Milne’s rationality conjecture and algebraic specialization

Proves Milne's rationality conjecture for abelian varieties over \overline{\mathbb Q} 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.

Manuscript

Milne's rationality conjecture for abelian varieties

October 7, 2026 · source · cite as \cite{OAI:Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026}

We prove Milne's rationality conjecture for abelian varieties, including residue characteristic 2. After good reduction, the pairing of a rational Hodge class with any complementary product of divisor classes on the reduction is the same rational number in every prime-to-p realization and in crystalline cohomology. Using the Hodge theorem for CM abelian varieties, we also show that every such specialized Hodge class is represented by a single rational algebraic cycle in all these realizations.

Result 001 · 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.