Homogeneous spaces over an abelian variety

Abstract

In this paper, we study a question of Colliot-Th\'el\`ene and Iyer concerning the existence of rational sections in families of homogeneous spaces over an abelian variety, after base change by a suitable \'etale isogeny of the abelian variety. Assuming characteristic zero and that the homogeneous spaces arise from connected reductive groups, the problem is reformulated in terms of torsors under reductive groups over an abelian variety $A$. Building on work of Moonen and Polishchuk, we construct a filtration on the motive of a Jacobian variety to analyze the action of isogenies on unramified cohomology and Witt groups. This approach allows for a positive response to the question for reductive groups whose root data do not contain a factor of type~$E_8$ when $\dim A > 2$ and $\mathrm{cd}(k) \leqslant 1$, and for all reductive groups when $\dim A = 2$ and $k$ is algebraically closed.