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abelian variety
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(Definition)
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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.
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"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 8 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 4171 times total.
Classification:
| AMS MSC: | 14K99 (Algebraic geometry :: Abelian varieties and schemes :: Miscellaneous) |
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Pending Errata and Addenda
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