Constructing Galois representations ramified at one prime

Abstract

Let $n>1$, $e\ge 0$ and a prime number $p\ge 2^{n+2+2e}+3$, such that the index of regularity of $p$ is $\le e$. We show that there are infinitely many irreducible Galois representations $\rho :Gal(\bar{\mathbb{Q}}/\mathbb{Q})\to {GL}_n({\mathbb{Q}}_p)$ unramified at all primes $l\ne p$. 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.

10 pages, final version