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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.
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 Andrew Archibald.
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