On curves of genus 2 with Jacobian of GL(2)-type

Ribet [Ri] has generalized the conjecture of Shimura-Taniyama-Weil to abeli an varieties defined over Q, giving rise to the study of abelian varieties of GL(2)-type. In this context, all curves over Q of genus one have Jacobia n variety of GL(2)-type. Our aim in this paper is to begin with the analysi s of which curves of genus 2 have Jacobian variety of GL(2)-type. To this e nd, we restrict our attention to curves with rational Rosenhain model and n on-abelian automorphism group, and use the embedding of this group into the endomorphism algebra of its Jacobian variety to determine if it is of GL(2 )-type.