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.

Manuscripts (2)

Étale covers with a prescribed exterior sheet

September 24, 2026 · source · cite as \cite{OAI:Etale-covers-with-a-prescribed-exterior-sheet-September-24-2026}

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.

The p-adic section conjecture

October 6, 2026 · source · cite as \cite{OAI:The-p-adic-section-conjecture-October-6-2026}

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.

Result 019 · Number theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.