Q1374007
type: Item
instance of: scholarly article
title: Propriétés du groupe tannakien des structures de Hodge \(p\)-adiques et torseur entre cohomologies cristalline et étale. (Properties of the Tannakian group of \(p\)-adic Hodge structures and torsor between crystalline and étale cohomologies.)
zbMATH Open document ID: 0888.14008
author: Jean-Pierre Wintenberger
publication date: 1997-12-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75264
review text: The paper consists of an introduction, notations and conventions, and four chapters: \((1)\) Construction of a fibre functor over \({\mathbb Q}_p\) on the category of weakly admissible filtered \((\phi,N)\)-modules; \((2)\) Weakly admissible filtered \((\phi,N)\)-modules and isogenies; \((3)\) Simple connectivity of the semi-simple part of the group \(\mathbf{H}_c\); \((4)\) Consequences for the torsor between the fibre functors \(\omega_c\) and \(\omega_{\text{Gal}}\). The introduction gives a very readible overview of this complicated matter. Let \(p\) be a prime number, \(K\) a local field of zero characteristic with algebraically closed residue field \(k\) of characteristic \(p\), \(W=W(k)\) the ring of Witt vectors with fraction field \(K_0\). Write \(\sigma\) for the Frobenius on \(k\), \(W\) and \(K_0\). \(K\) is a finite extension of \(K_0\) with algebraic closure \(\overline{K}\). One also writes \(G_K=\text{Gal}(\overline{K}/K)\) and \(G_{{\mathbb Q}_p}=\text{Gal}(\overline{\mathbb Q}_p/{\mathbb Q}_p)\). J.-M. Fontaine introduced filtered \((\phi,N)\)-modules over \(K\) as finite dimensional \(K_0\)-vector spaces equipped with a bijective \(\sigma\)-linear map \(\phi\) and a linear nilpotent map \(N\), and after extension of scalars to \(K\), a filtration. These \((\phi,N)\)-modules may satisfy a kind of stability condition in the sense of geometric invariant theory. In this case they are called \textit{weakly admissible}, and they form a Tannakian category \(\underline{\text{MF}}^{fa}_K (\phi,N)\) over \({\mathbb Q}_p\). This category can be considered as a \(p\)-adic analogue of the category of complex Hodge structures. On the other hand, one has the Tannakian category \(\mathbf{Rep}_{{\mathbb Q}_p}(G_K)\) of \(p\)-adic representations of \(G_K\), and its Tannakian subcategory \(\mathbf{Rep}_{{\mathbb Q}_p}^{ss}(G_K)\) of semi-stable representations. Fontaine constructed a \(\otimes\)-equivalence between \(\mathbf{Rep}_{{\mathbb Q}_p}^{ss}(G_K)\) and a T…
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 19E20
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 11S25
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14G20
zbMATH DE Number: 1092695
zbMATH Keywords: Tannakian category
zbMATH Keywords: \(p\)-adic Hodge structures
zbMATH Keywords: torsor
zbMATH Keywords: étale cohomology
zbMATH Keywords: crystalline cohomology
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
OpenAlex ID: W2005079189
cites work: Dualité sur un corps local à corps résiduel algébriquement clos
cites work: Q4882803
cites work: Complements à l'article: ''Groupes reductifs''
cites work: Reductive groups on a local field. II. Groups schemes. Existence of valuated root datum
cites work: Q5602064
cites work: Q3932925
cites work: La conjecture de Weil pour les surfaces K3
cites work: Q3349023
cites work: Q4866013
cites work: Q3866210
cites work: Q4325443
cites work: Construction de représentations $p$-adiques
cites work: Galois-Kohomologie halb einfacher algebraischer Gruppen über \(p\)-adischen Körpern. I, II
cites work: Q3729169
cites work: Q3977512
cites work: Q4361592
cites work: Groupes $p$-divisibles et modules filtrés : le cas peu ramifié
cites work: Q3941610
cites work: Q4200417
cites work: Q4896971
cites work: Period Spaces for "p"-divisible Groups (AM-141)
cites work: Categories tannakiennes
cites work: Q5590138
cites work: Galois Cohomology
cites work: Q3890896
cites work: Q4297065
cites work: Regular elements of semisimple algebraic groups
cites work: Q3050630
cites work: Tensor products in \(p\)-adic Hodge theory
cites work: Un scindage de la filtration de Hodge pour certaines variétés algébriques sur les corps locaux. (A splitting of the Hodge filtration for certain algebraic varieties over local fields)
cites work: Groupes Algebriques Associes a Certaines Representations p-adiques
cites work: The torsor between \(p\)-adic étale cohomology and crystalline cohomology; the abelian case
cites work: Lifting according to an isogeny of \(\ell\)-adic Galois representations associated to motives
DOI: 10.5802/AIF.1600
published in: Annales de l’institut Fourier
label: Propriétés du groupe tannakien des structures de Hodge \(p\)-adiques et torseur entre cohomologies cristalline et étale. (Properties of the Tannakian group of \(p\)-adic Hodge structures and torsor between crystalline and étale cohomologies.)
description: scientific article; zbMATH DE number 1092695