Fields of rationality of cusp forms

Abstract

In this paper, we prove that for any totally real field F, weight k, and nebentypus character χ, the proportion of Hilbert cusp forms over F of weight k and character χ with bounded field of rationality approaches zero as the level grows large. This answers, in the affirmative, a question of Serre. The proof has three main inputs: first, a lower bound on fields of rationality for admissible GL2 representations; second, an explicit computation of the (fixed-central-character) Plancherel measure for GL2; and third, a Plancherel equidistribution theorem for cusp forms with fixed central character. The equidistribution theorem is the key intermediate result and builds on earlier work of Shin and Shin–Templier and mirrors work of Finis–Lapid–Mueller by introducing an explicit bound for certain families of orbital integrals.