Quasiplatonic curves with symmetry group ${\mathbb Z}_{2}^{2} \rtimes {\mathbb Z}_{m}$ are definable over ${\mathbb Q}$

Abstract: It is well known that every closed Riemann surface $S$ of genus $g \geq 2$, admitting a group $G$ of conformal automorphisms so that $S/G$ has triangular signature, can be defined over a finite extension of ${\mathbb Q}$. It is interesting to know, in terms of the algebraic structure of $G$, if $S$ can in fact be defined over ${\mathbb Q}$. This is the situation if $G$ is either abelian or isomorphic to $A \rtimes {\mathbb Z}{2}$, where $A$ is an abelian group. On the other hand, as shown by Streit and Wolfart, if $G = {\mathbb Z}{p} \rtimes {\mathbb Z}{q}$ where $p,q>3$ are prime integers, then $S$ is not necessarily definable over ${\mathbb Q}$. In this paper, we observe that if $G={\mathbb Z}{2}^{2} \rtimes {\mathbb Z}_{m}$ with $m \geq 3$, then $S$ can be defined over ${\mathbb Q}$. Moreover, we describe explicit models for $S$, the corresponding groups of automorphisms and an isogenous decomposition of their Jacobian varieties as product of Jacobians of hyperelliptic Riemann surfaces.