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Milne's rationality conjecture for abelian varieties
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Abstract
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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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- J. S. Milne, Articles (abstracts), entry for "Abelian motives and Shimura varieties in nonzero characteristic" (jmilne.org, notes added 7 and 13 October 2026). In a note dated 7 October 2026, Milne writes "OpenAI has announced a proof of the rationality conjecture, so, it seems, the statements in the article are now proven"; a note dated 13 October says Madapusi has sent him an improved exposition of the proof.
- J. S. Milne, Abelian motives and Shimura varieties in nonzero characteristic, v2.4, Section 12 "Comments on OpenAI's proof of the rationality conjecture" (author's preprint v2.4, 9 October 2026). Milne summarizes the proof in three steps: the Weil-class dichotomy, the reduction from CM varieties to Weil classes (which he calls not new), and lifting to a CM variety. He identifies the dichotomy as the core of the proof and writes: "The immediate tasks for the mathematical community are to verify the proof of this statement and to construct a more comprehensible proof." [cites earlier version]
- Keerthi Madapusi, Parsing AI proofs: Milne's rationality conjecture (personal website, October 2026). Madapusi criticizes the original exposition but describes the underlying idea as simple and clever after exploring the argument with Astra 6.
Related prior literature mentioned by others
- J. S. Milne, The Tate conjecture over finite fields (AIM talk) (AIM workshop notes, version 2.2, 7 May 2008). The AI-generated exposition hosted by Madapusi identifies Question 10.6 and Proposition 10.7 as antecedents of the simultaneous-lifting strategy used for rationality.
- Mark Kisin and Rong Zhou, Strongly compatible systems associated to semistable abelian varieties (arXiv:2505.02165v1, 4 May 2025). The AI-generated exposition hosted by Madapusi identifies Section 2.2 as the input for its CM and prescribed-filtration lifting arguments.
- Michael Rapoport and Eva Viehmann, Towards a theory of local Shimura varieties (arXiv:1401.2849; Münster Journal of Mathematics 7, 2014). Not cited in the OpenAI manuscript; Proposition 4.1 of the AI-generated exposition hosted by Madapusi invokes Proposition 3.1 of this work for nonemptiness of the weakly admissible period domain used to lift additional tensors.
Possibly relevant literature (found by the agent, not checked)
- J. S. Milne, Points on Shimura varieties over finite fields: the conjecture of Langlands and Rapoport (arXiv:0707.3173, v4 November 2009). Its §6 has a subsection titled "The rationality conjecture", bearing on the introduction's account of the conjecture's origins in the Langlands–Rapoport conjecture, which the paper cites only through Milne's 2000 IHP notes.
- J. S. Milne, The Tate and Standard Conjectures for Certain Abelian Varieties (arXiv:2112.12815, v2 February 2022). Adapts the implication "Hodge for CM abelian varieties ⇒ Tate and standard conjectures over finite fields" to a single abelian variety, relevant to the paper's corollary on algebraicity of specialized Hodge classes, which depends on the CM Hodge theorem.
Digests and expositions
- ChatGPT and Claude, with questions, feedback and light editing by Keerthi Madapusi, Milne's rationality conjecture (Parsing AI Proofs, 13 October 2026; also on Hexagon as hexagon:2610.00226v1). This cleaned-up rewrite of the OpenAI preprint presents the reduction to balanced Weil classes, their Lefschetz dichotomy, and simultaneous lifting through a reductive centralizer, with an appendix detailing the two applications of Kisin–Zhou; the Hexagon page records "human understanding: all". [AI-generated per source]
Elsewhere
- Hexagon: Milne's rationality conjecture (AI-generated; human understanding: all)
- VibeMathed: Problem record
- Emergent Mind: family 001
Independent sites; their content is not reviewed here.
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}
}Versions
- Earlier edition: September 23, 2026
This version updates a citation to a revised manuscript in this collection; see the previous version.
Family 001: 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.
This is the only paper in this family.