Barsotti–Tate group

In algebraic geometry, Barsotti–Tate groups or p-divisible groups are similar to the points of order a power of p on an abelian variety in characteristic p. They were introduced by Barsotti (1962) under the name equidimensional hyperdomain and by Tate (1967) under the name p-divisible groups, and named Barsotti–Tate groups by Grothendieck (1971).

Definition

Grothendieck (1971) defined a Barsotti–Tate group G over a scheme S to be a fppf sheaf of commutative groups over S that is p-divisible, p-torsion, such that the points G(1) of order p of G are (represented by) a finite locally free scheme.

The group G(1) has rank p^d for some locally constant function d on S, called the rank or height of the group G. The subgroup G(n) of points of order pⁿ is a scheme of rank p^nd, and G is the direct limit of these subgroups.

References

Barsotti, Iacopo (1962), "Analytical methods for abelian varieties in positive characteristic", Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962), Librairie Universitaire, Louvain, pp. 77–85, MR 0155827
Demazure, Michel (1972), Lectures on p-divisible groups, Lecture Notes in Mathematics 302, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0060741, ISBN 978-3-540-06092-5, MR 0344261
Dolgachev, I.V. (2001), "P-divisible group", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Grothendieck, Alexander (1971), "Groupes de Barsotti-Tate et cristaux", Actes du Congrès International des Mathématiciens (Nice, 1970) 1, Gauthier-Villars, pp. 431–436, MR 0578496
de Jong, A. J. (1998), "Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998)", Documenta Mathematica II: 259–265, ISSN 1431-0635, MR 1648076 |chapter= ignored (help)
Messing, William (1972), The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics 264, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058301, MR 0347836
Serre, Jean-Pierre (1995) [1966], "Groupes p-divisibles (d'après J. Tate), Exp. 318", Séminaire Bourbaki 10, Paris: Société Mathématique de France, pp. 73–86, MR 1610452
Tate, John T. (1967), "p-divisible groups.", in Springer, Tonny A., Proc. Conf. Local Fields( Driebergen, 1966), Berlin, New York: Springer-Verlag, MR 0231827

This article is issued from Wikipedia - version of the Tuesday, August 20, 2013. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.