Galois representations attached to Q-curves and the generalized Fermat equation A^4 + B^2 = C^p
| dc.creator | Ellenberg, Jordan S. | |
| dc.date | 2003-11-27 | |
| dc.date.accessioned | 2026-07-07T05:03:20Z | |
| dc.date.available | 2026-07-07T05:03:20Z | |
| dc.description | 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. | |
| dc.description | 20pp. To appear, American Journal of Mathematics | |
| dc.identifier | https://arxiv.org/abs/math/0311497 | |
| dc.identifier | http://arxiv.org/abs/math/0311497 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/69372 | |
| dc.subject | Number Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 11G18; 11D41; 11G05; 11F67 | |
| dc.title | Galois representations attached to Q-curves and the generalized Fermat equation A^4 + B^2 = C^p | |
| dc.type | text |