Skip to main content
SpringerLink
Log in

A Stratification of a Moduli Space of Abelian Varieties

  • Chapter
Moduli of Abelian Varieties

Part of the book series: Progress in Mathematics ((PM,volume 195))

Abstract

In this paper we study the moduli space A0 Fp of polarized abelian varieties of dimensiongin positive characteristic. We construct a stratification of this space. The strata are indexed by isomorphism classes of group schemes killed byp;a polarized abelian variety (X, A) has its moduli point in a certain stratum if X [p] belongs to the isomorphism class given by a certain discrete invariant. We define these invariants by a numerical property of a filtration ofN= X [p].Passing from one stratum to a stratum in its boundary feels like “degenerating the p-structure”. The fact that these strata are all quasi-affine allows us to keep going in this process until we arrive at the unique zero-dimensional stratum, the superspecial locus. One can formulate this idea by saying that the ordinary locus has several “boundaries”, one where the abelian variety degenerates, one where the p-structure “becomes more special” (and an analogous idea for all non-zero-dimensional strata). This phenomenon, non-present in this form in characteristic zero, but available and powerful in positive characteristic, is ex-pected to have many applications.

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 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
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-Compactification of Siegel moduli schemes. London Math. Soc. Lect. Notes Ser. 107, Cambridge Univ. Press, 1985.

    Google Scholar 

  2. C.-L. Chai-Everyordinary symplectic isogeny class in positive characteristic is dense in the moduli.Invent. Math.121(1995), 439–479.

    MathSciNet  MATH  Google Scholar 

  3. C.-L. Chai & F. Oort -Density of Hecke orbits.[In preparation.]

    Google Scholar 

  4. P. Deligne&M. Rapoport -Les schemes de modules de courbes elliptiques.In: Modular functions of one variable, II (Ed.P.Deligne, W. Kuyk), Lect. Notes Math. 349, Springer-Verlag 1973; pp. 143–316 (pp. DeRa 1–174).

    Chapter  Google Scholar 

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

    Google Scholar 

  6. J. Dieudonne -Lie groups and Lie hyperalgebras over a field of characteristic p.II. Amer. Journ. Math.77(1955), 218–244.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Eichler -Zur Zahlentheorie der Quaternionen-Algebren.J. Reine Angew. Math.195(1955), 127–151.

    MathSciNet  Google Scholar 

  8. T. Ekedahl -On supersingular curves and abelian varieties.Math. Scand.60(1987), 151–178.

    MathSciNet  MATH  Google Scholar 

  9. G. Faltings -Arithmetische Kompaktifizierung des Modulraums der abelschen Varietdten.Arbeitstagung Bonn 1984(Ed.F. Hirzebruchet al.)Lect. Notes Math1111Springer-Verlag, 1985.

    Google Scholar 

  10. G. Faltings&C.-L. Chai-Degeneration of abelian varieties. Ergebn. Math. 3. Folge, Bd. 22, Springer-Verlag, 1990.

    Google Scholar 

  11. G. van der Geer -Cycles on the moduli space of abelian varieties.In: Moduli of curves and abelian varieties (the Dutch intercity seminar on moduli) (Eds C. Faber, E. Looijenga). Aspects of Math. E 33, Vieweg 1999; pp. 65–89.

    Chapter  Google Scholar 

  12. G. van der Geer & M. van der Vlugt -On the existence of supersingular curves of given genus.Journ. reine angew. Math.45(1995), 53–61.

    Google Scholar 

  13. E. Z. Goren & F. Oort -Stratification of Hilbert modular varieties I.Journ. Algebraic Geom.9(2000), 111–154.

    MathSciNet  MATH  Google Scholar 

  14. A. Grothendieck -Fondements de la geornetrie algebrique.[Extraits du Seminaire Bourbaki 1957–1962.] Exp. 212, Sem. Bourbaki13(1960/1961):Preschemas quotients.

    Google Scholar 

  15. A. Grothendieck&J. Dieudonne -Elements de geometric algebrique.Vol. II: Etude globale elementaire de quelques classes de morphismes. Vol. III1: Etude cohomologique des faisceaux coherents (Premiere Partie). Publ. Math.8, 11Inst. Hautes Et. Sc. 1961, 1961 [Cited as EGA II, EGA III]

    Google Scholar 

  16. H. Hasse & E. Witt -Zyklische unverzweigte Erweiterungskorper vom Primzahlgrade p iiber einem algebraischen Funktionenkorper der Charakteristik p.Monatsh. Math. Phys.43(1963), 477–492.

    Google Scholar 

  17. L. Illusie -Deformations de groupes de B ars otti- Tate. Exp.VI in: Seminaire sur les pinceaux arithmetiques: la conjecture de Mordell (Ed. L. Szpiro), Asterisque 127, Soc. Math. France, 1985.

