001. Milne’s rationality conjecture and algebraic specialization

Milne’s rationality conjecture and algebraic specialization

1 paper

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.

Watch

The problem

What is being asked, where it comes from, and why it is hard. No proof needed.

▶Introduction to the problemin preparation

The solution proposed by OpenAI

How the paper solves it, step by step.

Full explanation

▶The proposed solutionin preparation

Papers 1

Milne's rationality conjecture for abelian varieties

OpenAI · 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.

This version updates a citation to a revised manuscript in this collection; see the previous version.

Read PDF Download Source

Cite

@misc{OAI:Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026,
  author = {{OpenAI}},
  title = {{Milne's rationality conjecture for abelian varieties}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf}{OAI:Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026}},
  year = {2026}
}