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Milne's rationality conjecture

Primarily AI-generated textHuman understanding: all partsmath.NT — Number Theorymath.AG — Algebraic Geometry

Contributed by Keerthi Madapusi ↗

Version 1 / Oct 13, 2026 / CC BY 4.0

Abstract

We prove Milne's rationality conjecture for reductions of Hodge classes on abelian varieties: The cohomological realizations of such classes pair rationally with Lefschetz classes (that is, classes in the algebra generated by divisor classes) in complementary codimension. The proof proceeds by establishing a dichotomy for reductions of Weil classes (an expected consequence of the rationality conjecture): Either they are orthogonal to all Lefschetz classes or are themselves Lefschetz classes. The key new input are lifting results of Kisin--Zhou for Hodge-type Shimura varieties. This is a cleaned up version of a preprint produced and released by OpenAI on Oct 6, 2026.
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References

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Show all 16 referencesShow fewer references
  1. J. S. Milne, The Tate conjecture over finite fields (AIM talk) (https://www.jmilne.org/math/articles/2007e.pdf), notes, revised 2008.link
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  5. OpenAI, Milne's rationality conjecture for abelian varieties , OpenAI Math Release preprint (https://github.com/openai/math/blob/main/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf), 2026.link
  6. M. Rapoport and E. Viehmann, Towards a theory of local Shimura varieties (https://arxiv.org/abs/1401.2849), Münster Journal of Mathematics 7 (2014), 273--326.link

Version history

  1. v1Submitted by Keerthi MadapusiInitial depositCurrentOct 13, 2026