, Volume 98, Issue 1, pp 37-54

On curves of genus 2 with Jacobian of GL2-type

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Ribet [Ri] has generalized the conjecture of Shimura–Taniyama–Weil to abelian varieties defined over Q,giving rise to the study of abelian varieties of GL2-type. In this context, all curves over Q of genus one have Jacobian variety of GL2-type. Our aim in this paper is to begin with the analysis of which curves of genus 2 have Jacobian variety of GL2-type. To this end, we restrict our attention to curves with rational Rosenhain model and non-abelian automorphism group, and use the embedding of this group into the endomorphism algebra of its Jacobian variety to determine if it is of GL2-type.

Received: 31 March 1998 / Revised version: 29 June 1998