Fields of Rationality of Cusp Forms

TL;DR

This paper proves that as the level increases, the proportion of Hilbert cusp forms over a totally real field with bounded rationality fields tends to zero, confirming a question posed by Serre.

Contribution

It establishes the asymptotic behavior of rationality fields of Hilbert cusp forms, using new bounds and explicit measures, advancing understanding of automorphic forms over totally real fields.

Findings

Proportion of forms with bounded rationality fields approaches zero as level grows

Provides explicit bounds for orbital integrals in the context of automorphic forms

Develops an equidistribution theorem for cusp forms with fixed central character

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 $GL_2$ representations; second, an explicit computation of the (fixed-central-character) Plancherel measure for $GL_2$; 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.