Sign changes of Fourier coefficients of Siegel cusp forms of degree two on Hecke congruence subgroups

TL;DR

This paper establishes lower bounds and explicit upper bounds for the first sign change of Fourier coefficients of degree two Siegel cusp forms on Hecke congruence subgroups, advancing understanding of their oscillatory behavior.

Contribution

It provides new quantitative bounds on sign changes of Fourier coefficients for degree two Siegel cusp forms on specific subgroups, extending previous results to these settings.

Findings

Lower bound on the number of sign changes

Explicit upper bound for the first sign change

Extension of previous bounds to Hecke congruence subgroups

Abstract

In this article, the authors give a lower bound on the number of sign changes of Fourier coefficients of a non-zero degree two Siegel cusp form of even integral weight on a Hecke congruence subgroup. They also provide an explicit upper bound for the first sign change of Fourier coefficients of such Siegel cusp forms. Explicit upper bound on the first sign change of Fourier coefficients of a non-zero Siegel cusp form of even integral weight on the Siegel modular group for arbitrary genus were dealt in an earlier work of Choie, the first author and Kohnen.