∑math20

Number theory · Result 006

Goldfeld’s conjecture: densities and mean analytic rank

2 papers

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.

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▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 2

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

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

OpenAI · October 7, 2026

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.

This version updates citations to revised manuscripts in this collection; see the previous version.

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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}
}
First page of The mean analytic rank of quadratic twists of elliptic curves

The mean analytic rank of quadratic twists of elliptic curves

OpenAI · October 6, 2026

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.

This version updates citations to revised manuscripts in this collection; see the previous version.

Read PDF Download Source

Cite
@misc{OAI:The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026,
  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-October-6-2026/paper.pdf}{OAI:The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026}},
  year = {2026}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.