Skip to main content
SpringerLink
Log in

Foliations in Moduli Spaces of Abelian Varieties and Dimension of Leaves

  • Chapter
  • First Online:
Book cover Algebra, Arithmetic, and Geometry

Part of the book series: Progress in Mathematics ((PM,volume 270))

Summary

In moduli spaces of abelian varieties and of p-divisible groups in characteristic p we have various foliations and statifications. In this paper we compute the dimensions of central leaves. We give three different proofs of these results, where every proof presents a different flavour of this beautiful topic. Components of Newton polygon strata for one fixed Newton polygon may have various different dimensions, according to properties of the polarizations considered; we show which dimensions do appear for a given Newton polygon. Hence dimensions of isogeny leaves can be computed this way.

2000 Mathematics Subject Classifications: 11G15, 14L05, 14L15

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. C.-L. Chai – Hecke orbits on Siegel modular varieties. Progress in Math. 235, Birkhäuser, 2004, pp. 71–107.

    Article  MathSciNet  Google Scholar 

  2. C.-L. Chai – Canonical coordinates on leaves of p-divisible groups: the two-slope case. Manuscript 10-I-2005. [Submitted for publication.]

    Google Scholar 

  3. C.-L. Chai & F. Oort – Moduli of abelian varieties and p-divisble groups. Lectures at the conference on Arithmetic Geometry, Göttingen, July-August 2006. [to appear] See arXiv math.AG/0701479

    Google Scholar 

  4. C.-L. Chai & F. Oort – Monodromy and irreducibility. [To appear.]

    Google Scholar 

  5. C.-L. Chai & F. Oort – Hecke orbits. [In preparation]

    Google Scholar 

  6. M. Demazure – Lectures on p-divisible groups. Lect. Notes Math. 302, Springer–Verlag, Berlin 1972.

    Book  MATH  Google Scholar 

  7. V. G. Drinfeld – Coverings of p-adic symmetric domains. [Funkcional. Anal. i Priložen. 10 (1976), no. 2, 29–40.] Functional Analysis and its Applications, 10 (1976), 107–115.

    MathSciNet  Google Scholar 

  8. S. J. Edixhoven, B. J. J. Moonen, & F. Oort (Editors) – Open problems in algebraic geometry. Bull. Sci. Math. 125 (2001), 1–22.

    Article  MATH  MathSciNet  Google Scholar 

  9. G. Faltings & C.-L. Chai – Degeneration of abelian varieties. Ergebnisse Bd 22, Springer–Verlag, 1990.

    Google Scholar 

  10. A. Grothendieck & J. Dieudonné - Eléments de géométrie algébrique. Ch. III 1 : Etude cohomologique des faisceaux cohérents. Publ. Math. 11, IHES 1961.

    Google Scholar 

  11. A. Grothendieck – Groupes de Barsotti-Tate et cristaux de Dieudonné. Sém. Math. Sup. 45, Presses de l’Univ. de Montreal, 1970.

    Google Scholar 

  12. S. Harashita – Configuration of the central streams in the moduli of abelian varieties. Manuscript 33 pp.See: http://www.math.sci.hokudai.ac.jp/~harasita/

  13. L. Illusie – Déformations de groupes de Barsotti-Tate. Exp.VI in: Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell (L. Szpiro), Astérisque 127, Soc. Math. France 1985.

    Google Scholar 

  14. A. J. de Jong – Crystalline Dieudonné module theory via formal rigid geometry. Publ. Math. IHES 82 (1995), 5–96.

    MATH  Google Scholar 

  15. A. J. de Jong & F. Oort – Purity of the stratification by Newton polygons. Journ. Amer. Math. Soc. 13 (2000), 209–241. See: http://www.ams.org/jams

    Article  MATH  MathSciNet  Google Scholar 

  16. N. M. Katz – Appendix to Expose V. In: Surfaces algébriques (Ed. J. Giraud, L. Illusie, M. Raynaud). Lect. Notes Math. 868, Springer–Verlag, Berlin 1981; pp. 127 – 137.

    Google Scholar 

  17. N. M. Katz – Slope filtration of F–crystals. Journ. Géom. Alg. Rennes, Vol. I, Astérisque 63 (1979), Soc. Math. France, 113 - 164.

    Google Scholar 

  18. N. M. Katz – Serre-Tate local moduli. In: Surfaces algébriques (Ed. J. Giraud, L. Illusie, M. Raynaud). Lect. Notes Math. 868, Springer–Verlag, Berlin 1981; pp. 138–202.

    Google Scholar 

  19. K.-Z. Li & F. Oort - Moduli of supersingular abelian varieties. Lect. Notes Math. 1680, Springer–Verlag 1998.

    Google Scholar 

  20. J. Lubin, J-P. Serre, & J. Tate – Elliptic curves and formal groups. In: Lecture notes prepared in connection with the seminars held at the Summer Institute on Algebraic Geometry, Whitney Estate, Woods Hole, Massachusetts, July 6-July 31, 1964. Mimeographed notes.See http://www.ma.utexas.edu/users/voloch/lst.html

  21. Yu. I. Manin – The theory of commutative formal groups over fields of finite characteristic. Usp. Math. 18 (1963), 3–90; Russ. Math. Surveys 18 (1963), 1–80.

