Commentarii Mathematici Helvetici
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Published online: 2018-05-31
On the asymptotic Fermat’s last theorem over number fields
Mehmet Haluk Şengün[1] and Samir Siksek[2] (1) University of Sheffield, UK(2) University of Warwick, Coventry, UK
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 \[ \lambda+\mu=1, \quad \lambda,~\mu \in \mathcal O_S^\times, \] 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.
Keywords: Fermat equation, Bianchi modular forms, Galois representations
Şengün Mehmet Haluk, Siksek Samir: On the asymptotic Fermat’s last theorem over number fields. Comment. Math. Helv. 93 (2018), 359-375. doi: 10.4171/CMH/437