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abelian variety (Definition)
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.



"abelian variety" is owned by archibal.
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Cross-references: bibliography for algebraic geometry, abelian, isomorphic, elliptic curve, development, Jacobian, curve, abelian group, ring, implies, commutative, group, variety, group scheme, field
There are 7 references to this entry.

This is version 3 of abelian variety, born on 2004-04-05, modified 2004-04-06.
Object id is 5742, canonical name is AbelianVariety.
Accessed 3248 times total.

Classification:
AMS MSC14K99 (Algebraic geometry :: Abelian varieties and schemes :: Miscellaneous)

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