Milne's rationality conjecture for abelian varieties

Let $A$ be an abelian variety of dimension $d$ over $\overline{\mathbb Q}$ with good reduction $A_0$ at a $p$ -adic place, and $\gamma$ a rational Hodge class in $H^{2r}(A(\mathbb C),\mathbb Q(r))$ . Deligne's absolute-Hodge theorem gives compatible specializations of $\gamma$ to $\ell$ -adic cohomology of $A_0$ for $\ell\ne p$ and to crystalline cohomology, but the reduction may acquire divisors that do not lift. Milne conjectured that the specialized class nevertheless behaves rationally, which he showed equivalent (for CM abelian varieties) to the existence of a good theory of rational Tate classes. For every list of divisors $D_1,\dots,D_{d-r}$ on $A_0$ , is the intersection number of the specialized $\gamma$ with $D_1\cdots D_{d-r}$ one and the same rational number in every $\ell$ -adic and in the crystalline realization?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Arithmetic geometry: Hodge classes, reduction and rational Tate classes
Posed by: J. S. Milne (weak form in his 2000 Langlands-Rapoport notes; pairing form in 2007 AIM notes and Conjecture 4.1 of Rational Tate classes, Mosc. Math. J. 2009)
Year posed: 2000
Years open: 26y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 28 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Claims Theorem 1.1: for every abelian variety over $\overline{\mathbb Q}$ with good reduction at a $p$ -adic place, every residue characteristic $p$ including $2$ , every rational Hodge class $\gamma$ of codimension $r$ and every divisors $D_1,\dots,D_{d-r}$ on the reduction, the $\ell$ -adic traces ( $\ell\ne p$ ) and the crystalline trace of $\gamma_0\cdot D_1\cdots D_{d-r}$ equal one rational number. Also claims each specialized Hodge class lies in one fixed good theory of rational Tate classes, and (using the CM Hodge theorem) is represented by a rational algebraic cycle in all realizations. It does not prove the Hodge or Tate conjecture for general abelian varieties.

What the AI did

Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues. The main theorem is proved independently of the release's Hodge theorem for CM abelian varieties; only the algebraic-specialization corollary uses that theorem, which the README lists as produced outside the fixed procedure.

Verification

No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1, read against Milne's conjecture as cited (pairing form). The proof was not refereed. No Lean formalization exists for this family at the pinned commit (no lean/docs/001.md, no comparator challenge). Inputs the paper relies on include Kisin-Zhou and Pappas-Rapoport integral models and a 2026 uniformization result of Gleason, Lim and Xu. The corollary that every specialized Hodge class is represented by one rational algebraic cycle rests on the release's unrefereed Hodge theorem for CM abelian varieties (a README exception); Theorem 1.1 itself does not.

Milne's rationality conjecture for abelian varieties

What was actually shown

What the AI did

Verification

Sources

Discussion