View source for The p-adic section conjecture

'''The ''p''-adic section conjecture''' is the local analogue of Grothendieck's section conjecture: for a smooth proper geometrically connected curve of genus at least two over a finite extension ''k'' of '''Q'''<sub>''p''</sub>, it asserts that the rational points correspond bijectively to the conjugacy classes of sections of the projection from the arithmetic étale fundamental group of the curve to the absolute Galois group of ''k''. A manuscript in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), first dated 24 September 2026 and revised on 6 October 2026, claims this conjecture, together with consequences for Grothendieck's section conjecture over number fields, including the modular curves ''X''<sub>0</sub>(''N'') and ''X''<sub>1</sub>(''N'') of genus at least two over '''Q'''. The collection has no Lean formalization of these results. 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 manuscript:
** Grothendieck formulated the section conjecture in his 1983 letter to Faltings for fields finitely generated over the prime field; the local question keeps the full arithmetic and geometric fundamental groups while bringing in ''p''-adic valuations and reduction.
** Koenigsmann proved that every birational section over a finite extension of '''Q'''<sub>''p''</sub> lies in the decomposition group of a rational point, and Pop gave a birational criterion from sections of an elementary abelian quotient that lift to the elementary metabelian quotient. Pop and Stix localized every étale section at a valuation extending the ''p''-adic valuation, and Pajwani developed this valuative approach through étale homotopy and fixed points in Berkovich spaces.
** Bresciani introduced toric sections and proved the section conjecture for them, and Betts and Stix proved finiteness of the local image of global Selmer sections for curves over number fields containing no CM subfield. For the global corollaries the manuscript uses the descent results of Harari and Stix, Stix's finite-support theorem, and Stoll's finite-descent criteria.

== The original results ==
* '''Claim (local).''' Theorem 1.1 of ''The p-adic section conjecture'' states that for every prime ''p'', every finite extension ''k'' of '''Q'''<sub>''p''</sub> and every smooth proper geometrically connected curve ''X'' over ''k'' of genus at least two, the map from ''X''(''k'') to the set of sections of π<sub>1</sub>(''X'') → ''G''<sub>''k''</sub>, up to conjugation by the geometric fundamental group, is bijective. Theorem 1.2, the completion statement, states that at every rank-one norm on the function field extending the ''p''-adic norm, the separable closure lies in the completion of the compositum of the function fields of all connected finite étale covers.
* '''Claim (global).''' The manuscript states that the local theorem identifies the image of global sections under localization with a finite-cover descent locus, and that when this locus consists exactly of the rational points, every global section is a point section. It derives Grothendieck's section conjecture over '''Q''' for the modular curves ''X''<sub>0</sub>(''N'') and ''X''<sub>1</sub>(''N'') of genus at least two from Stoll's finite-descent criteria, and gives corollaries for curves mapping to suitable abelian varieties and to elliptic curves of Selmer corank zero, the latter using a companion manuscript of the collection on the Selmer converse.
* '''Companion.''' The theorem of ''Étale covers with a prescribed exterior sheet'' states that for a smooth proper connected curve of genus at least two over an algebraic closure of ''k'' and finitely many pairwise disjoint open disks in its Berkovich analytification, with exterior ''U'', there is a connected finite étale cover with a component above ''U'' mapping isomorphically to ''U'' and with ''p'' dividing the degree of every component above each disk.
* ''Method, as described:'' the completion statement compares the absolute and étale decomposition groups at every rank-one valuation extending the ''p''-adic norm, which supplies a birational lift at the valuations that must be excluded from the Pop–Stix localization; combined with Koenigsmann's birational theorem, this rules out sections localized at such valuations, and the remaining sections are point sections.
* '''Sources.''' The manuscript [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/The-p-adic-section-conjecture-October-6-2026 The p-adic section conjecture] dated 6 October 2026, whose version note states that it updates citations to revised manuscripts of the collection, its [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/The-p-adic-section-conjecture-September-24-2026 previous version] dated 24 September 2026, and the companion [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026 Étale covers with a prescribed exterior sheet], at commit <code>1581d4c</code>.
* '''Formalization.''' The repository has no Lean scope note for this family.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 20 October 2026)
|-
| 2026-10-16
| J. Jardine, Lattice24, referee reviews of [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_230.json the October edition of the main manuscript] and [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_502.json the companion manuscript] in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995])
| Review (AI-assisted audit)
| Both reviews report no gap.
* The review of the main manuscript states that the reviewer read every section in full. Its two minor notes concern external citations that it could not check against the sources: a theorem of Mochizuki and Tsujimura and a theorem of Amini, Baker, Brugallé and Rabinoff used in two lemmas on étale collars and stable edges, where a weaker form than stated would require adjusting the argument, and the hypotheses and numbering of the cited results of Koenigsmann and of Pop and Stix, whose use it describes as matching their known content.
* The review of the companion states that the reviewer read the whole paper and checked every step, with the same residual dependence on the cited theorem of Mochizuki and Tsujimura.
| Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0.
|-
| 2026-10-18 || M. Luo, T. Watanabe, ''The section conjecture for generic pointed curves'', [https://arxiv.org/abs/2610.24986v1 arXiv:2610.24986v1] || Remark (citation) || Proves the section conjecture, profinite and geometrically pro-ℓ, for the generic ''n''-pointed curve of genus ''g'' ≥ 3 over a field finitely generated over '''Q''', for every ''n'' ≥ 1. Its introduction states that, to the authors' knowledge, prior to the recent announcement of a proof of the ''p''-adic section conjecture by OpenAI, no examples had been established of smooth proper hyperbolic curves over fields finitely generated over '''Q''' with a nonempty set of rational points for which the full section conjecture holds; it cites the version of 6 October 2026. Its AI disclosure states that the authors used ChatGPT to assist with literature searches, their mathematical review, notation and exposition. || arXiv preprint, version 1 of 18 October 2026.
|}

== Notes ==
* The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]