j}], deg_{R} a(ξ, R_{i,j}) ≤ r\} ⊂ A[[q^{B_{m}}]][R_{i,j}] \end{matrix}

Consider also for any natural number a the A-modules

Q_{r,a} = {g = ∑_{ξ∈B_{m}} a(ξ, R_{i,j})q^{a^{−1}ξ}},

and put Q_{r} = ⋃_{a≥1} Q_{r,a}. Definition 1.2. Let ℳ_{r,k}^{m}(N, ψ, A) denote the A-submodule of A[[q^{B_{m}}]][R_{i,j}] generated by all F = ∑_{ξ∈B_{m}} a(ξ, R)q^{ξ} such that all a(ξ, R) ∈ A[R_{i,j}]_{i,j=1,...,m} are polynomials of degree ≤ r over A ⊂ ℂ_{p} which belong to

i_{p}(ℚ̄)[R_{i,j}]_{i,j=1,...,m}

and

F = ∑_{ξ∈B_{m}} i_{p}^{−1}(a(ξ, R))|_{R=(4πY)^{−1}}q^{ξ} ∈ ℳ_{r,k}^{m}(N), ψ, ℚ̄) ⊂ ℳ_{r,k}^{m}(N), ψ, ℂ)

is a 𝒞^{∞}-complex function on ℍ_{m} satisfying the following two conditions

∀γ = ({a \atop c} {b \atop d}) ∈ Γ_{0}^{m}(N) F |_{k}γ = ψ(det(d))F  (1.1)
∀γ ∈ Sp_{2m}(ℤ) F |_{k}γ ∈ Q_{r}  (1.2)

(a complex nearly holomorphic modular form of type r and weight k over ℚ̄ in the sense of Shimura, the second condition is necessary only for m = 1).

Put

ℳ_{r,k}^{m}(A) = ⋃_{v≥1}ℳ_{r,k}^{m}(Np^{v}; A)  (1.3)

(an A-module of nearly holomorphic Siegel modular forms of type r and level Np^{v} (v ≥ 0) with coefficients in A). 1.3. We may view a given cusp form

f = ∑_{ξ∈B_{m}} a(ξ; f)q^{ξ} ∈ A[[q^{B_{m}}]],

as an element of ℳ_{r,k}^{m}(A) ⊂ A[[q^{B_{m}}]][R_{i,j}] in the formal q-expansion ring A[[q^{B_{m}}]][R_{i,j}]).

Let f_{0} = ∑_{ξ∈B_{m},\underline{n}≥0} a(ξ, \underline{n}; f_{0})R\underline{n}q^{ξ} ∈ A[[q^{B_{m}}]][R_{ij}] denote an eigenfunction of U with the eigenvalue α ∈ A, (U(f_{0}) = αf_{0}) so that for any fixed ξ = p^{m} · ξ_{0} we have

p^{v|\underline{n}|}a(p^{v}ξ_{0}, \underline{n},; f_{0}) = α^{m}a(ξ_{0}, \underline{n}; f_{0}) ∈ A,

with |\underline{n}| = ∑_{ij} n_{ij}, (pR)^{v\underline{n}} = p^{v|\underline{n}|}R^{v\underline{n}}.

1.4. Define the action of the operator U = U(p) on

g = ∑_{ξ∈B_{m},\underline{n}≥0} a(ξ, \underline{n})R\underline{n}q^{ξ} ∈ A[[q^{B_{m}}]][R_{i,j}],

by

g|U(p) = ∑_{ξ∈B_{m},\underline{n}≥0} a(pξ, \underline{n})(pR)\underline{n}q^{ξ}

(over the complex numbers R = (4π)^{−1}Im(z)^{−1} and this notation corresponds to

f|U(p)(z) = ∑_{^{t}u=u∈M_{m}(ℤ) mod p} f|_{k} ( \begin{matrix}1_{m} & u \\ 0_{m} & p1_{m} \end{matrix} ) =

p^{−κm} ∑_{^{t}u=u∈M_{m}(ℤ) mod p} f((z + u)/p),

Im ((z + u)/p) = Im(z/p) = Im(z)/p = (4π)^{−1}(pR)^{−1}

An element α ∈ A is called a U- characteristic value on ℳ = ℳ_{r,k}^{m}(A) attached to a cusp form if there exists a cusp form f_{0} which is an eigenfunction of U with the eigenvalue α ∈ A (U(f_{0}) = αf_{0}).

Definition 1.5. Let α ∈ A. a) The α-proper submodule ℳ^{(α)} ⊂ ℳ of the operator U is the following A-submodule

ℳ^{(α)} = ℳ^{(α)}(A) = Ker(αI − U) ⊂ ℳ = ℳ(A).

b) The α-characteristic (primary) submodule of ℳ^{α} = ℳ^{α}(A) ⊂ ℳ of the operator U is the maximal A-submodule on which U − αI is nilpotent.

ℳ^{α}(A) = ⋃_{r≥1} Ker(U − αI)^{r} ⊂ ℳ(A).

c) For each fixed level Np^{v} put

ℳ^{(α)}(Np^{v}) = ℳ^{(α)}(Np^{v}; A) = ℳ^{(α)} ∩ ℳ(Np^{v}),
ℳ^{α}(Np^{v}) = ℳ^{α}(Np^{v}; A) = ℳ^{α} ∩ ℳ(Np^{v}).

Proposition 1.6. Suppose that α ∈ A^{×} is an invertible element of the algebra A then a) the A-linear map U^{v} takes ℳ(Np^{v+1}) to ℳ(Np);

b) assume that ℳ^{α}(Np^{v}; A) is a locally free A-module of finite rank then for all v ≥ 1 the A-linear map U is invertible on ℳ^{α}(Np^{v}) ⊂ ℳ(Np^{v});

c) assume that ℳ^{α}(Np^{v}; A) is a locally free A-module of finite rank then for all v ≥ 1 ℳ^{α}(Np^{v}; A) = ℳ^{α}(Np; A) is independent of v. Proof of 1.6: The statement a) is verified using a known relation of U^{v} with the trace operator on modular forms of weight k:

g|Tr_{Np}^{Np^{v+1}} = ∑_{γ∈Γ_{0}(Np^{v+1})\Γ_{0}(Np)} g|_{k}γ,

then

g|U^{v} = p^{\frac{vm(k−m−1)}{2}} g|W_{Np^{v+1}}Tr_{Np}^{Np^{v+1}}W_{Np},

where W_{M} = ( \begin{matrix}0 & −1 \\ M & 0 \end{matrix} ) is the standard involution.

This operator Tr_{Np}^{Np^{v+1}} acts on the space ℳ̃_{k}^{m}(Np^{v+1}, ψ) by

Tr_{Np}^{Np^{v+1}}(g) = ∑_{^{t}u=u∈M_{m}(ℤ) (mod p^{v})} g|_{k} ( \begin{matrix}1_{m} & 0_{m} \\ Npu & 1_{m} \end{matrix} ) .

One can write down the action of this operator on Fourier expansions using the operators U(p^{v}) acting by

F |U(p^{v})(z) = (p^{v})^{− \frac{m(m+1)}{2}} ∑_{^{t}u=u∈M_{m}(ℤ) (mod p^{v})} F ((z + u)/p^{v}).

One uses the following matrix identity:

\begin{matrix}( \begin{matrix}1_{m} & 0_{m} \\ Npu & 1_{m} \end{matrix} ) = \\ (−Np^{v+1})^{−1} ( \begin{matrix}0_{m} & −1_{m} \\ Np^{v+1}1_{m} & 0_{m} \end{matrix} ) ( \begin{matrix}1_{m} & −u \\ 0_{m} & p^{v}1_{m} \end{matrix} ) ( \begin{matrix}0_{m} & −1_{m} \\ Np1_{m} & 0_{m} \end{matrix} ) , \end{matrix}

which implies the relation:

Tr_{Np}^{Np^{v+1}}(g) = p^{− \frac{vm(k−m−1)}{2}} g|W (Np^{v+1})U(p^{v})W (Np).

In order to prove b) notice that the operator U acts on the A-module of finite rank ℳ^{α}(Np^{v}; A) ⊂ ℳ(Np^{v}; A) and its determinant is in A^{×} hence U is invertible on ℳ^{α}(Np^{v}; A).

The statement c) then directly follows from a) and b).

Definition 1.7. Suppose that α ∈ A^{×} is an invertible element of the algebra A.

a) Define the α-characteristic projection

π_{α,v} : ℳ(Np^{v}) → ℳ^{α}(Np^{v}) (v ≥ 1)

as the canonical projector to the α-characteristic submodule

ℳ^{α}(Np^{v}) = ⋃_{r≥1} Ker(U − αI)^{r}

of the U-operator with the kernel ⋂_{r≥1} Im(U − αI)^{r};

b) For any g ∈ ℳ put

π_{α}(g) = U^{−v} [π_{α,1}U^{v}(g)]

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

§2. Algebraic differential operators and Siegel-Eisenstein distributions

2.1. Consider the differential operator Δ_{m} (the Maass differential operator) of degree m, acting on complex C^{∞}-functions on ℍ_{m}, defined by the equality:

Δ_{m} = det(˜∂_{ij}),  ˜∂_{ij} = 2^{−1}(1 + δ_{ij})∂/∂_{ij}  (2.1)

For any f ∈ 𝒞^{∞}(ℍ_{m}, ℂ), one defines also the operators

M_{k}f(z) = det(z − z̄)^{κ−k}Δ [det(z − z̄)^{k−κ+1}f(z)] ,

δ_{k}f(z) = (−\frac{1}{4π} )^{m} det(y)^{−1}M_{k}f(z).

Due to H. Maass [Maa71], for an integer k and a Dirichlet character ψ modulo N, the differential operator δ_{k} acts on 𝒞^{∞}-Siegel modular forms, δ_{k} : ℳ̃_{k}^{m}(N, ψ) → ℳ̃_{k+2}^{m}(N, ψ). One defines then the Shimura differential operator as the composition

δ_{k}^{(r)} = δ_{k+2r−2} ∘ ⋯ ∘ δ_{k}.

In order to describe explicitely the action of δ_{k}^{(r)} on Fourier expansions consider, for an integer r ≥ 0 and for a complex number β, the polynomial

R_{m}(z; r, β) = (−1)^{mr}e^{tr(z)}det(z)^{r+β}Δ_{m}^{r} [e^{−tr(z)}det(z)^{−β}] .  (2.2)

According to its definition the degree of the polynomial R_{m}(z; r, β) is equal to mr and the term of the highest degree coincides with det(z)^{r}. We have also that for β ∈ ℚ the polynomial R_{m}(z; r, β) has rational coefficients.

Notice also that for m = 1 one has

R_{1}(z, r, β) = ∑_{t=0}^{r} \binom{r}{t}z^{r−t}β(β + 1) ⋯ (β + t − 1).

M.Courtieu has established in his PhD thesis the following more explicit expression for the function R_{m}(z; r, β) for arbitrary m (see [Cour], Chapter 3 and Theorem 2.6 of [PaAD]):

R_{m}(z; r, β) = ∑_{t=0}^{r} \binom{r}{t}det(z)^{r−t} ∑_{|L|≤mt−t} R_{L}(β)λ_{l_{1}}(z) ⋯ λ_{l_{t}}(z),  (2.3)

where 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] are polynomials in β of degree (mt − |L|) and with coefficients in the ring ℤ[1/2]. The polynomials λ_{i}(z) are defined by

det(t1_{m} + z) = ∑_{i=0}^{m} λ_{i}(z)t^{m−i}

(λ_{i}(z) is the sum of all diagonal minors of size i × i of the matrix z). 2.2. Let f ∈ ℳ_{k}^{m}(N, ψ) be a Siegel modular form whose Fourier expansion is given by f(z) = ∑_{ξ∈B_{m}} c(ξ)e_{m}(ξz), then we proved (Theorem 3.2 of [PaAD]) that the nearly holomorphic Siegel modular form that δ_{k}^{(r)}f(z) is given then by the following formal power series expansion

δ_{k}^{(r)}f(z) = ∑_{ξ∈B_{m}} c(ξ) ∑_{t=0}^{r} \binom{r}{t}det(ξ)^{r−t}

∑_{|L|≤mt−t} R_{L}(κ − k − r)tr (^{t}ρ_{m−l_{1}}(R)ρ_{l_{1}}^{⋆}(ξ)) ⋯ tr (^{t}ρ_{m−l_{t}}(R)ρ_{l_{t}}^{⋆}(ξ)))q^{ξ}

= ∑_{ξ∈B_{m}} c(ξ) ∑_{t=0}^{r} \binom{r}{t}det(ξ)^{r−t} ∑_{|L|≤mt−t} R_{L}(κ − k − r)Q_{L}(R, ξ)q^{ξ}
= ∑_{ξ∈B_{m}} c(ξ)Q(R, ξ)q^{ξ} ∈ ℚ(f)[[q^{B_{m}}]][R_{i,j}]_{i,j=1,···,m},

(2.4)

where ℚ(f) is the subfield of ℂ generated by the Fourier coefficients of f, L runs over all the multi-indices 0 ≤ l_{1} ≤ ⋯ ≤ l_{t} ≤ m as above,

\begin{matrix}Q(R, ξ) = Q(R, ξ; k, r) = ∑_{t=0}^{r} \binom{r}{t}det(ξ)^{r−t} ∑_{|L|≤mt−t} R_{L}(κ − k − r)Q_{L}(R, ξ)), \\ Q_{L}(R, ξ) = tr (^{t}ρ_{m−l_{1}}(R)ρ_{l_{1}}^{⋆}(ξ)) ⋯ tr (^{t}ρ_{m−l_{t}}(R)ρ_{l_{t}}^{⋆}(ξ))). \end{matrix}

For r = 1 this gives:

δ_{k}f(z) = ∑_{ξ∈B_{m}} c(ξ) ∑_{l=0}^{m}(−1)^{m−l}c_{m−l}(k + 1 − κ)tr (^{t}ρ_{m−l}(R) · ρ_{l}^{⋆}(ξ)) q^{ξ},

where

c_{m−l}(k + 1 − κ) = ∏_{j=0}^{m−l−1}(k + 1 − κ + j/2) = \frac{Γ_{m−l}(k + 1/2)}{Γ_{m−l}(k − 1)/2)} (l ≤ m − 1),

2.3. Siegel-Eisenstein series. We recall the definition of the Siegel-Eisenstein seriesE_{k}(z, s, χ) which come up in the integral representation of Rankin type for the standard L-function. We introduce some additional notation. Let us fix a positive integer m then a pair of integral matrices c, d ∈ M_{m}(ℤ) is called primitive if any matrix with rational coefficients γ ∈ M_{m}(ℚ) such that γc, γd ∈ M_{m}(ℤ) is integral: γ ∈ M_{m}(ℤ). The pair (c, d) is said symmetric if c·^{t}d = d·^{t}c. Two primitive symmetric pairs (c_{1}, d_{1}) and (c_{2}, d_{2}) are said equivalent if (c_{1}, d_{1}) = (uc_{2}, ud_{2}) for a matrix u ∈ SL_{m}(ℤ). Let D_{m} denote the set of equivalence classes of symmetric couples of coprime matrices. Now let k, N be positive integers, s a complex number and χ a Dirichlet character modulo N such that χ(−1) = (−1)^{k}. For z ∈ ℍ_{m} the Siegel-Eisenstein series is defined by

E(z, s; k, χ, N) = E(z, s) = det(y)^{s} ∑ χ(det(d))det(cz + d)^{−k−|2s|},  (2.5)

where the summation is taken over all (c, d) ∈ D_{m} with the condition c ≡ 0 (mod N) and we adopt the convenient notation by Deligne and Ribet [De-Ri]:

z^{−k−|2s|} =^{def} z^{−k}|z|^{−2s} for z ∈ ℂ^{∗}

The series (2.5) is absolutely convergent for k+2Re(s) > m+1 and it admits a meromorphic analytic continuation over the whole complex s-plane. Put j(α, z) = det(cz + d) for α = ({a \atop c} {b \atop d}) and z ∈ ℍ_{m}, then it follows from a known description of D_{m} that

E(z, s; k, χ, N) = det(y)^{s} ∑_{α∈P ∩Γ\Γ} χ(det(d)j(α, z)^{−k−|2s|},  (2.6)

where Γ = Γ_{0}^{m}(N), α = ({a \atop c} {b \atop d}), and P denotes the subgroup of P ⊂ G_{∞+}, consisting of elements α with the condition c = 0. The series (2.6) define 𝒞^{∞}-modular forms E_{k}(z, s, χ) ∈ ℳ̃_{m}^{k}(N, χ̄) and if s = 0 and k > m + 1, these forms are holomorphic: E_{k}(z, χ) ∈ ℳ_{k}^{m}(N, χ̄). This series has a Fourier expansion of the type

E_{k}(z, χ) = ∑_{ξ∈B_{m}} b_{k}(ξ)e_{m}(ξz).

Following the works of H. Maass (see [Maa], chap. 19), the series E_{k}(z, s, χ) are related to each other for different values of k and s via the Shimura differential operator:

E_{k+2}(z, s − 1, χ) = (−4π)^{m} \frac{Γ_{m}(k + s)}{Γ_{m}(k + s + 1)}δ_{k}E_{k}(z, s, χ).  (2.7)

One obtains by iteration of (2.7) r times

E_{k+2r}(z, s − r, χ) = (−4π)^{mr} \frac{Γ_{m}(k + s)}{Γ_{m}(k + s + r)}δ_{k}^{(r)}E_{k}(z, s, χ)

where δ_{k}^{(r)} = δ_{k}, ⋯, δ_{k+2r−2}. It follows from (2.3) that the series E_{k+2r}(z, −r, χ) is a nearly holomorphic Siegel modular form with the following Fourier expansion:

E_{k+2r}(z, −r, χ) = det(−y)^{−r} \frac{Γ_{m}(k)}{Γ_{m}(k + r)} ∑_{ξ∈B_{m}} b_{k}(ξ)R(4πξy; r, κ − k − r)e_{m}(ξz).

