Birch and Swinnerton-Dyer conjecture

The conjecture relates the number of rational points on an elliptic curve to the behaviour of its L-function near s = 1. It is a Millennium Prize problem and a central question in arithmetic geometry.

Partial results are known (e.g. for rank 0 and rank 1 curves under additional hypotheses), but the full conjecture remains open. A resolution would have major consequences for the study of Diophantine equations.