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

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 $| a(T) | ~\ll_{F, ε}~ \big( \det T \big)^{\frac{k-1}{2}+ε}$ 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.