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

الأصل · EN

In this paper we classify curves of genus 2 with group of automorphisms isomorphic to D₈ or D₁2 over an arbitrary field k (of characteristic different from 2 in the D₈ case and from 2 and 3 in the D₁₂ 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 D₈ and D₁₂ are precisely the Q-curves of degrees 2 and 3, respectively, and we determine which curves have jacobians of GL₂-type.