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The Birch and Swinnerton-Dyer formula in low Selmer corank

From OpenAI Aftermath

The Birch and Swinnerton-Dyer formula in low Selmer corank concerns elliptic curves over the rational numbers whose p-power Selmer group has corank zero or one for some prime p. Three manuscripts dated 24 September and 3 October 2026 in OpenAI's public mathematics collection (github.com/openai/math) claim, for every such curve, equality of the analytic rank, the Mordell–Weil rank and the Selmer corank, finiteness of the Tate–Shafarevich group, and the full Birch and Swinnerton-Dyer leading-term formula. One manuscript derives from this that every prime congruent to 4, 7 or 8 modulo 9 is a sum of two rational cubes. 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 3 October manuscript:
    • The modularity theorems of Wiles, Taylor–Wiles and Breuil–Conrad–Diamond–Taylor give the analytic continuation and functional equation of the L-function of every elliptic curve over the rationals. Gross–Zagier's height formula and Kolyvagin's Euler system give rank equality and finiteness of the Tate–Shafarevich group in analytic ranks zero and one, but do not identify every prime valuation of the leading term.
    • Integral refinements come from Iwasawa theory: Rubin's main conjecture for CM curves, Kato's zeta elements, Skinner–Urban in rank zero under residual and ramification hypotheses, and Jetchev–Skinner–Wan in rank one for semistable curves at good primes under irreducibility hypotheses. Burungale and Flach prove the full rank-zero formula for a class of CM curves.
  • As recounted in the section on cube sums of the 24 September converse manuscript: Dasgupta and Voight proved that primes congruent to 4 or 7 modulo 9 are sums of two rational cubes when 3 is not a cube modulo the prime; Yin removed this restriction; Burungale and Tian proved the remaining case 8 and stated the full prime cube-sum theorem; Kriz had stated the ramified-prime CM converse and its cube-sum consequence in an earlier preprint.

The original results

  • Selmer converse at every prime (24 September 2026). For every elliptic curve over the rationals and every prime p, if the p-power Selmer group has corank r equal to 0 or 1, then the analytic rank and the Mordell–Weil rank both equal r and the whole Tate–Shafarevich group is finite. As an application at the additive prime 3, for every prime ℓ congruent to 4, 7 or 8 modulo 9 the cubic X3 + Y3 = ℓZ3 has analytic and Mordell–Weil rank one and finite Tate–Shafarevich group, so every such prime is a sum of two rational cubes. The manuscript describes this deduction as a uniform one from its main theorem at 3, using Satgé's paired isogeny descent.
  • Two-primary formula (24 September 2026). The two-primary leading-term formula holds for every elliptic curve over the rationals whose two-power Selmer group has corank at most one. Combined with the distribution of Selmer groups in quadratic twist families, this gives the exact two-primary formula for a density-one set of signed squarefree twists of each fixed curve, with ranks zero and one each of density one half.
  • Full formula (3 October 2026). Theorem 1.1 of the manuscript states that if some prime q has q-power Selmer corank 0 or 1, then the ranks equal that corank, the Tate–Shafarevich group is finite, and the leading coefficient of the L-function at the center equals the product of the real period, the regulator, the order of the Tate–Shafarevich group and the Tamagawa factors, divided by the square of the order of the rational torsion. The manuscript states that there are no further hypotheses on reduction, rational torsion, isogenies, complex multiplication or residual Galois representations. It takes rank equality and finiteness from the converse manuscript and the two-primary equality from the companion, and describes the exact valuation at every odd prime as its new contribution. It also cites a twist-density theorem from the collection's manuscript on Goldfeld's conjecture and the rank-zero CM formula of Burungale and Flach.
  • With the collection's Goldfeld results. The collection's contents list states that, combined with its result on Goldfeld's conjecture, this gives the full Birch and Swinnerton-Dyer formula for a density-one set of quadratic twists of every elliptic curve over the rationals.
  • Sources. Commit 1581d4c of the openai/math repository, folders The Selmer converse for elliptic curves at every prime, The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one and Exact Birch–Swinnerton-Dyer formula from low Selmer corank.
  • 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)
2026-10-07 Ruihan Chen, arXiv:2610.09370 (v1), titled On the algebraic and analytic ranks of the twin-prime elliptic curve y² = x(x−2)(x−p) Use (conditional) For primes p at least 7 with p − 2 also prime, a two-Selmer calculation gives two-power Selmer corank one for p congruent to 3 or 5 modulo 8 and zero for p congruent to 7 modulo 8. Assuming the low-corank Birch and Swinnerton-Dyer statement announced in the OpenAI release, the abstract deduces equality of the analytic and algebraic ranks, finiteness of the Tate–Shafarevich group and the exact leading-term formula in these cases, and identifies the obstruction left when p is congruent to 1 modulo 8. arXiv v1.
2026-10-08 atimics/math2, Cross-family corollaries of openai/math, entries NT-8 and NT-9 (GitHub, commit 2c0f005) Use (conditional corollaries) Entry NT-8 combines the converse theorem at the prime 3 with a conditional corollary of Koymans and Smith (arXiv:2405.09311) on the 3-Selmer groups of the curves x3 + y3 = n, and concludes that at least 47.92% of positive integers are sums of two rational cubes. Entry NT-9 combines the full formula at the prime 5 with Bhargava and Shankar's bounds on 5-Selmer groups, and concludes that the full leading-term formula holds for at least 83.75% of elliptic curves over the rationals ordered by height. Repository catalogue; it states that every entry is conditional on the OpenAI claims it cites, that the entries are written derivations without formalization, and that they were rechecked by two independent agents.
2026-10-09 Hongbo Yin, arXiv:2610.08015 (v2), titled A concise proof of the 8 case of Sylvester's conjecture Related work A shorter proof of the case 8 of Sylvester's conjecture, building on the author's earlier work on the cases 4 and 7 and using the ordinary trace and the Kronecker congruence of modular functions. The paper does not mention the OpenAI manuscripts. arXiv v2; the paper states that the details of its realization were produced with AI assistance.
2026-10-16 J. Jardine, Lattice24, referee reviews of the converse, two-primary and full-formula manuscripts in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) Review (AI-assisted audit) The referee reviews report no gap in any of the three manuscripts. The review of the converse manuscript states that its automorphic half is too compressed to check line by line, and that the correctness of the main theorem rests on parts that the review could not verify. 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.
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