Z\'ero-cycles sur les fibrations au-dessus d'une courbe de genre quelconque

TL;DR

This paper proves conjectures related to zero-cycles on varieties over number fields, specifically for fibrations over curves with finite Tate-Shafarevich groups into rationally connected varieties satisfying weak approximation, under an abelian fibers assumption.

Contribution

It establishes the conjectures on the Chow group of zero-cycles for a new class of fibrations over curves with finite Tate-Shafarevich groups, assuming abelian singular fibers.

Findings

Proved conjectures for fibrations over curves with finite Tate-Shafarevich groups.

Validated the conjectures for rationally connected varieties satisfying weak approximation.

Demonstrated the role of abelian assumptions on singular fibers in these results.

Abstract

Let X be a smooth and proper variety over a number field k. Conjectures on the image of the Chow group of zero-cycles of X in the product of the corresponding groups over all completions of k were put forward by Colliot-Th\'el\`ene, Kato and Saito. We prove these conjectures for the total space of fibrations, over curves with finite Tate-Shafarevich group, into rationally connected varieties which satisfy weak approximation, under an abelianness assumption on the singular fibers. ---- Soit X une vari\'et\'e propre et lisse sur un corps de nombres k. Des conjectures sur l'image du groupe de Chow des z\'ero-cycles de X dans le produit des m\^emes groupes sur tous les compl\'et\'es de k ont \'et\'e propos\'ees par Colliot-Th\'el\`ene, Kato et Saito. Nous d\'emontrons ces conjectures pour l'espace total de fibrations en vari\'et\'es rationnellement connexes v\'erifiant l'approximation faible, au-dessus de courbes dont le groupe de Tate-Shafarevich est fini, sous une hypoth\`ese d'ab\'elianit\'e sur les fibres singuli\`eres.