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

This paper is an extension of Kim et al. (2020a), 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 "nongenuine forms", which come from nontrivial endoscopic contributions by Langlands functoriality classified by Arthur, are negligible.1. Introduction 993 2. Preliminaries 998 3. Asymptotics of Hecke eigenvalues 1000 4. Arthur classification of Siegel modular forms 1010 5.A notion of newforms in S k ..N // 1017 6. Equidistribution theorem of Siegel cusp forms; proof of Theorem 1.1 1019 7. Vertical Sato-Tate theorem for Siegel modular forms: proofs of Theorems 1.2 and 1.3 1020 8. Standard L-functions of Sp.2n/ 1021 9. `-level density of standard L-functions 1024 10.The order of vanishing of standard L-functions at s D 1 2