The Birch and Swinnerton‑Dyer Conjecture

This Millennium Problem connects elliptic curves (y² = x³ + ax + b) to L‑functions. It says that the rank of the elliptic curve (the number of independent rational points) equals the order of vanishing of its L‑function at s=1. If the L‑function has a zero of order r at s=1, then there are infinitely many rational points when r>0. This is one of the deepest unsolved problems in number theory. It has been verified numerically for thousands of elliptic curves. A proof would revolutionise Diophantine geometry.