Birch and Swinnerton-Dyer conjecture

The conjecture of Birch and Swinnerton - Dyer is one of the most important unsolved problems of modern mathematics and makes statements about number theory on elliptic curves.

Formulation

The conjecture says something about the rank of elliptic curves. Elliptic curves are given by equations of the third degree in x and y in the second degree, the " discriminant " D does not disappear. On these curves can be rational points after a by Henri Poincaré in 1901 examined " secant - tangent method " add so that the result is a rational point on the curve again. This " addition" is geometrically defined as follows: you put a line through two rational points P and Q. Cuts the line crosses the curve at a third point, one reflects it to the x- axis, which again provides a point on the curve, because it is symmetrical to the x-axis. The rational point thus obtained the curve is the sum of P Q As a neutral element "0", the point is at infinity ( projective plane ). The mirror point P on the curve is its inverse. In the case that the line through P, Q has no third intersection point on the curve, but the point is taken at infinity, and the addition is as follows: P = P 0 One can also form the P P, by taking the intersection of the tangent at P as the second item in the addition structure. Behind this construction is the fact that elliptic curves Riemann surfaces of the form of a torus have ( genus 1 ), ie geometric lattice and are thus are additive groups, which carries over to their behavior in the rational numbers or finite fields. The existence of such a strange kind of addition is also exploited in the so-called " elliptic curve" primality tests and " public key " encryption method in cryptography. For this you need curves with as many rational points and uses the difficulty of finding the initial data for the additive generation of large rational points of the curve. See elliptic curve cryptosystems.

If you added as a rational starting point P0 to himself, one obtains a sequence of points:

And so on.

Now, two cases may occur - of course on the same curve at different rational points:

In her assumption Bryan Birch and Peter Swinnerton - Dyer passed to a method as can be determined from the equation of the elliptic curve whose rank. It results from the consideration of the L- function L (E, s) is the function of the studied elliptic curve E and a complex variable s, the L- function is analogous to the Riemann zeta function defined, only one is now on the prime page - so the Euler product - in addition and encoded in the series, the number of solutions of the elliptic curve modulo a prime p:

With the number of solutions mod p. L (E, S) in the form of a real number of zeta function ( as the sum of the natural numbers ), it converges on the real parts of s ≥ 3 / second One can now investigate whether they can be continued analytically across s if it satisfies a functional equation, where their zeros lie, etc. As with the Riemann zeta function for the primes is obtained from L (E, s) information about the asymptotic distribution of solutions (mod p for large p). Birch and Swinnerton - Dyer examined the solutions in the 1960s with the computer and formulating their famous conjecture for the asymptotic distribution of the number N ( p) of points on E over finite fields F (p) mod p thus:

So you connect the product of local densities ( each finite field have a maximum of p elements ) over the primes with the asymptotic logarithmic distribution ( with an exponent r since r " natural numbers " are present). Translated into the language of the zeta function L (E, s), it says that the order of the zero of L (E, s) at s = 1 - if the function has a there - equal to the rank r of the group of rational points is. This of course has to be proved that L can be analytically continued to s = 1, so that L is there developable in a Taylor series. There is also a more detailed version = 1 sets the coefficients of the Taylor expansion at s related to arithmetic objects such as the order of the Tate - Shafarevich group, " local factors ", the real period of the curve and the order of the Torsionsgruppen.

From the conjecture of Birch - Swinnerton - Dyer are some more sentences of number theory, for example, Jerrold Tunnell, the problem of determination of congruent figures.

Status

The conjecture has been proved only in special cases:

For curves with groups that have a rank r > 1, nothing has been proven, but there is strong numerical arguments for the correctness of the assumption.

The proof of the conjecture of Birch still open and Swinnerton - Dyer was taken from the Clay Mathematics Institute in their list of the Millennium problems.