Klein forms and the generalized superelliptic equation

Abstract

If $F(x,y) \in \mathbb{Z}[x,y]$ is an irreducible binary form of degree $k \geq 3$, then a theorem of Darmon and Granville implies that the generalized superelliptic equation $$ F(x,y)=z^l $$ has, given an integer $l \geq \mathrm{max} \{ 2, 7-k \}$, 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.