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The p-adic section conjecture

OpenAI

For every prime p, finite extension k/Q_p and smooth proper geometrically connected curve X/k of genus at least two, the manuscript claims a bijection between X(k) and sections of the full arithmetic étale fundamental-group sequence modulo conjugation by the full geometric group.

Sources for this record

What was already known

Koenigsmann established the birational p-adic analogue; Pop–Stix localized étale sections at valuations, and Bresciani handled toric sections. Those results did not identify every étale section with a unique rational point. The local statement is distinct from the global conjecture over all number fields.

AI involvement AI-led

OpenAI credits its internal model with the manuscript. The release does not document an individual autonomous run or a named human mathematical checker.

OpenAI is credited as author and its release attributes the family manuscript to an internal model; individual-session independence is undocumented.

Importance Major

Would settle the local p-adic section conjecture for the stated curves if the proof is correct.

Limitations

No completed human or machine check of the completion-to-birational lifting step is documented. This local p-adic claim does not settle the global section conjecture over number fields.

Technical verification Reported

The paper states and argues the exact local section theorem, but the collection lists no matching Lean proof and no completed human check. Reported is the evidence default; proof availability alone is not verification.

Human understanding Few

No named person explains the new completion-to-birational lifting mechanism; Few is the evidence default.

Correction and publication history