Arithmetical differential operators on 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

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

The purpose of this paper is to describe the action of certain arithmetical differential operators on algebraically defined nearly holomorphic Siegel modular forms (in the sense of Shimura, see [Shi2000]) over an 𝒪-algebra A where 𝒪 is the ring of integers in a finite extension K of ℚ_{p}. Often we simply assume that A = ℂ_{p}.

We view such modular forms as certain formal expansions. The important property of these arithmetical differential operators is their commutation with Hecke operators (under an appropriate normalization, see Theorem 2.8). Their action produces natural families of distributions on a profinite group Y = \underleftarrow{lim} Y_{i} with values in these A-modules of nearly holomorphic Siegel modular forms inside a formal q-expansion ring like A[[q^{B}]][R_{i,j}] where B is an additive semi-group, q^{B} = {q^{ξ}|ξ ∈ B} the corresponding formally written multiplicative semi-group (for example B = B_{m} = {ξ = ^{t}ξ ∈ M_{m}(ℚ)|ξ ≥ 0, ξ half-integral} is the semi-group, important for the theory of Siegel modular forms), and the nearly holomorphic parameters (R_{i,j}) = R correspond to the matrix R = (4πIm(z))^{−1} in the Siegel modular case.

We discuss some applications of this theory to construction of certain new p-adic families of modular forms (families of Siegel-Eisenstein series, families of theta-series wth spherical polynomials…). Main sources of this theory are:

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

1) Serre’s theory of p-adic modular forms as certain formal q-expansions (J.-P. Serre, Formes modulaires et fonctions zêta p-adiques, LNM 350 (1973) 191-268) [Se73].

2) Shimura’s theory of arithmeticity for nearly holomorphic forms (Shimura, G., Arithmeticity in the theory of automorphic forms. AMS, 2000, [Sh2000]).

3) Hida’s theory of p-adic modular forms and p-adic Hecke algebras (H. Hida, Elementary theory of L-functions and Eisenstein series, Cambridge University Press, 1993 [Hi93]).

4) Construction of p-adic Siegel-Eisenstein series by the author, appeared in Israel Journal of Mathematics in 2000, [PaSE]. Nerarly holomorphic Siegel modular forms provide additional arithmetical parameters.

Moreover they admit an algebraic interpretation in terms of sections of certain de Rham cohomology sheaves on Siegel modular varieties (see [Ha2]), and a typical example of the arithmetical differential operators is given by the Gauss- Manin connection. On the other hand, differential operators preserving rational structures naturally come from the action of universal envelopping algebras of corresponding Lie algebras on automorphic forms.

Arithmetical nearly holomorphic Siegel modular forms where studied by G.Shimura [Shi2000] and they 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) and satisfying certain reciprocity laws at these points.

The first description motivates our choice of arithmetical variables: q^{ξ}, R = (R_{i,j}) = (4πIm(z))^{−1} = 2πi(z̄ − z)^{−1}. We describe the action of δ_{k}^{(r)} in terms of these arithmetical variables.

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}

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} : ℳ̃_{k}^{m}(N, ψ) → ℳ̃_{k+2r}^{m}(N, ψ).

In order to describe explicitely the action of δ_{k} 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}] .

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):

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),

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).

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).

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.

We prove (Theorem 3.2) then that the nearly holomorphic Siegel modular form δ_{k}^{(r)}f(z) is given 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},

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),

and in the elliptic modular case m = 1, κ = 1, the only possibilities are t = 0, 1 when c_{0}(k) = 1, c_{1}(k) = k, and one obtains again the classical formula

δ_{k}(∑_{n≥0} a_{n}q^{n}) = ∑_{n≥1} na_{n}q^{n} − \frac{k}{4πy} ∑_{n≥0} a_{n}q^{n} = ∑_{n≥1} na_{n}q^{n} − kR ∑_{n≥0} a_{n}q^{n}

in which

R = \frac{1}{4πy} , ∑_{n≥1} na_{n}q^{n} = \frac{1}{2πi} \frac{∂}{∂z} (∑_{n≥0} a_{n}q^{n}) = q \frac{d}{dq} (∑_{n≥0} a_{n}q^{n}) .

Note that the operators δ_{k} act on nearly holomorphic forms and they are very different from operators studied in [Ch-Eh] and [Eh-Ib] although the polynomials which come up in their action on Fourier expansions have similar properties.

Applications to constructions of p-adic families of nearly holomophic Siegel modular forms are given in Sections 4-6 (Theorems 4.4, 5.4, and 6.2). In order to define p-adic families of nearly holomorphic Siegel modular forms one uses the notion of the p-adic integration with values in spaces of (formal) nearly holomorphic Siegel modular forms.

For any profinite group Y = \underleftarrow{lim} Y_{i} the completed group ring 𝒪[[Y]] = \underleftarrow{lim} 𝒪[Y_{i}] is a compact ring which generalizes the Iwasawa algebra Λ_{𝒪} , and it may be viewed as the 𝒪algebra of all 𝒪-valued distributions on Y (Section 4). The natural multiplication in 𝒪[[Y]] corresponds to the convolution of distributions. In Section 5 we construct p-adic families of nearly holomophic Siegel-Eisenstein series obtained by applying δ_{k}^{(r)} to a holomorphic Siegel-Eisenstein series. In Section 6 we construct p-adic families of coming from theta

series δ_{k_{m,P}}^{(r)} (θ_{A,P}) = δ_{k_{m,P}}^{(r)} ( ∑_{x∈ℤ^{mn}} P (x))q^{A[x]}) where A ∈ M_{m×m}(ℚ) is a positive definite

half-integral symmetric matrix which determines a quadratic form Q_{A}(x) = ^{t}xAx =^{def}A[x] for x ∈ ℤ_{p}^{mn} and P run over homogeneous A-spherical polynomials which take integral values on the lattice X = ℤ^{mn} (we view P as a ℤ_{p}-valued continuous function on ℤ_{p}^{mn}). Content of the paper: §1. Nearly holomorphic Siegel modular forms. §2. Algebraical differential operators of Maass and Shimura. §3. Arithmetical variables of nearly holomorphic Siegel modular forms and diffrerntial

operators. §4. Arithmetical families of nearly holomorphic Siegel modular forms. §5. Families of distributions attached to Siegel-Eisenstein series. §6. Families of distributions attached to theta functions with spherical polynomials.

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. Main results about the action of the arithmetical differential operators on Fourier expansions where obtained during the visit of the author at MPIM in 2002.

Let these institutions be thanked for their kind hospitality.

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}]). (over an 𝒪-algebra A).

§2. Algebraic differential operators of Maass and Shimura

Now let us 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} : ℳ̃_{k}^{m}(N, ψ) → ℳ̃_{k+2r}^{m}(N, ψ).

In order to describe explicitely the action of δ_{k} 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)

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

det(y)^{β} = exp (β log[det(y)]) ,

for det(y) > 0, y ∈ Y ⊗ ℂ. According to definition (2.2) 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)^{n}. 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}β(β + 1) ⋯ (β + t − 1)z^{r−t}.

M.Courtieu has established in his PhD thesis in 2000 the following more explicit expression for the function R_{m}(z; r, β) for arbitrary m (see [Cour], Chapter 3)

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},  (2.4)

In other words λ_{i}(z) is the sum of all diagonal minors of size i × i of the matrix z.

The polynomial R_{m}(z; r, β).

Here we want to describe explicitely the polynomial R_{m}(z; r, β) defined by (2.2). We show that this is a polynomial invariant for the action of conjugation on the space of symetric matrices y ∈ V = {ξ ∈ M_{m}(ℝ) | ^{t}ξ = ξ}.

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. We consider the differential operators ρ_{r}(˜∂/∂z) and ρ_{r}^{⋆}(˜∂/∂z) which associate to each ℂ-valued function on ℍ_{m} a certain M_{t}(ℂ) valued function on ℍ_{m} with t = \binom{m}{r}. In particular, the Maass differential operator is given by

Δ = ρ_{m}(˜∂/∂z) = ρ_{m}^{⋆}(˜∂/∂z) = det(2^{−1}(1 + δ_{ij})˜∂/∂z_{ij}).

We define the invariant polynomials λ_{r}(z) = tr(ρ_{r}(z)). They are the coefficients of the characteristic polynomial of z

det(t1_{m} − z) = ∑_{j=0}^{m}(−1)^{j}λ_{j}(z)t^{m−j}.

The following differentiation rules are valid (see [Sh83], lemma 9.1):

Δ(fg) = ∑_{r=0}^{m} tr (^{t}ρ_{r}(˜∂/∂z)f · ρ_{m−r}^{⋆}(˜∂/∂z)g),  (2.5)
ρ_{r}(˜∂/∂z)det(z)^{α} = c_{r}(α)det(z)^{α−1}ρ_{m−r}^{⋆}(z),  (2.6)
ρ_{r}^{⋆}(˜∂/∂z)det(z)^{α} = c_{r}(α)det(z)^{α−1}ρ_{m−r}(z),  (2.7)
ρ_{r}(˜∂/∂z)e^{tr(uz)} = ρ_{r}(u)e^{tr(uz)},  (2.8)
Δ (e^{tr(uz)}det(z)^{α}) = e^{tr(uz)}det(z)^{α−1} ∑_{t=0}^{m} c_{m−t}(α)λ_{t}(uz),  (2.9)

for α ∈ ℂ with

c_{0}(α) = 1, c_{r}(α) = ∏_{j=0}^{r−1}(α + j/2) = \frac{Γ_{r}(α + (r + 1)/2)}{Γ_{r}(α + (r − 1)/2)} (r ≥ 1).  (2.10)

One can extend the formula (2.5) to the case of the action of Δ on n functions f_{1}, ⋯, f_{n}:

Lemma 2.1. Let f_{1}, ⋯, f_{n} : ℍ_{m} → ℂ be n 𝒞^{∞}-functions, then we get

Δ(f_{1} ⋯ f_{n}) = ∑_{|T |=m} ∑_{K,L}(−1)^{ε(σ_{K,L})}ρ_{t_{1}}(˜∂/∂z)_{K_{t_{1}}}^{L_{t_{1}}} (f_{1}) ⋯ ρ_{t_{n}}(˜∂/∂z)_{K_{tn}}^{L_{tn}} (f_{n}),  (2.11)

where T runs over the sets of multiplets T = {0 ≤ t_{1}, ⋯, t_{n} ≤ m} which satisfy |T | = t_{1} + ⋯ + t_{n} = m. The second sum is extended over the sets of all multi-indices K and L consisting of strictely ordered subsets of multi-indices K_{t_{j}} and L_{t_{j}} such that card(K_{t_{j}}) = card(L_{t_{j}}) and ∪_{j=1}^{n}K_{t_{j}} = ∪_{j=1}^{n}L_{t_{j}} = \underline{m} = \{0, ⋯, m\} are two partitions of \underline{m}. The permutation σ_{K,L} of \underline{m} is determined by the following rule: for any h ∈ \underline{m}, take t_{j} such that h = h_{i} ∈ K_{t_{j}} where i the rank of h in K_{t_{j}} (with respect to the natural order), then σ_{K,L}(h_{i}) = h′_{i} where h′_{i} is the element of rank i of L_{t_{j}}, and denote by ε(σ_{K,L}) the sign of the permutation σ_{K,L}.

Proof of 2.1: One proves this lemma by induction on the number n of functions. • If n = 2, this is the formula (2.5)

• We assume that (2.12) is true for (n − 1). By hypothesis we have

Δ(f_{1}, ⋯, f_{n}) = ∑_{|T |=m} ∑_{K,L}(−1)^{ε(σ_{K,L})}ρ_{t_{1}}(∂_{z})_{K_{t_{1}}}^{L_{t_{1}}} (f_{1}) ⋯ ρ_{t_{n−1}}(˜∂_{z})_{K_{t_{n−1}}}^{L_{t_{n−1}}} (f_{n−1}f_{n}). (2.12)

But the operator ρ_{t_{n−1}}(˜∂_{z})_{K_{t_{n−1}}}^{L_{t_{n−1}}} coincides with the operator Δ_{t_{n−1}}(z_{1}) (with respect to the variable z_{1} = (z_{ij})_{i∈K_{t_{n−1}} ,j∈L_{t_{n−1}}} ). Thus formula (2.5) gives

\begin{aligned}ρ_{t_{n−1}}(˜∂_{z})_{K_{t_{n−1}}}^{L_{t_{n−1}}} & (f_{n−1}f_{n}) \\ & = ∑_{t′_{n−1}+t′_{n}=t_{n−1}}(−1)^{ε(σ′_{K,L})}ρ_{t′_{n−1}}(˜∂_{z})_{K_{t{′ \atop n−1}}}^{L_{t{′ \atop n−1}}} (f_{n−1}) ⋯ ρ_{t′_{n}}(˜∂_{z})_{K_{t′_{n}}}^{L_{t′_{n}}} (f_{n}), \end{aligned}

with card(K_{t′_{n−1}}) = card(L_{t′_{n−1}}) = t′_{n−1}, card(K_{t′_{n}}) = card(L_{t′_{n}}) = t′_{n} and one has K_{t′_{n−1}} ∪ K_{t′_{n}} = K_{t_{n−1}}, L_{t′_{n−1}} ∪ L_{t′_{n}} = L_{t_{n−1}} and (2.12) follows.

