Linearization of finite subgroups of Cremona groups over non-closed fields

We study linearizability properties of finite subgroups of the Cremona group ${\mathrm{C}\mathrm{r}}_n(k)$ in the case where $k$ is a global field, with the focus on the local-global principle. For every global field $k$ of characteristic different from 2 and every $n\ge 3$ we give an example of a birational involution of ${\mathbb{P}}_k^n$ (=an element $g$ of order $2$ in ${\mathrm{C}\mathrm{r}}_n(k)$) such that $g$ is not $k$-linearizable but $g$ is $k_v$-linearizable in ${\mathrm{C}\mathrm{r}}_n(k_v)$ for all places $v$ of $k$. The main tool is a new birational invariant generalizing those introduced by Manin and Voskresenski\uı in the arithmetic case and by Bogomolov–Prokhorov in the geometric case. We also apply it to the study of birational involutions in real plane.