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We prove new cases of the Fontaine-Mazur conjecture, that a 2-dimensional p-adic representation of G ₐ, 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.
Mark Kisin (Wed,) studied this question.