Curves of genus 2 with group of automorphisms isomorphic to D_8 or D_12

Abstract

In this paper we classify curves of genus 2 with group of automorphisms isomorphic to D_8 or D_12 over an arbitrary field k (of characteristic different from 2 in the D_8 case and from 2 and 3 in the D_{12} case) up to k-isomorphism. As an application of the classification of curves of genus 2 obtained, we get precise arithmetic information on their elliptic quotients and on their jacobians. Over the field k=Q, we show that the elliptic quotients of the curves with automorphisms D_8 and D_{12} are precisely the Q-curves of degrees 2 and 3, respectively, and we determine which curves have jacobians of GL_2-type.