# On the asymptotic Fermat’s last theorem over number fields

### Mehmet Haluk Şengün

University of Sheffield, UK### Samir Siksek

University of Warwick, Coventry, UK

## 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 $T \ne \emptyset$, and moreover, that for every solution $(\lambda,\mu)$ to the $S$-unit equation

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