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

Pramath Anamby

paper

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

arXiv:2104.00401

Abstract

We prove that a non--zero Jacobi form of arbitrary level $N$ and square--free index $m_{1} m_{2}$ with $m_{1} ∣ N$ and $(N, m_{2}) = 1$ has a non--zero theta component $h_{μ}$ with either $(μ, 2 m_{1} m_{2}) = 1$ or $(μ, 2 m_{1} m_{2}) ∤ 2 m_{2}$ . 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 $N$ .

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