Fields of rationality of cuspidal automorphic representations

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2016.Cataloged from PDF version of thesis.Includes bibliographical references (pages 115-120).This thesis examines questions related to the growth of fields of rationality of cuspidal automorphic representations in families. Specifically, if F is a family of cuspidal automorphic representations with fixed central character, prescribed behavior at the Archimedean places, and such that the finite component [pi] [infinity] has a [Gamma]-fixed vector, we expect the proportion of [pi] [epsilon] F with bounded field of rationality to be close to zero if [Gamma] is small enough. This question was first asked, and proved partially, by Serre for families of classical cusp forms of increasing level. In this thesis, we will answer Serre's question affirmatively by converting the question to a question about fields of rationality in families of cuspidal automorphic GL2(A) representations. We will consider the analogous question for certain sequences of open compact subgroups F in UE/F(n). A key intermediate result is an equidistribution theorem for the local components of families of cuspidal automorphic representations.by John Binder.Ph. D