Lifting Galois representations to ramified coefficient fields

Abstract

Let $p>5$ be a prime integer and $K/\mathbb{Q}_p$ a finite ramified extension with ring of integers $\mathcal{O}$ and uniformizer $\pi$. Let $n>1$ be a positive integer and $\rho_n:G_\mathbb{Q} \to \text{GL}_2(\mathcal{O}/\pi^n)$ be a continuous Galois representation. In this article we prove that under some technical hypotheses the representation $\rho_n$ can be lifted to a representation $\rho:G_\mathbb{Q} \to \text{GL}_2(\mathcal{O})$. Furthermore, we can pick the lift restriction to inertia at any finite set of primes (at the cost of allowing some extra ramification) and get a deformation problem whose universal ring is isomorphic to $W(\mathbb{F})[[X]]$. The lifts constructed are "nearly ordinary" (not necessarily Hodge-Tate) but we can prove the existence of ordinary modular points (up to twist).