The defect of weak approximation for homogeneous spaces. II

Details The defect of weak approximation for homogeneous spaces. II The defect of weak approximation for homogeneous spaces. II

2008-04-30

arxiv.org/abs/0804.4767v2

Mathematics

Number Theory

10 pages. A section on metacyclic extensions is added

Scientific paper

Let X be a 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, recently introduced by Colliot-Th\'el\`ene, we can represent X in the form X=G/H, where G is a quasi-trivial k-group and H is a connected k-subgroup of G. Let S be a finite set of places of k. Applying results of [B2], we compute the defect of weak approximation for X with respect to S in terms of the biggest toric quotient T of H. In particular, we show that if T 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 T) has the real approximation property.

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