Automorphic representations with prescribed ramification for unitary groups

TL;DR

This paper establishes an asymptotic count for cuspidal automorphic representations of certain unitary groups with specified local behavior, extending previous methods from GL_2 to more general groups.

Contribution

It generalizes Weinstein's techniques to unitary groups over totally real fields, providing new asymptotic formulas for automorphic representations with prescribed ramification.

Findings

Asymptotic formula for automorphic representations with prescribed local factors

Extension of methods from GL_2 to unitary groups

Results applicable to totally real fields

Abstract

Let F be a totally real number field, n a prime integer, and G a unitary group of rank n defined over F that is compact at every infinite place. We prove an asymptotic formula for the number of cuspidal automorphic representations of G whose factors at finitely many places are prescribed up to inertia. The results and the methods used are a generalization to this setting of those used by Weinstein for GL_2.