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Ordinary \(p\)-adic representations. With an appendix by Luc Illusie: Ordinary semi-stable reduction, \(p\)-adic étale cohomology and de Rham cohomology after Bloch-Kato and Hyodo. (Représentations \(p\)-adiques ordinaires. Avec un appendice par Luc Illusie: Réduction semi-stable ordinaire, cohomologie étale \(p\)-adique et cohomologie de de Rham d’après Bloch-Kato et Hyodo.) (French) Zbl 1043.11532

Fontaine, Jean-Marc (ed.), Périodes \(p\)-adiques. Séminaire du Bures-sur-Yvette, France, 1988. Paris: Société Mathématique de France. Astérisque 223, 185-220; Appendix 209-220 (1994).
Introduction: Let \(k\) be a perfect field of characteristic \(p\), \(K_0\) a complete local field unramified over \(\mathbb Q_p\), of characteristic 0 and with residue class field \(k\), and \(K\) a finite extension of \(K_0\) totally ramified over \(K_0\). We choose an algebraic closure \(\overline K\) of \(K\) and an algebraic closure \(\overline k\) of \(k\), and we denote by \(G_K\) the Galois group of \(\overline K/K\). Let \(I_K\) be the inertia subgroup of \(G_K\), \(\sigma\) the absolute Frobenius endomorphism over \(K_0\) and \(k\), and \(P_0\) the quotient field of \(W(\overline k)\).
In this paper we are interested in the \(p\)-adic representations of \(G_K\) called ordinary \(p\)-adic representations, and in their complete description in terms of certain filtered \((\phi, N)\)-modules. These \(p\)-adic representations occur in algebraic geometry. In fact, Bloch and Kato, and then Hyodo, showed that under certain conditions on the variety \(X\), the \(p\)-adic representations given by the étale cohomology of \(X\) are ordinary (cf. L. Illusie’s appendix). From another point of view, R. Greenberg [Algebraic number theory, Adv. Stud. Pure Math. 17, 97–137 (1989; Zbl 0739.11045)] constructed an Iwasawa theory for ordinary \(p\)-adic representations that generalizes the known theory for ordinary abelian varieties and the cyclotomic Tate module. We will not comment further in these two directions. All of the results presented in this paper are due to J.-M. Fontaine.
For the entire collection see [Zbl 0802.00019].

MSC:

11G45 Geometric class field theory
14F30 \(p\)-adic cohomology, crystalline cohomology
11R23 Iwasawa theory

Citations:

Zbl 0739.11045
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