A semi-ordinary $p$-stabilization of Siegel Eisenstein series for symplectic groups and its $p$-adic interpolation (updated on 2023/02/23)

TL;DR

This paper constructs a $p$-stabilization of Siegel Eisenstein series for symplectic groups, derives explicit Fourier coefficient formulas, and establishes their $p$-adic interpolation, generalizing classical $ ext{GL}(2)$ results to higher genus cases.

Contribution

It introduces a new $p$-stabilization method for Siegel Eisenstein series and proves their $p$-adic interpolation, extending the theory to symplectic groups of arbitrary genus.

Findings

Explicit Fourier coefficient formulas for $p$-stabilized series

Existence of a $ ext{Lambda}$-adic form interpolating families of Eisenstein series

Generalization of ordinary $ ext{Lambda}$-adic Eisenstein series to higher genus

Abstract

For any rational prime $p$, we define a certain $p$-stabilization of holomorphic Siegel Eisenstein series for the symplectic group $\text{Sp}(2n)_{/\mathbb{Q}}$ of an arbitrary genus $n \ge 1$. In addition, we derive an explicit formula for the Fourier coefficients and conclude their $p$-adic interpolation problems. Consequently, for any odd prime $p$, we deduce the existence of a $\Lambda$-adic form (in the sense of A. Wiles, H. Hida and R.L. Taylor) such that after taking a suitable constant multiple, it interpolates $p$-adic analytic families of the above-mentioned $p$-stabilized Siegel Eisenstein series with nebentypus characters locally trivial at $p$ and Siegel Eisenstein series with nebentypus characters locally non-trivial at $p$ simultaneously. This can be viewed as a quite natural generalization of the ordinary $\Lambda$-adic Eisenstein series for $\text{GL}(2)_{/\mathbb{Q}}$.

Equations

$$E_{\kappa ,\chi }(z):={\left\{\frac{2G({\chi }^{-1})(-2\pi \sqrt{-1}{)}^{\kappa }}{M^{\kappa }(\kappa -1)!}\right\}}^{-1}\sum _{(c,d)\in {\mathbb{Z}}^2\setminus \{(0,0)\}}{\chi }^{-1}(d)(cMz+d{)}^{-\kappa },$$

$$E_{\kappa ,\chi }(z)=\frac{L(1-\kappa ,\chi )}{2}+\sum _{m=1}^{\infty }{\sigma }_{\kappa -1,\chi }(m)q^m,$$

$$L(s,E_{\kappa ,\chi })=\zeta (s)L(s-\kappa +1,\chi ).$$

$$E_{\kappa ,\chi }^{\ast }(z):=E_{\kappa ,\chi }(z)-\chi (p)p^{\kappa -1}{E}_{\kappa ,\chi }(pz).$$

$$E_{\kappa ,\chi }^{\ast }(z)=\frac{L^{\{p\}}(1-\kappa ,\chi )}{2}+\sum _{m=1}^{\infty }{\sigma }_{\kappa -1,\chi }^{\{p\}}(m)q^m,$$

$${\sigma }_{\kappa -1,\chi }^{\{p\}}(m):=\sum _{\genfrac{}{}{0ex}{}{0<d\mid m,}{\gcd (d,p)=1}}\chi (d)d^{\kappa -1}.$$

$$L(s,E_{\kappa ,\chi }^{\ast })=\zeta (s)L^{\{p\}}(s-\kappa +1,\chi ).$$

$${\mathcal{E}}_{\chi {\omega }^a}(X)=\sum _{m=0}^{\infty }{\mathcal{A}}_{\chi {\omega }^a}(m;X)q^m\in F_{\Lambda }[[q]],$$

$${\mathcal{E}}_{\chi {\omega }^a}((1+p{)}^{\kappa }-1)=E_{\kappa ,\chi }^{\ast }.$$

$${\mathcal{E}}_{\chi {\omega }^a}(\epsilon (1+p)(1+p{)}^{\kappa }-1)=E_{\kappa ,\chi {\omega }^{a-\kappa }\epsilon }$$

$$E_{\kappa ,\chi }^{\ast }=E_{\kappa ,\chi }{\Vert }_{\kappa }(U_p-\chi (p)p^{\kappa -1}).$$

$$\vspace*{-3mm}\begin{array}[]{lcl}\hskip 2.84526pt\mathbb{C}&\underset{\simeq}{\stackrel{{\scriptstyle\widehat{\iota}_{p}}}{{\to}}}&\hskip 2.84526pt\mathbb{C}_{p}\\ \hskip 5.69054pt\rotatebox{90.0}{$\hookrightarrow$}&&\hskip 5.69054pt\rotatebox{90.0}{$\hookrightarrow$}\\ \mathbb{Q}^{\rm alg}&\underset{\iota_{p}}{\hookrightarrow}&\mathbb{Q}_{p}^{\rm alg}\end{array}\vspace*{2mm}$$

$${\bf p}=\left\{\begin{array}[]{cl}4&\text{ if $p=2$},\\ p&\text{ otherwise}.\end{array}\right.$$

$$\begin{array}[]{ccc}\mathbb{Z}_{p}^{\times}&\stackrel{{\scriptstyle\sim}}{{\to}}&\mathbb{Z}_{p,{\rm tor}}^{\times}\times(1+{\bf p}\mathbb{Z}_{p})\\[2.84526pt] x&\mapsto&(\omega(x),\,\langle x\rangle).\end{array}\vspace*{-1mm}$$

$$1+\mathbf{p}{\mathbb{Z}}_p/(1+\mathbf{p}{\mathbb{Z}}_p{)}^{p^m}\simeq (1+\mathbf{p}{)}^{{\mathbb{Z}}_p}/(1+\mathbf{p}{)}^{p^m{\mathbb{Z}}_p}\simeq {\mathbb{Z}}_p/p^m{\mathbb{Z}}_p\simeq \mathbb{Z}/p^m\mathbb{Z}.$$

$$\mathrm{GSp}(2n):=\left\{g\in \mathrm{GL}(2n)\right|{}^{t}gJg=\nu (g)J\text{ for some }\nu (g)\in {\mathbb{G}}_m\},$$

$$1\to \mathrm{Sp}(2n)\to \mathrm{GSp}(2n)\stackrel{\nu }{\to }{\mathbb{G}}_m\to 1,$$

$${\mathfrak{H}}_n:=\left\{Z=X+\sqrt{-1}Y\in {\mathrm{Mat}}_{n\times n}(\mathbb{C})|{}^{t}Z=Z,Y>0(\text{positive-definite})\right\}$$

$$(F{\Vert }_{\kappa }g)(Z):=\nu (g{)}^{n\kappa -n(n+1)/2}\det (CZ+D{)}^{-\kappa }F(g⟨Z⟩).$$

$$\Gamma_{0}(N)^{(n)}\,({\rm resp.}\ \Gamma_{1}(N)^{(n)}):=\left\{\gamma\in{\rm Sp}(2n,\mathbb{Z})\left|\,\,\gamma\equiv\left[\begin{array}[]{cc}*&*\\ 0_{n}&*\end{array}\right]\left({\rm resp.}\left[\begin{array}[]{cc}*&*\\ 0_{n}&1_{n}\end{array}\right]\right)\!\!\!\pmod{N}\right.\right\}.$$

$$(F{\Vert }_{\kappa }\gamma )(Z)=\sum _{T\in {\mathrm{Sym}}_n^{\ast }(\mathbb{Z})}{A}_{F,\gamma }(T)e(\mathrm{tr}(TZ)),$$

$${\mathrm{Sym}}_n^{\ast }(\mathbb{Z}):=\{T=\left[t_{ij}\right]\in {\mathrm{Sym}}_n(\mathbb{Q})|t_{ii},2t_{ij}\in \mathbb{Z}(1\le i<j\le n)\},$$

$$F{\Vert }_{\kappa }\gamma =\chi (\det D)F\text{ for any }\gamma =\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]\in {\Gamma }_0(N{)}^{(n)}.$$

$${\mathbf{q}}^T:=e(\mathrm{tr}(TZ))=\prod _{i=1}^n{q}_{ii}^{t_{ii}}\prod _{i<j\le n}{q}_{ij}^{2t_{ij}},$$

