Sato–Tate equidistribution for families of automorphic representations through the stable trace formula

Dalal, Rahul

doi:10.2140/ant.2022.16.59

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Sato–Tate equidistribution for families of automorphic representations through the stable trace formula

Rahul Dalal

Vol. 16 (2022), No. 1, 59–137

DOI: 10.2140/ant.2022.16.59

Abstract

Shin and Templier studied families of automorphic representations with local restrictions: roughly, Archimedean components contained in a fixed $L$ -packet of discrete series and non-Archimedean components ramified only up to a fixed level. They computed limiting statistics of local components as either the weight of the $L$ -packet or level went to infinity. We extend their weight-aspect results to families where the Archimedean component is restricted to a single discrete-series representation instead of an entire $L$ -packet.

We do this by using a so-called “hyperendoscopy” version of the stable trace formula of Ferarri. The main technical difficulties are first, defining a version of hyperendoscopy that works for groups without simply connected derived subgroup and second, bounding the values of transfers of unramified functions. We also present an extension to noncuspidal groups of Arthur’s simple trace formula since it does not seem to appear elsewhere in the literature.

Keywords

automorphic representations, trace formula, limit multiplicity, stable trace formula, endoscopy

Mathematical Subject Classification 2010

Primary: 11F55

Secondary: 11F70, 11F72, 11F75, 22E50, 22E55

Milestones

Received: 29 November 2019

Revised: 18 February 2021

Accepted: 21 April 2021

Published: 22 February 2022

Authors

Rahul Dalal
Department of Mathematics University of California Berkeley, CA United States

Vol. 16, No. 1, 2022