Let be a number field, be the set of primes of above 2 and the subset of primes above 2 having inertial degree 1. Suppose that , and moreover, that for every solution to the -unit equation
there is some such that max. Assuming two deep but standard conjectures from the Langlands programme, we prove the asymptotic Fermat's last theorem over : there is some such that for all prime exponents the only solutions to with , , satisfy . We deduce that the asymptotic Fermat's last theorem holds for imaginary quadratic fields with 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. 359–375DOI 10.4171/CMH/437