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

TL;DR

This paper generalizes the description of Lie algebras associated with $p$-adic Galois representations to the case of imperfect residue fields, extending Sen's classical results using Brinon's operators.

Contribution

It provides a new framework to describe the Lie algebra of Galois groups in the imperfect residue field case using Brinon's operators, generalizing Sen's work.

Findings

Extended Lie algebra description to imperfect residue fields

Connected Brinon's operators with Galois Lie algebras

Generalized Sen's results to broader residue field cases

Abstract

Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$ with residue field $k_K$ such that $[k_K:k_K^p]=p^d<\infty$. Let $G_K$ be the absolute Galois group of $K$ and $\rho:G_K\to GL_h(\Q_p)$ a $p$-adic representation. When $k_K$ is perfect, Shankar Sen described the Lie algebra of $\rho(G_K)$ in terms of so-called Sen's operator $\Theta$ for $\rho$. When $k_K$ may not be perfect, Olivier Brinon defined $d+1$ operators $\Theta_0,...,\Theta_d$ for $\rho$, which coincides with Sen's operator $\Theta$ in the case of $d=0$. In this paper, we describe the Lie algebra of $\rho(G_K)$ in terms of Brinon's operators $\Theta_0,...,\Theta_d$, which is a generalization of Sen's result.