manuscripta mathematica

, Volume 98, Issue 1, pp 37-54

First online:

On curves of genus 2 with Jacobian of GL2-type

  • G. CardonaAffiliated withFacultat de Matemàtiques i Estadística, Universitat Politècnica de Catalunya, Pau Gargallo, 5, E-08028 Barcelona, Spain.¶e-mail: cardona@grec.upc.es
  • , J. GonzálezAffiliated withFacultat de Matemàtiques i Estadística, Universitat Politècnica de Catalunya, Pau Gargallo, 5, E-08028 Barcelona, Spain.¶e-mail: cardona@grec.upc.es
  • , J. C. LarioAffiliated withFacultat de Matemàtiques i Estadística, Universitat Politècnica de Catalunya, Pau Gargallo, 5, E-08028 Barcelona, Spain.¶e-mail: cardona@grec.upc.es
  • , A. RioAffiliated withFacultat de Matemàtiques i Estadística, Universitat Politècnica de Catalunya, Pau Gargallo, 5, E-08028 Barcelona, Spain.¶e-mail: cardona@grec.upc.es

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Abstract:

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.

Mathematics Subject Classification (1991):14H40, 11G10, 11F11