Number Theory Seminar

The study of eigenvarieties began with Coleman and Mazur, who constructed the first eigencurve, a rigid analytic space whose points are in bijection with $p$-adic modular Hecke eigenforms. Since then various authors have constructed eigenvarieties for automorphic forms on many other groups. We will state a structure theorem about Chenevier's eigenvarieties for definite unitary groups which generalizes slope bounds of Liu-Wan-Xiao for dimension $2$ 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.