Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Siegel--Eisenstein series of level $p$ and their $p$-adic properties

Siegfried Boecherer, Keiichi Gunji, Toshiyuki Kikuta

TL;DR

The paper addresses the $p$-adic interpolation of Siegel--Eisenstein series at level $p$ by constructing a level-$p$ Siegel--Eisenstein series with a quadratic character that is a $U(p)$-eigenfunction with eigenvalue $1$, and by computing its Fourier coefficients explicitly. It develops a framework in which level-$p$ forms arise as $p$-adic limits of level-$1$ series, using Gu's functional equation and $p$-adic $L$-functions to control coefficients. The authors also extend the analysis to Takemori’s nonquadratic-character case, showing $p$-adic interpolation persists and matches known complex-analytic limits under appropriate embeddings. The work provides explicit Fourier formulas, clarifies the relationship between classical and $p$-adic families of Siegel--Eisenstein series, and offers methods potentially applicable to higher-rank Eisenstein series and $p$-adic automorphic forms.

Abstract

We construct a Siegel--Eisenstein series of level $p$ with a quadratic character mod $p$ which is a $U(p)$-eigenfunction with eigenvalue $1$, and calculate its Fourier coefficients explicitly. We show that this Siegel--Eisenstein series is a $p$-adic Siegel--Eisenstein series, i.e., it is a $p$-adic limit of a sequence of Siegel--Eisenstein series of level $1$. We prove also that the Siegel--Eisenstein series with a nonquadratic character mod $p$ constructed by Takemori is also a $p$-adic Siegel--Eisenstein series.

On Siegel--Eisenstein series of level $p$ and their $p$-adic properties

TL;DR

The paper addresses the -adic interpolation of Siegel--Eisenstein series at level by constructing a level- Siegel--Eisenstein series with a quadratic character that is a -eigenfunction with eigenvalue , and by computing its Fourier coefficients explicitly. It develops a framework in which level- forms arise as -adic limits of level- series, using Gu's functional equation and -adic -functions to control coefficients. The authors also extend the analysis to Takemori’s nonquadratic-character case, showing -adic interpolation persists and matches known complex-analytic limits under appropriate embeddings. The work provides explicit Fourier formulas, clarifies the relationship between classical and -adic families of Siegel--Eisenstein series, and offers methods potentially applicable to higher-rank Eisenstein series and -adic automorphic forms.

Abstract

We construct a Siegel--Eisenstein series of level with a quadratic character mod which is a -eigenfunction with eigenvalue , and calculate its Fourier coefficients explicitly. We show that this Siegel--Eisenstein series is a -adic Siegel--Eisenstein series, i.e., it is a -adic limit of a sequence of Siegel--Eisenstein series of level . We prove also that the Siegel--Eisenstein series with a nonquadratic character mod constructed by Takemori is also a -adic Siegel--Eisenstein series.
Paper Structure (10 sections, 7 theorems, 67 equations)

This paper contains 10 sections, 7 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Construction of Siegel--Eisenstein series
  4. Siegel--Eisenstein series and the $U(p)$-operator
  5. The case of trivial character
  6. The case of quadratic character
  7. $p$-Adic limits of the Siegel--Eisenstein series and their Fourier coefficients
  8. $p$-Adic limits of Siegel--Eisenstein series as classical modular forms
  9. The case of quadratic character
  10. The case of nonquadratic character

Key Result

Theorem 2.1

For any positive integer $k$ such that $\psi(-1) = (-1)^k$, $E^{n,(0)}_{k,\psi}(Z,k/2)$ is a non-zero holomorphic Siegel modular form of level $p$, weight $k$ with character $\psi$. It is a $U(p)$-eigenfunction with eigenvalue $1$.

Theorems & Definitions (12)

  • Theorem 2.1
  • Proposition 2.2: Shi2
  • Theorem 2.3
  • Theorem 2.4
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Theorem 4.1
  • proof
  • Remark 4.2
  • ...and 2 more