Overconvergent modular forms and the Fontaine-Mazur conjecture

We prove a conjecture of Fontaine and Mazur on modularity of representations of G(Q) which are potentially semi-stable at p, for representations coming from finite slope, overconvergent eigenforms. We also give an application of our technique to a question of Gouvea on overconvergence of certain modular forms. If rho is a suitable representation of G(Q,S) on a two dimensional vector space over a finite field product of G(m) and the S a finite set of primes-we construct a rigid analytic subspace of the universal deformation space of rho, which is defined by a purely representation theoretic condition, and contains the eigencurve of Coleman-Mazur. Conjecturally, it is equal to this curve.