Siegel cusp forms of degree 2 are determined by their fundamental Fourier coefficients

TL;DR

This paper proves that Siegel cusp forms of degree 2 are uniquely determined by their Fourier coefficients at certain fundamental indices, extending known results to non-eigenforms and connecting to classical half-integral weight forms.

Contribution

It establishes that Siegel cusp forms of degree 2 are determined by Fourier coefficients at odd squarefree determinants, and generalizes the determination result for classical cusp forms of half-integral weight beyond Hecke eigenforms.

Findings

Siegel cusp forms are determined by Fourier coefficients at odd squarefree determinants.

Classical cusp forms of half-integral weight are determined by Fourier coefficients at odd squarefree integers.

Extension of known results from Hecke eigenforms to all cusp forms.

Abstract

We prove that a Siegel cusp form of degree 2 for the full modular group is determined by its set of Fourier coefficients a(S) with 4 det(S) ranging over odd squarefree integers. As a key step to our result, we also prove that a classical cusp form of half-integral weight and level 4N, with N odd and squarefree, is determined by its set of Fourier coefficients a(d) with d ranging over odd squarefree integers, a result that was previously known only for Hecke eigenforms.