Number Theory Seminar

We prove that the eigencurve associated to a definite quaternion algebra over Q satisfies the following properties, as conjectured by Buzzard-Kilford and others: (a) over the boundary annuli of the weight space, the eigencurve is a disjoint union of (countably) infinitely many connected components each finite and flat over the weight annuli, (b) the Up-slopes of points on each fixed connected component are proportional to the p-adic valuations of the parameter on the weight space, and (c) the sequence of the slope ratios form a union of finitely many arithmetic progressions with the same common difference. Joint work with Daqing Wan and Liang Xiao.