Modular elliptic curves over real abelian fields and the generalized Fermat equation $x^{2\ell }+y^{2m}=z^p$

Abstract

Using a combination of several powerful modularity theorems and class field theory we derive a new modularity theorem for semistable elliptic curves over certain real abelian fields. 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 non-trivial primitive solution of the generalized Fermat $x^{2\ell }+y^{2m}=z^p$ we associate a Frey elliptic curve defined over $\mathbb{Q}({\zeta }_p{)}^{+}$, and study its mod $\ell$ representation with the help of level lowering and our modularity result. We deduce the non-existence of non-trivial primitive solutions if $p\le 11$, or if $p=13$ and $\ell$, $m\ne 7$.

Introduction rewritten to emphasise the new modularity theorem. Paper revised in the light of referees' comments

References in corpus (2)

• Fermat's Last Theorem over some small real quadratic fields
• Shifted powers in binary recurrence sequences

Cited by in corpus (6)

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• On some Generalized Fermat Equations of the form $x^2+y^{2n}=z^p$
• Proof of the Tijdeman-Zagier Conjecture via Slope Irrationality and Term Coprimality