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

TL;DR

This paper classifies genus 2 curves with specific automorphism groups over various fields, providing detailed arithmetic insights into their elliptic quotients and Jacobians, including over the rationals.

Contribution

It offers a complete classification of genus 2 curves with automorphism groups D_8 or D_12 over arbitrary fields, extending understanding of their arithmetic properties.

Findings

Elliptic quotients are Q-curves of degrees 2 and 3.

Identifies which Jacobians are of GL_2-type.

Provides explicit classifications over different fields.

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.