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

TL;DR

This paper proves that quasiplatonic curves with symmetry group ${ m Z}_2^2 times { m Z}_m$ are definable over ${ m Q}$ and provides explicit models and decompositions for their Jacobians.

Contribution

It establishes that certain quasiplatonic curves with specific symmetry groups are definable over ${ m Q}$ and constructs explicit models and Jacobian decompositions.

Findings

Curves with group ${ m Z}_2^2 times { m Z}_m$ are definable over ${ m Q}$.

Provides explicit models for these curves and their automorphism groups.

Describes isogenous decompositions of their Jacobian varieties.

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.