The Fontaine-Mazur conjecture in the residually reducible case

Abstract

In this thesis, 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 also generalize works of Skinner-Wiles in the ordinary case.