齐性空间弱逼近的缺陷。II
The defect of weak approximation for homogeneous spaces. II
摘要：
设X是数域k上的连通线性代数群G'的齐性空间，其中x是k点.设G'中x的稳定子是连通的。利用Colliot-Thélène最近引入的拟平凡群的概念，我们可以用X=G/H的形式表示X，其中G是拟平凡k-群，H是G的连通k-子群。设S是k的位置的有限集。应用[B2]的结果，我们用H的最大复曲面商T来计算X关于S的弱逼近的亏度。特别地，我们证明了如果T在k的亚循环扩张上分裂，则X具有弱逼近性质。我们还证明了任何具有连通稳定子的齐性空间X（不对T作任何假设）具有实逼近性质。
英文摘要：
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-Thélène, 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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