Klein forms and the generalized superelliptic equation

Abstract

If F (x;y)2 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) = z l has, given an integer l maxf2; 7 kg, at most nitely 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 rst 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.