On Fields of rationality for automorphic representations

TL;DR

This paper investigates the field of rationality of automorphic representations, proving finiteness results for certain groups and showing unbounded growth of the degree of the field of rationality in specific families.

Contribution

It establishes finiteness of automorphic representations with bounded rationality degree and demonstrates unbounded growth of this degree in level aspect for classical groups.

Findings

Finiteness of automorphic representations with bounded field of rationality degree.

Unbounded growth of the degree of the field of rationality in level aspect for classical groups.

Results apply to general linear and classical groups.

Abstract

This paper proves two results on the field of rationality $\Q(\pi)$ for an automorphic representation $\pi$, which is the subfield of $\C$ fixed under the subgroup of $\Aut(\C)$ stabilizing the isomorphism class of the finite part of $\pi$. For general linear groups and classical groups, our first main result is the finiteness of the set of discrete automorphic representations $\pi$ such that $\pi$ is unramified away from a fixed finite set of places, $\pi_\infty$ has a fixed infinitesimal character, and $[\Q(\pi):\Q]$ is bounded. The second main result is that for classical groups, $[\Q(\pi):\Q]$ grows to infinity in a family of automorphic representations in level aspect whose infinite components are discrete series in a fixed $L$-packet under mild conditions.