Klein forms and the generalized superelliptic equation | Annals of Mathematics

Abstract If F(x,y)∈ℤ[x,y] is an irreducible binary form of degree k≥3, then a theorem of Darmon and Granville implies that the generalized superelliptic equation F(x,y)=zl has, given an integer l≥max{2,7−k}, at most finitely many solutions in coprime integers x,y and z. In this paper, for large classes of forms of degree k=3,4,6 and 12 (including, heuristically, “most” cubic forms), we extend this to prove a like result, where the parameter l is now taken to be variable. In the case of irreducible cubic forms, this provides the first examples where such a conclusion has been proven. The method of proof combines classical invariant theory, modular Galois representations, and properties of elliptic curves with isomorphic mod-n Galois representations.

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数学年刊(Annals of Mathematics)

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