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

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Abstract

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 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.

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