Result 019, Number theory

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 ℚ_p 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.

Proof

The bigger picture

Why it matters

Can a curve's rational points be recovered from the symmetries of its algebraic covers? These manuscripts claim this is possible over p-adic number fields, connecting solutions of equations with the groups attached to them.

What changes?

The manuscript reports a bijection for every smooth, proper, geometrically connected curve of genus at least two over any finite extension of the p-adic numbers. Points defined over that field correspond exactly to conjugacy classes of sections of the full arithmetic étale fundamental group. This group encodes finite unramified algebraic covers and field symmetries. A section is a homomorphism lifting the field's symmetry group back into this larger group, undoing its natural projection.

What does that help mathematicians do?

The claimed bijection rules out sections that do not come from rational points, while distinguishing different points. Combined with established finite-descent theorems, the manuscript reports the same correspondence over number fields when the curve's finite-cover descent locus equals its rational points. It specifically includes the modular curves X_0(N) and X_1(N) of genus at least two over the rational numbers. This is not an unconditional result for all curves over number fields.

Are there practical applications?

The immediate value is foundational: the result would make a curve's arithmetic fundamental group an exact detector of its rational points under the stated hypotheses. It also connects local p-adic information with global questions through finite descent. This is a structural correspondence, not a demonstrated algorithm for finding or listing rational points.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Étale covers with a prescribed exterior sheet

September 24, 2026 7 pages

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.

PDF LaTeX source

Cite (BibTeX)

@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}
}

The p-adic section conjecture

September 24, 2026 24 pages

We prove the local p-adic section conjecture for smooth proper geometrically connected curves of genus at least two, over every finite extension of ℚ_p. 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.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:The-p-adic-section-conjecture-September-24-2026,
  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-September-24-2026/main.pdf}{OAI:The-p-adic-section-conjecture-September-24-2026}},
  year = {2026}
}