Irreducible binary cubics and the generalized superelliptic equation over number fields

TL;DR

This paper extends results on the finiteness of solutions to generalized superelliptic equations involving irreducible binary cubic forms from integers to coefficients in the ring of integers of arbitrary number fields, leveraging modularity conjectures.

Contribution

It generalizes previous finiteness results for binary cubic forms over integers to those over arbitrary number fields, using modularity conjectures.

Findings

Finiteness of solutions for generalized superelliptic equations over number fields.

Extension of modularity-based methods to broader algebraic settings.

Framework for applying modularity conjectures to binary cubic forms over number fields.

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.