mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the arithmetic of abelian varieties is the study of the

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

of an

abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group ...

, or a family of abelian varieties. It goes back to the studies of

Pierre de Fermat Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...

on what are now recognized as

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...

s; and has become a very substantial area of

arithmetic geometry In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. ...

both in terms of results and conjectures. Most of these can be posed for an abelian variety ''A'' over a

number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...

''K''; or more generally (for

global field In mathematics, a global field is one of two types of fields (the other one is local fields) that are characterized using valuations. There are two kinds of global fields: *Algebraic number field: A finite extension of \mathbb *Global functio ...

s or more general finitely-generated rings or fields).

Integer points on abelian varieties

There is some tension here between concepts: ''integer point'' belongs in a sense to

affine geometry In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle. As the notion of '' parallel lines'' is one of the main properties that is i ...

, while ''abelian variety'' is inherently defined in

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...

. The basic results, such as Siegel's theorem on integral points, come from the theory of

diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated ...

Rational points on abelian varieties

The basic result, the

Mordell–Weil theorem 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, called the Mordell–Weil group. The case with A an ellip ...

Diophantine geometry In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study ...

, says that ''A''(''K''), the group of points on ''A'' over ''K'', is a

finitely-generated abelian group In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...

. A great deal of information about its possible

torsion subgroup In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group ...

s is known, at least when ''A'' is an elliptic curve. The question of the ''rank'' is thought to be bound up with

L-function In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may gi ...

s (see below). The

torsor In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a no ...

theory here leads to the

Selmer group In arithmetic geometry, the Selmer group, named in honor of the work of by , is a group constructed from an isogeny of abelian varieties. Selmer group of an isogeny The Selmer group of an abelian variety ''A'' with respect to an isogeny ''f'' ...

and

Tate–Shafarevich group In arithmetic geometry, the Tate–Shafarevich group of an abelian variety (or more generally a group scheme) defined over a number field consists of the elements of the Weil–Châtelet group \mathrm(A/K) = H^1(G_K, A), where G_K = \mathrm(K ...

, the latter (conjecturally finite) being difficult to study.

Heights

The theory of heights plays a prominent role in the arithmetic of abelian varieties. For instance, the canonical

Néron–Tate height In number theory, the Néron–Tate height (or canonical height) is a quadratic form on the Mordell–Weil group of rational points of an abelian variety defined over a global field. It is named after André Néron and John Tate. Definition and ...

is a

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...

with remarkable properties that appear in the statement of the

Birch and Swinnerton-Dyer conjecture In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory ...

Reduction mod ''p''

Reduction of an abelian variety ''A'' modulo a

prime ideal In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...

of (the integers of) ''K'' — say, a prime number ''p'' — to get an abelian variety ''A_p'' over a

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

, is possible for

almost all In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...

''p''. The 'bad' primes, for which the reduction degenerates by acquiring singular points, are known to reveal very interesting information. As often happens in number theory, the 'bad' primes play a rather active role in the theory. Here a refined theory of (in effect) a

right adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...

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.

L-functions

For abelian varieties such as A_''p'', there is a definition of

local zeta-function In mathematics, the local zeta function (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as :Z(V, s) = \exp\left(\sum_^\infty \frac (q^)^k\right) where is a non-singular -dimensional projective algeb ...

available. To get an L-function for A itself, one takes a suitable

Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard E ...

of such local functions; to understand the finite number of factors for the 'bad' primes one has to refer to the

Tate module In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group ''A''. Often, this construction is made in the following situation: ''G'' is a commutative group scheme over a field ''K'', '' ...

of A, which is (dual to) the

étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectu ...

group H¹(A), and the

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...

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 In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...

, are still conjectural – the Taniyama–Shimura conjecture (which was proven in 2001) was just a special case, so that's hardly surprising. 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.

Complex multiplication

Since the time of

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...

(who knew of the ''lemniscate function'' case) the special role has been known of those abelian varieties

A

with extra automorphisms, and more generally endomorphisms. In terms of the ring

(A)

, there is a definition of

abelian variety of CM-type 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''). The terminology here is from complex multiplication theory, which was deve ...

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 Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded do ...

required is all of the

Pontryagin duality In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), ...

type, rather than needing more general

automorphic representation In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset ...

s. That reflects a good understanding of their Tate modules as

Galois module In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring i ...

s. It also makes them ''harder'' to deal with in terms of the conjectural

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

(

Hodge conjecture In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties. In simple terms, the Hodge conjectur ...

and

Tate conjecture In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The ...

). In those problems the special situation is more demanding than the general. In the case of elliptic curves, the

Kronecker Jugendtraum Hilbert's twelfth problem is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. It is one of the 23 mathematical Hilbert problems and asks for analogues of the roots of unity th ...

was the programme

Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, abstract algebra and logic, and criticized Georg Cantor's work on set theory. Heinrich Weber quoted Kronecker as having said, ...

proposed, to use elliptic curves of CM-type to do

class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...

explicitly for

imaginary quadratic field In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 an ...

s – in the way that

roots of unity In mathematics, a root of unity is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group char ...

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 The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space \mathbb C^n, that is, -tuples of complex numbers. The name of the field dealing with the properties ...

Manin–Mumford conjecture

The Manin–Mumford conjecture of

Yuri Manin Yuri Ivanovich Manin (; 16 February 1937 – 7 January 2023) was a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics. Life an ...

and

David Mumford David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded th ...

, proved by

Michel Raynaud Michel Raynaud (; 16 June 1938 – 10 March 2018 Décès de Michel Raynaud
So ...

, states that a curve ''C'' in its

Jacobian variety In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelia ...

''J'' can only contain a finite number of points that are of finite order (a torsion point) in ''J'', unless ''C'' = ''J''. There are other more general versions, such as the Bogomolov conjecture which generalizes the statement to non-torsion points.

References

{{reflist Abelian varieties Diophantine geometry