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Local p-adic section conjecture

The local p-adic section conjecture predicts a bijection between X(k) and sections of the full arithmetic étale fundamental-group sequence modulo conjugation by the full geometric group, for finite k/Q_p and smooth proper geometrically connected genus-at-least-two curves X/k.

Why it matters and what counts as progress

Why it matters: It asks whether arithmetic fundamental-group sections recover all rational points and no additional nongeometric sections in the local p-adic setting.

What counts as progress: Prove the point-to-section bijection for all stated p-adic curves with the full geometric conjugacy relation, or produce a nongeometric section.

How the problem developed

  1. May 2003 · Partial result [1]

    Jochen Koenigsmann Proved a birational p-adic section theorem for curves, a weaker setting than the full arithmetic section problem for proper curves.

  2. November 2011 · Partial result [2]

    Florian Pop and Jakob Stix Established a valuative version for p-adic local fields, still short of the full arithmetic section conjecture for complete curves.

  3. 24 Sept 2026 · Theorem claim [3]

    OpenAI Claimed the local section correspondence for smooth proper geometrically connected curves of genus at least two over finite extensions of Qₚ, using the full arithmetic fundamental group.

AI contribution

Sources

  1. Jochen Koenigsmann, On the “Section Conjecture” in anabelian geometry
  2. Florian Pop and Jakob Stix, Arithmetic in the fundamental group of a p-adic curve: On the p-adic section conjecture for curves
  3. OpenAI, The p-adic section conjecture
  4. The p-adic section conjecture
  5. OpenAI local p-adic section conjecture manuscript
  6. OpenAI mathematics release account
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