    Google Scholar 

  22. W. Messing – The crystals associated to Barsotti-Tate groups: with applications to abelian schemes. Lect. Notes Math. 264, Springer–Verlag 1972.

    Google Scholar 

  23. B. Moonen – Group schemes with additional structures and Weyl group cosets. In: Moduli of abelian varieties. (Ed. C. Faber, G. van der Geer, F. Oort). Progress Math. 195, Birkhäuser Verlag 2001; pp. 255–298.

    Google Scholar 

  24. B. Moonen – Serre-Tate theory for moduli spaces of PEL type. Ann. Scient. Ec. Norm. Sup. 4e Série 37 (2004), 223–269.

    MATH  MathSciNet  Google Scholar 

  25. B. Moonen – A dimension formula for Ekedahl - Oort strata. Ann. Inst. Fourier 54 (2004), 666–698.

    MATH  MathSciNet  Google Scholar 

  26. B. Moonen & T. Wedhorn – Discrete invariants of varieties in positive characteristic. International Math. Research Notices 72 (2004), 3855–3903.

    Article  MathSciNet  Google Scholar 

  27. P. Norman & F. Oort – Moduli of abelian varieties. Ann. Math. 112 (1980), 413–439.

    Article  MathSciNet  Google Scholar 

  28. F. Oort – Commutative group schemes. Lect. Notes Math. 15, Springer–Verlag 1966.

    Google Scholar 

  29. F. Oort – Newton polygons and formal groups: conjectures by Manin and Grothendieck. Ann. Math. 152 (2000), 183–206.

    Article  MATH  MathSciNet  Google Scholar 

  30. F. Oort – A stratification of a moduli space of polarized abelian varieties. In: Moduli of abelian varieties. (Ed. C. Faber, G. van der Geer, F. Oort). Progress in Math. 195, Birkhäuser Verlag 2001; pp. 345–416.

    Google Scholar 

  31. F. Oort – Newton polygon strata in the moduli space of abelian varieties. In: Moduli of abelian varieties. (Ed. C. Faber, G. van der Geer, F. Oort). Progress in Math. 195, Birkhäuser Verlag 2001; pp. 417–440.

    Google Scholar 

  32. F. Oort – Foliations in moduli spaces of abelian varieties. Journ. Amer. Math. Soc. 17 (2004), 267–296.

    Article  MATH  MathSciNet  Google Scholar 

  33. F. Oort – Monodromy, Hecke orbits and Newton polygon strata. Talk Bonn, 24-II-2003. See: http://www.math.uu.nl/people/oort/

  34. F. Oort – Hecke orbits and stratifications in moduli spaces of abelian varieties. Talk at the Orsay / SAGA, 14-X-2003. See: http://www.math.uu.nl/people/oort/

  35. F. Oort – Hecke orbits in moduli spaces of abelian varieties and foliations. Talk at the ETH in Zürich, 2-IV-2004.See: http://www.math.uu.nl/people/oort/

  36. F. Oort – Minimal p-divisible groups. Annals of Math. 161 (2005), 1–16.

    Article  MathSciNet  Google Scholar 

  37. F. Oort – Simple p-kernels of p-divisible groups. Advances in Mathematics 198 (2005), 275–310.

    Article  MATH  MathSciNet  Google Scholar 

  38. F. Oort & T. Zink – Families of p-divisible groups with constant Newton polygon. Documenta Mathematica 7 (2002), 183–201.See: http://www.mathematik.uni-bielefeld.de/documenta/vol-07/09.html

    MATH  MathSciNet  Google Scholar 

  39. M. Rapoport & Th. Zink – Period spaces for p-divisible groups. Ann. Math. Studies 141, Princeton University Press, 1996.

    Google Scholar 

  40. E. Viehmann – Moduli spaces of p-divisible groups. Journal of Algebraic Geometry, 17 (2008), 341–374.

    MATH  MathSciNet  Google Scholar 

  41. E. Viehmann – The global structure of moduli spaces of polarized p-divisible groups. Document. Math. 13 (2008), 825–852.

    MATH  MathSciNet  Google Scholar 

  42. T. Wedhorn – The dimension of Oort strata of Shimura varieties of PEL-type.Moduli of abelian varieties. (Ed. C. Faber, G. van der Geer, F. Oort). Progress in Math. 195, Birkhäuser Verlag 2001; pp. 441–471.

    Google Scholar 

  43. T. Wedhorn – Specializations of F-Zips. Manuscript 22 pp., 20-VI-2005.

    Google Scholar 

  44. T. Zink – On the slope filtration. Duke Math. Journ. 109 (2001), 79–95.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Utrecht University, 80.010, NL - 3508, TA, Utrecht, The Netherlands

    Frans Oort

Authors
  1. Frans Oort
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Frans Oort .

Editor information

Editors and Affiliations

  1. Courant Inst. Mathematical, New York University, Mercer St. 251, New York, 10012, USA

    Yuri Tschinkel

  2. Eberly College of Science, Pennsylvania State University, University Park, 16802, USA

    Yuri Zarhin

Additional information

Dedicated to Yuri Manin on his seventieth birthday

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Oort, F. (2009). Foliations in Moduli Spaces of Abelian Varieties and Dimension of Leaves. In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry. Progress in Mathematics, vol 270. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4747-6_15

Download citation

Publish with us

Policies and ethics

Search

Navigation