Slopes in eigenvarieties for definite unitary groups

Abstract

We generalize bounds of Liu-Wan-Xiao for slopes in eigencurves for definite unitary groups of rank $2$, which formed the core of their proof of the Coleman-Mazur-Buzzard-Kilford conjecture about the decomposition of the eigencurve over the boundary of weight space, to eigenvarieties for definite unitary groups of any rank. We show that for a definite unitary group of rank $n$, the Newton polygon of the characteristic power series of the $U_p$ Hecke operator has exact polynomial growth rate $x^{1+\frac2{n(n-1)}}$, with constant proportional to the distance of the weight from the boundary of weight space. This improves a previous lower bound of $x^{1+\frac1{2^n-n-1}}$ of Chenevier (which applied only to the center of weight space). The proof goes through the classification of forms associated to principal series representations. We also give a consequence for the geometry of these eigenvarieties over the boundary of weight space.