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.

Abstract. Recent results by Abert, Bergeron, Biringer et al., Finis, Lapid and Mueller, and Shin and Templier have extended the limit multiplicity property to quite general classes of groups and sequences of level subgroups. Automorphic representations in the limit multiplicity problem are traditionally counted with multiplicity according to the number of fixed vectors of a level subgroup; our goal is to perform a slightly more refined analysis and count only automorphic representations with a given conductor with multiplicity 1.

This paper examines fields of rationality in families of cuspidal automorphic representations of unitary groups. Specifically, for a fixed A and a sufficiently large family F , a small proportion of representations π ∈ F will satisfy [Q(π) : Q] ≤ A. Like earlier work of Shin and Templier, the result depends on a Plancherel equidistribution result for the local components of representations in families. An innovation of our work is an upper bound on the number of discrete series GLn(L) representations with small field of rationality, counted with appropriate multiplicity, which in turn depends upon an asymptotic character expansion of Murnaghan and formal degree computations of Aubert and Plymen.