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In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. Also it is used for the generalization of this concept studied in algebraic geometry over base fields more general than the complex numbers. One-dimensional abelian varieties are elliptic curves. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...
Complex analysis is the branch of mathematics investigating holomorphic functions, i. ...
Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ...
In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ...
Geometry In geometry, a torus (pl. ...
In mathematics, a projective space is a fundamental construction from any vector space. ...
This article is about algebraic varieties. ...
The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...
In mathematics, an elliptic curve is a non-singular projective algebraic curve of genus 1 over a field K, together with a distinguished point defined over K. A more accessible (though less accurate) definition is that an elliptic curve is a plane curve defined by an equation of the form...
History and motivation The success in the early nineteenth century of the theory of elliptic functions in giving a basis for the theory of elliptic integrals left open an obvious avenue of research. The standard forms for elliptic integrals involved the square roots of cubic and quartic polynomials. When those were replaced by polynomials of higher degree, say quintics, what would happen? In the work of Niels Abel and Carl Jacobi, the answer was formulated: this would involve functions of two complex variables, having four independent periods (i.e. period vectors). This gave the first glimpse of an abelian variety of dimension 2 (an abelian surface): what would now be called the Jacobian of a hyperelliptic curve of genus 2. After Abel and Jacobi the theory of abelian functions was worked out mainly by Riemann, Weierstraß, Frobenius, Poincaré and Picard. In complex analysis, an elliptic function is, roughly speaking , a function defined on the complex plane which is periodic in two directions. ...
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Fagnano and Leonhard Euler. ...
Niels Henrik Abel (August 5, 1802–April 6, 1829), Norwegian mathematician, was born in Finnøy. ...
Karl Gustav Jacob Jacobi (Potsdam December 10, 1804 - Berlin February 18, 1851), was not only a great German mathematician but also considered by many as the most inspiring teacher of his time (Bell, p. ...
In algebraic geometry, a hyperelliptic curve (over the complex numbers) is an algebraic curve given by an equation of the form where f(x) is a polynomial of degree n > 4 with n distinct roots. ...
Bernhard Riemann. ...
Karl Theodor Wilhelm Weierstraß (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. (The letter ß may be transliterated as ss; one often writes Weierstrass. ...
Picture of Frobenius Ferdinand Georg Frobenius (October 26, 1849 - August 3, 1917) was a German mathematician, best-known for his contributions to the theory of differential equations and to group theory. ...
Henri Poincaré, photograph from the frontispiece of the 1913 edition of Last Thoughts Jules Henri Poincaré (April 29, 1854 – July 17, 1912) was one of Frances greatest mathematicians, theoretical scientists and a philosopher of science. ...
Charles Émile Picard (July 24, 1856 - December 11, 1941) was a leading French mathematician. ...
By the end of the 19th century mathematicians began to use geometric methods in the study of abelian functions. By the 20s, eventually, Lefschetz laid the basis for the study of abelian functions in terms of complex tori. Also he appears to be the first to use the name "abelian variety". It was Weil who gave the subject its modern foundations in the language of algebraic geometry. Solomon Lefschetz (3 September 1884-5 October 1972) was a US mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations. ...
André Weil (May 6, 1906 _ August 6, 1998) was one of the great mathematicians of the 20th century, a founding member of the influential Bourbaki group. ...
Today abelian varieties from 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. In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ...
Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ...
In mathematics, the Picard group of a ringed space is the group of isomorphism classes of invertible sheaves on , with the group operation being tensor product. ...
In mathematics, the Albanese variety is a construction of algebraic geometry, which for an algebraic variety V solves a universal problem for morphisms of V into abelian varieties. ...
Analytic theory
Definition A complex torus of dimension g is a torus of real dimension 2g that carries the structure of a complex manifold. It can always be obtained as the quotient of a g-dimensional complex vector space by a lattice of rank 2g. A complex abelian variety of dimension g is a complex torus of dimension g that is also a projective algebraic variety over the field of complex numbers. Since they are complex tori, abelian varieties carry the structure of a group. A morphism of abelian varieties is a morphism of the underlying algebraic varieties that preserves the identity element for the group structure. An isogeny is a finite-to-one morphism. Geometry In geometry, a torus (pl. ...
In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ...
For quotient spaces in linear algebra, see quotient space (linear algebra). ...
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
See lattice for other meanings of this term, both within and without mathematics. ...
In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ...
In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ...
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 set. ...
When a complex torus carries the structure of an algebraic variety, this structure is necessarily unique. In case n is 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 non-singular projective algebraic curve of genus 1 over a field K, together with a distinguished point defined over K. A more accessible (though less accurate) definition is that an elliptic curve is a plane curve defined by an equation of the form...
Bernhard Riemann. ...
Riemann conditions The following criterion by Riemann decides whether or not a given complex torus is an abelian variety, i.e. whether or not it can be embedded into a projective space. Let X be a g-dimensional torus given as X=V/L where V is a complex vector space of dimension g and L is a lattice in V. Then X is an abelian variety if and only if there exists a positive definite hermitian bilinear form on V whose imaginary part takes integral values on LxL. 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. In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ...
In mathematics, a sesquilinear form on a complex vector space V is a map V × V → C that is linear in one argument and antilinear in the other. ...
In mathematics, the imaginary part of a complex number z is the second element of the ordered pair of real numbers representing z, i. ...
The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ...
