The \'etale Brauer-Manin obstruction to strong approximation on homogeneous spaces

TL;DR

This paper extends the understanding of the étale Brauer-Manin obstruction, showing it is the only obstruction to strong approximation on homogeneous spaces with arbitrary stabilizers, under certain algebraic conditions.

Contribution

It proves the étale-Brauer-Manin obstruction's completeness for homogeneous spaces with arbitrary stabilizers, generalizing previous results that required connected stabilizers.

Findings

Étale-Brauer-Manin obstruction is the only obstruction under broad conditions.

Results apply to homogeneous spaces with arbitrary stabilizers.

Compatibility between Brauer-Manin pairing and abelianization morphisms established.

Abstract

It is known that, under a necessary non-compactness assumption, the Brauer-Manin obstruction is the only one to strong approximation on homogeneous spaces $X$ under a linear group $G$ (or under a connected algebraic group, under assumption of finiteness of a suitable Tate-Shafarevich group), provided that the geometric stabilizers of $X$ are connected. In this work we prove, under similar assumptions, that the \'etale-Brauer-Manin obstruction to strong approximation is the only one for homogeneous spaces with arbitrary stabilisers. We also deal with some related questions, concerning strong approximation outside a finite set of valuations. Finally, we prove a compatibility result, suggested to be true by work of Cyril Demarche, between the Brauer-Manin obstruction pairing on quotients $G/H$, where $G$ and $H$ are connected algebraic groups and $H$ is linear, and certain abelianization morphisms associated with these spaces.