Open Access
2013 The $p$-adic monodromy theorem in the imperfect residue field case
Shun Ohkubo
Algebra Number Theory 7(8): 1977-2037 (2013). DOI: 10.2140/ant.2013.7.1977


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.


Download Citation

Shun Ohkubo. "The $p$-adic monodromy theorem in the imperfect residue field case." Algebra Number Theory 7 (8) 1977 - 2037, 2013.


Received: 1 July 2012; Revised: 2 April 2013; Accepted: 2 May 2013; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 1312.11046
MathSciNet: MR3134041
Digital Object Identifier: 10.2140/ant.2013.7.1977

Primary: 11F80
Secondary: 11F85 , 11S15 , 11S20 , 11S25

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

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 8 • 2013
Back to Top