The following more delicate differentiation rule holds:

Lemma 2.2. Let u = ^{t}u be a real symmetric m × m-matrix, α a complex number, r, j two integers such that 0 ≤ r ≤ j ≤ m and z ∈ ℍ_{m} then one has the following equality:

ρ_{r}(˜∂) (λ_{j}(z)^{α}) = c_{r}(α)λ_{j}(z)^{α−1}μ_{r,j−r}^{⋆}(z).  (2.13)
Here μ_{r,j−r}^{⋆}(z) is a matrix of size \binom{m}{r} × \binom{m}{r} with the coefficients given by

μ_{r,j−r}^{⋆}(z)_{K}^{L} = ∑_{T ⊃K,L} ρ_{j−r}^{⋆}(z)\overline{T \L} ; \overline{T \K},

the sum is being taken over all the multi-indices T : 1 ≤ t_{1} < ⋯ < t_{j} ≤ m containing the multiindices K : 1 ≤ k_{1} < ⋯ < k_{r} ≤ m et L : 1 ≤ l_{1} < ⋯ < l_{r} ≤ m. The notation L̄ is used for the the complementary multi-index in \underline{m} = \{0, ⋯, m\} of the multi-index L. Recall that the matrix coefficient ρ_{m−r}^{⋆}(z)_{K}^{L} is equal (up to a sign) to the determinant of the complementary matrix in z for the determinant ρ_{r}(z)_{K}^{L} more precisely one has

ρ_{m−r}^{⋆}(z)_{K}^{L} = (−1)^{ε(σ_{K,L})}ρ_{m−r}(z)L̄ ; K̄,

where σ_{K,L} is the permutation of S_{m} given by the following rule: for every k_{i} ∈ K, σ_{K,L}(k_{i}) = l_{i} and for every k̄_{i} ∈ K̄, σ_{K,L}(k̄_{i}) = l̄_{i} with l̄_{i} ∈ L̄ and i denoting the rank of the index k_{i} (respectively l_{i}, k̄_{i}, l̄_{i}) with respect to the natural order in K (respectively L, K̄, L̄) (recall that ε(σ_{K,L}) denotes the signature of the permutation σ_{K,L}).

Proof of 2.2: is deduced from (2.8). Notice that λ_{j}(z) = ∑_{T} ρ_{j}(z)_{T}^{T} is the sum of all diagonal minors of size j in z. If one applies ρ_{r}(˜∂)_{K}^{L}, one gets non zero terms only for those diagonal minors ρ_{j}(z)_{T}^{T}, when the multi-index T contains the multi-indices K et L. The corresponding minor is then the determinant of a submatrix z, so that the computation is similar to that in (2.8) but for a matrix of a lower size. This gives us then

ρ_{r}(˜∂)_{K}^{L} (λ_{j}(z)^{α}) = c_{r}(α)λ_{j}(z)^{α−1} ∑_{T ⊃K,L} ρ_{j−r}^{⋆}(z)\overline{T \L} ; \overline{T \K},

and (2.13) follows. Action of the Shimura differential operator on formal Fourier expansions

We now apply the above results on R_{m}(z; r, β) in order to obtain an explicit dscripton of the action of the Shimura differential operator on formal Fourier expansions of nearly holomorphic Siegel modular forms. First of all we recall that the Shimura differential operator δ_{k} in the Siegel modular case is given as the twisted Maass operator Δ [Maa71]. Definition. For any f ∈ 𝒞^{∞}(ℍ_{m}, ℂ), define the operators

M_{k}f(z) = det(z − z̄)^{κ−k}Δ [det(z − z̄)^{k−κ+1}f(z)] ,  (2.14)
δ_{k}f(z) = (−\frac{1}{4π} )^{m} det(y)^{−1}M_{k}f(z).  (2.15)

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

We show now that the Fourier expansion of δ_{k}^{(r)}f, where f = ∑_{B_{m}} c(ξ)e_{m}(ξz) ∈ ℳ_{k}^{m}(N, ψ), is given by the formula

δ_{k}^{(r)}f = ∑_{ξ∈B_{m}} c(ξ)det(4πy)^{−r}R_{m}(4πξy; r, β)e_{m}(ξz).

In order to describe the action of δ_{k}^{(r)}f one establishes first the following

Lemma 2.3. Let α be a complex number, r a positive integer, u a real symmetric matrix and z ∈ ℍ_{m} then one has the following relation

det(z)^{−r}Δ (det(z)^{r+1}Δ^{r} (e^{tr(uz)}det(z)^{α})) = Δ^{r+1} (e^{tr(uz)}det(z)^{α+1}) .

Proof of 2.3: Let us use the formula (2.9) giving Δ (e^{tr(uz)}det(z)^{α}) in order to compare it with the action of the operator det(z)^{−r}Δ (det(z)^{r+1} · ) one has

det(z)^{−r}Δ(e^{tr(uz)}det(z)^{α+1}·

· ∑_{l_{1},···,l_{r}=0}^{m} c_{m−l_{1}}(α) ⋯ c_{m−l_{r}}(α − r + 1)P_{l_{1}···l_{r}}(u, z)) =

e^{tr(uz)}det(z)^{α−r}·

· ∑_{l_{1},···,l_{r+1}=0}^{m} c_{m−l_{1}}(α) ⋯ c_{m−l_{r}}(α − r + 1)c_{m−l_{r+1}}(α + 1)P_{l_{1}···l_{r}l_{r+1}}(u, z).

The polynomials P_{l_{1}···l_{r}} are defined recursively in such a way that

Δ^{r} (e^{tr(uz)}det(z)^{α}) =

e^{tr(uz)}det(z)^{α−r} ∑_{l_{1},···,l_{r}=0}^{m} c_{m−l_{1}}(α) ⋯ c_{m−l_{r}}(α − r + 1)P_{l_{1}···l_{r}}(u, z),

hence one has the following condition on P_{l_{1}···l_{r}}(u, z) :

P_{l_{1}···l_{r}}(u, z) = ∑_{i_{1}+i_{2}=m−l_{r}} ∑_{K,L}(−1)^{ε(σ_{K,L})}ρ_{i_{1}}(u)_{K_{i_{1}}}^{L_{i_{1}}} ρ_{m−l_{r}}^{⋆}(z)_{K_{lr}}^{L_{lr}} ρ_{i_{2}}(˜∂)_{K_{i_{2}}}^{L_{i_{2}}} (P_{l_{1}···l_{r−1}}(u, z)),

(a polynomial in matrix variables u and z independent of α and symmetric in l_{j}). The independence of P_{l_{1}···l_{r}}(u, z) on α results from the fact that the terms depending on α factorise out using (2.6). The symmetry in l_{j} results from the theorem of Schwarz on the independence of the mixed partial derivatives of 𝒞^{∞}-functions on the order of differentiaion. Here the choice of l_{1}, ⋯, l_{r} corresponds to another order of derivation applied to the factor det(z)^{⋆}. Replacing u by −1_{m} and α by −β, and using then the formula defining R(z; r, β), one finds that this is indeed a polynomial as the factors e^{tr(z)} and e^{−tr(z)} cancel, as well as the factors det(z)^{β} and det(z)^{−β}. One obtains then:

R(z; r, β) = ∑_{l_{1},···,l_{r}=0}^{m} c_{m−l_{1}}(−β) ⋯ c_{m−l_{r}}(−β − r + 1)P_{l_{1}···l_{r}}(−1_{m}, z).

In particular the polynomials P_{l_{1}···l_{r}} are symmetric in l_{j}, and one can replace the index of summation l′_{1} = l_{r+1}, l′_{2} = l_{1}, ⋯, l′_{r+1} = l_{r} so that one obtains the expression Δ^{r+1} (e^{tr(uz)}det(z)^{α+1}). This gives exactly the announced property.

In order to compute the Fourier expansion of δ_{k}^{(r)}f, one proceeds by induction on r. One computes the action of the Shimura differential operator on the test functions e_{m}(ξz), more precisely one shows that

δ_{k}^{(r)} (e_{m}(ξz)) = det(4πy)^{−r}R_{m}(4πξy; r, κ − k − r)e_{m}(ξz).

In fact if one takes f(z) = ∑_{ξ∈B_{m}} c(ξ)e_{m}(ξz), one obtains then by linearity the Fourier

expansion of δ_{k}^{(r)}f, resulting

δ_{k}^{(r)}f(z) = ∑_{ξ∈B_{m}} c(ξ)det(4πy)^{−r}R_{m}(4πξy; r, κ − k − r)e_{m}(ξz),

It remains to prove the following

Lemma 2.4. The action of the Shimura differential operator on the test functions e_{m}(ξz) (where ξ ∈ B_{m} and z ∈ ℍ_{m}) is given by the formula

δ_{k}^{(r)} (e_{m}(ξz)) = det(4πy)^{−r}R_{m}(4πξy; r, κ − k − r)e_{m}(ξz).  (2.16)

Proof of 2.4: One proves this lemma by induction on r :

• If r = 0, it is obvious.

• Suppose next that it holds for some r then if one applies the Shimura operator δ_{k+2r} one obtains

δ_{k}^{(r+1)}(e_{m}(ξz)) =(−1)^{m}det(4πy)^{−1}det(z − z̄)^{κ−k−2r}·
· Δ [det(z − z̄)^{k+2r+1−κ}det(4πy)^{−r}e_{m}(ξz) ·
· R_{m}(4πξy; r, κ − k − r)] .

Notice that 4πξy = −2iπξ(z − z̄). On the other hand z̄ is aniholomorphic hence it behaves as a constant under the action of Δ, and one can write, by putting u = −2iπξ, β = κ−k−r:

R_{m}(u(z − z̄); r, β) =
(−1)^{mr}e^{tr(u(z−z̄))}det(z − z̄)^{r+β}Δ^{r} (e^{−tr(u(z−z̄))}det(z − z̄)^{−β}) .

One replaces this expression in the first formula pulling out the factor e_{m}(ξz̄) = e^{tr(−uz̄)} which is an antiholomorphic function. One obtains then

δ_{k}^{(r+1)}(e_{m}(ξz)) = det(4πy)^{−(r+1)}e_{m}(ξz)(−1)^{m(r+1)}e^{tr(u(z−z̄))}det(z − z̄)^{β}·
·Δ [det(z − z̄)^{r+1}Δ^{r} (e^{−tr(u(z−z̄))}det(z − z̄)^{−β})].

Let us now apply lemma 2.3 which gives

δ_{k}^{(r+1)}(e_{m}(ξz)) =det(4πy)^{−(r+1)}e_{m}(ξz)(−1)^{m(r+1)}e^{tr(u(z−z̄))}·
· det(z − z̄)^{r+β}Δ^{r+1} (e^{−tr(u(z−z̄))}det(z − z̄)^{−β})
=det(4πy)^{−(r+1)}R_{m}(4πξy; r + 1, κ − k − (r + 1))e_{m}(ξz),

finishing the proof by induction.

Now we can write down the action of the Shimura operator on the Fourier expansions of Siegel modular forms in terms of the polynomial R_{m}(z; r, β) as follows:

Proposition 2.5. 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

δ_{k}^{(r)}f(z) = ∑_{ξ∈B_{m}} c(ξ)det(4πy)^{−r}R_{m}(4πξy; r, κ − k − r)e_{m}(ξz).

Let us now use the explicite expressions (2.3) of M.Courtieu for the function R_{m}(z; r, β) for arbitrary m in order to describe the action of δ_{k}^{(r)} on Fourier expansions.

Theorem 2.6. a) 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

\begin{aligned}δ_{k}^{(r)}f(z) = & ∑_{ξ∈B_{m}} c(ξ)det(4πy)^{−r} ∑_{t=0}^{r} \binom{r}{t}det(2πiξ(z̄ − z))^{r−t} \\ & ∑_{|L|≤mt−t} R_{L}(κ − k − r)λ_{l_{1}}(2πiξ(z̄ − z)) ⋯ λ_{l_{t}}(2πiξ(z̄ − z))e_{m}(ξz) \end{aligned} (2.17)

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] (recall that the polynomials λ_{i}(z) are defined by det(t1_{m} + z) = ∑_{i=0}^{m} λ_{i}(z)t^{m−i}.

b) In particular, if r = 1,

\begin{matrix}δ_{k}f(z) = & (−1)^{m} ∑_{ξ∈B_{m}} c(ξ)det(2πi(z̄ − z))^{−1} \\ ∑_{t=0}^{m} c_{m−t}(k + 1 − κ)λ_{t}(−2πiξ(z̄ − z))e_{m}(ξz) \end{matrix}  (2.18)

where

c_{0}(α) = 1, c_{r}(α) = ∏_{j=0}^{r−1}(α + j/2) = \frac{Γ_{r}(α + (r + 1)/2)}{Γ_{r}(α + (r − 1)/2)}.

