Equidistribution theorems for holomorphic Siegel modular forms for GSp4; Hecke fields and n-level density

• Published in: Mathematische Zeitschrift Vol. 295; no. 3-4; pp. 917 - 943
• Main Authors: Kim, Henry H., Wakatsuki, Satoshi, Yamauchi, Takuya
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 2020; Springer Nature B.V
• Subjects: Density; Eigenvalues; Mathematical analysis; Mathematics; Mathematics and Statistics; Subgroups; Theorems; 11F46; 11F70; Trace formula; 11R45; Siegel modular forms; Hecke fields; 22E55
• ISSN: 0025-5874; 1432-1823
• DOI: 10.1007/s00209-019-02378-7

This paper is a continuation of Kim et al. (J Inst Math Jussieu, 2018 ). We supplement four results on a family of holomorphic Siegel cusp forms for G S p 4 / 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 G S p 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 the conjecture on root numbers. Main tools are the equidistribution theorem in our previous work and Shin–Templier’s (Compos Math 150(12):2003–2053, 2014 ) work.