On Galois representations with large image

TL;DR

This paper constructs specific Galois representations over the rationals with large images, unramified outside a small set of primes, demonstrating their existence for various primes and dimensions.

Contribution

It proves the existence of continuous Galois representations with open image and controlled ramification for all primes p ≥ 3 and dimensions m ≥ 1, expanding understanding of Galois representations with large image.

Findings

Existence of Galois representations with open image for all p ≥ 3 and m ≥ 1.

Representations unramified outside specified small sets of primes depending on p mod 4.

Construction applies uniformly across different primes and dimensions.

Abstract

For every prime number $p\geq 3$ and every integer $m\geq 1$, we prove the existence of a continuous Galois representation $\rho: G_\mathbb{Q} \rightarrow Gl_m(\mathbb{Z}_p)$ which has open image and is unramified outside $\{p,\infty\}$ (resp. outside $\{2,p,\infty\}$) when $p\equiv 3$ mod $4$ (resp. $p \equiv 1$ mod $4$).