On sign changes of primitive Fourier coefficients of Siegel cusp forms

TL;DR

This paper investigates the sign changes of primitive Fourier coefficients of Siegel cusp forms, providing quantitative results and establishing non-vanishing properties of related Fourier-Jacobi and theta components.

Contribution

It introduces new quantitative results on sign changes of Fourier coefficients and proves non-vanishing of specific Fourier-Jacobi and theta components of Siegel cusp forms.

Findings

Sign changes in subsequences of primitive Fourier coefficients.

Non-vanishing of Fourier-Jacobi coefficients.

Sign changes of diagonal Fourier coefficients in degree two forms.

Abstract

In this article, we establish quantitative results for sign changes in certain subsequences of primitive Fourier coefficients of a non-zero Siegel cusp form of arbitrary degree over congruence subgroups. As a corollary of our result for degree two Siegel cusp forms, we get sign changes of its diagonal Fourier coefficients. In the course of our proofs, we prove the non-vanishing of certain type of Fourier-Jacobi coefficients of a Siegel cusp form and all theta components of certain Jacobi cusp forms of arbitrary degree over congruence subgroups, which are also of independent interest.