Birch and Swinnerton-Dyer Conjecture
As of 2005, it has been proved only in special cases, such as over certain quadratic fields (Henri Darmon, of McGill University). 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 recognised. It is one of the Clay Mathematics Institute's seven Millennium Prize Problems.
In 1922 Louis Mordell proved Mordell's Theorem: the group of rational points on an elliptic curve is finitely generated. This means that for any elliptic curve there is a finite subset 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.
can be defined for an elliptic curve E by constructing an Euler product from the number of points on the curve modulo each prime . This -function is analogous to the Riemann zeta function and the Dirichlet -series that is defined for a binary quadratic form. It is a special case of a Hasse-Weil -function.
The natural definition of only converges for values of in the complex plane with
Helmut Hasse conjectured that 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 Taniyama-Shimura theorem.
Finding rational points on a general elliptic curve is a difficult problem. Finding the points on an elliptic curve modulo a given prime is conceptually straightforward, as there are only a finite number of possibilities to check. However, for large primes it is computationally intensive.
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 (denoted by ) for a large number of primes on elliptic curves whose rank was known. From these numerical results Bryan Birch and Swinnerton-Dyer conjectured that for a curve with rank 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). Over time the numerical evidence stacked up.
This in turn led them to make a general conjecture about the behaviour of a curve's -function at , namely that it would have a zero of order at this point. This was a far-sighted conjecture for the time, given that the analytic continuation of there was only established for curves with complex multiplication, which were also the main source of numerical examples. (NB that the reciprocal of the -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 -function at . It is conjecturally given by a complex formula involving 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. 
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 with complex multiplication and is not 0 then 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 Arthaud.
2. In 1983 Benedict Gross and Don Zagier showed that if a modular elliptic curve has a first-order zero at then it has a rational point of infinite order; see Gross-Zagier theorem.
3. In 1990 Victor Kolyvagin showed that a modular elliptic curve for which is not zero has rank 0, and a modular elliptic curve for which has a first-order zero at has rank 1.
4. In 1991 Karl Rubin showed that for elliptic curves defined over an imaginary quadratic field with complex multiplication by , if the -series of the elliptic curve was not zero at , then the -part of the Tate-Shafarevich group had the order predicted by the Birch and Swinnerton-Dyer conjecture, for all primes .
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 the previous two results 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.