# P-divisible group

Barsotti–Tate group

A generalization of the concept of a commutative formal group of finite height. The homomorphism induced by multiplication by the prime number is an epimorphism for a -divisible group.

Let be a scheme and let be a prime number. A -divisible group of height is an inductive system of commutative finite group schemes of order (cf. Group scheme) such that the sequences

are exact (cf. Exact sequence; here is the homomorphism of multiplication by ). A morphism of -divisible groups is a morphism of the inductive systems. A -divisible group is said to be connected (respectively, étale) if all are connected (respectively, étale) group schemes. A connected -divisible group over a field of characteristic is a commutative formal group (considered as the inductive limit of the kernels of , i.e. of multiplication by ) for which multiplication by is an isogeny [6]. This fact is generalized to include the case of an arbitrary base scheme on which the homomorphism induced by multiplication by is locally nilpotent [4]. The category of étale -divisible groups is equivalent to the category of -adic representations of the fundamental group of the scheme . Each -divisible group over an Artinian scheme contains a maximal connected subgroup , which is known as the connected component of the identity, the quotient by which is an étale -divisible group. The dimension of the Lie algebra of any is said to be the dimension of the -divisible group .

Let be an Abelian variety over the field of dimension , let be the kernel of the homomorphism of multiplication by in and let be a natural inclusion. The inductive system is a -divisible group of height . Its connected component of the identity coincides with the formal completion of along the unit section, while the height of equals .

Let be a -divisible group of height , let denote the Cartier dual finite group schemes and let be the mapping dual to the mapping of multiplication by . The system is then a -divisible group of height and is called the Serre dual to the -divisible group . The sum of the dimensions of and is equal to .

As in formal groups, so, too, in -divisible groups, the concept of a Dieudonné module, which plays an important role in the deformation theory of -divisible groups [2], [3], [4] may be introduced.

If is the spectrum of an unequal-characteristic discrete valuation ring with residue field of characteristic , the structure of a -divisible group is closely connected with the structure of the completion of the algebraic closure of the quotient field of , considered as a module over the Galois group of the field [6].

#### References

 [1] I. Barsotti, "Analytic methods for abelian varieties in positive characteristic" , Coll. Théorie des Groupes Algébriques (Bruxelles, 1962) , Gauthier-Villars (1962) pp. 77–85 [2] A. Grothendieck, "Groupes de Barsotti–Tate et cristaux" , Proc. Internat. Congress Mathematicians (Nice, 1970) , 1 , Gauthier-Villars (1971) pp. 431–436 [3] B. Mazur, W. Messing, "Universal extensions and one-dimensional crystalline cohomology" , Springer (1974) [4] W. Messing, "The crystals associated to Barsotti–Tate groups: with applications to Abelian schemes" , Springer (1972) [5] J.-P. Serre, "Groupes -divisibles (d'après J. Tate)" , Sem. Bourbaki , 318 , Benjamin (1968) [6] J.T. Tate, "-divisible groups" T.A. Springer (ed.) et al. (ed.) , Proc. Conf. local fields (Driebergen, 1966) , Springer (1967) pp. 158–183