Proof of 2.6.: Let us write by (2.16)

δ_{k}^{(r)}f(z) = ∑_{ξ∈B_{m}} c(ξ)det(4πy)^{−r}R_{m}(4πξy; r, κ − k − r)e_{m}(ξz)

and substitute in it

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),

with 4πξy = 2πiξ(z̄ − z), β = κ − k − r:

\begin{aligned}δ_{k}^{(r)}f(z) = & ∑_{ξ∈B_{m}} c(ξ)det(4πy)^{−r} ∑_{t=0}^{r} \binom{r}{t}det(2πiξ(z̄ − z))^{r−t} \\ & ∑_{|L|≤mt−t} R_{L}(κ − k − r)λ_{l_{1}}(2πiξ(z̄ − z)) ⋯ λ_{l_{t}}(2πiξ(z̄ − z))e_{m}(ξz) \end{aligned}

Note also that by (2.9)

Δ (e^{tr(uz)}det(z)^{α}) = e^{tr(uz)}det(z)^{α−1} ∑_{t=0}^{m} c_{m−t}(α)λ_{t}(uz),

for α ∈ ℂ with

c_{0}(α) = 1, c_{r}(α) = ∏_{j=0}^{r−1}(α + j/2) = \frac{Γ_{r}(α + (r + 1)/2)}{Γ_{r}(α + (r − 1)/2)}

and by definition (2.2)

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

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

(−1)^{m}e^{tr(z)}det(z)^{1+β}e^{−tr(z)}det(z)^{−β−1} ∑_{t=0}^{m} c_{m−t}(−β)λ_{t}(−z)

giving

δ_{k}f(z) = ∑_{ξ∈B_{m}} c(ξ)det(4πy)^{−1}R_{m}(4πξy; 1, κ − k − r)e_{m}(ξz) =

∑_{ξ∈B_{m}} c(ξ)det(2πi(z̄ − z))^{−1}(−1)^{m}e^{tr(2πiξ(z̄−z))}det(2πiξ(z̄ − z))^{1+β}·

· e^{−tr(2πiξ(z̄−z))}det(2πiξ(z̄ − z))^{−β−1} ∑_{t=0}^{m} c_{m−t}(k + 1 − κ)λ_{t}(−2πiξ(z̄ − z))e_{m}(ξz) =

(−1)^{m} ∑_{ξ∈B_{m}} c(ξ)det(2πi(z̄ − z))^{−1} ∑_{t=0}^{m} c_{m−t}(k + 1 − κ)λ_{t}(−2πiξ(z̄ − z))e_{m}(ξz).

2.7. We conclude this section with a general result on commutation of the Shimura operator with Hecke operators (up to a normalization). Recall some basic fact about the action of Hecke algebras (see[And79]. Let q be a prime, q |̸ N,

\begin{matrix}Δ = Δ_{q}^{m}(N) = \\ \{γ = ({a \atop c} {b \atop d}) ∈ G_{ℚ}^{+} ∩ GL_{2m}(ℤ[q^{−1}]) | ν(γ)^{±} ∈ ℤ[q^{−1}], c ≡ 0_{m} (mod N)\} \end{matrix}

be a subgroup in G_{ℚ}^{+} containing Γ = Γ_{0}^{m}(N). The Hecke algebra

ℒ = ℒ_{q}^{m}(N) = D_{ℚ}(Γ, Δ)

over ℚ is then defined as a ℚ-linear space generated by the double cosets (g) = (ΓgΓ), g ∈ Δ of the group S with respect to the subgroup Γ, for which multiplication is defined by the standard rule (see [An7], [Shi1] and in 1.7 below). We recall the description of the structure of ℒ = ℒ_{q}^{m}(N), (q |̸ N): for each j, 1 ≤ j ≤ m let us denote by w_{j} an automorphism of the algebra ℚ[x_{0}^{±1}, x_{1}^{±1}, ⋯, x_{m}^{±1}] defined on its generators by the rule:

x_{0} ↦ x_{0}x_{j}, x_{j} ↦ x_{j}^{−1}, x_{i} ↦ x_{i} (1 ≤ i ≤ m, i ≠ j).

Then the automorphisms w_{j} and the permutation group Σ_{m} of the variables x_{i} (1 ≤ i ≤ m) generate together the Weyl group W = W_{m}, and there is the Satake isomorphism:

Sat : ℒ →^{∼} ℚ[x_{0}^{±1}, x_{1}^{±1}, ⋯, x_{m}^{±1}]^{W_{m}}  (2.19)

where W_{m} indicates the subalgebra of elements fixed by W_{m}. For any commutative ℚalgebra A the group W_{m} acts on the set (A^{×})^{m+1}, therefore any homomorphism of ℚalgebras λ : ℒ → A can be identified with some element

(α_{0}, α_{1}, ⋯, α_{m}) ∈ [(A^{×})^{m+1}]^{W_{m}}.  (2.20)

Any double coset (g) = (ΓgΓ) (g ∈ Δ = Δ_{q}^{m}(N)) can be represented is a disjoint union of left cosets:

(g) = ∪_{i=1}^{t(g)}Γg_{i},

therefore any element X ∈ ℒ of the Hecke algebra ℒ takes the form of a finite linear combination

X = ∑_{i=1}^{t(X)} v_{i}(Γg_{i}),

with v_{i} ∈ ℚ, g_{i} ∈ Δ. In order to define the Hecke operators we use the notation by Petersson and Andrianov: for any g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Δ

(f|_{k,ψ}g)(z) = (detg)^{k−(m+1)/2}ψ(deta)det(cz + d)^{−k}f(g(z))  (2.21)

(f|_{k,ψ}γ)(z) = f for all γ ∈ Γ = Γ_{0}^{m}(N).  (2.22)

In this case for any

X = ∑_{i=1}^{t(X)} v_{i}(Γg_{i}) ∈ ℒ

we have that the expression

f|X = ∑_{i=1}^{t(X)} v_{i}f|_{k,ψ} g_{i},  (2.23)

is well defined and f|X ∈ M_{k}^{m}(N, ψ) so that the formula (2.23) gives a representation of the Hecke algebra ℒ = ℒ_{q}^{m}(N) on the complex vector space M_{k}^{m}(N, ψ) (q |̸ N).

Action of the Hecke operators on Fourier expansions. Let

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

and for a fixed prime q

X = ∑_{i=1}^{t(X)} v_{i}(Γ_{0}g_{i}) ∈ ℒ_{0} (Γ_{0} = Γ_{0}^{m}, ℒ_{0} = ℒ_{0,q}^{m})

be an arbitrary element of the extended Hecke algebra ℒ_{0} with

g_{i} = ( \begin{matrix}q^{ν_{i}}d_{i}^{∗} & b_{i} \\ 0 & d_{i} \end{matrix} ) ∈ Δ_{0,q}^{m}.  (2.24)

Let coefficients (c|X)(ξ) be defined by the equality

(f|X)(z) = ∑_{ξ∈B_{m}}(c|X)(ξ)e_{m}(ξz).  (2.25)

Then we have that

\begin{matrix}(c|X)(ξ) & = ∑_{i=1}^{t(X)} v_{i}q^{ν_{i}(mk−l1m⟩)}(detd_{i})^{−k}× \\ ×ψ(q^{mν_{i}} & (detd_{i})^{−1})e_{m}(q^{−ν_{i}}ξ^{t}b_{i}d_{i})c(q^{−ν_{i}}d_{i}ξ^{t}d_{i}), \end{matrix}  (2.26)

(here we assume that c(ξ) = 0 for ξ ∉ B_{m} (see [An79], §3.2)).

As an example we consider the Frobenius elements (see [An7], §2.1):

Π_{−} = Π_{−}^{m}(q) = (Γ_{0} ( \begin{matrix}q1_{m} & 0 \\ 0 & 1_{m} \end{matrix} ) Γ_{0}),

Π_{+} = Π_{+}^{m}(q) = (Γ_{0} ( \begin{matrix}1_{m} & 0 \\ 0 & q1_{m} \end{matrix} ) Γ_{0}).

(2.27)

Then for f(z) = ∑_{ξ∈B_{m}} c(ξ)e_{m}(ξz) ∈ M_{m}^{k}(N, ψ) we get from (2.26)

(c|Π_{−})(ξ) = ψ(q)^{m}q^{mk−l1m⟩}c(q^{−1}ξ),  (2.28)
(c|Π_{+})(ξ) = c(qξ) (ξ ∈ B_{m} ⊂ A_{m})  (2.29)

Note that the operator Π_{+} = Π_{+}(q) is defined by the formula (2.29) also for q|N, when Π_{+}(q) sends M_{m}^{k}(N, ψ) to itself. Theorem 2.8. Let

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

and let

δ_{k}^{(r)} = δ_{k+2r−2} ∘ ⋯ ∘ δ_{k} : ℳ̃_{k}^{m}(N, ψ) → ℳ̃_{k+2r}^{m}(N, ψ)

be the Shimura differential operator defined by (2.15). Consider for a fixed prime q an arbitrary element

X = ∑_{i=1}^{t(X)} v_{i}(Γ_{0}g_{i}) ∈ ℒ_{0} (Γ_{0} = Γ_{0}^{m}, ℒ_{0} = ℒ_{0,q}^{m})

of the extended Hecke algebra ℒ_{0} with

g_{i} = ( \begin{matrix}q^{ν_{i}}d_{i}^{∗} & b_{i} \\ 0 & d_{i} \end{matrix} ) ∈ Δ_{0,q}^{m},

and assume X homogeneous (ν_{1} = ν_{2} = ⋯ ν_{t} = ν). Then

(δ_{k}^{(r)}f)|_{k+2r}X = q^{νrm}δ_{k}^{(r)}(f|_{k}X)  (2.30)

In particular

(δ_{k}^{(r)}f)|_{k+2r}Π_{+}^{m}(q) = q^{rm}δ_{k}^{(r)}(f|_{k}Π_{+}^{m}(q)), (δ_{k}^{(r)}f)|_{k+2r}Π_{−}^{m}(q) = q^{rm}δ_{k}^{(r)}(f|_{k}Π_{−}^{m}(q)) (2.31)

Proof of 2.8: 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 (δ_{k}f)|_{k+2}g_{i} = det(g_{i})δ_{k}(f|_{k}g_{i}), one immediately obtains (2.31) from the relation detg_{i} = q^{ν_{i}rm} taking into account the homogeneousity of X.

§3. Arithmetical variables of nearly holomorphic Siegel modular forms and differerntial operators.

3.1. Arithmetical nearly holomorphic Siegel modular forms where studied by G.Shimura [Shi2000] and they 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 = (4πIm(z))^{−1} and z ∈ ℍ_{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) and satisfying certain reciprocity laws at these points.

The first description motivates our choice of arithmetical variables: q^{ξ}, R = (R_{i,j}) = (4πIm(z))^{−1} = 2πi(z̄− z)^{−1}. Our task in this Section is to expliciteltly describe the action of δ_{k}^{(r)} in terms of these arithmetical variables.

Theorem 3.2. 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)q^{ξ}, then

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},

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}

If r = 1 one has

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).

Remark. In the elliptic modular case m = 1, κ = 1, the only possibilities are t = 0, 1 when c_{0}(k) = 1, c_{1}(k) = k, and one obtains again the classical formula

δ_{k}(∑_{n≥0} a_{n}q^{n}) = ∑_{n≥1} na_{n}q^{n} − \frac{k}{4πy} ∑_{n≥0} a_{n}q^{n} = ∑_{n≥1} na_{n}q^{n} − kR ∑_{n≥0} a_{n}q^{n}

in which

R = \frac{1}{4πy} , ∑_{n≥1} na_{n}q^{n} = \frac{1}{2πi} \frac{∂}{∂z} (∑_{n≥0} a_{n}q^{n}) = q \frac{d}{dq} (∑_{n≥0} a_{n}q^{n}) .

Proof of 3.2.: Let us change the variable by putting R equal (4πy)^{−1} = −2πiz−z̄)^{−1} and let us use the equality (2.9) with u = −2πiξ and the polynomial (2.2) with n = r, β = κ−k−r

δ_{k}f(z) = (−\frac{1}{4π} )^{m} det(y)^{−1}det(z − z̄)^{κ−k}Δ [det(z − z̄)^{k−κ+1}f(z)]

= det(2πi)^{−m}det(z − z̄)^{κ−k−1}Δ [det(z − z̄)^{k−κ+1}f(z)] . δ_{k}^{(r)} (e_{m}(ξz)) = det(4πy)^{−r}e_{m}(ξz)R(4πξy; r, κ − k − r) = det(2πi(z̄ − z))^{−r}e_{m}(ξz)R(2πiξ(z̄ − z); r, κ − k − r)

We have by Theorem 2.6

\begin{aligned}δ_{k}^{(r)}f(z) = & ∑_{ξ∈B_{m}} c(ξ)det(4πy)^{−r} ∑_{t=0}^{r} \binom{r}{t}det(2πiξ(z̄ − z))^{r−t} \\ & ∑_{|L|≤mt−t} R_{L}(κ − k − r)λ_{l_{1}}(2πiξ(z̄ − z)) ⋯ λ_{l_{t}}(2πiξ(z̄ − z))e_{m}(ξz) \end{aligned} (3.1)

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] (recall that the polynomials λ_{j}(z) are defined by det(t1_{m} + z) = ∑_{j=0}^{m} λ_{j}(z)t^{m−j}.

