Birch and Swinnerton-Dyer Conjecture

Establish the Birch and Swinnerton-Dyer Conjecture by proving that for elliptic curves the order of vanishing of the associated L-function L(s) at s = 1 equals the rank of the curve.

Background

Within a discussion of top-down mathematics and the role of pure, noiseless data in guiding discovery, the paper highlights historical examples where computational exploration led to major conjectures. In particular, after describing how Gauss’s data-driven insight led to the Prime Number Theorem, the paper turns to Birch and Swinnerton-Dyer’s computer-assisted investigations of elliptic curves.

The authors explicitly state the Birch and Swinnerton-Dyer Conjecture and emphasize its status and importance, noting that it is a Millennium Prize problem and central to modern mathematics. This contextualizes the conjecture as a canonical example of how data-driven intuition can yield profound, enduring open problems.