On the asymptotic Fermat’s last theorem over number fields

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 $T\ne \varnothing$, 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.