$$F(Z)=\sum _{\genfrac{}{}{0ex}{}{T\in {\mathrm{Sym}}_n^{\ast }(\mathbb{Z}),}{T\ge 0}}{A}_F(T){\mathbf{q}}^T,$$

$$R[[\mathbf{q}]{]}^{(n)}:=R[q_{ij}^{\pm 1}|1\le i<j\le n][[q_{11},\cdots ,q_{nn}]],$$

$$E_{\kappa ,\chi }^{(n)}(Z)$$

$$b_l(S;X):=\sum _{R\in {\mathrm{Sym}}_r({\mathbb{Q}}_l)/{\mathrm{Sym}}_r({\mathbb{Z}}_l)}{e}_l(\mathrm{tr}(SR))X^{{\mathrm{val}}_l({\mu }_R)},$$

$$(-1{)}^{r/2}{\mathfrak{D}}_S={\mathfrak{d}}_S{\mathfrak{f}}_S^2,$$

$$\xi_{l}(x)=\left\{\begin{array}[]{cl}1&\text{if\, }{\mathbb{Q}_{l}(x^{1/2})}={\mathbb{Q}_{l}},\\[2.13394pt] -1&\text{if\, }{\mathbb{Q}_{l}(x^{1/2})}/{\mathbb{Q}_{l}}\text{ is unramified},\\[2.13394pt] 0&\text{if\, }{\mathbb{Q}_{l}(x^{1/2})}/{\mathbb{Q}_{l}}\text{ is ramified}.\end{array}\right.$$

Full text

1. Introduction

Given a positive integer $M$, a Dirichlet character $\chi$ modulo $M$ and an integer $\kappa\geq 2$ with $\chi(-1)=(-1)^{\kappa}$, the classical (holomorphic) Eisenstein series E_{\kappa,\,\chi}\bigl{(}=E_{\kappa,\,\chi}^{(1)}(z)\bigr{)} is defined as follows: For each $z\in\mathfrak{H}_{1}=\{z\in\mathbb{C}\mid{\rm Im}\,z>0\}$, put

[TABLE]

where $G(\chi^{-1})=\sum_{n=1}^{M}\chi^{-1}(n)\exp(2\pi\sqrt{-1}\,n/M)$. As is well-known, $E_{\kappa,\,\chi}$ gives rise to a holomorphic modular form of weight $\kappa$ and nebentypus character $\chi$ for the congruence subgroup $\Gamma_{0}(M)$ of ${\rm SL}(2,\mathbb{Z})$ unless $\kappa=2$ and $\chi$ is trivial (or principal). If $\chi$ is trivial (i.e., $M=1$) or primitive (i.e., the conductor of $\chi$ equals $M>1$), then $E_{\kappa,\,\chi}$ possesses the Fourier expansion

[TABLE]

where $L(s,\chi)=\prod_{\scriptstyle l\,{\rm:\,prime}}(1-\chi(l)\,l^{-s})^{-1}$, $\sigma_{\kappa-1,\,\chi}(m)=\sum_{\scriptstyle 0<d\,\mid\,m}\chi(d)\,d^{\kappa-1}$ and $q=\exp(2\pi\sqrt{-1}z)$. This implies that $E_{\kappa,\,\chi}$ is a non-cuspidal Hecke eigenform such that the associated $L$-function $L(s,E_{\kappa,\,\chi})$ is taken of the form

[TABLE]

Let $p$ be a fixed prime number not dividing $M$, which we assume to be odd for simplicity. Put

[TABLE]

We easily see that $E_{\kappa,\,\chi}^{*}$ is also a non-cuspidal Hecke eigenform of weight $\kappa$ and nebentypus character $\chi$ for $\Gamma_{0}(Mp)\,(\subset\Gamma_{0}(M))$, and its Fourier expansion is taken of the form

[TABLE]

where $L^{\{p\}}(s,\chi):=(1-\chi(p)\,p^{-s})\,L(s,\chi)=\prod_{l\neq p}(1-\chi(l)\,l^{-s})^{-1}$ and

[TABLE]

It turns out that for $E_{\kappa,\,\chi}^{*}$, all the Hecke eigenvalues outside $p$ agree with $E_{\kappa,\,\chi}$, but the eigenvalue of Atkin’s $U_{p}$-operator, which is converted from the $p$-th Hecke operator $T_{p}$ (cf. [AL, Mi]), is $\sigma_{\kappa-1,\,\chi}^{\{p\}}(p)=1$. Namely, we have

[TABLE]

This type of normalization $E_{\kappa,\,\chi}\mapsto E_{\kappa,\,\chi}^{*}$ given by eliminating the $p$-part of every Fourier coefficient or equivalently, eliminating the latter half of the Euler factor at $p$ was firstly introduced by Serre [Se], and we refer to it as the ordinary111In general, a Hecke eigenform $f$ is said to be ordinary at $p$ if the eigenvalue of $U_{p}$ (or $T_{p}$) is a $p$-adic unit. $p$-stabilization. Here we should mention that every Fourier coefficient of $E_{\kappa,\,\chi}^{*}$ depends on the weight $\kappa$ $p$-adically. Indeed, it follows immediately from Fermat’s little theorem that for each positive integer $m$, the function $\kappa\mapsto\sigma_{\kappa-1,\,\chi}^{\{p\}}(m)$ can be extended to an analytic function defined on the ring of $p$-adic integers $\mathbb{Z}_{p}$. In addition, the constant term $L^{\{p\}}(1-\kappa,\chi)/2$ is also interpolated by the $p$-adic $L$-function in the sense of Kubota-Leopoldt or Deligne-Ribet [DR]. This fact turns out to be the following theorem due to Hida and Wiles:

Fact 1.1** (cf. Proposition 7.1.1 in [H1]; Proposition 1.3.1 in [W]).**

Suppose that a Dirichlet character $\chi$ modulo $M$ is either trivial or primitive, and $M$ is not divisible by $p$. Let $\Lambda=\mathbb{Z}_{p}[\chi][[X]]$ be the one-variable power series ring over $\mathbb{Z}_{p}[\chi]$ and $F_{\Lambda}$ the field of fractions of $\Lambda$, respectively. For each integer $a$ with $0\leq a<p-1$, there exists a formal Fourier expansion

[TABLE]

where $\omega:\mathbb{Z}_{p}^{\times}\twoheadrightarrow\mu_{p-1}=\{x\in\mathbb{Z}_{p}^{\times}\mid x^{p-1}=1\}$222Since $\mu_{p-1}\simeq(\mathbb{Z}/p\mathbb{Z})^{\times}$, $\omega$ can be identified with a Dirichlet character modulo $p$ in a natural way. denotes the Teichmüller character, such that for each integer $\kappa>2$ with $\chi(-1)=(-1)^{\kappa}$ and $\kappa\equiv a\pmod{p-1}$333The latter condition is regarded as if $\omega^{a-\kappa}$ corresponds to the trivial Dirichlet character. , we have

[TABLE]

Moreover, if $\varepsilon:1+p\mathbb{Z}_{p}\to(\mathbb{Q}_{p}^{\rm alg})^{\times}$ is a character of exact order $p^{r}$ for some integer $r\geq 0$444Namely, $\varepsilon$ can be also regarded as a primitive Dirichlet character of conductor $p^{r+1}$. , then for each integer $\kappa\geq 2$, we have

[TABLE]

*as long as $\omega^{a-\kappa}\varepsilon$ is non-trivial. *

Strictly speaking, it turns out that $\mathcal{E}_{\chi\omega^{a}}(X)\in\Lambda[[q]]$ unless $\chi\omega^{a}$ is trivial. However, we note that if $\chi\omega^{a}$ is trivial, then $\mathcal{E}_{\chi\omega^{a}}(X)$ (more precisely, the constant term $\mathcal{A}_{\chi\omega^{a}}(0;\,X)$) has a simple pole at $X=0$ and thus $\overline{\mathcal{E}}_{\chi\omega^{a}}(X):=X\cdot\mathcal{E}_{\chi\omega^{a}}(X)\in\Lambda[[q]]$. Hence the above-mentioned fact implies that $\mathcal{E}_{\chi\omega^{a}}(X)$ or $\overline{\mathcal{E}}_{\chi\omega^{a}}(X)$ according as $\chi\omega^{a}$ is non-trivial or trivial, is a $\Lambda$-adic form of level $Mp^{\infty}$ associated with character $\chi\omega^{a}$, which interpolates families of non-cuspidal ordinary Hecke eigenforms $\{E_{\kappa,\,\chi}^{*}\}$ and $\{E_{\kappa,\,\chi\omega^{a-\kappa}\varepsilon}\}$ or their constant multiples, given by varying the weight $\kappa$ $p$-adically analytically (cf. [W, H1]). In this context, we refer to it as the ordinary $\Lambda$-adic Eisenstein series of genus 1 and level $Mp^{\infty}$ associated with character $\chi\omega^{a}$.

