Short··0:21·Number theory
Grothendieck's Section Conjecture: OpenAI Claims the p-adic Case
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About this video
Grothendieck's section conjecture (1983) predicts that the rational points of a curve of genus at least two are exactly the sections of its arithmetic étale fundamental group, up to conjugacy. OpenAI claims a proof of the local p-adic section conjecture for every smooth proper geometrically connected curve of genus at least two over every finite extension of the p-adic numbers, and, combined with known descent results, the global conjecture over the rationals for the modular curves X0(N) and X1(N) of genus at least two. These are OpenAI's preprints, not yet peer reviewed. The animation shows a genus-2 surface and its real covers of degree 3, 9 and 27; the point and the thread are a picture of a section, not a computed p-adic point.
Papers (OpenAI, catalogue result 019):
- Étale covers with a prescribed exterior sheet: github.com/openai/math/blob/main/preprints/Etale-covers-with-…
- The p-adic section conjecture: github.com/openai/math/blob/main/preprints/The-p-adic-section…
The animation illustrates the result; it is not a proof.
Gradient Damage: mathematics you can see, equations you can hear.
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Every result explained: gradientdamage.com/p/019
About the problem
Number theory·OpenAI Math·Result 019
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 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.
Topics
section conjectureGrothendieck section conjecturep-adic section conjectureanabelian geometryGrothendieckétale fundamental groupGalois sectionsrational pointscurves of genus 2p-adic numbersmodular curvesX0(N)arithmetic geometrynumber theorycovering spacesOpenAIOpenAI mathAI mathematicsAI proofmathsmathematicsmath shorts3D animationetale fundamental group
