Endoscopy and cohomology of U(n,1)

TL;DR

This paper investigates the growth of cohomology in towers of locally symmetric spaces related to the unitary group U(n,1), assuming the ongoing endoscopic classification of automorphic representations, and conjectures sharp bounds in certain cases.

Contribution

It provides bounds on cohomology growth for U(n,1) spaces based on endoscopic classification, advancing understanding under current conjectural frameworks.

Findings

Bounded cohomology growth in congruence towers.

Conjecture of sharp growth exponents in Hermitian form cases.

Progress contingent on endoscopic classification assumptions.

Abstract

By assuming the endoscopic classification of automorphic representations on inner forms of unitary groups, which is currently work in progress by Kaletha, Minguez, Shin, and White, we bound the growth of cohomology in congruence towers of locally symmetric spaces associated to $U(n,1)$. In the case of lattices arising from Hermitian forms, we conjecture that the growth exponents we obtain are sharp in all degrees.