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

We study linearizability properties of finite subgroups of the Cremona group Crₙ(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 ≥ 3 we give an example of a birational involution of Pⁿₖ (=an element g of order $2$ in Crₙ(k)) such that g is not k-linearizable but g is kᵥ-linearizable in Crₙ(kᵥ) for all places v of k. The main tool is a new birational invariant generalizing those introduced by Manin and Voskresenskiı 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.