Curves of genus 2 with group of automorphisms isomorphic to D_8 or D_12, by Gabriel Cardona and Jordi Quer

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.

Gabriel Cardona and Jordi Quer <gabriel.cardona@uib.es, quer@ma2.upc.es>