Vol. 7, No. 8, 2013

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

Shun Ohkubo

Vol. 7 (2013), No. 8, 1977–2037

Let K be a complete discrete valuation field of mixed characteristic (0,p) and GK the absolute Galois group of K. In this paper, we will prove the p-adic monodromy theorem for p-adic representations of GK 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,GK)-module ˜rig+(V ) for a de Rham representation V , which is a generalization of Pierre Colmez’s ˜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 GK and the category of de Rham representations of an absolute Galois group of the canonical subfield of K. Finally, we compute H1 of some p-adic representations of GK, which is a generalization of Osamu Hyodo’s results.

$p$-adic Hodge theory, $p$-adic representations
Mathematical Subject Classification 2010
Primary: 11F80
Secondary: 11F85, 11S15, 11S20, 11S25
Received: 1 July 2012
Revised: 2 April 2013
Accepted: 2 May 2013
Published: 24 November 2013
Shun Ohkubo
Department of Mathematical Sciences
University of Tokyo
Tokyo 153-8914