We now express all terms of type λ_{j}(2πiξ(z̄− z)) in terms of our arithmetical variable R: 2πi(z̄ − z) = R^{−1} and λ_{j}(2πiξ(z̄ − z)) = λ_{j}(ξR^{−1})). Viewing symmetric matrices R and ξ as independent variables let us write the definition

det(tR + ξ) = det(R)det(t1_{m} + ξR^{−1}) = det(R)det(t1_{m} + ξR^{−1})

= ∑_{j=0}^{m} det(R)λ_{j}(ξR^{−1})t^{m−j} = t^{m}det(ξ)det(Rξ^{−1} + t^{−1}1_{m}) =

t^{m}det(ξ) ∑_{j=0}^{m} λ_{j}(Rξ^{−1})t^{−m+j} = ∑_{j=0}^{m} det(ξ)λ_{j}(Rξ^{−1})t^{j}

(3.2)

Let us show that the coefficients of the polynomial det(tR + ξ) admit the following very explicite expression

det(tR + ξ) = ∑_{r=0}^{m} t^{r}tr (^{t}ρ_{r}(R) · ρ_{m−r}^{⋆}(ξ)))  (3.3)

In order to prove (3.3) we use the differentiation rules (2.5) and (2.8)

Δ(fg) = ∑_{r=0}^{m} tr (^{t}ρ_{r}(˜∂/∂z)f · ρ_{m−r}^{⋆}(˜∂/∂z)g) ,

ρ_{r}(˜∂/∂z)e^{tr(uz)} = ρ_{r}(u)e^{tr(uz)},

applied for the functions f = e^{tr(tRz)} and g = e^{tr(ξz)}:

Δ(e^{tr(tR+ξ)z}) = Δ(e^{tr(tRz)} · e^{tr(ξz)}) = ∑_{r=0}^{m} tr (^{t}ρ_{r}(˜∂/∂z)e^{tr(tRz)} · ρ_{m−r}^{⋆}(˜∂/∂z)e^{tr(ξz)}) =

∑_{r=0}^{m} tr (^{t}ρ_{r}(tRz)e^{tr(tRz)} · ρ_{m−r}^{⋆}(ξz)e^{tr(ξz)})) = ∑_{r=0}^{m} tr (^{t}ρ_{r}(tRz) · ρ_{m−r}^{⋆}(ξz)) e^{tr((tR+ξ)z)},

(3.4)

on the other hand

Δ(e^{(tr(tR+ξ)z)}) = ρ_{m}(˜∂/∂z)e^{tr((tR+ξ)z)} =
ρ_{m}((tR + ξ)z)e^{tr((tR+ξ)z)} = det((tR + ξ)z)e^{tr((tR+ξ)z)}

(3.5)

and the substutution of z = 1_{m} to (3.4) and (3.5) implies (3.3).

It follows from (3.3) and (3.2) that

det(R)λ_{j}(ξR^{−1}) = tr (^{t}ρ_{m−j}(R) · ρ_{j}^{⋆}(ξ))  (3.6)

Let us substitute (3.6) to (3.1) writng 4πy = 2πi(z̄ − z) = R^{−1},

λ_{j}(2πiξ(z̄ − z)) = λ_{j}(ξR^{−1}) = det(R)^{−1}tr (^{t}ρ_{m−j}(R) · ρ_{j}^{⋆}(ξ))

\begin{aligned}δ_{k}^{(r)} & f(z) = ∑_{ξ∈B_{m}} c(ξ)det(R^{−1})^{−r} ∑_{t=0}^{r} \binom{r}{t}det(ξR^{−1})^{r−t} \\ & ∑_{|L|≤mt−t} R_{L}(κ − k − r)λ_{l_{1}}(ξR^{−1}) ⋯ λ_{l_{t}}(ξR^{−1})e_{m}(ξz) \end{aligned}

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

∑_{|L|≤mt−t} R_{L}(κ − k − r)λ_{l_{1}}(ξR^{−1}) ⋯ λ_{l_{t}}(ξR^{−1})e_{m}(ξz)

= ∑_{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)))e_{m}(ξz)

(3.7)

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 first statement of Theorem 3.2 follows.

In order to prove the second statement it suffices to specialize these formulae to the case of r = 1 using (2.18). 3.3. Note from (3.7) that the action of δ_{k}^{(r)} on the exponential q^{ξ} = e_{m}(ξz) written in variables q^{ξ}, R = (R_{i,j}) is given by

\begin{matrix}δ_{k}^{(r)}(q^{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}, \end{matrix} (3.8)

and one may view ξ also as an independent variable for which the notation

∂_{ξ} = (∂_{ξ,i,j}), ˜∂_{ξ} = (˜∂_{ξ,i,j}), ˜∂_{ξ,i,j} = 2^{−1}(1 + δ_{i,j})˜∂_{ξ,i,j}  (3.9)

will be used. 3.4. Our next task is to write down the formal action

δ_{k}^{(r)}(R_{i_{1},j_{1}}^{a_{1}} ⋯ R_{i_{n},j_{n}}^{a_{n}}q^{ξ})  (3.10)

of δ_{k}^{(r)} using the differentiaion over the formal parameter ξ of the equality (3.8). Let us show first that

R_{i_{1},j_{1}}^{a_{1}} ⋯ R_{i_{n},j_{n}}^{a_{n}}q^{ξ} = det(R)^{a_{1}+···+a_{n}}e^{tr(2πiξ(−z̄))} · D_{ξ}(e^{tr(2πiξ(z̄−z))})  (3.11)

for an appropriate differential polynomial on ˜∂_{ξ,i,j} which we now specify. Recall that (R_{i,j}) = (2πi(z̄ − z))^{−1} = ρ_{1}((2πi(z̄ − z))^{−1}) = det(2πi(z̄ − z))^{−1} ·^{t}ρ_{m−1}^{⋆}((2πi(z̄ − z))) on the other hand

tρ_{m−1}^{⋆}((2πi(z̄ − z)))e^{tr(2πiξ(z̄−z))} = ^{t}ρ_{m−1}^{⋆}((˜∂_{ξ}))e^{tr(2πiξ(z̄−z))}

hence

D_{ξ} = D_{ξ,i_{1},j_{1}}^{a_{1}} ⋯ D_{ξ,i_{n},j_{n}}^{a_{n}}  (3.12)

where

(D_{ξ,i,j}) = ^{t}ρ_{m−1}^{⋆}((˜∂_{ξ})).

(we used the commutation of ˜∂_{z,i,j} with ˜∂_{ξ,i,j} with respect to the independent formal variables z and ξ).

Let us compute δ_{k}^{(r)}(R_{i_{1},j_{1}}^{a_{1}} ⋯ R_{i_{n},j_{n}}^{a_{n}}q^{ξ}) = δ_{k}^{(r)} (det(R)^{a_{1}+···+a_{n}}e^{tr(2πiξ(−z̄))} · D_{ξ}(e^{tr(2πiξ(z̄−z))}))

using (3.11), (2.2), Lemma 2.4 and (3.8):

\begin{matrix}δ_{k}^{(r)}(q^{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}, \end{matrix}

δ_{k}^{(r)}(R_{i_{1},j_{1}}^{a_{1}} ⋯ R_{i_{n},j_{n}}^{a_{n}}q^{ξ})
= δ_{k}^{(r)} (det(R)^{a_{1}+···+a_{n}}e^{tr(2πiξz̄)} · D_{ξ}(e^{tr(2πiξ(z−z̄))}))
= e^{tr(2πi(ξz̄))}δ_{k}^{(r)} (det(R)^{a_{1}+···+a_{n}} · D_{ξ}(e^{tr(2πiξ(z−z̄))}))
= e^{tr(2πiξz̄))}D_{ξ} (δ_{k}^{(r)} (det(R)^{a_{1}+···+a_{n}} · e^{tr(2πiξ(z−z̄))}))

(3.13)

and our computation of (3.13) is reduced to computation of

δ_{k}^{(r)} (det(R)^{a} · e^{tr(2πiξ(z−z̄))}) = (−2πi)^{−am}e^{tr(−2πi(ξz̄))}δ_{k}^{(r)}(det(z − z̄)^{−a} · e^{tr(2πi(ξz))})

where a = a_{1} + ⋯ + a_{n}. Lemma 3.5. Let a ≥ 0 be a natural number then the action of the Shimura differential operator on the functions det(z̄ − z)^{−a}e_{m}(ξz) (where ξ ∈ B_{m} and z ∈ ℍ_{m}) is given by the formula

δ_{k}^{(r)} (det(z − z̄)^{−a}e_{m}(ξz))
= (−2πi)^{−rm}det(z − z̄)^{−r−a}e_{m}(ξz)R_{m}(2πiξ(z̄ − z); r, κ − k − r + a)
= (2πi)^{−rm}e^{tr(2πiξ(z̄))}det(z − z̄)^{κ−k−r}Δ^{r} (e^{tr(2πiξ(z−z̄))}det(z − z̄)^{−κ+k+r−a})

where δ_{k}^{(r)} = δ_{k+2r−2} ∘ ⋯ ∘ δ_{k},

δ_{k}f(z) = (−\frac{1}{4π} )^{m} det(y)^{−1}M_{k}f(z) = (2πi)^{−m}det(z − z̄)^{κ−k−1}Δ [det(z − z̄)^{−κ+k+1}f]

M_{k}f(z) = det(z − z̄)^{κ−k}Δ [det(z − z̄)^{k−κ+1}f(z)] ,
R_{m}(z; r, β) = (−1)^{mr}e^{tr(z)}det(z)^{r+β}Δ_{m}^{r} [e^{−tr(z)}det(z)^{−β}] .

Proof of 3.5: is very similar to that of 2.4. One proves this lemma by induction on r : • If r = 0, it is obviously true.

• Suppose next that it holds for some r then δ_{k}^{(r+1)} = δ_{k+2r} ∘ δ_{k}^{(r)}, and if one applies the Shimura operator δ_{k+2r} by induction one obtains

δ_{k}^{(r+1)}(det(z − z̄)^{−a}e_{m}(ξz)) = (−2πi)^{−m}det(z − z̄)^{κ−k−2r−1}·
· Δ[det(z − z̄)^{k+2r+1−κ}(2πi)^{−rm}det(z − z̄)^{−a−r}e_{m}(ξz)·
· R_{m}(2πiξ(z̄ − z); r, κ − k − r + a)]

(3.14)

(recall that 4πξy = −2iπξ(z −z̄)). On the other hand z̄ is aniholomorphic hence it behaves as a constant under the action of Δ, and it makes it possible to write, by putting u = −2iπξ, β = κ − k − r + a :

R_{m}(u(z − z̄); r, β) =
(−1)^{mr}e^{tr(u(z−z̄))}det(z − z̄)^{r+β}Δ^{r} (e^{−tr(u(z−z̄))}det(z − z̄)^{−β})

(3.15)

(compare with the proof of 2.4). One substituts (3.15) into (3.14) , then the factor e_{m}(ξz̄) = e^{tr(−uz̄)} which is an antiholomorphic function can be pulled out. One obtains then

δ_{k}^{(r+1)}(det(z − z̄)^{−a}e_{m}(ξz))
= (−2πi)^{−m}det(z − z̄)^{κ−k−2r−1}·
· Δ[det(z − z̄)^{k+2r+1−κ}(2πi)^{−rm}det(z − z̄)^{−a−r}e_{m}(ξz)·
· (−1)^{mr}e^{tr(u(z−z̄))}det(z − z̄)^{r+β}Δ^{r} (e^{−tr(u(z−z̄))}det(z − z̄)^{−β}) ]
= (2πi)^{−(r+1)m}(−1)^{m(r+1)}det(z − z̄)^{κ−k−2r−1}e_{m}(ξz̄))·
· Δ[det(z − z̄)^{k+2r+1−κ−a−r+r+β}Δ^{r} (e^{−tr(u(z−z̄))}det(z − z̄)^{−β}) ]
= (−2πi)^{−(r+1)m}det(z − z̄)^{κ−k−r−1}e_{m}(ξz̄))·
det(z − z̄)^{−r} · Δ[det(z − z̄)^{r+1}Δ^{r} (e^{−tr(u(z−z̄))}det(z − z̄)^{−β}) ]

(3.16)

Let us now apply lemma 2.3 which gives

det(z − z̄)^{−r}·Δ[det(z − z̄)^{r+1}Δ^{r} (e^{−tr(u(z−z̄))}det(z − z̄)^{−β+1}) ]
= Δ^{r+1} (e^{−tr(u(z−z̄))}det(z − z̄)^{−β+1})

and (3.16) transforms to

δ_{k}^{(r+1)}(det(z − z̄)^{−a}e_{m}(ξz))

= (−2πi)^{−(r+1)m}det(z − z̄)^{κ−k−r−1}e_{m}(ξz̄)Δ^{r+1} (e^{−tr(u(z−z̄))}det(z − z̄)^{−β+1})

= (−2πi)^{−(r+1)m}det(z − z̄)^{κ−k−(r+1)}e_{m}(ξz̄)· · Δ^{r+1} (e^{−tr(u(z−z̄))}det(z − z̄)^{−κ+k+r+1−a})

(3.17)

finishing the proof by induction.

