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Construction of \(p\)-adic representations. (Construction de représentations \(p\)-adiques.) (French) Zbl 0579.14037
Let \(K\) be a local field of characteristic 0 with residue field \(k\) of characteristic \(p>0\) and \(G=\text{Gal}(\bar K/K)\). In a recent work of the first of the authors [Astérisque 65, 3–80 (1979; Zbl 0429.14016)] equivalence has been established between a certain full subcategory \(\text{Rep}_{\text{cris}}(G)\) of the category \(\text{Rep}(G)\) of \(p\)-adic finite-dimensional representations of the group \(G\) and a certain full subcategory \({\mathcal M}\) of the category \({\mathcal M}F_ K^{\phi}\) of finite-dimensional filtered Dieudonné modules over \(K\). Unfortunately, there was no success in obtaining an explicit description of \({\mathcal M}.\)
In the present paper, the authors prove a series of results related both to necessary and to sufficient conditions for the embedding of a module into \({\mathcal M}\). They state the remarkable hypothesis that, for a scheme \(X\) over \(W(k)\), the \(p\)-adic representation in \(H^ i_{\text{ét}}(X_{\bar k}{\mathbb{Q}}_ p)\) lies in \(\text{Rep}_{\text{cris}}(G)\) and the Dieudonné module corresponding to it is the de Rham cohomology module \(H^ i_{\text{DR}}(X_ K)\). In the light of this hypothesis, the result of the paper is interesting concerning the fact that if \(\dim\, X<p\), then \(H^ i_{\text{DR}}(X_ K)\) lie in \({\mathcal M}\) and, for the \(p\)-adic representations obtained from \(H^ i_{\text{DR}}(X_ K)\), the Tate hypotheses and the Serre hypothesis on the “tame inertia” are fulfilled.

MSC:
14L05 Formal groups, \(p\)-divisible groups
14F30 \(p\)-adic cohomology, crystalline cohomology
14G20 Local ground fields in algebraic geometry
20G05 Representation theory for linear algebraic groups
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