| Version current |
Version 2 |
| \PMlinkescapeword{theory} |
\PMlinkescapeword{theory} |
| \PMlinkescapeword{properties} |
\PMlinkescapeword{properties} |
| \begin{defn} |
\begin{defn} |
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An \emph{abelian variety} over a field $k$ is a proper group scheme over $\operatorname{Spec} k$ that is a variety.
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An \emph{abelian variety} over an algebraically closed field $k$ is a proper group scheme over $\operatorname{Spec} k$ that is a variety.
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| \end{defn} |
\end{defn} |
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| This extremely terse definition needs some further explanation. |
This extremely terse definition needs some further explanation. |
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| \begin{prop} |
\begin{prop} |
| The group law on an abelian variety is commutative. |
The group law on an abelian variety is commutative. |
| \end{prop} |
\end{prop} |
| This implies that for every ring $R$, the $R$-points of an abelian variety form an abelian group. |
This implies that for every ring $R$, the $R$-points of an abelian variety form an abelian group. |
| \begin{prop} |
\begin{prop} |
| An abelian variety is projective. |
An abelian variety is projective. |
| \end{prop} |
\end{prop} |
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| 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 $C$ 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. |
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| If $E$ is an elliptic curve, then $E$ is an abelian variety (and in fact $E$ is naturally isomorphic to its Jacobian). |
If $E$ is an elliptic curve, then $E$ is an abelian variety (and in fact $E$ is naturally isomorphic to its Jacobian). |
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| See Mumford's excellent book \emph{Abelian Varieties}. The bibliography for algebraic geometry has details and other books. |
See Mumford's excellent book \emph{Abelian Varieties}. The bibliography for algebraic geometry has details and other books. |
