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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
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: |
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| 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 | ||||||
| Related URLs: | |||||||
| Open Access Version: |
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