View source for Milne's rationality conjecture

'''Milne's rationality conjecture''' for abelian varieties is the subject of a manuscript in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]), first dated 23 September 2026. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== Background ==
* An abelian variety can acquire new divisor classes after good reduction. J. S. Milne asked whether a rational Hodge class, after specialization, pairs with every complementary product of divisor classes on the reduction to the same rational number in every cohomology theory (Question 10.6 in his AIM talk, [https://www.jmilne.org/math/articles/2007e.pdf The Tate conjecture over finite fields]).

== The original result ==
* '''Claim.''' For an abelian variety over the algebraic numbers with 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, including residue characteristic 2. Combined with the collection's manuscript on the rational Hodge conjecture for CM abelian varieties, the summary states that every specialized Hodge class is represented by a single rational algebraic cycle in all these realizations.
* '''Sources''' (openai/math repository):
** the manuscript folder [https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-September-23-2026 Milnes-rationality-conjecture-for-abelian-varieties-September-23-2026] (commit <code>adc7f12</code>);
** a later edition, [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026 Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026] (commit <code>1581d4c</code>), whose README states that it updates a citation to a revised manuscript in the collection.
* '''Formalization.''' The repository has no Lean scope note for this result (checked 15 October 2026).

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 15 October 2026)
|-
| 2026-10-13 || Keerthi Madapusi, Hexagon record [https://hexagonmath.org/2610.00226v1 2610.00226v1], titled Milne's rationality conjecture || Revised version (by another author) || Proves Milne's rationality conjecture for reductions of Hodge classes on abelian varieties, by a dichotomy for reductions of Weil classes: they are either orthogonal to all Lefschetz classes or are themselves Lefschetz classes. The key new input is the lifting results of M. Kisin and R. Zhou for Hodge-type Shimura varieties ([https://arxiv.org/abs/2505.02165 arXiv:2505.02165]). The abstract describes the record as a cleaned-up version of the OpenAI preprint. || Hexagon labels: primarily AI-generated text; human understanding: all parts. Tools listed: Claude Opus 5.5 and ChatGPT Astra 6.
|}

== Notes ==
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]