Birational automorphism groups of Severi-Brauer surfaces over the field of rational numbers

TL;DR

This paper classifies the finite subgroups of the birational automorphism groups of non-trivial Severi-Brauer surfaces over the rationals, identifying specific cyclic groups and their embeddings depending on the field's characteristics.

Contribution

It determines the structure of finite subgroups of birational automorphism groups of Severi-Brauer surfaces over the rationals, revealing the presence of specific cyclic groups and their conditions.

Findings

Finite subgroups are limited to Z/3Z and (Z/3Z)^2.

(Z/3Z)^2 is always contained in the birational automorphism group.

(Z/3Z)^3 is contained when the field has a non-trivial cube root of unity.

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 $\mathrm{Bir}(V)$ 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 $\mathrm{Bir}(V)$ 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.