  We show that the mod p Galois representations attached to a Q-curve E of
degree d over an imaginary quadratic number field K are surjective for all p
larger than some constant M_{K,d}, if E has potentially multiplicative
reduction at any prime not dividing 6. The proof uses Mazur's formal immersion
method, a result of Darmon and Merel on non-split modular curves, and an
analytic argument showing that Jacobians of certain twisted modular curves
admit quotients with Mordell-Weil rank 0. In combination with a previous
theorem of the author and Skinner on modularity of Q-curves, the surjectivity
result allows one to show that the generalized Fermat equation A^4 + B^2 = C^p
has no nontrivial primitive solutions for p >= 211.
