Asymptotics of automorphic spectra and the trace formula

TL;DR

This survey explores the asymptotic behavior of automorphic spectra and the trace formula, focusing on the discrete spectrum's limits, Weyl law, limit multiplicity, and applications to arithmetic group cohomology.

Contribution

It provides a comprehensive overview of the asymptotic properties of automorphic spectra and their connections to the trace formula and arithmetic applications.

Findings

Analysis of the Weyl law and limit multiplicity in automorphic spectra

Connections between spectral asymptotics and cohomology of arithmetic groups

Discussion of analytic torsion and spectral limits

Abstract

This paper is a survey article on the limiting behavior of the discrete spectrum of the right regular representation in $L^2(\Gamma\bs G)$ for a lattice $\Gamma$ in a reductive group $G$ over a number field. We discuss various aspects of the Weyl law, the limit multiplicity problem, the analytic torsion, and applications to the cohomology of arithmetic groups.