Birch and Swinnerton-Dyer conjecture

It is named after mathematicians Bryan John Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation.

The modern formulation of the conjecture relates to arithmetic data associated with an elliptic curve E over a number field K to the behaviour of the Hasse–Weil L-function L(E, s) of E at s = 1.

More specifically, it is conjectured that the rank of the abelian group E(K) of points of E is the order of the zero of L(E, s) at s = 1.

The first non-zero coefficient in the Taylor expansion of L(E, s) at s = 1 is given by more refined arithmetic data attached to E over K (Wiles 2006).

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.

Helmut Hasse conjectured that L(E, s) could be extended by analytic continuation to the whole complex plane.

This conjecture was first proved by Deuring (1941) for elliptic curves with complex multiplication.

It was subsequently shown to be true for all elliptic curves over Q, as a consequence of the modularity theorem in 2001.

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.

In the early 1960s Peter Swinnerton-Dyer used the EDSAC-2 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 & 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.

This in turn led them to make a general conjecture about the behavior 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) 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[3] where the quantities on the right-hand side are invariants of the curve, studied by Cassels, Tate, Shafarevich and others (Wiles 2006):

is the Tamagawa number of E at a prime p dividing the conductor N of E. It can be found by Tate's algorithm.

the left side is now known to be well-defined and the finiteness of Ш(E) is known when additionally the analytic rank is at most 1, i.e., if

The Birch and Swinnerton-Dyer conjecture has been proved only in special cases: There are currently no proofs involving curves with a rank greater than 1.

is the following:[8]: 462 All of the terms have the same meaning as for elliptic curves, except that the square of the order of the torsion needs to be replaced by the product

A plot, in blue, of for the curve y 2 = x 3 − 5 x as X varies over the first 100000 primes. The X -axis is in log(log) scale - X is drawn at distance proportional to from 0- and the Y -axis is in a logarithmic scale, so the conjecture predicts that the data should tend to a line of slope equal to the rank of the curve, which is 1 in this case -that is, the quotient as , with C , r as in the text. For comparison, a line of slope 1 in (log(log),log)-scale -that is, with equation - is drawn in red in the plot.