Abelian function
For the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of real dimension 2n that is a complex manifold) that is also a projective algebraic variety of dimension n, i.e. can be defined in projective space by algebraic equations. In case n is 1 this notion is the same as that of elliptic curve, and every complex torus gives rise to such a curve; for n > 1 it has been known since Riemann that the algebraic variety condition imposes extra constraints on a complex torus. An abelian function is a meromorphic function on an abelian variety, which may be regarded therefore as a periodic function of n complex variables, having 2n independent periods; equivalently, it is a function in the function field of an abelian variety.
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 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 geometry, its function field of abelian functions is the fixed field of the symmetric group on g letters acting on the function field of Cg.
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 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 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 theta-functions.
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. See arithmetic of abelian varieties.
See also: abelian integral.
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