Equidistribution theorems for holomorphic Siegel modular forms for $GSp_4$; Hecke fields and $n$-level density

Abstract

This paper is a continuation of the author's previous wotk. We supplement four results on a family of holomorphic Siegel cusp forms for $GSp_4/\mathbb{Q}$. First, we improve the result on Hecke fields. Namely, we prove that the degree of Hecke fields is unbounded on the subspace of genuine forms which do not come from functorial lift of smaller subgroups of $GSp_4$ under a conjecture in local-global compatibility and Arthur's classification for $GSp_4$. Second, we prove simultaneous vertical Sato-Tate theorem. Namely, we prove simultaneous equidistribution of Hecke eigenvalues at finitely many primes. Third, we compute the $n$-level density of degree 4 spinor $L$-functions, and thus we can distinguish the symmetry type depending on the root numbers. This is conditional on certain conjecture on root numbers. Fourth, we consider equidistribution of paramodular forms. In this case, we can prove a result on root numbers. Main tools are the equidistribution theorem in our previous work and Shin-Templier's work.