Non-Surjectivity of the Universal Torsor Evaluation Map for Homogeneous Spaces

Abstract

Let $K$ be a field of characteristic zero, let $G$ be a connected linear $K$-algebraic group, and let $H$ be a connected closed subgroup of $G$. Let $X_c$ be a smooth compactification of $X=G/H$, and let $Y\overset{}{\longrightarrow}X_c$ be the universal $S$-torsor with trivial fibre over the class of the identity of $G$. Colliot-Th\'el\`ene and Kunyavski\u{\i} have shown that $S$ is a flasque torus, and that the evaluation map $X_c(K)\rightarrow \text{H}^1(K,S)$, induced by the universal torsor, is surjective when the field $K$ is 'good'; and the same is true when we restrict the evaluation map to the $K$-points of $X$. In this article, we establish that in cases where the field is not 'good', surjectivity may fail when the domain is $X(K)$. We provide two concrete examples: one over a field of cohomological dimension $2$ and the other over an arithmetic field, such as $\mathbb Q_7((t))$.