On fields of rationality for automorphic representations

Abstract

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

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