Goldfeld's conjecture on analytic ranks of quadratic twists

For an elliptic curve $E/\mathbb{Q}$ and squarefree $d$ let $E^{(d)}$ be the quadratic twist and $a(E^{(d)})=\mathrm{ord}_{s=1}L(E^{(d)},s)$ its analytic rank. Goldfeld (1979, Conjecture B) conjectured that the average analytic rank over the twist family is $1/2$ ; the density form predicts that analytic ranks 0 and 1 each occur for half of all twists. Known before: positive proportions for special curves (James, Vatsal, Kriz-Li for curves with a rational 3-isogeny), conditional means under GRH (Heath-Brown, Fiorilli), and Smith's theorem that 2-power Selmer coranks 0 and 1 each have density 1/2, giving Goldfeld's conjecture under BSD. For every $E/\mathbb{Q}$ , do analytic ranks 0 and 1 each have density $1/2$ among the twists, and does the mean analytic rank tend to $1/2$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Arithmetic of elliptic curves; arithmetic statistics
Posed by: Dorian Goldfeld, Conjectures on elliptic curves over quadratic fields, Number Theory Carbondale 1979, LNM 751, p. 113, Conjecture (B)
Year posed: 1979
Years open: 47y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 55 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Density paper: for every $E/\mathbb{Q}$ and $j\in\{0,1\}$ , the proportion of squarefree $0<|d|\le X$ with $a(E^{(d)})=j$ tends to $1/2$ ; on a density-one set analytic and Mordell-Weil ranks agree and Sha is finite. The main new input is the 2-converse: if the $2^\infty$ -Selmer corank is 0 or 1 it equals the analytic and Mordell-Weil ranks and Sha is finite. Mean paper: $\frac{1}{\#\mathcal D(Y)}\sum_{d}a(E^{(d)})\to1/2$ , via a tail bound on high-rank twists, plus algebraic-rank moments. Both use signed squarefree twists ordered by $|d|$ , not Goldfeld's discriminant ordering; no higher analytic-rank moments are claimed.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has two manuscripts dated 23 September 2026: the density paper (principal) and the mean analytic rank paper, which adds a tail estimate. The density paper's key input is the pointwise 2-converse, combined with Smith's human Selmer-distribution theorem. The README also cautions that unformalized results could have issues.

Verification

No independent mathematician has checked this yet. Checked here: the abstracts, introductions and main theorems of both manuscripts were read against Goldfeld's Conjecture (B) as the manuscripts cite it. The proofs were not refereed. No Lean formalization: neither manuscript is in lean/formalization.yaml and lean/docs/006.md does not exist at the pinned commit. Conditional inputs: the density theorem uses Smith's distribution of 2-power Selmer coranks (arXiv 2503.17619, cited as version 1); the mean uses the density theorem plus a new tail estimate. Scope the papers state: twists are counted over signed squarefree $d$ ordered by $|d|$ , whereas Goldfeld's Conjecture (B) orders quadratic-field discriminants; the mean paper says it resolves the conjecture 'in this counting convention'. The density manuscript cites companions dated 24 September and the mean paper cites a 'V2' of the density paper, so texts were revised after their dates.

Goldfeld's conjecture on analytic ranks of quadratic twists

What was actually shown

What the AI did

Verification

Sources

Discussion