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 Np) for a large number of primes p on elliptic curves whose rank was known. From these numerical results Birch and Swinnerton-Dyer (1965) conjectured that Np for a curve E with rank r obeys an asymptotic law
where C is a constant.
Initially this was based on somewhat tenuous trends in graphical plots; this 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
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 (Wiles 2006).
Read more about this topic: Birch And Swinnerton-Dyer Conjecture
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