2.4. Normaized Siegel-Eisenstein series. In order to obtain arithmetical nearly holomorphic forms whose Fourier coefficients depend nicely on the parameter s = −r ≤ 0 we introduce normalized Siegel-Eisenstein series starting from the series (2.5). Such series where studied in detail by Shimura (voir [Shi83]. Keeping the same notation as above define

E_{k}^{⋆}(z, s, χ) = E_{k}(−z^{−1}, s, χ)det(z)^{−k}.  (2.8)

(the involuted Siegel-Eisenstein series): E_{k}^{⋆}(z, s, χ) = E_{k}(z, s, χ)|_{k}J_{m}. The series (2.8) admit the following Fourier expansion

E_{k}^{⋆}(z, s, χ) = ∑_{ξ∈N^{−1}A_{m}} b(ξ, y, s, χ)e_{m}(ξz),  (2.9)

where the coefficients b(ξ, y, s, χ) are explicitely given in in [Shi83] (see also [PaLNM] (chap. 2, prop. 3.4.)) in terms of the confluent hypergeometric function.

We introduce here other three types of normalized Siegel – Eisenstein series in order to give a precise statement on their holomorphy properties with respect to the variable s, the properties of positivity of matrices ξ by which their Fourier coefficients are indexed, and also algebraic properties of these Fourier coefficients:

\begin{matrix}G^{∗}(z, s) = G^{∗}(z, s; k, χ, N) = \\ N^{m(k+s)}Γ̃(k, s)L_{N}(k + 2s, χ) ∏_{i=1}^{[m/2]} L_{N}(2k + 4s − 2i, χ^{2})E^{∗}(Nz, s), \end{matrix}  (2.10)

with E^{∗}(Nz, s) = E(−(Nz)^{−1}, s)det(Nz)^{−k} = N^{−km/2}E|W (N), Γ̃(k, s) =

i^{mk}2^{−m(k+1)}π^{−m(s+k)} × \{ \begin{matrix}Γ_{m}(k + s)Γ(s + (k − \frac{m}{2} + μ)/2), & if m is even; \\ Γ_{m}(k + s), & otherwise. \end{matrix}

(2.11)

and

L_{N}(s, χ) = ∏_{q |̸N}(1 − χ(q)q^{−s})^{−1}

is the Dirichlet series associated to the character χ and μ = ε(m/2 + k), with ε(r) ∈ {0, 1} such that ε(r) ≡ r (mod 2) for any integer r ∈ ℕ. If the character χ^{2} is non trivial, the function G_{k}^{⋆}(z, s, χ) is an entire function of the complex variable s (see [Fe], Th. 9.1, p. 49, and [PaLNM], Chap. 3, Th. 3.6). Proposition 2.5. Let N > 1 a positive integer such that 2|N then

(a) If χ^{2} ≠ 1 the series G_{k}^{⋆}(z, s, χ) is an entire function of the variable s.

(b) If χ^{2} is trivial then (b_{1}) if either 2k ≥ m and m odd or 2k ≥ m and m is even, but (m/2) + k is odd (i.e.

μ = ε((m/2) + k) = 1 ), then the function G^{∗}(z, s) is an entire function of the variable s;

(b_{2}) if 2k ≥ m and both numbers m and (m/2) + k are even (i.e. μ = ε(m) = 0) then the function G^{∗}(z, s) is an entire function of the variable s with the exclusion of a possible simple pole at the point s = (m + 2 − 2k)/4;

(b_{3}) if m > 2k ≥ 0 then the function G^{∗}(z, s) is an entire function of the variable s with possible exclusion of simple poles at those points s for which 2s is an integer and [(m − 2k + 3)/2] ≤ 2s ≤ (m + 1 − 2k)/2 ;

(b_{4}) if k = 0 then the function G^{∗}(z, s) has a simple pole at the point s = (m + 1)/2 iff χ is trivial, and in this case we have that the function

Res_{s=(m+1)/2}G^{∗}(z, s; 0, 1, N)

of the variable z is a non-zero constant.

Proof of 2.5: (see [Fe], Th. 9.1, p. 49).

If m is odd then we put G^{+}(z, s, χ) = G^{−}(z, s, χ) = G^{∗}(z, s, χ). If m is even then we define

G^{−}(z, s) = Γ((k + 2s − (m/2) + μ)/2)^{−1}G^{∗}(z, s),  (2.12)

\begin{matrix}G^{+}(z, s) = \frac{i^{μ}π^{κ−k−2s}}{Γ((1 − k − 2s + (m/2) + μ)/2)}G^{∗}(z, s) = \\ \frac{2i^{μ}Γ(k + 2s − (m/2)) cos(π(k + 2s − (m/2) − μ)/2)}{(2π)^{k+2s−(m/2)}}G^{−}(z, s). \end{matrix}  (2.13)

where μ = ε((m/2) + k), κ = (m + 1)/2.

The Fourier expansions of G_{k}^{⋆}(z, s, χ) and G_{k}^{±}(z, s, χ) are given by

\begin{aligned}G_{k}^{⋆}(z, s, χ) & = ∑_{ξ∈A_{m}} b^{⋆}(ξ, y, s, χ)e_{m}(ξz), \\ G_{k}^{±}(z, s, χ) & = ∑_{ξ∈A_{m}} b^{±}(ξ, y, s, χ)e_{m}(ξz). \end{aligned}

The Fourier coefficients b^{⋆}(ξ, y, s, χ), b^{±}(ξ, y, s, χ) are explicitely given in [PaLNM], (Chap. 2, 3.5) and in Theorem 2.6 below, and we show that for all crtical paire (s, χ) these series define arithmetical nearly holomorphic Siegel modular forms.

Next we define an additional quadratic Dirichlet character χ_{ξ} depending on ξ ∈ A_{m} and defined only for even rank r. Namely, for ξ = 0 let χ_{ξ} = χ_{0} be trivial; for ξ ≠ 0 we know that for some matrix u ∈ GL_{m}(ℚ)

tuξu = ( \begin{matrix}ξ_{1} & 0 \\ 0 & 0 \end{matrix} ) with det ξ_{1} ≠ 0,  (2.14)

then let χ_{ξ} denote the quadratic character attached to the quadratic field ℚ(\sqrt{detξ_{1}})/ℚ (this definition does not depend on choice of a matrix u).

Theorem 2.6. Let m is even, 2k > m. Then:

(a) If 2s is an integer, s ≤ 0, k + 2s ≥ 1 + (m/2) there is the following Fourier expansion

G^{+}(z, s) = ∑_{A_{m}∋ξ>0} b^{+}(ξ, y, s, χ)e_{m}(ξz),  (2.15)

where for s > (m + 2 − 2k)/4 in (2.15) non-zero terms only occur for positive definite ξ > 0, and for all s from (a) with ξ > 0, ξ ∈ A_{m} the following identity holds

b^{+}(ξ, y, s, χ) = W^{+}(y, ξ, s)L_{N}^{+}(k + 2s − (m/2), χχ_{ξ})M(ξ, χ, k + 2s),

with

L^{+}(s, χ) = \frac{2i^{δ}Γ(s) cos(π(s − δ)/2)}{(2π)^{s}}L_{N}(s, χ)

is the normalized Dirichlet L-function, δ = 0 or 1 according to χ(−1) = (−1)^{δ}, the factor M(ξ, χ, k + 2s) is a finite Euler product

M(ξ, χ, k + 2s) = ∏_{q∈P (ξ)} M_{q}(ξ, χ(q)q^{−k−2s})

with M_{q}(ξ, t) ∈ ℤ[t], extended over primes q in the set P (ξ) of prime divisors of the number N and of all elementary divisors of the matrix ξ,and the character χ_{ξ} is given by (2.14).

W^{+}(y, ξ, s) = 2^{−mκ}detξ^{k+2s−κ}det(4πy)^{s}R(4πξy; −s; κ − k + s),

provided s is an integer, where R(y; n, β) is defined by (2.2), and b^{+}(ξ, y, s, χ) = 0 otherwise (if s ∉ ℤ).

(b)If 2s is an integer with k + 2s ≤ m/2, k + s ≥ κ then there is the following Fourier expansion

G^{−}(z, s) = ∑_{A_{m}∋ξ≥0} b^{−}(ξ, y, s, χ)e_{m}(ξz),  (2.16)

and for all s from (b) with ξ > 0, ξ ∈ A_{m} the following identity holds

b^{−}(ξ, y, s, χ) = W^{−}(y, ξ, s)L_{N}^{−}(k + 2s − (m/2), χχ_{ξ})M(ξ, χ, k + 2s),

where

\begin{matrix}L^{−}(s, χ) = L_{N}(s, χ), \\ W^{−}(y, ξ, s) = 2^{−mκ}det(4πy)^{κ−k−s}R(4πξy; k + s − κ, −s), \end{matrix}

provided s + k − κ is an integer, and b^{−}(ξ, y, s, χ) = 0 otherwise.

Proof of 2.6.: (see [PaLNM], Chap. 2, Th. 3.8).

Let us now write down the action of the Shimura differential operator on these normailzed Siegel-Eisenstein series.

Lemma 2.7. The action of the Shimura differential operator on E_{k}^{⋆}(z, s, χ) is given by:

δ_{k}E_{k}^{⋆}(z, s, χ) = \frac{1}{(−4π)^{m}} \frac{Γ_{m}(k + s + 1)}{Γ_{m}(k + s)}E_{k+2}^{⋆}(z, s − 1, χ),  (2.17)
δ_{k}E_{k}^{⋆}(Nz, s, χ) = \frac{N^{m}}{(−4π)^{m}} \frac{Γ_{m}(k + s + 1)}{Γ_{m}(k + s)}E_{k+2}^{⋆}(Nz, s − 1, χ).  (2.18)

Proof of 2.7: We use the known property δ_{k}(f|_{k}γ) = (δ_{k}f)|_{k+2}γ for all γ ∈ Sp_{2m}(ℝ) and k ∈ ℕ. Taking into account E_{k}^{⋆}(z, s, χ) = E_{k}(z, s, χ)|_{k}J_{m}, one immediately obtains (2.17) from the relation (2.7) satisfied by E_{k}(z, s, χ). In order to prove (2.18), one changes the variable z′ = Nz. One obtains then

δ_{k}(z)E_{k}^{⋆}(Nz, s, χ) = det(N1_{m})δ_{k}(z′)E_{k}^{⋆}(z′, s, χ),

and (2.18) follows. Corollary 2.8. One has the following relations :

\begin{matrix}δ_{k}(G_{k}^{⋆}(z, s, χ)) = G_{k+2}^{⋆}(z, s − 1, χ), \\ δ_{k}(G_{k}^{+}(z, s, χ)) = G_{k+2}^{+}(z, s − 1, χ), \\ δ_{k}(G_{k}^{−}(z, s, χ)) = G_{k+2}^{−}(z, s − 1, χ). \end{matrix}  (2.19)

Proof of 2.8: One shows all the three equalities in the same way, so let us consider only the first one. One applies the Shimura operator to G_{k}^{⋆}(z, s, χ). The only factors in the formula (2.10) for G_{k}^{⋆}(z, s, χ) which are not invariant with respect to the change of variables k ↦ k + 2, s ↦ s − 1, are the following:

N^{m(s+k)}i^{mk}2^{−m(k+1)}π^{−m(s+k)}Γ_{m}(k + s).

These factors come up essentially in the factor Γ̃(k, s) (see (2.11)). One applies then the Shimura operator to G_{k}^{⋆}(z, s, χ) using (2.18). A simplification of the powers of N, π, 2, and i, and of the factor Γ_{m}(k + s) gives δ_{k}(G_{k}^{⋆}(z, s, χ)) = G_{k+2}^{⋆}(z, s − 1, χ). 2.9. Distributions with values in nearly holomorphic Siegel modular forms. Consider a profinite group Y = lim_{\overleftarrow{v}} Y_{v} for example

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 commutative ring, and let M be an A-module. Let 𝒞^{loc−const}(Y, A) denote the A-module of locally constant functions. A distribution on Y with values in M is an A-module homomorphism

Φ : 𝒞^{loc−const}(Y, A) → M.

We adopt the notations Φ(𝒰) := Φ(χ_{𝒰} ) for the characteristic function χ_{𝒰} of an open subset 𝒰 ⊂ Y, and (a)_{v} = a + (Np^{v}) for both an element of Y_{v} and the corresponding open compact subset of Y.

Recall an algebraic version of nearly holomorphic Siegel modular forms over an 𝒪algebra A where 𝒪 is the ring of integers in a finite extension K of ℚ_{p}. We assume that A is equipped with an augmentation e_{A} : A → 𝒪, and fix embeddings i_{∞}: ℚ̄ ↪ ℂ, i_{p}: ℚ̄ ↪ ℂ_{p}.

Let us fix another posivite integer N′ and let denote by g = ∑_{ξ∈B_{m}} a(ξ, R)q^{ξ} ∈ ℳ_{r,k}^{m}(A) an element of

ℳ_{r,k}^{m}(A) = ℳ_{r,k}^{m}(A; N′) = ⋃_{v≥1}ℳ_{r,k}^{m}(N′p^{v}; A)

(A-module of nearly holomorphic Siegel modular forms of type r and level N′p^{v} (v ≥ 0) with coefficients in A). Let M = ℳ_{r,k}^{m}(A) and let Φ : 𝒞^{loc−const}(Y, A) → M be a distribution on Y with values in M. Assume that A contains (φ(Np^{v}))^{−1} and all the values of Dirichlet characters χ mod Np^{v} viewed as homomorphisms Y → A^{×} then obviously

Φ(χ) = ∑_{a mod Np^{v}} χ(a)Φ((a)_{v}), Φ((a)_{v}) = \frac{1}{φ(Np^{v})} ∑_{χ mod Np^{v}} χ(a)^{−1}Φ((a)_{v}). (2.20)

Proposition 2.10 (Normaized Siegel-Eisenstein distributions).

a) Let N > 1 a positive integer such that 2|N and let k, s be integers such that 2k > m and κ − k ≤ s ≤ 0 then there exist distributions G_{k,s}^{⋆}, G_{k,s}^{+}, G_{k,s}^{−} on Y with values in ℳ_{r,k}^{m}(A) such that for any character χ mod Np^{v}

G_{k,s}^{⋆}(χ) = G_{k}^{⋆}(z, s, χ), G_{k,s}^{+}(χ) = G_{k}^{+}(z, s, χ), G_{k,s}^{−}(χ) = G_{k}^{−}(z, s, χ).  (2.21)

where r = m · max(−s, k − κ + s). b) Suppose that κ − k + 1 ≤ s ≤ 0 then the action of the Shimura differential operators on G_{k,s}^{⋆}, G_{k,s}^{+}, G_{k,s}^{−} is given by

\begin{matrix}δ_{k}(G_{k,s}^{⋆}(χ)) = G_{k+2,s−1}^{⋆}(χ), \\ δ_{k}(G_{k,s}^{+}(χ)) = G_{k+2,s−1}^{+}(χ), \\ δ_{k}(G_{k,s}^{−}(χ)) = G_{k+2,s−1}^{−}(χ). \end{matrix}  (2.22)

c) Moreover if t ≥ 0 is an integer such that κ − k + t ≤ s ≤ 0 then the action of δ_{k}^{(t)} on G_{k,s}^{⋆}, G_{k,s}^{+}, G_{k,s}^{−} is given by

\begin{matrix}δ_{k}^{(t)}(G_{k,s}^{⋆}(χ)) = G_{k+2t,s−t}^{⋆}(χ), \\ δ_{k}^{(t)}(G_{k,s}^{+}(χ)) = G_{k+2t,s−t}^{+}(χ), \\ δ_{k}^{(t)}(G_{k,s}^{−}(χ)) = G_{k+2t,s−t}^{−}(χ). \end{matrix}  (2.23)

Proof of Proposition 2.10: One deduces a) from (2.20) taking into account that for all integrs k, s such that 2k > m and κ − k ≤ s ≤ 0 the Fourier coefficients b^{⋆}(ξ, y, s, χ), b^{±}(ξ, y, s, χ)) are polynomials of the variables R_{i,j}(, j = 1, …, m), where R = (4πIm(z))^{−1}, of degree ≤ r = m·max(−s, k−κ+s) which belong to ℚ̄[R_{i,j}]_{i,j=1,...,m}. They are explicitely given in Theorem 2.6 above (2.15), (2.16), and in [PaLNM], (Chap. 2, 3.5) . Indeed let us consider the natural representation (0 ≤ r ≤ m)

\begin{aligned}ρ_{r} & : & GL_{m}(ℂ) & ⟶ & GL(Λ^{r}ℂ^{m}) \\ & & z & ⟼ & ρ_{r}(z) \end{aligned}

of the group GL_{m}(ℂ) on the vector space Λ^{r}ℂ^{m} with respect to the basis

{e_{i_{1}} ∧ ⋯ ∧ e_{i_{r}} : i_{1} < ⋯ < i_{r}}.

