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Irreducible binary cubics and the generalised superelliptic equation over number fields
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Ţurcaş, George C. (2020) Irreducible binary cubics and the generalised superelliptic equation over number fields. Acta Arithmetica, 192 . pp. 73-93. doi:10.4064/aa180814-2-5 ISSN 0065-1036.
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Official URL: https://doi.org/10.4064/aa180814-2-5
Abstract
For a large class (heuristically most) of irreducible binary cubic forms F(x,y)∈Z[x,y], Bennett and Dahmen proved that the generalized superelliptic equation F(x,y)=zl has at most finitely many solutions in x,y∈Z coprime, z∈Z and exponent l∈Z≥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 OK, the ring of integers of an arbitrary number field K, using by now well-documented modularity conjectures.
| Item Type: | Journal Article | ||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Algebraic fields, Number theory, Galois theory | ||||||||
| Journal or Publication Title: | Acta Arithmetica | ||||||||
| Publisher: | IMPAN | ||||||||
| ISSN: | 0065-1036 | ||||||||
| Official Date: | 2020 | ||||||||
| Dates: |
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| Volume: | 192 | ||||||||
| Page Range: | pp. 73-93 | ||||||||
| DOI: | 10.4064/aa180814-2-5 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Open Access (Creative Commons) | ||||||||
| Copyright Holders: | Instytut Matematyczny PAN, 2019 | ||||||||
| Date of first compliant deposit: | 28 October 2019 | ||||||||
| Date of first compliant Open Access: | 4 November 2019 | ||||||||
| RIOXX Funder/Project Grant: |
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| Is Part Of: | 1 |
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