We prove that the eigencurve associated to a definite quaternion algebra over satisfies the following properties, as conjectured by Coleman and Mazur as well as Buzzard and 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 -slopes of points on each fixed connected component are proportional to the -adic valuations of the parameter on weight space; and (c) the sequence of the slope ratios forms a union of finitely many arithmetic progressions with the same common difference. In particular, as a point moves toward the boundary on an irreducible connected component of the eigencurve, the slope converges to zero.
"The eigencurve over the boundary of weight space." Duke Math. J. 166 (9) 1739 - 1787, 15 June 2017. https://doi.org/10.1215/00127094-0000012X