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.

References

In the twentieth century, Birch and Swinnerton-Dyer plotted, using the earliest computers of the 1960s, ranks and other quantities for elliptic curves, and conjectured that the order of vanishing of the L-function $L(s)$ for the curve at $s \rightarrow 1$ equals to the rank. This observation is the the now celebrated BSD Conjecture that bears their name; it is a Millennium Prize problem and central to modern mathematics.

— A Triumvirate of AI Driven Theoretical Discovery  (2405.19973 - He, 2024) in Section: Top-Down Mathematics

One needs to bear in mind that two of the remaining six Millennium Problems - the Riemann Hypothesis and the Birch--Swinnerton-Dyer (BSD) Conjecture - arose from mathematical experimentation by listing the first zeroes or by plotting rank and conductor.

— Mathematics: the Rise of the Machines  (2511.17203 - He, 21 Nov 2025) in Section 2, Top-Down Mathematics

Therefore, in spite of various decades of high-level research, the conjecture is still widely open.

— Introduction to the Birch and Swinnerton-Dyer Conjecture  (2608.25975 - Ruffino, 26 Aug 2026) in Section 4, The Conjecture; current status in Section 4, subsection “Current Status”