Fermat's last theorem over some small real quadratic fields

Abstract

Using modularity, level lowering, and explicit computations with Hilbert modular forms, Galois representations, and ray class groups, we show that for $3\le d\le 23$, where $d\ne 5,17$ and is squarefree, the Fermat equation $x^n+y^n=z^n$ has no nontrivial solutions over the quadratic field $\mathbb{Q}\left(\sqrt{d}\right)$ for $n\ge 4$. Furthermore, we show that for $d=17$, the same holds for prime exponents $n\equiv 3,5\left(mod8\right)$.