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**Crystalline and semi-stable representations in the imperfect residue field case.**
*(English)*
Zbl 1305.11048

Let \(K\) be a \(p\)-adic local field with residue field \(k\). For any field \(L\), denote by \(G_L\) the Galois group \(\mathrm{Gal}(\bar{L}/L)\), where \(\bar{L}\) is an algebraic closure of \(L\). In this paper, the author deals with \(p\)-adic representations of \(G_K\) in the imperfect residue field case, showing that a representation \(V\) is potentially crystalline or potentially semi-stable if and only if it is so when restricted to a representation of \(G_{K^{\mathrm{pf}}}\), \(K^{\mathrm{pf}}\) being certain \(p\)-adic local field (constructed in a detailed way at the beginning of the paper) whose residue field is the smallest perfect field containing \(k\). Note that in this case the classical theory in the perfect residue field case can be applied.

The paper is organized in five sections, the first of them consisting of a general introduction. Then the author continues by recalling in the next section and a half the necessary concepts about \(p\)-adic representations, both in the perfect and imperfect residue field cases, and \(p\)-adic differential modules according to some of the references of the text. The second half of Section 3 is devoted to understand several special elements which behave well under the action of certain differential operators, and are used in the proof of the main theorem in Section 4. He ends in Section 5, by applying that theorem to prove Fontaine’s \(p\)-adic monodromy theorem in the imperfect residue field case.

The paper is organized in five sections, the first of them consisting of a general introduction. Then the author continues by recalling in the next section and a half the necessary concepts about \(p\)-adic representations, both in the perfect and imperfect residue field cases, and \(p\)-adic differential modules according to some of the references of the text. The second half of Section 3 is devoted to understand several special elements which behave well under the action of certain differential operators, and are used in the proof of the main theorem in Section 4. He ends in Section 5, by applying that theorem to prove Fontaine’s \(p\)-adic monodromy theorem in the imperfect residue field case.

Reviewer: Alberto Castaño Domínguez (Sevilla)