Q-curves and Galois representations

Let K be a number field, Galois over Q. A Q-curve over K is an elliptic curve over K which is isogenous to all its Galois conjugates. The current interest in Q-curves, it is fair to say, began with Ribet’s observation [27] that an elliptic curve over Q̄ admitting a dominant morphism from X1(N) must be a Q-curve. It is then natural to conjecture that, in fact, all Q-curves are covered by modular curves. More generally, one might ask: from our rich storehouse of theorems about elliptic curves over Q, which ones generalize to Q-curves? In this paper, we discuss recent progress towards several problems of this type, and some Diophantine applications. We will also state several open problems which seem both interesting and accessible to existing methods. Remark 1. Elliptic curves with complex multiplication supply a natural population of Q-curves. Indeed, the original use of the term “Q-curve”, by Gross [13], referred to CM curves exclusively. The arithmetic of CM curves is much more fully understood than that of curves without extra endomorphisms. For that reason, we will assume hereafter that our Q-curves are not CM. One might think of the class of Q-curves as the “mildest possible generalization” of the class of elliptic curves over Q. For many structures on elliptic curves over Q are invariant under isogeny. And since a Q-curve E/K has an isogeny class which is fixed by Gal(Q̄/Q), we should expect that any isogeny-invariant structure of elliptic curves over Q can be defined for Q-curves as well. The structure we have chiefly in mind is the `-adic Galois representation