Ternary Diophantine Equations via Galois Representations and Modular Forms

http://dx.doi.org/10.4153/CJM-2004-002-2
Canad. J. Math. 56(2004), 23-54

  Published:2004-02-01
 Printed: Feb 2004

In this paper, we develop techniques for solving ternary Diophantine equations of the shape $Ax^n + By^n = Cz^2$, 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 $A$, $B$ and $C$. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan--Nagell type.