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

Henry H. Kim, Satoshi Wakatsuki and Takuya Yamauchi

Vol. 18 (2024), No. 5, 993–1038
Abstract

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.

Keywords
trace formula, holomorphic Siegel modular forms, equidistribution theorems, standard $L$-functions
Mathematical Subject Classification
Primary: 11F46, 11F70, 22E55, 11R45
Milestones
Received: 26 October 2022
Revised: 17 April 2023
Accepted: 29 May 2023
Published: 16 April 2024
Authors
Henry H. Kim
Department of Mathematics
University of Toronto
Toronto, ON
Canada
Korea Institute for Advanced Study
Seoul
Korea
Satoshi Wakatsuki
Faculty of Mathematics and Physics
Institute of Science and Engineering
Kanazawa University
Ishikawa
Japan
Takuya Yamauchi
Mathematical Institute
Tohoku University
Sendai
Japan

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