Sengun, M.H. orcid.org/0000-0002-6210-6877 and Siksek, S. (2018) On the asymptotic Fermat’s last theorem over number fields. Commentarii Mathematici Helvetici, 93 (2). pp. 359-375. ISSN 0010-2571
Abstract
Let K be a number field, S be the set of primes of K above 2 and T the subset of primes above 2 having inertial degree 1. Suppose that T≠∅, and moreover, that for every solution (λ,μ) to the S-unit equation λ+μ=1,λ, μ∈O×S, there is some P∈T such that max{νP(λ),νP(μ)}≤4νP(2). Assuming two deep but standard conjectures from the Langlands programme, we prove the asymptotic Fermat's last theorem over K: there is some BK such that for all prime exponents p>BK the only solutions to xp+yp+zp=0 with x, y, z∈K satisfy xyz=0. We deduce that the asymptotic Fermat's last theorem holds for imaginary quadratic fields Q(−d−−−√) with −d≡ 2, 3 (mod) 4) squarefree.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2018 European Mathematical Society. This is an author produced version of a paper subsequently published in Commentarii Mathematici Helvetici. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Fermat equation; Bianchi modular forms; Galois representations |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 19 Jun 2018 13:23 |
Last Modified: | 19 Apr 2024 21:45 |
Status: | Published |
Publisher: | European Mathematical Society |
Refereed: | Yes |
Identification Number: | 10.4171/CMH/437 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:132166 |