The local p-adic section conjecture for curves of genus at least two
Let be a finite extension of and a smooth proper geometrically connected curve of genus at least two, with arithmetic etale fundamental group sequence . Each rational point gives a section of , well defined up to conjugation. The local p-adic section conjecture, the local analogue of the section conjecture in Grothendieck's 1983 letter to Faltings, asserts that this map from to conjugacy classes of sections is a bijection. Koenigsmann proved the birational analogue, Pop-Stix localized every section at a valuation, and Bresciani handled toric sections. Is every section of induced by a unique rational point of ?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Anabelian geometry; etale fundamental groups of p-adic curves
- Posed by
- Local analogue of Grothendieck's section conjecture (letter to Faltings, 27 June 1983); studied as the p-adic section conjecture by Koenigsmann (2005), Pop (2010) and Pop-Stix (2017)
- Year posed
- —
- Years open
- —
- 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
- 50 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claims Theorem 1.1: for every finite extension and every smooth proper geometrically connected curve of genus at least two, is bijective. Through Theorem 1.2 every section localized at a rank-one valuation lifts to a birational section. It does NOT prove the global section conjecture over number fields in general (see the separate entry for the cases it does reach), and says nothing about affine hyperbolic curves or curves over other local fields.
What the AI did
The release README says the manuscripts were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. Its named exceptions to that procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was human edited, and the Hodge conjecture for CM abelian varieties) do not concern this family. Both manuscripts of the family are credited to OpenAI with no human author named. The proof patches finite etale covers over annuli, using lifting theorems of Amini-Baker-Brugalle-Rabinoff and the resolution of nonsingularities of Mochizuki-Tsujimura as cited inputs.
Verification
No independent mathematician has checked this yet. Theorem 1.1 was read against the conjecture as posed: for every prime , every finite and every smooth proper geometrically connected curve of genus at least two, the point-to-section map is bijective, with conjugation by the full geometric fundamental group. The key new input is Theorem 1.2 (the universal etale field is dense in the separable closure at every rank-one valuation extending the p-adic norm), combined with Koenigsmann's birational theorem and the Pop-Stix localization. No Lean formalisation exists for this family. The release README warns that some unformalised results could have issues.