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

Abstract

Let $G$ be a split reductive group with $\dim Z(G)\le 1$. We show that for any prime $p$ that is large enough relative to $G$, there is a finitely ramified Galois representation $\rho :{\Gamma }_{\mathbb{Q}}\to G({\mathbb{Z}}_p)$ 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.

arXiv commentAccepted version, to appear in Journal of Number Theory