A note on Galois representations valued in reductive groups with open image

TL;DR

This paper constructs Galois representations valued in split reductive groups with open image, generalizing previous results and providing new methods for representations ramified at a single prime, under certain conditions.

Contribution

It proves the existence of finitely ramified Galois representations with open image in split reductive groups for large primes, extending Ray's theorem to broader contexts.

Findings

Existence of Galois representations with open image for large primes

Construction of ramified representations using lifting theorems

Generalization of Ray's theorem to new settings

Abstract

Let $G$ be a split reductive group with $\dim Z(G) \leq 1$. We show that for any prime $p$ that is large enough relative to $G$, there is a finitely ramified Galois representation $\rho \colon \Gamma_{\mathbb Q} \to G(\mathbb Z_p)$ with open image. We also show that for any given integer $e$, if the index of irregularity of $p$ is at most $e$ and if $p$ is large enough relative to $G$ and $e$, then there is a Galois representation $\Gamma_{\mathbb Q} \to G(\mathbb Z_p)$ ramified only at $p$ with open image, generalizing a theorem of A. Ray. The first type of Galois representation is constructed by lifting a suitable Galois representation into $G(\mathbb F_p)$ using a lifting theorem of Fakhruddin--Khare--Patrikis, and the second type of Galois representation is constructed using a variant of Ray's argument.