Browse › Number theory › 019 The local p-adic section conjecture and global consequences
Étale covers with a prescribed exterior sheet
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Abstract
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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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)
- Emmanuel Lepage, Resolution of non-singularities and the absolute anabelian conjecture (arXiv:2306.07058, June 2023). Uses resolution-of-nonsingularities covers of $p$-adic hyperbolic curves, together with their Berkovich geometry, in anabelian arguments; this is the same input that Theorem 1.1 here combines with exterior and disk conditions on a single finite étale cover.
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 blog post discusses both manuscripts' use of Mochizuki–Tsujimura's theorem and distinguishes the claimed local theorem from its conditional global consequences. [cites earlier version] [family-level]
Elsewhere
- Emergent Mind: family 019
Independent sites; their content is not reviewed here.
Cite
@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}
}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:
- The p-adic section conjecture · October 6, 2026