Certain Siegel Cusp Forms with Level are Determined by their Fundamental Fourier Coefficients

TL;DR

This paper proves that certain Siegel cusp forms and Jacobi forms are uniquely determined by their fundamental Fourier coefficients or primitive theta components, under specific level and discriminant conditions, advancing understanding of their structure.

Contribution

It establishes that vector-valued Siegel cusp forms with level are uniquely identified by fundamental Fourier coefficients, especially for odd square-free levels and genus 3, and extends to Jacobi forms.

Findings

Siegel cusp forms are determined by fundamental Fourier coefficients coprime to level N

For genus 3, determination extends to coefficients from maximal orders in quaternion algebras

Jacobi forms are determined by primitive theta components with discriminant coprime to N

Abstract

We prove that vector-valued Siegel cusp forms for $\Gamma_0^n(N)$ with certain nebentypus are determined by their fundamental Fourier coefficients with discriminants coprime to the level $N$, assuming $N$ is odd and square-free. In the case of genus $3$, we strengthen this to Fourier coefficients corresponding to maximal orders in quaternion algebras. We also prove that Jacobi forms of fundamental index with discriminant coprime to the odd level $N$ are determined by their primitive theta components.