Galois representations ramified at one prime and with suitably large image

TL;DR

This paper constructs infinitely many Galois representations unramified outside a prime p and infinity, with large images, using Galois theoretic lifting, under certain conditions related to Vandiver's conjecture.

Contribution

It provides a new Galois theoretic method to produce infinitely many Galois representations with large images, ramified only at p and infinity.

Findings

Existence of infinitely many such Galois representations for p ≥ 7.

Constructed representations have images containing large subgroups of SL_n(Z_p).

Results depend on a weak form of Vandiver's conjecture when p ≡ 1 mod 4.

Abstract

Let $p\geq 7$ be a prime and $n>1$ be a natural number. We show that there exist infinitely many Galois representations $\varrho:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_{n}(\mathbb{Z}_p)$ which are unramified outside $\{p, \infty\}$ with large image. More precisely, the Galois representations constructed have image containing the kernel of the mod-$p^t$ reduction map $SL_n(\mathbb{Z}_p)\rightarrow SL_n(\mathbb{Z}/p^t\mathbb{Z})$, where $t:=8(n^2-n)\left(3+\lfloor log_p(2^n+1)\rfloor\right)+8$. The results are proven via a purely Galois theoretic lifting construction. When $p\equiv 1\mod{4}$, our results are conditional since in this case, we assume a very weak version of Vandiver's conjecture.