Goldfeld's Conjecture: Half Rank 0, Half Rank 1

About this video

Goldfeld's conjecture says that if you take one elliptic curve and twist it by every squarefree number d, half of the twists have analytic rank 0, half have analytic rank 1, and the average rank tends to 1/2. OpenAI claims a proof for the quadratic twists of every elliptic curve over the rationals (twist parameters ordered by size), together with the mean analytic rank result; these are OpenAI's preprints, not yet peer reviewed. The picture is a real computation for the curve 11a1 (y² + y = x³ − x² − 10x − 20): each line is the completed L-function of one of its first 120 twists on the real segment from s = 1/2 to 3/2, and the counters are the true rank proportions of its first 1000 twists (44% / 50% / 6% with rank 2 or more, which the theorem says becomes negligible only slowly).

Papers (OpenAI, catalogue result 006):
- Goldfeld's analytic density conjecture and the 2-converse for elliptic curves: github.com/openai/math/blob/main/preprints/Goldfelds-analytic…
- The mean analytic rank of quadratic twists of elliptic curves: github.com/openai/math/blob/main/preprints/The-mean-analytic-…

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

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Every result explained: gradientdamage.com/p/006