Le principe de Hasse pour les espaces homog\`enes : r\'eduction au cas des stabilisateurs finis (The Hasse principle for homogeneous spaces: reduction to the case of finite stabilizers)

TL;DR

This paper demonstrates that certain properties of homogeneous spaces, including the Brauer-Manin obstruction conjecture, can be verified by examining only spaces with finite stabilizers, simplifying the proof process.

Contribution

It reduces the verification of broad properties of homogeneous spaces to the case of spaces with finite stabilizers, impacting major conjectures in arithmetic geometry.

Findings

Reduces verification of properties to finite stabilizer case

Applies to the Brauer-Manin obstruction conjecture

Extends to zero-cycle conjectures (conjecture (E))

Abstract

Nous montrons, pour une grande famille de propri\'et\'es $P$ des espaces homog\`enes, que $P$ vaut pour tout espace homog\`ene d'un groupe lin\'eaire connexe d\`es qu'elle vaut pour les espaces homog\`enes de $\mathrm{SL}_n$ \`a stabilisateur fini. Nous r\'eduisons notamment \`a ce cas particulier la v\'erification d'une importante conjecture de Colliot-Th\'el\`ene sur l'obstruction de Brauer-Manin au principe de Hasse et \`a l'approximation faible. Des travaux r\'ecents de Harpaz et Wittenberg montrent que le r\'esultat principal s'applique \'egalement \`a la conjecture analogue (dite conjecture (E)) pour les z\'ero-cycles. We prove, for a wide family of properties $P$ of homogeneous spaces, that if $P$ is satisfied for homogeneous spaces of $\mathrm{SL}_n$ with finite stabilizers, then $P$ is satisfied for all homogeneous spaces of linear connected groups. In particular, we reduce to this particular case the verification of an important conjecture by Colliot-Th\'el\`ene on the Brauer-Manin obstruction to the Hasse principle and to weak approximation. Recent work by Harpaz and Wittenberg show that our main result can also be applied to the analog conjecture on zero-cycles (known as conjecture (E)).