Summary
We sketch a proof of the Hecke orbit conjecture for the Siegel modular variety \(\mathcal{A}_{g,n}\) over \(\overline {\mathbb{F}_p }\), where p is a prime number, fixed throughout this article. We also explain several techniques developed for the Hecke orbit conjecture, including a generalization of the Serre-Tate coordinates.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
F. Andreatta and E. Z. Goren —Hilbert modular varieties of low dimension”, To appear in Geometric Aspects of Dwork’s Theory, A Volume in memory of Bernard Dwork, A. Adolphson, F. Baldassarri, P. Berthelot, N. Katz, and F. Loeser (Eds.), 62 pages.
C.-L. Chai —Canonical coordinates on leaves of p-divisible groups”, Part II. Cartier theory, preprint, 28 pp., September 2003.
—,Families of ordinary abelian varieties: canonical coordinates, p-adic mon-odromy, Tate-linear subvarieties and hecke orbits”, Preprint, 55 pp., July 2003, available from http://www.math.upenn.edu/~chai/.
—,Monodromy of Hecke-invariant subvarieties”, Preprint, 10 pp., April, 2003, available from http://www.math.upenn.edu/~chai/.
—,A rigidity result of p-divisible formal groups”, Preprint, 11 pp., July, 2003, available from http://www.math.upenn.edu/~chai/.
—,Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli”, Invent. Math. 121 (1995), no. 3, p. 439–479.
C.-L. Chai and F. Oort —Canonical coordinates on leaves of p-divisible groups”, Part I. General properties, preprint, 24 pp., August 2003.
P. Deligne and G. Pappas —Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant”, Compositio Math.90 (1994), no. 1, p. 59–79.
P. Deligne and K. A. Ribet —Values of abelian L-functions at negative integers over totally real fields”, Invent. Math.59 (1980), no. 3, p. 227–286.
E. Z. Goren and F. Oort —Stratifications of Hilbert modular varieties”, J. Algebraic Geom.9 (2000), no. 1, p. 111–154.
A. Grothendieck — Groupes de monodromie en géométrie algébrique. I, Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I), Lecture Notes in Math., vol. 288, Springer-Verlag, Berlin, 1972.
A. J. de Jong —Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic”, Invent. Math.134 (1998), no. 2, p. 301–333.
A. J. de Jong and F. Oort —Purity of the stratification by Newton polygons”, J. Amer. Math. Soc.13 (2000), no. 1, p. 209–241.
N. M. Katz —Slope filtration of F-crystals”, Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. I, Astérisque, vol. 63, Soc. Math. France, Paris, 1979, p. 113–163.
J. Lubin and J. Tate —Formal complex multiplication in local fields”, Ann. of Math. (2)81 (1965), p. 380–387.
Y. I. Manin —Theory of commutative formal groups over fields of finite characteristic”, Uspehi Mat. Nauk18 (1963), no. 6 (114), p. 3–90.
D. Mumford —Bi-extensions of formal groups”, Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford Univ. Press, London, 1969, p. 307–322.
T. Oda and F. Oort —Supersingular abelian varieties”, Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) (Tokyo), Kinokuniya Book Store, 1978, p. 595–621.
F. Oort —Hypersymmetric abelian varieties”, preliminary version, Feb. 12, 2004, available from http://www.math.uu.nl/people/oort/.
—,Minimal p-divisible groups”, to appear in Ann. of Math., available from http://www.math.uu.nl/people/oort/.
—,Monodromy, Hecke orbits and Newton polygon strata”, Seminar at MPI, Bonn, Feb. 14, 2004, available from http://www.math.uu.nl/people/oort/.
—,Some questions in algebraic geometry”, Preliminary version, June 1995, available from http://www.math.uu.nl/people/oort/.
—,The isogeny class of a CM-type abelian variety is defined over a finite extension of the prime field”, J. Pure Appl. Algebra 3 (1973), p. 399–408.
—,Newton polygons and formal groups: conjectures by Manin and Grothendieck”, Ann. of Math. (2) 152 (2000), no. 1, p. 183–206.
—,A stratification of a moduli space of abelian varieties”, Moduli of abelian varieties (Texel Island, 1999), Progr. Math., vol. 195, Birkhäuser, Basel, 2001, p. 345–416.
—,Foliations in moduli spaces of abelian varieties”, J. Amer. Math. Soc. 17 (2004), no. 2, p. 267–296 (electronic).
F. Oort and T. Zink —Families of p-divisible groups with constant Newton polygon”, appear in Documenta Mathematica.
K. A. Ribet —p-adic interpolation via Hilbert modular forms”, Algebraic geometry (Proc. Sympos. Pure Math., vol. 29, Humboldt State Univ., Arcata, Calif., 1974), Amer. Math. Soc., Providence, R. I., 1975, p. 581–592.
J. Tate —Classes d’isogeny de variétés abéliennes sur un corps fini (d’apès T. Honda”, Séminaire Bourbaki 1968/69, Exposé 352, Lecture Notes in Math., vol. 179, Springer, Berlin, 1971, p. 95–110.
J. Tate —Endomorphisms of abelian varieties over finite fields”, Invent. Math.2 (1966), p. 134–144.
C.-F. Yu —On reduction of Hilbert-Blumenthal varieties”, Ann. Inst. Fourier (Grenoble)53 (2003), no. 7, p. 2105–2154.
T. Zink —On the slope filtration”, Duke Math. J.109 (2001), no. 1, p. 79–95.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Birkhäuser Boston
About this chapter
Cite this chapter
Chai, CL. (2005). Hecke orbits on Siegel modular varieties. In: Bogomolov, F., Tschinkel, Y. (eds) Geometric Methods in Algebra and Number Theory. Progress in Mathematics, vol 235. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4417-2_4
Download citation
DOI: https://doi.org/10.1007/0-8176-4417-2_4
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-4349-2
Online ISBN: 978-0-8176-4417-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)