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 algebraic K-group, and let H be a connected closed subgroup of G. Let $X_c$ be a smooth compactification of $X=G/H$, and let $Y\stackrel{}{⟶}{X}_c$ be the universal S-torsor with trivial fibre over the class of the identity of G. Colliot-Thélène and Kunyavskiĭ have shown that S is a flasque torus, and that the evaluation map $X_c\left(K\right)\stackrel{}{⟶}{H}^1(K,S)$, 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 $X\left(K\right)$. We provide two concrete examples: one over a field of cohomological dimension 2 and the other over an arithmetic field, such as $Q_7\left(\right(t\left)\right)$.