Large images of reducible galois representations

Abstract

Given a reducible Galois representation $\overline{\rho }:G_{\mathbb{Q}}\to G{L}_2({\mathbb{F}}_q)$ we show there exists an irreducible deformation $\rho :G_{\mathbb{Q}}\to G{L}_2(\mathbb{W}[[T_1,T_2,..,T_r,....,]])$ 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.

To appear in Revista de la Unión Matemática Argentina