    Google Scholar 

  18. A. J. de Jong -Finite locally free group schemes in characteristic p and Dieudonne modules. Invent. Math.114(1993), 89–137.

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  20. T. Katsura & F. Oort -Families of supersingular abelian surfaces.Compos. Mat. 62 (1987), 107–167.

    MathSciNet  MATH  Google Scholar 

  21. T. Katsura & F. Oort-Supersingular abelian varieties of dimension two or three and class numbers. Algebraic Geometry, Sendai 1985 (ed. T. Oda); Adv. St. Pure Math. 10 (1987), Kinokuniya & North-Holland, 1987.

    Google Scholar 

  22. N. Katz -Serre-Tate local moduli.Exp. Vbis in: Surfaces algebriques, Sem. Geom. Alg. Orsay 1976–78 (Ed. J. Giraud, L. Illusie and M. Raynaud), Lect. Notes Math. 868, Springer-Verlag 1981, pp. 138–202.

    Google Scholar 

  23. N. M. Katz -Slope filtrations of F-crystals.Journ. Geom. Algebr. de Rennes, Vol. I; Asterisque 63, Soc. Math. France, 1979; pp. 113–163.

    MATH  Google Scholar 

  24. M. Kneser-Strong approximation. Proceed. Sympos. Pure Math., Vol. 9: Algebraic groups and discontinuous subgroups. A.M.S. 1966.

    Google Scholar 

  25. M.-A. Knus-Quadratic and hermitian forms over rings. Grundl. math. Wissensch. 294, Springer-Verlag 1991.

    Google Scholar 

  26. M.-A. Knus, A. Merkurjev, M. Rost & J.-P. Tignol -The book of involutions.AMS Colloq. Publ. 44, Amer. Math. Soc. 1998.

    Google Scholar 

  27. H.-P. Kraft -Kommutative algebraische p-Gruppen (mit anwendungen auf p-divisible Gruppen and abelsche Varietiiten).Sonderforsch. Bereich Bonn, September 1975. Ms. 86 pp.

    Google Scholar 

  28. H.-P. Kraft & F. Oort -Finite group schemes annihilated by p.[In preparation.]

    Google Scholar 

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

    Google Scholar 

  30. J. Lubin, J-P. Serre&J. Tate - Elliptic curves and formal groups. In: Lect. Notes, Summer Inst. Algebraic Geometry, Woods Hole, July 1964; 9 pp.

    Google Scholar 

  31. 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.

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  33. B. Moonen -Group schemes with additional structures and Weyl group cosets.This volume, pp. 255–298.

    Google Scholar 

  34. L. Moret-Bailly -Familles de courbes at de varietes abeliennes surP1.Familles de courbes et de varietes abeliennes.Sem. sur les pinceaux de courbes de genre aux moins deux. Ed. L. Szpiro. Asterisque 86, Soc. Math. France, 1981; pp. 109–140.

    Google Scholar 

  35. L. Moret-Bailly-Pinceaux de varietes abeliennnes. Asterisque 129, Soc. Math. France, 1985.

    Google Scholar 

  36. D. Mumford - Constructions of deformation of Dieudonne modules. Private communication, 1968.

    Google Scholar 

  37. D. Mumford-Abelian varieties. Tata Inst. Fund. Res. & Oxford Univ. Press, 1970 (2nd print. 1974).

    Google Scholar 

  38. P. Norman - Analgorithm for computing local moduli of abelian varieties.Ann. Math. 101 (1975), 499–509.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  40. T. Oda -The first de Rham cohomology group and Dieudonne modules.Ann. Sc. Ecole Norm. Sup. 2 (1969), 63–135.

    MathSciNet  MATH  Google Scholar 

  41. A. Ogus -Supersingular KS crystals.In: Journ. Geom. Algebr. de Rennes, Vol II; Asterisque 64, Soc. Math. France, 1979; pp. 3–86.

    MathSciNet  MATH  Google Scholar 

  42. F. Oort-Commutative group schemes. Lect. Notes Math. 15, Springer-Verlag 1966.

    Google Scholar 

  43. F. Oort -Subvarieties of moduli spaces.Invent. Math. 24 (1974), 95–119.

    Article  MathSciNet  MATH  Google Scholar 

  44. F. Oort -Which abelian surfaces are products of elliptic curves?Math. Ann. 214 (1975), 35–47.

    Article  MathSciNet  MATH  Google Scholar 

  45. F. Oort-Astratification of moduli spaces of polarized abelian varieties in positive characteristic.In: Moduli of curves and abelian varieties (the Dutch intercity seminar on moduli) (Eds C. Faber, E. Looijenga). Aspects of Math. E 33, Vieweg 1999; pp. 47–64.

