Abelian variety

In mathematics, particularly in algebraic geometry, complex analysis and number theory, an Abelian variety is a projective algebraic variety that is at the same time an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory.

An Abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined "over" that field. Historically the first Abelian varieties to be studied were those defined over the field of complex numbers. Such Abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localisation techniques lead naturally from Abelian varieties defined over number fields to ones defined over finite fields and various local fields.

Abelian varieties appear naturally as Jacobian varieties (the connected components of zero in Picard varieties) and Albanese varieties of other algebraic varieties. The group law of an Abelian variety is necessarily commutative and the variety is non-singular. An elliptic curve is an Abelian variety of dimension 1. Abelian varieties have Kodaira dimension 0.

History and motivation

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. 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, some of the most important contributors to the theory of Abelian functions were Riemann, Weierstrass, Frobenius, Poincaré and Picard. The subject was very popular at the time, already having a large literature.

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. He also 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 2"g" that carries the structure of a complex manifold. It can always be obtained as the quotient of a "g"-dimensional complex vector space by a lattice of rank 2"g".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 finite-to-one morphism.

When a complex torus carries the structure of an algebraic variety, this structure is necessarily unique. In the case "n" = 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 "g"-dimensional 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 form on "V" whose imaginary part takes integral values on "L"×"L". Such a form on "X" is usually called a (non-degenerate) 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" ≥ 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 "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". The Abelian variety "J" is called the Jacobian variety of "C", for any non-singular 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 2"n" 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:
* a connected and complete algebraic group over "k"
* a connected and projective algebraic group over "k"When the base is the field of complex numbers, these notions coincide with the previous definition. Over all bases, elliptic curves are Abelian varieties of dimension 1.

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).

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.

For C, 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 (Q/Z)^2"g". Hence, its "n"-torsion part is isomorphic to (Z/"n"Z)^2"g", i.e. the product of 2"g" copies of the cyclic group of order "n".

When the base field is an algebraically closed field of characteristic "p", the "n"-torsion is still isomorphic to (Z/"n"Z)^2"g" when "n" and "p" are coprime. 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 reduced scheme structure (i.e: looks only at points), one obtains a new invariant for varieties in characteristic "p" (the so-called "p"-rank when "n = p").

The group of "k"-rational points for a number field "k" is finitely generated by the Mordell-Weil 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".

Products

The product of an abelian variety "A" of dimension "m", and an abelian variety "B" of dimension "n", over the same field, is an abelian variety of dimension "m" + "n". An abelian variety is simple if it is not isogenous to a product of abelian varieties of lower dimension. Any abelian variety is isogenous to a product of simple abelian varieties.

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 "n"-torsion of an Abelian variety and the "n"-torsion of its dual are dual to each other when "n" is coprime to the characteristic of the base. In general - for all "n" - the "n"-torsion 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 principal polarization is an "isomorphism" between an Abelian variety and its dual. Jacobians of curves are naturally equipped with a principal polarization as soon as one picks an arbitrary base point on the curve, 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 scheme-theoretically and relative to a base. This allows for a uniform treatment of phenomena such as reduction mod "p" of Abelian varieties (see Arithmetic of abelian varieties), and parameter-families of Abelian varieties. An Abelian scheme 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, so one could think of an abelian scheme over S as being a family of abelian varieties parametrized by S.

emiabelian variety

A semiabelian variety is a commutative group variety which is an extension of an abelian variety by a torus.

See also

* Motives
* Timeline of abelian varieties

Further reading

*. A comprehensive treatment of the complex theory, with an overview of the history the subject.
*. Online course notes.
*
*. The first modern text on Abelian varieties. In French.