# Deformations of Certain Reducible Galois Representations, III

## TL;DR

This paper studies the deformation theory of certain reducible Galois representations into symplectic groups, demonstrating conditions under which these representations admit geometric lifts, extending previous results to higher dimensions.

## Contribution

It extends the deformation theory of reducible Galois representations from the case n=1 to higher dimensions, providing new conditions for geometric lifts.

## Key findings

- Representations have geometric lifts under certain hypotheses.
- Extension of deformation results from n=1 to higher dimensions.
- Conditions for unramified outside finite set of primes.

## Abstract

Let $p$ be an odd prime and $q$ a power of $p$. We examine the deformation theory of reducible and indecomposable Galois representations $\bar{\rho}:G_{\mathbb{Q}}\rightarrow \text{GSp}_{2n}(\mathbb{F}_q)$ that are unramified outside a finite set of primes $S$ and whose image lies in a Borel subgroup. We show that under some additional hypotheses, such representations have geometric lifts to the Witt vectors $\text{W}(\mathbb{F}_q)$. The main theorem extends that of Hamblen and Ramakrishna in which the $n=1$ case was treated.