The Fontaine-Mazur conjecture in the residually reducible case

We prove new cases of Fontaine-Mazur conjecture on two-dimensional Galois representations over \mathbb {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\geq 5.