Curves of Genus 2 with Group of Automorphisms Isomorphic to D8 or D12

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 threedimensional moduli variety M2. The locus of curves with group of automorphisms isomorphic to one of the dihedral groupsD8 orD12 is a one-dimensional subvariety. In this paper we classify these curves over an arbitrary perfect field k of characteristic char k = 2 in the D8 case and char k = 2, 3 in the D12 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 M2; 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(Gk,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 GL2(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. 1. Preliminaries on hyperelliptic curves and curves of genus 2 This section contains basic definitions, notation, and some well-known facts on hyperelliptic curves and curves of genus 2. References are [2], [5], [6]. Throughout the paper, k is a perfect field of characteristic different from 2, and Gk is the Galois group of an algebraic closure k/k. The Galois action on the elements of any Gk-set will be denoted exponentially on the left: (σ, a) → a for σ ∈ Gk and a in a Gk-set. Whenever a cohomology group or set H(Gk, A) is considered, we mean Galois cohomology, where cocycles are continuous with respect to the discrete topology on A and the Krull topology on Gk. We refer the reader to [7] for definitions and basic results on nonabelian Galois cohomology. Some results are stated in terms of elements of Br2(k) H(Gk, {±1}), the 2-torsion of the Brauer group of the field k. We denote by (a, b) the class of the quaternion algebra with basis 1, i, j, ij and multiplication defined by i = a, j = b, ji = −ij. Received by the editors November 24, 2003 and, in revised form, June 7, 2005. 2000 Mathematics Subject Classification. Primary 11G30, 14G27.