The Hasse principle for homogeneous spaces with nilpotent stabilizer (Le principe de Hasse pour les espaces homog\`enes \`a stabilisateur fini)

TL;DR

This paper proves that the Brauer-Manin obstruction is the only obstacle to the Hasse principle for homogeneous spaces with nilpotent stabilizers, extending previous results to broader classes including abelian stabilizers.

Contribution

It generalizes recent work by Harpaz and Wittenberg to nilpotent stabilizers and shows these spaces satisfy real approximation, broadening the understanding of obstructions in arithmetic geometry.

Findings

Brauer-Manin obstruction is the only obstruction for nilpotent stabilizers.

Homogeneous spaces with abelian stabilizers satisfy real approximation.

Results extend classical theorems to more general stabilizer types.

Abstract

We prove that the Brauer-Manin obstruction is the only obstruction to the Hasse principle for homogeneous spaces with nilpotent stabilizer. We thus generalize recent results by Harpaz and Wittenberg on finite "hyper-solvable" stabilizers. In particular, this result is true for abelian stabilizers without any hypotheses on the ambient group, which generalizes classic results by Borovoi on the subject. We prove moreover that these spaces have the real approximation property. On montre que l'obstruction de Brauer-Manin est la seule obstruction au principe de Hasse pour les espaces homog\`enes \`a stabilisateur nilpotent. On g\'en\'eralise ainsi les r\'esultats r\'ecents de Harpaz et Wittenberg sur les stabilisateurs finis "hyper-r\'esolubles". En particulier, ce r\'esultat vaut pour les espaces homog\`enes \`a stabilisateur ab\'elien sans hypoth\`ese sur le groupe ambiant, ce qui g\'en\'eralise des r\'esultats d\'ej\`a classiques de Borovoi sur le sujet. On d\'emontre au passage que ces espaces poss\`edent la propri\'et\'e d'approximation r\'eelle.