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Crystalline representations and F-crystals

  • Mark Kisin
Part of the Progress in Mathematics book series (PM, volume 253)

Summary

Following ideas of Berger and Breuil, we give a new classification of crystalline representations. The objects involved may be viewed as local, characteristic 0 analogues of the “shtukas” introduced by Drinfeld. We apply our results to give a classification of p-divisible groups and finite flat group schemes, conjectured by Breuil, and to show that a crystalline representation with Hodge-Tate weights 0, 1 arises from a p-divisible group, a result conjectured by Fontaine.

Keywords

Group Scheme Full Subcategory Divided Power Coherent Sheaf Unique Lift 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [An]
    Y. André, Différentielles non commutatives et théorie de Galois différentielle ou aux différences, Ann. Sci. École Norm. Sup., 34 (2001), 685–739.MATHGoogle Scholar
  2. [Ba]
    F. Baldassarri, Comparaison entre la cohomologie algébrique et la cohomologie p-adique rigide à coefficients dans un module différentiel II: Cas des singularités régulières à plusieurs variables, Math. Ann., 280 (1998), 417–439.CrossRefMathSciNetGoogle Scholar
  3. [BBM]
    P. Berthelot, L. Breen, and W. Messing, Théorie de Dieudonné cristalline II, Lecture Notes in Mathematics, Vol. 930, Springer-Verlag, Berlin, New York, Heidelberg, 1982.MATHGoogle Scholar
  4. [Be 1]
    L. Berger, Équations différentielles p-adiques et (ϕ, N)-modules filtrés, preprint, 2004.Google Scholar
  5. [Be 2]
    L. Berger, Limites des représentations cristallines, Compositio Math., 140 (2004), 1473–1498.MATHGoogle Scholar
  6. [Be 3]
    L. Berger, Représentations p-adiques et équations différentielles, Invent. Math., 148-2 (2002), 219–284.MATHCrossRefMathSciNetGoogle Scholar
  7. [Br 1]
    C. Breuil, Une application du corps des normes, Compositio Math., 117 (1999), 189–203.MATHCrossRefMathSciNetGoogle Scholar
  8. [Br 2]
    C. Breuil, Groupes p-divisibles, groupes finis et modules filtrés, Ann. Math., 152 (2000), 489–549.MATHCrossRefMathSciNetGoogle Scholar
  9. [Br 3]
    C. Breuil, Integral p-adic Hodge theory, in S. Usui, ed., Algebraic Geometry 2000, Azumino, Advanced Studies in Pure Mathematics, Vol. 36, Mathematical Society of Japan, Tokyo, 2002, 51–80.Google Scholar
  10. [Br 4]
    C. Breuil, Schémas en groupes et corps des normes, unpublished manuscript, 1998.Google Scholar
  11. [CF]
    P. Colmez and J. M. Fontaine, Construction des représentations p-adiques semistables, Invent. Math., 140-1 (2000), 1–43.MATHCrossRefMathSciNetGoogle Scholar
  12. [Co]
    P. Colmez, Espaces de Banach de dimension finie, J. Inst. Math. Jussieu, 1 (2002), 331–439.MATHCrossRefMathSciNetGoogle Scholar
  13. [deJ]
    A. de Jong, Crystalline Dieudonné module theory via formal and rigid geometry, Inst. Études Sci. Publ. Math., 82 (1995), 5–96.MATHCrossRefGoogle Scholar
  14. [De]
    P. Deligne, Equations Différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin, Heidelberg, New York, 1970.MATHGoogle Scholar
  15. [Fa]
    G. Faltings, Integral crystalline cohomology over very ramified valuation rings, J. Amer. Math. Soc., 12 (1999), 117–144.MATHCrossRefMathSciNetGoogle Scholar
  16. [Fo 1]
    J.-M. Fontaine, Représentations p-adiques des corps locaux, in P. Carier, ed., Grothendieck Festschrift, Vol. 2, Progress in Mathematics, Vol. 87, Birkhäuser, Basel, 1991, 249–309.Google Scholar
  17. [Fo 2]
    J. M. Fontaine, Représentations p-adiques semi-stables, in Périodes p-adiques, Astérisque, Vol. 223, Société Mathématique de France, Paris, 1994, 113–184.Google Scholar
  18. [Fo 3]
    J.-M. Fontaine, Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate, in Journées de Géométrie Algébrique de Rennes (Orsay 1978), Astérisque, Vol. 65, Société Mathématique de France, Paris, 1979, 3–80.Google Scholar
  19. [GL]
    A. Genestier and V. Lafforgue, Théories de Fontaine et Fontaine-Laffaille en égale caractéristique, preprint, 2004.Google Scholar
  20. [La]
    G. Laffaille, Groupes p-divisibles et modules filtrés: Le cas peu ramifié, Bull. Soc. Math. France, 108 (1980), 187–206.MATHMathSciNetGoogle Scholar
  21. [Laz]
    M. Lazard, Les zéros des fonctions analytiques d’une variable sur un corps valué complet, Inst. Hautes Études Sci. Publ. Math., 14 (1962), 47–75.CrossRefMathSciNetGoogle Scholar
  22. [Ka]
    D. Kazhdan, An introduction to Drinfeld’s “Shtuka,” in A. Borel and W. Casselman, eds., Automorphic Forms, Representations and L-Functions II, Proceedings of Symposia in Pure Mathematics, Vol. 33, American Mathematical Society, Providence, RI, 1979, 347–356.Google Scholar
  23. [Kat]
    N. Katz, Serre-Tate local moduli, in J. Giraud, ed., Surfaces Algébriques: Séminaire de Géométrie Algébriques d’Orsay 1976–1978, Lecture Notes in Mathematics, Vol. 868, Springer-Verlag, Berlin, 1981, 128–202.Google Scholar
  24. [Kat 2]
    N. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Études Sci. Publ. Math., 39 (1970), 175–232.MATHCrossRefGoogle Scholar
  25. [Ke 1]
    K. S. Kedlaya, Ap-adic local monodromy theorem, Ann. Math., 160-1 (2004), 93–184.MATHMathSciNetCrossRefGoogle Scholar
  26. [Ke 2]
    K. S. Kedlaya, Slope filtrations revisited, math.NT/0504204, 2005; Documenta Math., to appear.Google Scholar
  27. [Ki]
    M. Kisin, Moduli of finite flat group schemes and modularity, preprint, 2004.Google Scholar
  28. [Me]
    W. Messing, The Crystals Associated to Barsotti-Tate Groups: With Applications to Abelian Schemes, Lecture Notes in Mathematics, Vol. 264, Springer-Verlag, Berlin, New York, Heidelberg, 1972.MATHGoogle Scholar
  29. [MM]
    B. Mazur and W. Messing, Universal Extensions and One Dimensional Crystalline Cohomology, Lecture Notes in Mathematics, Vol. 370, Springer, Berlin, New York, Heidelberg, 1974.MATHGoogle Scholar

Copyright information

© Birkhäuser Boston 2006

Authors and Affiliations

  • Mark Kisin
    • 1
  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA

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