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 under the name equidimensional hyperdomain and by under the name p-divisible groups, and named Barsotti–Tate groups by .

Definition

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, 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