Abelian variety

# Abelian variety

Overview
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, particularly in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, complex analysis
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 useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, an abelian variety is a projective algebraic variety that is also an algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...

, i.e., has a group law that can be defined by regular function
Regular function
In mathematics, a regular function is a function that is analytic and single-valued in a given region. In complex analysis, any complex regular function is known as a holomorphic function...

s. 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.
Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, particularly in algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, complex analysis
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 useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, an abelian variety is a projective algebraic variety that is also an algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...

, i.e., has a group law that can be defined by regular function
Regular function
In mathematics, a regular function is a function that is analytic and single-valued in a given region. In complex analysis, any complex regular function is known as a holomorphic function...

s. 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
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

; 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
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...

that can be embedded into a complex projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localization
Localization of a ring
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...

techniques lead naturally from abelian varieties defined over number fields to ones defined over finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

s and various local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...

s.

Abelian varieties appear naturally as Jacobian varieties
Jacobian variety
In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...

(the connected components of zero in Picard varieties) and Albanese varieties
Albanese variety
In mathematics, the Albanese variety A, named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve, and is the abelian variety generated by a variety V. In other words there is a morphism from the variety V to its Albanese variety A, such that any morphism from V to an...

of other algebraic varieties. The group law of an abelian variety is necessarily commutative and the variety is non-singular. An elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...

is an abelian variety of dimension 1. Abelian varieties have Kodaira dimension
Kodaira dimension
In algebraic geometry, the Kodaira dimension κ measures the size of the canonical model of a projective variety V.The definition of Kodaira dimension, named for Kunihiko Kodaira, and the notation κ were introduced in the seminar.-The plurigenera:...

0.

## History and motivation

In the early nineteenth century, the theory of elliptic function
Elliptic function
In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...

s succeeded in giving a basis for the theory of elliptic integral
Elliptic integral
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler...

s, and this left open an obvious avenue of research. The standard forms for elliptic integrals involved the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

s 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
Carl Gustav Jakob Jacobi
Carl Gustav Jacob Jacobi was a German mathematician, widely considered to be the most inspiring teacher of his time and is considered one of the greatest mathematicians of his generation.-Biography:...

, 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
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....

, Weierstrass
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass was a German mathematician who is often cited as the "father of modern analysis".- Biography :Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia....

, Frobenius
Ferdinand Georg Frobenius
Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of differential equations and to group theory...

, Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...

and Picard
Charles Émile Picard
Charles Émile Picard FRS was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie Française in 1924.- Biography :...

. 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
Solomon Lefschetz
Solomon Lefschetz was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.-Life:...

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
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...

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
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

s (more specifically in the study of Hamiltonian system
Hamiltonian system
In physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....

s), and in algebraic geometry (especially Picard varieties and Albanese varieties
Albanese variety
In mathematics, the Albanese variety A, named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve, and is the abelian variety generated by a variety V. In other words there is a morphism from the variety V to its Albanese variety A, such that any morphism from V to an...

).

### Definition

A complex torus of dimension g is a torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

of real dimension 2g that carries the structure of a complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

. It can always be obtained as the quotient
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...

of a g-dimensional complex vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

by a lattice
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...

of rank 2g.
A complex abelian variety of dimension g is a complex torus of dimension g that is also a projective algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...

over the field of complex numbers. Since they are complex tori, abelian varieties carry the structure of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

. A morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

of abelian varieties is a morphism of the underlying algebraic varieties that preserves the identity element
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...

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 g = 1, the notion of abelian variety is the same as that of elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...

, and every complex torus gives rise to such a curve; for g > 1 it has been known since Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....

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
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

values on L×L. Such a form on X is usually called a (non-degenerate) Riemann form
Riemann form
In mathematics, a Riemann form in the theory of abelian varieties and modular forms, is the following data:* A lattice Λ in a complex vector space Cg.* An alternating bilinear form α from Λ to the integers satisfying the following two conditions:...

. 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 (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...

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
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

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
Birational geometry
In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian...

, its function field
Function field
Function field may refer to:*Function field of an algebraic variety*Function field...

is the fixed field of the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

on g letters acting on the function field of Cg.

### Abelian functions

An abelian function is a meromorphic function
Meromorphic function
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...

on an abelian variety, which may be regarded therefore as a periodic function of n complex variables, having 2n 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
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in  x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...

an isogeny.

## Algebraic definition

Two equivalent definitions of abelian variety over a general field k are commonly in use:
• a connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

and complete algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...

over k
• a connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

and projective
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...

over k.

