The success in the early nineteenth century of the theory of elliptic functions in giving a basis for the theory of elliptic integrals left open an obvious avenue of research. The standard forms for elliptic integrals involved the square roots of cubic and quartic polynomials. When those were replaced by polynomials of higher degree, say quintics, what would happen? In the work of Niels Abel and Carl Jacobi, the answer was formulated: this would involve functions of two complex variables, having four independent periods (i.e. period vectors). This gave the first glimpse of an abelian variety of dimension 2: what would now be called the Jacobian of a hyperelliptic curve of genus 2.
In a more geometric language, every algebraic curve C of genus g which is at least 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 group structure (commutative), and the image of C generates J as a group. More accurately, J is covered by C added to itself g times: any point in J comes from a gtuple of points in C. The study of differential forms on C, which give rise to the abelian integrals with which the theory starts, can be derived from the simpler, translationinvariant theory of differentials on J. For example there was much interest in the case of hyperelliptic integrals that may be expressed in terms of elliptic integrals: this comes down to asking that J is a product of elliptic curves, up to a finitetoone mapping.
This theory was much later put on an axiomatic basis, in which abelian varieties are by definition the connected groups in the category of projective algebraic varieties. That is, they are one part of the theory of algebraic groups[?]. The Jacobian varieties of curves generalise to the Albanese varieties of varieties in general.
The explicit equations defining abelian varieties are in general complex: their properties involve the detailed theory of thetafunctions[?].
For the purposes of number theory the foundations of the theory of abelian varieties are developed over any field, and in fact using a commutative ring, in order to control the process of reduction mod p.
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