Foliations in moduli spaces of abelian varieties

Oort, Frans

doi:10.1090/S0894-0347-04-00449-7

Foliations in moduli spaces of abelian varieties
HTML articles powered by AMS MathViewer

by Frans Oort

J. Amer. Math. Soc. 17 (2004), 267-296

DOI: https://doi.org/10.1090/S0894-0347-04-00449-7

Published electronically: January 7, 2004

PDF | Request permission

Abstract:

We study moduli spaces of polarized abelian varieties in positive characteristic. Our final goal will be to understand Hecke orbits in such spaces. This paper provides one of the tools. For a given $p$-divisible group, all abelian varieties which give rise to this group have moduli points in a locally closed subset of the moduli space; we call an irreducible component of this subset a central leaf. Newton polygon strata are foliated by such leaves. Moreover, iterated $\alpha _p$-isogenies give a second leaf structure, which was already known under the name of Rapoport-Zink spaces. Any Newton polygon stratum is, up to a finite morphism, isomorphic to a product of an isogeny leaf and a finite cover of a central leaf. We conjecture that any Hecke-$\ell$-orbit is dense in the corresponding central leaf.

References

Canonical coordinates on leaves of $p$-divisible groups.

Gerd Faltings and Ching-Li Chai, Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22, Springer-Verlag, Berlin, 1990. With an appendix by David Mumford. MR 1083353, DOI 10.1007/978-3-662-02632-8
Hermann Weyl, Invariants, Duke Math. J. 5 (1939), 489–502. MR 30
Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001. With an appendix by Vladimir G. Berkovich. MR 1876802
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
A. J. de Jong, Crystalline Dieudonné module theory via formal and rigid geometry, Inst. Hautes Études Sci. Publ. Math. 82 (1995), 5–96 (1996). MR 1383213
A. J. de Jong and F. Oort, Purity of the stratification by Newton polygons, J. Amer. Math. Soc. 13 (2000), no. 1, 209–241. MR 1703336, DOI 10.1090/S0894-0347-99-00322-7
Toshiyuki Katsura and Frans Oort, Supersingular abelian varieties of dimension two or three and class numbers, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 253–281. MR 946242, DOI 10.2969/aspm/01010253
P. Deligne, Cristaux ordinaires et coordonnées canoniques, Algebraic surfaces (Orsay, 1976–78) Lecture Notes in Math., vol. 868, Springer, Berlin-New York, 1981, pp. 80–137 (French). With the collaboration of L. Illusie; With an appendix by Nicholas M. Katz. MR 638599
Nicholas M. Katz, Slope filtration of $F$-crystals, Journées de Géométrie Algébrique de Rennes (Rennes, 1978) Astérisque, vol. 63, Soc. Math. France, Paris, 1979, pp. 113–163. MR 563463
Ke-Zheng Li and Frans Oort, Moduli of supersingular abelian varieties, Lecture Notes in Mathematics, vol. 1680, Springer-Verlag, Berlin, 1998. MR 1611305, DOI 10.1007/BFb0095931
Ju. I. Manin, Theory of commutative formal groups over fields of finite characteristic, Uspehi Mat. Nauk 18 (1963), no. 6 (114), 3–90 (Russian). MR 0157972

On certain unitary group Shimura varieties.

Peter Norman and Frans Oort, Moduli of abelian varieties, Ann. of Math. (2) 112 (1980), no. 3, 413–439. MR 595202, DOI 10.2307/1971152
Tadao Oda, The first de Rham cohomology group and Dieudonné modules, Ann. Sci. École Norm. Sup. (4) 2 (1969), 63–135. MR 241435
F. Oort, Commutative group schemes, Lecture Notes in Mathematics, vol. 15, Springer-Verlag, Berlin-New York, 1966. MR 0213365
Frans Oort, Newton polygons and formal groups: conjectures by Manin and Grothendieck, Ann. of Math. (2) 152 (2000), no. 1, 183–206. MR 1792294, DOI 10.2307/2661381
Frans Oort, A stratification of a moduli space of abelian varieties, Moduli of abelian varieties (Texel Island, 1999) Progr. Math., vol. 195, Birkhäuser, Basel, 2001, pp. 345–416. MR 1827027, DOI 10.1007/978-3-0348-8303-0_{1}3
Frans Oort, Newton polygon strata in the moduli space of abelian varieties, Moduli of abelian varieties (Texel Island, 1999) Progr. Math., vol. 195, Birkhäuser, Basel, 2001, pp. 417–440. MR 1827028, DOI 10.1007/978-3-0348-8303-0_{1}4

Minimal $p$-divisible groups.

Irreducibility of Newton polygon strata.

Frans Oort and Thomas Zink, Families of $p$-divisible groups with constant Newton polygon, Doc. Math. 7 (2002), 183–201. MR 1938119
M. Rapoport and Th. Zink, Period spaces for $p$-divisible groups, Annals of Mathematics Studies, vol. 141, Princeton University Press, Princeton, NJ, 1996. MR 1393439, DOI 10.1515/9781400882601
I. Reiner, Maximal orders, London Mathematical Society Monographs, No. 5, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1975. MR 0393100
Thomas Zink, On the slope filtration, Duke Math. J. 109 (2001), no. 1, 79–95. MR 1844205, DOI 10.1215/S0012-7094-01-10913-7