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The local p-adic section conjecture and global consequences

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Proves that rational points on every smooth proper geometrically connected curve of genus at least two over a finite extension of ℚp 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.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

Two current papers; one replaced by a later edition.

The paper states

Theorem 1.1

In The p-adic section conjecture, 6 October 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

The other possible matches are Theorem 1.2 in the same paper () and Theorem 1.1 in Étale covers with a prescribed exterior sheet ().

For every prime p, every finite extension k/\mathbb Q_p, and every smooth proper geometrically connected curve X/k of genus at least two, the map

X(k)\longrightarrow \left\{\begin{gathered} s:G_k\to\pi_1(X)\text{ a continuous group homomorphism}\\ \text{such that }(\pi_1(X)\to G_k)\circ s=\mathrm{id}_{G_k} \end{gathered}\right\}\big/\pi_1(X_{\bar k})

is bijective. The quotient is by conjugation by the full geometric étale fundamental group.

Reported proof verification
The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations
Two current papers, each readable here as a page beside its PDF.

Papers

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Étale covers with a prescribed exterior sheet

24 September 2026, current edition.

Read it herePDF

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.

The p-adic section conjecture

6 October 2026, current edition.

Read it herePDF

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

Earlier editions: 24 September 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 ℚp. 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.

Previous editions

Lean statements

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Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

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What these papers cite in the release, and what cites them

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Papers elsewhere in the release that this family’s papers cite

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

References outside the release

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.

    Author not stated. Source

  • Daebeom Choi, Geometrically Abelian Sections on Higher-Dimensional Varieties, 8 October 2026. Source

    Choi cites OpenAI’s p-adic section conjecture manuscript as motivation for asking whether analogous phenomena hold for higher-dimensional varieties.

Questions awaiting review

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Identify the theorem the catalogue describes

  • Family 019 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

State the result in Lean

  • Family 019 The release has no Lean statement for this family. Open the form