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} (r = 0, 1, ⋯, m),

in other words, ρ_{m−r}^{⋆}(z) is the matrix representing the action of z on Λ^{m−r}ℂ^{m} with respect to the basis dual to the above basis of Λ^{r}ℂ^{m}. 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. Then by Lemma 2.4 and (3.8) of [PaAD] we have that if r, k − κ − r ≥ 0,

det(4πy)^{−r}R_{m}(4πξy; r, κ − k − r) = ∑_{t=0}^{r} \binom{r}{t}det(ξ)^{r−t}

∑_{|L|≤mt−t} R_{L}(κ − k − r)tr (^{t}ρ_{m−l_{1}}(R)ρ_{l_{1}}^{⋆}(ξ)) ⋯ tr (^{t}ρ_{m−l_{t}}(R)ρ_{l_{t}}^{⋆}(ξ))),

(2.24)

where 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] are polynomials in β of degree (mt − |L|) and with coefficients in the ring ℤ[1/2], and the polynomials λ_{i}(z) are defined by

det(t1_{m} + z) = ∑_{i=0}^{m} λ_{i}(z)t^{m−i}

(λ_{i}(z) is the sum of all diagonal minors of size i × i of the matrix z). It follows that in (2.15) one has for an integer r = −s ≥ 0:

det(4πy)^{s}R(4πξy; −s; κ − k + s) = ∑_{t=0}^{r} \binom{r}{t}det(ξ)^{r−t}

∑_{|L|≤mt−t} R_{L}(κ − k − r)tr (^{t}ρ_{m−l_{1}}(R)ρ_{l_{1}}^{⋆}(ξ)) ⋯ tr (^{t}ρ_{m−l_{t}}(R)ρ_{l_{t}}^{⋆}(ξ))),

(2.25)

and in (2.16) one has for an integer r′ = k − κ + s ≥ 0:

det(4πy)^{κ−k−s}R(4πξy; k + s − κ, −s) = ∑_{t=0}^{r′} \binom{r′}{t}det(ξ)^{r′−t}

∑_{|L|≤mt−t} R_{L}(κ − k − r′)tr (^{t}ρ_{m−l_{1}}(R)ρ_{l_{1}}^{⋆}(ξ)) ⋯ tr (^{t}ρ_{m−l_{t}}(R)ρ_{l_{t}}^{⋆}(ξ))).

(2.26)

These are polynomials of degree ≤ m · max(−s, k − κ + s) with rational coefficients , hence the Fourier coefficients b^{⋆}(ξ, y, s, χ), b^{±}(ξ, y, s, χ)) define for each ξ distributions with values in ℚ[R_{i,j}]_{i,j=1,...,m} (evaluated at χ).

In order to deduce b) and c) (by iteration) it suffices to use then Corollary 2.8. 2.11. Convolutions of distributions with values in nearly holomorphic Siegel modular forms. Consider again the profinite group

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

Let M = ℳ_{r,k}^{m}(A) and let Φ_{i} : 𝒞^{loc−const}(Y, A) → M be two distributions on Y with values in M_{i} = ℳ_{r_{i},k_{i}}^{m}(A), (i = 1, 2). Assume that A contains (φ(Np^{v}))^{−1} and all the values of Dirichlet characters χ mod Np^{v} viewed as homomorphisms Y → A^{×} then obviously

Φ^{i}(χ) = ∑_{a mod Np^{v}} χ(a)Φ^{i}((a)^{v}), Φ^{i}((a)^{v}) = \frac{1}{φ(Np^{v})} ∑_{χ mod Np^{v}} χ(a)^{−1}Φ^{i}(χ). (2.27)

Let us fix a Dirichlet character ψ on Y with values in A then there exists a unique distribution Φ = Φ_{1} ∗_{ψ} Φ_{2} on Y with values in M = ℳ_{r_{1}+r_{2},k_{1}+k_{2}}^{m}(A) such that for all Dirichlet characters χ on Y with values in A one has Φ(χ) = Φ_{1}(χ) · Φ_{2}(ψχ^{−1}). One defines Φ on basic open subsets by

Φ((a)_{v}) = ∑_{y∈Y_{v}} ψ(y)Φ_{1}(ya)Φ_{2}(y).  (2.28)

for any v such that ψ is defined modNp^{v}.

Indeed,

Φ((a)_{v}) = ∑_{y∈Y_{v}} ψ(y)Φ_{1}(ya)Φ_{2}(y) =

\frac{1}{(φ(Np^{v}))^{2}} ∑_{y∈Y_{v}} ψ(y) ∑_{χ′ mod Np^{v}} χ′(ya)^{−1}Φ_{1}(χ′) ∑_{χ′′ mod Np^{v}} χ′′(y)^{−1}Φ_{2}(χ′′)

hence

Φ(χ) = ∑_{a∈Y_{v}} χ(a) ∑_{y∈Y_{v}} ψ(y)Φ_{1}(ya)Φ_{2}(y) = \frac{1}{(φ(Np^{v}))^{2}} ∑_{a∈Y_{v}} χ(a) ∑_{y∈Y_{v}} ψ(y)

∑_{χ′ mod Np^{v}} χ′(ya)^{−1}Φ_{1}(χ′) ∑_{χ′′ mod Np^{v}} χ′′(y)^{−1}Φ_{2}(χ′′) = Φ_{1}(χ) · Φ_{2}(ψχ^{−1}) =

\frac{1}{(φ(Np^{v}))^{2}} ∑_{a∈Y_{v}}(χχ′^{−1})(a) ∑_{y∈Y_{v}}(ψχ′^{−1}χ′′^{−1})(y) ∑_{χ′ mod Np^{v}} Φ_{1}(χ′) ∑_{χ′′ mod Np^{v}} Φ_{2}(χ′′)

= Φ_{1}(χ) · Φ_{2}(ψχ^{−1})

because of the orthogonality relations which give χ′ = χ, χ′′ = ψχ^{−1}.

§3. A general result on admissible measures with values in nearly holomorphic Siegel modular forms.

3.1. Consider a profinite group Y = lim_{\overleftarrow{v}} Y_{v} for example

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 topological ring over ℤ_{p} with the norm | · | = | · |_{p,A}, and M be a normed A-module with the norm | · |_{p,M}.

Definition 3.2. a) For h ∈ ℕ, h ≥ 1 let C^{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,

C^{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 continous functions so that

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

b) For a given positive integer h an h-admissible measure on Y with values in M is an A-module homomorphism

Φ̃ : C^{h}(Y, A) → M

such that for fixed 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.

3.3. We wish now to construct an h-admissible measure Φ̃^{α} : C^{h}(Y, A) → ℳ_{r^{∗}}(A) out of a sequence of distributions

Φ_{j}^{α} : C^{1}(Y, A) → ℳ_{r^{∗}}

with values in an A-module M = ℳ_{r^{∗}}(A) of nearly holomorphic modular forms over A of type r^{∗} (for all j ∈ ℕ with j ≤ h − 1 where r^{∗} ∈ ℕ be a natural number, and A an algebra over 𝒪 as in Section 1). For this purpose we define

Φ̃^{α} : C^{h}(Y, A) → ℳ_{r^{∗}}

by

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

where Φ_{j} : C^{1}(Y, A) → ℳ_{r^{∗}}(A) are certain ℳ_{r^{∗}}(A)-valued distributions on Y for all j = 0, 1, …, h − 1 , and Φ_{j}^{α}((a)_{v}) their α-characteristic projections given by

Φ_{j}^{α}((a)_{v}) = U^{−v′} [π_{α,1}U^{v′}Φ_{j}((a)_{v})] = π_{α,v′+1}(Φ_{j}((a)_{v})

for any v′ ≥ κv. Note first of all that the definition

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

of the linear form Φ̃^{α} : C^{h}(Y, A) → ℳ_{r^{∗}}(A) is undependent on the choice of the level: for any v′ ≥ κv we have by Proposition 1.6 the following comutative diagram

ℳ_{r^{∗}}(Np^{v′+1}; A) ⟶^{π_{α,v′+1}} ℳ_{r^{∗}}^{α}(Np^{v′+1}; A)
U^{v′} ↓  ↓≀ U^{v′}
ℳ_{r^{∗}}(Np; A) ⟶^{π_{α,1}}  ℳ_{r^{∗}}^{α}(Np; A)

in which the right vertical arrow is an A-isomorphism by Proposition 1.6 b), and by linear algebra the A-linear endomorphism U commutes with the characteristic projectors π_{α,v′+1}, π_{α,1} hence

U^{−κv} [π_{α,1}U^{κv}Φ_{j}((a)_{v})] = U^{−v′} [π_{α,1}U^{v′}Φ_{j}((a)_{v})] = π_{α}(Φ_{j}((a)_{v})).

Theorem 3.4. Let α ∈ A be an element of A whose absolute value |α|_{p} satisfies 0 < |α|_{p} < 1 and put h = [v_{p}(α)] + 1. Suppose that there exists a positive integer κ such that for any (a)_{v} ⊂ Y the following two conditions are satisfied:

Φ_{j}((a)_{v}) ∈ ℳ_{r^{∗}}(Np^{κv}),  (level)

|U^{κv}(∑_{j′=0}^{j} \binom{j}{j′}(−a_{p}^{0})^{j−j′}Φ_{j′}((a)_{v})))|_{p} ≤ Cp^{−vj}  (growth)

for all j = 0, 1, …, κh − 1.

Then there exists an h^{∗}-admissible measure Φ̃^{α} : C^{h}(Y, A) → ℳ_{r^{∗}} with h^{∗} = hκ such that for any ((a)_{v}) ∈ Y

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

for all j = 0, 1, …, h^{∗} − 1 where

Φ_{j}^{α}((a)_{v}) = U^{−κv} [π_{α,1}U^{κv}Φ_{j}((a)_{v})] .

and

π_{α,v} : ℳ_{r^{∗}}(Np^{v}) → ℳ_{r^{∗}}^{α}(Np^{v}) (v ≥ 1)

denotes the α-characteristic projector to the α-characteristic submodule of the U-operator

ℳ_{r^{∗}}^{α}(Np^{v}) = ⋃_{n≥1} Ker(U − αI)^{n}

(with the kernel ⋂_{n≥1} Im(U − αI)^{n}, see definition 1.5) (where

U^{κv}Φ_{j}((a)_{v}) ∈ ℳ_{r^{∗}}^{α}(Np) = ℳ_{r^{∗}}^{α}(Np; A)

because of the inclusion

U^{κv−1}(ℳ_{r^{∗}}(Np^{κv}; A)) ⊂ ℳ_{r^{∗}}(Np; A) (for all v ≥ 1)).

(see 1.6 a)) Proof of 3.5: We need to check the h^{∗}-growth condition for the linear form

Φ̃^{α} : C^{h^{∗}}(Y, A) → ℳ_{r^{∗}}

(given by the condition of the theorem 3.4), which says that for fixed a, b ∈ Y and for v → ∞

|∫_{(a)_{v}}(y_{p} − a_{p})^{j} dΦ̃^{α}|_{p,ℳ_{r^{∗}}} = o(p^{−v(j−hκ)})

for all j = 0, 1, …, hκ − 1 where a_{p} = y_{p}(a).

Let us develop the definition of Φ̃^{α} using the binomial formula:

∫_{(a)_{v}}(y_{p} − a_{p})^{j} dΦ̃^{α} =

∑_{j′=0}^{j} \binom{j}{j′}(−a_{p})^{j−j′}Φ_{j′}^{α}((a)_{v}) = α^{−vκ}·

α^{vκ} · U^{−vκ} [π_{α,1}U^{κv}(∑_{j′=0}^{j} \binom{j}{j′}(−a_{p})^{j−j′}Φ_{j′}((a)_{v}))]

First we note that all the operators

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

are uniformely bounded for v → ∞ by a positive constant C_{1} (where U = αI + Z and Z^{n} = 0 for n = rk_{A}ℳ_{r^{∗}}(Np; A)) because n does not depend on v and the binomial coefficients \binom{−vκ}{t} are all ℤ_{p}-integers.

On the other hand by the condition (growth) of the theorem (for the distributions Φ_{j})

|U^{κv}(∑_{j′=0}^{j} \binom{j}{j′}(−a_{p})^{j−j′}Φ_{j′}((a)_{v}))|_{p,ℳ_{r^{∗}}} ≤ Cp^{−vj}

for all j = 0, 1, …, κh − 1. If we apply to this estimate the previous bounded operators π_{α,1}U^{κv} we get the following inequality

|∫_{(a)_{v}}(y_{p} − a_{p})^{j} dΦ̃^{α}|_{p,ℳ_{r^{∗}}} ≤ C · C_{1}|α^{−vκ}|_{p} · p^{−vj} = o(p^{−v(j−hκ)})

because of the estimate

|α^{−vκ}|_{p} = (p^{v_{p}(α)})^{vκ} = o(p^{vhκ})

§4. The standard L-function of a Siegel cusp eigenform and its critical values.

4.1. The standard L function. Let f ∈ M_{k}^{m}(C, ψ) be an eigenfunction of all Hecke operators, and let

(α_{0}, α_{1}, ⋯ α_{m}) = (α_{0,f}(q), α_{1,f}(q), ⋯ α_{m,f}(q)) ∈ [(ℂ^{×})^{m+1}]^{W_{m}}  (4.1)

be the Satake q-parameters of the modular form f (for all primes q not dividing the level C).

The standard L function D(s, f, χ) of f ∈ M_{m}^{k}(C, ψ) is the product D(s, f, χ) = ∏_{q |̸N} D^{(q)}(s, f, χ) with

D^{(q)}(s, f, χ) = (1 − χ(q)ψ(q)q^{−s})^{−1}R_{f,q}(χ(q)ψ(q)q^{−s})^{−1},

that is

\begin{matrix}D(s, f, χ) = \\ = ∏_{q} \{(1 − \frac{χ(q)ψ(q)}{q^{s}}) ∏_{i=1}^{m} (1 − \frac{χ(q)ψ(q)α_{i}(q)^{−1}}{q^{s}} ) (1 − \frac{χ(q)ψ(q)α_{i}(q)}{q^{s}})\}^{−1} , \end{matrix} (4.2)

where χ is a Dirichlet character mod M. Analytic properties of the standard zeta functions were investigated by A. N. Andrianov and V. L. Kalinin [An79] in the case of even degree m, and S. Böcherer [Bö85] extended these results to the case of arbitrary degree using a different approach (see also [Bö-Sch] where a p-adic version of his construction was developped). For m = 1 and a normalized cusp eigenform f(z) = ∑_{n=0}^{∞} a(n)e(nz) ∈ M_{1}^{k}(C, ψ) we have that

D(s, f, χ) = L_{2,f}(s + k − 1, χ),

where

L_{2,f}(s, χ) = L_{CM}(2s − 2k + 2, χ^{2}ψ^{2}) ∑_{n=1}^{∞} χ(n)a(n^{2})n^{−s}

is the symmetric square of the modular form f.

4.2. Theta series (see [An79], [St2]). Let F ∈ 2C_{m} be an even symmetric positive definite matrix, and q_{0} its level (i.e. the smallest positive integer such that q_{0}F^{−1} ∈ 2C_{m}, and χ a Dirichlet character modulo Q (not necesarily primitive). Put ν = 1 or 0 and define the theta function

θ(χ) =θ_{F}^{(ν)}(z; χ) = ∑_{ξ∈M_{m}(ℤ)} χ(detξ)detξ^{ν}e_{m}(zF [ξ]/2).  (4.2)

Proposition 4.3. (a) If

γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}^{m}(q_{0}Q^{2}),

then the following transformation formula holds

θ_{F}^{(ν)}(γ(z); χ) = χ(det d)χ_{Q^{2}F}^{(m)} (γ)det(cz + d)^{(m/2)+ν}θ_{F}^{(ν)}(z; χ),  (4.3)

where χ_{F}^{(m)}(γ) is a root of unity of the eighth degree, and if m is even, then

χ_{F}^{(m)}(γ) = (\frac{(−1)^{m/2}det(F)}{det d}) .  (4.4)

(b) Let J(M) denote for M > 0 the matrix ( \begin{matrix}0_{m} & −1_{m} \\ M1_{m} & 0_{m} \end{matrix} ). If χ is primitive modulo Q then the action of the involution J(Q^{2}q_{0}) on (4.3) is given by

\begin{matrix}θ_{F}^{(ν)}( & J(Q^{2}q_{0})z; χ) = \\ χ(−1)^{m}Q^{mν}G_{Q}(1_{m}, χ)det(F)^{(m/2)+ν}[det(−iz)]^{(m/2)+ν}θ(ν) ; F̂(z; \overline{χ}), \end{matrix}  (4.5)

with F̂ = q_{0}F^{−1} and

G_{Q}(ξ, χ) = ∑_{h∈M_{m}(ℤ) mod Q} χ(det h) e_{m}(^{t}ξh/Q)

is the Gauss sum of degree m of the character χ.

4.4. The Rankin zeta function. Let f and g be two holomorphic modular forms of weights k and l on the congruence subgroup γ_{0}^{m}(N) ⊂ Γ^{m}, and with Dirichlet characters ψ and ω. More precisely we suppose that

f(z) = ∑_{ξ∈B_{m}} a(ξ)e_{m}(ξz) ∈ 𝒮_{m}^{k}(N, ψ),  (4.6)

g(z) = ∑_{ξ∈C_{m}} b(ξ)e_{m}(ξz) ∈ ℳ_{m}^{l}(N, ψ),  (4.7)

with B_{m} being the set of half integral non negative matrices of size m × m, and C_{m} ⊂ B_{m} the subset of all positive definite matrices, see §1. Define an equivalence relation on B_{m} by ξ_{1} ∼ ξ_{2} iff ξ_{1} = ^{t}uξ_{2}u for some matrix u ∈ SL_{m}(ℤ). Then the Rankin zeta function (the convolution of two Siegel modular forms) is defined as the series:

L(s, f, g) = ∑_{ξ∈C̃_{m}} ε(ξ)^{−1}a(ξ)b(ξ)det(ξ)^{−s} (s ∈ ℂ),  (4.8)

where ε(ξ) = |Aut(ξ)| is the order of the unit group

Aut(ξ) = {η ∈ SL_{m}(ℤ)| ^{t}ηξη = ξ[η] = ξ}.

