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 $3 \le d \le 23$ squarefree, $d \ne 5$, $17$, the Fermat equation $x^n+y^n=z^n$ has no non-trivial solutions over the quadratic field $\mathbb{Q}(\sqrt{d})$ for $n \ge 4$. Furthermore, we show for $d=17$ that the same holds for prime exponents $n \equiv 3$, $5 \pmod{8}$.