varieties where is a Jacobian. This theorem remains true if the ground field is infinite. Algebraic definition Two equivalent definitions of abelian variety over a general field k are commonly in use: a connected and complete algebraic group over k a connected and projective algebraic group over k. When the base is the field of complex numbers, these notions coincide with the previous definition. Over all bases, elliptic curves are abelian varieties of dimension 1. In the early 1940s, Weil used the first definition (over an arbitrary base field) but could not at first prove that it implied the second. Only in 1948 did he prove that complete algebraic groups can be embedded into projective space. Meanwhile, in order to make the proof of the Riemann hypothesis for curves over finite fields that he had announced in 1940 work, he had to introduce the notion of an abstract variety and to rewrite the foundations of algebraic geometry to work with varieties without projective embeddings (see also the history section in the Algebraic Geometry article). Structure of the group of points By the definitions, an abelian variety is a group variety. Its group of points can be proven to be commutative. For C, and hence by the Lefschetz principle for every algebraically closed field of characteristic zero, the torsion group of an abelian variety of dimension g is isomorphic to (Q/Z)2g. Hence, its n-torsion part is isomorphic to (Z/nZ)2g, i.e. the product of 2g copies of the cyclic group of order n. When the base field is an algebraically closed field of characteristic p, the n-torsion is still isomorphic to (Z/nZ)2g when n and p are coprime. When n and p are not coprime, the same result can be recovered provided one interprets it as saying that the n-torsion defines a finite flat group scheme of rank 2g. If instead of looking at the full scheme structure on the n-torsion, one considers only the geometric points, one obtains a new invariant for varieties in characteristic p (the so-called p-rank when n = p). The group of k-rational points for a global field k is finitely generated by the Mordell-Weil theorem. Hence, by the structure theorem for finitely generated abelian groups, it is isomorphic to a product of a free abelian group Zr and a finite commutative group for some non-negative integer r called the rank of the abelian variety. Similar results