Deformations of Reducible Galois Representations to Hida-Families

TL;DR

This paper investigates the deformation theory of reducible Galois representations, demonstrating their lift to Hida families with weights in a specific congruence class, using a Galois-theoretic approach to construct p-adic families.

Contribution

It establishes the lift of residually reducible Galois representations to Hida families with weights in a congruence class, advancing the understanding of their deformation theory.

Findings

Lifts to Hida lines with weights in a congruence class are constructed.

Purely Galois theoretic approach enables lifting actual representations, not just semisimplifications.

Provides a framework for p-adic families of Galois representations.

Abstract

The global deformation theory of residually reducible Galois representations with fixed auxiliary conditions is studied. We show that $\bar{\rho}:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}_2(\bar{\mathbb{F}}_p)$ lifts to a Hida line for which the weights range over a congruence class modulo-$p^2$. The advantage of the purely Galois theoretic approach is that it allows us to construct $p$-adic families of Galois representations lifting the actual representation $\bar{\rho}$, and not just the semisimplification.