Birch And Swinnerton-Dyer Conjecture Background History

In 1922 Louis Mordell proved that the group of rational points on an elliptic curve has a finite basis. This means that for any elliptic curve there is a finite sub-set of the rational points on the curve from which all other rational points may be generated.

If the number of rational points on a curve is infinite then some points in a finite basis must have infinite order. The number of basis points with infinite order is called the rank of the curve, and is an important invariant property of an elliptic curve.

If the rank of an elliptic curve is 0 then the curve has only a finite number of rational points. On the other hand, if the rank of the curve is greater than 0, then the curve has an infinite number of rational points.

Although the Mordell's theorem shows that the rank of an elliptic curve is always finite, it does not give an effective method for calculating the rank of every curve. The rank of certain elliptic curves can be calculated using numerical methods but these cannot be generalised to handle all curves.

An L-function L(E,s) can be defined for an elliptic curve E by constructing an Euler product from the number of points on the curve modulo each prime p. This L-function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example, the distance between two points in three-dimensional Euclidean space is found by taking the square root of a quadratic form involving six var.

The natural definition of L(E,s) only converges for values of s in the complex plane with Re(s) > 3/2. Helmut HasseHelmut Hasse (pronounced HAHS uh) ( 25 August 1898- 26 December 1979) was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local classfield theory an conjectured that L(E,s) could be extended by analytic continuationIn complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Suppose f is an analytic function defined on the open subset U of the complex plane C . If V is an open sub to the whole complex plane. This conjecture was first proved by Max Deuring for elliptic curves with complex multiplicationIn mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense (it r. It was subsequently shown to be true for all elliptic curves, as a consequence of the Taniyama-Shimura theoremAlgebraic curves Riemann surfaces Modular forms Theorems 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 functio.

Finding rational points on a general elliptic curve is a difficult problem. Finding the points on an elliptic curve modulo a given prime p is conceptually straightforward, as there are only a finite number of possibilities to check. However, for large primes it is computationally intensive.

2 History

In the early 1960s Bryan Birch and Peter Swinnerton-Dyer used the EDSACThe EDSAC E lectronic D elay S torage A utomatic C omputer ran its first program May 6, 1949, and was constructed by Maurice Wilkes and his team at the University of Cambridge Mathematical Laboratory, inspired by the EDVAC design report by John von Neuman computer at the University of Cambridge Computer LaboratoryThe Computer Laboratory is Cambridge University's computer science department. For much of its history it also provided the University's Computing Service. On 2 December, 1936 the University's General Board published a Report on the Establishment of a Com to calculate the number of points modulo p (denoted by N_p) for a large number of primes p on elliptic curves whose rank was known. From these numerical results they conjectured that N_p for a curve E with rank r obeys an asymptotic law

Initially this was on the basis of somewhat tenuous trends in graphical plots; which induced a measure of scepticism in J. W. S. Cassels (Birch's Ph.D. advisor).

This in turn led them to make a conjecture about the behaviour of a curve's L-function L(E,s) at s = 1, namely that it would have a zero of order r at this point. This was a particularly far-sighted conjecture because at the time the analytic continuation of L(E,s) to the point s = 1 was only established for curves with complex multiplication.

The conjecture was subsequently extended to include the prediction of the residue of the zero at s = 1, which is conjectured to be a function of invariants of the curve studied by Cassels , Tate, Shafarevich and others.

3 Current status

The Birch and Swinnerton-Dyer conjecture has been proved only in special cases :-

In 1976 John Coates and Andrew Wiles proved that if E is a curve with complex multiplication and L(E,1) is not 0 then E has only a finite number of rational points, in the case of class number 1. This was extended to all imaginary quadratic fields by Nicole Artaud.
In 1983 B. Gross and D. Zagier showed that if a modular elliptic curve has a first-order zero at s = 1 then it has a rational point of infinite order.
In 1990 V. Kolyvagin showed that a modular elliptic curve E for which L(E,1) is not zero has rank 0, and a modular elliptic curve E for which L(E,1) has a first order zero at s = 1 has rank 1.
In 1999 Andrew Wiles, Christophe Breuil , Brian Conrad , Fred Diamond and Richard Taylor proved that all elliptic curves are modular (the Taniyama-Shimura theorem), which extends the previous two results to all elliptic curves.

Nothing has been proved for curves with rank greater than 1, although there is extensive numerical evidence for the truth of the conjecture.

The Birch and Swinnerton-Dyer conjecture is one of the seven Millennium Problems selected by the Clay Mathematics Institute, which is offering a prize of $1 million for a proof of the whole conjecture.

4 External links

Mathworld, http://mathworld.wolfram.com/Swinnerton-DyerConjecture.html
Clay Mathematics Institute, http://www.claymath.org/millennium/Birch_and_Swinnerton-Dyer_Conjecture/
PlanetMath.org, http://planetmath.org/encyclopedia/BirchAndSwinnertonDyerConjecture.html

<Hide>

Home > Birch and Swinnerton-Dyer conjecture

1 Background

2 History

3 Current status

4 External links

Topics: Birch And Swinnerton-Dyer Conjecture, Background, History...