Goldfeld's conjecture for quadratic twists

Goldfeld's conjecture for quadratic twists predicts that, among the quadratic twists of a fixed elliptic curve over the rational numbers, analytic ranks zero and one each occur with density one half, and that the average analytic rank is one half. Two manuscripts dated 23 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claim both forms for every elliptic curve over the rationals, ordering signed squarefree twist parameters by absolute value, together with a low-corank converse theorem at the prime 2. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

Background

• As recounted in the introduction of the density manuscript:
  • Goldfeld's 1979 formulation asks for the mean analytic order of vanishing in a family of quadratic twists; the density formulation predicts that analytic ranks zero and one each occur half the time.
  • Analytic work on quadratic twists produced many nonvanishing central values and simple central zeros: Bump–Friedberg–Hoffstein, Murty–Murty, Iwaniec, Perelli–Pomykała, Ono–Skinner (a lower bound of order X divided by log X), and more recently Kumar–Mallesham–Sharma–Singh and Kriz–Nordentoft. These do not by themselves give a positive proportion for every fixed curve.
  • Positive proportions were obtained in special families: James for particular curves, Vatsal for both ranks for X₀(19), Kriz–Li for every curve with a rational 3-isogeny, and Tian, Tian–Yuan–Zhang and Smith in the congruent-number family.
  • Smith's work on the distribution of 2-power Selmer groups culminated in a theorem giving Selmer coranks zero and one with density one half each for every elliptic curve over the rationals; its analytic corollary assumes the Birch and Swinnerton-Dyer conjecture.

The original results

• Analytic density and the 2-converse (23 September 2026). For every elliptic curve over the rationals and each of the ranks zero and one, the quadratic twists with that analytic rank have density one half among signed squarefree twist parameters ordered by absolute value, so twists of analytic rank at least two have density zero. For a density-one set of parameters, the analytic and Mordell–Weil ranks agree and the whole Tate–Shafarevich group is finite.
  • Method, as described: the density theorem combines Smith's Selmer distribution theorem with a pointwise converse at 2: if the 2-power Selmer corank of an elliptic curve over the rationals is zero or one, it equals the analytic and Mordell–Weil ranks, and the Tate–Shafarevich group is finite. The manuscript describes the pointwise converse as its main work.
• Mean analytic rank (23 September 2026). For every elliptic curve over the rationals, the average analytic rank of its quadratic twists tends to one half, in the same ordering, which the manuscript describes as resolving Goldfeld's mean analytic-rank conjecture in this counting convention.
• Use in the collection. The collection's manuscripts on the Birch and Swinnerton-Dyer formula cite these results; see The Birch and Swinnerton-Dyer formula in low Selmer corank.
• Sources. Commit 1581d4c of the openai/math repository, folders Goldfeld's analytic density conjecture and the 2-converse for elliptic curves and The mean analytic rank of quadratic twists of elliptic curves.
• Formalization. The collection has no Lean scope note in lean/docs/ for this family.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 17 October 2026)
2026-10-16	J. Jardine, Lattice24, referee reviews of the density and mean-rank manuscripts in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	The referee reviews report no gap in either manuscript. The review of the density manuscript describes itself as a partial audit of a manuscript of about 9,400 lines. The review of the mean-rank manuscript notes that passing from its tail bound to the exact mean one half depends on the companion density theorem.	Part of a completed audit of the collection's manuscripts without Lean; the reviews are published under CC BY 4.0.

Notes

• The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.