Distribution of Satake parameters of GL₂ holomorphic cuspidal representations

Dr. Charles, Chun Che LI

Academia Sinica, Institute of Mathematics, Taiwan

Abstract

We prove that for a fixed non-archimedean  place v of a totally real number field F, the traces of the associated Langlands' conjugacy classes of holomorphic cuspidal representations of GL₂(A_F) of prime levels is equidistributed with respect to the measure

where q_v is the norm of v and  is the Sato-Tate measure. This generalizes a result of Serre on the distribution of Hecke eigenvalues of modular forms. The proof involves establishing a trace formula for the Hecke operators. While not explicit, this trace formula can be used as a starting point for generalizing the Eichler-Selberg trace formula to totally real number fields.