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The p-adic section conjecture
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Abstract
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.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- Magnus Carlson, Minhyong Kim, Padmavathi Srinivasan and Jakob Stix, The p-adic section conjecture (following OpenAI): an online seminar (online seminar, first meeting 2026-10-20). The seminar aims to understand the claimed proof, with Benjamin Collas's opening overview relating the valuative approach to the resolution-of-nonsingularities geometry used in this family. [family-level]
Related prior literature mentioned by others
- Shinichi Mochizuki and Shota Tsujimura, Resolution of Nonsingularities, Point-theoreticity, and Metric-admissibility for p-adic Hyperbolic Curves (RIMS Preprint 1974, 2023). The October 8 article by ぶちょう identifies this work, already cited in both manuscripts, as their shared geometric input for constructing suitable finite étale covers. [family-level]
Possibly relevant literature (found by the agent, not checked)
- L. Alexander Betts, Theresa Kumpitsch, Martin Lüdtke, Chabauty–Kim, finite descent, and the Section Conjecture for locally geometric sections (arXiv:2305.09462, 2023; v2 April 2026). Studies the section conjecture for locally geometric sections over number fields via finite descent and Chabauty–Kim, which is close to the finite-descent route to the global consequences (Theorem 8.2, $X_0(N)$, $X_1(N)$) in this paper.
- Florian Pop, Generalizations and minimalistic refinements of the t-birational Section Conjecture (arXiv:2604.26131, April 2026). Recent work on birational versions of the section conjecture, which bears on the paper's step of lifting localized étale sections to birational sections and applying Koenigsmann's and Pop's birational results.
News and blogs
- ぶちょう, Grothendieck's 'Section Conjecture', AI claims proof of p-adic version using Shinichi Mochizuki's tools—OpenAI and the Kyoto lineage (note.com, 2026-10-08). This AI-assisted Japanese blog post discusses both manuscripts' use of Mochizuki–Tsujimura's theorem and distinguishes the claimed local theorem from its conditional global consequences; it reports that as of 7 October no anabelian-geometry experts had publicly announced a verification. [cites earlier version] [family-level]
Elsewhere
- VibeMathed: Problem record
- VibeMathed: Problem record
- Emergent Mind: family 019
Independent sites; their content is not reviewed here.
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}
}Versions
- Earlier edition: September 24, 2026
This version updates citations to revised manuscripts in this collection; see the previous version.
Family 019: The local p-adic section conjecture and global consequences
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.
Other papers in this family:
- Étale covers with a prescribed exterior sheet · September 24, 2026