Galois representations attached to Q-curves and the generalized Fermat equation A^4 + B^2 = C^p
Abstract
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 M_{K,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 A^4 + B^2 = C^p has no nontrivial primitive solutions for p >= 211.
Keywords
Cite
@article{arxiv.math/0311497,
title = {Galois representations attached to Q-curves and the generalized Fermat equation A^4 + B^2 = C^p},
author = {Jordan S. Ellenberg},
journal= {arXiv preprint arXiv:math/0311497},
year = {2007}
}
Comments
20pp. To appear, American Journal of Mathematics