    Chapter  Google Scholar 

  46. F. Oort- Newton polygons and formal groups: conjectures by Manin and Grothendieck.[To appear: Ann. Math. 152 (2000).

    Google Scholar 

  47. F. Oort-Newton polygon strata in the moduli space of abelian varieties.This volume, pp. 417–440.

    Google Scholar 

  48. F. Oort -Foliations of moduli spaces.[In preparation.]

    Google Scholar 

  49. V. Platonov & A. Rapinchuk Algebraic groups and number theory. Pure and appl math. Vol 139, Academic Press 1994.

    Google Scholar 

  50. M. Poletti- Differenziali esatti di prima specie su variety abeliane.Ann. Scuola Norm. Sup. Pisa 21 (1967), 107–110.

    MathSciNet  MATH  Google Scholar 

  51. G. Prasad- Strong approximation for semi-simple groups over function fields.Ann. Math. 105 (1977), 553–572.

    Article  MATH  Google Scholar 

  52. I. Reiner-Maximal orders. Academis Press 1975.

    Google Scholar 

  53. W. Scharlau-Quadratic and hermitian forms. Grundl. math. Wissensch. 270, Springer-Verlag 1985.

    Google Scholar 

  54. Seminaire de geometric algebrigue1967–1969, SGA 7:Groupes de monodromie en geometric algebrigue.(A. Grothendieck, M. Raynaud, D. S. Rim.) Vol. I, Lect. Notes Math. 288, Springer-Verlag, 1972.

    Google Scholar 

  55. J-P. Serre -Cohomologie galoisienne.Cinquieme edition, revisee et completee. Lect. Notes Math. 5, Springer-Verlag 1994.

    Google Scholar 

  56. J-P. Serre -Sur la topologie des varietes algebrigues en caracteristigue p.Symp. Int. Top. Alg. Mexico (1985), pp. 24–53 (see Collected works, Vol. I, pp. 501–530).

    Google Scholar 

  57. G. Shimura -Arithmetic of alternating forms and quaternion hermitian forms.Journ. Math. Soc. Japan 15 (1963), 33–65.

    Article  MathSciNet  MATH  Google Scholar 

  58. T. Shioda -Supersingular K3 surfaces.In: Algebraic Geometry, Copenhagen 1978 (Ed. K. Lonsted). Lect. Notes Math. 732, Springer-Verlag, 1979; pp. 564–591.

    Google Scholar 

  59. T. A. Springer - Linear algebraic groups. Second edition. Progress Math. 9, Birkhauser, 1998.

    Google Scholar 

  60. L. Szpiro -Proprietes numerigues du faisceau dualisant relatif.Sem. sur les pinceaux de courbes de genre aux moms deux. Ed. L. Szpiro. Asterisque 86, Soc. Math. France, 1981; pp. 44–78.

    Google Scholar 

  61. T. Wedhorn -The dimension of Oort strata of Shimura varieties of PEL-type.This volume, pp. 441–471.

    Google Scholar 

  62. Yu. G. Zarhin -Endomorphisms of abelian varieties and points of finite order in characteristic p.Math. Notes Acad. Sci. USSR 21 (1977), 734–744.

    MathSciNet  Google Scholar 

  63. T. Zink-Cartiertheorie kommutativer formaler Gruppen. Teubner-Texte Math. 68, Teubner, Leipzig, 1984.

    Google Scholar 

  64. T. Zink-The display of a formal p-divisible group, University of Bielefeld, Preprint 98–017, February 1998.

    Google Scholar 

  65. T. Zink-A Dieudonne theory for p-divisible groups. Manuscript, 23 pp., September 1998.

    Google Scholar 

Download references

Authors
  1. Frans Oort
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institutionen för Matematik, Kungliga Tekniska Högskolan (KTH), Stockholm, 10044, Sweden

    Carel Faber

  2. Korteweg de Vries Instituut, Universiteit van Amsterdam, Plantage Muidergracht 24, Amsterdam, 1018 TV, The Netherlands

    Gerard van der Geer

  3. Mathematisch Instituut, Universiteit Utrecht, Budapestlaan 6, Utrecht, 3508 TA, The Netherlands

    Frans Oort

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer Basel AG

About this chapter

Cite this chapter

Oort, F. (2001). A Stratification of a Moduli Space of Abelian Varieties. In: Faber, C., van der Geer, G., Oort, F. (eds) Moduli of Abelian Varieties. Progress in Mathematics, vol 195. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8303-0_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8303-0_13

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9509-5

  • Online ISBN: 978-3-0348-8303-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation