Modularity of nearly ordinary 2-adic residually dihedral Galois representations

TL;DR

This paper proves the modularity of certain 2-adic Galois representations over totally real fields, specifically those that are nearly ordinary and residually dihedral, using advanced patching and Hida family techniques.

Contribution

It introduces a novel approach combining Skinner-Wiles strategy with 2-adic patching to establish modularity for these Galois representations.

Findings

Proves modularity of nearly ordinary residually dihedral 2-adic Galois representations.

Deduces modularity of specific elliptic curves over totally real fields.

Employs advanced techniques like Hida families and 2-adic patching.

Abstract

We prove modularity of some two dimensional, 2-adic Galois representations over totally real fields that are nearly ordinary and that are residually dihedral. We do this by employing the strategy of Skinner and Wiles, using Hida families, together with the 2-adic patching method of Khare and Wintenberger. As an application we deduce modularity of some elliptic curves over totally real fields that have good ordinary or multiplicative reduction at places above 2.