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).
Definitionedit
Tate (1967) defined a p-divisible group of height h (over a scheme S) to be an inductive system of groups Gn for n≥0, such that Gn is a finite group scheme over S of order phn and such that Gn is (identified with) the group of elements of order divisible by pn in Gn+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 ph 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 pnh, and G is the direct limit of these subgroups.
Exampleedit
Take Gn 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 Gn to be the group scheme of pnth roots of 1. This is a p-divisible group of height 1.
Take Gn 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.
Referencesedit
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
Grothendieck, Alexander (1971), "Groupes de Barsotti-Tate et cristaux", , 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
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
April 10, 2024
barsotti, tate, group, algebraic, geometry, divisible, groups, similar, points, order, power, abelian, variety, characteristic, they, were, introduced, barsotti, 1962, under, name, equidimensional, hyperdomain, tate, 1967, under, name, divisible, groups, named. 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 editTate 1967 defined a p divisible group of height h over a scheme S to be an inductive system of groups Gn for n 0 such that Gn is a finite group scheme over S of order phn and such that Gn is identified with the group of elements of order divisible by pn in Gn 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 ph 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 pnh and G is the direct limit of these subgroups Example editTake Gn 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 Gn to be the group scheme of pnth roots of 1 This is a p divisible group of height 1 Take Gn 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 editBarsotti Iacopo 1962 Analytical methods for abelian varieties in positive characteristic Colloq Theorie des Groupes Algebriques 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 Congres International des Mathematiciens 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 apres J Tate Exp 318 Seminaire Bourbaki vol 10 Paris Societe Mathematique 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 Retrieved from https en wikipedia org w index php title Barsotti Tate group amp oldid 1045235360, wikipedia, wiki, book, books, library,