This series is well defined due to basic properties of Fourier coefficients of Siegel modular forms; C̃_{m} = C_{m}mod ∼ denotes the orbit space with respect to ∼ and Re(s) is supposed to be large enough.

Proposition 4.5. (Integral representation for the Rankin zeta function) (see [St2], proposition 6).

For s with Re(s) ≫ 0 there is the following integral representation.

(4π)^{−ms}Γ_{m}(s)L(s, f, g) = ⟨f^{ρ}, gE(z, s − k + (m + 1)/2; k − l, ψω, N)⟩_{N}  (4.10)

where the Petersson inner product,

f^{ρ}(z) = ∑_{ξ∈B_{m}} \overline{a(ξ)}e_{m}(ξz) ∈ 𝒮_{m}^{k}(N, ψ),

and Γ_{m}(s) denotes the Γ-function of degree m, i.e.

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

4.6. The standard zeta function D(s, f, χ) as the Rankin convolution. Now let

f(z) = ∑_{ξ∈C_{m}} \overline{a(ξ)}e_{m}(ξz) ∈ 𝒮_{m}^{k}(N, ψ)

be a cusp form of even degree m which is an eigenfunction of the Hecke algebra ℒ_{q}^{m}(N) for q not dividing N, and let

(α_{0}, α_{1}, ⋯, α_{m}) = (α_{0}(q), α_{1}(q), ⋯, α_{m}^{q})) ∈ [(A^{×})^{m+1}]^{W_{m}}

be the corresponding (m + 1)-tuple of the Satake q-parameters of f. Fix a matrix ξ_{0} ∈ C_{m} such that a(ξ_{0}) ≠ 0 and consider the primitive quadratic Dirichlet character χ_{ξ_{0}} definied for positive integers d with (d, 2det(2ξ_{0})) = 1 by the formula

χ_{ξ_{0}}(d) = (\frac{(−1)^{m/2}det(2ξ_{0})}{d}) ,

and when det(2ξ_{0}) is odd, then by definition χ_{ξ_{0}}(2) = 1 or −1 according as to which of the two following quadratic forms

x_{1}x_{2} + x_{3}x_{4} + … x_{m−1}x_{m}

or

x_{1}x_{2} + … + x_{m−3}x_{m−2} + x_{m−1}^{2} + x_{m−1}x_{m} + x_{m}^{2}

our quadratic form ξ_{0} is equivalent over the field F_{2} of two elements. Assume also that χ mod M is chosen so that (−1)^{ν} = χ(−1). The main result of the A.N.Andrianov’s work [An6] can be stated as a certain identity expressing the standard zeta function D(s, f, χ) as a Rankin zeta function, namely the convolution of the given form f and a theta function with the Dirichlet character χ mod M. The precise statement of the result is given in the following proposition.

Proposition 4.7. Under the notation and assumptions as above for the sufficiently large values of Re(s) the following identity holds

\begin{matrix}a(ξ_{0}) & R(s, f, χ) = \\ 2^{−1}det ξ_{0}^{(s+k−1+ν)/2}L((s + k − 1 + ν)/2, f, θ_{2ξ_{0}}^{(ν)}(z; χ)), \end{matrix}  (4.11)

where the function R(s, f, χ) is defined by the following equality

D(s, f, χ) = L(s + (m/2), ψχ_{ξ_{0}}χ) ∏_{i=0}^{m−1} L(2s + 2i, ψ^{2}χ^{2})R(s, f, χ),  (4.12)

and it is assumed that the modulus M of the character χ is divisible by all prime divisors of the number Ndet2ξ_{0}, where θ_{2ξ_{0}}^{(ν)}(z; χ) is the theta function as in4.2.

More explicitly, the right hand side of (4.11) can be represented as the series

(detξ_{0})^{(s+k−1+ν)/2} ∑_{ξ} χ(detξ)detξ^{ν}a(ξ_{0}[ξ])(detξ_{0}[ξ])^{−(s+k−1+ν)/2} =

∑_{ξ} χ(detξ)a(^{t}ξξ_{0}ξ)(detξ)^{−(s+k−1+ν)/2},

(4.13)

where the summation is taken over the set of equivalence classes ξ ∈ SL_{m}(ℤ)\M_{m}^{+}(ℤ) of the form SL_{m}(ℤ)ξ.

Now we set q_{0} to be equal to the level of the quadratic form with the matrix 2ξ_{0} (see 4.2). In order to get the integral representation of the standard zeta function we apply to it the result of 2.6. For this purpose we put in the notation of 2.6 l = (m/2) + ν, ω = χχ_{ξ_{0}} with χ being a Dirichlet character modulo M, and take the number Cq_{0}M^{2} as N. We note also that both parts of (2.27) and (2.26) converge absolutely for Re(s) > m (see also [An79], p. 133).

Proposition 4.8 (Integral representation of the standard zeta function). With the notation and assumption as above for Re(s) the following equality holds:

\begin{matrix}2a(ξ_{0}) & (detξ_{0})^{−(s+k−1+ν)/2}(4π)^{−ms} × Γ_{m}((s + k − 1 + ν)/2) R(s, f, χ) = \\ ⟨f^{ρ}, θ_{2ξ_{0}}^{(ν)}(z; χ)E(z, (s − k + m + ν)/2)⟩_{NM^{2}q_{0}}, \end{matrix}  (4.14)

where the Siegel – Eisenstein series in the right hand side

E(z, s) = E(z, s; k − ν − (m/2), χχ_{ξ_{0}}ψ, NM^{2}q_{0})

is defined by (2.6) .

4.9. Algebraic properties of the special values of normalized distributions. Consider only the case of even m. In order to give a precise statements on algebraic properties of the standard zeta functions and of the corresponding distributions, it is convenient to make some additional normalization of these values because these properties look different for the integral points to the left and to the right of the critical line Re(s) = \frac{1}{2} (the same effect happens in the case of the Riemann zeta function).

Recall the following three types of the normalized zeta function: for κ = (m + 1)/2, s ∈ ℂ

\begin{matrix}D^{∗}(s, f, χ) = \\ (2π)^{−m(s+k−κ)}Γ((s + δ)/2) ∏_{j=1}^{m} Γ(s + k − j)D(s, f, χ), \end{matrix}  (4.15)

\begin{matrix}D^{−}(s, f, χ) = Γ((s + δ)/2)^{−1}D^{∗}(s, f, χ) = \\ (2π)^{−m(s+k−κ)} ∏_{j=1}^{m} Γ(s + k − j)D(s, f, χ), \end{matrix}  (4.16)

\begin{matrix}D^{+}(s, f, χ) = \frac{2 i^{δ}Γ(s) cos(π(s − δ)/2)}{(2πi)^{s}}D^{−}(s, f, χ) = \\ \frac{i^{δ}π^{(1−2s)/2}}{Γ((1 + δ − s)/2)}D^{∗}(s, f, χ), \end{matrix}  (4.17)

where δ = 0 or 1 according as ψχ(−1) = (−1)^{δ} with f ∈ S_{k}^{m}(C, ψ) be an eigenfunction of the global Hecke algebra ℒ^{m}(C) = ⊗_{q |̸C}ℒ_{q}^{m}(C) with the eigenvalue given as a homomorphism λ_{f} : ℒ^{m}(C) → ℂ.

Now we turn to the normalizations (4.16) and (4.17). Their convenience is illustrated by the following result on algebraicity of the special values of (4.16) and (4.17) which is proven in this section. Theorem 4.10 (Algebraic properties of the special values of the standard zeta function). Assume that the cusp eigenform f ∈ S_{k}^{m}(C, ψ) satisfies the condition a(ξ_{0}) = 1 for some ξ_{0} ∈ A_{m}, ξ_{0} > 0. Then

a) For all integers s satisfying 1 ≤ s ≤ k − δ − m and s ≠ 1 if the character χ^{2}ψ^{2} is trivial we have that

⟨f, f⟩^{−1} D^{+}(s, f, χ) ∈ K = ℚ(f, λ_{f}, ψ, χ),  (4.18)

where K = ℚ(f, λ_{f}, ψ, χ) denote the field generated by Fourier coefficients of f, by the eigenvalues λ_{f}(X) of Hecke operators X on f, and by the values of the characters χ and ψ. Moreover,

D^{+}(s, f, χ) = 0 for s ≢ δ(mod2).

b) For all integers s satisfying 1 − k + δ + m ≤ s ≤ 0 we have that

⟨f, f⟩^{−1} D^{−}(s, f, χ) ∈ K,  (4.19)

and

D^{−}(s, f, χ) = 0 for s ≡ δ(mod2), s ∈ Z, s ≤ 0.

Proof of the theorem 4.10 (see [PaLNM], Chapter 3). One uses the identity (2.29) of chapter 2 of [PaLNM], which says that for all non-trivial Dirichlet characters χ mod p^{v}:

\begin{matrix}2a(ξ_{0}) & ((4π)^{m}det(ξ_{0}))^{−(s+k−1+ν)/2}Γ_{m}((s + k − 1 + ν)/2) D(s, f, χ) = \\ L_{N}(s + \frac{m}{2}, ψχ_{ξ_{0}}χ) ∏_{i=0}^{(m/2)−1} L_{N}(2s + 2i, ψ^{2}χ^{2}))× \\ × ⟨f^{ρ}(z), θ_{2ξ_{0}}^{(ν)}(z; χ)E(z, (s − k + m + ν)/2)⟩_{N}, \end{matrix}  (4.20)

E(z, s) = E(z, s; k − (m/2) − ν, ψχ_{ξ_{0}}χ, N)

is the Siegel – Eisenstein series of the weight k − (m/2) − ν, the level N = 4q_{0}CM^{2}, and the Dirichlet character ψχ_{ξ_{0}}χ is viewed modN.

§5. Algebraic linear forms on arithmetical nearly holomorphic Siegel modular forms.

5.1. Convolutions of theta distributions and Eisenstein distributions with values in nearly holomorphic Siegel modular forms. Consider again the profinite group

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

Let M = ℳ_{r,k}^{m}(A) and let Φ_{i} : 𝒞^{loc−const}(Y, A) → M be two distributions on Y with values in M_{i} = ℳ_{r_{i},k_{i}}^{m}(A), (i = 1, 2). Assume that A contains (φ(Np^{v}))^{−1} and all the values of Dirichlet characters χ mod Np^{v} viewed as homomorphisms Y → A^{×} then obviously

Φ_{i}(χ) = ∑_{a mod Np^{v}} χ(a)Φ_{i}((a)_{v}), Φ_{i}((a)_{v}) = \frac{1}{φ(Np^{v})} ∑_{χ mod Np^{v}} χ(a)^{−1}Φ_{i}(χ).

Let us fix a Dirichlet character ψ on Y with values in A then there exists a unique distribution Φ = Φ_{1} ∗_{ψ} Φ_{2} on Y with values in M = ℳ_{r_{1}+r_{2},k_{1}+k_{2}}^{m}(A) such that for all Dirichlet characters χ on Y with values in A one has Φ(χ) = Φ_{1}(χ) · Φ_{2}(ψχ^{−1}) (Φ_{1} ∗_{ψ} Φ_{2} is called the twisted convolution of Φ_{1} and Φ_{2}.

Let us put M = p^{v}, N = 4q_{0}C and take as Φ_{1} the theta distribution on Y defined on Dirichlet characters χ mod M by

Φ_{1}(χ) = θ_{2ξ̂_{0}}^{(ν)}(χ)|V (C)  (5.1)

and let us take as Φ_{2,s}^{±} the normalized Siegel-Eisenstein distribution (2.21) defined on Dirichlet characters ω by

\begin{matrix}Φ_{2,s}^{±}(ω) = G^{±}(z, (s − k + m + ν)/2); k − (m/2) − ν, ω, NM′) = \\ G_{′s−k+m+ν)/2),k−(m/2)−ν}^{±}(ω), NM′) = δ^{(r^{±})}Ψ_{2,s}^{±}(ω) \end{matrix}  (5.2)

(we use it for the Dirichlet character ω = ψχ_{ξ_{0}} χ̄ ) where M′ = M^{2},

\begin{matrix}Ψ_{2,s}^{+}(ω) = G_{m/2+s,0}^{+}(ω), \\ Ψ_{2,s}^{−}(ω) = G_{m/2+1−s,s−(1/2)}^{−}(ω) \end{matrix}

are two holomorphic Siegel-Eisenstein series.

Next define Φ_{s}^{±} = Φ_{1} ∗_{ψχ_{ξ_{0}}} Φ_{2,s}^{±} (on basic open subsets one has
Φ_{s}^{±}((a)_{v}) = ∑_{y∈Y_{v}} ψχ_{ξ_{0}}(y)Φ_{1}(ya)Φ_{2,s}^{±}(y).  (5.3)

for any v such that ψχ_{ξ_{0}} is defined modNp^{v}).

We show that for all critical pairs (s, χ) as Theorem 4.10 the function Φ_{s}^{±}(χ) is an arithmetical nearly holomorphic Siegel modular form in ℳ_{r,k}^{m}(Γ_{1}^{m}(Np^{v}), ℚ̄), and for an appropriate linear form ℒ_{f,α} on ℳ_{r,k}^{m}(Γ_{1}(Np^{v}), ℚ̄) (defined over ℚ̄) the value ℒ_{f,α}(Φ_{s}^{±}(χ)) coincides, up to a multiplicative constant in ℚ̄^{×}, with the corresponding value D_{s,χ}^{±} of the normalized distribution constructed in [PaLNM], Chapter 3 (which produces the critical value D^{±}(s, f, χ̄))).

Let us consider (as in Proposition 1.6) the canonical projection operator

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

attached to a non-zero eigenvalue α of U = Π^{+}(p) acting on

ℳ_{r,k}^{m}(N, p; ℂ) = ⋃_{v≥1}ℳ_{r,k}^{m}(Γ_{1}^{m}(Np^{v}); ℂ) :

for any g ∈ ℳ = ℳ_{r,k}^{m}(ℂ) put

π_{α}(g) = U^{−v} [π_{α,1}U^{v}(g)]

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

Proposition 5.2. Let α ∈ ℚ̄ ⊂ ℂ be a non-zero eigenvalue of U = Π^{+}(p) on ℳ_{r,k}^{m}(ℂ) associated with a cusp eigenform f ∈ 𝒮_{k}(Γ_{0}^{m}(N), ψ) and let f_{0} = f_{0,α} be a corresponding eigenfunction (f_{0}|U = αf_{0}), let us define f^{0} = f_{0}^{ρ}|W_{N_{0}}, f_{0}^{ρ} = ∑_{ξ} \overline{a_{ξ}(f_{0})}q^{ξ}. Then

a) U^{∗} = W_{N_{0}}^{−1}UW_{N_{0}}, the adjoint operator of U in the hermitian vector space 𝒮_{r,k}(Γ_{1}^{m}(N_{0}), ℂ) with respect to the Petersson scalar product.

b) One has f^{0}|U^{∗} = ᾱf^{0}, and for all ”good” Hecke operators T ∈ ℒ(Np^{v}) one has T f^{0} = λ_{f}(T)f^{0}.

c) The linear form ℓ_{f,α} : g ↦ ⟨f^{0}, g⟩ on ℳ_{r,k}^{m}(Γ_{1}^{m}(N_{0}), ℂ) vanishes on Kerπ_{α,1}, where π_{α,1} : ℳ_{r,k}^{m}(Γ_{1}^{m}(N_{0}), ℂ) → ℳ_{r,k}^{m}(Γ_{1}^{m}(N_{0}), ℂ)^{α} (the projector onto the α-primary subspace with the kernel Kerπ_{α,0} = Im(U − αI)^{n}) hence

⟨f^{0}, g⟩ = ⟨f^{0}, π_{α,1}(g)⟩ = ⟨f^{0}, π_{α}(g)⟩.
d) If g ∈ ℳ^{m}(Np^{v+1}, \overline{ℚ}) = ℳ^{m}(N_{0}p^{v}, \overline{ℚ}) and α ≠ 0, one has
⟨f^{0}, g^{α}⟩ = ⟨f^{0}, π_{α}(g)⟩ = α^{−v}⟨f^{0}, g | U^{v}⟩

where

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

is the α-primary part of g (the image of g under the canonical projection π_{α}): e) Let us define

ℒ_{f,α}(g) = \frac{⟨f^{0}, α^{−v}g|U^{v}⟩_{N_{0}}}{⟨f^{0}, f_{0}⟩_{N_{0}}} = \frac{⟨f^{0}, π_{α}(g)⟩_{N_{0}}}{⟨f^{0}, f_{0}⟩_{N_{0}}} ,

hence ℒ_{f,α} : ℳ_{r,k}^{m}(N, p; ℚ) → \overline{ℚ} is the linear form ℒ_{f,α} on ℳ_{r,k}^{m}(N, p; ℂ) which is defined over \overline{ℚ} and factorizes through the finite dimensional projection

π_{α} : ℳ_{r,k}^{m}(N, p; ℂ) → ℳ_{r,k}^{m}(N, p; ℂ)^{α} ⊂ ℳ_{r,k}^{m}(Γ_{1}^{m}(Np); ℂ);

There exists a unique ℂ_{p}-linear form ℓ_{f,α} ∈ (ℳ_{r,k}^{m}(N_{0}, ℂ_{p})^{α})^{∗} such that

ℓ_{f,α}(i_{p}(π_{α}(g))) = i_{p} (ℒ_{f,α}(g)) = i_{p} (\frac{⟨f^{0}, α^{−v}U^{v}(g)⟩_{N_{0}}}{⟨f^{0}, f_{0}⟩_{N_{0}}} )

(g ∈ ℳ_{r,k}^{m}(N, p; ℚ̄) with coefficients in ℚ̄).

