OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
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.
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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.
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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…
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.
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