Galois representations attached to ℚ-curves and the generalized fermat equation a4 + B2 = CP

Abstract

We prove that the equation A4 + B2 = Cp has no solutions in coprime positive integers when p ≥ 211. The main step is to show that, for all sufficiently large primes p, every ℚ-curve over an imaginary quadratic field K with a prime of potentially multiplicative reduction greater than 6 has a surjective mod p Galois representation. The bound on p depends on K and the degree of the isogeny between E and its Galois conjugate, but is independent of the choice of E. The proof of this theorem combines geometric arguments due to Mazur, Momose, Darmon, and Merel with an analytic estimate of the average special values of certain L-functions.