Mark Kisin

Title: Overconvergent modular forms and Galois representations.

Abstract: The theory of p-adic modular forms grew out of the study of congruences between classical modular forms. The most interesting type of modular forms are eigenforms. Several years ago Coleman and Mazur showed that classical eigenforms with non-zero U_p eigenvalue could be p-adically interpolated into a rigid analytic curve

- the "Eigencurve".

The points on this curve correspond to so called "overconvergent eigenforms". There is a growing body of evidence that these forms deserve to be regarded as (examples of) the p-adic analogues of motives. In particular attached to such a form is a p-adic representation of the absolute Galois group of Q. I will explain how a study of these Galois representations yields new insights and results on questions which include - Modularity of Galois representations - Finiteness of certain Selmer groups - The structure of unrestricted deformations rings.