Fields of definition of Q-curves

Let C be a Q-curve with no complex multiplication. In this note we characterize the number fields K such that there is a curve C′ isogenous to C having all the isogenies between its Galois conjugates defined over K, and also the curves C′ isogenous to C defined over a number field K such that the abelian variety ResK/Q(C/K) obtained by restriction of scalars is a product of abelian varieties of GL2-type. 1 Definitions, notation and basic facts We work in the category of abelian varieties up to isogeny. Endk(A) will denote the Q-algebra of endomorphisms defined over a field k of an abelian variety A. For a Galois (profinite) group G all the G-modules are discrete and we always assume that the corresponding cohomological objects are continuous. A Q-curve is an elliptic curve defined over a number field that is isogenous to all of its Galois conjugates. An abelian variety of GL2-type is an abelian variety A defined over Q whose Q-algebra of endomorphisms EndQ(A) is a number field of degree equal to its dimension (these are the primitive abelian varieties of GL2-type of Ribet’s definition in 1.1). Both families appear in generalizations of the Shimura-Taniyama conjecture: the Q-curves are conjecturally the elliptic curves C/Q for which there is a nontrivial morphism X1(N)→ C; the abelian varieties of GL2-type are conjecturally the varieties Q-isogenous to a Q-simple factor of some J1(N). The two conjectures are equivalent as a consequence of the following Theorem 1.1 (Ribet [4]) An elliptic curve over Q is a Q-curve if, and only if, it is a quotient of some abelian variety of GL2-type. We will only consider Q-curves with no complex multiplication, the study of the complex multiplication case requiring different techniques. Let C/Q be a Q-curve. We will say that C is completely defined over a number field K if all the Galois conjugates of C and the isogenies between them are defined over K. ∗Research partially supported by DGICYT PB96-0970-C02-02 grant