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

TL;DR

This paper studies the fields of rationality of automorphic representations of unitary groups, showing that only a small proportion have a bounded degree of rationality in large families, using advanced harmonic analysis and representation theory techniques.

Contribution

It introduces new bounds on the number of discrete series representations with small fields of rationality, extending previous work with novel character expansion and formal degree methods.

Findings

Most automorphic representations have large fields of rationality.

New bounds on discrete series with small rationality fields.

Dependence on Plancherel equidistribution and character expansion techniques.

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 $\mathcal{F}$, a small proportion of representations $\pi\in \mathcal{F}$ will satisfy $[\mathbb{Q}(\pi):\mathbb{Q}] \leq A$. 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 $GL_n(L)$ 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.