Abelian scheme

A smooth group scheme over a base scheme $S$, the fibres of which are Abelian varieties (cf. Abelian variety). The following is an equivalent definition: An Abelian scheme over $S$, or an Abelian $S$-scheme, is a proper smooth group $S$-scheme all fibres of which are geometrically connected. Intuitively, an Abelian $S$-scheme may be understood as a family of Abelian varieties parametrized by the scheme $S$. A number of fundamental properties of Abelian varieties carry over to Abelian schemes. For instance, an Abelian $S$-scheme $A$ is a commutative group $S$-scheme [1], and if $S$ is a normal scheme, $A$ is projective over $S$, [2].

Abelian schemes are used in the context of moduli schemes of Abelian varieties with various auxiliary structures, and also in the theory of reduction of Abelian varieties (cf. Néron model).

References

[1]	D. Mumford, "Geometric invariant theory" , Springer (1965)
[2]	M. Raynaud, "Faisceaux amples sur les schémas en groupes et les espaces homogénes" , Springer (1970)