Result 006, Number theory

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.

Proof

The bigger picture

Why it matters

The manuscripts report a universal pattern in families of elliptic curves: asymptotically, half have analytic rank zero and half have analytic rank one. They also claim that rare higher ranks do not raise the limiting average.

What changes?

For every elliptic curve over the rational numbers, the claims concern quadratic twists, related curves indexed by signed squarefree integers, which are divisible by no prime square. Counting both signs together by increasing absolute value, the proportions with analytic ranks zero and one each tend to one-half, and the average analytic rank tends to one-half. Analytic rank is the order of vanishing at the central point of the curve's L-function, an analytic function encoding arithmetic information.

What does that help mathematicians do?

Together, the claims imply that twists of rank at least two have density zero and contribute a vanishing amount to the overall average. This second conclusion does not follow from density zero alone: an increasingly sparse set could still carry increasingly large ranks. Researchers would therefore gain control over both the frequency of higher analytic ranks and their aggregate contribution, without any claimed bound on individual ranks.

Are there practical applications?

The immediate value is foundational: the results would give an exact statistical benchmark for studying L-functions across every quadratic-twist family over the rational numbers. These functions are central to questions about rational solutions on elliptic curves, but the stated density and mean claims concern analytic rank, not an algorithm for finding those solutions.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Goldfeld's analytic density conjecture and the 2-converse for elliptic curves

September 23, 2026 130 pages

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∞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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026,
  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-September-23-2026/paper.pdf}{OAI:Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026}},
  year = {2026}
}

The mean analytic rank of quadratic twists of elliptic curves

September 23, 2026 41 pages

For every elliptic curve over ℚ, we prove that the average analytic rank of its quadratic twists tends to 1/2 when signed squarefree twist parameters are ordered by absolute value. This resolves Goldfeld's mean analytic-rank conjecture in this counting convention.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-September-23-2026,
  author = {{OpenAI}},
  title = {{The mean analytic rank of quadratic twists of elliptic curves}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-September-23-2026/paper.pdf}{OAI:The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-September-23-2026}},
  year = {2026}
}

Posts about this result

Aran Nayebi @aran_nayebi, Assistant Professor, Carnegie Mellon

Ok update, yes *this* updates my timelines! github.com/openai/math/blob/main/overview.pdf Rational Hodge over CM abelian varieties is true (032) and BSD for a density-one set (002 & 006). Also Hilbert's tenth problem over Q is false (004), just as everyone expected, but we didn't have a proof for!

Quoting @aran_nayebi: If either Hodge or BSD are proven to be *true* by AI (thereby likely using deep mathematical techniques), then this would update my timelines. This may also mean the Riemann Hypothesis is not far off. But if it's a be...

Oct 6, 2026, 6:40 PM ET

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.