AbelianVariety 2004-04-05 23:36:29 2004-04-06 00:04:10 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newtheorem{theorem}{Theorem} \newtheorem{defn}{Definition} \newtheorem{prop}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{cor}{Corollary} \PMlinkescapeword{theory} \PMlinkescapeword{properties} \begin{defn} An \emph{abelian variety} over a field $k$ is a proper group scheme over $\operatorname{Spec} k$ that is a variety. \end{defn} This extremely terse definition needs some further explanation. \begin{prop} The group law on an abelian variety is commutative. \end{prop} This implies that for every ring $R$, the $R$-points of an abelian variety form an abelian group. \begin{prop} An abelian variety is projective. \end{prop} 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 \emph{Abelian Varieties}. The bibliography for algebraic geometry has details and other books.