Eigencurve was introduced by Coleman and Mazur to parametrize
modular forms varying p-adically. It is a rigid analytic curve such that
each point corresponds to an overconvegent eigenform. In this talk, we
discuss a result on the geometry of the eigencurve: over the boundary
annuli of the weight space, the eigencurve breaks up into infinite disjoint
union of connected components and the weight map is finite and flat on each
component. This was first observed by Buzzard and Kilford by an explicit
computation in the case of p=2 and tame level 1. We will explain a
generalization to the definite quaternion case with no restriction on p and
the tame level. This is a joint work with Ruochuan Liu and Daqing Wan,
based on an idea of Robert Coleman.