I’m about to give a talk on how to conclude lots of cases of Fontaine-Mazur for 2-dimensional Gal(\bar{\mathbb{Q}}/\mathbb{Q})-representations from Emerton’s study of completed cohomology. For this to work, one needs Bockle’s arguments after Gouvea-Mazur and Coleman to deduce “big” R=T theorems from “small” ones which are not too difficult given the success of the original Taylor-Wiles-(Diamond?)-Kisin approach. I attach a set of notes saying roughly how this works and giving references to the literature.

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