Distinguishing Siegel modular forms

2025/06/27 by Arvind Kumar, Kumar, Arvind, Ariel Weiss +1
Mathematics · #11f46 #11f80 #11r45 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2506.22264

openalex publication_date 2025/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Local Intertwining Relations and Co-tempered A-packets of Classical Groups
• Arthur's multiplicity formula for GSp₄ and restriction to Sp₄
• Distinguishing Galois representations by their normalized traces
• Twisting of Siegel paramodular forms
• Locally potentially equivalent two dimensional Galois representations and Frobenius fields of elliptic curves
• Multiplicity one for L-functions and applications
• Distinguishing Hecke eigenforms
• Existence of Whittaker models related to four dimensional symplectic Galois representations
• Siegel Modular Forms
• Generic transfer from GSp(4) to GL(4)
• Recovering modular forms and representations from tensor and symmetric powers
• On the images of the Galois representations attached to genus 2 Siegel modular forms
• Galois representations associated to Siegel modular forms of low weight

Related

• On the Lang--Trotter conjecture for Siegel modular forms
• Paramodular Cusp Forms
• Siegel modular forms (mod p) and algebraic modular forms
• Siegel modular forms of genus 2 attached to elliptic curves
• Hecke eigenvalues of Siegel modular forms (mod p) and of algebraic modular forms