The Tate conjecture for abelian varieties over finite fields

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. The companion on split abelian eightfolds linked here was withdrawn on 6 October 2026 for a sign error; the release names only the Kuga-Satake and K3-product papers as depending on it, and this result is not withdrawn.

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. The companion on split abelian eightfolds linked here was withdrawn on 6 October 2026 for a sign error; the release names only the Kuga-Satake and K3-product papers as depending on it, and this result is not withdrawn.