Overconvergent modular forms and the Fontaine-Mazur conjecture

Abstract

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 ρ 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 ρ, which is defined by a purely representation theoretic condition, and contains the eigencurve of Coleman-Mazur. Conjecturally, it is equal to this curve.