#### Vol. 10, No. 6, 2016

 Recent Issues
 The Journal Cover Editorial Board Editors' Addresses Editors' Interests About the Journal Scientific Advantages Submission Guidelines Submission Form Subscriptions Editorial Login Contacts Author Index To Appear ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print)
Modular elliptic curves over real abelian fields and the generalized Fermat equation $x^{2\ell}+y^{2m}=z^p$

### Samuele Anni and Samir Siksek

Vol. 10 (2016), No. 6, 1147–1172
##### Abstract

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 nonempty proper subset $S\subset {S}_{5}$ there is a totally positive unit $u\in {\mathsc{O}}_{K}$ such that

$\prod _{\mathfrak{q}\in S}{Norm}_{{\mathbb{F}}_{\mathfrak{q}}∕{\mathbb{F}}_{5}}\left(u\phantom{\rule{0.2em}{0ex}}mod\phantom{\rule{0.2em}{0ex}}\mathfrak{q}\right)\ne \stackrel{̄}{1}.$

We prove that every semistable elliptic curve over $K$ is modular, using a combination of several powerful modularity theorems and class field theory. We deduce that if $K$ is a real abelian field of conductor $n<100$, with $5\nmid n$ and $n\ne 29,87,89$, then every semistable elliptic curve $E$ over $K$ is modular.

Let $\ell ,m,p$ be prime, with $\ell ,m\ge 5$ and $p\ge 3$. To a putative nontrivial primitive solution of the generalized Fermat equation ${x}^{2\ell }+{y}^{2m}={z}^{p}$ we associate a Frey elliptic curve defined over $ℚ{\left({\zeta }_{p}\right)}^{+}$, and study its mod $\ell$ representation with the help of level lowering and our modularity result. We deduce the nonexistence of nontrivial primitive solutions if $p\le 11$, or if $p=13$ and $\ell ,m\ne 7$.

##### Keywords
elliptic curves, modularity, Galois representation, level lowering, irreducibility, generalized Fermat, Fermat–Catalan, Hilbert modular forms
##### Mathematical Subject Classification 2010
Primary: 11D41, 11F80
Secondary: 11G05, 11F41