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

Let G be a split reductive group with dim ⁡ Z (G) ≤ 1. We show that for any prime p that is large enough relative to G , there is a finitely ramified Galois representation ρ : Γ Q → G (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 ρ : Γ Q → G (Z p) ramified only at p with open image, generalizing a theorem of Ray [8]. The first type of Galois representation is constructed by lifting a suitable Galois representation into G (F p) using a lifting theorem of Fakhruddin–Khare–Patrikis [4] , and the second type of Galois representation is constructed using a variant of the argument in Ray's work [8]. [ABSTRACT FROM AUTHOR]