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, MR0155827 
• 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, MR0344261 
• Dolgachev, I.V. (2001), "P-divisible group", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/p/p071030.htm 
• Grothendieck, Alexander (1971), "Groupes de Barsotti-Tate et cristaux", Actes du Congrès International des Mathématiciens (Nice, 1970), 1, Gauthier-Villars, pp. 431–436, MR0578496, http://mathunion.org/ICM/ICM1970.1/ 
• de Jong, A. J. (1998), Barsotti-Tate groups and crystals, "Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998)", Documenta Mathematica II: 259–265, ISSN 1431-0635, MR1648076, http://mathunion.org/ICM/ICM1998.2/ 
• 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, MR0347836 
• 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, MR1610452, http://www.numdam.org/item?id=SB_1966-1968__10__73_0 
• Tate, John T. (1967), "p-divisible groups.", in Springer, Tonny A., Proc. Conf. Local Fields( Driebergen, 1966), Berlin, New York: Springer-Verlag, MR0231827