# 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) |

**How to Cite This Entry:**

Abelian scheme.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Abelian_scheme&oldid=31904