The aim of the present article is to formulate a similar statement in the case of Siegel modular forms, that is, automorphic forms on the symplectic group ${\rm Sp}(2n)_{/\mathbb{Q}}$ of an arbitrary genus $n\geq 1$. Let us explain how it goes briefly: Given a positive integer $\kappa>n+1$ and a Dirichlet character $\chi$ modulo a positive integer $M$, let $E_{\kappa,\,\chi}^{(n)}$ be the classical (holomorphic) Siegel Eisenstein series of weight $\kappa$ and nebentypus character $\chi$ for $\Gamma_{0}(M)^{(n)}\subset{\rm Sp}(2n,\mathbb{Z})$, to be described in the subsequent §2, whenever either

1. (i)

$M=1$, and thus, $\chi$ is trivial or 2. (ii)

$M>1$ is odd, $\chi$ is primitive and $\chi^{2}$ is locally non-trivial at every prime $l\mid M$555Namely, if we factor $\chi$ as $\chi=\prod_{l\,\mid\,M}\chi_{l}$, then for each prime factor $l$ of $M$, $\chi_{l}$ is not a quadratic character. .

For an arbitrary prime number $p\nmid M$, we define, in §4, a certain $p$-stabilization map $E_{\kappa,\,\chi}^{(n)}\mapsto(E_{\kappa,\,\chi}^{(n)})^{*}$ by means of the action of a linear combination of $(U_{p,n})^{i}$ ’s for $i=0,1,\cdots,n$, where $U_{p,n}$ denotes a generalized Atkin $U_{p}$-operator to be defined in §3 below, so that the associated eigenvalue of $U_{p,n}$ is $1$. This formulation is natural from the viewpoint of which in the case where $n=1$, $E_{\kappa,\,\chi}^{*}=(E_{\kappa,\,\chi}^{(1)})^{*}$ can be easily written in terms of Atkin’s original operator $U_{p}=U_{p,1}$ as

[TABLE]

Following in the tradition of Skinner-Urban [SU], in the case where $n>1$, we call $(E_{\kappa,\,\chi}^{(n)})^{*}$ semi-ordinary at $p$. Moreover, we derive an explicit formula for all Fourier coefficients of $(E_{\kappa,\,\chi}^{(n)})^{*}$, which can be regarded as a natural generalization of the equation (2) (cf. Theorem 4.2 below). As a consequence of the above issues, we first show in Theorem 4.4 below the existence of a formal Fourier expansion with coefficients in $F_{\Lambda}$ associated to $\chi\omega^{a}$ with some nonnegative integer $a$, whose specialization at $X=(1+p)^{\kappa}-1$ coincides with the semi-ordinary $p$-stabilized Siegel Eisenstein series $(E_{\kappa,\,\chi}^{(n)})^{*}$ for any $\kappa$ taken as above. Finally in §5, for a fixed odd prime number $p$, we show that a suitable constant multiplication of the above-mentioned formal Fourier expansion gives rise to a $\Lambda$-adic form of genus $n$ and level $Mp^{\infty}$ (or tame level $M$ in the sense of Taylor [Ta]) associated with character $\chi\omega^{a}$, which can be viewed as a satisfactory generalization of Fact 1.1 (cf. Theorem 5.4 below).

Notation*.*

We summarize here some notation we will use in the sequel. We denote by $\mathbb{Z},\,\mathbb{Q},\,\mathbb{R},$ and $\mathbb{C}$ the ring of integers, fields of rational numbers, real numbers and complex numbers, respectively. Let $\mathbb{Q}^{\rm alg}$ denote the algebraic closure of $\mathbb{Q}$ sitting inside $\mathbb{C}$. We put $e(x)=\exp(2\pi\sqrt{-1}x)$ for $x\in\mathbb{C}$. Given a prime number $p$, we denote by $\mathbb{Q}_{p}$, $\mathbb{Z}_{p}$ and $\mathbb{Z}_{p}^{\times}$ the field of $p$-adic numbers, the ring of $p$-adic integers and the group of $p$-adic units, respectively. Hereinafter, given a prime number $p$, we fix an algebraic closure $\mathbb{Q}_{p}^{\rm alg}$ of $\mathbb{Q}_{p}$ and an embedding $\iota_{p}:\mathbb{Q}^{\rm alg}\hookrightarrow\mathbb{Q}_{p}^{\rm alg}$ once for all. Let ${\rm val}_{p}$ denote the $p$-adic valuation on $\mathbb{Q}_{p}^{\rm alg}$ normalized so that ${\rm val}_{p}(p)=1$, and $|*|_{p}$ the corresponding norm on $\mathbb{Q}_{p}^{\rm alg}$, respectively. Let $e_{p}$ be the continuous additive character of $\mathbb{Q}_{p}^{\rm alg}$ such that $e_{p}(x)=e(x)$ for all $x\in\mathbb{Q}$. Let $\mathbb{C}_{p}$ be the completion of the normed space $(\mathbb{Q}_{p}^{\rm alg},|*|_{p})$. We also fix, once for all, an isomorphism $\widehat{\iota}_{p}:\mathbb{C}\stackrel{{\scriptstyle\sim}}{{\to}}\mathbb{C}_{p}$ such that the diagram

[TABLE]

is commutative. Given a prime number $p$, put

[TABLE]

Let $\mathbb{Z}_{p,{\rm tor}}^{\times}$ be the torsion subgroup of $\mathbb{Z}_{p}^{\times}$, that is, $\mathbb{Z}_{2,{\rm tor}}^{\times}=\{\pm 1\}$ and $\mathbb{Z}_{p,{\rm tor}}^{\times}=\mu_{p-1}$ if $p\neq 2$. We define the Teichmüller character $\omega:\mathbb{Z}_{p}^{\times}\twoheadrightarrow\mathbb{Z}_{p,{\rm tor}}^{\times}$ by putting $\omega(x)=\pm 1$ according as $x\equiv\pm 1\pmod{4\mathbb{Z}_{2}}$ if $p=2$, and $\omega(x)=\displaystyle\lim_{i\to\infty}x^{p^{i}}$ for $x\in\mathbb{Z}_{p}^{\times}$ if $p\neq 2$. In addition, for each $x\in\mathbb{Z}_{p}^{\times}$, put $\langle x\rangle:=\omega(x)^{-1}x\in 1+{\bf p}\mathbb{Z}_{p}$. Then we have a canonical isomorphism

[TABLE]

We note that the maximal torsion-free subgroup $1+{\bf p}\mathbb{Z}_{p}$ of $\mathbb{Z}_{p}^{\times}$ is topologically cyclic, that is, $1+{\bf p}\mathbb{Z}_{p}=(1+{\bf p})^{\mathbb{Z}_{p}}$. As already mentioned in §1, the Teichmüller character $\omega$ gives rise to a Dirichlet character $\omega:(\mathbb{Z}/{\bf p}\mathbb{Z})^{\times}\simeq\mathbb{Z}_{p,{\rm tor}}^{\times}\hookrightarrow\mathbb{C}_{p}^{\times}\simeq\mathbb{C}^{\times}$. Let $\varepsilon:1+{\bf p}\mathbb{Z}_{p}\to(\mathbb{Q}_{p}^{\rm alg})^{\times}$ be a character of finite order. More precisely, if $\varepsilon$ has exact order $p^{m}$ for some nonnegative integer $m$, then $\varepsilon$ optimally factors through

[TABLE]

Since $(\mathbb{Z}/p^{m}{\bf p}\mathbb{Z})^{\times}\simeq(\mathbb{Z}/{\bf p}\mathbb{Z})^{\times}\times\mathbb{Z}/p^{m}\mathbb{Z}$, we may naturally regard $\varepsilon$ as a Dirichlet character of conductor $p^{m}{\bf p}$. By abuse of notation, we will often identify such $p$-adic characters $\omega$ and $\varepsilon$ with the corresponding Dirichlet characters whose conductors are powers of $p$ in the sequel.

