The defect of weak approximation for homogeneous spaces.~II

M. V. Borovoi

Dalʹnevostočnyj matematičeskij žurnal, Tome 9 (2009) no. 1, pp. 15-23

Résumé

Let $X$ be a right homogeneous space of a connected linear algebraic group $G'$ over a number field $k$, containing a $k$-point $x$. Assume that the stabilizer of $x$ in $G'$ is connected. Using the notion of a quasi-trivial group introduced by Colliot-Thélène, we can represent $X$ in the form $X=H\setminus G$, where $G$ is a quasi-trivial $k$-group and $H\subset G$ is a connected $k$-subgroup. Let $S$ be a finite set of places of $k$. We compute the defect of weak approximation for $X$ with respect to $S$ in terms of the biggest toric quotient $H^{\rm tor}$ of $H$. In particular, we show that if $H^{\rm tor}$ splits over a metacyclic extension of $k$, then $X$ has the weak approximation property. We show also that any homogeneous space $X$ with connected stabilizer (without assumptions on $H^{\rm tor}$) has the real approximation property.