The Fontaine-Mazur conjecture in the residually reducible case

TL;DR

This paper advances the understanding of the Fontaine-Mazur conjecture by proving new cases for residually reducible Galois representations over Q, employing local-global compatibility and Taylor-Wiles methods.

Contribution

It introduces a novel approach combining semi-simple local-global compatibility with Taylor-Wiles patching for residually reducible cases, extending previous work to cover more instances.

Findings

Proves new cases of Fontaine-Mazur conjecture for reducible residual representations.

Generalizes Skinner-Wiles work to the residually reducible case.

Achieves complete proof of the conjecture in the regular case for p ≥ 5.

Abstract

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.