On the fibration method for zero-cycles and rational points

Abstract

Conjectures on the existence of zero-cycles on arbitrary smooth projective varieties over number fields were proposed by Colliot-Thélène, Sansuc, Kato and Saito in the 1980’s. We prove that these conjectures are compatible with fibrations, for fibrations into rationally connected varieties over a curve. In particular, they hold for the total space of families of homogeneous spaces of linear groups with connected geometric stabilisers. We prove the analogous result for rational points, conditionally on a conjecture on locally split values of polynomials which a recent work of Matthiesen establishes in the case of linear polynomials over the rationals.

Authors

Yonatan Harpaz

Institute for Mathematics, Astrophysics and Particle Physics, Radboud University, Nijmegen, The Netherlands

Current address:

Département de mathématiques et applications, École normale supérieure, Paris, France Olivier Wittenberg

Département de mathématiques et applications, École normale supérieure, Paris, France