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Higher-level canonical subgroups for p-divisible groups

Part of: Algebraic groups Arithmetic algebraic geometry

Published online by Cambridge University Press:  13 December 2011

Joseph Rabinoff
Joseph Rabinoff
Affiliation:
Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA ([email protected])
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Abstract

Let R be a complete rank-1 valuation ring of mixed characteristic (0, p), and let K be its field of fractions. A g-dimensional truncated Barsotti–Tate group G of level n over R is said to have a level-n canonical subgroup if there is a K-subgroup of G ⊗RK with geometric structure (Z/pnZ)g consisting of points ‘closest to zero’. We give a non-trivial condition on the Hasse invariant of G that guarantees the existence of the canonical subgroup, analogous to a result of Katz and Lubin for elliptic curves. The bound is independent of the height and dimension of G.

Keywords

abelian varietiesp-divisible groupscanonical subgroupsautomorphic formstropical geometry

MSC classification

Secondary: 11G10: Abelian varieties of dimension $> 1$ 14L05: Formal groups, $p$-divisible groups
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 11 , Issue 2 , April 2012 , pp. 363 - 419
Copyright
Copyright © Cambridge University Press 2012

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References

1.Abbes, A. and Mokrane, A., Sous-groupes canoniques et cycles évanescents p-adiques pour les variétés abéliennes, Publ. Math. IHES 99 (2004), 117–162.CrossRefGoogle Scholar
2.Andreatta, F. and Gasbarri, C., The canonical subgroup for families of abelian varieties, Compositio Math. 143(3) (2007), 566–602.CrossRefGoogle Scholar
3.Artin, M., Bertin, J. E., Demazure, M., Gabriel, P., Grothendieck, A., Raynaud, M. and Serre, J.-P., Schémas en groupes., Séminaire de Géométrie Algébrique de l'Institut des Hautes études Scientifiques (SGA 3) (Institut des Hautes études Scientifiques, Paris, 1962–1964).Google Scholar
4.Barvinok, A., A course in convexity, Graduate Studies in Mathematics, Volume 54 (American Mathematical Society, Providence, RI, 2002).Google Scholar
5.Bosch, S., Güntzer, U. and Remmert, R., Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften, Volume 261 (Springer, 1984).CrossRefGoogle Scholar
6.Buzzard, K., Analytic continuation of overconvergent eigenforms, J. Am. Math. Soc. 16(1) (2003), 29–55.CrossRefGoogle Scholar
7.Buzzard, K. and Taylor, R., Companion forms and weight one forms, Annals Math. (2) 149(3) (1999), 905–919.CrossRefGoogle Scholar
8.Coleman, R. F., Classical and overconvergent modular forms, Invent. Math. 124(1–3) (1996), 215–241.CrossRefGoogle Scholar
9.Coleman, R. F., Classical and overconvergent modular forms of higher level, J. Théorie Nombres Bordeaux 9(2) (1997), 395–403.CrossRefGoogle Scholar
10.Conrad, B., Modular curves and rigid-analytic spaces, Pure Appl. Math. Q. 2(1) (2006), 29–110.CrossRefGoogle Scholar
11.Conrad, B., Higher-level canonical subgroups in abelian varieties, preprint (available at http://math.stanford.edu/~conrad/papers/subgppaper.pdf).Google Scholar
12.de Jong, A. J., Crystalline Dieudonné module theory via formal and rigid geometry, Publ. Math. IHES 82 (1995), 5–96.CrossRefGoogle Scholar
13.Fargues, L. and Tian, Y., La filtration canonique des points de torsion des groupes p-divisibles, preprint (2009).Google Scholar
14.Fulton, W., Intersection theory, 2nd edn (Springer, 1998).CrossRefGoogle Scholar
15.Goren, E. Z. and Kassaei, P. L., The canonical subgroup: a ‘subgroup-free’ approach, Comment. Math. Helv. 81(3) (2006), 617–641.CrossRefGoogle Scholar
16.Grothendieck, A., Éléments de géométrie algébrique, III, Étude cohomologique des faisceaux cohérents, I, Publ. Math. IHES 11 (1961), 167.Google Scholar
17.Grothendieck, A., Éléments de géométrie algébrique, IV, Étude locale des schémas et des morphismes de schémas, III, Publ. Math. IHES 28 (1966), 255.Google Scholar
18.Hazewinkel, M., Formal groups and applications, Pure and Applied Mathematics, Volume 78 (Academic Press, 1978).Google Scholar
19.Illusie, L., Déformations de groupes de Barsotti-Tate (d'après A. Grothendieck), Astérisque 127 (1985), 151–198.Google Scholar
20.Kassaei, P. L., -adic modular forms over Shimura curves over totally real fields, Compositio Math. 140(2) (2004), 359–395.CrossRefGoogle Scholar
21.Kassaei, P. L., A gluing lemma and overconvergent modular forms, Duke Math. J. 132(3) (2006), 509–529.CrossRefGoogle Scholar
22.Katz, N. M., p-adic properties of modular schemes and modular forms, Modular functions of one variable, III, in Proc. Int. Summer School, Univ. Antwerp, Antwerp, 1972, Lecture Notes in Mathematics, Volume 350, pp. 69–190 (Springer, 1973).Google Scholar
23.Kisin, M. and Lai, K. F., Overconvergent Hilbert modular forms, Am. J. Math. 127(4) (2005), 735–783.CrossRefGoogle Scholar
24.Lang, S., Algebra, 2nd edn (Addison-Wesley, Reading, MA, 1984).Google Scholar
25.Lau, E., Displays and formal p-divisible groups, Invent. Math. 171(3) (2008), 617–628.CrossRefGoogle Scholar
26.Matsumura, H., Commutative ring theory (transl. from Japanese by Reid, M.), 2nd edn, Cambridge Studies in Advanced Mathematics, Volume 8 (Cambridge University Press, 1989).Google Scholar
27.Messing, W., The crystals associated to Barsotti–Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, Volume 264 (Springer, 1972).CrossRefGoogle Scholar
28.Rabinoff, J., Tropical analytic geometry, newton polygons, and tropical intersections, preprint (arXiv:1007.2665).Google Scholar
29.Tate, J. T., p-divisible groups, in Proc. Conf. Local Fields, Driebergen, 1966, pp. 158–183 (Springer, 1967).Google Scholar
30.Tian, Y., Canonical subgroups of Barsotti–Tate groups, Annals Math. 172(2) (2010), 955–988.CrossRefGoogle Scholar
31.Zink, T., Cartiertheorie kommutativer formaler Gruppen, Teubner-Texte zur Mathematik, Volume 68 (B. G. Teubner, Leipzig, 1984).Google Scholar
32.Zink, T., The display of a formal p-divisible group, Astérisque 278 (2002), 127–248.Google Scholar