Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for research on other topics in algebraic geometry and number theory.
Historically the first abelian varieties to be studied were those defined over the field of complex numbers.
In the early nineteenth century, the theory of elliptic functions succeeded in giving a basis for the theory of elliptic integrals, and this left open an obvious avenue of research.
The standard forms for elliptic integrals involved the square roots of cubic and quartic polynomials.
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).
After Abel and Jacobi, some of the most important contributors to the theory of abelian functions were Riemann, Weierstrass, Frobenius, Poincaré, and Picard.
By the end of the 19th century, mathematicians had begun to use geometric methods in the study of abelian functions.
Eventually, in the 1920s, Lefschetz laid the basis for the study of abelian functions in terms of complex tori.
It was André Weil in the 1940s who gave the subject its modern foundations in the language of algebraic geometry.
It can always be obtained as the quotient of a g-dimensional complex vector space by a lattice of rank 2g.
By invoking the Kodaira embedding theorem and Chow's theorem, one may equivalently define a complex abelian variety of dimension g to be a complex torus of dimension g that admits a positive line bundle.
Since they are complex tori, abelian varieties carry the structure of a group.
it has been known since Riemann that the algebraic variety condition imposes extra constraints on a complex torus.
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.
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 form on V whose imaginary part takes integral values on
is associated with an abelian variety J of dimension g, by means of an analytic map of C into J.
:[1] 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.
In the early 1940s, Weil used the first definition (over an arbitrary base field) but could not at first prove that it implied the second.
Only in 1948 did he prove 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 embeddings (see also the history section in the Algebraic Geometry article).
, and hence by the Lefschetz principle for every algebraically closed field of characteristic zero, the torsion group of an abelian variety of dimension g is isomorphic to
When n and p are not coprime, the same result can be recovered provided one interprets it as saying that the n-torsion defines a finite flat group scheme of rank 2g.
If instead of looking at the full scheme structure on the n-torsion, one considers only the geometric points, one obtains a new invariant for varieties in characteristic p (the so-called p-rank when
The group of k-rational points for a global field k is finitely generated by the Mordell-Weil theorem.
and a finite commutative group for some non-negative integer r called the rank of the abelian variety.
correspond to line bundles of degree 0 on A, so there is a natural group operation on
given by tensor product of line bundles, which makes it into an abelian variety.
A polarisation of an abelian variety is an isogeny from an abelian variety to its dual that is symmetric with respect to double-duality for abelian varieties and for which the pullback of the Poincaré bundle along the associated graph morphism is ample (so it is analogous to a positive-definite quadratic form).
Not all principally polarised abelian varieties are Jacobians of curves; see the Schottky problem.
Viktor Abrashkin [ru][3] and Jean-Marc Fontaine[4] independently proved that there are no nonzero abelian varieties over