Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T01:36:08.801Z Has data issue: false hasContentIssue false
  • English
  • Français

Tate modules of universal p-divisible groups

Part of: Algebraic groups

Published online by Cambridge University Press:  23 November 2009

Eike Lau
Eike Lau*
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A p-divisible group over a complete local domain determines a Galois representation on the Tate module of its generic fibre. We determine the image of this representation for the universal deformation in mixed characteristic of a bi-infinitesimal group and for the p-rank strata of the universal deformation in positive characteristic of an infinitesimal group. The method is a reduction to the known case of one-dimensional groups by a deformation argument based on properties of the stratification by Newton polygons.

Keywords

p-divisible groupslocal p-adic monodromyNewton stratificationdeformation theory

MSC classification

Secondary: 14L05: Formal groups, $p$-divisible groups
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 1 , January 2010 , pp. 220 - 232
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Achter, J. and Norman, P., Local monodromy of p-divisible groups, Preprint (2006), arXiv:math/0402460v2.Google Scholar
[2]Berthelot, P. and Messing, W., Théorie de Dieudonné cristalline III, in The Grothendieck Festschrift, Vol. I, Progress in Mathematics, vol. 86 (Birkhäuser, Boston, 1990), 173–247.Google Scholar
[3]Chai, C.-L., Local monodromy for deformations of one-dimensional formal groups, J. Reine Angew. Math. 524 (2000), 227–238.Google Scholar
[4]Chai, C.-L., Methods for p-adic monodromy, J. Inst. Math. Jussieu 7 (2008), 247–268.CrossRefGoogle Scholar
[5]Conway, J. H., Three lectures on exceptional groups, in Finite simple groups (Academic Press, New York, 1971).Google Scholar
[6]de Jong, J. A., Crystalline Dieudonné module theory via formal and rigid geometry, Publ. Math. Inst. Hautes Études Sci. 82 (1995), 5–96.CrossRefGoogle Scholar
[7]de Jong, A. J. and Oort, F., Purity of the stratification by Newton polygons, J. Amer. Math. Soc. 13 (2000), 209–241.CrossRefGoogle Scholar
[8]Grothendieck, A., Groupes de Barsotti–Tate et Cristaux de Dieudonné (Les presses de l’Université de Montréal, Montréal, 1974).Google Scholar
[9]Igusa, J., On the algebraic theory of elliptic modular functions, J. Math. Soc. Japan 20 (1968), 96–106.CrossRefGoogle Scholar
[10]Illusie, L., Déformations de groupes de Barsotti–Tate (d’après A. Grothendieck), in Seminar on arithmetic bundles: the Mordell conjecture, Astérisque 127 (1985), 151–198.Google Scholar
[11]Katz, N. M., p-adic properties of modular schemes and modular forms, in Modular functions of one variable, III, Lecture Notes in Mathematics, vol. 350 (Springer, Berlin, 1973), 69–190.CrossRefGoogle Scholar
[12]Lau, E., A duality theorem for Dieudonné displays, Ann. Sci. École Norm. Sup. (4) 42 (2009), 241–259.CrossRefGoogle Scholar
[13]Messing, W., The crystals associated to Barsotti–Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, vol. 264 (Springer, Berlin, 1972).CrossRefGoogle Scholar
[14]Oort, F., Newton polygons and formal groups: conjectures by Manin and Grothendieck, Ann. of Math. (2) 152 (2000), 183–206.CrossRefGoogle Scholar
[15]Oort, F., Newton polygon strata in the moduli space of abelian varieties, in Moduli of abelian varieties, Progress in Mathematics, vol. 195 (Birkhäuser, Boston, 2001), 417–440.CrossRefGoogle Scholar
[16]Raynaud, M., Schémas en groupes de type (p,…,p), Bull. Soc. Math. France 102 (1974), 241–280.CrossRefGoogle Scholar
[17]Rosen, M. and Zimmermann, K., Torsion points of generic formal groups, Trans. Amer. Math. Soc. 311 (1989), 241–253.CrossRefGoogle Scholar
[18]Strauch, M., Galois actions on torsion points of universal one-dimensional formal modules, Preprint (2007), arXiv:0709.3542, J. Number Theory, to appear.Google Scholar
[19]Strauch, M., Deformation spaces of one-dimensional formal modules and their cohomology, Adv. Math. 217 (2008), 889–951.CrossRefGoogle Scholar
[20]Tian, Y., p-adic monodromy of the universal deformation of an elementary Barsotti–Tate group, Preprint (2007), arXiv:0708.2022.Google Scholar
[21]Zimmermann, K., Torsion points of generic formal groups defined over ℤ2[[t 1,…,t h−1]], Arch. Math. (Basel) 54 (1990), 119–124.CrossRefGoogle Scholar
[22]Zink, Th., Cartiertheorie kommutativer formaler Gruppen (Teubner, Leipzig, 1984).Google Scholar