In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, particularly in
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
,
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
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 ob ...
, an abelian variety is a smooth
projective algebraic variety
In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, the de ...
that is also an
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Man ...
, i.e., has a
group law that can be defined by
regular function
In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a reg ...
s. 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.
An abelian variety can be defined by equations having coefficients in any
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
; 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
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
. Such abelian varieties turn out to be exactly those
complex tori that can be
holomorphically 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
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field (mathematics), field of rational numbers such that the field extension K / \mathbb has Degree of a field extension, finite degree (and hence ...
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 (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
s and various
local field
In mathematics, a field ''K'' is called a non-Archimedean local field if it is complete with respect to a metric induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. In general, a local field is a locally compact t ...
s. Since a number field is the fraction field of a
Dedekind domain
In mathematics, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily un ...
, 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
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
and the variety is
non-singular
Singular may refer to:
* Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms
* Singular or sounder, a group of boar, see List of animal names
* Singular (band), a Thai jazz pop duo
*'' 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. If the ...
is an abelian variety of dimension 1. Abelian varieties have
Kodaira dimension
In algebraic geometry, the Kodaira dimension measures the size of the canonical model of a projective variety .
Soviet mathematician Igor Shafarevich in a seminar introduced an important numerical invariant of surfaces with the notation . ...
0.
History and motivation
In the early nineteenth century, the theory of
elliptic function
In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Those integrals are ...
s succeeded in giving a basis for the theory of
elliptic integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising i ...
s, 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 y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
s of
cubic
Cubic may refer to:
Science and mathematics
* Cube (algebra), "cubic" measurement
* Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex
** Cubic crystal system, a crystal system w ...
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 de ...
s. When those were replaced by polynomials of higher degree, say
quintics, what would happen?
In the work of
Niels Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
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 In mathematics, an abelian surface is a 2-dimensional abelian variety.
One-dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic via the Riemann bi ...
): what would now be called the ''Jacobian of a
hyperelliptic curve
In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form
y^2 + h(x)y = f(x)
where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dis ...
of genus 2''.
After Abel and Jacobi, some of the most important contributors to the theory of abelian functions were
Riemann
Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
,
Weierstrass
Karl Theodor Wilhelm Weierstrass (; ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the " father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school t ...
,
Frobenius Frobenius is a surname. Notable people with the surname include:
* Ferdinand Georg Frobenius (1849–1917), mathematician
** Frobenius algebra
** Frobenius endomorphism
** Frobenius inner product
** Frobenius norm
** Frobenius method
** Frobenius g ...
,
Poincaré
Poincaré is a French surname. Notable people with the surname include:
* Henri Poincaré
Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philos ...
, and
Picard
Picard may refer to:
Places
* Picard, Quebec, Canada
* Picard, California, United States
* Picard (crater), a lunar impact crater in Mare Crisium
People and fictional characters
* Picard (name), a list of people and fictional characters with th ...
. 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
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was one of the most influential mathematicians of the twentieth century. His influence is du ...
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 (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
s (more specifically in the study of
Hamiltonian system
A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can ...
s), and in algebraic geometry (especially
Picard varieties and
Albanese varieties).
Analytic theory
Definition
A
complex torus
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'', whe ...
of dimension ''g'' is a
torus
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
of real dimension 2''g'' that carries the structure of a
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...
. It can always be obtained as the
quotient
In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
of a ''g''-dimensional complex
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
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 solution set, set of solutions of a system of polynomial equations over the real number, ...
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 group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
. A
morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
of abelian varieties is a morphism of the underlying algebraic varieties that preserves the
identity element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
for the group structure. An
isogeny
In mathematics, particularly 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 underlyi ...
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
, 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. If the ...
, and every complex torus gives rise to such a curve; for
it has been known since
Riemann
Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
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
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'', whe ...
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
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
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of w ...
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 map, linear in each of its arguments, but a sesquilinear f ...
on ''V'' whose
imaginary part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
takes
integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
values on
. 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 (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
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
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
structure, and the image of ''C'' generates ''J'' as a group. More accurately, ''J'' is covered by
: 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 integral
In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form
:\int_^z R(x,w) \, dx,
where R(x,w) is an arbitrary rational function of the two variables x and w, wh ...
s'' 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
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying Map (mathematics), mappings that are gi ...
, its
function field is the fixed field of the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
on ''g'' letters acting on the function field of
.
