LIFTING TORSION GALOIS REPRESENTATIONS

Abstract

Let $p\geqslant 5$ be a prime, and let ${\mathcal{O}}$ be the ring of integers of a finite extension $K$ of $\mathbb{Q}_{p}$ with uniformizer ${\it\pi}$. Let ${\it\rho}_{n}:G_{\mathbb{Q}}\rightarrow \mathit{GL}_{2}\left({\mathcal{O}}/({\it\pi}^{n})\right)$ have modular mod-${\it\pi}$ reduction $\bar{{\it\rho}}$, be ordinary at $p$, and satisfy some mild technical conditions. We show that ${\it\rho}_{n}$ can be lifted to an ${\mathcal{O}}$-valued characteristic-zero geometric representation which arises from a newform. This is new in the case when $K$ is a ramified extension of $\mathbb{Q}_{p}$. We also show that a prescribed ramified complete discrete valuation ring ${\mathcal{O}}$ is the weight-$2$ deformation ring for $\bar{{\it\rho}}$ for a suitable choice of auxiliary level. This implies that the field of Fourier coefficients of newforms of weight 2, square-free level, and trivial nebentype that give rise to semistable $\bar{{\it\rho}}$ of weight 2 can have arbitrarily large ramification index at $p$.