On the automorphy of potentially semi-stable deformation rings

Using p-adic local Langlands correspondence for GL2(Qp) and an ordinary R = T theorem, we prove that the support of patched modules for quaternionic forms meet every irreducible component of the potentially semi-stable deformation ring. This gives a new proof of the Breuil-Mézard conjecture for 2-dimensional representations of the absolute Galois group of Qp, which is new in the case p=2 or p=3 and rbar a twist of an extension of the trivial character by the mod p cyclotomic character. As a consequence, a local restriction in Kisin's proof of Fontaine-Mazur conjecture is removed.



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