Abelian variety - Citizendia

In mathematics, particularly in algebraic geometry, complex analysis and number theory, an Abelian variety is a projective algebraic variety that is at the same time an algebraic group, i. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety In Algebraic geometry, an algebraic group (or group variety) is a group that is an Algebraic variety, such that the multiplication and inverse e. , has a group law that can be defined by regular functions. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Complex analysis, see Holomorphic function. In Mathematics, a regular function in the sense of Algebraic geometry 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. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Historically the first Abelian varieties to be studied were those defined over the field of complex numbers. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Such Abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. In Mathematics, a complex torus is a particular kind of Complex manifold M, for which (ignoring the Complex structure) the underlying Smooth In Mathematics a projective space is a set of elements constructed from a vector space such that a distinct element of the projective space consists of all non-zero vectors which Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. In Mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) Field extension of the Localisation techniques lead naturally from Abelian varieties defined over number fields to ones defined over finite fields and various local fields. In Abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements 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

Abelian varieties appear naturally as Jacobian varieties (the connected components of zero in Picard varieties) and Albanese varieties of other algebraic varieties. In Mathematics, the Jacobian variety of a non-singular Algebraic curve C of genus g &ge 1 is a particular Abelian variety In Mathematics, the Picard group of a Ringed space X is the group of Isomorphism classes of Invertible sheaves on X, with In Mathematics, the Albanese variety is a construction of Algebraic geometry, which for an Algebraic variety V solves a Universal problem The group law of an Abelian variety is necessarily commutative and the variety is non-singular. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, a singular point of an Algebraic variety V is a point P that is 'special' (so singular in the geometric sense that V An elliptic curve is an Abelian variety of dimension 1. In Mathematics, an elliptic curve is a smooth, projective Algebraic curve of genus one on which there is a specified point O Abelian varieties have Kodaira dimension 0. In Mathematics, the pluricanonical ring of an Algebraic variety V (which is Non-singular) or of a Complex manifold, is the graded

1 History and motivation
2 Analytic theory
3 Algebraic definition
4 Structure of the group of points
5 Products
6 Polarization and dual Abelian variety
7 Abelian scheme
8 See also
9 Further reading

History and motivation

For more details on this topic, see History of manifolds and varieties. The study of Manifolds combines many important areas of Mathematics: it generalizes concepts such as Curves and Surfaces as well as ideas from Linear

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 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar In Complex analysis, an elliptic function is a function defined on the Complex plane which is periodic in two directions (a Doubly-periodic In Integral calculus, elliptic integrals originally arose in connection with the problem of giving the Arc length of an Ellipse. The standard forms for elliptic integrals involved the square roots of cubic and quartic polynomials. In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. A quartic function is a function of the form f(x=ax^4+bx^3+cx^2+dx+e \ with nonzero a; or in other words a Polynomial 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. In Mathematics, a quintic equation is a Polynomial Equation of degree five Niels Henrik Abel (August 5 1802 &ndash April 6 1829 was a noted Norwegian Mathematician who proved the impossibility of solving the Quintic equation Carl Gustav Jacob Jacobi ( December 10, 1804 - February 18, 1851) was a Prussian Mathematician, widely considered to be The theory of functions of several complex variables is the branch of Mathematics dealing with functions f ( z1 z2 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. In Algebraic geometry, a hyperelliptic curve (over the Complex numbers) is an Algebraic curve given by an equation of the form y^2 = f(x

After Abel and Jacobi, some of the most important contributors to the theory of Abelian functions were Riemann, Weierstrass, Frobenius, Poincaré and Picard. Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician Ferdinand Georg Frobenius ( October 26, 1849 – August 3, 1917) was a German Mathematician, best-known for his contributions Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician Charles Émile Picard (usually referred to simply as Émile Picard) ( July 24, 1856 - December 12, 1941) was a leading French 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. The 1920s is sometimes referred to as the " Jazz Age " or the " Roaring Twenties " when speaking about the United States and Canada Solomon Lefschetz ( 3 September 1884 – 5 October 1972) was an American Mathematician who did fundamental work on He also appears to be the first to use the name "Abelian variety". It was Weil in the 1940s who gave the subject its modern foundations in the language of algebraic geometry. André Weil should not be confused with two other mathematicians with similar names Hermann Weyl (1885-1955 who made substantial contributions The 1940s decade ran from 1940 to 1949 Events and trends The 1940s was a period between the radical 1930s and the conservative 1950s which also leads the period to be

