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

TL;DR

The paper determines the finite subgroups of the birational automorphism group of a non-trivial Severi–Brauer surface $V$ over $\mathbb{Q}$, proving that only $\mathbb{Z}/3\mathbb{Z}$ and $(\mathbb{Z}/3\mathbb{Z})^2$ can occur as subgroups of $\mathrm{Bir}(V)$, with $(\mathbb{Z}/3\mathbb{Z})^2$ always realizable over fields of characteristic not equal to $2,3$ that contain a non-trivial cube root of unity, and $(\mathbb{Z}/3\mathbb{Z})^3$ realizable under additional cube-root-of-unity assumptions. The approach reduces birational questions to birational actions on cubic surfaces obtained by blowing up six points on $V$, leveraging the Fermat cubic and the Weyl group $W(E_6)$ to constrain possible $3$-groups. Central simple algebras provide the automorphism-structure framework $\mathrm{Aut}(V) \simeq A^*/\mathbf{F}^*$, and Galois actions on the Picard group guide which group_actions lift to regular automorphisms. The results extend Shramov’s classifications by giving explicit realizability criteria tied to cube roots of unity in the ground field and by detailing the obstruction to larger $3$-groups in unextended settings.

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.

Theorems & Definitions (60)

• Definition 1.1
• Theorem 1.2: Shramov1
• Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• Remark 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• proof