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Le principe de Hasse pour les espaces homogènes : réduction au cas des stabilisateurs finis

Part of: Algebraic groups Forms and linear algebraic groups

Published online by Cambridge University Press:  04 July 2019

Cyril Demarche  and
Giancarlo Lucchini Arteche
Cyril Demarche
Affiliation:
Sorbonne Université, Institut de Mathématiques de Jussieu – Paris Rive Gauche, UMR 7586, CNRS, Université Paris Diderot, 75005 Paris, France Département de mathématiques et applications, École normale supérieure, CNRS, PSL Research University, 45 rue d’Ulm, 75005 Paris, France email cyril.demarche@imj-prg.fr
Giancarlo Lucchini Arteche
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Las Palmeras 3425, Ñuñoa, Santiago, Chile email luco@uchile.cl
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Abstract

Nous montrons, pour une grande famille de propriétés des espaces homogènes, qu’une telle propriété vaut pour tout espace homogène d’un groupe linéaire connexe dès qu’elle vaut pour les espaces homogènes de $\text{SL}_{n}$ à stabilisateur fini. Nous réduisons notamment à ce cas particulier la vérification d’une importante conjecture de Colliot-Thélène sur l’obstruction de Brauer–Manin au principe de Hasse et à l’approximation faible. Des travaux récents de Harpaz et Wittenberg montrent que le résultat principal s’applique également à la conjecture analogue (dite conjecture (E)) pour les zéro-cycles.

For a large family of properties of homogeneous spaces, we prove that such a property holds for all homogeneous spaces of connected linear algebraic groups as soon as it holds for homogeneous spaces of $\text{SL}_{n}$ with finite stabilisers. As an example, we reduce to this particular case an important conjecture by Colliot-Thélène about the Brauer–Manin obstruction to the Hasse principle and to weak approximation. A recent work by Harpaz and Wittenberg proves that the main result also applies to the analogous conjecture (known as conjecture (E)) for zero-cycles on homogeneous spaces.

Keywords

espaces homogènesgerbesprincipe de Hasseobstruction de Brauer–Manin

MSC classification

Primary: 11E72: Galois cohomology of linear algebraic groups 14L30: Group actions on varieties or schemes (quotients)
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 8 , August 2019 , pp. 1568 - 1593
Copyright
© The Authors 2019 

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