Page "Abelian variety" Paragraph 12

Every algebraic curve C of genus g ≥ 1 is associated with an abelian variety J of dimension g, by means of an analytic map of C into J.

As a torus, J carries a commutative group structure, and the image of C generates J as a group.

More accurately, J is covered by C < sup > g </ sup >: any point in J comes from a g-tuple of points in C. The study of differential forms on C, which give rise to the abelian integrals with which the theory started, can be derived from the simpler, translation-invariant theory of differentials on J.

The abelian variety J is called the Jacobian variety of C, for any non-singular curve C over the complex numbers.

From the point of view of birational geometry, its function field is the fixed field of the symmetric group on g letters acting on the function field of C < sup > g </ sup >.