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

TL;DR

This paper investigates local-global principles and weak approximation for homogeneous spaces over $p$-adic function fields, utilizing Galois cohomology duality, and demonstrates that any finite abelian group can be realized as a Galois group over such fields.

Contribution

It extends the understanding of Galois groups and local-global principles for homogeneous spaces over $p$-adic function fields, including new results on the inverse Galois problem.

Findings

Any finite abelian group is a Galois group over $K$

Reconfirmation of the abelian inverse Galois problem over $ ext{Q}_p(t)$

Coarser results for higher-dimensional local fields

Abstract

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-Th\'el\`ene, 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.