Advertisement

Mazur’s inequality and Laffaille’s theorem

  • Christophe Cornut
Article
  • 26 Downloads

Abstract

We look at various questions related to filtrations in p-adic Hodge theory, using a blend of building and Tannakian tools. Specifically, Fontaine and Rapoport used a theorem of Laffaille on filtered isocrystals to establish a converse of Mazur’s inequality for isocrystals. We generalize both results to the setting of (filtered) G-isocrystals and also establish an analog of Totaro’s \(\otimes \)-product theorem for the Harder–Narasimhan filtration of Fargues.

Mathematics Subject Classification

14F30 20G25 

References

  1. 1.
    Bridson, M.R., Haefliger, A.: Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999)Google Scholar
  2. 2.
    Chaoha, P., Phon-on, A.: A note on fixed point sets in CAT(0) spaces. J. Math. Anal. Appl. 320(2), 983–987 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Chen, M., Viehmann, E.: Affine Deligne–Lusztig varieties and the action of \(J\). J. Algebraic Geom. 27(2), 273–304 (2018)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Colmez, P., Fontaine, J.-M.: Construction des représentations \(p\)-adiques semi-stables. Invent. Math. 140(1), 1–43 (2000)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cornut, C.: Filtrations and Buildings. To appear in Memoirs of the AMSGoogle Scholar
  6. 6.
    Cornut, C.: A fixed point theorem in Euclidean buildings. Adv. Geom. 16(4), 487–496 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Cornut, C., Nicole, M.-H.: Cristaux et immeubles. Bull. Soc. Math. France 144(1), 125–143 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dat, J.-F., Orlik, S., Rapoport, M.: Period domains over finite and \(p\)-adic fields. Cambridge Tracts in Mathematics, vol. 183. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
  9. 9.
    Faltings, G.: Mumford-Stabilität in der algebraischen Geometrie. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pp. 648–655. Birkhäuser, Basel (1995)CrossRefGoogle Scholar
  10. 10.
    Fargues, L.: Théorie de la réduction pour les groupes p-divisibles (Preprint) Google Scholar
  11. 11.
    Fontaine, J.-M., Laffaille, G.: Construction de représentations \(p\)-adiques. Ann. Sci. École Norm. Sup. (4) 15(4), 547–608 (1983). 1982zbMATHGoogle Scholar
  12. 12.
    Fontaine, J.-M., Rapoport, M.: Existence de filtrations admissibles sur des isocristaux. Bull. Soc. Math. France 133(1), 73–86 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Gashi, Q.R.: On a conjecture of Kottwitz and Rapoport. Ann. Sci. Éc. Norm. Supér. (4) 43(6), 1017–1038 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Kapovich, M.: Generalized triangle inequalities and their applications. In: International Congress of Mathematicians. Vol. II, pp. 719–741. Eur. Math. Soc., Zürich (2006)Google Scholar
  15. 15.
    Kottwitz, R.E.: Isocrystals with additional structure. Compos. Math. 56(2), 201–220 (1985)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kottwitz, R.E.: On the Hodge-Newton decomposition for split groups. Int. Math. Res. Not. 26, 1433–1447 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Kumar, S.: A survey of the additive eigenvalue problem. Transform. Groups 19(4), 1051–1148 (2014). (With an appendix by M. Kapovich) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Laffaille, G.: Groupes \(p\)-divisibles et modules filtrés: le cas peu ramifié. Bull. Soc. Math. France 108(2), 187–206 (1980)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Landvogt, E.: Some functorial properties of the Bruhat–Tits building. J. Reine Angew. Math. 518, 213–241 (2000)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Rapoport, M., Richartz, M.: On the classification and specialization of \(F\)-isocrystals with additional structure. Compos. Math. 103(2), 153–181 (1996)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Rapoport, M., Zink, T.H.: Period spaces for \(p\)-divisible groups. Annals of Mathematics Studies, vol. 141. Princeton University Press, Princeton (1996)zbMATHGoogle Scholar
  22. 22.
    Rapoport, M., Zink, T.: A finiteness theorem in the Bruhat–Tits building: an application of Landvogt’s embedding theorem. Indag. Math. (N.S.) 10(3), 449–458 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Tits, J.: Reductive groups over local fields. In: Automorphic Forms, Representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1. Proc. Sympos. Pure Math., XXXIII, pp. 29–69. Amer. Math. Soc., Providence, R.I. (1979)Google Scholar
  24. 24.
    Totaro, B.: Tensor products in \(p\)-adic Hodge theory. Duke Math. J. 83(1), 79–104 (1996)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Vollaard, I., Wedhorn, T.: The supersingular locus of the Shimura variety of GU\((1, n-1)\) II. Invent. Math. 184, 591–627 (2011)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CNRS, Sorbonne Université, Université Paris Diderot, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRGParisFrance

Personalised recommendations