Let M 2 be the moduli space that classifies genus 2 curves. If a curve C is defined over a field k, the corresponding moduli point P = [C] ∈ M 2 is defined over k. In [7], Mestre solves the converse problem for curves with Aut(C) ≃ C 2 . Given a moduli point defined over k, Mestre finds an obstruction to the existence of a corresponding curve defined over k, that is an element in Br 2 (k) not always trivial. In this paper we prove that for all the other possibilities of Aut(C), every moduli point defined over k is represented by a curve defined over k. We also give an explicit construction of such a curve in terms of the coordinates of the moduli point.

IntroductionA Q-curve is an elliptic curve de®ned over Q that is isogenous to all its Galois conjugates. The term Q-curve was ®rst used by Gross to denote a special class of elliptic curves with complex multiplication having that property, and later generalized by Ribet to denote any elliptic curve isogenous to its conjugates. In this paper we deal only with Q-curves with no complex multiplication; the complex multiplication case requires different techniques.An abelian variety of GL 2 -type is an abelian variety A de®ned over Q such that the Q-algebra of Q-endomorphisms E Q End Q A is a number ®eld of degree equal to the dimension of the variety; the reason for the name is that the Galois action on the ,-adic Tate module of the variety gives rise to a representation of G Q with values in GL 2 E Q , . The main source of abelian varieties of GL 2 -type is a construction by Shimura (see [12, Theorem 7.14]) of abelian varieties A f attached to newforms f for the congruence subgroups G 1 N .Recent interest in Q-curves with no complex multiplication has been motivated by the works of Elkies [1] and Ribet [9] on the subject. In [1], Elkies shows that every isogeny class of Q-curves with no complex multiplication contains a curve whose j-invariant corresponds to a rational non-cusp non-CM point of the modular curve X Ã N quotient of the curve X 0 N by all the Atkin±Lehner involutions, for some squarefree integer N . In [9], Ribet characterizes Q-curves as the elliptic curves de®ned over Q that are quotients of some abelian variety of GL 2 -type. In the same paper, he gives evidence for the conjecture that the varieties A f constructed by Shimura exhaust (up to isogeny) all the abelian varieties of GL 2type; in particular, and as a consequence, one has the conjectural characterization of Q-curves as those elliptic curves over Q that are quotients of some J 1 N .The condition of being a Q-curve is invariant by isogeny. It is then natural to investigate some of their arithmetic properties up to isogeny; in particular their ®elds of de®nition. In [10] Ribet attaches to a Q-curve C a two-cocycle class yC P H 2 G Q ; Q Ã , which is in fact an invariant of the isogeny class of C, and characterizes the ®elds of de®nition for the curve up to isogeny in terms of yC. Then he uses that characterization to show that there is always a smallest such ®eld, which is a ®eld of type 2; . . . ; 2.Many aspects of the arithmetic of a Q-curve appear when we place it in a ®eld over which not only the curve is de®ned but also all its Galois conjugates and the

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 automor-phisms 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 char k = 2 in the D 8 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. 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 G k is the Galois group of an algebraic closure k/k. The Galois action on the elements of any G k-set will be denoted exponentially on the left: (σ, a) → σ a for σ ∈ G k and a in a G k-set. Whenever a cohomology group or set H i (G k , A) is considered, we mean Galois cohomology, where cocycles are continuous with respect to the discrete topology on A and the Krull topology on G k. We refer the reader to [7] for definitions and basic results on nonabelian Galois cohomology. Some results are stated in terms of elements of Br 2 (k) H 1 (G k , {±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 2 = a, j 2 = b, ji = −ij.

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