Statistics of Cohomological Automorphic Representations on Unitary Groups via the Endoscopic Classification

TL;DR

This paper computes the asymptotic distribution and statistics of cohomological automorphic representations on unitary groups using endoscopic classification, with implications for cohomology growth, equidistribution, and density hypotheses.

Contribution

It provides exact leading terms for counts and averages of automorphic representations on unramified unitary groups, extending previous bounds and establishing new asymptotic results.

Findings

Exact asymptotics for representation counts

New upper bounds for cohomological representations

Verification of Sarnak-Xue density hypothesis

Abstract

Consider the family of automorphic representations on a unitary group with cohomological factor $\pi_0$ at infinity and given split level. We compute statistics of this family as the level goes to infinity. For unramified unitary groups and a large class of $\pi_0$, we use the endoscopic classification of representations to compute the exact leading term for counts of representations and averages of Satake parameters. The bounds on our error terms are similar to previous work by Shin-Templier who studied the case of discrete series at infinity. We also prove new upper bounds for all cohomological representations. This has many corollaries: new exact asymptotics on the growth of cohomology in certain towers of locally symmetric spaces, an averaged Sato-Tate equidistribution law for spectral families with specific non-tempered cohomological components at infinity, and the Sarnak-Xue density hypothesis for cohomological representations at infinity on all unitary groups of rank $\geq 5$.