Proof of Proposition 5.2 a) See [Miy], Th. 4.5.5 for m = 1 (the general case is very similar). b) By definition, f^{0}|U^{∗} = f_{0}^{ρ} | W_{Np}W_{Np}^{−1}UW_{Np} = ᾱf_{0}^{ρ} | W_{Np} = ᾱf^{0}.

c) For any function

g_{1} = (U − αI)^{n}g ∈ Kerπ_{α,0} = Im(U − αI)^{n}

one has

⟨f^{0}, g_{1}⟩ = ⟨f^{0}, (U − αI)^{n}g⟩ = ⟨(U^{∗} − ᾱI)f^{0}, (U − αI)^{n−1}g⟩ = 0

hence for g_{1} = g − π_{α,0}(g) we get

⟨f^{0}, g⟩ = ⟨f^{0}, π_{α,0}(g) + (g − g^{α})⟩ = ⟨f^{0}, π_{α,0}(g)⟩ + ⟨f^{0}, g_{1}⟩ = ⟨f^{0}, π_{α,0}(g)⟩
d) Let us use directly the equality (U^{∗})^{v}f^{0} = ᾱ^{v}f^{0} of b):
α^{v} · ⟨f^{0}, g^{α}⟩ = ⟨(U^{∗})^{v}f^{0}, U^{−v}π_{α,1}(g | U^{v})⟩ = ⟨f^{0}, π_{α,1}(g | U^{v})⟩ = ⟨f^{0}, g | U^{v}⟩

by c) because g | U^{v} ∈ ℳ^{m}(Np).

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

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

which admits a \overline{ℚ}-rational basis because it is stable by the action of for all ”good” Hecke operators T ∈ ℒ(Np^{v}):

⟨f^{0}, g⟩ = 0 ⟹ ⟨f^{0}, g|T ⟩ = ⟨f^{0}|T^{∗}, g⟩ = 0

and one obtains such a basis by the diagonalisation of the action of all the T (a commutative family of normal operators) and e) follows.

5.3. Our next task is to compute the value

ℒ_{f,α}(Φ_{s}^{±}(χ)) = ℒ_{f,α}(Φ_{1}(χ) · Φ_{2,s}^{±}(ψχ_{ξ_{0}}χ^{−1})) = ; ℒ_{f,α}(θ_{2ξ̂_{0}}^{(ν)}(χ)|V (C)G^{±}(z, (s − k + m + ν)/2; k − (m/2) − ν, ψχ_{ξ_{0}} \overline{χ}, NM′)) = ; \frac{α^{−2v}⟨f^{0}, (Φ_{1}(χ) · Φ_{2,s}^{±}(ψχ_{ξ_{0}}χ^{−1}))|U^{2v}⟩_{Np}}{⟨f^{0}, f_{0}⟩_{Np}} ; \frac{α^{−2v}⟨f^{0}, (θ_{2ξ̂_{0}}^{(ν)}(χ)|V (C)G^{±}(z, \frac{s−k+m+ν}{2} ; k − \frac{m}{2} − ν, ψχ_{ξ_{0}} \overline{χ}, NM′))|U^{2v}⟩_{Np}}{⟨f^{0}, f_{0}⟩_{Np}} ; \frac{α^{−2v}⟨f_{0}^{ρ}|W_{Np}, (θ_{2ξ̂_{0}}^{(ν)}(χ)|V (C)G^{±}(z, \frac{s−k+m+ν}{2} ; k − \frac{m}{2} − ν, ψχ_{ξ_{0}} \overline{χ}, NM′))|U^{2v}⟩_{Np}}{⟨f^{0}, f_{0}⟩_{Np}} (5.4)

where χ mod N_{0}p^{v} is a Dirichlet character of conductor C_{χ} = p^{v}. Let us unfold the integral (5.4) using the relation (given in Proposition 1.6) of U^{2v} with the trace operator

g|U^{2v} = p^{vm(k−m−1)}g|W_{Np^{2v+1}}Tr_{Np}^{Np^{2v+1}}W_{Np} :

ℒ_{f,α}(Φ_{s}^{±}(χ)) = \frac{(−1)^{mk}p^{vm(k−m−1)}}{α^{2v}} ; \frac{⟨f_{0}^{ρ}, (Φ_{1}(χ) · Φ_{2,s}^{±}(ψχ^{−1}))|W_{Np^{2v+1}}Tr_{Np}^{Np^{2v+1}}⟩_{Np}}{⟨f^{0}, f_{0}⟩_{Np}} ; = \frac{(−1)^{mk}p^{vm(k−m−1)}}{α^{2v}} ; \frac{⟨f_{0}^{ρ}, (θ_{2ξ̂_{0}}^{(ν)}(χ)|V (C)G^{±}(z, \frac{s−k+m+ν}{2} ; k − \frac{m}{2} − ν, ψχ_{ξ_{0}} \overline{χ}, Np^{2v+1}))|W_{Np^{2v+1}}⟩_{Np^{2v+1}}}{⟨f^{0}, f_{0}⟩_{Np}} ; = \frac{(−1)^{mk}p^{vm(k−m−1)}}{α^{2v}⟨f^{0}, f_{0}⟩_{Np}} · ⟨f_{0}^{ρ}, Φ_{1}(χ)|W_{Np^{2v+1}}Φ_{2,s}^{±}(ψχ_{ξ_{0}}χ^{−1})|W_{Np^{2v+1}}⟩_{Np^{2v+1}} (5.5)

Let us compute separately

Φ_{1}(χ)|W_{Np^{2v+1}} = θ_{2ξ̂_{0}}^{(ν)}(χ)|V (C)W_{Np^{2v+1}}  (5.6)

and

Φ_{2,s}^{±}(ψχ_{ξ_{0}}χ^{−1})|W_{Np^{2v+1}} = G^{±}(z, \frac{s − k + m + ν}{2} ; k − \frac{m}{2} − ν, ψχ_{ξ_{0}} \overline{χ}, NM′)|W_{Np^{2v+1}} (5.7)

using the action of the main involution on theta series, and using the definitions (2.12) and (2.13) of the normalized Siegel-Eisenstein series.

Now let us put N_{0}M′ = 4q_{0}C_{χ}^{2}N′ with N′ = NM_{0}^{m̃−1}M′C_{χ}^{−2} so that the following transformation formula holds:

\begin{matrix}θ_{2ξ̂_{0}}^{(ν)} & (χ)|W (N_{0}M′) = N^{′m(m+2ν)/4}θ_{2ξ̂_{0}}^{(ν)}(χ)|W (4q_{0}C_{χ}^{2})V (N′) \\ = & (NM_{0}^{m̃−1}M′)^{m(m+2ν)/4} \frac{G_{C_{χ}}(1_{m}, χ)}{(iC_{χ})^{m^{2}/2}C_{χ}^{−m(m+2ν)/2}} · \\ · det(q_{0}^{−1/2}ξ_{0})^{m/2+ν}θ_{2ξ_{0}}^{(ν)}(χ̄)|V (N′) \end{matrix}  (5.8)

Recall the definition of the normalized Siegel-Eisenstein series of the level N_{0}M′ = Np^{2v+1}:

G^{−}(z, s) = Γ((k + 2s − (m/2) + μ)/2)^{−1}G^{∗}(z, s),

\begin{matrix}G^{+}(z, s) = \frac{i^{μ}π^{κ−k−2s}}{Γ((1 − k − 2s + (m/2) + μ)/2)}G^{∗}(z, s) = \\ \frac{2i^{μ}Γ(k + 2s − (m/2)) cos(π(k + 2s − (m/2) − μ)/2)}{(2π)^{k+2s−(m/2)}}G^{−}(z, s). \end{matrix}

where μ = ε((m/2) + k), and

\begin{matrix}G^{∗}(z, s) = G^{∗}(z, s; k, χ, N_{0}M′) = \\ (N_{0}M′)^{m(k+s)}Γ̃(k, s)L_{N_{0}}(k + 2s, χ) ∏_{i=1}^{[m/2]} L_{N_{0}}(2k + 4s − 2i, χ^{2})E^{∗}(N_{0}M′z, s), \end{matrix} (5.9)

with E^{∗}(N_{0}M′z, s) = E(−(N_{0}M′z)^{−1}, s)det(N_{0}M′z)^{−k} = (N_{0}M′)^{−km/2}E|W (N_{0}M′), Γ̃(k, s) =

i^{mk}2^{−m(k+1)}π^{−m(s+k)} × \{ \begin{matrix}Γ_{m}(k + s)Γ(s + (k − \frac{m}{2} + μ)/2), & if m is even; \\ Γ_{m}(k + s), & otherwise. \end{matrix}

(5.10)

and

L_{N}(s, χ) = ∏_{q |̸N}(1 − χ(q)q^{−s})^{−1}

is the Dirichlet series associated to the character χ and μ = ε(m/2 + k), with ε(r) ∈ {0, 1} such that ε(r) ≡ r (mod 2) for any integer r ∈ ℕ.

Let us introduce the Gamma-factors Γ^{±}(k, s) by the equality

\begin{matrix}G^{±}(z, s) = G^{±}(z, s; k, χ, N_{0}M′) \\ = (N_{0}M′)^{m(k+s)}Γ^{±}(k, s) \\ L_{N_{0}}(k + 2s, χ) ∏_{i=1}^{[m/2]} L_{N_{0}}(2k + 4s − 2i, χ^{2})E^{∗}(N_{0}M′z, s) \\ = (N_{0}M′)^{m((k/2)+s)}Γ^{±}(k, s) \\ L_{N_{0}}(k + 2s, χ) ∏_{i=1}^{[m/2]} L_{N_{0}}(2k + 4s − 2i, χ^{2})E|W (N_{0}M′), \end{matrix}  (5.11)

so that the change of variables s := (s − k + m + ν)/2, k := k − (m/2) − ν in (5.11) gives k + 2s := s + (m/2) = (s + 2m)/2 and

\begin{matrix}G^{±} (z, \frac{s − k + m + ν}{2} ; k − \frac{m}{2} − ν, ψχ_{ξ_{0}} \overline{χ}, N_{0}M′) |W (N_{0}M′) \\ = (N_{0}M′)^{m((s+2m)/4}Γ^{±}(k − \frac{m}{2} − ν, \frac{s − k + m + ν}{2} ) \\ L_{N_{0}}((2s + m)/2, ψχ_{ξ_{0}} \overline{χ}) ∏_{i=1}^{[m/2]} L_{N_{0}}(2s + m − 2i, \overline{χ}^{2}ψ^{2}) \\ E(z, (s − k + m + ν)/2; k − (m/2) − ν, ψχ_{ξ_{0}} \overline{χ}, N_{0}M′), \end{matrix}  (5.12)

and substitute it to (4.14) with χ replaced by the Dirichlet character \overline{χ}

\begin{matrix}2a(ξ_{0}) & (detξ_{0})^{−(s+k−1+ν)/2}(4π)^{−ms} × Γ_{m}((s + k − 1 + ν)/2) R(s, f, \overline{χ}) = \\ ⟨f_{0}^{ρ}, θ_{2ξ_{0}}^{(ν)}(z; \overline{χ})E(z, (s − k + m + ν)/2; k − (m/2) − ν, ψχ_{ξ_{0}} \overline{χ})⟩_{NM^{2}q_{0}}, \end{matrix}  (5.13)

where

\begin{matrix}a(ξ_{0}) & R(s, f, \overline{χ}) = \\ 2^{−1}det ξ_{0}^{(s+k−1+ν)/2}L((s + k − 1 + ν)/2, f_{0}, θ_{2ξ_{0}}^{(ν)}(z; \overline{χ})). \end{matrix}  (5.14)

The function R(s, f, \overline{χ}) is related to D(s, f, \overline{χ}) by the following equality

D(s, f, \overline{χ}) = L(s + (m/2), ψχ_{ξ_{0}} \overline{χ}) ∏_{i=0}^{(m/2)−1} L(2s + 2i, ψ^{2}\overline{χ}^{2}ψ^{2})R(s, f, \overline{χ})  (5.15)

therefore (5.13) takes the form

\begin{matrix}2a(ξ_{0}) & (detξ_{0})^{−(s+k−1+ν)/2}(4π)^{−ms} × Γ_{m}((s + k − 1 + ν)/2)D(s, f, \overline{χ}) = \\ L(s + (m/2), ψχ_{ξ_{0}} \overline{χ}) ∏_{i=0}^{(m/2)−1} L(2s + 2i, ψ^{2}\overline{χ}^{2}ψ^{2}) \\ ⟨f_{0}^{ρ}, θ_{2ξ_{0}}^{(ν)}(z; \overline{χ})E(z, (s − k + m + ν)/2; k − (m/2) − ν, ψχ_{ξ_{0}} \overline{χ})⟩_{NM^{2}q_{0}}, \end{matrix}  (5.16)

We see that the integral in (5.5) transforms to the following

\begin{matrix}⟨f_{0}^{ρ}, Φ_{1}(χ)|W_{Np^{2v+1}}Φ_{2,s}^{±}(ψχ^{−1})|W_{Np^{2v+1}}⟩_{Np^{2v+1}} = \\ ⟨f_{0}^{ρ}, θ_{2ξ̂_{0}}^{(ν)}(χ)|V (C)W_{N_{0}M′} \\ G^{±}(z, \frac{s − k + m + ν}{2} ; k − \frac{m}{2} − ν, ψχ_{ξ_{0}} \overline{χ}, N_{0}M′)|W_{N_{0}M′}⟩_{N_{0}M′} \end{matrix}  (5.17)

and (5.17) becomes

\begin{matrix}⟨f_{0}^{ρ}, & Φ_{1}(χ)|W_{Np^{2v+1}}Φ_{2,s}^{±}(ψχ_{ξ_{0}}χ^{−1})|W_{Np^{2v+1}}⟩_{Np^{2v+1}} = \\ (NM_{0}^{m̃−1}M′)^{m(m+2ν)/4} \frac{G_{C_{χ}}(1_{m}, χ)}{(iC_{χ})^{m^{2}/2}C_{χ}^{−m(m+2ν)/2}} det(q_{0}^{−1/2}ξ_{0})^{m/2+ν} \\ (N_{0}M′)^{m((s+2m)/4}Γ^{±}(k − \frac{m}{2} − ν, \frac{s − k + m + ν}{2} ) \\ L_{N_{0}}((2s + m)/2, ψχ_{ξ_{0}} \overline{χ}) ∏_{i=1}^{[m/2]} L_{N_{0}}(2s + m − 2i, \overline{χ}^{2}ψ^{2}) \\ ⟨f_{0}^{ρ}, θ_{2ξ_{0}}^{(ν)}(χ̄)|V (N′)E(z, \frac{s − k + m + ν}{2} ; k − \frac{m}{2} − ν, ψχ_{ξ_{0}} \overline{χ}, N_{0}M′)⟩_{N_{0}M′} \end{matrix}  (5.18)

Let us now compare the value of the linear form (5.5) expliciely given by (5.18), with the integral representation (5.16) for the standard L-function:

\begin{matrix}ℒ_{f,α}(Φ_{s}^{±}(χ)) = \\ = \frac{(−1)^{mk}p^{vm(k−m−1)}}{α^{2v}⟨f^{0}, f_{0}⟩_{Np}} · ⟨f_{0}^{ρ}, Φ_{1}(χ)|W_{Np^{2v+1}}Φ_{2,s}^{±}(ψχ_{ξ_{0}}χ^{−1})|W_{Np^{2v+1}}⟩_{Np^{2v+1}} \end{matrix}

\begin{matrix}= \frac{(−1)^{mk}p^{vm(k−m−1)}}{α^{2v}⟨f^{0}, f_{0}⟩_{Np}} \\ (NM_{0}^{m̃−1}M′)^{m(m+2ν)/4} \frac{G_{C_{χ}}(1_{m}, χ)}{(iC_{χ})^{m^{2}/2}C_{χ}^{−m(m+2ν)/2}} det(q_{0}^{−1/2}ξ_{0})^{m/2+ν} \\ (N_{0}M′)^{m((s+2m)/4}Γ^{±}(k − \frac{m}{2} − ν, \frac{s − k + m + ν}{2} ) \\ L_{N_{0}}((2s + m)/2, ψχ_{ξ_{0}} \overline{χ}) ∏_{i=1}^{[m/2]} L_{N_{0}}(2s + m − 2i, \overline{χ}^{2}ψ^{2}) \\ ⟨f_{0}^{ρ}, θ_{2ξ_{0}}^{(ν)}(χ̄)|V (N′)E(z, \frac{s − k + m + ν}{2} ; k − \frac{m}{2} − ν, ψχ_{ξ_{0}} \overline{χ}, N_{0}M′)⟩_{N_{0}M′} \end{matrix}  (5.19)

Let us remove from the Euler products D(s, f, \overline{χ}) and L(s, χ) all the factors corresponding to the prime divisors of N_{0}:

\begin{matrix}2a(ξ_{0}) & (detξ_{0})^{−(s+k−1+ν)/2}(4π)^{−ms} × Γ_{m}((s + k − 1 + ν)/2)D_{N_{0}}(s, f, \overline{χ}) = \\ L_{N_{0}}(s + (m/2), ψχ_{ξ_{0}} \overline{χ}) ∏_{i=0}^{(m/2)−1} L_{N_{0}}(2s + 2i, ψ^{2}\overline{χ}^{2}ψ^{2}) \\ ⟨f_{0}^{ρ}, θ_{2ξ_{0}}^{(ν)}(z; \overline{χ})E(z, (s − k + m + ν)/2; k − (m/2) − ν, ψχ_{ξ_{0}} \overline{χ})⟩_{NM^{2}q_{0}}, \end{matrix}  (5.20)

Let χ be a non-trivial Dirichlet character modulo Np^{v} of conductor C_{χ} such that C_{χ}|p^{v}).

