Fields of rationality of automorphic representations: The case of unitary groups

Abstract

This paper examines fields of rationality in families of cuspidal automorphic representations of unitary groups. Specifically, for a fixed A and a sufficiently large family F , a small proportion of representations π ∈ F will satisfy [Q(π) : Q] ≤ A. Like earlier work of Shin and Templier, the result depends on a Plancherel equidistribution result for the local components of representations in families. An innovation of our work is an upper bound on the number of discrete series GLn(L) representations with small field of rationality, counted with appropriate multiplicity, which in turn depends upon an asymptotic character expansion of Murnaghan and formal degree computations of Aubert and Plymen.