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On pseudo-real finite subgroups of $operatorname{PGL}_3(mathbb{C})$ | | Eslam Badr
; Ahmad Elguindy
; | | Date: |
2 Jan 2023 | | 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
$exists,phiinoperatorname{PGL}_3(mathbb{C})$ 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
$G’subsetoperatorname{PGL}_3(mathbb{R})$. 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
$G=mathbb{Z}/nmathbb{Z}$ in $operatorname{PGL}_3(mathbb{C})$ has {a real
field of moduli} and we provide a necessary and sufficient condition for
$G=mathbb{Z}/nmathbb{Z}$ 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 $ngeq3$ 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. | | Source: | arXiv, 2301.00543 | | Services: | Forum | Review | PDF | Favorites | |
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