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On distinguishing Siegel cusp forms of degree two

Zhining Wei, Shaoyun Yi

TL;DR

This work establishes several results on distinguishing Siegel cusp forms of degree two and can also distinguish two Hecke eigenforms of level one by using $L$-functions.

Abstract

In this work, we establish several results on distinguishing Siegel cusp forms of degree two. In particular, a Hecke eigenform of level one can be determined by its second Hecke eigenvalue under a certain assumption. Moreover, we can distinguish two Hecke eigenforms of level one by using $L$-functions.

On distinguishing Siegel cusp forms of degree two

TL;DR

This work establishes several results on distinguishing Siegel cusp forms of degree two and can also distinguish two Hecke eigenforms of level one by using -functions.

Abstract

In this work, we establish several results on distinguishing Siegel cusp forms of degree two. In particular, a Hecke eigenform of level one can be determined by its second Hecke eigenvalue under a certain assumption. Moreover, we can distinguish two Hecke eigenforms of level one by using -functions.
Paper Structure (6 sections, 12 theorems, 62 equations)

This paper contains 6 sections, 12 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['distinguish-weight']}
  4. Proof of Theorem \ref{['disti-maeda']}
  5. Distinguishing Hecke eigenforms by using L-functions
  6. Archimedean factors associated to certain L-functions

Key Result

Theorem 1.1

Let $k_1,k_2$ be distinct positive integers. Let $F\in\mathcal{S}_{k_1}(\Gamma_0(N))$ and $G\in\mathcal{S}_{k_2}(\Gamma_0(N))$ be Hecke eigenforms. Then we can find $n$ satisfying such that $\lambda_F(n)\neq\lambda_G(n)$.

Theorems & Definitions (23)

  • Theorem 1.1
  • Remark 1
  • Theorem 1.2
  • Remark 2
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['1.1']}
  • ...and 13 more