Fields of rationality of cusp forms

Abstract

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 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.

Refined Limit Multiplicity for Varying Conductor

“…We will also prove a fixed-central-character analog of the limit multiplicity result in this case, following the author's previous work ([Bin15]). Let χ : A × → C × be an automorphic character of conductor f, and let χ S , χ S be its components at S and away from S respectively; write f S for the conductor of χ S .…”

“…7.3 of [Sau97]. The fixed-central-character statement follows by the same logic as in Lemma 11.2.7 of [Bin15].…”

“…The existence of a fixed-central-character measure is known to the experts but to our knowledge is not written down fully. In Proposition 6.2.8 and Subsection 11.2 of [Bin15], the author constructs such a measure from the non-fixed character case using abelian Fourier analysis in the p-adic GL 2 case; the same proof goes through in the situation here.…”

Fields of rationality of automorphic representations: The case of unitary groups

“…To our knowledge, the construction of the fixed central character Plancherel measure has not been written down explicitly. However, the construction follows from abelian Fourier analysis and the non-fixed central character Plancherel measure as in [Bin15].…”

“…Serre posited that his theorem could be extended to arbitrary sequences of levels; this was completed by the author in [Bin15], following work of Shin and Templier in [ST14]. In their paper, Shin and Templier considered fields of rationality of representations of classical groups in families more generally.…”