Birational Automorphism Groups of Severi–Brauer Surfaces Over the Field of Rational Numbers

Abstract We prove that the only non-trivial finite subgroups of birational automorphism group of non-trivial Severi–Brauer surfaces over the field of rational numbers are $\mathbb{Z}/3\mathbb{Z}$ and $(\mathbb{Z}/3\mathbb{Z})^{2}.$ Moreover, we show that $(\mathbb{Z}/3\mathbb{Z})^{2}$ is contained in $\textrm{Bir}(V)$ for any Severi–Brauer surface $V$ over a field of characteristic different from $2$ and $3$, and $(\mathbb{Z}/3\mathbb{Z})^{3}$ is contained in $\textrm{Bir}(V)$ for any Severi–Brauer surface $V$ over a field of characteristic different from $2$ and $3$, which contains a non-trivial cube root of unity.