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 $n\equiv 3$, $5(\mathrm{mod}8)$.

15 pages

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