Constructing Galois representations ramified at one prime

TL;DR

This paper proves the existence of infinitely many irreducible Galois representations unramified outside a prime p with large image, constructed via lifting residual representations with unobstructed deformation problems.

Contribution

It establishes the existence of infinitely many such Galois representations with large image, using deformation theory and residual representations in the diagonal torus.

Findings

Infinitely many irreducible Galois representations with large image.

Representations are unramified outside prime p.

Construction via lifting residual representations with unobstructed deformation problems.

Abstract

Let $n>1$, $e\geq 0$ and a prime number $p\geq 2^{n+2+2e}+3$, such that the index of regularity of $p$ is $\leq e$. We show that there are infinitely many irreducible Galois representations $\rho: Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow {GL}_n(\mathbb{Q}_p)$ unramified at all primes $l\neq p$. Furthermore, these representations are shown to have image containing a fixed finite index subgroup of ${SL}_n(\mathbb{Z}_p)$. Such representations are constructed by lifting suitable residual representations $\bar{\rho}$ with image in the diagonal torus in ${GL}_n(\mathbb{F}_p)$, for which the global deformation problem is unobstructed.