A smooth group scheme over a base scheme , the fibres of which are Abelian varieties (cf. Abelian variety). The following is an equivalent definition: An Abelian scheme over , or an Abelian -scheme, is a proper smooth group -scheme all fibres of which are geometrically connected. Intuitively, an Abelian -scheme may be understood as a family of Abelian varieties parametrized by the scheme . A number of fundamental properties of Abelian varieties carry over to Abelian schemes. For instance, an Abelian -scheme is a commutative group -scheme , and if is a normal scheme, is projective over , .
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).
|||D. Mumford, "Geometric invariant theory" , Springer (1965)|
|||M. Raynaud, "Faisceaux amples sur les schémas en groupes et les espaces homogénes" , Springer (1970)|
Abelian scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abelian_scheme&oldid=11728