The eigencurve over the boundary of weight space

We prove that the eigencurve associated to a definite quaternion algebra over $\text{QQ}$ 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 © 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.