Abelian functions
An abelian function is a
meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are ''poles'' of the 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 integral
In mathematics, ''differential of the first kind'' is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential ...
s that may be expressed in terms of elliptic integrals. This comes down to asking that ''J'' is a product of elliptic curves,
up to Two Mathematical object, mathematical objects and are called "equal up to an equivalence relation "
* if and are related by , that is,
* if holds, that is,
* if the equivalence classes of and with respect to are equal.
This figure of speech ...
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
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
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
In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Man ...
over ''k''
* a
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
and
projective algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Man ...
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. If the ...
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 pure ...
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), ''Curve'' (album), a 2012 album by Our Lady Peace
* Curve ( ...
over
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
s 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
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
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
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
.
For the field
, and hence by the
Lefschetz principle
In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic variety, algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces ...
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 . In other words, a field is algebraically closed if the fundamental theorem of algebra ...
of
characteristic zero, the
torsion group
In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which every element has finite order. The exponent of such a group, if it exists, is the least common multiple of the orders of the elements.
For exam ...
of an abelian variety of dimension ''g'' is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...
to
. Hence, its ''n''-torsion part is isomorphic to
, i.e., the product of 2''g'' copies of the
cyclic group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
of order ''n''.
When the base field is an algebraically closed field of characteristic ''p'', the ''n''-torsion is still isomorphic to
when ''n'' and ''p'' are
coprime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...
. 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
).
The group of
''k''-rational points for a
global field
In mathematics, a global field is one of two types of fields (the other one is local fields) that are characterized using valuations. There are two kinds of global fields:
*Algebraic number field: A finite extension of \mathbb
*Global functio ...
''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, n ...
s, it is isomorphic to a product of a
free abelian group
In mathematics, a free abelian group is an abelian group with a Free module, basis. Being an abelian group means that it is a Set (mathematics), set with an addition operation (mathematics), operation that is associative, commutative, and inverti ...
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
. 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
(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 organis ...
''L'' on
such that
# for all ''t'' in ''T'', the restriction of ''L'' to
is a degree 0 line bundle,
# the restriction of ''L'' to
is a trivial line bundle (here 0 is the identity of ''A'').
Then there is a variety
and a family of degree 0 line bundles ''P'', the Poincaré bundle, parametrised by
such that a family ''L'' on ''T'' is associated a unique morphism
so that ''L'' is isomorphic to the pullback of ''P'' along the morphism
. Applying this to the case when ''T'' is a point, we see that the points of
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.
This association is a duality in the sense that it is
contravariant functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...
ial, i.e., it associates to all morphisms
dual morphisms
in a compatible way, and there is a
natural isomorphism
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natura ...
between the double dual
and
(defined via the Poincaré bundle). 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 scheme
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups hav ...
s of dual abelian varieties are
Cartier duals of each other. This generalises the
Weil pairing
In mathematics, the Weil pairing is a pairing (bilinear form, though with multiplicative notation) on the points of order dividing ''n'' of an elliptic curve ''E'', taking values in ''n''th roots of unity. More generally there is a similar Weil ...
for elliptic curves.
Polarisations
A polarisation of an abelian variety is an ''
isogeny
In mathematics, particularly 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 underlyi ...
'' 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
In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of ...
(so it is analogous to a
positive definite quadratic form). Polarised abelian varieties have finite
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
s. 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
. Not all principally polarised abelian varieties are Jacobians of curves; see the
Schottky problem. A polarisation induces a
Rosati involution on the
endomorphism ring
In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in ...
of ''A''.
Polarisations over the complex numbers
Over the complex numbers, a polarised abelian variety can be defined as an abelian variety ''A'' together with a choice of a
Riemann form ''H''. Two Riemann forms
and
are called
equivalent
Equivalence or Equivalent may refer to:
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*Album-equivalent unit, a measurement unit in the music industry
*Equivalence class (music)
*'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*'' Equiva ...
if there are positive integers ''n'' and ''m'' such that
. A choice of an equivalence class of Riemann forms on ''A'' is called a polarisation of ''A''; over the complex number this is equivalent to the definition of polarisation given above. A morphism of polarised abelian varieties is a morphism
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-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
In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has b ...
), and parameter-families of abelian varieties. An abelian scheme over a base scheme ''S'' of relative dimension ''g'' is a
proper,
smooth group scheme
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups hav ...
over ''S'' whose
geometric fibers are
connected
Connected may refer to:
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* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
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
, the group of ''n''-torsion points forms a
finite flat group scheme. The union of the
-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