Today, Abelian varieties form an important tool in number theory, in dynamical systems (more specifically in the study of Hamiltonian systems), and in algebraic geometry (especially Picard varieties and Albanese varieties). The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position In Classical mechanics, a Hamiltonian system is a Physical system in which Forces are Velocity invariant In Mathematics, the Picard group of a Ringed space X is the group of Isomorphism classes of Invertible sheaves on X, with In Mathematics, the Albanese variety is a construction of Algebraic geometry, which for an Algebraic variety V solves a Universal problem

Analytic theory

Definition

A complex torus of dimension g is a torus of real dimension 2g that carries the structure of a complex manifold. In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar In Differential geometry, a complex manifold is a Manifold with an atlas of charts to the open unit disk in C n, It can always be obtained as the quotient of a g-dimensional complex vector space by a lattice of rank 2g. In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, especially in Geometry and Group theory, a lattice in R n is a Discrete subgroup of A complex Abelian variety of dimension g is a complex torus of dimension g that is also a projective algebraic variety over the field of complex numbers. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety Since they are complex tori, Abelian varieties carry the structure of a group. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element A morphism of Abelian varieties is a morphism of the underlying algebraic varieties that preserves the identity element for the group structure. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that An isogeny is a finite-to-one morphism.

When a complex torus carries the structure of an algebraic variety, this structure is necessarily unique. In the case n = 1, the notion of Abelian variety is the same as that of elliptic curve, and every complex torus gives rise to such a curve; for n > 1 it has been known since Riemann that the algebraic variety condition imposes extra constraints on a complex torus. In Mathematics, an elliptic curve is a smooth, projective Algebraic curve of genus one on which there is a specified point O

Riemann conditions

The following criterion by Riemann decides whether or not a given complex torus is an Abelian variety, i. e. whether or not it can be embedded into a projective space. Let X be a g-dimensional torus given as X = V/L where V is a complex vector space of dimension g and L is a lattice in V. Then X is an Abelian variety if and only if there exists a positive definite hermitian form on V whose imaginary part takes integral values on L×L. In Mathematics, a definite bilinear form is a Bilinear form B such that B ( x, x) has a fixed In Mathematics, a sesquilinear form on a Complex vector space V is a map V × V &rarr C that is linear In Mathematics, the imaginary part of a Complex number z is the second element of the ordered pair of Real numbers representing z The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Such a form on X is usually called a (non-degenerate) Riemann form. In Mathematics, a Riemann form in the theory of Abelian varieties and Modular forms, is the following data A lattice Λ in Choosing a basis for V and L, one can make this condition more explicit. There are several equivalent formulations of this; all of them are known as the Riemann conditions.

The Jacobian of an algebraic curve

Every algebraic curve C of genus g ≥ 1 is associated with an Abelian variety J of dimension g, by means of an analytic map of C into J. In Mathematics, genus has a few different but closely related meanings Topology Orientable surface As a torus, J carries a commutative group structure, and the image of C generates J as a group. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element 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. 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 In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying

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 2n independent periods; equivalently, it is a function in the function field of an Abelian variety. In Complex analysis, a meromorphic function on an open subset D of the Complex plane is a function that is holomorphic For example, in the nineteenth century there was much interest in hyperelliptic integrals that may be expressed in terms of elliptic integrals. In Mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally Complex manifolds This comes down to asking that J is a product of elliptic curves, up to an isogeny. In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose

See also: Abelian integral. In Mathematics, an abelian integral in Riemann surface theory is a function related to the Indefinite integral of a Differential of the first kind

Algebraic definition

Two equivalent definitions of Abelian variety over a general field are commonly in use:

a connected and complete algebraic group over k
a connected and projective algebraic group over k

When the base is the field of complex numbers, these notions coincide with the previous definition. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Mathematics, in particular in Algebraic geometry, a complete algebraic variety is an Algebraic variety X, such that for any variety In Algebraic geometry, an algebraic group (or group variety) is a group that is an Algebraic variety, such that the multiplication and inverse In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Algebraic geometry, an algebraic group (or group variety) is a group that is an Algebraic variety, such that the multiplication and inverse Over all bases, elliptic curves are Abelian varieties of dimension 1. In Mathematics, an elliptic curve is a smooth, projective Algebraic curve of genus one on which there is a specified point O

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. Year 1948 ( MCMXLVIII) was a Leap year starting on Thursday (link will display the 1948 calendar of the Gregorian calendar. Meanwhile, in order to make the proof of the Riemann hypothesis for curves over finite fields that he had announced in 1940 work, he had to introduce the notion of an abstract variety and to rewrite the foundations of algebraic geometry to work with varieties without projective embeddings (see also the history section in the Algebraic Geometry article). The Riemann hypothesis is one of the most important Conjectures in Mathematics. In Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements Year 1940 ( MCMXL) was a Leap year starting on Monday (link will display the full 1940 calendar of the Gregorian calendar. 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 Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with

