Ternary Diophantine Equations via Galois Representations and Modular Forms
- 1 February 2004
- journal article
- Published by Canadian Mathematical Society in Canadian Journal of Mathematics
- Vol. 56 (1), 23-54
- https://doi.org/10.4153/cjm-2004-002-2
Abstract
In this paper, we develop techniques for solving ternary Diophantine equations of the shape Axn + Byn = Cz2, based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters A, B andC. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan–Nagell type.Keywords
This publication has 30 references indexed in Scilit:
- Modularity of Fibres in Rigid Local SystemsAnnals of Mathematics, 1999
- Sur l'équation a3+ b3= cpExperimental Mathematics, 1998
- Majorations Effectives Pour L’ Équation de Fermat GénéraliséeCanadian Journal of Mathematics, 1997
- On Deformation Rings and Hecke RingsAnnals of Mathematics, 1996
- On the diophantine equations x2 + 74 = y5 and x2 + 86 = y5Glasgow Mathematical Journal, 1996
- On the Equations zm
= F (x, y ) and Axp
+ Byq
= Czr
Bulletin of the London Mathematical Society, 1995
- Sur les représentations modulaires de degré 2 de Gal(Q¯/Q)Duke Mathematical Journal, 1987
- On the Diophantine equation $x\sp {2n}-\mathcal{D}y^2=1$Proceedings of the American Mathematical Society, 1986
- Rational isogenies of prime degreeInventiones Mathematicae, 1978
- On the Diophantine equationCx2+D=ynPacific Journal of Mathematics, 1964