Ternary Diophantine Equations via Galois Representations and Modular Forms

[摘要] In this paper, we develop techniques for solving ternary Diophantineequations of the shape $Ax^n + By^n = Cz^2$, based upon the theory ofGalois representations and modular forms. We subsequently utilizethese methods to completely solve such equations for various choicesof the parameters $A$, $B$ and $C$. We conclude with an applicationof our results to certain classical polynomial-exponential equations,such as those of Ramanujan--Nagell type.

[学科分类] 数学（综合）