Irreducible binary cubics and the generalised superelliptic equation over number fields
Volume 192 / 2020
Acta Arithmetica 192 (2020), 73-93
MSC: Primary 11G05, 11F80; Secondary 11F03, 11F75.
DOI: 10.4064/aa180814-2-5
Published online: 18 October 2019
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.
