abelian variety

Definition 1.

An abelian variety over a field $k$ is a proper group scheme over $\mathrm{Spec}\mathit{}k$ 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
Canonical name	AbelianVariety
Date of creation	2013-03-22 14:17:17
Last modified on	2013-03-22 14:17:17
Owner	archibal (4430)
Last modified by	archibal (4430)
Numerical id	6
Author	archibal (4430)
Entry type	Definition
Classification	msc 14K99