# abelian variety

###### Definition 1.

An *abelian variety ^{}* over a field $k$ is a proper group scheme over $\mathrm{Spec}\mathit{}k$ 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.

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 |