Large images of reducible galois representations

TL;DR

This paper constructs large, ramified deformations of reducible Galois representations over infinite-dimensional rings, extending Ramakrishna's work from irreducible to reducible cases.

Contribution

It demonstrates the existence of irreducible deformations of reducible Galois representations with infinite ramification, generalizing prior results to a broader class.

Findings

Existence of irreducible deformations ramified at infinitely many primes

Extension of Ramakrishna's irreducible case results to reducible representations

Construction over infinite-dimensional Witt vector rings

Abstract

Given a reducible Galois representation $\overline{\rho}: G_{\mathbb{Q}} \rightarrow GL_2( \mathbb{F}_q)$ we show there exists an irreducible deformation $\rho : G_{\mathbb{Q}} \rightarrow GL_2 (\mathbb{W} [[T_1, T_2,.., T_r,....,]])$ of $\overline{\rho}$ ramified at infinitely many primes, where $\mathbb{W}$ denotes the ring of Witt vectors of $\mathbb{F}_q$. This is a modification of Ramakrishna's result for the irreducible case.