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Galois representations attached to [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]-curves and the generalized Fermat equation A 4 + B 2 = C P

Galois representations attached to [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]-curves and the generalized Fermat equation A 4 + B 2 = C P

We prove that the equation A 4 + B 2 = C p has no solutions in coprime positive integers when p ≥ 211. The main step is to show that, for all sufficiently large primes p , every [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]-curve over an imaginary quadratic field K with …