Q-Curves and Abelian Varieties of GL2-Type

The relation between Q-curves and certain abelian varietiesof GL₂-type was established by Ribet (‘Abelian varietiesover Q and modular forms’, Proceedings of the KAIST MathematicsWorkshop (1992) 53–79) and generalized to building blocks,the higher-dimensional analogues of Q-curves, by Pyle in herPhD Thesis (University of California at Berkeley, 1995). Inthis paper we investigate some aspects of Q-curves with no complexmultiplication and the corresponding abelian varieties of GL₂-type,for which we mainly use the ideas and techniques introducedby Ribet (op. cit. and ‘Fields of definition of abelianvarieties with real multiplication’, Contemp.\ Math. 174(1994) 107–118). After the Introduction, in Sections 2and 3 we obtain a characterization of the fields where a Q-curveand all the isogenies between its Galois conjugates can be definedup to isogeny, and we apply it to certain fields of type (2,...,2).In Section 4 we determine the endomorphism algebras of all theabelian varieties of GL₂-type having as a quotient a given Q-curvein easily computable terms. Section 5 is devoted to a particularcase of Weil's restriction of scalars functor applied to a Q-curve,in which the resulting abelian variety factors over Q up toisogeny as a product of abelian varieties of GL₂-type. Finally,Section 6 contains examples: we parametrize the Q-curves comingfrom rational points of the modular curves X*N having genuszero, and we apply the results of Sections 2–5 to someof the curves obtained. We also give results concerning theexistence of quadratic Q-curves. 1991 Mathematics Subject Classification:primary 11G05; secondary 11G10, 11G18, 11F11, 14K02.