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The Birch-Swinnerton-Dyer formula in Selmer corank at most one

For an elliptic curve E/QE/\mathbb{Q} of rank rr, the Birch and Swinnerton-Dyer conjecture predicts ords=1L(E,s)=r\mathrm{ord}_{s=1}L(E,s)=r, finiteness of the Tate-Shafarevich group, and the leading-term formula L(r)(E,1)/r!=ΩE RegE #Sha(E/Q)∏ℓcℓ(E)/#E(Q)tors2L^{(r)}(E,1)/r! = \Omega_E\,\mathrm{Reg}_E\,\#Sha(E/\mathbb{Q})\prod_\ell c_\ell(E)/\#E(\mathbb{Q})_{tors}^2. Gross-Zagier and Kolyvagin gave the rank equality and finiteness of Sha when the analytic rank is at most one, but the exact leading-term formula in that range was known only at particular primes and under reduction, residual or CM hypotheses (Skinner-Urban, Jetchev-Skinner-Wan, Rubin, Burungale-Flach). Does the full leading-term formula, at every prime, hold for every elliptic curve over Q\mathbb{Q} whose q∞q^\infty-Selmer group has corank 00 or 11 for some prime qq?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Arithmetic of elliptic curves; Iwasawa theory
Posed by
Bryan Birch and Peter Swinnerton-Dyer (the conjecture); Tate's 1966 Bourbaki account fixed the leading-term formula
Year posed
1965
Years open
61y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
78 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

For every E/QE/\mathbb{Q} and prime qq with sq(E)=corank Selq∞(E/Q)∈{0,1}s_q(E)=\mathrm{corank}\,\mathrm{Sel}_{q^\infty}(E/\mathbb{Q})\in\{0,1\}: rank E(Q)=ords=1L(E,s)=sq(E)\mathrm{rank}\,E(\mathbb{Q})=\mathrm{ord}_{s=1}L(E,s)=s_q(E), Sha(E/Q)Sha(E/\mathbb{Q}) is finite, and the full BSD leading-term formula holds at every prime, with no hypotheses on reduction, torsion, isogenies, CM or residual representations. By Gross-Zagier-Kolyvagin this covers every curve of analytic rank at most one (curator's reading). The new step is the exact pp-adic valuation at every odd pp; p=2p=2 comes from the two-primary companion. With family 006 it gives full BSD for a density-one set of quadratic twists of each curve. It says nothing about analytic rank ≥2\geq 2.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The principal manuscript (October 3, 2026) builds on two companion manuscripts in this family and on the release's Goldfeld-density manuscript (family 006).

Verification

No independent mathematician has checked this yet. Checked here: the main theorem of the October 3 manuscript was read against the posed BSD leading-term formula, including its period, regulator, Tamagawa and torsion normalizations. The proof was not refereed. No Lean formalization is listed in lean/formalization.yaml for any manuscript in this family. The result is conditional on unreviewed companion inputs from the same release: the Selmer converse at every prime and the two-primary formula (this family), and the Goldfeld analytic-density theorem (family 006). The rank-zero CM case uses Burungale-Flach. The paper states the low-Selmer-corank hypothesis is essential to its scope.

Sources

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