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On the asymptotic Fermat’s last theorem over number fields

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Sengun, Mehmet Haluk and Siksek, Samir (2018) On the asymptotic Fermat’s last theorem over number fields. Commentarii Mathematici Helvetici, 93 (2). pp. 359-375. doi:10.4171/CMH/437

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Official URL: http://dx.doi.org/10.4171/CMH/437

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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.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH): Fermat's last theorem, Number theory
Journal or Publication Title: Commentarii Mathematici Helvetici
Publisher: European Mathematical Society Publishing House
ISSN: 0010-2571
Official Date: 31 May 2018
Dates:
DateEvent
31 May 2018Published
16 September 2017Accepted
Date of first compliant deposit: 20 June 2019
Volume: 93
Number: 2
Page Range: pp. 359-375
DOI: 10.4171/CMH/437
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
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