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 or order pⁿ and such that G_n is (identified with) the group of elements of order pⁿ in G_n+1.

More generally, 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^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 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, 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), "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, 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