Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-23T08:21:46.806Z Has data issue: false hasContentIssue false
  • English
  • Français

Structures de Hodge–Pink pour les φ/𝔖-modules de Breuil et Kisin

Part of: Algebraic number theory: local and $p$-adic fields (Co)homology theory Algebraic groups

Published online by Cambridge University Press:  03 May 2012

Alain Genestier  and
Vincent Lafforgue
Alain Genestier
Affiliation:
Institut Élie Cartan, Université Henri Poincaré-Nancy 1, B.P. 70239, F-54506 Vandoeuvre-lès-Nancy cedex, France (email: alain.genestier@iecn.u-nancy.fr)
Vincent Lafforgue
Affiliation:
CNRS, MAPMO - Université d’Orléans, Bâtiment de mathématiques, Rue de Chartres, B.P. 6759, F-45067 Orléans cedex 2, France (email: vlafforg@math.jussieu.fr)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this article, we apply the methods of our work on Fontaine’s theory in equal characteristics to the φ/𝔖-modules of Breuil and Kisin. Thanks to a previous article of Kisin, this yields a new and rather elementary proof of the theorem ‘weakly admissible implies admissible’ of Colmez and Fontaine.

Résumé

Dans cet article, nous appliquons les méthodes de notre travail sur la théorie de Fontaine en égales caractéristiques aux φ/𝔖-modules de Breuil et Kisin. Grâce à un article précédent de Kisin, cela fournit une nouvelle démonstration assez élémentaire du théorème ‘faiblement admissible implique admissible’ de Colmez et Fontaine.

Keywords

Fontaine theoryBreuil–Kisin modulesDieudonné modulesHodge structures

MSC classification

Secondary: 11S20: Galois theory 14F30: $p$-adic cohomology, crystalline cohomology 14L05: Formal groups, $p$-divisible groups
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 3 , May 2012 , pp. 751 - 789
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[Ber02]Berger, L., Représentations p-adiques et équations différentielles, Invent. Math. 148 (2002), 219–284.CrossRefGoogle Scholar
[Ber04]Berger, L., An introduction to the theory of p-adic representations, in Geometric aspects of Dwork theory (Walter de Gruyter, Berlin, 2004), 255–292.CrossRefGoogle Scholar
[Ber08]Berger, L., Équations différentielles p-adiques et (φ,N)-modules filtrés, Astérisque 319 (2008), 13–38.Google Scholar
[Bre97a]Breuil, C., Construction de représentations p-adiques semi-stables, Ann. Sci. Éc. Norm. Supér. 31 (1997), 281–327.CrossRefGoogle Scholar
[Bre97b]Breuil, C., Représentations p-adiques semi-stables et transversalité de Griffiths, Math. Ann. 307 (1997), 191–224.Google Scholar
[Bre98]Breuil, C., Schémas en groupes et corps des normes (1998), disponible àhttp://www.ihes.fr/∼breuil/publications.html.Google Scholar
[Bre99a]Breuil, C., Une application du corps des normes, Compositio Math. 117 (1999), 189–203.Google Scholar
[Bre99b]Breuil, C., Représentations semi-stables et modules fortement divisibles, Invent. Math. 136 (1999), 89–122.CrossRefGoogle Scholar
[Bre02]Breuil, C., Integral p-adic Hodge theory, Adv. Stud. Pure Math. 36 (2002), 51–80.Google Scholar
[CL09]Caruso, X. and Liu, T., Quasi-semi-stable representations, Bull. Soc. Math. France 137 (2009), 185–223.Google Scholar
[CF00]Colmez, P. and Fontaine, M., Construction des représentations p-adiques semi-stables, Invent. Math. 140 (2000), 1–43.CrossRefGoogle Scholar
[Fal99]Faltings, G., Integral crystalline cohomology over very ramified valuation rings, J. Amer. Math. Soc. 12 (1999), 117–144.Google Scholar
[Fal02]Faltings, G., Group schemes with strict 𝒪-action, Mosc. Math. J. 2 (2002), 249–279.CrossRefGoogle Scholar
[Fon90]Fontaine, J.-M., Représentations p-adiques des corps locaux, I, in The Grothendieck festschrift. Vol. II, Progress in Mathematics, vol. 87 (Birkhäuser, Boston, Basel, Berlin, 1990), 249–309.Google Scholar
[Fon94]Fontaine, J.-M., Représentations p-adiques semi-stables, Astérisque, vol. 223 (Soc. Math. France, Paris, 1994), 113–184, with an appendix by Pierre Colmez.Google Scholar
[GL10]Genestier, A. and Lafforgue, V., Théorie de Fontaine en égales caractéristiques. A paraître aux Annales de l’ENS. Disponible à l’adresse http://www.math.jussieu.fr/∼vlafforg/fontaine.pdf.Google Scholar
[Ked04]Kedlaya, K., A p-adic local monodromy theorem, Ann. of Math. (2) 160 (2004), 93–184.CrossRefGoogle Scholar
[Ked05]Kedlaya, K., Slope filtrations revisited, Doc. Math. 10 (2005), 447–525.CrossRefGoogle Scholar
[Kis06]Kisin, M., Crystalline representations and F-crystals, Progr. Math. 253 (2006), 459–496, Drinfeld’s 50th birthday volume.Google Scholar
[Kis08]Kisin, M., Potentially semi-stable deformation rings, J. Amer. Math. Soc. 21 (2008), 513–546.CrossRefGoogle Scholar
[KR09]Kisin, M. and Ren, W., Galois representations and Lubin–Tate groups, Doc. Math. 14 (2009), 441–461.CrossRefGoogle Scholar
[Pin97]Pink, R., Hodge structures over function fields, Preprint (1997), disponible à l’adresse http://www.math.ethz.ch/∼pink.Google Scholar
[Pin00]Pink, R., Uniformisierung von t-motiven. Exposé, 12 juilllet 2000 (notes non publiées).Google Scholar
[RZ96]Rapoport, M. and Zink, T., Period spaces for p-divisible groups, Annals of Mathematics Studies, vol. 141 (Princeton University Press, Princeton, NJ, 1996).Google Scholar
[RZ99]Rapoport, M. and Zink, T., A finiteness theorem in the Bruhat–Tits building : an application of Landvogt’s embedding theorem, Indag. Math. 10 (1999), 449–458.CrossRefGoogle Scholar
[Ser68]Serre, J. P., Corps locaux (Hermann, Paris, 1968).Google Scholar
[Ser95]Serre, J. P., Classes des corps cyclotomiques (d’après K. Iwasawa), Séminaire Bourbaki 1958/59, Exp. No. 174, vol. 5 (Société Mathématique de France, Paris, 1995), 83–93.Google Scholar