GALOIS REPRESENTATIONS ATTACHED TO Q-CURVES AND THE GENERALIZED FERMAT EQUATION A 4 +B 2 =C p

By Jordan S. Ellenberg. 2003

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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 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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/67c8c086f69b48593aaf275eb4566c1960194775

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Citations in the corpus

Ternary Diophantine Equations via Galois Representations and Modular Forms (2004, background): “While it does not seem possible to fully deal with this equation by these techniques (but see Darmon [17]), a number of partial (or even complete) results are available in case (p; q; r) = (p; p; 2) (see [19] for p 5), (p; q; r) = (p; p; 3) (see [19] for p 5), (3; 3; p) (see [30] for 3 p 22 ], [32]”
Generalized Fermat equations: A miscellany (2015, methodology): “(p, q, r) reference(s) (n, n, n), n ≥ 3 Wiles [68], Taylor-Wiles [66] (n, n, 2), n ≥ 4 Darmon-Merel [33], Poonen [57] (n, n, 3), n ≥ 3 Darmon-Merel [33], Poonen [57] (2n, 2n, 5), n ≥ 2 Bennett [1] (2, 4, n), n ≥ 4 Ellenberg [39], Bennett-Ellenberg-Ng [5], Bruin [15] (2, 6, n), n ≥ 3 Bennett-Chen [2]”