On Fourier coefficients and Hecke eigenvalues of Siegel cusp forms of degree 2

TL;DR

This paper studies Fourier coefficients and Hecke eigenvalues of Siegel cusp forms of degree 2, providing bounds, sign change results, and confirming conjectures for special cases, advancing understanding of their analytic properties.

Contribution

It offers new bounds for Hecke eigenvalues, demonstrates infinite sign changes, and confirms a conjecture on Fourier coefficient bounds for Saito-Kurokawa forms.

Findings

Improved bounds for the smallest prime with negative Hecke eigenvalue.

Proven infinite sign changes among Hecke eigenvalues in arithmetic progressions.

Established bounds on Fourier coefficients for Saito-Kurokawa type forms.

Abstract

We investigate some key analytic properties of Fourier coefficients and Hecke eigenvalues attached to scalar-valued Siegel cusp forms $F$ of degree 2, weight $k$ and level $N$. First, assuming that $F$ is a Hecke eigenform that is not of Saito-Kurokawa type, we prove an improved bound in the $k$-aspect for the smallest prime at which its Hecke eigenvalue is negative. Secondly, we show that there are infinitely many sign changes among the Hecke eigenvalues of $F$ at primes lying in an arithmetic progression. Third, we show that there are infinitely many positive as well as infinitely many negative Fourier coefficients in any ``radial" sequence comprising of prime multiples of a fixed fundamental matrix. Finally we consider the case when $F$ is of Saito--Kurokawa type, and in this case we prove the (essentially sharp) bound $| a(T) | ~\ll_{F, \epsilon}~ \big( \det T \big)^{\frac{k-1}{2}+\epsilon}$ for the Fourier coefficients of $F$ whenever $\gcd(4 \det(T), N)$ is squarefree, confirming a conjecture made (in the case $N=1$) by Das and Kohnen.