Consider fhe complex valued distributions ℒ_{f,α}(Φ_{s}^{±}(χ)) defined for the critical pairs (s, χ) as in Theorem 4.10 by

ℒ_{f,α}(Φ_{s}^{±}(χ)) = ℒ_{f,α}(Φ_{1}(χ) · Φ_{2,s}^{±}(ψχ_{ξ_{0}}χ^{−1})).

Proposition 5.4. Assume that the cusp eigenform f ∈ S_{k}^{m}(C, ψ) satisfies the condition a(ξ_{0}) = 1 for some ξ_{0} ∈ A_{m}, ξ_{0} > 0. Then

a) For all integers s satisfying 1 ≤ s ≤ k − δ − m and s ≠ 1 if the character χ^{2}ψ^{2} is trivial we have that the value

ℒ_{f,α}(Φ_{s}^{+}(χ)) ∈ K = ℚ(f, λ_{f}, ψ, χ),

where K = ℚ(f, λ_{f}, ψ, χ) denotes the field generated by Fourier coefficients of f, by the eigenvalues λ_{f}(X) of Hecke operators X on f, and by the values of the characters χ and ψ. Moreover,

ℒ_{f,α}(Φ_{s}^{+}(χ)) = 0 for s ≢ δ(mod2).

b) For all integers s satisfying 1 − k + δ + m ≤ s ≤ 0 we have that

ℒ_{f,α}(Φ_{s}^{−}(χ)) ∈ K,

and

ℒ_{f,α}(Φ_{s}^{−}(χ)) = 0 for s ≡ δ(mod2), s ∈ ℤ, s ≤ 0.

c) Suppose moreover that N is divisible by all of the prime divisors of 2det(2ξ_{0}) then there is the following equality

ℒ_{f,α}(Φ_{s}^{±}(χ)) = t_{±} \frac{G_{C_{χ}}(1_{m}, χ)}{(iC_{χ})^{m^{2}/2}} \frac{C_{χ}^{m(2s+2k−2−m)/4}}{α_{0}(C_{χ})^{2}} D^{±}(s, f, χ̄),

where t_{±} is a multiplicative constant in ℚ̄^{×}, independent of the critical pair (s, χ), D^{±}(s, f, χ̄) denotes the normalized standard L-function as in Theorem 4.10,

G_{C_{χ}}(1_{m}, χ) = ∑_{ξ′∈M_{m}(Z) (mod C_{χ})} χ(detξ′)e_{m}(^{t}ξξ′/C_{χ}),

is the Gauss sum of degree m of χ.

Proof of Proposition 5.4: is directly implied by the equalities (5.19) and (5.20) taking into account Theorem 4.10 on the critical calues of the standard L function.

§6. Congruences between nearly holomorphic modular forms and proof of the

Main theorem.

6.1. Regularized distributions with values in nearly holomorphic Siegel modular forms. We now outline the method of the proof of Main Theorem 0.7 and show the existence of the h–admissible measures attached to the standard zeta functions D(s, f, χ). This proof is based on congruences for the Fourier coefficients of nearly holomorphic Siegel modular forms. Consider again the profinite group

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

and consider fhe complex valued distributions ℒ_{f,α}(Φ_{s}^{±}(χ)) defined for the critical pairs (s, χ) as in Theorem 4.10 and Proposition 5.4 by

ℒ_{f,α}(Φ_{s}^{±}(χ)) = ℒ_{f,α}(Φ_{1}(χ) · Φ_{2,s}^{±}(ψχ_{ξ_{0}}χ^{−1})).  (6.1)

The proof is based on a regularization of the distributions ℒ_{f,α}(Φ_{s}^{+}(χ)) (for 1 ≤ s ≤ k − ν − m) and ℒ_{f,α}(Φ_{s}^{−}(χ)) (for 1 − k + ν + m ≤ s ≤ 0) (with values in ℚ̄).

Recall that by Proposition 5.2 the linear form ℒ_{f,α} factorizes through the canonical projection π_{α}:

ℒ_{f,α}(Φ_{s}^{±}(χ)) = ℒ_{f,α}(π_{α}(Φ_{s}^{±}(χ))).

Consider a positive integer c such that (c, N_{0}) = 1 and N_{0} = 4q_{0}NM_{0}^{m̃−1}. One supposes always that m is even and that k > 2m + 2. Hence there exist regularized distributions Φ_{s}^{c+}, Φ_{s}^{c−} on Y with values in the module

ℳ_{r,k}^{m}(ℚ̄) = ⋃_{v≥1}ℳ_{r,k}^{m}(Γ_{1}^{m}(Np^{v}); ℚ̄) :

of arithmetical nearly holomorphic Siegel modular forms. These regularized distributions are uniquely determined by their values as follows:

\begin{matrix}Φ_{s}^{c+}(χ) = & C_{χ̄ψ}^{s}G(χ̄ψ)^{−1} (1 − (χ̄ψ)^{2}(c)c^{−2s}) Φ_{s}^{+}(χ)· \\ · ∏_{q|N_{0}} [(1 − (χψ̄)_{0}(q)q^{s−1})(1 − (χ̄ψ)_{0}(q)q^{−s})^{−1}] , \\ Φ_{s}^{c−}(χ) = & (1 − (χ̄ψ̄)^{2}(c)c^{2(s−1)}) Φ_{s}^{−}(χ̄). \end{matrix}  (6.2)

Next consider the ℂ_{p}-linear forms D^{c±} : 𝒞^{k−m}(Y; ℂ_{p}) → ℂ_{p} defined on the local monomials x_{p}^{j} for j = 0, 1, ⋯, k − m − 1 by:

\begin{matrix}∫_{a+(M)} x_{p}^{j}D^{c+} = ∫_{a+(M)} ℒ_{f,α}(Φ_{j+1}^{c+}) = ∫_{a+(M)} ℒ_{f,α}(π_{α}(Φ_{j+1}^{c+})), \\ ∫_{a+(M)} x_{p}^{j}D^{c−} = ∫_{a+(M)} ℒ_{f,α}(Φ_{−j}^{c−}) = ∫_{a+(M)} ℒ_{f,α}(π_{α}(Φ_{−j}^{c−})). \end{matrix}  (6.3)

It follows that there exist the ℂ_{p}-linear maps Φ̃_{α}^{c±} : 𝒞^{k−m}(Y; ℂ_{p}) → ℳ_{r,k}^{m}(ℂ_{p}) defined on the local monomials x_{p}^{j} for j = 0, 1, ⋯, k − m − 1 by:

\begin{matrix}∫_{a+(M)} x_{p}^{j}Φ̃_{α}^{c±} = ∫_{a+(M)} π_{α}(Φ_{j+1}^{c+})), \\ ∫_{a+(M)} x_{p}^{j}Φ̃_{α}^{c±} = ∫_{a+(M)} π_{α}(Φ_{−j}^{c−}). \end{matrix}  (6.4)

In order to prove that ℂ_{p}-linear forms D^{c±} : 𝒞^{k−m}(Y; ℂ_{p}) → ℂ_{p} define in fact certain admissible measures we observe that

D^{c±} = ℒ_{f,α}(Φ̃_{α}^{c±})

where the linear forms Φ̃_{α}^{c±} depend on the choice of α (but not on the choice of f), and in fact they take values in a finite dimensional ℂ_{p}-vector space (the image of the finite dimensional projection:

π_{α} : ℳ_{r,k}^{m}(N, p; ℂ) → ℳ_{r,k}^{m}^{α}(N, p; ℂ) ⊂ ℳ_{r,k}^{m}(Γ_{1}^{m}(Np); ℂ),

see Proposition 5.2 (d)) hence it suffuces to prove the admissibility of the linear maps Φ̃_{α}^{c±} with values in spaces of arithmetical nearly holomorphic Siegel modular forms. For this purpose we use a general result on admissible measures with values in nearly holomorphic Siegel modular forms given in Theorem 3.4.

We have for the positive critical values

\begin{matrix}∫_{a+(M)}(x − a)^{r}dD^{c+} = ∑_{j=0}^{r} \binom{r}{j}(−a)^{r−j} ∫_{a+(M)} dD_{j+1}^{c+} \\ = ∑_{j=0}^{r} \binom{r}{j}(−a)^{r−j} \frac{1}{φ(M)} ∑_{χmodM} χ^{−1}(a)D_{j+1}^{c+}(χ) \\ = ∑_{j=0}^{r} \binom{r}{j}(−a)^{r−j} \frac{1}{φ(M)} ∑_{χmodM} χ^{−1}(a)ℒ_{f,α}(Φ_{j+1}^{c+}(χ)). \end{matrix}  (6.5)

similarly for D^{c−} we have

\begin{matrix}∫_{a+(M)}(x − a)^{r}dD^{c−} = ∑_{j=0}^{r} \binom{r}{j}(−a)^{r−j} ∫_{a+(M)} dD_{−j}^{c−} \\ = ∑_{j=0}^{r} \binom{r}{j}(−a)^{r−j} \frac{1}{φ(M)} ∑_{χmodM} χ^{−1}(a)D_{−j}^{c−}(χ) \\ = ∑_{j=0}^{r} \binom{r}{j}(−a)^{r−j} \frac{1}{φ(M)} ∑_{χmodM} χ^{−1}(a)ℒ_{f,α}(Φ_{−j}^{c−}(χ)). \end{matrix}  (6.6)

6.2. Sufficient conditions for admissibility of measures with values in nearly holomorphic Siegel modular forms. In order to prove the admissibility property D^{c±} of we need to verify the following growth condition for D^{c}:

sup_{a∈Y} |∫_{a+(m)}(x − a_{p})^{r}dD^{c±}|_{p} = o (|M|_{p}^{r−2ord_{p}α_{0}(p)}) ,  (6.7)

for r = 0, 1, …, k − m − 1, which will be implied by

sup_{a∈Y} |∫_{a+(m)}(x − a_{p})^{r}dΦ̃_{α}^{c±}|_{p} = o (|M|_{p}^{r−2ord_{p}α_{0}(p)}) ,  (6.8)

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 3.4): with κ = 2, h = [2ord_{p}α_{0}(p)] + 1. Then we need to prove that

|U^{2v}(∑_{j′=0}^{j} \binom{j}{j′}(−a_{p}^{0})^{j−j′}Φ_{1+j′}^{c+}((a)_{v})))|_{p} ≤ Cp^{−vj}  (6.9)

and

|U^{2v}(∑_{j′=0}^{j} \binom{j}{j′}(−a_{p}^{0})^{j−j′}Φ_{−j′}^{c−}((a)_{v})))|_{p} ≤ Cp^{−vj}  (6.10)

for all j = 0, 1, …, k − m − 1. 6.3. Fourier expansions of distributions with values in nearly holomorphic Siegel modular forms. Let us use that fact that

\begin{matrix}δ_{m/2+s}^{(r^{+})}G_{m/2+s}^{+}(z, 0, χ̄ψχ_{h_{0}}) = G_{k′}^{+}(z, s′, χ̄ψχ_{h_{0}}), \\ δ_{m/2+1−s}^{(r^{−})}G_{m/2+1−s}^{−}(z, s − 1/2, χ̄ψχ_{h_{0}}) = G_{k′}^{−}(z, s′, χ̄ψχ_{h_{0}}), \end{matrix}  (6.11)

where k′ = k − m/2 − ν, s′ = (s − k + m + ν)/2, r^{+} = r^{−} = (k − m − ν − j − 1)/2 ∈ ℕ and if j + 1 ≡ δ (mod 2) (otherwise the series vanishes).

The Fourier expansions of these series are given by

\begin{matrix}G_{m/2+s}^{+}(z, 0, χ̄ψχ_{ξ_{0}}) = ∑_{ξ∈C_{m}} b^{+}(ξ)e_{m}(ξz), \\ G_{m/2+1−s}^{−}(z, s − 1/2, χ̄ψχ_{ξ_{0}}) = ∑_{ξ∈C_{m}} b^{−}(ξ)e_{m}(ξz), \end{matrix}  (6.12)

where

\begin{matrix}b^{+}(ξ) = 2^{−mκ}det(ξ)^{m/2+s−κ}L_{N_{1}}^{+}(s, ωχ_{ξ})M(ξ, ω, m/2 + s), \\ b^{−}(ξ) = 2^{−mκ}L_{N_{1}}^{−}(s, ωχ_{ξ})M(ξ, ω, m/2 + s), \end{matrix}  (6.13)

ω = ψχ_{ξ_{0}} χ̄ is a Dirichlet character,

M(ξ, ω, m/2 + s) = ∏_{q∈P (ξ)} M_{q}(ξ, ω(q)q^{−m/2−s}),

is a finite Euler product such that M_{q}(ξ, t) ∈ ℤ[t]. The normalized Dirichlet L-functions are given by

\begin{matrix}L_{N_{1}}^{+}(s, ωχ_{ξ}) = i^{δ}π^{(1−2s)/2} \frac{Γ(s + δ)/2)}{Γ((1 − s + δ)/2}L_{N_{1}}(s, ωχ_{ξ}), \\ L_{N_{1}}^{−}(s, ωχ_{ξ}) = L_{N_{1}}(s, ωχ_{ξ}). \end{matrix}  (6.14)

Let us use these Fourier expansions in order to compute the following nearly holmorphic Fourier expansion

\begin{matrix}Φ_{s}^{±}(χ) = Φ_{1}(χ)Φ_{2,s}^{±}(ψχ_{ξ_{0}} χ̄) \\ Φ_{1}(χ)δ^{(r^{±})}Ψ_{2,s}^{±}(ω) = ∑_{ξ∈C_{m}} v^{±}(R, ξ, s, χ)e_{m}(ξz), \end{matrix}

\begin{aligned}Φ_{s}^{±}(χ)|U(M′) & = ∑_{ξ∈C_{m}} v^{±}(M′R, M′ξ, s, χ)e_{m}(ξz) \\ & = ∑_{ξ∈C_{m}} ∑_{Nξ̂_{0}[ξ_{1}]+ξ_{2}=M′ξ} d^{±}(s, M′R, ξ_{1}, ξ_{2})e_{m}(ξz) \end{aligned}  (6.15)

where M′ = p^{2v},

d^{+}(s, M′R, ξ_{1}, ξ_{2}) = (q_{0}N)^{−m(s−1+k−m−ν)/2}det(ξ_{0})^{(s−1+k−m−ν)/2}·
· χ(det(ξ_{1}))det(ξ_{1})^{ν}det(ξ_{2})^{s−1/2}L_{N_{1}}^{+}(s, ωχ_{ξ_{2}})·
· M(ξ_{2}, ω, s + m/2)Q(M′R, ξ_{2}; r^{+}, β^{+}),

(6.16)

under the condition s ≡ δ (mod 2) (otherwise d^{+}(s, M′Rξ_{1}, ξ_{2}) = 0). Recall the notation ω = χ̄ψχ_{ξ_{0}}, β^{+} = (1 − s − k + ν + m)/2 and r^{+} = (k − ν − m − s)/2. The coefficients d^{−}(s, M′R, ξ_{1}, ξ_{2}) are given by

d^{−}(s, M′R, ξ_{1}, ξ_{2}) = (q_{0}N)^{−m(s−1+k−m−ν)/2}det(ξ_{0})^{(s−1+k−m−ν)/2}·
· χ_{M}(det(ξ_{1}))det(ξ_{1})^{ν}L_{N_{1}}^{−}(s, ωχ_{ξ_{2}})·
· M(ξ_{2}, ω, s + m/2)Q(M′R, ξ_{2}; r^{−}, β^{−}),

(6.17)

under the condition s ≢ δ (mod 2) (otherwise d^{−}(s, ξ_{1}, ξ_{2}) = 0. Recall also that β^{−} = (s − k + ν + m)/2 and that r^{−} = (k − ν − m + s − 1)/2.

6.4. Fourier expansions of regularized distributions. Let us apply the above formulas for the normalized distributions:

\begin{aligned}Φ_{s}^{c±}(χ)|U(M′) & = ∑_{ξ∈C_{m}} v^{c±}(M′R, M′ξ, s, χ)e_{m}(ξz) \\ & = ∑_{ξ∈C_{m}} ∑_{Nξ̂_{0}[ξ_{1}]+ξ_{2}=M′ξ} d^{c±}(s, M′R, ξ_{1}, ξ_{2})e_{m}(ξz) \end{aligned}  (6.18),

where M′ = p^{2v},

d^{c+}(s, M′R, ξ_{1}, ξ_{2}) = (q_{0}N)^{−m(s−1+k−m−ν)/2}det(ξ_{0})^{(s−1+k−m−ν)/2}·

· χ(det(ξ_{1}))det(ξ_{1})^{ν}det(ξ_{2})^{s−1/2}Q(M′R, ξ_{2}; r^{+}, β^{+})· · M(ξ_{2}, χ̄ψχ_{ξ_{0}}, s + m/2)L_{N_{1}}^{+}(s, χ̄ψχ_{ξ_{0}}χ_{ξ_{2}})·

· \frac{C_{χ̄ψ}^{s}}{G(χ̄ψ)}(1 − (χ̄ψ)^{2}(c)c^{−2s}) ( ∏_{q|N_{0}} \frac{(1 − (χψ̄)_{0}(q)q^{s−1})}{(1 − (χ̄ψ)_{0}(q)q^{−s})}) .

