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

Tate (1967) defined a p-divisible group of height h (over a scheme S) to be an inductive system of groups G_n for n≥0, such that G_n is a finite group scheme over S of order p^hn and such that G_n is (identified with) the group of elements of order divisible by pⁿ in G_n+1.

More generally, Grothendieck (1971) defined a Barsotti–Tate group G over a scheme S to be an 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^h for some locally constant function h 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^nh, and G is the direct limit of these subgroups.

Example

Take G_n to be the cyclic group of order pⁿ (or rather the group scheme corresponding to it). This is a p-divisible group of height 1.
Take G_n to be the group scheme of pⁿth roots of 1. This is a p-divisible group of height 1.
Take G_n to be the subgroup scheme of elements of order pⁿ of an abelian variety. This is a p-divisible group of height 2d where d is the dimension of the Abelian variety.

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, vol. 302, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0060741, ISBN 978-3-540-06092-5, MR 0344261
Dolgachev, I.V. (2001) [1994], "P-divisible group", Encyclopedia of Mathematics, EMS Press
Grothendieck, Alexander (1971), "Groupes de Barsotti-Tate et cristaux", Actes du Congrès International des Mathématiciens (Nice, 1970), vol. 1, Gauthier-Villars, pp. 431–436, MR 0578496, archived from the original on 2017-11-25, retrieved 2010-11-25
de Jong, A. J. (1998), "Barsotti-Tate groups and crystals", Documenta Mathematica, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), II: 259–265, ISSN 1431-0635, MR 1648076
Messing, William (1972), The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics, vol. 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, vol. 10, Paris: Société Mathématique de France, pp. 73–86, MR 1610452
Tate, John T. (1967), "p-divisible groups.", in Springer, Tonny A. (ed.), Proc. Conf. Local Fields( Driebergen, 1966), Berlin, New York: Springer-Verlag, MR 0231827