Quasiplatonic curves with symmetry group Z22⋊Zm are definable over Q

It is well known that every closed Riemann surface S of genus g⩾2, admitting a group G of conformal automorphisms so that S/G has triangular signature, can be defined over a finite extension of Q. It is interesting to know, in terms of the algebraic structure of G, if S can in fact be defined over Q. This is the situation if G is either abelian or isomorphic to A⋊Z2, where A is an abelian group. On the other hand, as shown by Streit and Wolfart, if G≅Zp⋊Zq, where p,q>3 are prime integers, then S is not necessarily definable over Q. In this paper, we observe that if G≅Z22⋊Zm with m⩾3, then S can be defined over 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.