# Barsotti–Tate group

# Barsotti–Tate group

In algebraic geometry, **Barsotti–Tate groups** or * p* 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**n* and such that *G**n* is (identified with) the group of elements of order divisible by *p**n* 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 *p**n* 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*p**n*(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*p**n*th roots of 1. This is a*p*-divisible group of height 1.Take

*G**n*to be the subgroup scheme of elements of order*p**n*of an abelian variety. This is a*p*-divisible group of height 2*d*where*d*is the dimension of the Abelian variety.

## References

*Actes du Congrès International des Mathématiciens (Nice, 1970)*

*Actes du Congrès International des Mathématiciens (Nice, 1970)*