Short··0:21·Number theory
Zilber–Pink Conjecture: OpenAI Claims It for Abelian Varieties
Watch on YouTubeAbout the problem →
About this video
The Zilber–Pink conjecture says "unlikely intersections" are rare: a subvariety of an abelian variety meets the special subvarieties more than dimension counting allows only finitely often. OpenAI claims a proof of the abelian Zilber–Pink conjecture over the algebraic numbers, and of the full curve case in the Siegel threefold A2; these are OpenAI's preprints, not yet peer reviewed.
The animation shows the simplest case exactly: the real points of E×E for E: y² = x³ + x + 1, a curve x(P) + x(Q) = c, and all 7,776 torsion points of order up to 30. Only 4 of them lie on the curve (checked up to order 60).
Papers (OpenAI, catalogue result 016):
- The abelian Zilber–Pink conjecture: github.com/openai/math/blob/main/preprints/The-Abelian-Zilber…
- The E×CM component of Zilber–Pink for curves in A2: github.com/openai/math/blob/main/preprints/The-E-times-CM-Com…
- Quaternionic division points on curves in the Siegel threefold: github.com/openai/math/blob/main/preprints/Quaternionic-Divis…
- Elliptic squares and Zilber–Pink for curves in A2: github.com/openai/math/blob/main/preprints/Elliptic-Squares-a…
The animation illustrates the result; it is not a proof.
Gradient Damage: mathematics you can see, equations you can hear.
youtube.com/@gradientdamage
Every result explained: gradientdamage.com/p/016
About the problem
Number theory·OpenAI Math·Result 016
Zilber–Pink in abelian varieties and the Siegel threefold
Proves the abelian Zilber–Pink conjecture over ℚ̅: every irreducible subvariety has finitely many maximal atypical subvarieties relative to its smallest containing torsion coset. It also proves the full curve case in the Siegel threefold 𝒜2 for Hodge-generic curves defined over ℚ̅, without boundary or reduction assumptions.
Topics
Zilber-Pink conjectureZilber Pinkunlikely intersectionsabelian varietiesManin-Mumfordtorsion pointselliptic curvesSiegel threefoldSiegel modular varietyA2complex multiplicationdiophantine geometryarithmetic geometrynumber theoryOpenAIOpenAI mathAI mathematicsAI proofmathsmathematicsmath shorts3D animation
