Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

Abstract

For a large class (heuristically most) of irreducible binary cubic forms $F(x,y) \in \mathbb Z [x,y]$, Bennett and Dahmen proved that the generalized superelliptic equation $F(x,y)=z^l$ has at most finitely many solutions in $x,y \in \mathbb Z$ coprime, $z \in \mathbb Z$ and exponent $l \in \mathbb Z _{\geq 4}$. Their proof uses, among other ingredients, modularity of certain mod $l$ Galois representations and Ribet’s level lowering theorem. The aim of this paper is to treat the same problem for binary cubics with coefficients in $\mathcal O _K$, the ring of integers of an arbitrary number field $K$, using by now well-documented modularity conjectures.