Grothendieck's section conjecture for the modular curves X0(N) and X1(N) over Q
Let be a smooth proper geometrically connected curve of genus at least two over a number field . Grothendieck's section conjecture (letter to Faltings, 1983) asserts that the map from to conjugacy classes of sections of is a bijection. Before this work it was known only in restricted or conditional forms (for instance via the finite-descent framework of Harari-Stix and Stix, or for curves with no adelic points), and no standard family such as the modular curves was covered. Does the section conjecture hold for every smooth proper curve of genus at least two over a number field, in particular for the modular curves and over ?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Anabelian geometry; rational points on curves
- Posed by
- Alexander Grothendieck, letter to G. Faltings (27 June 1983)
- Year posed
- 1983
- Years open
- 43y
- 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
- 42 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claims Theorem 8.2: the global section conjecture holds for every smooth proper geometrically connected curve of genus at least two over a number field whose finite-cover descent locus equals its rational points; by Stoll's criteria this includes curves mapping nonconstantly to an abelian variety with finitely many rational points and trivial divisible Sha, and the modular curves , of genus at least two over . It does NOT prove the section conjecture for arbitrary curves over number fields; the descent-locus equality remains a separate hypothesis in general.
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. The manuscript is credited to OpenAI with no human author named. The global statements are deduced in its Section 8 from the local theorem together with established descent results (Harari-Stix, Stix, Stoll), which are credited to their authors.
Verification
No independent mathematician has checked this yet. Section 8 (Proposition 8.1, Theorem 8.2) was read against the global conjecture: given the local theorem, the image of global sections under localization equals the finite-cover descent locus, and when that locus is exactly the rational points every section is a point section. Stoll's finite-descent criteria are cited to cover and of genus at least two over . The result therefore depends on the unformalised local theorem and on the cited descent theorems. No Lean formalisation exists. The release README warns that some unformalised results could have issues.