Distinguishing Siegel modular forms

Abstract

Let and be genus 2 cuspidal Siegel paramodular newforms. We prove that if their Hecke eigenvalues and satisfy a non‐trivial polynomial relation for a set of primes of positive density, then is a scalar multiple of a quadratic twist of . This result extends the strong multiplicity one theorem, which handles the case , to arbitrary polynomial relations. Our proof analyses the image of the product Galois representation attached to the pair : We show that this image is as large as possible, unless is a twist of . 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 ‐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.