Corollary 3.6. Let a ≥ 0 be a natural number then the action of the Shimura differential operator on the functions det(2πi(z̄ − z))^{−a}e_{m}(ξz) (where ξ ∈ B_{m} and z ∈ ℍ_{m}) is given by the formula

δ_{k}^{(r)} (det(R)^{a}q^{ξ})

= det(R)^{r+a}R_{m}(ξR^{−1}; r, κ − k − r + a)q^{ξ}

= det(R)^{a}q^{ξ} ∑_{t=0}^{r} \binom{r}{t}det(ξ)^{r−t}

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

(3.18)

where δ_{k}^{(r)} = δ_{k+2r−2} ∘ ⋯ ∘ δ_{k}, and 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].

Proof of 3.6.: Let us substitute into (2.3) 4πξy = 2πiξ(z̄−z) = ξR^{−1} as z, β = κ−k−r+a:

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)

giving

R_{m}(ξR^{−1}; r, κ − k − r + a)

= ∑_{t=0}^{r} \binom{r}{t}det(ξR^{−1})^{r−t} ∑_{|L|≤mt−t} R_{L}(κ − k − r + a)λ_{l_{1}}(ξR^{−1}) ⋯ λ_{l_{t}}(ξR^{−1})
= ∑_{t=0}^{r} \binom{r}{t}det(ξ)^{r−t}det(R)^{−r}

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

using again (3.6): λ_{j}(ξR^{−1}) = det(R)^{−1}tr (^{t}ρ_{m−j}(R) · ρ_{j}^{⋆}(ξ)), and (3.18) follows.

In a more general case let us apply Lemma 3.5 to (3.13):

δ_{k}^{(r)}(R_{i_{1},j_{1}}^{a_{1}} ⋯ R_{i_{n},j_{n}}^{a_{n}}q^{ξ})

= e^{tr(2πiξz̄))}D_{ξ} (δ_{k}^{(r)} (det(R)^{a_{1}+···+a_{n}} · e^{tr(2πiξ(z−z̄))})) = e^{tr(2πiξz̄))}D_{ξ} ((−2πi)^{−am}e^{tr(−2πi(ξz̄))}δ_{k}^{(r)}(det(z − z̄)^{−a} · e^{tr(2πi(ξz))})) = (−2πi)^{−am}D_{ξ} (δ_{k}^{(r)}(det(z − z̄)^{−a} · e^{tr(2πi(ξz))})) = (−2πi)^{−am−rm}D_{ξ}(det(z − z̄)^{−r−a}e_{m}(ξz)

R_{m}(2πiξ(z̄ − z); r, κ − k − r + a))

(3.19)

where a = a_{1} + ⋯ + a_{n}. Let us use again the explicit expression (2.3) for the polynomial

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),

where L runs over all the multi-indices 0 ≤ l_{1} ≤ ⋯ ≤ l_{t} ≤ m, such that |L| = l_{1} +⋯+l_{t} ≤ mt − t. Let us substitute (3.6) to (3.19) writng 4πy = 2πi(z̄ − z) = R^{−1},

λ_{j}(2πiξ(z̄ − z)) = λ_{j}(ξR^{−1}) = det(R)^{−1}tr (^{t}ρ_{m−j}(R) · ρ_{j}^{⋆}(ξ)) :
δ_{k}^{(r)}(R_{i_{1},j_{1}}^{a_{1}} ⋯ R_{i_{n},j_{n}}^{a_{n}}q^{ξ})

= (−2πi)^{−am−rm}D_{ξ}(det(z − z̄)^{−r−a}q^{ξ}R_{m}(2πiξ(z̄ − z); r, κ − k − r + a))

= (−2πi)^{−am−rm}D_{ξ}(det(z − z̄)^{−r−a}q^{ξ} ∑_{t=0}^{r} \binom{r}{t}det(2πiξ(z̄ − z))^{r−t}

∑_{|L|≤mt−t} R_{L}(β)λ_{l_{1}}(2πiξ(z̄ − z)) ⋯ λ_{l_{t}}(2πiξ(z̄ − z)))

= D_{ξ}(det((2πi)(z̄ − z))^{−r−a}q^{ξ} ∑_{t=0}^{r} \binom{r}{t}det(2πiξ(z̄ − z))^{r−t}

∑_{|L|≤mt−t} R_{L}(β)λ_{l_{1}}(2πiξ(z̄ − z)) ⋯ λ_{l_{t}}(2πiξ(z̄ − z)))

= det(R)^{a}D_{ξ}(q^{ξ} ∑_{t=0}^{r} \binom{r}{t}det(ξ)^{r−t}

∑_{|L|≤mt−t} R_{L}(β)tr (^{t}ρ_{m−l_{1}}(R) · ρ_{l_{1}}^{⋆}(ξ)) ⋯ tr (^{t}ρ_{m−l_{t}}(R) · ρ_{l_{t}}^{⋆}(ξ))))

(3.20)

§4. Arithmetical families of nearly holomorphic Siegel modular forms.

A preliminary definition of a p-adic family of Siegel modular forms is the following: {f_{k}} is an infinite set of Siegel modular forms

f_{k} = ∑_{ξ∈B_{m}} a_{k}(ξ)q^{ξ} ∈ ℚ̄[[q^{B_{m}}]]

(parametrized by the weight k), whose Fourier coefficients depend p-adic analytically on the weight k. If we fix embeddings i_{∞}: ℚ̄ ↪ ℂ, i_{p}: ℚ̄ ↪ ℂ_{p} (ℂ_{p} = ˆ ℚ̄_{p} is the Tate field) then this amounts to analyticity of all maps k ↦ a_{k}(ξ) (where k is viewed as a p-adic parameter).

There are various approaches to construction of such families (at least for m = 1):

1) The technique of Hida and Wiles of Λ-adic modular forms where Λ = ℤ_{p}[[T]] is the Iwasawa algebra: each element A(T) = ∑_{n} a_{n}T^{n} ∈ Λ converges on the unit disc U = {t ∈ ℂ_{p} | |t|_{p} < 1} to a bounded ℂ_{p}-analytic function with a finite number of zeroes (in view of the p-adic Weierstrass preparation theorem, see [Ko]). We have therefore for each k ≥ 1 a specialization map: Λ ∋ A(T) ↦ A_{k} = A((1 + p)^{k} − 1) ∈ ℤ_{p}; (a conceptual definition of this map uses the description of Λ in terms of p-adic distributions μ on ℤ_{p} with values in ℤ_{p}: one can attach to such a μ a unique power series A_{μ}(T) = ∫_{ℤ_{p}}(1 + T)^{x}μ(x) ∈ Λ and this correspondence is a bijection. Then the above specialization A_{μ}((1 + p)^{k} − 1) = ∫_{ℤ_{p}}(1 + p)^{kx}μ(x) is the image at A_{μ}(T) of the ℤ_{p}-ring homomorphism κ_{k}: Λ → ℤ_{p} given by κ_{k}(1 + T) = (1 + p)^{k}). In order to introduce Λ-adic modular form consider the ring Λ[[q^{B_{m}}]] of all formal Siegel q-expansions with coefficients in Λ, and for each element F = ∑_{ξ∈B_{m}} A(ξ, T)q^{ξ} ∈ Λ[[q^{B_{m}}]] and for each k ≥ 1 we have a ring homomorphism κ_{k}: Λ[[q^{B_{m}}]] → ℤ_{p}[[q^{B_{m}}]], defined by

κ_{k}(F) = ∑_{ξ∈B_{m}} A_{k}(ξ)q^{ξ},  (4.1)

where

A_{k}(ξ = A(ξ, (1 + p)^{k} − 1) ∈ ℤ_{p} .

Let χ be a Dirichlet character modulo N and let us fix a finite extension 𝒪 of ℤ_{p} containing the values of χ, and put Λ_{𝒪} = 𝒪[[T]]. Consider the ring

Λ_{𝒪} [[q^{B_{m}}]][R_{i,j}]_{i,j=1,···,m} of all formal nearly holomorphic q-expansions (or (q, R)-expansions) with coefficients in Λ_{𝒪} . Definition 4.1. An element F = ∑_{ξ∈B_{m}} A(ξ, T)q^{ξ} ∈ Λ_{𝒪} [[q^{B_{m}}]] is called a Λ_{𝒪} -adic Siegel modular form of level N and character χ if for all k ≫ 0 the element

F_{k} = ∑_{ξ∈B_{m}} A(ξ, (1 + p)^{k} − 1)q^{n} ∈ 𝒪[[q^{B_{m}}]]

represents the Fourier expansion of a complex nearly holomorphic Siegel modular form: F_{k} ∈ ℳ_{k}^{m}(Γ_{0}(Np), χω^{−k}), where ω: (ℤ/pℤ)^{∗} → μ_{p−1} ⊂ ℚ̄^{∗} ↪ ℂ^{∗} is the Teichmüller character.

Let us denote by ℳ^{m}(N, χ, Λ_{𝒪} ) the Λ_{𝒪} -submodule of Λ_{𝒪} [[q^{B_{m}}]] generated by all such elements.

Note that such a family F = {F_{k}} depends only on one parameter T, and in the above definition one has to use a variable Dirichlet character χω^{−k} (which depends on k).

2) In order to define p-adic families of nearly holomorphic Siegel modular forms one uses the notion of the p-adic integration with values in spaces of (formal) nearly holomorphic Siegel modular forms.

Consider any profinite group Y = \underleftarrow{lim} Y_{i} (for example, Y = ℤ_{p}, Y = Γ = 1 + pℤ_{p} = \overline{⟨1 + p⟩}, Y = Gal(ℚ̄/ℚ) etc.).

The completed group ring 𝒪[[Y]] = \underleftarrow{lim} 𝒪[Y_{i}] is a compact ring which generalizes the Iwasawa algebra Λ_{𝒪} , and it may be viewed as the 𝒪-algebra of all 𝒪-valued distributions on Y; for the natural projection homomorphisms π_{i}: Y → Y_{i} put U_{ij} = π_{i}^{−1}(y_{j}) (y_{j} ∈ Y_{i}) then U_{ij} generate the profinite topology of Y and a distribution μ on Y corresponds to an element f = {f_{i}} ∈ 𝒪[[Y]] where f_{i}(y_{j}) = μ(U_{ij}). Recall that a distribution μ on Y with values in 𝒪 is a finite-additive map on open-compact subsets U ⊂ Y:

μ: \{ \begin{matrix}open-compact sets \\ U ⊂ Y \end{matrix} \} ⟶ 𝒪

The natural multiplication in 𝒪[[Y]] corresponds to the convolution of distributions:

(μ_{1} ∗ μ_{2})(π_{i}^{−1}(y)) = ∑_{y=z·w∈Y_{i}} μ_{1}(π_{i}^{−1}(z)) · μ_{2}(π_{i}^{−1}(w))

Examples. a) Λ_{𝒪} = 𝒪[[T]] ≅ 𝒪[[Γ]], 1 + T ↦ 1 + p ∈ Γ.

b) Λ_{𝒪} = 𝒪[[ℤ_{p}^{∗}]] ≅ Λ_{𝒪} ⊗_{𝒪} 𝒪[Δ] where ℤ_{p}^{∗} = Γ × Δ (a direct product, where Δ = ℤ_{p}^{∗tors}).

For a continuous group homomorphism κ: Y → 𝒪^{∗} we obtain an 𝒪-algebra homomorphism κ: 𝒪[[Y]] → 𝒪 which provides a group-theoretical interpretation of the above specialization maps: κ_{k}(x) = x^{k} for all x ∈ ℤ_{p}^{∗}.

In order to define more general p-adic families of nearly holomorphic Siegel modular forms, consider a family 𝒫 = {P } ⊂ 𝒞(Y, 𝒪) of continuous functions on Y with values in 𝒪.

Definition 4.2. A p-adic family of nearly holomorphic Siegel modular forms parametrized by 𝒫 is an element F of the ring 𝒪[[Y]][[q^{B_{m}}]][R_{i,j}] = Distr (Y, 𝒪[[q^{B_{m}}]][R_{i,j}]) such that if

F = ∑_{ξ∈B_{m},\underline{n}≥0} μ_{ξ,\underline{n}}R\underline{n}q^{ξ} ∈ 𝒪[[Y]][[q^{B_{m}}]][R_{i,j}],  (4.2)

then for almost all P ∈ 𝒫 the integral

∫_{Y} P dF = ∑_{ξ∈B_{m},\underline{n}≥0}( ∫_{Y} P dμ_{ξ,\underline{n}})R\underline{n}q^{ξ} ∈ 𝒪[[q^{B_{m}}]][R_{i,j}]

represents the Fourier expansion ((q, R)-expansion) of a complex nearly holomorphic Siegel modular form of weight k = k(P), type r = r(P), level N = N(P) and character χ = χ_{P}. Here

R = (R_{i,j}) = (4πIm(z))^{−1} = 2πi(z̄ − z)^{−1}, \underline{n} = (n_{ij})_{i,j=1,···,m,i≥j} ∈ ℕ^{m(m+1)/2},

R\underline{n} = ∏_{i,j=1,···,m,i≥j} R_{i,j}^{n_{i,j}}.

