Lifting Galois representations to ramified coefficient fields

TL;DR

This paper proves that certain mod p^n Galois representations over ramified extensions can be lifted to characteristic zero, with controlled ramification, and constructs related deformation rings with explicit structure.

Contribution

It establishes conditions under which Galois representations over ramified coefficient fields can be lifted to characteristic zero, extending previous results to ramified cases.

Findings

Lifts of Galois representations exist under technical hypotheses.

Deformation rings are isomorphic to a power series ring over Witt vectors.

Existence of ordinary modular points up to twist.

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).