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

Abstract

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 \mathcal M_2 . The locus of curves with group of automorphisms isomorphic to one of the dihedral groups D 8 D_8 or D 12 D_{12} is a one-dimensional subvariety. In this paper we classify these curves over an arbitrary perfect field k k of characteristic char ⁡ k ≠ 2 \operatorname {char} k\neq 2 in the D 8 D_8 case and char ⁡ k ≠ 2 , 3 \operatorname {char} k\neq 2,3 in the D 12 D_{12} case. We first parameterize the k ¯ \overline k -isomorphism classes of curves defined over k k by the k k -rational points of a quasi-affine one-dimensional subvariety of M 2 \mathcal M_2 ; then, for every curve C / k C/k representing a point in that variety we compute all of its k k -twists, which is equivalent to the computation of the cohomology set H 1 ( G k , Aut ⁡ ( C ) ) 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 GL 2 ⁡ ( k ¯ ) \operatorname {GL}_2(\overline k) . In particular, we give two generic hyperelliptic equations, depending on several parameters of k k , that by specialization produce all curves in every k k -isomorphism class.