Zéro-cycles sur les espaces homogènes et problème de Galois inverse

Harpaz, Yonatan; Wittenberg, Olivier

doi:10.1090/jams/943

Zéro-cycles sur les espaces homogènes et problème de Galois inverse
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by Yonatan Harpaz and Olivier Wittenberg

J. Amer. Math. Soc. 33 (2020), 775-805

DOI: https://doi.org/10.1090/jams/943

Published electronically: March 10, 2020

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Abstract:

Soit $X$ une compactification lisse d’un espace homogène d’un groupe algébrique linéaire $G$ sur un corps de nombres $k$. Nous établissons la conjecture de Colliot-Thélène, Sansuc, Kato et Saito sur l’image du groupe de Chow des zéro-cycles de $X$ dans le produit des mêmes groupes sur tous les complétés de $k$. Lorsque $G$ est semi-simple et simplement connexe et que le stabilisateur géométrique est fini et hyper-résoluble, nous montrons que les points rationnels de $X$ sont denses dans l’ensemble de Brauer–Manin. Pour les groupes finis hyper-résolubles, en particulier pour les groupes finis nilpotents, cela donne une nouvelle preuve du théorème de Shafarevich sur le problème de Galois inverse et résout en même temps, pour ces groupes, le problème de Grunwald.

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-Thélène, 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.

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