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Ternary Diophantine Equations via Galois Representations and Modular Forms

  Published:2004-02-01
 Printed: Feb 2004
  • Michael A. Bennett
  • Chris M. Skinner
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Abstract

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.
MSC Classifications: 11D41, 11F11, 11G05 show english descriptions Higher degree equations; Fermat's equation
Holomorphic modular forms of integral weight
Elliptic curves over global fields [See also 14H52]
11D41 - Higher degree equations; Fermat's equation
11F11 - Holomorphic modular forms of integral weight
11G05 - Elliptic curves over global fields [See also 14H52]
 

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