On Fields of rationality for automorphic representations

Abstract

This paper proves two results on the field of rationality $\Q(π)$ for an automorphic representation $π$, which is the subfield of $\C$ fixed under the subgroup of $\Aut(\C)$ stabilizing the isomorphism class of the finite part of $π$. For general linear groups and classical groups, our first main result is the finiteness of the set of discrete automorphic representations $π$ such that $π$ is unramified away from a fixed finite set of places, $π_\infty$ has a fixed infinitesimal character, and $[\Q(π):\Q]$ is bounded. The second main result is that for classical groups, $[\Q(π):\Q]$ 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.