On Fields of rationality for automorphic representations

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.