Arithmetic of abelian varieties
Encyclopedia
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 curve
s; 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 field
s or more general finitely-generated rings or fields).
, while abelian variety is inherently defined in projective geometry
. The basic results proving that elliptic curve
s have finitely many integer points come out of diophantine approximation
.
) 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-function
s (see below).
The torsor theory here leads to the Selmer group and Tate–Shafarevich group, the latter (conjecturally finite) being difficult to study.
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.
of (the integers of) K — say, a prime number p — to get an abelian variety Ap 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 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 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.
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 (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.
(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.
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
).
, proved by Michel 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.
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...
, the arithmetic of abelian varieties is the study of the 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...
of an abelian variety
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
, or family of those. It goes back to the studies of Fermat on what are now recognised as 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; 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 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...
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 geometryAffine geometry
In mathematics affine geometry is the study of geometric properties which remain unchanged by affine transformations, i.e. non-singular linear transformations and translations...
, while abelian variety is inherently defined in projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
. The basic results proving that 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 have finitely many integer points come out of diophantine approximation
Diophantine approximation
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 theoremMordell–Weil theorem
In mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the group A of K-rational points of A is a finitely-generated abelian group, called the Mordell-Weil group...
) 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-function
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
s (see below).
The torsor theory here leads to the Selmer group and Tate–Shafarevich group, the latter (conjecturally finite) being difficult to study.
Heights
There is a canonical Néron–Tate heightNéron–Tate height
In number theory, the Néron–Tate height is a quadratic form on the Mordell-Weil group of rational points of an abelian variety defined over a global field...
function, which is a quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
; 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.
Reduction mod p
Reduction of an abelian variety A modulo a prime idealPrime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
of (the integers of) K — say, a prime number p — to get an abelian variety Ap over a 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...
, is possible for almost all
Almost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....
p. The 'bad' primes, for which the reduction degenerates
Degeneracy (mathematics)
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....
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 to reduction mod p — the Néron model
Néron model
In algebraic geometry, a Néron model for an abelian variety AK defined over a local field K is the "best possible" group scheme AO defined over the ring of integers R of the local field K that becomes isomorphic to AK after base change from R to K.They were introduced by André Néron...
— cannot always be avoided. In the case of an elliptic curve there is an algorithm of John Tate
John Tate
John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.-Biography:...
describing it.
L-functions
For abelian varieties such as Ap, there is a definition of local zeta-functionLocal zeta-function
In number theory, a local zeta-functionis a function whose logarithmic derivative is a generating functionfor the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F.-Formulation:...
available. To get an L-function for A itself, one takes a suitable Euler product
Euler product
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The name arose from the case of the Riemann zeta-function, where such a product representation was proved by Leonhard Euler.-Definition:...
of such local functions; to understand the finite number of factors for the 'bad' primes one has to refer to the Tate module
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, Ks is the separable closure of K, and A = G...
of A, which is (dual to) the étale cohomology
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 conjectures...
group H1(A), and the Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
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
Functional equation
In mathematics, a functional equation is any equation that specifies a function in implicit form.Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional...
, 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 GaussCarl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
(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
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. The terminology here is from complex multiplication theory, which was developed for elliptic curves in the nineteenth century...
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
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
required is all of the Pontryagin duality
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:...
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
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...
(Hodge conjecture
Hodge conjecture
The Hodge conjecture is a major unsolved problem in algebraic geometry which relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they...
and Tate conjecture
Tate conjecture
In mathematics, the Tate conjecture is a 1963 conjecture of John Tate linking algebraic geometry, and more specifically the identification of algebraic cycles, with Galois modules coming from étale cohomology...
). In those problems the special situation is more demanding than the general.
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
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
explicitly for imaginary quadratic fields – in the way that roots of unity
Root of unity
In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
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
Several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...
).
Manin–Mumford conjecture
The Manin–Mumford conjecture of Yuri Manin and David MumfordDavid Mumford
David Bryant Mumford is an American mathematician known for distinguished 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 the National Medal of Science...
, proved by Michel Raynaud, states that a curve C in its Jacobian variety
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...
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.