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. Most of these can be posed for an abelian variety A over a number field K; or more generally (for global fields or more general finitely-generated rings or fields). Main article: History of mathematics The evolution of mathematics can be seen to be an ever increasing series of abstractions. ... Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... 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. ... Pierre de Fermat Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French lawyer at the Parliament of Toulouse and a mathematician who is given credit for the development of modern calculus. ... In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... 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... The term global field refers to either of the following: a number field, i. ...

Integer points on abelian varieties

There is some tension here between concepts: integer point belongs in a sense to affine geometry, while abelian variety is inherently defined in projective geometry. The basic results proving that elliptic curves have finitely many integer points come out of diophantine approximation. In geometry, affine geometry occupies a place intermediate between Euclidean geometry and projective geometry. ... Projective geometry can be thought of informally as the geometry which arises from placing ones eye at a point. ... In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers. ...

Rational points on abelian varieties

The basic result (Mordell-Weil theorem) says that A(K), the group of points on A over K, is a finitely-generated abelian group. A great deal of information about its possible torsion subgroups is known, at least when A is an elliptic curve. The question of the rank is thought to be bound up with L-functions (see below). 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. ... The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary number theory. ...

The torsor theory here leads to the Selmer group and Tate-Shafarevich group, the latter (conjecturally finite) being difficult to study. In mathematics, a principal homogeneous space, or G-torsor, for a group G is a set X on which G acts freely and transitively. ... In mathematics, the Weil-Châtelet group of an abelian variety A defined over a field K is the abelian group of principal homogeneous spaces for A, defined over K. It is named for André Weil, who introduced the general group operation in it, and F. Châtelet. ... In mathematics, the Weil-Châtelet group of an abelian variety A defined over a field K is the abelian group of principal homogeneous spaces for A, defined over K. It is named for André Weil, who introduced the general group operation in it, and F. Châtelet. ...

Heights

There is a canonical Tate-Néron height function, which is a quadratic form; it has some remarkable properties, amongst all height functions designed to pick of finite sets in A(K) of points of height (roughly, logarithmic size of co-ordinates) at most h. In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

Reduction mod p

Reduction of an abelian variety A modulo a prime ideal of (the integers of)K - say, a prime number p - to get an abelian variety A_p, is over a finite field, is possible for almost all p. The 'bad' primes, for which the reduction degenerates by acquiring singular points, are known to conceal very interesting information. As often happens in number theory, the 'bad' primes play a rather active role in the theory. In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ... In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ... In mathematics, the phrase almost all has a number of specialised uses. ... In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class. ...

Here a refined theory of (in effect) a right adjoint to reduction mod p - the Néron model - cannot always be avoided. In the case of an elliptic curve there is an algorithm of John Tate describing it. The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ... You may be looking for John Tate (boxer) John Torrence Tate, born March 13, 1925 in Minneapolis, Minnesota, is an American mathematician, distinguished for many fundamental contributions in algebraic number theory and related areas in algebraic geometry. ...

L-functions

For abelian varieties such as A_p, there is a definition of local zeta-function available. To get an L-function for A itself, one takes a suitable Euler product of such local functions; to understand the finite number of factors for the 'bad' primes one has to refer to the Tate module of A, which is (dual to) the étale cohomology group H¹(A), and the Galois group action on it. In this way one gets a respectable definition of Hasse-Weil L-function for A. In general its properties, such as functional equation, are still conjectural - the Taniyama-Shimura conjecture was just a special case, so that's hardly surprising. In number theory, a local zeta-function is a generating function Z(t) for the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F. The analogy with the Riemann zeta function comes via consideration of the logarithmic derivative . ... In mathematics, an Euler product is an infinite product expansion, indexed by prime numbers p, of a Dirichlet series. ... In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ... In mathematics, the Hasse-Weil zeta function attached to an algebraic variety V defined over a number field K is one of the most important types of L-function. ... In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. ... The Taniyama-Shimura theorem establishes an important connection between elliptic curves, which are objects from algebraic geometry, and modular forms, which are certain periodic holomorphic functions investigated in number theory. ...

It is in terms of this L-function that the conjecture of Birch and Swinnerton-Dyer is posed. It is just one particularly interesting aspect of the general theory about values of L-functions L(s) at integer values of s, and there is much empirical evidence supporting it. In mathematics, the Birch and Swinnerton-Dyer conjecture relates the rank of the abelian group of points over a number field of an elliptic curve E to the order of zero of the associated L-function L(E,s) at s = 1. ...

Complex multiplication

Since the time of Gauss (who knew of the lemniscate function case) the special role has been known of the A with extra automorphisms, and more generally endomorphisms. In terms of the ring End(A) there is a definition of abelian variety of CM-type that singles out the richest class. These are special in their arithmetic. This is seen in their L-functions in rather favourable terms - the harmonic analysis required is all of the Pontryagin duality type, rather than needing more general automorphic representations. That reflects a good understanding of their Tate modules as Galois modules. It also makes them harder to deal with in terms of the conjectural algebraic geometry (Hodge conjecture and Tate conjecture). In those problems the special situation is more demanding than the general. Johann Carl Friedrich Gauss Johann Carl Friedrich Gauss (Gauß) (April 30, 1777 – February 23, 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ... In mathematics, an abelian variety A defined over a field K is said to have CM_type if it has a large enough commutative subring in its endomorphism ring End(A). ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... 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, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... The Hodge conjecture is a major unsolved problem of algebraic geometry. ...

In the case of elliptic curves, the Kronecker Jugendtraum was the programme Kronecker proposed, to use elliptic curves of CM-type to do class field theory explicitly for imaginary quadratic fields - in the way that roots of unity allow one to do this for the field of rational numbers. This generalises, but in some sense with loss of explicit information (as is typical of several complex variables). Hilberts twelfth problem, of the 23 Hilberts problems, is the extension of Kroneckers theorem on abelian extensions of the rational numbers, to any base number field. ... Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the natural numbers; all else is the work of man (Bell 1986, p. ... Class field theory is a branch of algebraic number theory, including most of the major results that were proved in the period about 1900-1950. ... In mathematics, a quadratic field is a field extension K/Q of the form where d is a non-zero rational number. ... 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. ... The theory of functions of several complex variables is the branch of mathematics dealing with functions f(z1, z2, ... , zn) on the space Cn of n-tuples of complex numbers. ...

Manin-Mumford conjecture

The Manin-Mumford conjecture, proved by Raynaud, states that a curve C in its Jacobian variety J can only contain a finite number of points that are of finite order in J, unless C=J. There are more general statements; this one is most clearly motivated by the Mordell conjecture, where such a curve C should intersect J(K) only in finitely many points. There is now a general 'Manin-Mumford' theory. In number theory, the Mordell conjecture stated a basic result regarding the rational number solutions to Diophantine equations. ...