{"id":"hexagon/2610.00226","pdf":"https://hexagonmath.org/pdf/2610.00226v1","url":"https://hexagonmath.org/abs/2610.00226v1","tags":["math.AG","msc:11G15","msc:14K99","Hodge cycles","Shimura varieties","abelian varieties","algebraic cycles"],"score":0,"title":"Milne's rationality conjecture","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Keerthi Madapusi"],"reviews":0,"subject":"math.NT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus","ChatGPT"],"discussed":0,"published":"2026-10-13","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"all"}