Local to Global Compatibility on the Eigencurve

We generalise Coleman’s construction of Hecke operators to define an action of GL2(Ql) on the space of finite slope overconvergent p-adic modular forms (l 6= p). In this way we associate to any Cp-valued point on the tame level N Coleman-Mazur eigencurve an admissible smooth representation of GL2(Ql) extending the classical construction. Using the Galois theoretic interpretation of the eigencurve we associate a 2-dimensional Weil-Deligne representation to such points and show that away from a discrete set they agree under the Local Langlands correspondence.