Sur les espaces homog\`enes de Borovoi-Kunyavski\u{\i}

TL;DR

This paper proves the Hasse principle and weak approximation for specific homogeneous spaces of SL_n with nilpotent stabilizers of class 2, confirming a conjecture related to Brauer-Manin obstruction for rationally connected varieties.

Contribution

It establishes the Hasse principle and weak approximation for certain homogeneous spaces of SL_n with nilpotent stabilizers, confirming Colliot-Thélène's conjecture on Brauer-Manin obstruction.

Findings

Proved Hasse principle for these spaces.

Confirmed weak approximation property.

Validated Colliot-Thélène's conjecture for these cases.

Abstract

We establish the Hasse principle and the weak approximation property for certain homogeneous spaces of $\mathrm{SL}_n$ whose geometric stabilizer is of nilpotency class 2, which were constructed by Borovoi and Kunyavski\u{\i}. These homogeneous spaces verify thus a conjecture of Colliot-Th\'el\`ene concerning Brauer-Manin obstruction for geometrically rationally connected varieties. -- Nous \'etablissons le principe de Hasse et l'approximation faible pour certains espaces homog\`enes de $\mathrm{SL}_n$ \`a stabilisateur g\'eom\'etrique nilpotent de classe 2, construits par Borovoi et Kunyavski\u{\i}. Ces espaces homog\`enes v\'erifient donc une conjecture de Colliot-Th\'el\`ene concernant l'obstruction de Brauer-Manin pour les vari\'et\'es g\'eom\'etriquement rationnellement connexes.