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.