JournalscmhVol. 93, No. 2pp. 359–375

On the asymptotic Fermat’s last theorem over number fields

  • Mehmet Haluk Şengün

    University of Sheffield, UK
  • Samir Siksek

    University of Warwick, Coventry, UK
On the asymptotic Fermat’s last theorem over number fields cover

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Abstract

Let KK be a number field, SS be the set of primes of KK above 2 and TT the subset of primes above 2 having inertial degree 1. Suppose that TT \ne \emptyset, and moreover, that for every solution (λ,μ)(\lambda,\mu) to the SS-unit equation

λ+μ=1,λ, μOS×,\lambda+\mu=1, \quad \lambda,~\mu \in \mathcal O_S^\times,

there is some PT\mathfrak P \in T such that max{νP(λ),νP(μ)}4νP(2)\{ \nu_\mathfrak P(\lambda),\nu_\mathfrak P(\mu)\} \le 4 \nu_\mathfrak P(2). Assuming two deep but standard conjectures from the Langlands programme, we prove the asymptotic Fermat's last theorem over KK: there is some BKB_K such that for all prime exponents p>BKp > B_K the only solutions to xp+yp+zp=0x^p+y^p+z^p=0 with xx, yy, zKz \in K satisfy xyz=0xyz=0. We deduce that the asymptotic Fermat's last theorem holds for imaginary quadratic fields Q(d)\mathbb Q(\sqrt{-d}) with d-d \equiv 2, 3 (mod) 4) squarefree.

Cite this article

Mehmet Haluk Şengün, Samir Siksek, On the asymptotic Fermat’s last theorem over number fields. Comment. Math. Helv. 93 (2018), no. 2, pp. 359–375

DOI 10.4171/CMH/437