Equidistribution theorems for holomorphic Siegel cusp forms of general degree: the level aspect

Abstract

This paper is an extension of [26] and we prove equidistribution theorems for families of holomorphic Siegel cusp forms of general degree in the level aspect. Our main contribution is to estimate unipotent contributions for general degree in the geometric side of Arthur's invariant trace formula in terms of Shintani zeta functions in a uniform way. Several applications including the vertical Sato-Tate theorem and low-lying zeros for standard L-functions of holomorphic Siegel cusp forms are discussed. We also show that the "non-genuine forms" which come from non-trivial endoscopic contributions by Langlands functoriality classified by Arthur are negligible.

A remark on conductor, depth and principal congruence subgroups

“…in some principal congruence subgroups. Our study suggests to consider new forms with respect to principal congruence subgroups introduced in Definition 5.1 in [8]. This might be helpful to guarantee Hypothesis 11.4,p.129 of [16].…”

“…kind of results to, for example, equidistribution theorems as in [16], [8] in the level aspects, we need to know an explicit form of the fundamental lemma for Hecke elements. In most cases, it is done for the characteristic functions for principal congruence subgroups.…”

“…A motivation in our study comes up to authors in an estimation of upper and lower bounds of the conductor of automorphic families in the setting of low lying zeros [16], [8] (see also Remark 4.2).…”

Isogeny graphs on superspecial abelian varieties: Eigenvalues and Connection to Bruhat-Tits buildings

“…Section 3.9 of [Gan08]). However, such forms are expected to be negligible among all forms when p goes to infinity and this is in fact true for Siegel modular forms on GSp g (see [KWY20], [KWY21]).…”