The rational Hodge conjecture for CM abelian varieties

Follow-up works

• Pol van Hoften and Jack Sempliner, The Hodge Conjecture for CM abelian varieties (hexagon:2610.00259v2, v1 15 Oct 2026, v2 18 Oct 2026). The authors present this as a proof of the Hodge conjecture for CM abelian varieties "following the recent work [Ope26]" (this paper), using theta lifts and the doubling method. The text is described as human-written, with ChatGPT used among other things to rework [Ope26, Section 2] via Hazama's work.
• Antti Hippeläinen, Algebraicity of Weil classes on non-split abelian sixfolds of Weil type over $\mathbb{Q}(\sqrt{-3})$ with Hermitian determinant $-2$ (hexagon:2610.00371v1, 20 Oct 2026). Treats a non-split sixfold case outside the split-form results and cites this paper and two companion papers in the family, without saying how they are used. [AI-generated per source]
• Benjamin Severn, A transfer theorem for Weil classes on Prym–Tyurin factors of Galois covers (hexagon:2610.00386v1, Oct 2026). Proves a transfer theorem for Weil classes on Prym–Tyurin factors; its reference list cites this paper together with the abelian-powers and Gale companions. [AI-generated per source] [cites earlier version]

Comments and discussion

• Elliot Glazer (@ElliotGlazer), post on X (6 Oct 2026). Calls the CM result "very, very good mathematics" while stressing that it does not cover all abelian varieties, contrary to earlier rumors.
• Elliot Glazer (@ElliotGlazer), post on X (8 Oct 2026). States that the CM result was not Lean-formalized and expresses confidence in it because it "was achieved independently by Ant and by OAI" and trusted people report the ideas make sense.

Related prior literature mentioned by others

• James S. Milne, Hodge classes on abelian varieties (arXiv:2010.08857v2, 11 Nov 2020). Cited by van Hoften–Sempliner and discussed by Agentic Economy; explains the Deligne–André reduction to divisor and split Weil classes that precedes this paper's cycle construction.
• James S. Milne, Lefschetz Motives and the Tate Conjecture (Compositio Mathematica 117, 45–76, 1999). Van Hoften–Sempliner and Agentic Economy identify Theorem 7.1 as the implication from CM Hodge to Tate over finite fields used for this paper's finite-field consequence.
• Thomas Agugliaro, Standard conjecture of Hodge type for powers of abelian varieties (arXiv:2510.21562, Oct 2025). Cited in van Hoften–Sempliner's reference list; proves the Hodge standard conjecture for powers of abelian threefolds and some cases over finite fields, related to this paper's Hodge-standard corollary.
• Bert van Geemen and Antonio Rapagnetta, Hyperkähler sixfolds, abelian fourfolds of Weil type and a Hodge class (arXiv:2607.18341, Jul 2026). Cited in van Hoften–Sempliner's reference list; gives another proof of the Hodge conjecture for general Weil fourfolds of trivial discriminant.

Possibly relevant literature (found by the agent, not checked)

• J. S. Milne, The Tate and Standard Conjectures for Certain Abelian Varieties (arXiv:2112.12815, 2021/2022). Reworks the Milne 1999/2002 deductions, which this paper uses for its Tate and Hodge-standard corollaries, so that they apply to a single abelian variety.
• Ningyi Li, The Tate conjecture for powers of abelian fourfolds and the Hodge conjecture for powers of CM fourfolds (arXiv:2609.27916, v1 10 Sep 2026, withdrawn in v2 on 8 Oct 2026). Claimed an overlapping special case, the Hodge conjecture for powers of CM fourfolds; the withdrawal comment cites "recent work on the Hodge and Tate conjectures for abelian varieties" without naming it.

News and blogs

• Agentic Economy, OpenAI's Hodge claims: constructing the missing geometry (aecon.ai, 8 Oct 2026). Explains this paper's four-factor construction and arithmetic source identification from the pre-withdrawal repository snapshot, alongside the K3 claims, and disclaims independent proof certification. [cites earlier version]
• Kingy.ai, OpenAI's 722 Math Manuscripts: The Results, Proofs, Compute and Costs (Kingy.ai blog, Oct 2026). Summarizes the family 032 claims and proof roadmap, and reports that the K3 papers in the family were withdrawn on 7 Oct while the CM paper remains listed. [family-level]