Note that the parameter P could include all classical parameters: the weight k, the type r, and the character ε mod p^{α}.

Let us use the notation M_{𝒫} (𝒪[[Y]]) for the 𝒪[[Y]]-submodule of 𝒪[[Y]][[q^{B_{m}}]][R_{i,j}] generated by all such F.

As a version of this definition one can also consider, instead of an integral valued distribution F ∈ Distr (Y, 𝒪[[q^{B_{m}}]][R_{i,j}]), growing measures (or “algebraic distributions”) with values in 𝒪[[q^{B_{m}}]][R_{i,j}]⊗_{𝒪} ℂ_{p}. Then such a construction provides us with more general families (for example, in the non-ordinary case, [CoMa]), see also [Colm 98].

One verifies that in the case of Y = Γ, Λ_{𝒪} = 𝒪[[Γ]], and

𝒫 = {P = P_{k,ε}: ⟼ ε(u)u^{k}, ε mod p^{α}, u ∈ Γ} ,

this leads to the first definition (we assume that 𝒪 contains the values of all Dirichlet characters ε).

An advantage of this alternative approach is a possibility to use the technique of construction of families F as distributions using the abstract Kummer congruences (which will be explained in more detail in this section).

Another advantage is a possibility to avoid the use of the variable Dirichlet character χω^{−k} in the definition 4.1: if we replace Γ by ℤ_{p}^{∗} = Γ × Δ, Δ = μ_{p−1} = (ℤ_{p}^{∗})^{tors} and put

F_{k} = ∫_{ℤ_{p}^{∗}} ψ ⟨x⟩^{k} dF = ∫_{ℤ_{p}^{∗}} ψω^{−k}(x)x^{k}dF

then we may simply require that ∫_{ℤ_{p}^{∗}} ψx^{k}dF ∈ M_{k}(Np, χψ) for all characters ψ: ℤ_{p}^{∗} → 𝒪^{∗} of finite order, including ψω^{−k} (we use the decomposition x^{k} = ω^{k}(x) · ⟨x⟩^{k}, where ω(x) =

lim_{n→∞} x^{p^{n}} ∈ Δ and ⟨x⟩ = x/ω(x) = (1 + p)^{s}, s ∈ ℤ_{p}).

In order to recall the technique of abstract Kummer congruences, consider a complete p-adic ring A = \begin{matrix}lim \\ ⟵ \\ N \end{matrix} A/p^{N}A and consider any family \{φ_{i}\}_{i∈I} ⊂ 𝒞(Y, ℤ_{p}) which generate a dense ℚ_{p}-linear subspace.

Theorem 4.1. Let {a_{i}} ⊂ A a collection of elements of A parametrized by the same set of indices I that we used for {φ_{i}}. Then the existence of a unique distribution μ ∈ Distr (Y, A) with given integrals a_{i} = ∫_{Y} φ_{i}dμ is equivalent to the validity of the following congruences: for ∀ N ∈ ℕ and for each finite linear combination ∑_{i} b_{i}φ_{i}, b_{i} ∈ ℤ_{p}

(∀ y ∈ Y ∑_{i} b_{i}φ_{i}(y) ≡ 0 mod p^{N}A ⟹ ∑_{i} b_{i}a_{i} ≡ 0 mod p^{N})  (4.1)

Proof: [Ka78] 1) If μ exists then ∫_{Y} ∑ b_{i}φ_{i}dμ = ∑ b_{i}a_{i} ≡ 0(p^{N}A) because ∑ b_{i}φ_{i} is divisible by p^{N} in the ring 𝒞(Y, A).

2) In order to construct μ out of a_{i} ∈ A, we use the equivalent definition of an Adistribution as an A-measure (a ℤ_{p}-linear form on 𝒞(Y, ℤ_{p}) with values in A): we need to construct such a linear form

μ: 𝒞(Y, ℤ_{p}) → A with the condition ∀i ∈ I ∫_{Y} φ_{i}dμ = a_{i}  (4.2)

For any φ ∈ 𝒞(Y, ℤ_{p}) let us choose an approximation ∑ b_{i}φ_{i}. Then φ = lim ∑_{i} b_{i}φ_{i} ⇒ ∫ φdμ = lim ∑_{i} b_{i}a_{i} ∈ A. We see that the limit is well defined in view of the congruences (4.1).

The first example of p-adic families (given by Serre [Se73] in the elliptic modular case) is related to Eisenstein series. Let k > m + 1, and let χ be a Dirichlet character modulo N. Consider in the Siegel modular case the series

E_{k}^{(m)}(z, χ) = ∑_{\binom{a b}{c d}∈P^{(m)}∩Γ_{0}^{(m)}(N)∖Γ_{0}^{(m)}(N)} χ(det d)det(cz + d)^{−k}  (4.3)

where

P^{(m)} = \{γ = ({a \atop 0_{m}} {b \atop d}) ∈ Γ^{(m)}\} ,

so that for m = N = 1, k ≥ 4 one has

\begin{matrix}E_{k}(z) = E_{k}^{(1)}(z) = ∑_{γ∈P \Γ} 1|_{k}γ = ∑_{{(c,d)=1,c>0 \atop c=0,d=1}}(cz + d)^{−k} = 1 + \frac{2}{ζ(1 − k)} ∑_{n=1}^{∞} σ_{k−1}(n)q^{n} , \\ q = exp(2πiz), σ_{k−1}(n) = ∑_{d|n} d^{k−1}, ζ(s) = ∑_{n=1}^{∞} n^{−s} = ∏_{ℓ}(1 − ℓ^{−s})^{−1} (Re(s) > 1) \end{matrix}

ζ(1 − k) = −\frac{B_{k}}{k} , ∑_{k=0}^{∞} B_{k} \frac{t^{k}}{k!} = \frac{t}{e^{t} − 1}

E_{4}^{(1)}(z) = 1 + 240 ∑_{n=1}^{∞} σ_{3}(n)q^{n}, E_{6}^{(1)} = 1 − 504 ∑_{n=1}^{∞} σ_{5}(n)q^{n}, …

Note also that (see [Nag2], p.408)

\begin{matrix}E_{4}^{(2)}(z) = & 1 + 240q_{11} + 240q_{22} + 2160q_{11}^{2} + (240q_{12}^{−2} + 13440q_{12}^{−1} \\ 30240 + 13440q_{12} + 240q_{12}^{2})q_{11}q_{22} + 2160q_{22}^{2} + … \end{matrix}

\begin{matrix}E_{6}^{(2)}(z) = & 1 − 504q_{11} − 504q_{22} − 16632q_{11}^{2} + (−540q_{12}^{−2} + 44352q_{12}^{−1} \\ 166320 + 44352q_{12} − 504q_{12}^{2})q_{11}q_{22} − 16632q_{22}^{2} + …. \end{matrix}

Let us normalize, for m = 1, the Eisenstein series E_{k}^{(1)}(z) by

G_{k} = G_{k}^{(1)} = \frac{ζ(1 − k)}{2}E_{k}^{(1)} = −\frac{B_{k}}{2k} + ∑_{n=1}^{∞} σ_{k−1}(n)q^{n}

and put

G_{k}^{∗}(z) = G_{k}(z) − p^{k−1}G_{k}(pz) then
G_{k}^{∗} = \frac{ζ^{∗}(1 − k)}{2} + ∑_{n=1}^{∞} σ_{k−1}^{∗}(n)q^{n}, σ_{k−1}^{∗}(n) = ∑_{{d|n \atop (d,p)=1}} d^{k−1}, where
ζ^{∗}(s) = ζ(s)(1 − p^{−s}) = ∑_{{n=1 \atop (p,n)=1}} n^{−s} denotes the Riemenn zeta function with p-Euler factor removed.

Theorem 4.2. a) Let k ≡ k′ mod (p − 1)p^{N−1} then for all k ≢ 0(p − 1) we have that G_{k}^{∗} ≡ G_{k′}^{∗} mod p^{N} in ℚ[[q]].

b) Let k ≡ k′ mod (p − 1)p^{N−1} then for any c ∈ ℤ, (c, p) = 1, c > 1 we have that (1 − c^{k})G_{k}^{∗} ≡ (1 − c^{k′})G_{k′}^{∗} mod p^{N} (without restriction on k).

c) f_{k} = (1 − c^{k})G_{k}^{∗} is a p-adic family on ℤ_{p}^{∗} parametrized by the set 𝒫 = {y ↦ y^{k}, k ≥ 4} of p-adic characters of ℤ_{p}^{∗}.

Before giving a proof of Theorem 4.2, we explain how to avoid the use of a regularizing integer c > 1, c ∈ ℤ_{p}^{∗} in 4.2. Consider Y = ℤ_{p}^{∗}, 𝒪 ⊃ ℤ_{p}, ℒ_{𝒪} = Quot Λ_{𝒪} , \underline{ℒ}_{𝒪} = Quot \underline{Λ}_{𝒪} , the complete quotient ring of the completed group ring \underline{Λ}_{𝒪} = 𝒪[[ℤ_{p}^{∗}]]. Let us denote by M(N, χ, \underline{Λ}_{𝒪} ) the \underline{Λ}_{𝒪} -submodule of \underline{Λ}_{𝒪} [[q]] generated by all F = ∑_{n ≥ 0} μ_{n}q^{n} ∈ \underline{Λ}_{𝒪} [[q]] such

that for all k ≫ 0 ∫_{ℤ_{p}^{∗}} x^{k}dF = ∑_{n ≥ 0}(∫_{ℤ_{p}^{∗}} x^{k}μ_{n})q^{n} ∈ 𝒪[[q]] is in M_{k}(N_{p}, χ, A) where A is a

subring of ℚ̄ such that i_{p}(A) ⊂ 𝒪.

Define ℳ(N, χ, \underline{ℒ}_{𝒪} ) = ℳ(N, χ, \underline{Λ}_{𝒪} ) ⊗_{\underline{Λ}_{𝒪}} \underline{ℒ}_{𝒪} then the series f of Theorem 4.2 is an element of ℳ(1, 1, \underline{Λ}_{ℤ_{p}}) depending on c, and 1 − c^{k} = ∫_{ℤ_{p}^{∗}} x^{k}(δ_{1} − δ_{c}) (δ_{c} denotes the Dirac

distribution at c ∈ ℤ_{p}^{∗}), δ_{c}(U) = \{ \begin{matrix}1, & if c ∈ U \\ 0, & if c ∉ U \end{matrix}. The element (δ_{1} − δ_{c})^{−1} ∈ \underline{ℒ}_{ℤ_{p}} and G^{∗} = f ⊗ (δ_{1} − δ_{c})^{−1} is an element of ℳ(1, 1, \underline{ℒ}_{ℤ_{p}}) independent on the choice of c. It has the property κ_{k}(G^{∗}) = G_{k}^{∗} where κ_{k} is a unique extension of κ_{k}: \underline{Λ}_{ℤ_{p}} → ℤ_{p} to \underline{ℒ}_{ℤ_{p}} (we apply κ_{k} coefficient-by-coefficient to G^{∗} ∈ ℳ(1, 1, \underline{ℒ}_{ℤ_{p}})). Proof of 4.2: The statements a) and b) follow from c). In order to show c) we put

f_{k} = ∑_{n ≥ 0} a_{k}(n)q^{n} then for n > 0 a_{k}(n) = (1 − c^{k}) ∑_{{d|n \atop (d,p)=1}} d^{k−1} = ∫_{ℤ_{p}} x^{k}dμ_{n,c}(x) where

μ_{n,c} = (δ_{1} − δ_{c}) ∗ ∑_{{d|n \atop (d,p)=1}} d^{−1} · δ_{d}, δ_{c}(U) = \{ \begin{matrix}1, & if c ∈ U, \\ 0, & if c ∉ U. \end{matrix}

(note that for any μ^{(1)}, μ^{(2)} ∈ Distr (ℤ_{p}^{∗}, 𝒪), and for any character P ∈ Hom_{cont}(ℤ_{p}^{∗}, 𝒪^{∗})

we have that ∫_{ℤ_{p}^{∗}} P d(μ^{(1)} ∗ μ^{(2)}) = ∫_{ℤ_{p}^{∗}} P dμ^{(1)} · ∫_{ℤ_{p}^{∗}} P dμ^{(2)}.)

