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.

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      PUBLISHER = {North-Holland},
      ADDRESS = {Amsterdam},
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      NOTE={pp.~46--188, {P}art {I}: \mr{0244269},
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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