Irreducible binary cubics and the generalised superelliptic equation over number fields

DOI10.4064/AA180814-2-5zbMATH Open1469.11175arXiv1808.04726OpenAlexW3098415251WikidataQ127000414 ScholiaQ127000414MaRDI QIDQ5207280

George C. Ţurcaş

Publication date: 19 December 2019

Published in: Acta Arithmetica (Search for Journal in Brave)

Abstract: For a large class (heuristically most) of irreducible binary cubic forms $F(x,y)inmathbbZ[x,y]$, Bennett and Dahmen proved that the generalized superelliptic equation $F(x,y)=z^l$ has at most finitely many solutions in $x,yinmathbbZ$ coprime, $zinmathbbZ$ and exponent $linmathbb{Z}_{geq4}$. 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.

Full work available at URL: https://arxiv.org/abs/1808.04726

zbMATH Keywords

generalized superelliptic equationmod \(l\) Galois representationsRibet's level lowering theorem

Mathematics Subject Classification ID

Elliptic curves over global fields (11G05) Modular and automorphic functions (11F03) Galois representations (11F80) Cohomology of arithmetic groups (11F75)

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