The eigencurve over the boundary of weight space

TL;DR

This paper proves detailed structural properties of the eigencurve over boundary regions of weight space, confirming conjectures about its geometry, slopes, and arithmetic progressions, with implications for p-adic modular forms.

Contribution

The paper establishes the geometric and slope properties of the eigencurve over boundary annuli, confirming conjectures by Coleman--Mazur and Buzzard--Kilford.

Findings

Eigencurve over boundary annuli decomposes into infinitely many finite flat components.

U_p-slope ratios are proportional to p-adic valuations of weight parameters.

Slope ratios form finitely many arithmetic progressions.

Abstract

We prove that the eigencurve associated to a definite quaternion algebra over $\QQ$ satisfies the following properties, as conjectured by Coleman--Mazur and Buzzard--Kilford: (a) over the boundary annuli of weight space, the eigencurve is a disjoint union of (countably) infinitely many connected components each finite and flat over the weight annuli, (b) the $U_p$-slopes of points on each fixed connected component are proportional to the $p$-adic valuations of the parameter on weight space, and (c) the sequence of the slope ratios form a union of finitely many arithmetic progressions with the same common difference. In particular, as a point moves towards the boundary on an irreducible connected component of the eigencurve, the slope converges to zero.