Browse › Number theory › 006 Goldfeld’s conjecture: densities and mean analytic rank
Goldfeld's analytic density conjecture and the 2-converse for elliptic curves
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
Abstract
We prove Goldfeld's analytic density conjecture: for every elliptic curve E over ℚ, the quadratic twists of E with analytic rank zero and one each have density 1/2 among signed squarefree twist parameters ordered by absolute value. We also prove the low-corank 2-converse: if the $2^\infty$-Selmer corank of E is zero or one, then it equals the analytic and Mordell–Weil ranks, and the Tate–Shafarevich group is finite.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- OpenAI, The Selmer converse for elliptic curves at every prime (OpenAI Math Release companion manuscript, 7 October 2026). This companion uses the density paper’s 2-converse for the prime two and its analytic twist-density theorem to select auxiliary fields in the proposed odd-prime argument. [AI-generated per source]
- OpenAI, Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank (OpenAI Math Release companion manuscript, 7 October 2026). This companion invokes the analytic twist-density theorem to choose two imaginary quadratic fields in its proposed proof of the full BSD formula from low Selmer corank. [AI-generated per source]
Comments and discussion
- holimart, Report: mod-8 congruence (eq. pw:split-mod8) in Lemma pw:split-witness fails numerically — Goldfeld / 2-converse manuscript (GitHub issue #2, 8 October 2026). This report gives numerical failures of the displayed congruence while acknowledging that it is derived under an additional branch assumption and that the computations are not a counterexample to Theorem 1.1. [cites earlier version]
- borism, Check of holimart/math#2: mod-8 congruence in Goldfeld 2-converse manuscript (numerics real, not a disproof) (GitHub issue #1 and discussion, 8 October 2026). This Claude-assisted check reproduces the congruence failures but reports equality witnesses for every explicitly named pair, and withdraws proposed local counterexamples because its prime-matching data did not capture the proof’s full finite quotient. [AI-generated per source] [cites earlier version]
Formalization
- mo271 (google-deepmind/formal-conjectures), Wikipedia/EllipticCurveRank:
goldfeld_conjecture.variants.squarefreeis true (issue #6916, opened 2026-10-07, open, label "solution found"). Proposes marking the squarefree Goldfeld statement in the Lean library as solved, citing the companion two-primary BSD manuscript, Smith's distribution theorem and this paper; maintainers reply (2026-10-08) that a status change requires consensus that the proof works, e.g. via formalization, peer review or a digest. [AI-generated per source] - google-deepmind/formal-conjectures, FormalConjectures/Wikipedia/EllipticCurveRank.lean (Lean file, main branch, read 2026-10-21). Contains
goldfeld_conjectureandgoldfeld_conjecture.variants.squarefree, both taggedresearch open, stating the 1/2–1/2 density of ranks 0 and 1 for twists ordered by |d|; thetwo_levariant is marked solved via Smith (2025), with a docstring saying the analytic-rank version remains open.
Related prior literature mentioned by others
- Alexander Smith, The Birch and Swinnerton-Dyer conjecture implies Goldfeld’s conjecture (arXiv:2503.17619, 22 March 2025). Formal-conjectures issue #6916 and the AI Proved What? family exposition both highlight this theorem on the half-densities of 2-power Selmer coranks zero and one as the statistical input combined with the 2-converse; the paper cites it as its Theorem 1.1 input.
Possibly relevant literature (found by the agent, not checked)
- Ashay Burungale and Ye Tian, The even parity Goldfeld conjecture: congruent number elliptic curves (arXiv:2104.06732, 2021). An expository survey of progress on the analytic-rank-zero half of Goldfeld's conjecture in the congruent-number family, the CM case discussed in the paper's introduction alongside Burungale–Tian (2026).
- Jeffrey Hatley and Anwesh Ray, Iwasawa theory and ranks of elliptic curves in quadratic twist families (arXiv:2412.07308, December 2024). Uses 2-adic Iwasawa invariants to produce rank-zero and rank-one quadratic twists with lower bounds close to the conjectured proportions, a 2-adic approach related to the paper's 2-converse and density results.
Digests and expositions
- AI Proved What?, Half and half: Goldfeld’s conjecture for elliptic curve twists (episode 006, October 2026; status checked 19 October 2026). This AI-assisted video and transcript explain the claimed 2-converse, the separate large-rank tail estimate needed for the mean, the signed-squarefree counting convention, and the numerical discussion of the density proof. [AI-generated per source] [family-level]
News and blogs
- Azmat (BinaryVerse AI), Goldfeld’s Conjecture: Why OpenAI’s 50–50 Claim Could Change Number Theory (blog post, 10 October 2026). A non-technical exposition of both manuscripts, separating Smith’s Selmer statistics, the proposed interpolation argument for the 2-converse, and the tail estimate required for mean analytic rank; the author reports finding no completed independent verification as of that date. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 006
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026,
author = {{OpenAI}},
title = {{Goldfeld's analytic density conjecture and the $2$-converse for elliptic curves}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf}{OAI:Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026}},
year = {2026}
}Versions
- Earlier edition: September 23, 2026
This version updates citations to revised manuscripts in this collection; see the previous version.
Family 006: Goldfeld’s conjecture: densities and mean analytic rank
Proves Goldfeld's conjecture for quadratic twists of every elliptic curve over ℚ: analytic ranks zero and one each have density 1/2, and the mean analytic rank tends to 1/2. Both statements order signed squarefree twist parameters by absolute value.
Other papers in this family:
- The mean analytic rank of quadratic twists of elliptic curves · October 6, 2026