论文标题
$ p $ adiC功能字段的均匀空间的算术
Arithmetics of homogeneous spaces over $p$-adic function fields
论文作者
论文摘要
令$ k $为$ \ mathbb {q} _p $的有限扩展名，是平滑的投射几何积分曲线的功能字段。遵循Harari，Scheiderer，Szamuely，Izquierdo和Tian的作品，我们研究了$ \ textrm {sl} _ {n，k} $均质{sl} _ {n，k} $均质空间的局部 - 全球和弱近似问题，并使用Unipotent组的几何学稳定器扩展了几何学稳定剂。使用的工具是Galois共同体中算术（本地和全局）二元定理，结合了类似于Harari，Szamuely，Colliot-Thélène，Sansuc和Skorobogogatov使用的技术。结果，我们表明，任何有限的Abelian集团都是$ k $的Galois集团，重新发现了对$ \ Mathbb {q} _p（t）$的Abelian case obelian案例的积极答案。如果在较高维的本地字段中定义曲线，而不是$ \ mathbb {q} _p $的有限扩展，也会给出更粗糙的结果。
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 $\textrm{SL}_{n,K}$ 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.
