A Q-curve is an elliptic curve over \bar{Q} which is isogenous to its Galois conjugates. Ribet has conjectured that every Q-curve is a quotient of some J_0(N)/\bar{Q}. We prove, in joint work with Chris Skinner, that this conjecture is true under certain local conditions at 3. As a consequence, we show that the equation A^4 + B^2 = C^p in coprime A,B,C has only the trivial solutions, under certain 2-adic conditions on A,B,C, and p.