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Distinguishing Siegel modular forms

Arvind Kumar, Ariel Weiss

TL;DR

This work addresses when two genus 2 Siegel paramodular newforms are determined by partial Hecke data. By embedding Hecke relations into Galois representations valued in $\mathrm{GSp}_4$ and analyzing the joint image of the product representation, the authors prove that a nontrivial polynomial relation among Hecke data for a positive-density set of primes forces the automorphic representations to be twists of each other, in particular quadratic twists when central characters are trivial. Their method yields a unified approach that also applies to elliptic modular forms, enabling a broad set of “distinguish by data” results across Hecke eigenvalues, Satake parameters, $L$-functions, and Sato–Tate angles, without assuming the Sato–Tate conjecture. The results strengthen strong multiplicity one by showing that algebraic relations among Hecke data determine forms up to twist, and they recover and extend several existing modular-forms distinguishability theorems while providing new generalizations to higher-level Siegel paramodular forms. Overall, the paper connects deep Galois-representation images with explicit Hecke-data criteria to distinguish modular forms in a robust, twist-aware framework with wide-ranging applications.

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.

Distinguishing Siegel modular forms

TL;DR

This work addresses when two genus 2 Siegel paramodular newforms are determined by partial Hecke data. By embedding Hecke relations into Galois representations valued in and analyzing the joint image of the product representation, the authors prove that a nontrivial polynomial relation among Hecke data for a positive-density set of primes forces the automorphic representations to be twists of each other, in particular quadratic twists when central characters are trivial. Their method yields a unified approach that also applies to elliptic modular forms, enabling a broad set of “distinguish by data” results across Hecke eigenvalues, Satake parameters, -functions, and Sato–Tate angles, without assuming the Sato–Tate conjecture. The results strengthen strong multiplicity one by showing that algebraic relations among Hecke data determine forms up to twist, and they recover and extend several existing modular-forms distinguishability theorems while providing new generalizations to higher-level Siegel paramodular forms. Overall, the paper connects deep Galois-representation images with explicit Hecke-data criteria to distinguish modular forms in a robust, twist-aware framework with wide-ranging applications.

Abstract

Let and be genus 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.

Paper Structure

This paper contains 22 sections, 33 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Galois representations attached to Siegel modular forms
  3. Images of Galois representations
  4. Results for a single Siegel modular form
  5. Results for elliptic modular forms
  6. Convention and Notation
  7. Applications: New Results and Extensions
  8. Distinguishing eigenforms by their Hecke eigenvalues
  9. Distinguishing eigenforms by their normalised eigenvalues
  10. Distinguishing eigenforms by the absolute values of their Hecke eigenvalues
  11. Distinguishing eigenforms by their Sato--Tate angles
  12. Distinguishing eigenforms by their Satake parameters
  13. Distinguishing eigenforms by their $L$-functions
  14. Distinguishing eigenforms by the distances of their eigenvalues
  15. Images of Galois representations
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $P(a,a')\in \overline{\mathbf Q}[a, a']$ be any non-zero polynomial. Suppose that for a set of primes $p$ of positive upper density Then $(k_1, k_2) = (k_1', k_2')$ and there is Dirichlet character $\chi$ such that $\Pi\simeq\Pi'\otimes\chi$. Hence, if $f,f'$ are paramodular newforms then $f$ is a scalar multiple of a quadratic twist of $f'$.

Theorems & Definitions (61)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 51 more