Birch–Swinnerton–Dyer Conjecture

Prove the Birch–Swinnerton–Dyer Conjecture for elliptic curves, as formulated in the Millennium Problems list, establishing the precise relationship between the rank of an elliptic curve and the order of vanishing of its L-function at s = 1.

Background

The paper highlights that some of the most important unsolved problems in mathematics emerged from experimental, data-driven insights. In this context, it notes that the Birch–Swinnerton–Dyer (BSD) Conjecture is one of the remaining Millennium Problems and arose from computational experimentation.

By invoking the BSD Conjecture explicitly in the discussion of top-down (experimental) mathematics, the author underscores its continued status as a central open problem and an exemplar of how empirical patterns can lead to deep theoretical questions.