Given a positive integer $n$, let ${\rm GSp}(2n)$ be the group of symplectic similitudes over $\mathbb{Q}$, that is,

[TABLE]

where $J=\left[\begin{smallmatrix}0_{n}&-1_{n}\\ 1_{n}&0_{n}\end{smallmatrix}\right]$ with the $n\times n$ unit (resp. zero) matrix $1_{n}$ (resp. $0_{n}$), and ${\rm Sp}(2n)$ the derived group of ${\rm GSp}(2n)$ characterized by the exact sequence

[TABLE]

respectively. Namely, ${\rm GSp}(2)={\rm GL}(2)$ and ${\rm Sp}(2)={\rm SL}(2)$ in this setting. We note that every real point $g=\left[\begin{smallmatrix}A&B\\ C&D\end{smallmatrix}\right]\in{\rm GSp}(2n,\mathbb{R})$ with $\nu(g)>0$, where $A,\,B,\,C,\,D\in{\rm Mat}_{n\times n}(\mathbb{R})$, acts on the Siegel upper-half space

[TABLE]

of genus $n$ via the linear transformation $Z\mapsto g\langle Z\rangle=(AZ+B)(CZ+D)^{-1}$. If $F$ is a function on $\mathfrak{H}_{n}$, then for each $\kappa\in\mathbb{Z}$, we define the slash action of $g$ on $F$ by

[TABLE]

For each positive integer $N$, we shall consider the following congruence subgroups of level $N$ for the full-modular group ${\rm Sp}(2n,\mathbb{Z})$:

[TABLE]

For each $\kappa\in\mathbb{Z}$, let us denote by $\mathscr{M}_{\kappa}(\Gamma_{1}(N))^{(n)}$ the space of (holomorphic) Siegel modular forms of genus $n$, weight $\kappa$ and level $N$, that is, $\mathbb{C}$-valued holomorphic functions $F$ on $\mathfrak{H}_{n}$ satisfying the following two conditions:

1. (i)

$F\,\|_{\kappa}\,\gamma=F$   for any $\gamma\in\Gamma_{1}(N)^{(n)}$. 2. (ii)

For each $\gamma\in{\rm Sp}(2n,\mathbb{Z})$, the function $F\,\|_{\kappa}\,\gamma$ possesses a Fourier expansion of the form

[TABLE] 3. where ${\rm Sym}_{n}^{*}(\mathbb{Z})$ is the set of all half-integral symmetric matrices of degree $n$ over $\mathbb{Z}$, namely,

[TABLE]

and ${\rm tr}(*)$ denotes the trace. Then it is satisfied that

$A_{F,\gamma}(T)=0$ unless $T\geq 0$ (positive-semidefinite)

for all $\gamma\in{\rm Sp}(2n,\mathbb{Z})$.

A Siegel modular form $F\in\mathscr{M}_{\kappa}(\Gamma_{1}(N))^{(n)}$ is said to be cuspidal if it is satisfied that $A_{F,\gamma}(T)=0$ unless $T>0$ for all $\gamma\in{\rm Sp}(2n,\mathbb{Z})$. We denote by $\mathscr{S}_{\kappa}(\Gamma_{1}(N))^{(n)}$ the subspace of $\mathscr{M}_{\kappa}(\Gamma_{1}(N))^{(n)}$ consisting of all cuspidal forms. Given a Dirichlet character $\chi:(\mathbb{Z}/N\mathbb{Z})^{\times}\to\mathbb{C}^{\times}$, we denote by $\mathscr{M}_{\kappa}(\Gamma_{0}(N),\,\chi)^{(n)}$ (resp. $\mathscr{S}_{\kappa}(\Gamma_{0}(N),\,\chi)^{(n)}$) the subspace of $\mathscr{M}_{\kappa}(\Gamma_{1}(N))^{(n)}$ (resp. $\mathscr{S}_{\kappa}(\Gamma_{1}(N))^{(n)}$) consisting of all forms $F$ with nebentypus character $\chi$, that is,

[TABLE]

In particular, whenever $\chi$ is trivial, we naturally write $\mathscr{M}_{\kappa}(\Gamma_{0}(N))^{(n)}=\mathscr{M}_{\kappa}(\Gamma_{0}(N),\,{\rm triv})^{(n)}$ and ${\mathscr{S}_{\kappa}(\Gamma_{0}(N))^{(n)}=\mathscr{S}_{\kappa}(\Gamma_{0}(N),\,{\rm triv})^{(n)}}$, respectively.

For a given pair of $Z=[\,z_{ij}\,]\in\mathfrak{H}_{n}$ and $T=[\,t_{ij}\,]\in{\rm Sym}_{n}^{*}(\mathbb{Z})$, put

[TABLE]

where $q_{ij}=e(z_{ij})\,(1\leq i\leq j\leq n)$. Since $\left[\begin{smallmatrix}1_{n}&S\\ 0_{n}&1_{n}\end{smallmatrix}\right]\in\Gamma_{1}(N)^{(n)}\subset\Gamma_{0}(N)^{(n)}$ for each $S\in{\rm Sym}_{n}(\mathbb{Z})$, we easily see that if $F\in\mathscr{M}_{\kappa}(\Gamma_{1}(N))^{(n)}$ (or $\mathscr{M}_{\kappa}(\Gamma_{0}(N),\chi)^{(n)}$), then $F$ possesses a Fourier expansion of the form

[TABLE]

which is regarded as belonging to the ring $\mathbb{C}[\,q_{ij}^{\pm 1}\,|\,1\leq i<j\leq n\,][[q_{11},\cdots,q_{nn}]]$. Given a ring $R$, we write

[TABLE]

in a similar fashion to the notation of the ring of formal $q$-expansions $R[[q]]$. In particular, if $F\in\mathscr{M}_{\kappa}(\Gamma_{1}(N))^{(n)}$ is a Hecke eigenform (i.e., a simultaneous eigenfunction of all Hecke operators whose similitude is coprime to $N$), then it is well-known that the field $K_{F}$ obtained by adjoining all Fourier coefficients (or equivalently, all Hecke eigenvalues) of $F$ to $\mathbb{Q}$ is an algebraic number field. Thus, by virtue of the presence of $\iota_{p}$ and $\widehat{\iota}_{p}$, we may regard $F\in K_{F}[[{\bf q}]]^{(n)}$ as sitting inside $\mathbb{C}[[{\bf q}]]^{(n)}$ and $\mathbb{C}_{p}[[{\bf q}]]^{(n)}$ interchangeably. For further details on the basic theory of Siegel modular forms set out above, see [AZ] or [Fr]. In particular, a comprehensive introduction to the theory of elliptic modular forms and Hecke operators can be found in [Mi].

2. Siegel Eisenstein series for symplectic groups

In this section, we review some elementary facts on the Siegel Eisenstein series defined for ${\rm Sp}(2n)_{/\mathbb{Q}}$ of an arbitrary genus $n\geq 1$. In particular, we describe a explicit form of its Fourier expansion according to some previous works of Shimura (e.g., [S1, S2]), which is the starting point for the subsequent arguments.

Let $N$ be a positive integer and $\chi:(\mathbb{Z}/N\mathbb{Z})^{\times}\to\mathbb{C}^{\times}$ a Dirichlet character, respectively. As mentioned in §1, for simiplicity, we restrict ourselves to either of the following cases:

1. (i)

$N=1$, that is, $\chi$ is trivial; 2. (ii)

$N>1$ is odd, $\chi$ is primitive and $\chi^{2}$ is locally non-trivial at every prime $l\mid N$.

