Q-curves and abelian varieties of GL2-type

A Q-curve is an elliptic curve defined over Q that is isogenous to all its Galois conjugates. The term Q-curve was first used by Gross to denote a special class of elliptic curves with complex multiplication having that property, and later generalized by Ribet to denote any elliptic curve isogenous to its conjugates. In this paper we deal only with Q-curves with no complex multiplication, the complex multiplication case requiring different techniques. An abelian variety of GL2-type is an abelian variety A defined over Q such that the Q-algebra of Q-endomorphisms E = Q ⊗ EndQ(A) is a number field of degree equal to the dimension of the variety; the reason for the name is that the Galois action on the `-adic Tate module of the variety gives rise to a representation of GQ with values in GL2(E ⊗ Q`). The main source of abelian varieties of GL2-type is a construction by Shimura (see [13, Theorem 7.14]) of abelian varieties Af attached to newforms f for the congruence subgroups Γ1(N). Recent interest in Q-curves with no complex multiplication has been motivated by the works of Elkies [1] and Ribet [10] on the subject. In [1], Elkies shows that every isogeny class of Q-curves with no complex multiplication contains a curve whose j-invariant corresponds to a rational noncusp non-CM point of the modular curve X∗(N) quotient of the curve X0(N) by all the Atkin-Lehner involutions, for some squarefree integer N . In [10], Ribet characterizes Q-curves as the elliptic curves defined over Q that are quotients of some abelian variety of GL2type. In the same paper, he gives evidence for the conjecture that the varieties Af constructed by Shimura exhaust (up to isogeny) all the abelian varieties of GL2-type; in particular, and as a consequence, one has the conjectural characterization of Q-curves as those elliptic curves over Q that are quotients of some J1(N).