Let $K$ be the function field of a smooth projective geometrically integral curve over a finite extension of ${\mathbb{Q}}_p$. Following the works of Harari, Scheiderer, Szamuely, Izquierdo, and Tian, we study the local–global and weak approximation problems for homogeneous spaces of ${\mathrm{SL}}_{n,K}$ with geometric stabilizers extension of a group of multiplicative type by a unipotent group. The tools used are arithmetic (local and global) duality theorems in Galois cohomology, in combination with techniques similar to those used by Harari, Szamuely, Colliot-Thélène, Sansuc, and Skorobogatov. As a consequence, we show that any finite abelian group is a Galois group over $K$, rediscovering the positive answer to the abelian case of the inverse Galois problem over ${\mathbb{Q}}_p(t)$. In the case where the curve is defined over a higher dimensional local field instead of a finite extension of ${\mathbb{Q}}_p$, coarser results are also given.

中文翻译：

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p-进函数域上的齐次空间的算术

让$K$是有限延伸上的平滑射影几何积分曲线的函数场${\mathbb{问}}_p$。继 Harari、Scheiderer、Szamuely、Izquierdo 和 Tian 的工作之后，我们研究了齐次空间的局部全局和弱逼近问题${\mathrm{SL}}_{n,K}$使用几何稳定剂通过单能群扩展乘法型群。使用的工具是伽罗瓦上同调中的算术（局部和全局）对偶定理，并结合了与 Harari、Szamuely、Colliot-Thélène、Sansuc 和 Skorobogatov 使用的类似技术。因此，我们证明任何有限交换群都是伽罗瓦群$K$，重新发现逆伽罗华问题的阿贝尔情况的肯定答案${\mathbb{问}}_p（t）$。在曲线是在更高维局部场而不是有限延伸上定义的情况下${\mathbb{问}}_p$，还给出了更粗略的结果。