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 )| ≪F,ǫ det T k−1 2 +ǫ 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.