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

2022/05/01 by Tang, Shiang
#11F80 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2205.00502

Abstract

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\colon Γ_{\mathbb Q} → 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 Γ_{\mathbb Q} → 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.

Related

• On Galois representations with large image
• Lifting G-Valued Galois Representations when ℓ ≠ p
• Large images of reducible galois representations
• Comparing Galois representations in the residually reducible case
• Explicit computation of Galois p-groups unramified at p