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

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

2011-05-04

arxiv.org/abs/1105.0846v2

Mathematics

Number Theory

Scientific paper

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

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