Lifting torsion Galois representations

TL;DR

This paper develops new methods to lift certain Galois representations from mod $\pi^n$ to characteristic zero, especially in ramified cases, extending modularity results for $p$-adic lifts over ramified DVRs.

Contribution

It introduces techniques to lift ramified weight two Galois representations and constructs deformation rings as prescribed DVRs, broadening the scope of modularity lifting.

Findings

Successfully lifts ramified Galois representations to characteristic zero.

Constructs deformation rings as prescribed DVRs with suitable auxiliary levels.

Extends modularity proofs to cases involving ramified DVRs.

Abstract

Typos in the abstract have been corrected. Let $\rho_n$ be an ordinary weight two representation of absolute Galois group of the rationals to $GL_2(\mathcal O/\pi^n)$. 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 $\mathcal O$.