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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 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 chark ≠ 2 in the Ds case and char k ≠ 2,3 in the D 12 case. We first parameterize the k-isomorphism classes of curves defined over k by the k-rational points of a quasi-affine one-dimensional subvariety of 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, 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 GL 2 (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.

Curves of genus 2 with group of automorphisms isomorphic to 𝐷₈ or 𝐷₁₂