Galois representations attached to Q-curves and the generalized Fermat equation A⁴ + B² = Cᵖ

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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 Mₖ,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 A⁴ + B² = Cᵖ has no nontrivial primitive solutions for p >= 211.