On the generalized Fermat equation $x^{2\ell}+y^{2m}=z^p$ for $3 \le p \le 13$

Samuele Anni, Samir Siksek

Published 2015-06-09Version 1

We show that the generalized Fermat equation $x^{2\ell}+y^{2 m}=z^p$ has no non-trivial primitive solutions for primes $\ell$, $m \ge 5$, and $3 \le p \le 13$. This is achieved by relating a putative solution to a Frey curve over a real subfield of the $p$-th cyclotomic field, and studying its mod $\ell$ representation using modularity and level lowering. Along the way we prove the following modularity theorem. Let $K$ be a real abelian field of odd class number in which $5$ is unramified. Let $S_5$ be the set of places of $K$ above $5$. Suppose for every non-empty proper subset $S \subset S_5$ there is a totally positive unit $u \in \mathcal{O}_K$ such that $\prod_{\mathfrak{q} \in S} \mathrm{Norm}_{\mathbb{F}_\mathfrak{q}/\mathbb{F}_5}(u \bmod{\mathfrak{q}}) \ne \overline{1}$. Then every semistable elliptic curve over $K$ is modular.