math.NT — Number Theory
Milne's rationality conjecture
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.
Primarily AI-generated textHuman understanding: all partsHodge cyclesShimura varietiesabelian varietiesalgebraic cycles