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. Specifically, it is conjectured that the Taylor expansion of L(E, s) at s = 1 is

L(C,s) = c(s-1)^r + \text{higher order terms} \, ,

where c is not zero and r is the rank of E over the field of rational numbers.^[1]

As of 2007, it has been proved only in special cases, all of rank less than or equal to 1. It has been an open problem for around 40 years, and has stimulated much research; its status as one of the most challenging mathematical questions has become widely recognized. It is one of the Millennium Prize Problems listed by the Clay Mathematics Institute, which has offered a USD 1,000,000 prize for the first correct proof.

Background

In 1922 Louis Mordell proved Mordell's theorem: 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 further rational points may be generated.

If the number of rational points on a curve is infinite then some point in a finite basis must have infinite order. The number of independent 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 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 (in the current state of knowledge) 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 form. It is a special case of a Hasse-Weil L-function.

The natural definition of L(E, s) only converges for values of s in the complex plane with Re(s) > 3/2. Helmut Hasse conjectured that L(E, s) could be extended by analytic continuation to the whole complex plane. This conjecture was first proved by Max Deuring for elliptic curves with complex multiplication. It was subsequently shown to be true for all elliptic curves, as a consequence of the modularity theorem.

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.

History

In the early 1960s Peter Swinnerton-Dyer used the EDSAC computer at the University of Cambridge Computer Laboratory 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 Bryan Birch and Swinnerton-Dyer conjectured that N_p for a curve E with rank r obeys an asymptotic law

\prod_{p<x} \frac{N_p}{p} \approx \log(x)^r \mbox{ as } x \rightarrow \infty.

Initially this was on the basis of somewhat tenuous trends in graphical plots; which induced a measure of skepticism in J. W. S. Cassels (Birch's Ph.D. advisor). Over time the numerical evidence stacked up.

This in turn led them to make a general 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 far-sighted conjecture for the time, given that the analytic continuation of L(E, s) there was only established for curves with complex multiplication, which were also the main source of numerical examples. (NB that the reciprocal of the L-function is from some points of view a more natural object of study; on occasion this means that one should consider poles rather than zeroes.)

The conjecture was subsequently extended to include the prediction of the precise leading Taylor coefficient of the L-function at s = 1. It is conjecturally given by

\frac{L^{(r)}(E,1)}{r!} = \frac{\#Sha(E)\Omega_E R_E \prod_{p|N}c_p}{(\#E_{Tor})^2}

where the quantities on the right hand side are invariants of the curve, studied by Cassels, Tate, Shafarevich and others: these include the order of the torsion group, the order of the Tate-Shafarevich group, and the canonical heights of a basis of rational points..^[1]

Current status

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

1. In 1976 John Coates and Andrew Wiles proved that if E is a curve over a number field F with complex multiplication by an imaginary quadratic field K of class number 1, F=K or Q, and L(E,1) is not 0 then E has only a finite number of rational points. This was extended to the case where F is any finite abelian extension of K by Nicole Arthaud-Kuhman, who shared an office with Wiles when both were students of Coates at Stanford.
2. In 1983 Benedict Gross and Don 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; see Gross-Zagier theorem.
3. In 1990 Victor 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.
4. In 1991 Karl Rubin showed that for elliptic curves defined over an imaginary quadratic field K with complex multiplication by K, if the L-series of the elliptic curve was not zero at s=1, then the p-part of the Tate-Shafarevich group had the order predicted by the Birch and Swinnerton-Dyer conjecture, for all primes p > 7.
5. In 1999 Andrew Wiles, Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor proved that all elliptic curves defined over the rational numbers are modular (the Taniyama-Shimura theorem), which extends results 2 and 3 to all elliptic curves over the rationals.

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

Clay Mathematics Institute Prize

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 the first proof of the whole conjecture.^[2]

References

External links

• The Birch and Swinnerton-Dyer Conjecture: An Interview with Professor Henri Darmon by Agnes F. Beaudry
• QEDen Millennium Prize Problems Wiki

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