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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 (an abelian surface): 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 g-tuple 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, 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 a field. From the point of view of birational geometryGeometry In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension 2, the birational geometry of algebrai, its function field of abelian functions is the fixed field of the symmetric groupIn mathematics, the symmetric group on a set X denoted by S or Sym X , is the group whose underlying set is the set of all bijective functions from X to X in which the group operation is that of composition of functions, i. two such functions f and g can on g letters acting on the function field of C^g.

For example there was much interest in the case of hyperelliptic integral s 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 finite-to-one mapping (called an isogeny of abelian varieties).

Abstract theory

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 groupIn algebraic geometry, two important classes of algebraic group arise, that for the most part are studied separately. The general definition of algebraic group is the expected one: a group in the category of algebraic varieties; or, more simply, a group ws. 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 theta-functions.

For the purposes of number theoryTraditionally, number theory is that branch of pure mathematics concerned with the properties of integers and contains many open problems that are easily understood even by non-mathematicians. More generally, the field has come to be concerned with a wide the foundations of the theory of abelian varieties are developed over any fieldIn abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are famil, and in fact using a commutative ring, in order to control the process of reduction mod p. See arithmetic of abelian varietiesIn mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or family of those. It goes back to the studies of Fermat on what are now recognised as elliptic curves; and has become a very substantial area bo.

See also: abelian integralIn mathematics, an abelian integral in Riemann surface theory is a function related to the indefinite integral of a differential of the first kind. Suppose given a Riemann surface S and on it a differential 1-form ω that is everywhere on S holomorph.