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arXiv · 1808.04726

Irreducible binary cubics and the generalized superelliptic equation over number fields

George Catalin Turcas
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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.

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George Catalin Turcas. 2019-05-02. Irreducible binary cubics and the generalized superelliptic equation over number fields. https://doi.org/10.4064/aa180814-2-5

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