The Jacobian of an algebraic curve Every algebraic curve C of genus g which is at least 1 is associated with an abelian variety J of dimension g, by means of an analytic map of C into J. As a torus, J carries a commutative group structure, and the image of C generates J as a group. More accurately, J is covered by Cg: any point in J comes from a g-tuple of points in C. The study of differential forms on C, which give rise to the abelian integrals with which the theory started, can be derived from the simpler, translation-invariant theory of differentials on J. The abelian variety J is called the Jacobian variety of C, for any non-singular curve C over the complex numbers. From the point of view of birational geometry, its function field is the fixed field of the symmetric group on g letters acting on the function field of Cg. In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ...
In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
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 algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions. ...
In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...
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. For example, in the nineteenth century there was much interest in hyperelliptic integrals that may be expressed in terms of elliptic integrals. This comes down to asking that J is a product of elliptic curves, up to an isogeny. A meromorphic function is a function that is holomorphic on an open subset of the complex number plane C (or on some other connected Riemann surface) except at points in a set of isolated poles, which are certain well-behaved singularities. ...
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 geometry), for everywhere_regular differential 1_forms. ...
In mathematics, the term up to xxxx is used to describe a situation in which members of an equivalence class can 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. When the base is the field of complex numbers, the notion coincides with the above definition. Over all bases, elliptic curves are abelian varieties of dimension 1. In mathematics, an elliptic curve is a non-singular projective algebraic curve of genus 1 over a field K, together with a distinguished point defined over K. A more accessible (though less accurate) definition is that an elliptic curve is a plane curve defined by an equation of the form...
In the early 40s, Weil used the former definition but could at first not prove that it implied the second. Only in 1948 he proved that complete algebraic groups can be embedded into projective space. Meanwhile, he had to introduce the notion of an abstract variety and to rewrite the foundations of algebraic geometry to work with varieties without projective embedding (see also the history section in the Algebraic Geometry article). In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism X × Y → Y is a closed map, i. ...
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. ...
In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...
Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...
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. ...
1948 is a leap year starting on Thursday (link will take you to calendar). ...
Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...
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, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...
Over C, and by the Lefschetz principle over every algebraically closed field of characteristic zero, its torsion group is isomorphic to (Q/Z)2g. So its n-torsion part is isomorphic to (Z/nZ)^2g, i.e. the product of 2g copies of the cyclic group of order n. As a group scheme, the n-torsion is isomorphic to (Z/nZ)g×(μn)g, the product of g copies of the constant group and g copies of the group of n-th roots of unity. Its free part is uncountable, since every algebraically closed field of characteristic zero is uncountable. In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. ...
In mathematics, a group scheme is a group object (some would prefer to say just group) in the category of schemes. ...
In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ...
In mathematics, an uncountable set is a set which is not countable. ...
When the base field is an algebraically closed field of characteristic p then the n-torsion is still isomorphic to (Z/nZ)^2g under the condition that n and p be coprime. In mathematics, the integers a and b are said to be coprime or relatively prime if and only if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ...
The group of k-rational points for a number field k is finitely generated by the Mordell-Weil theorem. Hence by the structure theorem for finitely generated abelian groups it is isomorphic to a product of a free abelian group Zr and a finite commutative group for some positive integer r called the rank of the abelian variety. Similar results hold for other classes of fields k. In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days...
In mathematics, the Mordell-Weil theorem states that for an abelian variety A over a number field K, the group A(K) of K-rational points of A is a finitely-generated abelian group. ...
In abstract algebra, an abelian group (G,+) 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 form x = n1x1 + n2x2 + ... + nsxs with integers n1,...,ns. ...
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. ...
Polarization and dual abelian variety
Dual abelian variety To an abelian variety A (over C, over a field, or over a general base), one associates a dual abelian variety Av (over the same base). This association is a duality in the sense that there is a natural isomorphism between the double dual Avv and A and that it is contravariant functorial, i.e. it associates to a morphism f:A->B a dual morphism fv:Av->Bv. 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 are Cartier dual to each other. In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...
For functors in computer science, see the function object article. ...
In mathematics, in particular in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform. ...
In mathematics, a group scheme is a group object (some would prefer to say just group) in the category of schemes. ...
Polarizations A polarization of an abelian variety is an isogeny of an abelian variety to its dual. Polarized abelian varieties have finite automorphism groups. A principle polarization is an isomorphism between an abelian variety and its dual. Jacobians of curves are naturally equipped with a principal polarization and the curve can be reconstructed from its polarized Jacobian. Not all principally polarized abelian varieties are Jacobians of curves, see the Schottky Problem. In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...
Polarizations over the complex numbers Over the complex numbers, a polarized abelian variety can also be defined as an abelian variety A together with a choice of a Riemann form H. Two Riemann forms H1 and H2 are called equivalent if there are positive integers n and m such that nH1=mH2. A choice of an equivalence class of Riemann forms on A is called a polarization of A. A morphism of polarized abelian varieties is a morphism A->B of abelian varieties such that the pullback of the Riemann form on B to A is equivalent to the chosen form on A. Polarized abelian varieties have finite automorphism groups. In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...
This article discusses the pullback in differential geometry. ...
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...
Abelian scheme One can also give a scheme-theoretic definition of an abelian variety which is relative to a base. This allows for a uniform treatment of phenomena as reduction mod p of abelian varieties (see Arithmetic of abelian varieties) and parameter-families of abelian varieties. An abelian scheme over a base scheme S of relative dimension g is a proper smooth group scheme over S whose geometric fibers are connected and of dimension g. The fibers of an abelian scheme are abelian varieties. In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...
In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or family of those. ...
This is a glossary of scheme theory. ...
In mathematics, a group scheme is a group object (some would prefer to say just group) in the category of schemes. ...
In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...
See also In algebraic geometry the idea of a motive intuitively refers to some essential part of an algebraic variety. Mathematically, the theory of motives is then the conjectural universal cohomology theory for such objects. ...
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