Kisin, Mark2015-03-132009Kisin, Mark. 2009. “The Fontaine-Mazur Conjecture for {GL}_2.” Journal of the American Mathematical Society 22, no. 3: 641–690.0894-0347http://nrs.harvard.edu/urn-3:HUL.InstRepos:14168860We prove new cases of the Fontaine-Mazur conjecture, that a 2 -dimensional p -adic representation rho of G_{{Q}, 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-Mezard 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.en-USJournal Article2015-03-1310.1090/S0894-0347-09-00628-6