Rational sections in families of homogeneous spaces over abelian varieties

In this talk, we study a question of Colliot-Thélène and Iyer concerning the existence of rational sections in families of homogeneous spaces over an abelian variety, after base change by a suitable étale isogeny of the abelian variety. Assuming characteristic zero and that the homogeneous spaces arise from connected reductive groups whose root datum contains no factor of type E8, we give a positive answer to the question posed by Colliot-Thélène and Iyer. Our approach relies on cohomological invariants such as the Milnor invariant and the Rost invariant. This leads us to a closer analysis of the action of the multiplication-by-n map on the unramified cohomology of abelian varieties, as well as on their motives with integral coefficients.