Birch and Swinnerton-Dyer Conjecture

Unsolved Problem
When the solution to a Diophantine equation in polynomials are the points of an Abelian variety, the order of the group of rational points is related to the behavior of an associated $\zeta$ (zeta) function $\map \zeta s$ near $s = 1$.

In particular:
 * if $\map \zeta 1 = 0$ then there is an infinite set of rational points
 * if $\map \zeta 1 \ne 0$ then there is a finite set of rational points.

Progress
As of now, this conjecture has been confirmed only for special cases.

Also known as
This is also presented as the Birch–Swinnerton-Dyer conjecture, where the first hyphen is longer than the second.

This style of presentation is not endorsed on, as the long dash is not simple to implement.