Curves of genus 2 with group of automorphisms isomorphic to $D_8$ or $D_{12}$

The classification of curves of genus 2 over an algebraically closed field was studied by Clebsch and Bolza using invariants of binary sextic forms, and completed by Igusa with the computation of the corresponding three-dimensional moduli variety $\mathcal M_2$. The locus of curves with group of automorphisms isomorphic to one of the dihedral groups $D_8$ or $D_{12}$ is a one-dimensional subvariety.

In this paper we classify these curves over an arbitrary perfect field $k$ of characteristic $\operatorname{char} k\neq2$ in the $D_8$ case and $\operatorname{char} k\neq2,3$ in the $D_{12}$ case. We first parameterize the $\overline k$-isomorphism classes of curves defined over $k$ by the $k$-rational points of a quasi-affine one-dimensional subvariety of $\mathcal M_2$; then, for every curve $C/k$ representing a point in that variety we compute all of its $k$-twists, which is equivalent to the computation of the cohomology set $H^1(G_k,\operatorname{Aut}(C))$.

The classification is always performed by explicitly describing the objects involved: the curves are given by hyperelliptic models and their groups of automorphisms represented as subgroups of $\operatorname{GL}_2(\overline k)$. In particular, we give two generic hyperelliptic equations, depending on several parameters of $k$, that by specialization produce all curves in every $k$-isomorphism class.