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
fields more general than the
complex numbers. One-dimensional abelian varieties are
elliptic curves.
In the early
nineteenth century of 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).
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 bilinear 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
Cg: 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
Cg.
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.
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
kWhen 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 40s, 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).
By the definitions, an abelian variety is a group variety. Its group of points can be proven to be
commutative.
By the
Lefschetz principle, for every
algebraically closed field of
characteristic zero (and in particular, for
C), the
torsion group of an abelian variety of dimension
g is
isomorphic to (
Q/
Z)
2g. Hence, its
n-torsion part is isomorphic to (
Z/
nZ)
2g, i.e. the product of 2
g 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, the
n-torsion is still isomorphic to (
Z/
nZ)
2g when
n and
p are
coprime.
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 Zr 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.
Dual abelian variety
To an abelian variety
A over a field
k, one associates a
dual abelian variety Av (over the same field). This association is a duality in the sense that there is a
natural isomorphism between the double dual
Avv and
A and that it is
contravariant functorial, i.e. it associates to all morphisms
f:
A →
B dual morphisms
fv:
Bv →
Av 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 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
H1 and
H2 are called
equivalent if there are positive integers
n and
m such that
nH1=
mH2. 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.
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, sometimes called an 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.
*
Motives
*
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** Online course notes.
