Birch and Swinnerton-Dyer Conjecture (BSD): OpenAI Claims Full Formula, Rank 0 and 1

About this video

The Birch and Swinnerton-Dyer conjecture (BSD conjecture), one of the Millennium Prize problems, says the number of rational points on an elliptic curve is controlled by its L-function: y² = x³ − x + 1 has infinitely many (rank 1), while y² = x³ − x has only four (rank 0). OpenAI claims a proof of the full BSD leading-term formula for every elliptic curve over ℚ whose q-power Selmer group has corank 0 or 1 for some prime q, including finiteness of the Tate–Shafarevich group; with a companion result this gives full BSD for a density-one set of quadratic twists of every curve. This is OpenAI's claim and has not yet been peer reviewed; ranks 2 and higher remain open.

Every point shown is computed exactly: the multiples of (0, 1), their chords, and their positions on the torus by elliptic logarithm.

The animation illustrates the result; it is not a proof.

Papers:
- Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank: github.com/openai/math/blob/main/preprints/Exact-Birch-Swinne…
- The Selmer converse for elliptic curves at every prime: github.com/openai/math/blob/main/preprints/The-Selmer-convers…
- The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one: github.com/openai/math/blob/main/preprints/The-two-primary-Bi…

Original visuals (three.js) and synthesized sound: every chime is a rational point, pitched by its height.

Gradient Damage: mathematics you can see, equations you can hear.
youtube.com/@gradientdamage