Distinguishing Siegel modular forms

TL;DR

This paper extends the strong multiplicity one theorem for genus 2 Siegel modular forms, showing that polynomial relations among Hecke eigenvalues imply the forms are related by a twist, and applies similar ideas to elliptic forms.

Contribution

It generalizes the strong multiplicity one theorem to arbitrary polynomial relations among Hecke eigenvalues for Siegel and elliptic modular forms, using Galois representation analysis.

Findings

Polynomial relations among Hecke eigenvalues imply forms are twists of each other.

The image of the product Galois representation is maximal unless forms are twists.

Unified method for distinguishing elliptic and Siegel modular forms based on Hecke data.

Abstract

Let $f$ and $f'$ be genus $2$ cuspidal Siegel paramodular newforms. We prove that if their Hecke eigenvalues $a_p$ and $a_p'$ satisfy a non-trivial polynomial relation $P(a_p, a_p') = 0$ for a set of primes $p$ of positive density, then $f$ is a scalar multiple of a quadratic twist of $f'$. This result extends the strong multiplicity one theorem, which handles the case $P(x,y) = x - y$, to arbitrary polynomial relations. Our proof analyses the image of the product Galois representation attached to the pair $(f, f')$: we show that this image is as large as possible, unless $f$ is a twist of $f'$. Our results also apply to elliptic modular forms. They therefore provide a unified method for distinguishing both elliptic and Siegel modular forms based on their Hecke data, including their Hecke eigenvalues, Satake parameters, Sato--Tate angles, and the coefficients of their $L$-functions. We apply our methods to recover and generalise a range of existing results and to prove new ones in both the elliptic and Siegel settings.