Given a positive integer $n$, if $\kappa$ is an integer with $\kappa>n+1$ and $\chi(-1)=(-1)^{\kappa}$, then the (holomorphic) Siegel Eisenstein series of genus $n$, weight $\kappa$ and level $N$ with nebentypus character $\chi$ is defined as follows: For each $Z\in\mathfrak{H}_{n}$, put

[TABLE]

where $L(s,\,\psi)$ denotes Dirichlet’s $L$-function associated with some character $\psi$, and $P_{2n}$ the Siegel parabolic subgroup of ${\rm Sp}(2n)$ consisting of all matrices $g=\left[\begin{smallmatrix}*&*\\ 0_{n}&*\end{smallmatrix}\right]$, respectively.

Let $r$ be a positive integer. For each rational prime $l$, let ${\rm Sym}_{r}^{*}(\mathbb{Z}_{l})$ denote the set of all half-integral symmetric matrices of degree $r$ over $\mathbb{Z}_{l}$. Given a nondegenerate $S\in{\rm Sym}_{r}^{*}(\mathbb{Z}_{l})$, we define a formal power series $b_{l}(S;\,X)$ in $X$ by

[TABLE]

where $\mu_{R}=[\mathbb{Z}_{l}^{r}+\mathbb{Z}_{l}^{r}R:\mathbb{Z}_{l}^{r}]$. Put $\mathfrak{D}_{S}:=2^{2[r/2]}\det S$. We note that if $r$ is even, then $(-1)^{r/2}\mathfrak{D}_{S}\equiv 0\text{ or }1\pmod{4}$, and thus, we may decompose it into the form

[TABLE]

where $\mathfrak{d}_{S}$ is the fundamental discriminant of the quadratic field extension $\mathbb{Q}_{l}(\{(-1)^{r/2}\mathfrak{D}_{S}\}^{1/2})/\mathbb{Q}_{l}$ and $\mathfrak{f}_{S}=\{(-1)^{r/2}\mathfrak{D}_{S}/\mathfrak{d}_{S}\}^{1/2}\in\mathbb{Z}_{l}$. Let $\xi_{l}:\mathbb{Q}_{l}^{\times}\to\{\pm 1,\,0\}$ denote the character defined by

[TABLE]

As shown in [Ki1, Fe, S2], for each nondegenerate $S\in{\rm Sym}_{r}^{*}(\mathbb{Z}_{l})$, there exists a polynomial $F_{l}(S;\,X)\in\mathbb{Z}[X]$ whose constant term is $1$ such that $b_{l}(S;\,X)$ is decomposed as follows:

[TABLE]

(cf. Proposition 3.6 in [S2]). We note that $F_{l}(S;\,X)$ satisfies the functional equation

[TABLE]

where

[TABLE]

in terms of the Hasse invariant $h_{l}(S)$ in the sense of Kitaoka [Ki1] and the Hilbert symbol $(*,*)_{l}$ defined over $\mathbb{Q}_{l}$ (cf. Theorem 3.2 in [K1]). Thus, it turns out that $F_{l}(S;\,X)$ has degree $2{\rm val}_{l}(\mathfrak{f}_{S})$ or ${\rm val}_{l}(\mathfrak{D}_{S})$ according as $r$ is even or odd. We easily see that $F_{l}(uS;\,X)=F_{l}(S;\,X)$ for each $u\in\mathbb{Z}_{l}^{\times}$, and that if $S$, $T\in{\rm Sym}_{r}^{*}(\mathbb{Z}_{l})$ are equivalent over $\mathbb{Z}_{l}$, that is, $T={}^{t}USU$ for some $U\in{\rm GL}(r,\mathbb{Z}_{l})$, then $F_{l}(S;\,X)=F_{l}(T;\,X)$. For further details on the above-mentioned issues, see [K1].

Lemma 2.1**.**

Let $n$, $\kappa$, $N$ and $\chi$ be taken as above.

• (I)

$E_{\kappa,\,\chi}^{(n)}\in\mathscr{M}_{\kappa}(\Gamma_{0}(N),\,\chi)^{(n)}$* and it is a Hecke eigenform, that is, a simultaneous eigenfunction of Hecke operators defined at least for all primes not dividing the level $N$.*

• (II)

Let us consider a Fourier expansion of $E_{\kappa,\,\chi}^{(n)}$ taken of the form

[TABLE]

Then every coefficient $A_{\kappa,\,\chi}(T)$, which is invariant under $T\mapsto{}^{t}UTU$ for $U\in{\rm GL}(n,\mathbb{Z})$, is described as follows:

• (II a)

For $T=0_{n}\in{\rm Sym}_{n}^{*}(\mathbb{Z})$, we have

[TABLE]

Therefore $E_{\kappa,\,\chi}^{(n)}$ is not cuspidal.

• (II b)

If $T\in{\rm Sym}_{n}^{*}(\mathbb{Z})$ is taken of the form

[TABLE]

*for some nondegenerate $T^{\prime}\in{\rm Sym}_{r}^{*}(\mathbb{Z})$ with $0<r\leq n$ (i.e., ${\rm rank}\,T={\rm rank}\,T^{\prime}=r$), then *

$\displaystyle{A_{\kappa,\,\chi}(T)}$ $\displaystyle=$ $\displaystyle 2^{[(r+1)/2]-[(n+1)/2]}\prod_{i=[r/2]+1}^{[n/2]}L(1-2\kappa+2i,\chi^{2})$

\displaystyle\times\left\{\begin{array}[]{ll}L(1-\kappa+r/2,\left({\mathfrak{d}_{T^{\prime}}\over*}\right)\chi)\displaystyle\prod_{l\,\mid\,\mathfrak{f}_{T^{\prime}}}F_{l}(T^{\prime};\,\chi(l)\,l^{\kappa-r-1})&\textrm{ if ris even},\\ \displaystyle\prod_{l\,\mid\,\mathfrak{D}_{T^{\prime}}}F_{l}(T^{\prime};\,\chi(l)\,l^{\kappa-r-1})&\textrm{ ifr is odd},\end{array}\right.

**

where $\left({\,\mathfrak{d}\,\over*}\right)$ denotes the Kronecker symbol.

Remark 2.2**.**

For the convenience in the sequel, we make the convention for $r=0$, that $\mathfrak{D}_{S}=\mathfrak{d}_{S}=\mathfrak{f}_{S}=1$, and $F_{l}(S;\,X)=1$ for all primes $l$. This enables us to regard (II a) as (II b) for $r=0$.

Proof.

The assertion (I) is well-known. The assertion (II) can be obtained by exploiting an idea of Shimura [S2] as follows: Let $\mathbb{A}$ be the ring of adeles over $\mathbb{Q}$, $G_{2n}(\mathbb{A})={\rm Sp}(2n,\mathbb{A})$, $P_{2n}(\mathbb{A})=M_{2n}(\mathbb{A})N_{2n}(\mathbb{A})$ a Levi decomposition of the Siegel parabolic subgroup, where

[TABLE]

and $\boldsymbol{\chi}:{\mathbb{A}^{\times}/\mathbb{Q}^{\times}}\to\mathbb{C}^{\times}$ the unitary Hecke character corresponding to $\chi$, respectively. For each $s\in\mathbb{C}$, let ${\rm Ind}_{P_{2n}(\mathbb{A})}^{G_{2n}(\mathbb{A})}(\boldsymbol{\chi}\cdot|*|_{\mathbb{A}}^{s})$ denote the normalized smooth representation induced from the character of ${\rm GL}(n,\mathbb{A})\simeq M_{2n}(\mathbb{A})$ defined by $A\mapsto\boldsymbol{\chi}(\det A)\,|\det A|_{\mathbb{A}}^{s}$, where $|*|_{\mathbb{A}}$ denotes the norm on $\mathbb{A}$. Choosing a suitable section $\varphi^{(s)}\in{\rm Ind}_{P_{2n}(\mathbb{A})}^{G_{2n}(\mathbb{A})}(\boldsymbol{\chi}\cdot|*|_{\mathbb{A}}^{s})$, we define the Eisenstein series ${\bf E}(\varphi^{(s)})$ on $G_{2n}(\mathbb{A})$ by

[TABLE]

which converges absolutely for ${\rm Re}(s)\gg 0$, and ${\bf E}(\varphi^{(\kappa-(n+1)/2)})$ evaluates

[TABLE]

