# Eigencurve over the boundary of the weight space

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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.