Ternary Diophantine Equations via Galois Representations and Modular Forms

Michael A. Bennett; Chris M. Skinner

doi:10.4153/CJM-2004-002-2

Ternary Diophantine Equations via Galois Representations and Modular Forms

Michael A. Bennett
, Chris M. Skinner

Research output: Contribution to journal › Article › peer-review

170 Scopus citations

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 and C. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan-Nagell type.

Original language	English (US)
Pages (from-to)	23-54
Number of pages	32
Journal	Canadian Journal of Mathematics
Volume	56
Issue number	1
DOIs	https://doi.org/10.4153/CJM-2004-002-2
State	Published - Feb 2004
Externally published	Yes

All Science Journal Classification (ASJC) codes

General Mathematics

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10.4153/CJM-2004-002-2

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title = "Ternary Diophantine Equations via Galois Representations and Modular Forms",

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 and C. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan-Nagell type.",

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AB - 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 and C. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan-Nagell type.

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Ternary Diophantine Equations via Galois Representations and Modular Forms

Abstract

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