Arithmetics of homogeneous spaces over p$p$‐adic function fields

Abstract

Abstract Let be the function field of a smooth projective geometrically integral curve over a finite extension of . Following the works of Harari, Scheiderer, Szamuely, Izquierdo, and Tian, we study the local–global and weak approximation problems for homogeneous spaces of 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 , rediscovering the positive answer to the abelian case of the inverse Galois problem over . In the case where the curve is defined over a higher dimensional local field instead of a finite extension of , coarser results are also given.