abelian variety

Definition 1.

An abelian varietyMathworldPlanetmath over a field k is a proper group scheme over Speck that is a varietyPlanetmathPlanetmath.

This extremely terse definition needs some further explanation.

Proposition 1.

The group law on an abelian variety is commutativePlanetmathPlanetmathPlanetmath.

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 curveMathworldPlanetmath, then E is an abelian variety (and in fact E is naturally isomorphicPlanetmathPlanetmathPlanetmathPlanetmath 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