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. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the

For C, and hence by the Lefschetz principle for every algebraically closed field of characteristic zero, the torsion group of an Abelian variety of dimension g is isomorphic to (Q/Z)^2g. In Mathematics, algebraic geometry and analytic geometry are two closely related subjects In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's In Group theory in Mathematics, a periodic group or a torsion group is a group in which each element has finite In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective Hence, its n-torsion part is isomorphic to (Z/nZ)^2g, i. e. the product of 2g copies of the cyclic group of order n. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an

When the base field is an algebraically closed field of characteristic p, the n-torsion is still isomorphic to (Z/nZ)^2g when n and p are coprime. In Mathematics, the Integers a and b are said to be coprime or relatively prime if they have no common factor other than When n and p are not coprime, the same result can be recovered provided one interprets it as saying that the n-torsion defines a finite flat group scheme of rank 2g. If instead of looking at the full scheme structure on the n-torsion, one considers only the reduced scheme structure (i. e: looks only at points), one obtains a new invariant for varieties in characteristic p (the so-called p-rank when n = p).

The group of k-rational points for a number field k is finitely generated by the Mordell-Weil theorem. 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. In Mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) Field extension of the In Mathematics, the Mordell–Weil theorem states that for an Abelian variety A over a Number field K, the group A ( Hence, by the structure theorem for finitely generated Abelian groups, it is isomorphic to a product of a free Abelian group Z^r and a finite commutative group for some positive integer r called the rank of the Abelian variety. In Abstract algebra, an Abelian group ( G,+ is called finitely generated if there exist finitely many elements x 1 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 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. Spectroscopy was originally the study of the interaction between Radiation and Matter as a function of Wavelength (λ Any abelian variety is isogenous to a product of simple abelian varieties.

Polarization and dual Abelian variety

Dual Abelian variety

To an Abelian variety A over a field k, one associates a dual Abelian variety A^v (over the same field). This association is a duality in the sense that there is a natural isomorphism between the double dual A^vv and A and that it is contravariant functorial, i. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories e. it associates to all morphisms f: A → B dual morphisms f^v: B^v → A^v in a compatible way. The n-torsion of an Abelian variety and the n-torsion of its dual are dual to each other when n is coprime to the characteristic of the base. In general - for all n - the n-torsion group schemes of dual Abelian varieties are Cartier duals of each other. In Mathematics, a group scheme is a Group object in the Category of schemes. This generalizes the Weil pairing for elliptic curves. 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

Polarizations

A polarization of an Abelian variety is an isogeny from an Abelian variety to its dual. Polarized Abelian varieties have finite automorphism groups. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself A principal polarization is an isomorphism between an Abelian variety and its dual. Jacobians of curves are naturally equipped with a principal polarization as soon as one picks an arbitrary base point on the curve, and the curve can be reconstructed from its polarized Jacobian. Not all principally polarized Abelian varieties are Jacobians of curves; see the Schottky problem. In Mathematics, the Schottky problem is a classical question of Algebraic geometry, asking for a characterisation of Jacobian varieties amongst Abelian

Polarizations over the complex numbers

Over the complex numbers, a polarized Abelian variety can also be defined as an Abelian variety A together with a choice of a Riemann form H. Two Riemann forms H₁ and H₂ are called equivalent if there are positive integers n and m such that nH₁=mH₂. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" A choice of an equivalence class of Riemann forms on A is called a polarization of A. A morphism of polarized Abelian varieties is a morphism A → B of Abelian varieties such that the pullback of the Riemann form on B to A is equivalent to the given form on A. The notion of pullback in Mathematics is a fundamental one It refers to two different but related processes precomposition and fiber-product

Abelian scheme

One can also define Abelian varieties scheme-theoretically and relative to a base. In Mathematics, a scheme is an important concept connecting the fields of Algebraic geometry, Commutative algebra and Number theory. 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. In Mathematics, the arithmetic of abelian varieties is the study of the Number theory of an Abelian variety, or family of those An Abelian scheme, sometimes called an Abelian variety, over a base scheme S of relative dimension g is a proper, smooth group scheme over S whose geometric fibers are connected and of dimension g. In Algebraic geometry, a proper morphism between schemes is an analogue of a Proper map between Topological spaces Definition A This is a glossary of scheme theory. For an introduction to the theory of schemes in Algebraic geometry, see Affine scheme, Projective space, sheaf In Mathematics, a group scheme is a Group object in the Category of schemes. This is a glossary of scheme theory. For an introduction to the theory of schemes in Algebraic geometry, see Affine scheme, Projective space, sheaf In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of The fibers of an Abelian scheme are Abelian varieties.

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