Result 001, Number theory

Milne’s rationality conjecture and algebraic specialization

Proves Milne's rationality conjecture for abelian varieties over Q‾\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.

Proof

The bigger picture

Why it matters

Different mathematical ways of measuring the same geometry should tell a consistent story. This result establishes one precise form of that agreement.

What changes?

Start with an abelian variety, a geometric object with an addition law, and reduce its equations modulo a prime where the geometry remains well behaved. The manuscript proves that certain intersection measurements involving Hodge classes and divisors give the same rational number across the cohomology theories it considers. In plain language, these different mathematical measuring systems agree on those measurements.

What does that help mathematicians do?

It resolves Milne's rationality conjecture in this setting and gives researchers a compatibility statement they can use when moving between cohomology theories. Combined with result 032, the manuscript goes further: each specialized Hodge class is represented by one rational algebraic cycle across all the stated realizations. That connects abstract cohomological information to geometric cycles.

Are there practical applications?

The immediate payoff is in pure mathematics. The result does not itself provide a faster algorithm, a new cryptographic system, or an engineering tool. Its value is a stronger foundation for studying how algebraic geometry behaves when passing from characteristic zero to finite characteristic.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Milne's rationality conjecture for abelian varieties

September 23, 2026 36 pages

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.

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Cite (BibTeX) 
@misc{OAI:Milnes-rationality-conjecture-for-abelian-varieties-September-23-2026,
  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-September-23-2026/paper.pdf}{OAI:Milnes-rationality-conjecture-for-abelian-varieties-September-23-2026}},
  year = {2026}
}

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An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.