In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
,
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
and
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic o ...
, an abelian variety is a
projective algebraic variety that is also an
algebraic group, i.e., has a
group law
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. The ...
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
In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', w ...
that can be embedded into a complex
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
.
Abelian varieties defined over
algebraic number fields are a special case, which is important also from the viewpoint of number theory.
Localization techniques lead naturally from abelian varieties defined over number fields to ones defined over
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s and various
local fields. Since a number field is the fraction field of a
Dedekind domain, for any nonzero prime of your
Dedekind domain, there is a map from the Dedekind domain to the quotient of the Dedekind domain by the prime, which is a finite field for all finite primes. This induces a map from the fraction field to any such finite field. Given a curve with equation defined over the number field, we can apply this map to the coefficients to get a curve defined over some finite field, where the choices of finite field correspond to the finite primes of the number field.
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
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. I ...
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 root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
s of
cubic and
quartic polynomial
In algebra, a quartic function is a function of the form
:f(x)=ax^4+bx^3+cx^2+dx+e,
where ''a'' is nonzero,
which is defined by a polynomial of degree four, called a quartic polynomial.
A ''quartic equation'', or equation of the fourth deg ...
s. 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
André 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 system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
s (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
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does n ...
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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
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
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
over the field of complex numbers. 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. A
morphism of abelian varieties is a morphism of the underlying algebraic varieties that preserves the
identity element for the group structure. An
isogeny In mathematics, in particular, in algebraic geometry, an isogeny is a morphism of algebraic groups (also known as group varieties) that is surjective and has a finite kernel.
If the groups are abelian varieties, then any morphism of the underlyin ...
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 ''g'' = 1, the notion of abelian variety is the same as that of
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. I ...
, and every complex torus gives rise to such a curve; for ''g'' > 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 In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite fu ...
hermitian form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allow ...
on ''V'' whose
imaginary part takes
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
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
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
''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.
Important Theorems
One important structure theorem of abelian varieties is Matsusaka's theorem. It states that over an algebraically closed field every abelian variety
is the quotient of the Jacobian of some curve; that is, there is some surjection of abelian varieties
where
is a Jacobian. This theorem remains true if the ground field is infinite.
Algebraic definition
Two equivalent definitions of abelian variety over a general field ''k'' are commonly in use:
* a
connected and
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
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 curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. I ...
s 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
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in p ...
for
curves
A curve is a geometrical object in mathematics.
Curve(s) may also refer to:
Arts, entertainment, and media Music
* Curve (band), an English alternative rock music group
* ''Curve'' (album), a 2012 album by Our Lady Peace
* "Curve" (song), a ...
over
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s that he had announced in 1940 work, he had to introduce the notion of an
abstract variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
and to rewrite the foundations of algebraic geometry to work with varieties without projective embeddings (see also the history section in the
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
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
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
of
characteristic zero, the
torsion group of an abelian variety of dimension ''g'' is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
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 2''g''. 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 ''n'' = ''p'').
The group of
''k''-rational points for a
global field In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields:
*Algebraic number field: A finite extension of \mathbb
*Global function f ...
''k'' is
finitely generated by the
Mordell-Weil theorem. Hence, by the structure theorem for
finitely generated abelian group
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, ...
s, it is isomorphic to a product of a
free abelian group Z
''r'' and a finite commutative group for some non-negative 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.
Polarisation 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), which is the solution to the following
moduli problem. A family of degree 0 line bundles parametrised by a ''k''-variety ''T'' is defined to be a
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the '' tangent bundle'' is a way of organisi ...
''L'' on
''A''×''T'' such that
# for all ''t'' in ''T'', the restriction of ''L'' to ''A''× is a degree 0 line bundle,
# the restriction of ''L'' to ×''T'' is a trivial line bundle (here 0 is the identity of ''A'').
Then there is a variety ''A''
v and a family of degree 0 line bundles ''P'', the Poincaré bundle, parametrised by ''A''
v such that a family ''L'' on ''T'' is associated a unique morphism ''f'': ''T'' → ''A''
v so that ''L'' is isomorphic to the pullback of ''P'' along the morphism 1
A×''f'': ''A''×''T'' → ''A''×''A''
v. Applying this to the case when ''T'' is a point, we see that the points of ''A''
v correspond to line bundles of degree 0 on ''A'', so there is a natural group operation on ''A''
v given by tensor product of line bundles, which makes it into an abelian variety.
This association is a duality in the sense that there is a
natural isomorphism between the double dual ''A''
vv and ''A'' (defined via the Poincaré bundle) 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 dual In mathematics,
Cartier duality is an analogue of Pontryagin duality for commutative group schemes. It was introduced by .
Definition using characters
Given any finite flat commutative group scheme ''G'' over ''S'', its Cartier dual is the group o ...
s of each other. This generalises the
Weil pairing for elliptic curves.
Polarisations
A polarisation of an abelian variety is an ''
isogeny In mathematics, in particular, in algebraic geometry, an isogeny is a morphism of algebraic groups (also known as group varieties) that is surjective and has a finite kernel.
If the groups are abelian varieties, then any morphism of the underlyin ...
'' 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). Polarised abelian varieties have finite
automorphism groups. A principal polarisation is a polarisation that is an isomorphism. Jacobians of curves are naturally equipped with a principal polarisation as soon as one picks an arbitrary rational base point on the curve, and the curve can be reconstructed from its polarised Jacobian when the genus is > 1. Not all principally polarised abelian varieties are Jacobians of curves; see the
Schottky problem. A polarisation induces a
Rosati involution on the
endomorphism ring of ''A''.
Polarisations over the complex numbers
Over the complex numbers, a polarised 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 polarisation of ''A''. A morphism of polarised abelian varieties is a morphism ''A'' → ''B'' of abelian varieties such that the
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of 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 A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
-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 parametrised by ''S''.
For an abelian scheme ''A'' / ''S'', the group of ''n''-torsion points forms a
finite flat group scheme. The union of the ''p''
''n''-torsion points, for all ''n'', forms a
p-divisible group.
Deformations of abelian schemes are, according to the
Serre–Tate theorem, governed by the deformation properties of the associated ''p''-divisible groups.
Example
Let
be such that
has no repeated complex roots. Then the discriminant
is nonzero. Let