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The Tate conjecture for abelian varieties over finite fields

Let AA be an abelian variety over a finite field Fq\mathbb F_q, Aˉ\bar A its base change to F‾q\overline{\mathbb F}_q, and ℓ\ell a prime different from the characteristic. The Tate conjecture predicts that the Gal(F‾q/Fq)\mathrm{Gal}(\overline{\mathbb F}_q/\mathbb F_q)-invariant classes in Het2r(Aˉ,Qℓ(r))H^{2r}_{et}(\bar A,\mathbb Q_\ell(r)) are spanned by classes of algebraic cycles defined over Fq\mathbb F_q, and that numerical and ℓ\ell-adic homological equivalence agree. It is known for divisors (Tate) and in some further cases. Milne proved in 1999 that the Hodge conjecture for complex CM abelian varieties would imply it for all abelian varieties over finite fields. Does the Tate conjecture hold for every abelian variety over every finite field, in every codimension?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Arithmetic geometry: algebraic cycles and l-adic cohomology
Posed by
John Tate, for the general conjecture; the abelian-variety case over finite fields was isolated by Milne (1999)
Year posed
—
Years open
—
Solved
2026-09-10
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
60 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims the full rational Tate conjecture, in every codimension, for every abelian variety over every finite field, together with equality of numerical and l-adic homological equivalence, as a corollary of the paper's Hodge conjecture for CM abelian varieties via Milne 1999. It does not treat varieties other than abelian varieties, finitely generated fields other than finite fields, or integral coefficients. The paper notes that Milne's implication does not lift each Tate class to a Hodge class on one CM lift.

What the AI did

The OpenAI math release (github.com/openai/math) states that its results were produced by an unreleased internal OpenAI model, with on average about three hours of ChatGPT Pro thinking compute per result, under one fixed procedure applied to roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. The manuscript is credited to OpenAI alone and names no human author. The README names the proof of the Hodge conjecture for CM abelian varieties as one of two exceptions to that fixed procedure. It does not say what was done differently or whether people intervened, so the three-hour figure does not apply to this result and the human role is undisclosed. This entry is a corollary in that same manuscript, obtained by feeding its Hodge theorem into Milne's published implication.

Verification

No independent mathematician has checked this yet. Checked here: Corollary (Tate conjecture for abelian varieties over finite fields) in the TeX source, read against the posed problem: surjectivity of CHr(A)⊗QℓCH^r(A)\otimes\mathbb Q_\ell onto the Galois invariants for every A/FqA/\mathbb F_q, every rr and every ℓ≠p\ell\neq p, plus agreement of numerical and homological equivalence. The deduction is short and rests on Milne's Theorem 7.1 (1999) with Milne's own published correction, applied to the paper's CM Hodge theorem. The conclusion therefore stands or falls with the unrefereed CM theorem. No Lean formalization is listed.

Sources

Changelog1 change

Discussion