Q-curves and Galois representations

Introduction and definitions 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 Q ̄ 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 modula