Lifting torsion Galois representations

Typos in the abstract have been corrected. Let ${\rho }_n$ be an ordinary weight two representation of absolute Galois group of the rationals to $G{L}_2(\mathcal{O}\mathrm{/}{\pi }^n)$. Here $\mathcal{O}$ is a ramified DVR with uniformiser $\pi$. If ${\rho }_n$ satisfies mild hypotheses we lift it to a characteristic zero $\mathcal{O}$-valued geometric weight two representation. The earlier methods could handle only the unramified case. We show that the deformation ring of a residual representation arising from a newform can be arranged to be a prescribed DVR provided we choose a suitable auxiliary level. We extend earlier proofs of modularity of $p$-adic lifts of modular residual representations, via $p$-adic approximations, to cover cases when the lift is defined over ramified DVR’s $\mathcal{O}$.