(6.19)

The coefficients d^{c−}(s, M′R, ξ_{1}, ξ_{2}) are given by

d^{c−}(s, M′R, ξ_{1}, ξ_{2}) = (q_{0}N)^{−m(s−1+k−m−ν)/2}det(ξ_{0})^{(s−1+k−m−ν)/2}·
· χ_{M}(det(ξ_{1}))det(ξ_{1})^{ν}(1 − (χ̄ψ)^{2}(c)c^{−2s})L_{N_{1}}^{−}(s, ωχ_{ξ_{2}})·
· M(ξ_{2}, ω, s + m/2)Q(M′R, ξ_{2}; r^{−}, β^{−})

(6.19)

under the condition s ≡ δ (mod 2) (otherwise d^{+}(s, M′Rξ_{1}, ξ_{2}) = 0). Recall the notation ω = χ̄ψχ_{ξ_{0}}, β^{+} = (1−s−k +ν +m)/2 and r^{+} = (k −ν −m−s)/2, β^{−} = (s−k +ν +m)/2 and that r^{−} = (k − ν − m + s − 1)/2.

6.5. Main congruences for the Fourier expansions of regularized distributions. Let us use the orthogonality relations for Dirichlet characters in order to prove (6.9) and (6.10) using the Fourier expansions (6.18).

|∑_{j′=0}^{r} \binom{j}{j′}(−a_{p}^{0})^{j−j′} \frac{1}{φ(M)} ∑_{χmodM} χ^{−1}(a)v^{c+}(M′R, M′ξ, 1 + j′, χ)|_{p} ≤ Cp^{−vj} (6.21)

and

|∑_{j′=0}^{r} \binom{j}{j′}(−a_{p}^{0})^{j−j′} \frac{1}{φ(M)} ∑_{χmodM} χ^{−1}(a)v^{c−}(M′R, M′ξ, −j′, χ)|_{p} ≤ Cp^{−vj}  (6.22)

for all j = 0, 1, …, k − m − 1, M = p^{v}, M′ = p^{2v}.

6.6. Kummer congruences and Mazur’s measure. Non–Archimedean integration and admissible measures. Let S be a finite set of primes containing p and all the prime divisors of N. Consider the ℂ_{p}–adic analytic Lie groups

X_{S} = Hom_{contin}(ℤ_{S}^{×}, ℂ_{p}^{×}), X = X_{N} = Hom_{contin}(Y, ℂ_{p}^{×})

where

ℤ_{S}^{×} = lim_{\overleftarrow{Q}}(ℤ/QZ)^{×}

The projection homomorphism ℤ_{S}^{×} → Y induces the analytic embedding X ↪ X_{S}. Recall the notion of the h–admissible measures on ℤ_{S}^{×} and properties of their Mellin transform. These Mellin transform are certain p–adic analytic functions on the ℂ_{p}–analytic Lie group X_{S} . The canonical ℂ_{p}–analytic structure on X_{S} is obtained by shifts from the obvious ℂ_{p}–analytic structure on the subgroup

Homcontin(Z_{p}^{×}, ℂ_{p}^{×}) ⊂ X_{S}.

We view the elements of finite order χ ∈ X tors_{S} as Dirichlet characters whose conductor C(χ) contain only primes in S, by means of the decomposition

χ : ℤ_{S}^{×} → \overline{ℚ}^{×} ⟶^{i_{∞}} ℂ^{×},

where i_{∞} is the fixed embedding. The characters χ ∈ X tors_{S} form a discrete subgroup X tors_{S} ⊂ X_{S}. We use also the following natural homomorphism

x_{p} : ℤ_{S}^{×} → ℤ_{p}^{×} → ℂ_{p}^{×}, x_{p} ∈ X_{S}.

so that all integers k ∈ ℤ may be viewed as characters of the type x_{p}^{k} : y ↦ y^{k}.

Recall that a p–adic measure on ℤ_{S}^{×} may be viewed as a bounded ℂ_{p}–linear form μ on the space C(ℤ_{S}^{×}) of all continuous ℂ_{p}–valued functions

φ → μ(φ) = ∫_{ℤ_{S}^{×}} dμ ∈ ℂ_{p}, φ ∈ C(ℤ_{S}^{×}),

which is uniquely determined by its restriction to the subspace C^{1}(ℤ_{S}^{×}) of locally constant functions. We denote by μ(a + (Q)) the value of μ on the characteristic function of the set

a + (Q) = {x ∈ ℤ_{S}^{×} | x ≡ a mod Q} ⊂ ℤ_{S}^{×}.

The Mellin transform L_{μ} of μ is a bounded analytic function

L_{μ} : X_{S} → ℂ_{p}, L_{μ}(χ) = ∫_{ℤ_{S}^{×}} dμ ∈ ℂ_{p}, χ ∈ X_{S},

on X_{S}, which is uniquely determined by its values L_{μ}(χ) for the characters χ ∈ X tors_{S}.

A more delicate notion of an h–admissible measure was introduced by Amice–Velu and Višik (see [Am-V], [V1]). Let C^{h}(ℤ_{S}^{×}) denote the space of ℂ_{p}–valued functions which can be locally represented by polynomials of degree less than a natural number h of the variable x_{p} ∈ X_{S} introduced above.

Definition. A ℂ_{p}–linear form

μ : C^{h}(ℤ_{S}^{×}) → ℂ_{p}

is called h–admissible measure if for all a ∈ ℤ_{S}^{×} and for all r = 0, 1, …, h − 1 the following growth condition is satisfied

| sup_{a∈ℤ_{S}^{×}} ∫_{a+(Q)}(x_{p} − a_{p})^{r}dμ |= o(| Q |_{p}^{r−h})

We know (essentially due to Amice–Velu and Višik) that each h–admissible measure can be uniquely extended to a linear form on the ℂ_{p}–space of all locally analytic functions so that one can associate to its Mellin transform

L_{μ} : X_{S} → ℂ_{p}, L_{μ}(χ) = ∫_{ℤ_{S}^{×}} χdμ ∈ ℂ_{p}, χ ∈ X_{S},

which is a ℂ_{p}–analytic function on X_{S} of the type o(log x_{p}^{h}). Moreover, the measure μ is uniquely determined by the special values of the type

L_{μ}(χx_{p}^{r}) (χ ∈ X_{S}^{tors}, r = 0, 1, …, h − 1).

Let ω mod A be a fixed primitive Dirichlet character such that (A, M_{0}) = 1 with M_{0} = ∏_{q∈S} q. Put \overline{S} = S ∪ S(A), \overline{M} = ∏_{q∈\overline{S}} q. Then for any positive integer c with (c, \overline{M}) = 1, c > 1 there exist ℂ_{p}-measures μ^{+}(c, ω), μ^{−}(c, ω) on ℤ_{S}^{×} which are uniquely determined by the following conditions: for s ∈ ℤ, s > 0

\begin{matrix}i_{p}^{−1} (∫_{ℤ_{S}^{×}} χ x_{p}^{s} dμ^{+}(c, ω)) = (1 − \overline{χ}ω(c)c^{−s})\frac{C_{ω\overline{χ}}}{G(ω\overline{χ})}× \\ × ∏_{q∈S\S(χ)} \{(1 − χ\overline{ω}(q)q^{s−1})/(1 − \overline{χ}ω(q)q^{−s})\}L_{M_{0}}^{+}(s, \overline{χ}ω), \end{matrix}  (6.23)

and for s ∈ ℤ, s ≤ 0

i_{p}^{−1} (∫_{ℤ_{S}^{×}} χ x_{p}^{s} dμ^{−}(c, ω)) = (1 − χ\overline{ω}(c)c^{s−1})L_{M_{0}}^{−}(s, \overline{χ}, ω),  (6.24)

where

L_{M_{0}}^{+}(s, \overline{χ}ω) = L\overline{M}(s, \overline{χ}ω)\frac{2i^{δ}Γ(s) cos(π(s − δ)/2)}{(2π)^{s}},  (6.25)
L_{M_{0}}^{+}(s, \overline{χ}ω) = L\overline{M}(s, \overline{χ}ω)  (6.26)

are the normalized Dirichlet L-functions with δ = 0, 1, (−1)^{δ} = \overline{χ}ω(−1). The functions (6.23) and (6.24) satisfy the following functional equation

L_{M_{0}}(1 − s, χ\overline{ω}) = ∏_{q∈S\S(χ)} \{(1 − χ\overline{ω}(q)q^{s−1})/(1 − \overline{χ}ω(q)q^{−s})\}L_{M_{0}}^{+}(s, \overline{χ}ω).  (6.27)

Recall that by the definition of the \overline{S}-adic Mazur measure μ^{c} on ℤ× ; \overline{S} (6.23) and (6.24) are given by

∫_{ℤ_{S}^{×}} x dμ^{−}(c, ω) = ∫_{ℤ× ; \overline{S}} x^{−1}ω dμ^{c},

∫_{ℤ_{S}^{×}} dμ^{+}(c, ω) = ∫_{ℤ× ; \overline{S}} x x_{p}^{−1}ω^{−1} dμ^{c},

where x ∈ X_{S}, and X\overline{S} is regarded as a subgroup of X\overline{S}.

6.7. Reduction of the Main congruence to congruences for partial sums. Recall from (2.4) that

δ_{k}^{(r)}f(z) = ∑_{ξ∈B_{m}} c(ξ)Q(R, ξ)q^{ξ} ∈ ℚ(f)[[q^{B_{m}}]][R_{i,j}]_{i,j=1,···,m},

where ℚ(f) is the subfield of ℂ generated by the Fourier coefficients of f, L runs over all the multi-indices 0 ≤ l_{1} ≤ ⋯ ≤ l_{t} ≤ m as above,

\begin{matrix}Q(R, ξ) = Q(R, ξ; k, r) = ∑_{t=0}^{r} \binom{r}{t}det(ξ)^{r−t} ∑_{|L|≤mt−t} R_{L}(κ − k − r)Q_{L}(R, ξ), \\ Q_{L}(R, ξ) = tr (^{t}ρ_{m−l_{1}}(R)ρ_{l_{1}}^{⋆}(ξ)) ⋯ tr (^{t}ρ_{m−l_{t}}(R)ρ_{l_{t}}^{⋆}(ξ)) . \end{matrix}

Hence combining all the above remarks we deduce that the coefficients

i_{p}(v^{c±}(M′R, M′ξ, s, χ_{M}))

are linear combinations with coefficients independent of M, M′and s of the following expressions:

B^{+} = χ_{M}(det(ξ_{1}))det(ξ_{1})^{ν}det(ξ_{2})^{s−1/2}det(ξ_{2})^{r^{+}} ∑_{t=0}^{r^{+}} \binom{r^{+}}{t} ∑_{|L|≥t} c_{L}(β^{+})·

· Q_{L}(M′R, ξ_{2}) ((q_{0}N)^{−m}det(ξ_{0}))^{r^{+}+s−1/2} \frac{χ̄ψ(τ)}{τ^{s−1/2}} ∫_{ℤ_{S}^{×}} χx_{p}^{s}dμ^{c+}(ω)

(6.28)

B^{−} = χ̄_{M}(det(ξ_{1}))det(ξ_{1})^{ν}det(ξ_{2})^{r^{−}} ∑_{t=0}^{r^{−}} \binom{r^{−}}{t} ∑_{|L|≥t} c_{L}(β^{−})·

· Q_{L}(M′R, ξ_{2}) ((q_{0}N)^{−m}det(ξ_{0}))^{r^{−}} ∫_{ℤ_{S}^{×}} χ̄x_{p}^{s}dμ^{c−}(ω),

(6.29)

with ω = ψχ_{ξ_{0}}χ_{ξ_{2}} and det(2ξ_{2})det(2ξ_{0}) = τa^{2} for a, τ ∈ ℤ such that τ is square free. Hence C_{χ̄ω} = τC_{χ̄ψ} and G(χ̄ω) = χ̄ψ(τ)\sqrt{τ}G(χ̄ψ) and using Nξ̂_{0}[ξ_{1}] + ξ_{2} = M′ξ we show the following congruence det(ξ_{2})det(ξ_{0}) ≡ (q_{0}N)^{m}det(ξ_{1})^{2} (mod M′). Moreover m is even, and one deduces then that det(ξ_{2})det(ξ_{0}) is a square modulo q for all q|N, then we simply obtain (6.28), (6.29) from (6.23) and (6.24). Concider M′ and M large enough such that M^{2}|M′ and q_{0}^{m}|M and recall that N ∈ ℤ_{S}^{×} hence on can find

det(ξ_{2})(q_{0}N)^{−m}det(ξ_{0}) ≡ det(ξ_{1})^{2} (mod M).

Thus the coefficients χ_{M}(det(ξ_{1})) do not vanish only if det(ξ_{1}) ∈ ℤ_{S}^{×}. We see then that det(ξ_{2})(q_{0}N)^{−m}det(ξ_{0}) is a square in ℤ_{S}^{×} and we put

λ = λ^{+} = τ^{−1}\sqrt{det(ξ_{2})(q_{0}N)^{−m}det(ξ_{0})} ∈ ℤ_{S}^{×} in the case +,
λ = λ^{−} = \sqrt{det(ξ_{2})(q_{0}N)^{−m}det(ξ_{0})} ∈ ℤ_{S}^{×} in the case −.

It follows that the expressions B^{+} and B^{−} become

B^{+} = ψ(τ)τ^{k−m−ν−1/2}det(ξ_{1})^{ν}χ(λ)λ^{s−1+k−m−ν} ∑_{t=0}^{[(k−m−1)/2]} \binom{r^{+}}{t}·

· ∑_{|L|≥t} c_{L}(β^{+})Q_{L}(M′R, ξ_{2}) ∫_{ℤ_{S}^{×}} χx_{p}^{s}dμ^{c+}(ω),

B^{−} = det(ξ_{1})^{ν} χ̄(λ)λ^{s−1+k−m−ν} ∑_{t=0}^{[(k−m−1)/2]} \binom{r^{−}}{t} ∑_{|L|≥t} c_{L}(β^{−})·

· Q_{L}(M′R, ξ_{2}) ∫_{ℤ_{S}^{×}} χ̄x_{p}^{s}dμ^{c−}(ω).

(6.30)

In this way A^{±} is a linear combination with coefficients independent on M, M′ and j of type

\begin{matrix}\underline{A}^{+} = & det(ξ_{1})^{ν}λ^{k−m−ν−1−l} ∑_{t=0}^{[(k−m−1)/2]} ∑_{|L|≥t} Q_{L}(M′R, ξ_{2}) ∑_{j=0}^{l} \binom{r_{j}^{±}}{t}· \\ · c_{L}(β_{j}^{±})lj(−aλ^{−1})^{l−j} ∫_{ℤ_{S}^{×}} \frac{1}{φ(M)} ∑_{χmodM} χ(xa^{−1})x^{j+1}dμ^{c±}(ω). \end{matrix}  (6.31)

Note that c_{L}(β_{j}^{+}) is a polynomial of degree |L| in β_{j}^{+} hence of degree |L| in j because β_{j}^{+} = (−j − k + ν + m)/2 and \binom{r_{j}^{+}}{t} is a polynomial of degree t in r_{j}^{+} thus of degree tin j in view of r_{j}^{+} = (k − m − ν − j − 1)/2. One can therefore write

\binom{r_{j}^{+}}{t}c_{L}(β_{j}^{+}) = ∑_{n=0}^{t+|L|} μ_{n} \frac{(j + n + 1)!}{(j + 1)!}.

Here the coefficients μ_{n} are certain fixed rational numbers.

Using again the orthogonality relations we see that the sum on j denoted by C^{+} takes the form

C^{+} = ∫_{x≡aλ^{−1}modM} ∑_{n=0}^{t+|L|} μ_{n} \underbrace{∑_{j=0}^{l} \binom{l}{j}(−aλ^{−1})^{l−j} \frac{(j + n + 1)!}{(j + 1)!}x^{j+1}}_{x^{−n} \frac{∂^{n}}{∂x^{n}} (x^{n+1}(x − aλ^{−1})^{l})} dμ^{+}(c, .)

Therefore (x − aλ^{−1})^{l} ≡ 0 (mod M^{l}) giving

\begin{aligned}2^{ν_{m,k}}C^{+} & ≡ 0 (mod M^{l−n}) \\ & ≡ 0 (mod M^{l−t−|L|}). \end{aligned}  (6.32)

6.8. Proof of the Main congruence. Now the expression (6.31) transforms to

\underline{A}^{+} = det(ξ_{1})^{ν}λ^{k−m−ν−1+l} ∑_{t=0}^{[(k−m−1)/2]} ∑_{|L|≥t} Q_{L}(M′R, ξ_{2})C^{+},  (6.33)

where Q_{L}(M′R, ξ_{2}) is a homogeneuos polynomial of degree |L| in the variables M′R_{ij} implying the congruence

\begin{aligned}Q_{L}(M′R, ξ_{2}) & ≡ 0 (mod M^{′|L|}) \\ & ≡ 0 (mod M^{2|L|}). \end{aligned}  (6.34)

We deduce that

\underline{A}^{+} = o (|M|_{p}^{2|L|+l−t−|L|}) .  (6.35)

On the other hand we know from the description of the polynomial

\begin{matrix}Q(R, ξ) = Q(R, ξ; k, r) = ∑_{t=0}^{r} \binom{r}{t}det(ξ)^{r−t} ∑_{|L|≤mt−t} R_{L}(κ − k − r)Q_{L}(R, ξ)), \\ Q_{L}(R, ξ) = tr (^{t}ρ_{m−l_{1}}(R)ρ_{l_{1}}^{⋆}(ξ)) ⋯ tr (^{t}ρ_{m−l_{t}}(R)ρ_{l_{t}}^{⋆}(ξ))) \end{matrix}

that |L| ≥ t so we obtain finally the congruence

\underline{A}^{+} = o (|M|_{p}^{l}) .  (6.36)

This shows that D^{c+} is an h-admissible measure by the sufficient condition (6.9) (implying (6.7)).

