Endoscopy and cohomology of a quasi-split U(4)

TL;DR

This paper establishes asymptotic upper bounds for the $L^2$ Betti numbers of 8-dimensional locally symmetric spaces related to quasi-split U(4), using automorphic representation classification, with conjectures on sharpness in degree 3.

Contribution

It provides the first asymptotic bounds for Betti numbers in this setting, leveraging endoscopic classification of automorphic representations.

Findings

Upper bounds for Betti numbers in degrees 2 and 3

Behavior in other degrees is well understood

Conjecture that bounds in degree 3 are sharp

Abstract

We prove asymptotic upper bounds for the $L^2$ Betti numbers of the locally symmetric spaces associated to a quasi-split $U(4)$. These manifolds are 8-dimensional, and we prove bounds in degrees 2 and 3, with the behaviour in the other degrees being well understood. In degree 3, we conjecture that these bounds are sharp. Our main tool is the endoscopic classification of automorphic representations of U(N) by Mok.