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The local p-adic section conjecture for curves of genus at least two

Claims Theorem 1.1: for every finite extension $k/\mathbb Q_p$ and every smooth proper geometrically connected curve $X/k$ of genus at least two, $X(k)\to\{\text{sections of }\pi_1(X)\to G_k\}/\pi_1(X_{\bar k})$ 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.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The local p-adic section conjecture for curves of genus at least two · p-adic section conjecture

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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Unreleased internal OpenAI model
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VibeMathed record: The local p-adic section conjecture for curves of genus at least two

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No independent mathematician has checked this yet. Theorem 1.1 was read against the conjecture as posed: for every prime $p$ , every finite $k/\mathbb Q_p$ 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.

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The local p-adic section conjecture for curves of genus at least two

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Companion: Etale covers with a prescribed exterior sheet

paper · pending

paper link supplied by the VibeMathed registry.

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Grothendieck, letter to Faltings (27 June 1983), in Geometric Galois Actions 1

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problem-record link supplied by the VibeMathed registry.

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Earlier edition (September 24, 2026)

paper · pending

paper link supplied by the VibeMathed registry.

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The local p-adic section conjecture for curves of genus at least two parent of this event

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Companion: Etale covers with a prescribed exterior sheet evidence for this event

VibeMathed record: The local p-adic section conjecture for curves of genus at least two evidence for this event

Grothendieck, letter to Faltings (27 June 1983), in Geometric Galois Actions 1 evidence for this event

Earlier edition (September 24, 2026) evidence for this event

The local p-adic section conjecture for curves of genus at least two evidence for this event

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