The \(p\)-adic monodromy theorem in the imperfect residue field case. (English) Zbl 1312.11046

Summary: 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 \((\phi,G_K\))-module \(\tilde{\mathbb N}_{\mathrm{rig}}^{\nabla +}(V)\) for a de Rham representation \(V\), which is a generalization of P. Colmez’s \(\tilde{\mathbb N}_{\mathrm{rig}}^+(V)\) [in: Représentation \(p\)-adiques de groupes \(p\)-adiques I. Représentations galoisiennes et \((\varphi, \Gamma)\)-modules. Paris: Société Mathématique de France. 117–186 (2008; Zbl 1168.11021)]. 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 O. Hyodo’s results [Adv. Stud. Pure Math. 12, 287–314 (1987; Zbl 0649.12011)].


11F80 Galois representations
11F85 \(p\)-adic theory, local fields
11S15 Ramification and extension theory
11S20 Galois theory
11S25 Galois cohomology
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