Thus, the desired equation can be obtained from an explicit formula for Fourier coefficients of ${\bf E}(\varphi^{(s)})$, more precisely, local Whittaker functions ${\rm Wh}_{T}(\varphi_{v}^{(s)})$ defined on $G_{2n}(\mathbb{Q}_{v})$ for all places $v$ of $\mathbb{Q}$, where $\mathbb{A}=\prod_{v}\mathbb{Q}_{v}$, $\boldsymbol{\chi}=\prod_{v}\boldsymbol{\chi}_{v}$ and $\varphi^{(s)}=\prod_{v}\varphi_{v}^{(s)}$. Whenever $v$ is archimedean (i.e., $v=\infty$) or non-archimedean at which $\boldsymbol{\chi}_{v}$ is unramified (i.e., $v$ is a prime $l$ not dividing $N$), it has been proved by Shimura (cf. Equations 4.34-35K in [S1] and Proposition 7.2 in [S2]). Whenever $v$ is a non-archimedean place at which $\boldsymbol{\chi}_{v}$ is ramified and $\boldsymbol{\chi}_{v}^{2}$ is non-trivial, the local Whittaker function ${\rm Wh}_{T}(\varphi_{v}^{(s)})$ is described by Takemori [T] in which the key argument relies upon a functional equation of ${\rm Wh}_{T}(\varphi_{v}^{(s)})$ due to Ikeda [Ik] (generalizing [Sw]). ∎

3. generalized Atkin $U_{p}$-operator

In this section, we recall the theory of Atkin’s $U_{p}$-operator and its generalization. For further details on the facts set out below, see, for instance, [AZ, Bö, Ta].

Let $p$ be a prime number and $N$ a positive integer, respectively. If $p\mid N$, the coset decomposition

[TABLE]

induces the following linear operator $U_{p}=U_{p,1}$ on $\mathscr{M}_{\kappa}(\Gamma_{0}(N),\,\chi)^{(1)}$: For each $f\in\mathscr{M}_{\kappa}(\Gamma_{0}(N),\,\chi)^{(1)}$, put

[TABLE]

We easily see that it is written in terms of Fourier expansions as

[TABLE]

and this is still valid even if $p\nmid N$, however it maps from $\mathscr{M}_{\kappa}(\Gamma_{0}(N),\,\chi)^{(1)}$ to $\mathscr{M}_{\kappa}(\Gamma_{0}(Np),\,\chi)^{(1)}$ in this case. We refer to the operator $U_{p}$, regardless of whether $p\mid N$ or not, as Atkin’s $U_{p}$-operator.

Remark 3.1**.**

Obviously, $U_{p}$ coincides with the usual Hecke operator $T_{p}$ if $p\mid N$. However, $U_{p}$ is slightly different from $T_{p}$ in general:

[TABLE]

if $p\nmid N$.

Similarly, if $n>1$ and $p\mid N$, the following $n$ double-coset operators at $p$ are relevant for Siegel modular forms of genus $n$ and level $N$:

[TABLE]

We note that if $N$ is divisible by $p$, these operators $U_{p,1},\,\cdots,\,U_{p,n-1}$ and $U_{p,n}$ generate the dilating Hecke algebra at $p$ acting on $\mathscr{M}_{\kappa}(\Gamma_{0}(N),\,\chi)^{(n)}$. In particular, we are interested in the operator $U_{p,n}$ which plays a central role among them. Namely, we define the operator $U_{p,n}$ on $\mathbb{C}_{p}[[{\bf q}]]^{(n)}$ by

[TABLE]

Indeed, we easily see that if $F\in\mathscr{M}_{\kappa}(\Gamma_{0}(N),\,\chi)^{(n)}$ with some positive integers $\kappa$, $N$ and a Dirichlet character $\chi$, then

[TABLE]

and the action of $U_{p,n}$ commutes with those of the Hecke operators defined outside $Np$.

4. Semi-ordinary $p$-stabilization of Siegel Eisenstein series

Let us fix an odd integer $M$ and a prime number $p\nmid M$ (including $p=2$) once for all. In this section, for a pair of positive integers $n$, $\kappa$ and a Dirichlet character $\chi$ modulo $M$ taken as in §2, that is,

[TABLE]

we introduce a certain $p$-stabilization of the Siegel Eisenstein series $E_{\kappa,\,\chi}^{(n)}\in\mathscr{M}_{\kappa}(\Gamma_{0}(M),\,\chi)$ which can be viewed as a natural generalization of the ordinary $p$-stabilization $E_{\kappa,\,\chi}^{(1)}\mapsto(E_{\kappa,\,\chi}^{(1)})^{*}$ (cf. Equations (1) and (1’)). Moreover, we derive an explicit form of the associated Fourier expansion

[TABLE]

which is expressed in a similar fashion to (2), and conclude its $p$-adic interpolation problem.

To begin with, we introduce the following two polynomials in $X$ and $Y$:

[TABLE]

In addition, let us denote by $\widetilde{\mathcal{R}}_{p}^{(n)}(X,\,Y)$ the reflected polynomial of $\mathcal{R}_{p}^{(n)}(X,\,Y)$ with respect to $Y$, that is,

[TABLE]

Remark 4.1**.**

Whenever $n=1$ and $2$, a straightforward calculation yields

[TABLE]

We note that if $n>2$, then $\mathcal{P}_{p}^{(n)}(X,\,Y)\neq\mathcal{R}_{p}^{(n)}(X,\,Y)$, however, $\mathcal{P}_{p}^{(n)}(X,\,1)$ divides $\mathcal{R}_{p}^{(n)}(X,\,1)$ in general. For readers’ convenience, we reveal the origins of these two polynomials here: Obviously, the former $\mathcal{P}_{p}^{(n)}(X,\,Y)$ is relevant to the local factor of the Fourier coefficient $A_{\kappa,\,\chi}(0_{n})$ described in Lemma 2.1 (II a) at $p$:

[TABLE]

The latter $\mathcal{R}_{p}^{(n)}(X,\,Y)$ has been introduced by Kitaoka [Ki2] and Böcherer-Sato [BS] to describe the denominator of the formal power series $\sum_{m=0}^{\infty}F_{p}(p^{m}S;\,X)Y^{m}$ for each nondegenerate $S\in{\rm Sym}_{n}^{*}(\mathbb{Z}_{p})$ (cf. Equation (11) below).

Now, we introduce a $p$-stabilization of the Siegel Eisenstein series $E_{\kappa,\,\chi}^{(n)}\in\mathscr{M}_{\kappa}(\Gamma_{0}(M),\,\chi)^{(n)}$ in terms of a linear combination of $(U_{p,n})^{i}=\underbrace{U_{p,n}\circ\cdots\circ U_{p,n}}_{i}$ for $i=0,1,\cdots,n$ as follows:

Theorem 4.2**.**

For a pair of $n$, $\kappa$ and $\chi$ taken as above, put

[TABLE]

Then we have

1. (I)

$(E_{\kappa,\,\chi}^{(n)})^{*}\in\mathscr{M}_{\kappa}(\Gamma_{0}(Mp),\,\chi)^{(n)}$* and it is a Hecke eigenform such that all the eigenvalues outside $Mp$ agree with those of $E_{\kappa,\,\chi}^{(n)}\in\mathscr{M}_{\kappa}(\Gamma_{0}(M),\,\chi)^{(n)}$.* 2. (II)

If $0\leq T\in{\rm Sym}_{n}^{*}(\mathbb{Z})$ is taken of the form

[TABLE]

for some nondegenerate $T^{\prime}\in{\rm Sym}_{r}^{*}(\mathbb{Z})$ with $0\leq r\leq n$, then the $T$-th Fourier coefficient of $(E_{\kappa,\,\chi}^{(n)})^{*}$ is taken of the following form:

[TABLE]

Therefore we have $(E_{\kappa,\,\chi}^{(n)})^{*}\,\|_{\kappa}\,U_{p,n}=(E_{\kappa,\,\chi}^{(n)})^{*}.$

The preceding theorem totally insists that the assignment $E_{\kappa,\,\chi}^{(n)}\mapsto(E_{\kappa,\,\chi}^{(n)})^{*}$ can be regarded as a $p$-stabilization which generalizes the ordinary $p$-stabilization of $E_{\kappa,\,\chi}=E_{\kappa,\,\chi}^{(1)}$ explained in §1. Whenever $n=2$, Skinner-Urban [SU] has already dealt with a similar type of $p$-stabilization for some Siegel modular forms of genus $2$ so that the associated eigenvalue of $U_{p,2}$ is a $p$-adic unit as well. Accordingly, in the same context, we may call $(E_{\kappa,\,\chi}^{(n)})^{*}$ semi-ordinary at $p$ if $n\geq 2$.

