Arithmetic Geometry and Number Theory RTG Seminar: Slopes in eigenvarieties for definite unitary groups

Seminar | November 5 | 3:10-5 p.m. | 748 Evans Hall

 Lynnelle Ye, Harvard

 Department of Mathematics

Pre-talk: 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 many other kinds of automorphic forms. We will define automorphic forms on definite unitary groups and explain Chenevier's construction of eigenvarieties for these forms.

Main talk: we will state a structure theorem about Chenevier's eigenvarieties for definite unitary groups which generalizes slope bounds of Liu-Wan-Xiao for dimension $2$, which they used to prove the Coleman-Mazur-Buzzard-Kilford conjecture, to any dimension $n$. The theorem says that the Newton polygon for the eigenvariety over a fixed weight has growth rate proportional to $x^{1+\frac 2{n(n-1)}}$ with constant proportional to distance from the boundary of weight space. Then we will discuss the ideas of the proof, which goes through the classification of automorphic representations that are principal series at $p$, and a geometric consequence.