Ekedahl-Oort strata contained in the supersingular locus and Deligne-Lusztig varieties

Harashita, Shushi

doi:10.1090/S1056-3911-09-00519-0

Author: Shushi Harashita
Journal: J. Algebraic Geom. 19 (2010), 419-438
DOI: https://doi.org/10.1090/S1056-3911-09-00519-0
Published electronically: August 17, 2009
MathSciNet review: 2629596
Full-text PDF

Abstract | References | Additional Information

Abstract: We study the moduli space of principally polarized abelian varieties over fields of positive characteristic. In this paper we describe certain unions of Ekedahl-Oort strata contained in the supersingular locus in terms of Deligne-Lusztig varieties. As a corollary we show that each Ekedahl-Oort stratum contained in the supersingular locus is reducible except possibly for small $p$.

References

Cédric Bonnafé and Raphaël Rouquier, On the irreducibility of Deligne-Lusztig varieties, C. R. Math. Acad. Sci. Paris 343 (2006), no. 1, 37–39 (English, with English and French summaries). MR 2241956, DOI https://doi.org/10.1016/j.crma.2006.04.014
Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012
Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR 1890629
Roger W. Carter, Simple groups of Lie type, John Wiley & Sons, London-New York-Sydney, 1972. Pure and Applied Mathematics, Vol. 28. MR 0407163

C.-L. Chai and F. Oort: Monodromy and irreducibility of leaves. 30 pages, preprint. See http://www.math.uu.nl/people/oort/.

P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103–161. MR 393266, DOI https://doi.org/10.2307/1971021

Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197–272 (German). MR 5125, DOI https://doi.org/10.1007/BF02940746

Carel Faber, Gerard van der Geer, and Frans Oort (eds.), Moduli of abelian varieties, Progress in Mathematics, vol. 195, Birkhäuser Verlag, Basel, 2001. Papers from the 3rd Conference held on Texel Island, April 1999. MR 1827014

U. Görtz and C.-F. Yu: Supersingular Kottwitz-Rapoport strata and Deligne-Lusztig varieties. Preprint. arXiv: 0802.3260v1

A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 217086

Shushi Harashita, Ekedahl-Oort strata and the first Newton slope strata, J. Algebraic Geom. 16 (2007), no. 1, 171–199. MR 2257323, DOI https://doi.org/10.1090/S1056-3911-06-00451-6

M. Hoeve: Ekedahl-Oort strata in the supersingular locus. Preprint. arXiv: 0802.4012

Tomoyoshi Ibukiyama, Toshiyuki Katsura, and Frans Oort, Supersingular curves of genus two and class numbers, Compositio Math. 57 (1986), no. 2, 127–152. MR 827350

Jun-ichi Igusa, Class number of a definite quaternion with prime discriminant, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 312–314. MR 98728, DOI https://doi.org/10.1073/pnas.44.4.312

A. J. de Jong, Finite locally free group schemes in characteristic $p$ and Dieudonné modules, Invent. Math. 114 (1993), no. 1, 89–137. MR 1235021, DOI https://doi.org/10.1007/BF01232664

Toshiyuki Katsura and Frans Oort, Families of supersingular abelian surfaces, Compositio Math. 62 (1987), no. 2, 107–167. MR 898731

Ke Zheng Li, Classification of supersingular abelian varieties, Math. Ann. 283 (1989), no. 2, 333–351. MR 980602, DOI https://doi.org/10.1007/BF01446439

K.-Z. Li and F. Oort: Moduli of Supersingular Abelian Varieties. Lecture Notes in Math. 1680 (1998).

Ben Moonen, Group schemes with additional structures and Weyl group cosets, Moduli of abelian varieties (Texel Island, 1999) Progr. Math., vol. 195, Birkhäuser, Basel, 2001, pp. 255–298. MR 1827024

Ben Moonen and Torsten Wedhorn, Discrete invariants of varieties in positive characteristic, Int. Math. Res. Not. 72 (2004), 3855–3903. MR 2104263, DOI https://doi.org/10.1155/S1073792804141263

David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1970. MR 0282985

D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906

Tadao Oda, The first de Rham cohomology group and Dieudonné modules, Ann. Sci. École Norm. Sup. (4) 2 (1969), 63–135. MR 241435

Arthur Ogus, Supersingular $K3$ crystals, Journées de Géométrie Algébrique de Rennes (Rennes, 1978) Astérisque, vol. 64, Soc. Math. France, Paris, 1979, pp. 3–86. MR 563467

Frans Oort, Subvarieties of moduli spaces, Invent. Math. 24 (1974), 95–119. MR 424813, DOI https://doi.org/10.1007/BF01404301

Frans Oort, Which abelian surfaces are products of elliptic curves?, Math. Ann. 214 (1975), 35–47. MR 364264, DOI https://doi.org/10.1007/BF01428253

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

Gopal Prasad, Volumes of $S$-arithmetic quotients of semi-simple groups, Inst. Hautes Études Sci. Publ. Math. 69 (1989), 91–117. With an appendix by Moshe Jarden and the author. MR 1019962

Tetsuji Shioda, Supersingular $K3$ surfaces, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Math., vol. 732, Springer, Berlin, 1979, pp. 564–591. MR 555718

I. Vollaard: The supersingular locus of the Shimura variety for $GU(1,s)$. To appear in Canad. J. Math. arXiv:0509067v2

I. Vollaard and T. Wedhorn: The supersingular locus of the Shimura variety of $GU(1,n-1)$ II. Preprint. arXiv: 0804.1522v1

Additional Information

Shushi Harashita
Affiliation: Institute for the Physics and Mathematics of the Universe, The University of Tokyo, 5-1-5 Kashiwanoha Kashiwa-shi Chiba 277-8582 Japan
Email: harasita@ms.u-tokyo.ac.jp

Received by editor(s): March 10, 2008
Received by editor(s) in revised form: June 13, 2008
Published electronically: August 17, 2009
Additional Notes: This research is partially supported by JSPS Research Fellowship for Young Scientists.
Dedicated: Dedicated to Professor Tomoyoshi Ibukiyama on his 60th birthday