Fields of Rationality of Cusp Forms

Abstract

In this paper, we prove that for any totally real field $F$, weight $k$, and nebentypus character $\chi$, the proportion of Hilbert cusp forms over $F$ of weight $k$ and character $\chi$ 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 $G{L}_2$ representations; second, an explicit computation of the (fixed-central-character) Plancherel measure for $G{L}_2$; and third, a Plancherel equidsitribution 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.

41 pages

References in corpus (2)

• Notes on the arithmetic of Hilbert modular forms
• An approximation principle for congruence subgroups