Far Eastern Mathematical Journal

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The defect of weak approximation for homogeneous spaces. II

M. V. Borovoi

2009, issue 1-2, Ñ. 15–23

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-The?le?ne, 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 Htor) has the real approximation property.

linear algebraic groups, homogeneous spaces, weak approximation

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