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

Kim, Henry H.; Wakatsuki, Satoshi; Yamauchi, Takuya

doi:10.2140/ant.2024.18.993

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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

DOI: 10.2140/ant.2024.18.993

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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Vol. 18, No. 5, 2024