The case n = 0: a_{k}(0) = (1 − c^{k})ζ^{∗}(1 − k) will be treated using the classical Kummer congruences given by the following Theorem 4.3 (Kummer). Put ζ_{(p)}^{(c)}(−k) = (1 − c^{k+1})(1 − p^{k})ζ(−k), k ≥ 0, and let h(x) = ∑_{i} α_{i}x^{i} ∈ ℤ[x] such that h(a) ≡ 0 mod p^{N} for all α ∈ ℤ_{p}^{∗}. Then we have that ∑_{i} α_{i}ζ_{(p)}^{(c)}(−i) ≡ 0 mod p^{N}. Corollary 1 (p-adic continuity of ζ_{(p)}^{(c)}(−k) in a progression mod (p − 1)). If h(x) = x^{k} −x^{k′}, k ≡ k′ mod (p − 1) p^{N−1} then

ζ_{(p)}^{(c)}(−k) ≡ ζ_{(p)}^{(c)}(−k′) mod p^{N} .

Proof of Corollary 1: We have that h(a) ≡ 0 mod p^{N} by Euler’s theorem: a^{φ(p^{N})} ≡ 1 (mod p^{N}), (a, p) = 1. Corollary 2 (Mazur). There exists a unique measure μ^{(c)} on ℤ_{p}^{∗} such that for all k ≥ 1

∫_{ℤ_{p}^{∗}} x^{k}dμ^{(c)} = ζ_{(p)}^{(c)}(1 − k) = (1 − c^{k})(1 − p^{k−1})ζ(1 − k).

Proof of Corollary 2: is implied by Theorem 4.1: the values

∫_{ℤ_{p}^{∗}} x^{k+1}dμ^{(c)} = ζ_{(p)}^{(c)}(−k)

satisfy the abstract Kummer congruences in view of Theorem 4.3.

Let us show that Theorem 4.3 implies Theorem 4.2 c): putf

f = \frac{1}{2}μ^{(c)} + ∑_{n ≥ 0} μ_{n,c}q^{n} ∈ \underline{Λ}[[q]], \underline{Λ} = Distr (ℤ_{p}^{∗}, ℤ_{p}) .

Then

∫_{ℤ_{p}^{∗}} x^{k}df = \frac{1}{2} ∫_{ℤ_{p}^{∗}} x^{k}dμ^{(c)} + ∑_{n ≥ 0}(∫_{ℤ_{p}^{∗}} x^{k}dμ_{n,c})q^{n} = (1 − c^{k})G_{k}^{∗} .

Proof of Theorem 4.3 [Ka78] : uses the sums

S_{k}(M) = ∑_{n=1}^{M−1} n^{k} = \frac{1}{k + 1}[B_{k+1}(M) − B_{k+1}]

where

S_{k}^{∗}(p^{N}) = ∑_{{n=1 \atop p∤n}}^{p^{N} −1} n^{k} = S_{k}(p^{N}) − p^{k}S_{k}(p^{N−1}) .

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)q^{ξ}. As in §1, we fix embeddings i_{∞}: ℚ̄ ↪ ℂ, i_{p}: ℚ̄ ↪ ℂ_{p} and assume that c(ξ) ∈ ℚ̄ and i_{p}(c(ξ)) ∈ 𝒪 ⊂ ℂ_{p}.

Let us construct p-adic families of nearly holomorphic Siegel modular forms starting from the action on a given form f of the arithmetical differential operator δ_{k}^{(r)}:

\begin{aligned}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} \\ & ∈ ℚ(f)[[q^{B_{m}}]][R_{i,j}]_{i,j=1,···,m} \end{aligned} (4.4)

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. Let us define the family of polynomials

\begin{matrix}P_{r,k}(ξ, 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))) \end{matrix}  (4.5)

Theorem 4.4. Put A_{m,p} = A_{m} ⊗_{ℤ} ℤ_{p} then there exists a p-adic measure μ_{f} on the profinite group Y = A_{m,p} with values in 𝒪[[q^{B_{m}}]][R_{i,j}]_{i,j=1,···,m} such that for each k > m + 1

∫_{A_{m,p}} P_{r,k}(ξ, R)μ_{f}(ξ) = δ_{k}^{(r)}f ∈ 𝒪[[q^{B_{m}}]][R_{i,j}]_{i,j=1,···,m} .  (4.6)

In this case 𝒫 = {P_{r,k}}, k(P_{r,k}) = k + 2r, N(P_{r,k}) = N, r(P_{r,k}) = r.

Proof of Theorem 4.4: Let δ_{ξ} ∈ Distr (A_{m,p}, 𝒪) = 𝒪[[A_{m,p}]] denote the Dirac distribution on A_{m,p} concentrated in ξ. Then we define

μ_{f} ∈ Distr (A_{m,p}, 𝒪[[q^{B_{m}}]][R_{i,j}]_{i,j=1,···,m})

as the following formal series

μ_{f} = ∑_{ξ∈B_{m}} c(ξ)δ_{ξ}q^{ξ}.

Let us compute the integral

∫_{A_{m,p}} P_{r,k}(ξ, R)μ_{f}(ξ) = ∫_{A_{m,p}} ∑_{ξ∈B_{m}} c(ξ)P_{r,k}(η, R)q^{ξ}δ_{ξ}(η) = ∑_{ξ∈B_{m}} c(ξ)P_{r,k}(ξ, R))q^{ξ}. (4.7)

If we take into account (4.4) and the definition (4.5) then the equality (4.6) follows from (4.7).

§5. Families of distributions attached to Siegel-Eisenstein series.

Recall that in the elliptic modular case m = 1, for a Dirichlet character χ mod N, and k > 2, χ(−1)^{k}, the following Eisenstein series is defined

G_{k}(z, χ) = G_{k}^{(1)}(z, χ) = \frac{L(1 − k, χ)}{2} + ∑_{n=1}^{∞} σ_{k−1,χ}(n)q^{n} ∈ ℳ_{k}(Γ_{0}(N), χ, ℚ(χ)) .

If χ is primitive, then

G_{k}(z, χ) = \frac{L(1 − k, χ)}{2} ∑_{γ∈P ∖Γ_{0}(N)} \overline{χ(d)}(cz + d)^{−k}, γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ_{0}(N) .

Let N > 1 then the action of the involution τ_{N} = \binom{0 −1}{N 0} brings G_{k}(z, χ) to a similar series but without a constant term:

G′_{k}(z, χ) = ∑_{n=1}^{∞} σ′_{k−1,χ̄}(n)q^{n} ∈ ℳ_{k}(Γ_{0}(N), χ̄, ℚ(χ̄)) ,

where

σ′_{k−1,χ̄}(n) = ∑_{d|n} χ̄(d) (\frac{n}{d})^{k−1} = n^{k−1} ∏_{p|n}( ∑_{i=1}^{ord_{p}n} χ(p)^{−i}p^{−i(k−1)})

is a finite Euler product.

In the Siegel modular case Γ^{m} = Sp_{2m}(ℤ) ⊃ Γ_{0}^{m}(N), and there is an analogue

E_{k}(z, χ) = ∑_{\binom{a b}{c d}∈P^{(m)}∩Γ_{0}^{(m)}(N)∖Γ_{0}^{(m)}(N)} χ(det d)det(cz + d)^{−k}

of G_{k}(z, χ). However, for N > 1, the Fourier expansion is known only for the involuted series E_{k}|τ_{N}, where τ_{N} = \binom{0_{m} −1_{m}}{N·1_{m} 0_{m}} (for N = 1 both series coincide). Here z ∈ ℍ_{m} is in the Siegel upper half plane:

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

is the Siegel parabolic subgroup.

Example. Involuted Siegel-Eisenstein series: let k > m+1, and let χ be a Dirichlet character modulo N. Let us define

E = E(k, χ) = ∑_{γ∈P ∖Γ_{0}^{m}(N)} χ(det d)det(cz + d)^{−k} ∈ ℳ_{k}(Γ_{0}(N), χ̄) ,  (5.1)

where γ = \binom{a b}{c d} ∈ Γ_{0}^{m}(N).

Let us also define

G^{+}(k, χ) = L^{+}(k, χ)E(k, χ)|τ_{N}, τ_{N} = ( \begin{matrix}0_{m} & −1_{m} \\ N · 1_{m} & 0_{m} \end{matrix} )  (5.2)
E|τ_{N} = N^{km/2}(det Nz)^{−k}E(−(Nz)^{−1}) ,
L^{+}(k, χ) = Γ(k, m)L_{N}(k, χ) ∏_{i=1}^{[m/2]} L_{N}(2k − 2i, χ^{2}) ,
Γ(k, m) = i^{mk}2^{−m(k+1)}π^{−mk}Γ_{m}(k), Γ_{m}(s) = π^{m(m−1)/4} ∏_{j=0}^{m−1} Γ(s − (j(2))) .

and recall that κ = (m + 1)/2.

Theorem 5.1 (Siegel, Shimura [Sh83], P. Feit [Fe]). If m is odd, k > m + 1, N > 1, then

G^{+}(k, χ) = ∑_{0<ξ∈B_{m}} b^{+}(ξ, k, χ)q^{ξ}, b^{+}(ξ, k, χ) = det ξ^{k−κ}M(ξ, k, χ) ,  (5.3)

where M(ξ, k, χ) = ∏_{b|det(2ξ)} M_{q}(ξ, χ(ℓ)ℓ^{−k}), M_{ℓ}(ξ, x) ∈ ℤ[x] (a finite Euler product, e.g.

for m = 1 M_{ℓ}(n, x) = ∑_{i=0}^{ord_{ℓ}n} x^{i}).

In order to construct a family, put G_{p}^{+}(k, χ) =^{det} ∑_{{ξ>0 \atop p∤det(2ξ)}} b^{+}(ξ, k, χ)q^{ξ}.

Theorem 5.2. see [PaSE] Let χ mod Np be a Dirichlet character, 𝒪 ⊃ ℤ_{p}[x] then there exists E_{λ_{𝒪}} (χ, i) ∈ Λ_{𝒪} [[q^{B_{m}}]] such that for all k ≫ 0, k ≡ i mod (p − 1)

E_{Λ_{𝒪}} (χ, i) _{T =(1+p)^{k}−1} = G_{p}^{+}(k, χ^{−1}ω^{k})  (5.4)

E_{Λ}(χ) is called the Λ-adic Siegel-Eisenstein series.

Proof: For any δ ∈ Δ = ℤ_{p}^{∗tors} let us introduce the notation

G_{p,δ}^{+}(k, χ) = ∑_{{ξ>0 \atop detξ≡δ mod p}} b^{+}(ξ, k, χ)q^{ξ}  (5.5)

then

G_{p}^{+}(k, χ^{−1}ω^{k})q^{ξ} = ∑_{δ∈Δ} ∑_{{ξ>0 \atop detξ≡δ mod p}} detξ^{k−((m+1)/2)} · M(ξ, k, χ^{−1}ω^{k})q^{ξ}

= ∑_{δ∈Δ} ∑_{{ξ>0 \atop detξ≡δ(mod p)}}(detξ)^{−(m+1)/2}

· ⟨detξ⟩^{k} ∏_{ℓ|det(2ξ)} M_{ℓ}(ξ, χ^{−1}ω^{k}(ℓ)ℓ^{−k})q^{ξ}

(5.6)

Note that ω^{k}(ℓ)ℓ^{−k} = ⟨ℓ⟩^{−k} hence the coefficient of q^{ξ} in (5.6) is a finite linear combination

∑_{j} α_{j}β_{j}^{k} where α_{j} ∈ 𝒪, β_{j} = (1 + p)^{γ_{j}} ∈ Γ.

It follows that we can define

E_{Λ_{𝒪}} (χ, i) = ∑_{δ∈Δ} δ^{i} ∑_{{ξ>0 \atop detξ≡δ mod p}} ∑_{j} α_{j}(1 + T)^{γ_{j}} · q^{ξ}  (5.7)

because if we substitute T = (1 + p)^{k} − 1 to (5.7) we obtain

∑_{δ∈Δ} δ^{i} ∑_{{ξ>0 \atop detξ≡δ mod p}} ∑_{j} α_{j}(1 + p)^{γ_{j}k} · q^{ξ} = G_{p}^{+}(k, χ^{−1}ω^{k}) .

Theorem 5.3. Let A_{m} = {ξ = ^{t}ξ|ξ half-integral} be the lattice in M_{m}(ℚ) dual to {η ∈ M_{m}(ℤ)|^{t}η = η} with respect to the pairing (ξ, η) ↦ tr(ξη). Put A_{m,p} = A_{m} ⊗_{ℤ} ℤ_{p} then there exists a p-adic measure μ_{Siegel−Eisenstein} = μ_{SE} on the profinite group Y = A_{m,p} × ℤ_{p}^{∗} with values in 𝒪[[q^{B_{m}}]] such that for each k > m + 1 and for each Dirichlet character χ: ℤ_{p}^{∗} → 𝒪^{∗}

∫_{A_{m,p}×ℤ_{p}^{∗}} detξ^{k−((m+1)/2)}χ(x)x^{k}μ_{SE}(ξ, x) = G_{p}^{+}(k, χ^{−1}) .  (5.8)

Proof: see [PaSE].

