On Lie algebras arising from p-adic representations in the imperfect residue field case

Abstract

Let K be a complete discrete valuation field of mixed characteristic (0,p) with residue field kK such that [kK:kKp]=pd<∞. Let GK be the absolute Galois group of K and ρ:GK→GLh(Qp) a p-adic representation. When kK is perfect, Shankar Sen described the Lie algebra of ρ(GK) in terms of so-called Sen's operator Θ for ρ. When kK may not be perfect, Olivier Brinon defined d+1 operators Θ0,…,Θd for ρ, which reduce to Sen's operator Θ in the case of d=0. In this paper, we describe the Lie algebra of ρ(GK) in terms of Brinon's operators Θ0,…,Θd, which is a generalization of Sen's result.