When the base is the field of complex numbers, these notions coincide with the previous definition. Over all bases, elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...

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 for curves
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...

over finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...

s that he had announced in 1940 work, he had to introduce the notion of an abstract variety
Abstract variety
In mathematics, in the field of algebraic geometry, the idea of abstract variety is to define a concept of algebraic variety in an intrinsic way. This followed the trend in the definition of manifold independent of any ambient space by some years, the first notions being those of Oscar Zariski and...

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
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

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
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

.

For C, and hence by the Lefschetz principle for every algebraically closed field
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...

of characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...

zero, 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 2g copies of the cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

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)2g when n and p are coprime
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...

. 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 n = p).

The group of k-rational points
Rational point
In number theory, a K-rational point is a point on an algebraic variety where each coordinate of the point belongs to the field K. This means that, if the variety is given by a set of equationsthen the K-rational points are solutions ∈Kn of the equations...

for a global field
Global field
In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...

k is finitely generated by the Mordell-Weil theorem. Hence, by the structure theorem for finitely generated abelian group
Finitely generated abelian group
In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the formwith integers n1,...,ns...

s, it is isomorphic to a product of a free abelian group
Free abelian group
In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...

Zr 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
Isogeny
In mathematics, an isogeny is a morphism of varieties between two abelian varieties that is surjective and has a finite kernel....

to a product of abelian varieties of lower dimension. Any abelian variety is isogenous to a product of simple abelian varieties.

### Dual abelian variety

To an abelian variety A over a field k, one associates a dual abelian variety Av (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
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 organising these...

L on
A×T such that
1. for all , the restriction of L to A×{t} is a degree 0 line bundle,
2. the restriction of L to {0}×T is a trivial line bundle (here 0 is the identity of A).

Then there is a variety Av and a family of degree 0 line bundles P, the Poincaré bundle, parametrised by Av such that a family L on T is associated a unique morphism f: TAv so that L is isomorphic to the pullback of P along the morphism 1A×f: A×TA×Av. Applying this to the case when T is a point, we see that the points of Av correspond to line bundles of degree 0 on A, so there is a natural group operation on Av 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 Avv and A (defined via the Poincaré bundle) and that it is contravariant functorial, i.e. it associates to all morphisms f: AB dual morphisms fv: BvAv in a compatible way. The n-torsion of an abelian variety and the n-torsion of its dual are dual
Pontryagin duality
In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact groups, such as R, the circle or finite cyclic groups.-Introduction:...

to each other when n is coprime to the characteristic of the base. In general - for all n - the n-torsion group scheme
Group scheme
In mathematics, a group scheme is a type of algebro-geometric object 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 have group scheme structure, but group schemes are not...

s of dual abelian varieties are Cartier duals of each other. This generalises the Weil pairing
Weil pairing
In mathematics, the Weil pairing is a construction of roots of unity by means of functions on an elliptic curve E, in such a way as to constitute a pairing on the torsion subgroup of E...

for elliptic curves.

### Polarisations

A polarisation of an abelian variety is an isogeny
Isogeny
In mathematics, an isogeny is a morphism of varieties between two abelian varieties that is surjective and has a finite kernel....

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
Schottky problem
In mathematics, the Schottky problem, named after Friedrich Schottky, is a classical question of algebraic geometry, asking for a characterisation of Jacobian varieties amongst abelian varieties.-Geometric formulation:...

.

### 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 H1 and H2 are called equivalent
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...

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 polarisation of A. A morphism of polarised abelian varieties is a morphism AB of abelian varieties such that the pullback
Pullback
Suppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N to the space of 1-forms on M. This linear map is known as the pullback , and is frequently denoted by φ*...

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
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...

-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
Arithmetic of abelian varieties
In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or family of those. It goes back to the studies of Fermat on what are now recognised as elliptic curves; and has become a very substantial area both in terms of results and conjectures...

), and parameter-families of abelian varieties. An abelian scheme over a base scheme S of relative dimension g is a proper, smooth group scheme
Group scheme
In mathematics, a group scheme is a type of algebro-geometric object 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 have group scheme structure, but group schemes are not...

over S whose geometric fibers are connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

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.

## Semiabelian variety

A semiabelian variety is a commutative group variety which is an extension of an abelian variety by a torus
Algebraic torus
In mathematics, an algebraic torus is a type of commutative affine algebraic group. These groups were named by analogy with the theory of tori in Lie group theory...

.