Klein forms and the generalized superelliptic equation


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.


Michael A. Bennett

Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2

Sander R. Dahmen

Mathematisches Instituut, Universiteit Utrecht, P. O. Box 80 010, 3508 TA Utrecht ,The Netherlands