Non-vanishing of theta components of Jacobi forms with level and an application

TL;DR

This paper proves that non-zero Jacobi forms with specific level and index conditions have non-zero theta components and applies this to show that certain Siegel cusp forms are determined by fundamental Fourier coefficients.

Contribution

It establishes non-vanishing of specific theta components of Jacobi forms with level and index constraints, and uses this to determine Siegel cusp forms by fundamental Fourier coefficients.

Findings

Non-zero Jacobi forms have non-zero theta components under given conditions.

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

Application of theta component non-vanishing to Siegel modular forms.

Abstract

We prove that a non--zero Jacobi form of arbitrary level $N$ and square--free index $m_1m_2$ with $m_1|N$ and $(N,m_2)=1$ has a non--zero theta component $h_\mu$ with either $(\mu,2m_1m_2)=1$ or $(\mu,2m_1m_2)\nmid 2m_2$. As an application, we prove that a non--zero Siegel cusp form $F$ of degree $2$ and an odd level $N$ in the Atkin--Lehner type newspace is determined by fundamental Fourier coefficients up to a divisor of $N$.