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The Fontaine-Mazur conjecture in the residually reducible case
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We prove new cases of Fontaine-Mazur conjecture on two-dimensional Galois representations over Q when the residual representation is reducible. Our approach is via a semi-simple local-global compatibility of the completed cohomology and a Taylor-Wiles patching argument for the completed homology in this case. As a key input, we generalize the work of Skinner-Wiles in the ordinary case. In addition, we also treat the residually irreducible case at the end of the paper. Combining with people's earlier work, we can prove the Fontaine-Mazur conjecture completely in the regular case when p is at least 5.
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On the modularity of 2-adic potentially semi-stable deformation rings
For p=2, the support of patched modules meets every irreducible component of the potentially semi-stable deformation ring, yielding the Breuil-Mezard conjecture in the case where the residual representation is a twist...
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