Fields of rationality of cusp forms

• Main Author: Binder, John (Author)
• Format: Article
• Language: English
• Published: The Hebrew University Magnes Press, 2018-04-09T19:48:27Z.
• Subjects: Article
• Online Access: Get fulltext

 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.