# Arithmetic of abelian varieties

﻿
Arithmetic 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. 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).

## 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.

## 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).

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 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.

## 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 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.

## L-functions

For abelian varieties such as Ap, 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 (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 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.

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).

## Manin–Mumford conjecture

The Manin–Mumford conjecture of Yuri Manin and David Mumford, 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.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Timeline of abelian varieties — This is a timeline of the theory of abelian varieties in algebraic geometry, including elliptic curves.Early history* c. 1000 Al Karaji writes on congruent numbers [ [http://www.cms.math.ca/Events/summer05/abs/pdf/hm.pdf PDF] ] eventeenth… …   Wikipedia

• Abelian variety — 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.e., has a group law that can be defined by regular… …   Wikipedia

• Arithmetic dynamics — is a new field that is an amalgamation of two areas of mathematics, dynamical systems and number theory. The subject can be viewed as the transfer of previous results in the theory of Diophantine equations to the setting of discrete dynamical… …   Wikipedia

• Glossary of arithmetic and Diophantine geometry — This is a glossary of arithmetic and Diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of… …   Wikipedia

• 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 mdash; that is, it has a multiplication defined algebraically with… …   Wikipedia

• List of mathematics articles (A) — NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A …   Wikipedia

• Weil restriction — In mathematics, restriction of scalars (also known as Weil restriction ) is a functor which, for any finite extension of fields L/k and any algebraic variety X over L , produces another variety Res L / k X , defined over k . It is useful for… …   Wikipedia

• 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 elliptic curve… …   Wikipedia

• Weil–Châtelet group — In mathematics, particularly in arithmetic geometry, 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… …   Wikipedia

• List of number theory topics — This is a list of number theory topics, by Wikipedia page. See also List of recreational number theory topics Topics in cryptography Contents 1 Factors 2 Fractions 3 Modular arithmetic …   Wikipedia