Remarks. a) For any k we can specialize E_{Λ_{𝒪}} (χ, i) to T = (1 + p)^{k} − 1:

E_{Λ_{𝒪}} (χ, i) _{T =(1+p)^{k}−1} = ∑_{δ∈Δ} δ^{i−k} ∑_{{ξ>0 \atop detξ≡δ}} (detξ)^{k−((m+1)/2)}M(ξ, k, χ^{−1}ω^{k})q^{ξ}
= G_{p}^{+}(k, χ^{−1}ω^{k})_{ω^{i−k}} ,

where the subscript ω^{i−k} denotes the twist of G_{p}^{+}(k, χ^{−1}ω^{k}) ∈ ℳ_{k}^{m}(Γ_{0}(N_{p}), χω^{−k}) with the Dirichlet character ω^{i−k}.

b) Theorem 5.2 and Theorem 5.3 can be extended also to even m, but in this case the formula for b^{+}(ξ, k, χ) contains also the factor L(k − (m/2), χω_{ξ}), where ω_{ξ} is a quadratic character associated to ξ,ω_{ξ}(d) = (\frac{(−1)^{m/2}det(2ξ)}{d}). Note that we need to assume χ(−1) = (−1)^{k}, and this additional factor is interpreted by the Kubota-Leopoldt theorem, see §1.

In order to obtain more general families of nearly holomorphic Eisenstein series one can apply to (5.8) the arithmetical differential operator δ_{k}^{(r)}. Let us use again the family of polynomials (4.5).

Theorem 5.4. Put A_{m,p} = A_{m} ⊗_{ℤ} ℤ_{p} then there exists a p-adic measure μ_{SE}^{(n.h)} on the profinite group Y = A_{m,p} × ℤ_{p}^{∗} with values in 𝒪[[q^{B_{m}}]][R_{i,j}]_{i,j=1,···,m} such that for each k > m + 1 and for each Dirichlet character χ: ℤ_{p}^{∗} → 𝒪^{∗}

∫_{A_{m,p}×ℤ_{p}^{∗}} detξ^{k−κ}χ(x)x^{k}P_{r,k}(ξ, R)dμ_{SE}^{(n.h)}(ξ, x) = δ_{k}^{(r)}(G_{p}^{+}(k, χ^{−1})) .  (5.9)

In this case

𝒫 = {Q_{r,k}(ξ, R, x) = detξ^{k−κ}χ(x)x^{k}P_{r,k}(ξ, R)},

k(Q_{r,k}) = k + 2r, N(Q_{r,k}) = N, r(Q_{r,k}) = r.

Proof of Theorem 5.4: is very similar to that of Theorem 4.4, but it can be also easily deduced from the abstract Kummer congruences (Theorem 4.1) applied to the family 𝒫.

§6. Families of distributions attached to theta functions with spherical polynomials.

In this section we describe certain p-adic families of Siegel modular forms whose parameter space is Y = X_{p} = X ⊗ ℤ_{p} where X = ℤ^{mn} is a lattice. Let A ∈ M_{m×m}(ℚ) be a positive definite half-integral symmetric matrix which determines a quadratic form

Q_{A}(x) = ^{t}xAx =^{def}A[x] for x ∈ X.

Theta-distribution attached to A is an element θ_{A} ∈ Distr (Y, 𝒪[[B_{n}]]) defined by

θ_{A}(U) = ∑_{x∈U∩X} q^{A[x]} U a compact open subset of X_{p}  (6.1)

In other words, θ_{A}(U) is a partial theta-series given by congruence conditions defining U, (see also [Hi85] where the elliptic modular case was treated).

Let Λ_{X} = Distr (X_{p}, 𝒪) = Λ_{Y,𝒪} be the generalized Iwasawa algebra (which is isomorphic to 𝒪[[T_{11}, …, T_{mn}]]: for every μ ∈ Distr (X_{p}, 𝒪) the corresponding power series is given by the Amice transform

A_{μ}(T_{11}, …, T_{mn}) = ∫_{ℤ_{p}^{mn}} (1 + T_{11})^{x_{11}} ⋯ (1 + T_{mn})^{x_{mn}}μ(x),  (6.2)

where (1 + T)^{x} = ∑_{j ≥ 0} \binom{x}{j}T^{j}, \binom{x}{j} = \frac{x(x − 1) ⋯ (x − j + 1)}{j!}.

One observes directly from definition (6.1) that

θ_{A} = ∑_{x∈ℤ^{mn}} δ_{x} · q^{A[x]} ∈ Λ_{X}[[q^{B_{n}}]] .  (6.3)

Using the Amice transform, we then rewrite (6.3) in the form of a power series

\begin{matrix}θ_{A}(T_{11}, …, T_{mn}; q) = ∫_{ℤ_{[}^{mn}} (1 + T_{11})^{x_{11}} ⋯ (1 + T_{mn})^{x_{mn}}dθ_{A}(x) = \\ ∑_{k_{11},...,k_{mn}=0}^{∞} T_{11}^{k_{11}} ⋯ T_{mn}^{k_{mn}} ∫_{ℤ_{p}^{mn}} \binom{x_{11}}{k_{11}} ⋯ \binom{x_{mn}}{k_{mn}}dθ_{A}(x) = \\ ∑_{ξ∈B_{n}} ∑_{k_{11},...,k_{mn}=0}^{∞} ∑_{{x∈ℤ^{mn} \atop A[x]=ξ}} T_{11}^{k_{11}} ⋯ T_{mn}^{k_{mn}} \binom{x_{11}}{k_{11}} ⋯ \binom{x_{mn}}{k_{mn}}q^{ξ} = \\ ∑_{x∈ℤ^{mn}}(1 + T_{11})^{x_{11}} ⋯ (1 + T_{mn})^{x_{mn}}q^{A[x]}. \end{matrix}  (6.4)

Theorem 6.1. Let us consider the family 𝒫 = {P: X → ℤ} of homogeneous A-spherical polynomials which take integral values on the lattice X = ℤ^{mn}. Then we may view P as a ℤ_{p}-valued continuous function on X_{p} = ℤ_{p}^{mn} and for each such P

∫_{X_{p}} P dθ_{A}(x) = θ_{A,P} = ∑_{x∈ℤ} P (x)q^{A[x]}  (6.5)

is a Siegel modular form of degree n and of weight \frac{m}{2} + deg P (we assume that m is even).

Proof of Theorem 6.1: is very similar to that of Theorem 4.4, but it can be also easily deduced from the abstract Kummer congruences (Theorem 4.1) applied to the family 𝒫 = {P }.

In order to obtain more general families of nearly holomorphic Eisenstein series one can apply to (5.8) the arithmetical differential operator δ_{k}^{(r)}. Let us use again the family of polynomials (4.5).

Theorem 6.2. Let us consider the family 𝒫 = {P: X → ℤ} of homogeneous A-spherical polynomials which take integral values on the lattice X = ℤ^{mn}. Then we may view P as a ℤ_{p}-valued continuous function on X_{p} = ℤ_{p}^{mn}. There exists a p-adic measure θ_{A}^{(n.h)} on X_{p} with values in 𝒪[[q^{B_{n}}]][R_{i,j}]_{i,j=1,···,n} such that for each P

∫_{X_{p}} P (x) · P_{r,k_{m,P}} (A[x], R)dθ_{A}^{(n.h)}(x) = δ_{k_{m,P}}^{(r)} (θ_{A,P})  (6.6)

(where k_{m,P} = \frac{m}{2} + deg P) is a Siegel modular form of degree n and of weight k_{m,P} + 2r = \frac{m}{2} + deg P + 2r (we assume that m is even). In this case

𝒫 = {H = P (x)P_{r,k_{m,P}} (A[x], R)}.

k(H) = k_{m,P} + 2r, N(H) = N, r(H) = r.

Proof of Theorem 6.2: is deduced from the equality

δ_{k_{m,P}}^{(r)} (θ_{A,P}) = ∑_{x∈ℤ^{mn}} P (x)P_{r,k_{m,P}} (A[x], R)q^{A[x]}  (6.7)

and very similar to that of Theorem 4.4. It can be also easily deduced from the abstract Kummer congruences (Theorem 4.1) applied to the family 𝒫.

The modular forms (6.5) can be obtained as certain specialization of the power series (6.4), and we obtain a p-adic family of theta series with spherical polynomials.

Note that

\begin{aligned}θ_{A,P} & = ∑_{x∈ℤ^{mn}} P (x)(1 + T_{11})^{x_{11}} ⋯ (1 + T_{mn})^{x_{mn}}q^{A[x]} _{T_{11},...,T_{mn}=0} \\ & = ∂_{P} θ_{A}(T_{11}, …, T_{mn}; q) _{T_{11},...,T_{mn}=0} \end{aligned}  (6.6)

where

∂_{P} = ((1 + T_{11})\frac{∂}{∂T_{11}} , …, (1 + T_{mn})\frac{∂}{∂T_{mn}} )

is a differential operator such that

∂_{P} (1 + T_{11})^{x_{11}} ⋯ (1 + T_{mn})^{x_{mn}} = P (x)(1 + T_{11})^{x_{11}} ⋯ (1 + T_{mn})^{x_{mn}}  (6.7)

The map F ↦ ∂_{P} F _{T_{11},...,T_{mn}=0} defines the following homomorphisms κ_{P}: Λ_{X} → ℤ_{p}, and

κ_{P}: Λ_{X}[[q^{B_{n}}]] → ℤ_{p}[[q^{B_{n}}]].

Let us consider the Laplace transform

L_{μ}(T_{11} …, T_{mn}) = ∑_{k_{11},...,k_{mn}=0}^{∞} \frac{T_{11}^{k_{11}} ⋯ T_{mn}^{k_{mn}}}{k_{11}! ⋯ k_{mn}!} ∫_{ℤ_{p}^{mn}} x_{11}^{k_{11}} ⋯ x_{mn}^{k_{mn}}dμ(x)

of a p-adic measure μ ∈ Distr (ℤ_{p}^{mn}, 𝒪) with values in a complete ring 𝒪. Then

\begin{aligned}L_{μ}(T_{11}, …, T_{mn}) & = A_{μ}(e^{T_{P 11}} − 1, …, e^{T_{m,n}} − 1), \\ L_{μ}(T_{11}, …, T_{mn}) & = ∫_{ℤ_{p}^{mn}} e^{T_{11}x_{11}+···+T_{mn}x_{mn}}dμ(x) \end{aligned}  (6.8)

Note that the coefficients of the power series (6.8) may contain denominators (k_{11}! ⋯ k_{mn}!) so that

L_{μ}(T_{11}, …, T_{mn}) ∈ K_{𝒪} [[T_{11}, …, T_{mn}]]

where K_{𝒪} = Quot 𝒪.

If

P = ∑_{k_{11},...,k_{mn}} b_{k_{11},...,k_{mn}}x_{11}^{k_{11}} ⋯ x_{mn}^{k_{mn}}

is the spherical polynomial in (6.5), then the integral

∫_{ℤ_{p}^{mn}} P dθ_{A}(x) = ∑_{k_{11},...,k_{mn}} b_{k_{11},...,k_{mn}} ∫_{ℤ_{p}^{mn}} x_{11}^{k_{11}} ⋯ x_{mn}^{k_{mn}}dθ_{A}

is related to the Laplace transform by a more simple formula than (6.6), namely

\begin{aligned}L_{θ_{A}}(T_{11}, …, T_{mn}) & = ∑_{x} e^{T_{11}x_{11}+···+T_{mn}x_{mn}}q^{A[x]}, \\ θ_{A,P} & = P (\frac{∂}{∂T_{11}} , …, \frac{∂}{∂T_{mn}} ) L_{θ} _{T_{11}=...=T_{mn}=0} \end{aligned}  (6.9)

A nice arithmetical example of a p-adic family of Hecke cusp eigenforms which are theta functions with spherical polynomials come from Hecke grössencharacters of an imaginary quadratic field K = ℚ (\sqrt{−D} ).

Recall that such a χ of conductor (dividing) a given integral ideal 𝔪 of K is by definition a homomorphism

{prime-to-𝔪 fractional ideals of K} → ℚ̄^{×}

such that for appropriate integers n = n(λ)

χ((α)) = ∏_{λ} λ(α))^{n(λ)} = σ(α)^{n(σ)}\overline{σ}(α)^{n(\overline{σ})} if α ≡ 1 mod^{×}𝔪

where λ: K ↪ ℚ̄, λ = σ, \overline{σ}, σ: K ↪ ℚ̄ is a fixed embedding, and \overline{σ} is its composition with the nontrivial Galois automorphism of K.

In order to define L(s, χ) one uses a fixed complex embedding of ℚ̄. If n(σ) > 0 and n(\overline{σ}) = 0 then there exists an elliptic cups eigenform f_{χ} ∈ S_{k}(Γ_{0}(N(𝔪)), ψ) of weight k = n(σ) + 1 with a Dirichlet character ψ (depending on χ) such that L(s, χ) = L(s, f_{χ}) (the Mellin transform of f_{χ}). Moreover f_{χ} is a linear combination of binary theta-series with spherical polynomials defined by the values of the grössencharacters χ (see [Bl86]).

The function χ ↦ f_{χ} provides us with a p-adic family of Hecke cusp eigenforms (see [Hi85], [Ka78],).

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