On pseudo-real finite subgroups of $\operatorname{PGL}_3(\mathbb{C})$

Abstract

Let $G$ be a finite subgroup of $\operatorname{PGL}_3(\mathbb{C})$, and let $\sigma$ be the generator of $\operatorname{Gal}(\mathbb{C}/\mathbb{R})$. We say that $G$ has a \emph{real field of moduli} if $^{\sigma}G$ and $G$ are $\operatorname{PGL}_3(\mathbb{C})$-conjugates, that is, if $\exists\,\phi\in\operatorname{PGL}_3(\mathbb{C})$ such that $\phi^{-1}\,G\,\phi=\,^{\sigma}G$. Furthermore, we say that $\mathbb{R}$ is \emph{a field of definition for $G$} or that \emph{$G$ is definable over $\mathbb{R}$} if $G$ is $\operatorname{PGL}_3(\mathbb{C})$-conjugate to some $G'\subset\operatorname{PGL}_3(\mathbb{R})$. In this situation, we call $G'$ \emph{a model for $G$ over $\mathbb{R}$}. If $G$ has $\mathbb{R}$ as a field of definition but is not definable over $\mathbb{R}$, then we call $G$ \emph{pseudo-real}. In this paper, we first show that any finite cyclic subgroup $G=\mathbb{Z}/n\mathbb{Z}$ in $\operatorname{PGL}_3(\mathbb{C})$ has {a real field of moduli} and we provide a necessary and sufficient condition for $G=\mathbb{Z}/n\mathbb{Z}$ to be definable over $\mathbb{R}$; see Theorems 2.1, 2.2, and 2.3. We also prove that any dihedral group $\operatorname{D}_{2n}$ with $n\geq3$ in $\operatorname{PGL}_3(\mathbb{C})$ is definable over $\mathbb{R}$; see Theorem 2.4. Furthermore, we study all six classes of finite primitive subgroups of $\operatorname{PGL}_3(\mathbb{C})$, and show that all of them except the icosahedral group $\operatorname{A}_5$ are pseudo-real; see Theorem 2.5, whereas $\operatorname{A}_5$ is definable over $\mathbb{R}$. Finally, we explore the connection of these notions in group theory with their analogues in arithmetic geometry; see Theorem 2.6 and Example 2.7.