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 $p$ is an epimorphism for a $p$- divisible group.

Let $S$ be a scheme and let $p$ be a prime number. A $p$- divisible group of height $h$ is an inductive system $G = ( G _ {n} , i _ {n} )$ of commutative finite group schemes $G _ {n}$ of order $p ^ {nh}$( cf. Group scheme) such that the sequences

$$0 \rightarrow G _ {n} \rightarrow ^ { {i _ n} } G _ {n+} 1 \rightarrow ^ { {\phi _ n} } \ G _ {n+} 1$$

are exact (cf. Exact sequence; here $\phi _ {n}$ is the homomorphism of multiplication by $p ^ {n}$). A morphism of $p$- divisible groups is a morphism of the inductive systems. A $p$- divisible group is said to be connected (respectively, étale) if all $G _ {n}$ are connected (respectively, étale) group schemes. A connected $p$- divisible group over a field of characteristic $p$ is a commutative formal group (considered as the inductive limit of the kernels of $\phi _ {n}$, i.e. of multiplication by $p ^ {n}$) for which multiplication by $p$ is an isogeny [6]. This fact is generalized to include the case of an arbitrary base scheme $S$ on which the homomorphism induced by multiplication by $p$ is locally nilpotent [4]. The category of étale $p$- divisible groups is equivalent to the category of $p$- adic representations of the fundamental group of the scheme $S$. Each $p$- divisible group $G$ over an Artinian scheme $S$ contains a maximal connected subgroup $G ^ {0}$, which is known as the connected component of the identity, the quotient by which is an étale $p$- divisible group. The dimension of the Lie algebra of any ${( G ^ {0} ) } _ {n}$ is said to be the dimension of the $p$- divisible group $G$.

Let $A$ be an Abelian variety over the field $k$ of dimension $d$, let $A ( n)$ be the kernel of the homomorphism of multiplication by $p ^ {n}$ in $A$ and let $i _ {n} : A ( n) \rightarrow A ( n + 1 )$ be a natural inclusion. The inductive system $A ( \infty ) = ( A ( n) , i _ {n} )$ is a $p$- divisible group of height $2d$. Its connected component of the identity $A ( \infty ) ^ {0}$ coincides with the formal completion of $A$ along the unit section, while the height of $A ( \infty ) ^ {0}$ equals $2 \mathop{\rm dim} A$.

Let $G = ( G _ {n} , i _ {n} )$ be a $p$- divisible group of height $h$, let ${\widehat{G} } _ {n}$ denote the Cartier dual finite group schemes and let $\widehat{i} _ {n} : \widehat{G} _ {n} \rightarrow \widehat{G} _ {n+} 1$ be the mapping dual to the mapping of multiplication by $p : G _ {n+} 1 \rightarrow G _ {n}$. The system $\widehat{G} = ( \widehat{G} _ {n} , \widehat{i} _ {n} )$ is then a $p$- divisible group of height $h$ and is called the Serre dual to the $p$- divisible group $G$. The sum of the dimensions of $G$ and $\widehat{G}$ is equal to $h$.

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

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