Remark 4.3**.**

It should be mentioned that if $n>1$, $(E_{\kappa,\,\chi}^{(n)})^{*}$ may not  satisfy the ordinary condition at $p$ in the sense of Hida (cf. [H2, H3]). This disparity is inevitable at least for Siegel Eisenstein series of higher genus in general. For instance, in the case where $M=1$, because of the shape of associated Satake parameters at $p$, there is no way to produce from $E_{\kappa}^{(n)}=E_{\kappa,\,{\rm triv}}^{(n)}\in\mathscr{M}_{\kappa}(\Gamma_{0}(1))^{(n)}$ to a Hecke eigenform of level $p$ such that the associated eigenvalues of $U_{p,1},\cdots,\,U_{p,n-1}$ and $U_{p,n}$ are $p$-adic units simultaneously. However, as mentioned in [SU], it turns out that the semi-ordinary condition concerning only on the eigenvalue of $U_{p,n}$ is sufficient to adapt Hida’s ordinary theory with some modification. (See also [Pi, BPS].)

Proof of Theorem 4.2.

The assertion (I) is obvious from Lemma 2.1 (I) and the properties of $U_{p,n}$ explained in §3. Since $U_{p,n}$ does not effect on the Fourier coefficient $A_{\kappa,\,\chi}(0_{n})$, the assertion (II) for $r=0$ follows immediately from Lemma 2.1 (II a), Equations (8) and (9). Hereinafter, we suppose that $r>0$. It follows by the definition of $\widetilde{\mathcal{R}}_{p}^{(n)}(X,Y)$ (cf. Equation (7)) that

[TABLE]

where $s_{m}(\{X_{1},\cdots,X_{n}\})$ denotes the $m$-th elementary symmetric polynomial in $X_{1},\,\cdots,\,X_{n}$. Thus, by Lemma 2.1 (II b) and Equation (9), we have

[TABLE]

Here we note that

[TABLE]

Thus, to prove the assertion (II), it suffices to show that the following equation holds valid for each nondegenerate $T^{\prime}\in{\rm Sym}_{r}^{*}(\mathbb{Z}_{p})$ with $0<r\leq n$:

[TABLE]

(Indeed, the preceding equation (10) yields

[TABLE]

and hence, by evaluating this at $X=\chi(p)\,p^{\kappa-n-1}$, we obtain the desired equation. ) On the other hand, Theorem 1 in [Ki2] (resp. Theorem 6 in [BS]) states that for each nondegenerate $T^{\prime}\in{\rm Sym}_{r}^{*}(\mathbb{Z}_{p})$, the equation

[TABLE]

holds for some polynomial $\mathcal{S}_{p}(T^{\prime};\,X,\,Y)\in\mathbb{Z}[X,\,Y]$ if $p\neq 2$ (resp. $p=2$). Since the preceding equation yields

[TABLE]

we may interpret Equation (10) as

[TABLE]

Whenever $r=1$, we easily see that $F_{p}(t;\,X)=\sum_{i=0}^{{\rm val}_{p}(t)}(pX)^{i}$ for each $t\in\mathbb{Z}_{p}\smallsetminus\{0\}$. Thus, we have

[TABLE]

Whenever $r>1$, for a given $T$, let $\mathfrak{i}(T)$ denote the least integer $m$ such that $p^{m}T^{-1}\in{\rm Sym}_{r}^{*}(\mathbb{Z}_{p})$. It is known that if $r=2$, then for each nondegenerate $T\in{\rm Sym}_{2}^{*}(\mathbb{Z}_{p})$, the polynomial $F_{p}(T;\,X)$ admits the explicit form

[TABLE]

(cf. [K1]). Thus a simple calculation yields that

[TABLE]

and hence, Equation (12) also holds for $r=2$. Now, we suppose that $r>2$. We note that every nondegenerate $T\in{\rm Sym}_{r}^{*}(\mathbb{Z}_{p})$ is equivalent, over $\mathbb{Z}_{p}$, to a canonical form

T=\left[\begin{array}[]{c|c}T_{1}&{}\hfil\\ \hline\cr{}\hfil&T_{2}\end{array}\right]

for some $T_{1}\in{\rm Sym}_{2}^{*}(\mathbb{Z}_{p})$ and $T_{2}\in{\rm Sym}_{r-2}^{*}(\mathbb{Z}_{p})\cap{\rm GL}(r-2,\mathbb{Q}_{p})$. It follows from Theorems 4.1 and 4.2 in [K1] that

[TABLE]

if $r$ is even, and

[TABLE]

if $r$ is odd. (See also [K2, §3]. ) Thus, it is proved by induction on $r$ that Equation (12) (and thus, (10) ) holds in general. This completes the proof. ∎

As a straightforward conclusion of Theorem 4.2 above, we have

Theorem 4.4**.**

Let $\chi$ be a Dirichlet character modulo $M$, where $M$ is odd and coprime to $p$, taken as above and $\omega^{a}$ a power of the Teichmüller character with $0\leq a<\varphi({\bf p})$666Here $\varphi$ denotes Euler’s totient function., respectively. For each $n\geq 1$, there exists a formal Fourier expansion

[TABLE]

where $F_{\Lambda}$ is the field of fractions of $\Lambda=\mathbb{Z}_{p}[\chi][[X]]$, such that for each positive integer $\kappa>n+1$ with $\chi(-1)=(-1)^{\kappa}$ and $\kappa\equiv a\pmod{\varphi({\bf p})}$, we have

[TABLE]

Moreover, put

[TABLE]

Then  $\overline{\mathcal{E}}_{\chi\omega^{a}}^{\,(n)}(X):=\mathcal{B}^{(n)}(X)\cdot\mathcal{E}_{\chi\omega^{a}}^{(n)}(X)$ belongs to $\Lambda[[{\bf q}]]^{(n)}$.

Proof.

As is well-known by Deligne-Ribet [DR] (generalizing the previous work of Kubota-Leopoldt), associated to a quadratic Dirichlet character $\xi$, a Dirichlet character $\chi$ taken as above, and a power of the Teichmüller character $\omega^{a}$ with $0\leq a<\varphi({\bf p})$, there exists $\overline{\mathcal{L}}(\xi\chi\omega^{a};\,X)\in\Lambda$ such that for each positive integer $k>1$, we have

[TABLE]

Accordingly, put

[TABLE]

We easily see that $\mathcal{L}(\xi\chi\omega^{a};\,(1+{\bf p})^{k}-1)=L^{\{p\}}(1-k,\,\xi\chi\omega^{a-k})$ for each $k>1$. More generally, it turns out that if $\varepsilon:1+{\bf p}\mathbb{Z}_{p}\to(\mathbb{Q}_{p}^{\rm alg})^{\times}$ is a character of finite order, then

[TABLE]

for any $k>1$. On the other hand, for each $x\in 1+{\bf p}\mathbb{Z}_{p}$, put $s(x):=\log_{p}(x)/\log_{p}(1+{\bf p})$, where $\log_{p}$ is the $p$-adic logarithm function in the sense of Iwasawa, and thus we have $s:1+{\bf p}\mathbb{Z}_{p}\stackrel{{\scriptstyle\sim}}{{\to}}\mathbb{Z}_{p}$. Then for each T=\left[\begin{array}[]{c|c}T^{\prime}&\\ \hline\cr&0_{n-r}\end{array}\right]\in{\rm Sym}_{n}^{*}(\mathbb{Z}) with $T^{\prime}\in{\rm Sym}_{r}^{*}(\mathbb{Z})\cap{\rm GL}(r,\mathbb{Q})$ and $0\leq r\leq n$, we define $\mathcal{A}_{\chi\omega^{a}}(T;\,X)\in F_{\Lambda}$ as follows:

