Linearization of finite subgroups of Cremona groups over nonclosed fields

Abstract We study linearizability properties of finite subgroups of the Cremona group Cr n ( k ) $\operatorname {{\mathrm {Cr}}}_n(k)$ upper C r Subscript n Baseline left parenthesis k right parenthesis 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 $n \ge 3$ n greater than or equals 3 , we give an example of a birational involution of P k n $\mathbb P^n_k$ double struck upper P Subscript k Superscript n (= an element g of order 2 $2$ 2 in Cr n ( k ) $\operatorname {{\mathrm {Cr}}}_n(k)$ upper C r Subscript n Baseline left parenthesis k right parenthesis ) such that g is not k -linearizable but g is k v $k_v$ k Subscript v -linearizable in Cr n ( k v )