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The p-converse to Gross-Zagier-Kolyvagin in Selmer corank at most one, at every prime

Let E/QE/\mathbb{Q} be an elliptic curve, pp a prime, and sp(E)s_p(E) the Zp\mathbb{Z}_p-corank of the full p∞p^\infty-Selmer group. Gross-Zagier and Kolyvagin show that analytic rank r≤1r\le 1 forces rank E(Q)=r\mathrm{rank}\,E(\mathbb{Q})=r and finite Sha. The pp-converse predicts ords=1L(E,s)=sp(E)\mathrm{ord}_{s=1}L(E,s)=s_p(E) (Keller-Yin, Conjecture A). Known cases (Skinner-Urban, Skinner, Zhang, Burungale-Castella-Skinner, Castella-Wan, Keller-Yin and others) need ordinary or supersingular reduction, residual irreducibility or Eisenstein conditions, CM, or p>3p>3. If sp(E)=r∈{0,1}s_p(E)=r\in\{0,1\}, does it follow for every EE and every pp that the analytic rank and Mordell-Weil rank equal rr and Sha is finite?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Arithmetic of elliptic curves; Heegner points
Posed by
Converse problem developed by Skinner-Urban, Skinner and Wei Zhang; stated as the p-converse conjecture (Conjecture A) by Timo Keller and Mulun Yin
Year posed
2014
Years open
12y
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
45 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

For every elliptic curve E/QE/\mathbb{Q}, every prime pp and r∈{0,1}r\in\{0,1\}: sp(E)=rs_p(E)=r implies ords=1L(E,s)=rank E(Q)=r\mathrm{ord}_{s=1}L(E,s)=\mathrm{rank}\,E(\mathbb{Q})=r and the whole of Sha(E/Q)Sha(E/\mathbb{Q}) is finite, with no restriction on reduction type, CM, residual representation, torsion or isogenies. It does not address coranks ≥2\geq 2 and does not evaluate the BSD leading term (that is the companion entry). As an application it rederives that X3+Y3=ℓZ3X^3+Y^3=\ell Z^3 has rank one for primes ℓ≡4,7,8(mod9)\ell\equiv 4,7,8 \pmod 9.

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 prime p = 2 is taken from the release's Goldfeld-density manuscript (family 006).

Verification

No independent mathematician has checked this yet. Checked here: the main theorem of the Selmer converse manuscript was read against the converse question as the manuscript and Keller-Yin state it, for coranks zero and one only. The proof was not refereed. No Lean formalization is listed for this manuscript. The case p = 2 and the auxiliary twist-density input come from another unreviewed release manuscript (family 006). The cube-sum corollary (Sylvester's primes 4, 7, 8 mod 9) was already proved by Yin and by Burungale-Tian, as the paper says, so it is not counted here.

Sources

Changelog1 change

Discussion