The Fontaine-Mazur conjecture for GL₂

We prove new cases of the Fontaine-Mazur conjecture, that a $2$-dimensional p-adic representation ρ of GQ, S which is potentially semi-stable at p with distinct Hodge-Tate weights arises from a twist of a modular eigenform of weight k≥ 2. Our approach is via the Breuil-Mézard conjecture, which we prove (many cases of) by combining a global argument with recent results of Colmez and Berger-Breuil on the p-adic local Langlands correspondence.