The local p-adic section conjecture and global consequences

What OpenAI claims

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 X₀(N) and X₁(N) of genus at least two.

Papers (2)

1. Étale covers with a prescribed exterior sheet

   24 September 2026 · Preprint, not peer reviewed · PDF

   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.

2. The p-adic section conjecture

   6 October 2026 · Preprint, not peer reviewed · PDF

   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 X₀(N) and X₁(N) of genus at least two over ℚ.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.