# Slopes in eigenvarieties for definite unitary groups

-
Lynnelle Ye, Harvard University
IAS - Simonyi Hall Seminar Room SH-101

The study of eigenvarieties began with Coleman and Mazur, who constructed the first eigencurve, a rigid analytic space parametrizing p-adic modular Hecke eigenforms. Since then various authors have constructed eigenvarieties for automorphic forms on many other groups. We will give bounds on the eigenvalues of the U_p Hecke operator appearing in Chenevier's eigenvarieties for definite unitary groups. These bounds generalize ones of Liu-Wan-Xiao for dimension 2, which they used to prove a conjecture of Coleman-Mazur-Buzzard-Kilford in that setting, to all dimensions. We will then discuss the ideas of the proof, which goes through the classification of automorphic representations that are principal series at p, and a geometric consequence.