Fermat’s Last Theorem on totally real fields

Abstract

[en] Fermat's Last Theorem states the equation $$ a^n+b^n+c^n=0 $$ has only trivial solutions, i.e $a b c=0$, for $n>2$ and $a, b, c$ integers. The idea of the proof is to attach the Frey Curve $$ E_{a^p, b^p, c^p}: y^2=x\left(x-a^p\right)\left(x+b^p\right), $$ of course we assume $a, b, c$ are coprime integers with $a \equiv-1 \bmod 4$ and $2 \mid b$. The conductor of this curve is $$ N_{a^p, b^p, c^p}=\prod_{\ell \mid a b c, \ell \text { prime }} \ell . $$ The curve is semistable and so modular by Wile's Theorem, since the conductor is of the form $2 N$ for some odd integer $N$, we can apply Ribet's Theorem to show there is a weight 2 newform $g$ of level 2 such that $\bar{\rho}_g \cong$ of level 2.

The first section is devoted to introduce the concepts needed to understand in more extense this proof. So, Galois representations, modular forms and Elliptics are introduced and some results stated. At the end, a more detailed proof is given.

In the second section we consider solutions over some real quadratic feilds $K$. We show a non-trivial solution in $K$ gives rise to a non-trivial solution.