Crystalline and semi-stable representations in the imperfect residue field case

Let $K$ be a $p$-adic local field with residue field $k$ such that $[k : k^p] = p^e \lt \infty$ and $V$ be a $p$-adic representation of $\mathrm{Gal}(\overline{K} / K)$. Then, by using the theory of $p$-adic differential modules, we show that $V$ is a potentially crystalline (resp. potentially semi-stable) representation of $\mathrm{Gal}(\overline{K} / K)$ if and only if $V$ is a potentially crystalline (resp. potentially semi-stable) representation of $\mathrm{Gal}(\overline{K^\mathrm{pf}} / K^\mathrm{pf})$ where $K^\mathrm{pf} / K$ is a certain p-adic local field whose residue field is the smallest perfect field $k^\mathrm{pf}$ containing $k$. As an application, we prove the p-adic monodromy theorem of Fontaine in the imperfect residue field case.

Keywords

$p$-adic Galois representation, $p$-adic cohomology, $p$-adic differential equation

2010 Mathematics Subject Classification

11F80, 12H25, 14F30

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Published 13 May 2014