Approximation forte pour les vari\'et\'es avec une action d'un groupe lin\'eaire

TL;DR

This paper proves strong approximation with Brauer-Manin condition for certain algebraic varieties with group actions over number fields, extending previous results to more general cases.

Contribution

It generalizes strong approximation results to smooth G-varieties containing G-homogeneous spaces with connected stabilizers, under specific conditions on the set of places.

Findings

Strong approximation holds off archimedean places for these varieties.

Strong approximation also holds off finite sets of places under certain invertibility conditions.

The proof extends previous work on G-homogeneous spaces to more general G-varieties.

Abstract

Let $G$ be a connected linear algebraic group over a number field. Let $U \hookrightarrow X$ be a $G$-equivariant open embedding of a $G$-homogeneous space with connected stabilizers into a smooth $G$-variety. We prove that $X$ satisfies strong approximation with Brauer-Manin condition off a set $S$ of places of $k$ under either of the following hypotheses : (i) $S$ is the set of archimedean places; (ii) $S$ is a nonempty finite set and $\bar{k}^{\times}= \bar{k}[X]^{\times}$. The proof builds upon the case $X=U$, which has been the object of several works.