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Congruences for Siegel modular forms of nonquadratic nebentypus mod $p$

Siegfried Boecherer, Toshiyuki Kikuta

Abstract

We prove that weights of two Siegel modular forms of nonquadratic nebentypus should satisfy some congruence relations if these modular forms are congruent to each other. Applying this result, we prove that there are no mod $p$ singular forms of nonquadratic nebentypus. Here we consider the case where the Fourier coefficients of the modular forms are algebraic integers, and we emphasize that $p$ is a rational prime. Moreover, we construct some examples of mod $\frak{p}$ singular forms of nonquadratic nebentypus using the Eisenstein series studied by Takemori.

Congruences for Siegel modular forms of nonquadratic nebentypus mod $p$

Abstract

We prove that weights of two Siegel modular forms of nonquadratic nebentypus should satisfy some congruence relations if these modular forms are congruent to each other. Applying this result, we prove that there are no mod singular forms of nonquadratic nebentypus. Here we consider the case where the Fourier coefficients of the modular forms are algebraic integers, and we emphasize that is a rational prime. Moreover, we construct some examples of mod singular forms of nonquadratic nebentypus using the Eisenstein series studied by Takemori.
Paper Structure (11 sections, 9 theorems, 42 equations)

This paper contains 11 sections, 9 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Siegel modular forms
  4. Congruences for modular forms
  5. An Eisenstein series congruent to a constant
  6. Main results and their proofs
  7. Statements of main results
  8. Proof of Theorem \ref{['thm:M']}
  9. Proofs of Corollary \ref{['cor:M']} and Proposition \ref{['prop:M']}
  10. Eisenstein series of nonquadratic nebentypus
  11. Mod $p$-power singular forms of nonquadratic nebentypus

Key Result

Theorem 2.1

Let $N$ be a natural number with $p\nmid N$ and $F_i\in M_{k_i}^1(\Gamma _0(p)\cap \Gamma _1(N))({\mathcal{O}}_K)$. If $F_1\equiv F_2$ mod $\frak{p}^r$ and $F_1\not \equiv 0$ mod $\frak{p}$, then we have

Theorems & Definitions (21)

  • Theorem 2.1: Rasmussen Ra Theorem 2.16
  • Remark 2.2
  • Theorem 2.3: Lang Lan Theorem 1.2, p.250, Rasmussen Ra, p.12
  • Remark 2.4
  • Theorem 3.1
  • Remark 3.2
  • Corollary 3.3
  • Remark 3.4
  • Proposition 3.5
  • Remark 3.6
  • ...and 11 more