On the asymptotic Fermat’s last theorem over number fields

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.