[TABLE]

It follows from Theorem 4.2 (II) that if $\kappa>n+1$, $\chi(-1)=(-1)^{\kappa}$ and further if $\kappa\equiv a\pmod{\varphi({\bf p})}$ (or equivalently, $\omega^{a-\kappa}$ is trivial), then

[TABLE]

Thus we obtain the former assertion. Moreover, it follows directly from Equation (14) that $\mathcal{B}^{(n)}(X)\cdot\mathcal{A}_{\chi\omega^{a}}(T;\,X)\in\Lambda$ for all $T$. Therefore $\overline{\mathcal{E}}_{\chi\omega^{a}}^{\,(n)}(X)\in\Lambda[[{\bf q}]]^{(n)}$. This completes the proof. ∎

5. $\Lambda$-adic Siegel Eisenstein series

In this section, we fix an odd  prime number $p$. We show that for a Dirichlet character $\chi$ modulo $M$ with $p\nmid M$ and an integer $a$ with $0\leq a<p-1$, the formal Fourier expansion $\mathcal{E}_{\chi\omega^{a}}^{(n)}(X)$ (resp. $\overline{\mathcal{E}}_{\chi\omega^{a}}^{\,(n)}(X)$) with coefficients in $F_{\Lambda}$ (resp. $\Lambda=\mathbb{Z}_{p}[\chi][[X]]$) defined in Theorem 4.4, give rise to classical Siegel modular forms via the specialization at $X=\varepsilon(1+p)(1+p)^{\kappa}-1$, where $\kappa$ is a positive integer sufficiently large and $\varepsilon:1+p\mathbb{Z}_{p}\to(\mathbb{Q}_{p}^{\rm alg})^{\times}$ is a character of finite order.

First, from Lemma 2.1 (II), we deduce the following:

Theorem 5.1**.**

If $\kappa$ is a positive integer with $\kappa>n+1$ and further if $\varepsilon:1+p\mathbb{Z}_{p}\to(\mathbb{Q}_{p}^{\rm alg})^{\times}$ is a character of exact order $p^{m}$ for some nonnegative integer $m$, then

[TABLE]

as long as $\omega^{2a-2\kappa}\varepsilon^{2}$ is non-trivial.

Proof.

For each T=\left[\begin{array}[]{c|c}T^{\prime}&\\ \hline\cr&0_{n-r}\end{array}\right]\in{\rm Sym}_{n}^{*}(\mathbb{Z}) with $T^{\prime}\in{\rm Sym}_{r}^{*}(\mathbb{Z})\cap{\rm GL}(r,\mathbb{Q})$ and $0\leq r\leq n$, the equation (14) is specialized as

[TABLE]

Thus, if $\omega^{2a-2\kappa}\varepsilon^{2}$ is non-trivial, then Lemma 2.1 (II) yields

[TABLE]

This completes the proof. ∎

On the other hand, by exploiting the vertical control theorem for $p$-adic Siegel modular forms in the sense of Hida [H2], we may deduce a slight weaker version of the preceding theorem as follows:

Theorem 5.2**.**

If $\kappa$ is a positive integer with ${\kappa>n(n+1)/2}$, and further if $\varepsilon:1+p\mathbb{Z}_{p}\to(\mathbb{Q}_{p}^{\rm alg})^{\times}$ is a character of exact order $p^{m}$ for some nonnegative integer $m$, then

[TABLE]

as long as $\omega^{a-\kappa}\varepsilon$ is non-trivial.

Proof.

It follows from Theorem 4.4 that $\mathcal{E}_{\chi\omega^{a}}^{(n)}((1+p)^{\kappa^{\prime}}-1)\in\mathscr{M}_{\kappa^{\prime}}(\Gamma_{0}(Mp),\,\chi)^{(n)}$ for infinitely many integers $\kappa^{\prime}$ with $\kappa^{\prime}>n+1$, $\chi(-1)=(-1)^{\kappa^{\prime}}$ and $\kappa^{\prime}\equiv a\pmod{p-1}$. Thus, Théorème 1.1 in [Pi] (generalizing [H2] in more general settings) turns out that if an integer $\kappa$ is sufficiently large, then the specialization of $\mathcal{E}_{\chi\omega^{a}}^{(n)}(X)$ at $X=\varepsilon(1+p)(1+p)^{\kappa}-1$ gives rise to an overconvergent $p$-adic Siegel modular form of weight $\kappa$, tame level $M$ and Iwahori level $p^{m+1}$ with character $\chi\omega^{a-\kappa}\varepsilon$ (for the precise definition, see [SU, Pi]). In addition, the explicit formula for Fourier coefficients of $\mathcal{E}_{\chi\omega^{a}}^{(n)}(\varepsilon(1+p)(1+p)^{\kappa}-1)$ also yields that it is an eigenform of $U_{p,n}$ whose eigenvalue is $1$. Therefore, if $\kappa$ satisfies the condition $\kappa>n(n+1)/2$ (cf. Hypothèse 4.5.1 in [BPS]), then the above-mentioned overconvergent form is indeed classical, that is,

[TABLE]

(cf. Théorème 5.3.1 in [ibid]). Thus we obtain the assertion. ∎

Remark 5.3**.**

Although Theorem 4.4 remains valid even for $p=2$, we may not deduce similar statements to Theorems 5.1 and 5.2 in the case where $p=2$, since there are some technical difficulties in understanding $\mathcal{E}_{\chi\omega^{a}}^{(n)}(\varepsilon(1+{\bf p})(1+{\bf p})^{k}-1)$ from automorphic and geometric viewpoints.

Now, as a summary of Theorems 4.4, 5.1 and 5.2 above, we have the following statement by which $\overline{\mathcal{E}}_{\chi\omega^{a}}^{\,(n)}(X)$ is said to be the semi-ordinary $\Lambda$-adic Siegel Eisenseiten series of genus $n$ and level $Mp^{\infty}$ associated with character $\chi\omega^{a}$:

Theorem 5.4**.**

Let $M$ be an odd integer and $\chi$ a Dirichlet character modulo $M$ taken as follows:

• (i)

if $M=1$, then $\chi$ is trivial (or principal);

• (ii)

if $M>1$, then $\chi$ is primitive and $\chi^{2}$ is locally non-trivial at every prime $l\mid M$;

For each integer $a$ with $0\leq a<p-1$, there exists a formal Fourier expansion  $\overline{\mathcal{E}}_{\chi\omega^{a}}^{\,(n)}(X)\in\Lambda[[{\bf q}]]^{(n)}$ such that if $\kappa>n(n+1)/2$ and further if $\varepsilon:1+p\mathbb{Z}_{p}\to(\mathbb{Q}_{p}^{\rm alg})^{\times}$ is a character of exact order $p^{m}$ with some $m\geq 0$, then

[TABLE]

is a Hecke eigenform whose eigenvalue of $U_{p,n}$ is $1$. In particular,

[TABLE]

where $C_{\kappa,\,\varepsilon}^{(n)}=\mathcal{B}^{(n)}(\varepsilon(1+p)(1+p)^{\kappa}-1)$, that is,

[TABLE]

Remark 5.5**.**

The preceding theorem for $n>1$ can be regarded as a satisfactory generalization of Fact 1.1 to some extent. The only exception is our lack of knowledge about the specialization $\overline{\mathcal{E}}_{\chi\omega^{a}}^{\,(n)}(\varepsilon(1+p)(1+p)^{\kappa}-1)$ in the case where $\omega^{a-\kappa}\varepsilon$ is non-trivial, but $\omega^{2a-2\kappa}\varepsilon^{2}$ is trivial. As shown in Theorem 5.2, it has been proved that if $\omega^{a-\kappa}\varepsilon$ is non-trivial, regardless of whether $\omega^{2a-2\kappa}\varepsilon^{2}$ is trivial or not, $\overline{\mathcal{E}}_{\chi\omega^{a}}^{\,(n)}(\varepsilon(1+p)(1+p)^{\kappa}-1)\in\mathscr{M}_{\kappa}(\Gamma_{0}(Mp^{m+1}),\chi\omega^{a-\kappa}\varepsilon)^{(n)}$ is a Hecke eigenform which is semi-ordinary at $p$.