In the case D^{c−} the proof is similar but one needs to replace χ by χ̄. Thus A^{−} is also a linear combination with coefficients independent of M, M′ and j of type

\begin{matrix}\underline{A}^{−} = det(ξ_{1})^{ν}λ^{k−m−ν−1} ∑_{t=0}^{[(k−m−1)/2]} ∑_{|L|≥t} Q_{L}(M′R, ξ_{2}) ∑_{j=0}^{l} \binom{r_{j}^{−}}{t}· \\ · c_{L}(β_{j}^{−})\binom{l}{j}(−aλ)^{l−j} ∫_{ℤ_{S(N_{0})}^{×}} \frac{1}{φ(M)} ∑_{χmodM} χ̄(aλx)x^{−j}dμ^{−}(c). \end{matrix}  (6.37)

Let us use again the orthogonality for the Dirichlet characters and the fact that \binom{r_{j}^{−}}{t}c_{L}(β_{j}^{−}) is a polynomial of degree t + |L| en j. One represents this polynomial in the form

\binom{r_{j}^{−}}{t}c_{L}(β_{j}^{−}) = ∑_{n=0}^{t+|L|} μ_{n} \frac{(j + 1 + n)!}{(j + 1)!}.  (6.38)

One obtains the following expression for the sum C^{−} over j in the formula giving \underline{A}^{−}:

\begin{matrix}C^{−} \\ = ∫_{x^{−1}≡aλmodM} ∑_{n=0}^{t+|L|} μ_{n} \underbrace{∑_{j=0}^{l} \binom{l}{j}(−aλ)^{l−j} \frac{(j + 1 + n)!}{(j + 1)!}x^{−j}}_{x_{1}^{n} \frac{∂^{n}}{∂x_{1}^{n}} (x_{1}^{n+1}(aλ − x_{1})^{l})} dμ^{−}(c, .), \end{matrix}  (6.39)

where x_{1} = x^{−1}. Then we use the same argument as for \underline{A}^{+} in (6.36) in order to prove that

\underline{A}^{−} = o (|M|_{p}^{l}) = o (|M|_{p}^{l−h}) .  (6.40)

It follows from (6.40) that the linear form D^{c−} is an admissible measure.

Theorem 6.9. Let f ∈ S_{m}^{k}(N, ψ) be a Siegel cusp eigenform of even degree m and of weight k > 2m+2 satisfying the nonvanishing condition (*) of Theorem 0.7. For a positive integer c > 1, (c, Np) = 1 there exist two ℂ_{p}-analytic functions L^{c±} : X_{S} → ℂ_{p} such that:

(i) For all pairs (s, χ) with s ∈ ℤ, 0 ≤ s ≤ k − m − 2 and a primitive non-trivial Dirichlet character χ ∈ X_{S}^{tors} of conductor C_{χ} = p^{v} one has

•L^{c+}(χx_{p}^{s}) = \frac{G_{C_{χ}}(1_{m}, χ)C_{χ}^{m(2s+2k−m)/4}}{(iC_{χ})^{m^{2}/2}α_{0}(C_{χ}^{2})} \frac{C_{χ̄ψ}^{s+1}}{G(χ̄ψ)}(1 − (χ̄ψ)^{2}(c)c^{−2s−2})·

· ∏_{q|N} \frac{(1 − (χψ̄)_{0}(q)q^{s})}{(1 − (χ̄ψ)_{0}(q)q^{−s−1})} \frac{D^{+}(s + 1, f, χ̄)}{⟨f, f⟩_{N}}.

•L^{c−}(χx_{p}^{s}) = \frac{G_{C_{χ}}(1_{m}, χ̄)C_{χ}^{m(2−2s+2k−m)/4}}{(iC_{χ})^{m^{2}/2}α_{0}(C_{χ}^{2})} (1 − (χ̄ψ̄)^{2}(c)c^{−2s−2})·

· \frac{D^{−}(−s, f, χ)}{⟨f, f⟩_{N}}.

These values are algebraic numbers ℚ̄ viewed as elements of the Tate field ℂ_{p} (via the fixed embedding i_{p} : ℚ̄ → ℂ_{p}). On the other hand the ℂ_{p}-analytic functions vanish at the following points

L^{c+}(χx_{p}^{s}) = L^{c−}(χx_{p}^{s}) = 0 if s ≡ 0 (mod 2).

(ii) If ord_{p}(α_{0}(p)) = 0 (i.e. f is p-ordinary), then the holomorphic functions L^{c+} et L^{c−} are bounded ℂ_{p}-analytic functions.

(iii) In the general case the functions L^{c+} and L^{c−} are of type o(log(x_{p}^{h})) where h = [2ord_{p}(α_{0}(p))] + 1 and they are Mellin transforms of certain h-admissible measuress.

(iv)If h ≤ k − m − 1, then the functions L^{c±} are uniquely determined by (i).

Proof of Theorem 6.9: The assertion (ii) (i.e. the case ord_{p}(α_{0}(p)) = 0) which was proved in [PaLNM], also follows easily from the Main congruence (6.9) and (6.10) taking into account the description of the canonical projection operator in Proposition 1.6: for α = α_{0}(p), Φ^{α} = π_{α}(Φ̃^{c±}) let us develop the definition of Φ̃^{α} using the binomial formula:

∫_{(a)_{v}}(y_{p} − a_{p})^{j} dΦ̃^{α} =

∑_{j′=0}^{j} \binom{j}{j′}(−a_{p})^{j−j′}Φ_{j′}^{α}((a)_{v}) = α^{−vκ}·

α^{vκ} · U^{−vκ} [π_{α,1}U^{κv}(∑_{j′=0}^{j} \binom{j}{j′}(−a_{p})^{j−j′}Φ_{j′}((a)_{v}))]

First we note that all the operators

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

are uniformely bounded for v → ∞ by a positive constant C_{1} (where U = αI + Z and Z^{n} = 0 for n = rk_{A}ℳ_{r^{∗}}(Np; A)) because n does not depend on v and the binomial coefficients \binom{−vκ}{t} are all ℤ_{p}-integers.

On the other hand by the Main congruences (6.9) and (6.10) with κ = 2 (for the distributions Φ_{j} = Φ_{j+1}^{c+} or Φ_{j} = Φ_{−j}^{c−})

|U^{κv}(∑_{j′=0}^{j} \binom{j}{j′}(−a_{p})^{j−j′}Φ_{j′}((a)_{v}))|_{p,ℳ_{r^{∗}}} ≤ Cp^{−vj}

for all j = 0, 1, …, k − m − 1. If we apply to this estimate the previous bounded operators π_{α,1}U^{κv} we get the following inequality

|∫_{(a)_{v}}(y_{p} − a_{p})^{j} dΦ̃^{α}|_{p,ℳ_{r^{∗}}} ≤ C · C_{1}|α^{−vκ}|_{p} · p^{−vj} = CC_{1}p^{−vj}

because |α^{−vκ}|_{p} = 1. In particular, for j = 0 it implies that Φ̃^{α} = π_{α}(Φ_{0}) is a bounded measure, and for all j = 0, 1, …, k − m − 1 we have moreover that

∫_{(a)_{v}} dπ_{α}(Φ_{j}) = ∫_{(a)_{v}} y_{p}^{j} dπ_{α}(Φ_{0}) = ∫_{(a)_{v}} y_{p}^{j} dΦ̃^{α}

Remark. We see in this case that all distributions π_{α}(Φ_{j}) are bounded but we do not claim that the distributions Φ_{j} are bounded themselves.

In the general case h > k−m−1 the integer j run over {0, ⋯, h−1} and one can extend the values of our functions L^{c±} by the equality L^{c±}(χx_{p}^{s}) = 0 (for all χ ∈ X_{S}^{tors} of a conductor divisible by all the prime divisors of Np) for j > k −m−1−ν (but keeping the same values for 0 ≤ s ≤ k − m − 1 − ν). These are the factors \binom{r^{±}}{t} which vanish in the above formulas (6.30) for the distributions B^{±} and the verification of the h-admissibility goes without change in this situation. Also, one obtains again the h-admissible measures D^{c±} with h = [2ord_{p}(α_{0}(p))]+1. The functions L^{c±} coincide therefore with the Mellin transforms of these h-admissible measures. Finally, if h ≤ k − m − 1 then the conditions in (i) uniquely determine the analytic functions L^{c±} of type o(log(x_{p}^{h})) by their values following a general property of admissible measures [Am-Ve], [Vi]. In the case h > k −m−1, there exist many analytic functions L^{c±} verifying the conditions in (i) which depend on a choice of analytic continuation (interpolation) for the values L^{c±}(χx_{p}^{j}) if j > k − m − 1 − ν but one shows in the theorem 6.9 that there exists at least one such continuation (for example the one which was described in the proof of (iii)).

References

[And74] Andrianov, A.N., Euler products attached to Siegel modular forms of degree 2, Uspekhi Mat. Nauk 29 (1974) 44–109 (in Russian).

[And79] Andrianov, A.N., Multiplicative Arithmetic of Siegel Modular Forms, Uspekhi Mat. Nauk 34 (1979), 67-135.

[And87] Andrianov, A.N., Quadratic Forms and Hecke operators, Springer-Verlag, Berlin-Heidelberg-New-York (1987)

[AndPa] Andrianov A.N., Panchishkin A.A., Singular Frobenius operators on Siegel modular forms with characters and zeta functions, St.-Petersbourg Math.J., vol. 12, no. 2 (2001) 233-257

[Bö85] Böcherer, S., Über die Funktionalgleichung automorpher L-Funktionen zur Sie-

gelschen Modulgruppe. J. Reine Angew. Math. 362 (1985), 146-168.

[Bö-Sch] Böcherer S., Schmidt C.-G., p-adic measures attached to Siegel modular forms

Ann. Inst. Fourier 50 (2000), no. 5, 1375–1443.

[Ch-Eh] Choie, Young Ju, Eholzer, W., Rankin-Cohen operators for Jacobi and Siegel

forms. J. Number Theory 68 (1998), no. 2, 160–177.

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

[Co-PeRi] Coates J., Perrin-Riou, B., On p-adic L-functions attached to motives over ℚ,

Advanced Studies in Pure Math. 17 (1989), 23–54.

[Colm98] Colmez, P., Fonctions L p-adiques, Séminaire Bourbaki, 51 ème année, 1998–99, n^{∘} 851. Novembre 1998.

[Co-Ma] Coleman, R., Mazur, B., The eigencurve. Galois representations in arithmetic algebraic geometry (Durham, 1996), 1–113, London Math. Soc. Lecture Note Ser., 254, Cambridge Univ. Press, Cambridge, 1998.

[Cour] Courtieu M., Familles d’opérateurs sur les formes modulaires de Siegel et fonctions L p-adiques http://www-fourier.ujf-grenoble.fr/THESE/ps/t101.ps.gz, Thèse de Doctorat, Institut Fourier, 2000, 1–122.

[Eh-Ib] Eholzer, W., Ibukiyama, T, Rankin-Cohen type differential operators for Siegel

modular forms. Internat. J. Math , 9 (1998) no. 4, 443–463.

[Fe] Feit, P., Poles and residues of Eisenstein series for symplectic and unitary groups. Memoir AMS 61 (1986) N 346, iv.+89p.

[Ge-PSh] Gelbart S., Piatetski-Shapiro, I.I., Rallis, S., Explicit constructions of automor-

phic L-functions, Springer-Verlag, Lect. Notes in Math. N 1254 (1987) 152p.

[Ha1] Harris, M., The rationality of holomorphic Eisenstein series, Invent. Math. 63 (1981), no. 2, 305–310.

[Ha2] Harris, M., Special values of zeta functions attached to Siegel modular forms., Ann. Sci. Ecole. Norm. Sup. 4 (1981), no. 1, 77–120.

[Hi85] Hida, H., A p-adic measure attached to the zeta functions associated with two elliptic modular forms,I, Invent. Math. 79 (1985), no. 1, 159–1.

[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.

[Hi91] Hida H., On p-adic L-functions of GL(2) × GL(2) over totally real fields, Ann. Inst. Fourier, (Grenoble) 40, no.2 (1991) 311–391.

[Hi94] Hida, H., p-adic ordinary Hecke algebras for GL(2), Ann. Inst. Fourier (Grenoble) 44 (1994), no. 5, 1289–1322.

[Iw] Iwasawa K., Lectures on p-adic L-functions, Ann. of Math. Studies, N 74. Princeton University Press, 1972.

[Ka78] Katz, N. M., p-adic L-functions for CM fields, Invent. Math. 49 (1978), no. 3, 199–297.

[Ku-Le] Kubota, T., Leopoldt, H.-W, Eine p-adische Theorie der Zetawerte, J. reine

angew. math. 214/215 (1964) 328–339.

[Ko-Za] Kohnen, W., Zagier, D., Modular forms with rational periods, In: Modul. Forms

Symp., Durham, 30 June–10 July 1983. Chichester (1984) 197–249.

[LBP] Le Bras, Y.-H. , Panchishkin A.A., Sur les produits triples Λ-adiques, Communications in Algebra, 29 (2001) No. 9, 3727-3740.

[Nag1] Nagaoka S., A remark on Serre’s example of p-adic Eisenstein series, Math. Z. 235 (2000) 227-250.

[Nag2] Nagaoka S., Note on mod p Siegel modular forms, Math. Z. 235 (2000) 405-420.

[Man] Manin, Yu. I., Non-Archimedean integration and Jacquet-Langlands p-adic Lfunctions, Russ. Math. Surveys 31, N 1 (1976) 5–57.

[PaLNM] Panchishkin, A. A., Non-Archimedean L-functions of Siegel and Hilbert modular forms, Lecture Notes in Math., 1471, Springer-Verlag, 1991, 166p.

[PaAdm] 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.

[PaIF] Panchishkin, A. A., Motives over totally real fields and p-adic L-functions, Annales Inst. Fourier (Grenoble), 44 (1994) 989–1023.

[PIAS] Panchishkin, A.A., On p-adic integration in spaces of modular forms and its applications, Preprint of IAS, January (2000) 1-31.

[PaSE] Panchishkin, A. A., On the Siegel-Eisenstein measure and its applications, Israel Journal of Mathemetics, 120, Part B (2000) 467-509.

[PaAD] Panchishkin, A. A., Arithmetical differential operators on nearly holomorphic Siegel modular forms, MPIM, April 2002, 1-49.

[PaNM] Panchishkin, A. A., A new method of constructing p-adic L-functions associated with modular forms, Prépublication de l’IF, no 553 (2001) (to appear in Moscow Matheletical Journal in 2002).

[PaTV] Panchishkin, A. A., Two variable p-adic L functions attached to eigenfamilies of positive slope , DFG - Forschergruppe Arithmetik , Preprintserie, Nr. 08 (2001) 1-63

[AnPa] Andrianov, A.N.,, Panchishkin, A. A., Singular Frobenius operators on Siegel modular forms with characters and zeta functions, St.-Petersbourg Math.J., vol. 12, no. 2 (2001), pp.233-257

[PSh-R] Piatetski-Shapiro, I. I., Rallis, S., L-functions of automorphic forms on simple classical groups, In: Modul. Forms Symp., Durham, 30 June–10 July 1983, Chichester, 1984, 251–261.

[Se73] Serre, J.-P., Formes modulaires et fonctions zêta p-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972) 191–268, Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973.

[Shi83] Shimura, G., On Eisenstein series, Duke Math. J. 50 (1983) 417–476.

[Shi95] Shimura, G., Eisenstein series and zeta functions on symplectic groups, Invent. Math. 119 (1995), no. 3, 539–584.

[Shi2000] Shimura, G., Arithmeticity in the theory of automorphic forms, Mathematical surveys and monographs, 82 AMS, Providence, RI 2000, 302 p.

[Sie] Siegel C.-L., Über die analytische Theorie der quadratische Formen, Ann. of Math. 36 (1935) 527-606.

[St1] Sturm J., The Critical Values of Zeta-Functions associated to the Symplectic Group, Duke Math. J. 48 (1981), p. 327-350.

[St2] Sturm J., Special Values of Zeta Functions and Eisenstein Series of half-integer Weight, Amer. J. Math. 102 (1981), p. 219-240.

[Til-U] Tilouine, J., Urban, E., Several variable p-adic families of Siegel-Hilbert cusp eigenforms and their Galois representations, Ann. scient. Éc. Norm. Sup. 4^{e} série, 32 (1999) 499–574.

[Vi] Višik, M.M., Non-Archimedean Measures associated with Dirichlet Series, Mat. Sbornik 99 (1976), p. 248-260.

[Wi90] Wiles, A., The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), no. 3, 493–540.

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

[We56] Weil, André, 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.