The local p-adic section conjecture and global consequences

2 papers

Proves that rational points on every smooth proper geometrically connected curve of genus at least two over a finite extension of ℚ_p correspond bijectively to conjugacy classes of sections of its full arithmetic étale fundamental group. It also proves Grothendieck's section conjecture over ℚ for the modular curves \(X_0(N)\) and \(X_1(N)\) of genus at least two.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 2

Étale covers with a prescribed exterior sheet

OpenAI · September 24, 2026

Let X be a smooth proper hyperbolic curve over an algebraic closure of a p-adic local field. Given finitely many disjoint small open disks, we construct one connected finite étale cover with a sheet isomorphic to their entire exterior and with degree divisible by p on every connected component above each disk.

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@misc{OAI:Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026,
  author = {{OpenAI}},
  title = {{\'{E}tale covers with a prescribed exterior sheet}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026/main.pdf}{OAI:Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026}},
  year = {2026}
}

The p-adic section conjecture

OpenAI · October 6, 2026

We prove the local p-adic section conjecture for smooth proper geometrically connected curves of genus at least two, over every finite extension of ℚ_p. Combined with established finite-descent theorems, this also yields the global section conjecture for smooth proper geometrically connected curves of genus at least two over number fields when their finite-cover descent locus equals their rational points; this includes \(X_0(N)\) and \(X_1(N)\) of genus at least two over ℚ.

This version updates citations to revised manuscripts in this collection; see the previous version.

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Cite

@misc{OAI:The-p-adic-section-conjecture-October-6-2026,
  author = {{OpenAI}},
  title = {{The p-adic section conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-p-adic-section-conjecture-October-6-2026/main.pdf}{OAI:The-p-adic-section-conjecture-October-6-2026}},
  year = {2026}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.