Short··0:20·Number theory
Milne's Rationality Conjecture: OpenAI Claims a Proof
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About this video
Milne's rationality conjecture says that when an abelian variety is reduced mod p, pairing a Hodge class with products of divisors gives one and the same rational number in every cohomology theory. The animation shows the classical case with divisors: on the curve y² = x³ + 2, the 9 points with −2x = x appear both on the complex torus and among the 117 points mod 103. OpenAI claims a proof of the conjecture for abelian varieties with good reduction, including residue characteristic 2; this is OpenAI's preprint, not yet peer reviewed.
Papers (OpenAI, catalogue result 001):
- Milne's rationality conjecture for abelian varieties: github.com/openai/math/blob/main/preprints/Milnes-rationality…
The animation illustrates the result; it is not a proof.
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Every result explained: gradientdamage.com/p/001
About the problem
Number theory·OpenAI Math·Result 001
Milne’s rationality conjecture and algebraic specialization
Proves Milne's rationality conjecture for abelian varieties over ℚ̅ with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology theory. Together with result 032, every specialized Hodge class is represented by a single rational algebraic cycle simultaneously in all prime-to-p and crystalline realizations, for every residue characteristic p.
Topics
Milne rationality conjectureMilne's conjectureJames MilneHodge classesHodge conjectureabelian varietieselliptic curvesgood reductionfinite fieldscrystalline cohomologyetale cohomologyalgebraic cyclesTate conjecturearithmetic geometrynumber theoryOpenAIOpenAI mathAI mathematicsAI proofmathsmathematicsmath shorts3D animation
