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Mathematische Zeitschrift

, Volume 291, Issue 1–2, pp 473–497 | Cite as

On the Hodge–Newton filtration for p-divisible groups of Hodge type

  • Serin HongEmail author
Article

Abstract

A p-divisible group, or more generally an F-crystal, is said to be Hodge–Newton reducible if its Newton polygon and Hodge polygon have a nontrivial contact point. Katz proved that Hodge–Newton reducible F-crystals admit a canonical filtration called the Hodge–Newton filtration. The notion of Hodge–Newton reducibility plays an important role in the deformation theory of p-divisible groups; the key property is that the Hodge–Newton filtration of a p-divisible group over a field of characteristic p can be uniquely lifted to a filtration of its deformation. We generalize Katz’s result to F-crystals that arise from an unramified local Shimura datum of Hodge type. As an application, we give a generalization of Serre–Tate deformation theory for local Shimura data of Hodge type.

Notes

Acknowledgements

I would like to sincerely thank my advisor E. Mantovan for her continuous encouragement and advice. I also thank T. Wedhorn for his helpful comments on a preliminary version of this paper. Finally, I sincerely thank the anonymous referee for their suggestions which greatly helped in improving and clarifying the manuscript.

References

  1. 1.
    de Jong, A.J.: Crystalline Dieudonné module theory via formal and rigid geometry. Inst. Hautes Études Sci. Publ. Math. 82, 5–96 (1995)CrossRefzbMATHGoogle Scholar
  2. 2.
    Demazure, M., Grothendieck, A., et al., Séminaire de Géometrie Algébrique du Bois Marie- Schémas en groupes (SGA 3), Lecture notes in Mathematics. Springer, New York (1970)Google Scholar
  3. 3.
    Faltings, G.: Integral crystalline cohomology over very ramified valuation rings. J. Am. Math. Soc. 12(1), 117–144 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gashi, Q.: On a conjecture of Kottwitz and Rapoport. Ann. Sci. Éc. Norm. Supeér. 43, 1017–1038 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hong, S.: Harris-Viehmann conjecture for Hodge-Newton reducible Rapoport-Zink spaces (preprint). arXiv:1612.08475 (2016)
  6. 6.
    Howard, B., Pappas, G.: Rapoport–Zink spaces for spinor groups. Comput. Math. 153, 1050–1118 (2017)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Katz, N.: Slope filtration of \(F\)-crystals. Astérisque 63, 113–164 (1979)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Kim, W.: Rapoport–Zink spaces of Hodge type (preprint). arXiv:1308.5537 (2013)
  9. 9.
    Kisin, M.: Integral models for Shimura varieties of abelian type. J. Am. Math. Soc. 23(4), 967–1012 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kottwitz, R.: Isocrystals with additional structure. Comput. Math. 56, 201–220 (1985)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kottwitz, R.: Isocrystals with additional structure II. Comput. Math. 109, 255–339 (1997)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kottwitz, R., Rapoport, M.: On the existence of \(F\)-crystals. Comment. Math. Helv. 78, 153–184 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lubin, J., Serre, J.-P., Tate, J.: Elliptic Curves and Formal Groups. Woods Hole Summer Institute (1964)Google Scholar
  14. 14.
    Lucarelli, C.: A converse to Mazurs inequality for split classical groups. J. Inst. Math. Jussieu 3, 165–183 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Manin, Y.: The theory of commutative formal groups over fields of positive characteristic. Russ. Math. Surv. 18, 1–83 (1963)CrossRefzbMATHGoogle Scholar
  16. 16.
    Mantovan, E.: On non-basic Rapoport–Zink spaces. Ann. Sci. Éc. Norm. Supeér. 41(5), 671–716 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mantovan, E., Viehmann, E.: On the Hodge–Newton filtration for \(p\)-divisible \(\cal{O}\)-modules. Math. Z. 266, 193–205 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Messing, W.: The crystals associated to Barsotti–Tate groups: with applications to abelian schemes. Lecture Notes in Mathematics, vol. 264. Springer, Berlin (1972)Google Scholar
  19. 19.
    Moonen, B.: Models of Shimura Varieties in Mixed Characteristics, Galois Representations in Arithmetic Algebraic Geometry. Cambridge University Press, Cambridge, pp. 267–350 (1998)Google Scholar
  20. 20.
    Moonen, B.: Serre–Tate theory for moduli spaces of PEL-type. Ann. Sci. Éc. Norm. Supeér. 37, 223–269 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rapoport, M., Richartz, M.: On the classification and specialization of \(F\)-isocrystals with additional structure. Comput. Math. 103, 153–181 (1996)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Rapoport, M., Viehmann, E.: Towards a theory of local Shimura varieties. M’unster. J. Math. 7, 273–326 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Serre, J.-P.: Groupes de Grothendieck des schémas en groupes réductifs déployés. Inst. Hautes Études Sci. Publ. Math., 37–52 (1968)Google Scholar
  24. 24.
    Shankar, A., Zhou, R.: Serre–Tate theory for Shimura varieties of Hodge type (preprint). arXiv:1612.06456 (2016)
  25. 25.
    Shen, X.: On the Hodge–Newton filtration for \(p\)-divisible groups with additional structures. Int. Math. Res. Notices 13, 3582–3631 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Vasiu, A.: Generalized Serre–Tate Ordinary Theory. International Press, Boston (2013)zbMATHGoogle Scholar
  27. 27.
    Wedhorn, T.: The dimension of Oort strata of Shimura varieties of PEL-type. Prog. Math. 195, 441–471 (2001)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Wortmann, D.: The \(\mu \)-ordinary locus for Shimura varieties of Hodge type (preprint). arXiv:1310.6444 (2013)

Copyright information

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

Authors and Affiliations

  1. 1.Department of MathematicsCalifornia Institute of TechnologyPasadenaUSA

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