# abelian variety

###### Definition 1.

An abelian variety over a field $k$ is a proper group scheme over $\operatorname{Spec}k$ that is a variety.

This extremely terse definition needs some further explanation.

###### Proposition 1.

The group law on an abelian variety is commutative.

This implies that for every ring $R$, the $R$-points of an abelian variety form an abelian group.

###### Proposition 2.

An abelian variety is projective.

If $C$ is a curve, then the Jacobian of $C$ is an abelian variety. This example motivated the development of the theory of abelian varieties, and many properties of curves are best understood by looking at the Jacobian.

If $E$ is an elliptic curve, then $E$ is an abelian variety (and in fact $E$ is naturally isomorphic to its Jacobian).

See Mumford’s excellent book Abelian Varieties. The bibliography for algebraic geometry has details and other books.

Title abelian variety AbelianVariety 2013-03-22 14:17:17 2013-03-22 14:17:17 archibal (4430) archibal (4430) 6 archibal (4430) Definition msc 14K99