Abelian variety

In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a complex torus that can be embedded into projective space. Also it is used for the generalization of this concept studied in algebraic geometry over base fields more general than the complex numbers. Onedimensional abelian varieties are elliptic curves.
Contents 
History and motivation
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. After Abel and Jacobi the theory of abelian functions was worked out mainly by Riemann, Weierstrass, Frobenius, Poincaré and Picard.
By the end of the 19th century mathematicians began to use geometric methods in the study of abelian functions. By the 1920s, eventually, Lefschetz laid the basis for the study of abelian functions in terms of complex tori. Also he appears to be the first to use the name "abelian variety". It was Weil in the 1940s who gave the subject its modern foundations in the language of algebraic geometry.
Today abelian varieties form an important tool in number theory, in dynamical systems, more specifically in the study of Hamiltonian systems and in algebraic geometry, especially Picard varieties and Albanese varieties.
Analytic theory
Definition
A complex torus of dimension g is a torus of real dimension 2g that carries the structure of a complex manifold. It can always be obtained as the quotient of a gdimensional complex vector space by a lattice of rank 2g. A complex abelian variety of dimension g is a complex torus of dimension g that is also a projective algebraic variety over the field of complex numbers. Since they are complex tori, abelian varieties carry the structure of a group. A morphism of abelian varieties is a morphism of the underlying algebraic varieties that preserves the identity element for the group structure. An isogeny is a finitetoone morphism.
When a complex torus carries the structure of an algebraic variety, this structure is necessarily unique. In case n is 1 the notion of abelian variety 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.
Riemann conditions
The following criterion by Riemann decides whether or not a given complex torus is an abelian variety, i.e. whether or not it can be embedded into a projective space. Let X be a gdimensional torus given as X=V/L where V is a complex vector space of dimension g and L is a lattice in V. Then X is an abelian variety if and only if there exists a positive definite hermitian bilinear form on V whose imaginary part takes integral values on LxL. Such a form on X is usually called a (nondegenerate) Riemann form. Choosing a basis for V and L, one can make this condition more explicit. There are several equivalent formulations of this, all of them are known as the Riemann conditions.
The Jacobian of an algebraic curve
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 commutative group structure, and the image of C generates J as a group. More accurately, J is covered by C^{g}: 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 started, can be derived from the simpler, translationinvariant theory of differentials on J. The abelian variety J is called the Jacobian variety of C, for any nonsingular 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^{g}.
Abelian functions
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. For example, in the nineteenth century there was much interest in 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 an isogeny.
See also: abelian integral.
Algebraic definition
Two equivalent definitions of abelian variety over a general field are commonly in use. When the base is the field of complex numbers, the notion coincides with the above definition. Over all bases, elliptic curves are abelian varieties of dimension 1.
 a connected and complete algebraic group over k
 a connected and projective algebraic group over k
In the early 40s, Weil used the former definition but could at first not prove that it implied the second. Only in 1948 he proved 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 embedding (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.
Over C, and by the Lefschetz principle over 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 ntorsion part is isomorphic to (Z/nZ)^{2g}, i.e. the product of 2g copies of the cyclic group of order n. Its free part is uncountable, since every algebraically closed field of characteristic zero is uncountable.
When the base field is an algebraically closed field of characteristic p then the ntorsion is still isomorphic to (Z/nZ)^{2g} under the condition that n and p be coprime.
The group of krational points for a number field k is finitely generated by the MordellWeil theorem. Hence by the structure theorem for finitely generated abelian groups it is isomorphic to a product of a free abelian group Z^{r} and a finite commutative group for some positive integer r called the rank of the abelian variety. Similar results hold for some other classes of fields k.
Polarization and dual abelian variety
Dual abelian variety
To an abelian variety A over a field k, one associates a dual abelian variety A^{v} (over the same field). This association is a duality in the sense that there is a natural isomorphism between the double dual A^{vv} and A and that it is contravariant functorial, i.e. it associates to all morphisms f:A>B dual morphisms f^{v}:B^{v}>A^{v} in a compatible way. The ntorsion of an abelian variety and the ntorsion of its dual are dual to each other when n is coprime to the characteristic of the base. In general  for all n  the ntorsion group schemes of dual abelian varieties are Cartier duals of each other. This generalizes the Weil pairing for elliptic curves.
Polarizations
A polarization of an abelian variety is an isogeny from an abelian variety to its dual. Polarized abelian varieties have finite automorphism groups. A principle polarization is an isomorphism between an abelian variety and its dual. Jacobians of curves are naturally equipped with a principal polarization and the curve can be reconstructed from its polarized Jacobian. Not all principally polarized abelian varieties are Jacobians of curves, see the Schottky Problem.
Polarizations over the complex numbers
Over the complex numbers, a polarized abelian variety can also be defined as an abelian variety A together with a choice of a Riemann form H. Two Riemann forms H_{1} and H_{2} are called equivalent if there are positive integers n and m such that nH_{1}=mH_{2}. A choice of an equivalence class of Riemann forms on A is called a polarization of A. A morphism of polarized abelian varieties is a morphism A>B of abelian varieties such that the pullback of the Riemann form on B to A is equivalent to the given form on A.
Abelian scheme
One can also define abelian varieties schemetheoretically and relative to a base. This allows for a uniform treatment of phenomena as reduction mod p of abelian varieties (see Arithmetic of abelian varieties) and parameterfamilies of abelian varieties. An abelian scheme, sometimes called abelian variety, over a base scheme S of relative dimension g is a proper smooth group scheme over S whose geometric fibers are connected and of dimension g. The fibers of an abelian scheme are abelian varieties.
See also
Further reading
 David Mumford: Abelian Varieties, 1970
 H. Lange and Ch. Birkenhake, Complex Abelian Varieties, 1992, ISBN 0387547479
 A comprehensive treatment of the complex theory, with an overview of the history the subject.
 André Weil: Courbes algébriques et variétés abéliennes, 1948
 The first modern text on abelian varieties. In French.
 James Milne: Abelian Varieties (http://www.jmilne.org/math/CourseNotes/math731.html)
 Online course notes.