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arXiv · 1307.8107

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

Shun Ohkubo
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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 $ρ:G_K\to GL_h(\Q_p)$ a $p$-adic representation. When $k_K$ is perfect, Shankar Sen described the Lie algebra of $ρ(G_K)$ in terms of so-called Sen's operator $Θ$ for $ρ$. When $k_K$ may not be perfect, Olivier Brinon defined $d+1$ operators $Θ_0,...,Θ_d$ for $ρ$, which coincides with Sen's operator $Θ$ in the case of $d=0$. In this paper, we describe the Lie algebra of $ρ(G_K)$ in terms of Brinon's operators $Θ_0,...,Θ_d$, which is a generalization of Sen's result.

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Shun Ohkubo. 2013-07-31. On Lie algebras arising from $p$-adic representations in the imperfect residue field case. https://arxiv.org/abs/1307.8107

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