The p-adic monodromy theorem in the imperfect residue field case

Ohkubo, Shun

doi:10.2140/ant.2013.7.1977

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The $p$-adic monodromy theorem in the imperfect residue field case

Shun Ohkubo

Vol. 7 (2013), No. 8, 1977–2037

DOI: 10.2140/ant.2013.7.1977

Abstract

Let $K$ be a complete discrete valuation field of mixed characteristic $(0, p)$ and $G_{K}$ the absolute Galois group of $K$ . In this paper, we will prove the $p$ -adic monodromy theorem for $p$ -adic representations of $G_{K}$ without any assumption on the residue field of $K$ , for example the finiteness of a $p$ -basis of the residue field of $K$ . The main point of the proof is a construction of $(φ, G_{K})$ -module ${\tilde{ℕ}}_{rig}^{\nabla +} (V)$ for a de Rham representation $V$ , which is a generalization of Pierre Colmez’s ${\tilde{ℕ}}_{rig}^{+} (V)$ . In particular, our proof is essentially different from Kazuma Morita’s proof in the case when the residue field admits a finite $p$ -basis.

We also give a few applications of the $p$ -adic monodromy theorem, which are not mentioned in the literature. First, we prove a horizontal analogue of the $p$ -adic monodromy theorem. Secondly, we prove an equivalence of categories between the category of horizontal de Rham representations of $G_{K}$ and the category of de Rham representations of an absolute Galois group of the canonical subfield of $K$ . Finally, we compute $H^{1}$ of some $p$ -adic representations of $G_{K}$ , which is a generalization of Osamu Hyodo’s results.

Keywords

$p$-adic Hodge theory, $p$-adic representations

Mathematical Subject Classification 2010

Primary: 11F80

Secondary: 11F85, 11S15, 11S20, 11S25

Milestones

Received: 1 July 2012

Revised: 2 April 2013

Accepted: 2 May 2013

Published: 24 November 2013

Authors

Shun Ohkubo
Department of Mathematical Sciences University of Tokyo Tokyo 153-8914 Japan

Vol. 7, No. 8, 2013