Quasiplatonic curves with symmetry group ${\mathbb{Z}}_2^2⋊{\mathbb{Z}}_m$ are definable over $\mathbb{Q}$

Abstract

It is well known that every closed Riemann surface $S$ of genus $g\ge 2$, 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 $A⋊{\mathbb{Z}}_2$, where $A$ is an abelian group. On the other hand, as shown by Streit and Wolfart, if $G={\mathbb{Z}}_p⋊{\mathbb{Z}}_q$ where $p,q>3$ are prime integers, then $S$ is not necessarily definable over $\mathbb{Q}$. In this paper, we observe that if $G={\mathbb{Z}}_2^2⋊{\mathbb{Z}}_m$ with $m\ge 3$, 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.

References in corpus (1)

• On The Group Algebra Decomposition of a Jacobian Variety

Cited by in corpus (9)

• On the one-dimensional family of Riemann surfaces of genus $q$ with $4q$ automorphisms
• A note on large automorphism groups of compact Riemann surfaces
• On Jacobians with group action and coverings
• On families of Riemann surfaces with automorphisms
• On Riemann surfaces of genus $g$ with $4g-4$ automorphisms
• On $pq$-fold regular covers of the projective line
• On large prime actions on Riemann surfaces
• Nilpotent groups of automorphisms of families of Riemann surfaces
• A note on Jacobians of quasiplatonic Riemann surfaces with complex multiplication