Lifting residual Galois representations with the same semi-simplification

TL;DR

This paper investigates when two non-semi-simple residual Galois representations with the same semi-simplification can be lifted to representations attached to the same modular form, addressing an inverse problem in number theory.

Contribution

It provides conditions under which two non-semi-simple residual Galois representations with identical semi-simplifications originate from the same modular form.

Findings

Identifies mild conditions for lifting residual representations to modular forms.

Shows that non-semi-simple residual representations with the same semi-simplification can come from the same eigenform.

Addresses an inverse problem initially posed by Gee and Pozzi.

Abstract

If a $p$-adic Galois representation $\rho_{f,\nu}:\Gamma_{\mathbb Q} \to \GL_2(E_{f,\nu})$ 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 $\overline{\rho_1},\overline{\rho_2}:\Gamma_{\mathbb Q} \to \GL_2(k)$, 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 $\varepsilon$.