Modularity of 2-dimensional Galois representations

On the Modularity of Certain 2-adic Galois Representations

We prove some results of the form “r residually irreducible and residually modular implies r is modular,” where r is a suitable continuous odd 2-dimensional 2-adic representation of the absolute Galois group of Q. These results are analogous to those obtained by A. Wiles, R. Taylor, F. Diamond, and others for p-adic representations in the case when p is odd; some extra work is required to overc...

Modularity of Nearly Ordinary 2-adic Residually Dihedral Galois Representations

We prove modularity of some two dimensional, 2-adic Galois representations over a totally real field that are nearly ordinary at all places above 2 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...

Serre’s Conjecture on 2-dimensional Galois representations

Modularity lifting theorems for ordinary Galois representations

We generalize the results of [CHT08] and [Tay08] by proving modularity lifting theorems for ordinary l-adic Galois representations of any dimension of a CM or totally real number field F . The main theorems are obtained by establishing an R = T theorem over a Hida family. A key part of the proof is to construct appropriate ordinary lifting rings at the primes dividing l and to determine their i...

Deformation of Outer Representations of Galois Group

To a hyperbolic smooth curve defined over a number-field one naturally associates an "anabelian" representation of the absolute Galois group of the base field landing in outer automorphism group of the algebraic fundamental group. In this paper, we introduce several deformation problems for Lie-algebra versions of the above representation and show that, this way we get a richer structure than t...