Lifting residual Galois representations with the same semi-simplification

Abstract

If a $p$-adic Galois representation $ρ_{f,ν}:Γ_{\mathbb Q} \to \GL_2(E_{f,ν})$ attached to some eigenform $f$ is residually reducible it will have 2 non-isomorphic reductions, which have the same semi-simplification. In this paper, we answer a version of the inverse question, first brought up by Toby Gee and Alice Pozzi. Starting with two modulo $p$ non-semi-simple representations $\overline{ρ_1},\overline{ρ_2}:Γ_{\mathbb Q} \to \GL_2(k)$, which have the same semi-simplification we show that under some mild conditions that they are reductions of representations attached to newforms of the same weight $r\ge 2$, the same level $N\ge 1$, and the same Neben character $\epsilon$.

50 pages. Comments are welcomed!