Inventiones mathematicae

, Volume 121, Issue 1, pp 439–479

Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli

  • Ching-Li Chai
Article

Abstract

We prove that any ordinary symplectic separable isogeny class in the moduli space of principally polarized abelian varieties over a field of positive characteristic is dense in the Zariski topology.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Ash, D. Mumford, M. Rapoport and Y. Tai, Smooth Compactification of Locally Symmetric Spaces, Math. Sci. Press 1975.Google Scholar
  2. 2.
    A. Borel, Properties and linear representations of Chevalley groups, Seminar on algebraic groups and related finite groups, Lecture Notes in Math. 131, 1970, pp. 1–55.Google Scholar
  3. 3.
    C.-L. Chai, Compactification of Siegel Moduli Schemes, Lecture Notes Series 107, London Math. Soc., London, 1985.Google Scholar
  4. 4.
    C.-L. Chai, Arithmetic minimal compactification of Hilbert-Blumenthal moduli spaces, Appendix to Andrew Wiles “The lwasawa conjecture for totally real fields. Annals of Math.131 (1990) 541–554.Google Scholar
  5. 5.
    C.-L. Chai, The group action on the closed fiber of the Lubin-Tate moduli space, preprint 1994, to appear in Duke Math. J.Google Scholar
  6. 6.
    P. Deligne, Variétés de Shimura: Interprétation modulair, et techniques de construction de modèles canoniques, Automorphic Forms, Representations, and L-functions (A. Borel and W. Casselman, eds.), Proc. Symp. Pure Math. 33, part 2, AMS, 1979, pp. 247–290.Google Scholar
  7. 7.
    P. Deligne and G. Pappas, Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant. Compos. Math.90 (1994) 59–79.Google Scholar
  8. 8.
    T. Ekedahl, On supersingular curves and abelian varieties, Math. Scand60 (1987) 151–178.Google Scholar
  9. 9.
    T. Ededahl, The action of monodromy on torsion points of jacobians, Arithmetic Algebraic Geometry, Texel 1989, Eds. G. van der Geer, F. Oort, J. Steenbrink, Progress in Math. 89, Birkhäuser, 1991, pp. 41–49.Google Scholar
  10. 10.
    T. Ekedahl and F. Oort, Connected subsets of a moduli space of abelian varieties, preprint (1994).Google Scholar
  11. 11.
    G. Faltings, Arithmetische Kompaktifizierung des Modulraums der abelschen Varietäten, Lecture Notes in Math. 1111, Springer-Verlag, 1985, pp. 321–383.Google Scholar
  12. 12.
    G. Faltings and C.-L. Chai, Degeneration of Abelian Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 22, Springer-Verlag, 1990.Google Scholar
  13. 13.
    B. Gross, On canonical and quasi-canonical liftings, Inven. Math.84 (1986) 321–326.Google Scholar
  14. 14.
    B.H. Gross and M.J. Hopkins, Equivariant vector bundles on the Lubin-Tate moduli space, Topology and Representation Theory (Evanston, IL, 1992); Contemp. Math.158 (1994) 23–88Google Scholar
  15. 15.
    A. Grothendieck, Groupes de Barsotti-Tate et Cristaux de Dieudonné, Les Presses de l'Université de Montréal, 1974.Google Scholar
  16. 16.
    A.J. de Jong, The moduli space of polarized abelian varieties, Math. Ann.295 (1993) 485–503.Google Scholar
  17. 17.
    N.M. Katz, Travaux de Dwork, Séminaire Bourbaki 1971/72, exposé 409, Lecture Notes in Math. 317, Springer-Verlag, 1973, pp. 69–190.Google Scholar
  18. 18.
    N.M. Katz, Serre-Tate local moduli, Séminaire de Géométrie Algébrique d'Orsay 1976–78, Expposé Vbis, Surface Algébriques, Lecture Notes in Math. 868, Springer-Verlag, 1981, pp. 138–202.Google Scholar
  19. 19.
    N. Koblitz, P-adic variation of the zeta-function over families of varieties defined over finite fields. Compos. Math.31 (1975) 119–218.Google Scholar
  20. 20.
    R. Langlands and M. Rapoport, Shimuravarietäten und Gerben, J. reine angew. Math.378 (1987) 113–220.Google Scholar
  21. 21.
    J. Lubin and J. Tate, Formal moduli for one-parameter formal Lie groups, Bull. Soc. Math. France94 (1966) 49–60.Google Scholar
  22. 22.
    W. Messing, The Crystals Associated to Barsotti-Tate Groups: with Applications to Abelian Schemes, Lecture Notes in Math. 370, Springer-Verlag, 1972.Google Scholar
  23. 23.
    J.S. Milne, The points on a Shimura variety modulo a prime of good reduction, The Zeta Function of Picard Modular Surfaces (R. Langlands and D. Ramakrishnan, eds.), Les Publications CRM, Montréal, 1992, pp. 153–255.Google Scholar
  24. 24.
    J.S. Milne, Shimura varieties and motives, Proc. Symp. Pure Math. (1994).Google Scholar
  25. 25.
    L. Moret-Bailly, Pinceaux de Variétés Abéliennes, Astérisque 129, Soc. Math. France, 1985.Google Scholar
  26. 26.
    D. Mumford, Abelian Varieties, Tata Inst., Studies in Math. 5, Oxford University Press, 1974.Google Scholar
  27. 27.
    P. Norman, An algorithm for computing local moduli of abelian varieties, Ann. Math.101 (1975), 499–509.Google Scholar
  28. 28.
    P. Norman and F. Oort, Moduli of abelian varieties, Annals of Math.112 (1980) 413–439.Google Scholar
  29. 29.
    T. Oda and F. Oort, Supersingular abelian varieties, Proc., Kyoto Univ., Kyoto, 1977, Kinokuniya Book Store, Tokyo, 1978, pp. 595–621.Google Scholar
  30. 30.
    F. Oort, The isogeny class of a CM-type abelian variety is defined over a finite extension of the prime field, J. Pure Appl. Algebra3 (1973), 399–408.Google Scholar
  31. 31.
    M. Rapoport, Compactifications de l'espace de modules de Hilbert-Blumenthal, Compo. Math.36 (1978) 255–335.Google Scholar
  32. 32.
    G. Shimura, On analytic families of polarized abelian varieties and automorphic functions. Ann. of Math.78 (1963) 149–192.Google Scholar
  33. 33.
    J. Tate, Classes d'isogeny de variétés abéliennes sur un corps fini (d'apès T. Honda), Sém. Bourbaki Exp. 352 (1968/69), Lecture Notes in Math. 179, Springer Verlag, 1971.Google Scholar
  34. 34.
    S.P. Wang, On density properties of S-subgroups of locally compact groups, Annals of Math.94 (1971) 325–329.Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Ching-Li Chai
    • 1
  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA

Personalised recommendations