Z\'ero-cycles sur les espaces homog\`enes et probl\`eme de Galois inverse

TL;DR

This paper proves a conjecture relating zero-cycles on certain algebraic varieties over number fields and demonstrates the density of rational points in the Brauer-Manin set for specific groups, with applications to inverse Galois and Grunwald problems.

Contribution

It establishes the conjecture of Colliot-Thélène, Sansuc, Kato, and Saito for zero-cycles on compactifications of homogeneous spaces and applies these results to solve classical inverse Galois and Grunwald problems for finite supersolvable groups.

Findings

Confirmed the conjecture on Chow groups of zero-cycles for these varieties.

Proved density of rational points in the Brauer-Manin set under certain conditions.

Provided new proofs for inverse Galois and Grunwald problems for specific groups.

Abstract

Let X be a smooth compactification of a homogeneous space of a linear algebraic group G over a number field k. We establish the conjecture of Colliot-Th\'el\`ene, Sansuc, Kato and Saito on the image of the Chow group of zero-cycles of X in the product of the same groups over all the completions of k. When G is semisimple and simply connected and the geometric stabiliser is finite and supersolvable, we show that rational points of X are dense in the Brauer-Manin set. For finite supersolvable groups, in particular for finite nilpotent groups, this yields a new proof of Shafarevich's theorem on the inverse Galois problem, and solves, at the same time, Grunwald's problem, for these groups. ----- Soit X une compactification lisse d'un espace homog\`ene d'un groupe alg\'ebrique lin\'eaire G sur un corps de nombres k. Nous \'etablissons la conjecture de Colliot-Th\'el\`ene, Sansuc, Kato et Saito 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. Lorsque G est semi-simple et simplement connexe et que le stabilisateur g\'eom\'etrique est fini et hyper-r\'esoluble, nous montrons que les points rationnels de X sont denses dans l'ensemble de Brauer-Manin. Pour les groupes finis hyper-r\'esolubles, en particulier pour les groupes finis nilpotents, cela donne une nouvelle preuve du th\'eor\`eme de Shafarevich sur le probl\`eme de Galois inverse et r\'esout en m\^eme temps, pour ces groupes, le probl\`eme de Grunwald.