English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Using a combination of several powerful modularity theorems and class fieldtheory we derive a new modularity theorem for semistable elliptic curves overcertain real abelian fields. We deduce that if $K$ is a real abelian field ofconductor $n<100$, with $5 \nmid n$ and $n \ne 29$, $87$, $89$, then everysemistable elliptic curve $E$ over $K$ is modular.  Let $\ell$, $m$, $p$ be prime, with $\ell$, $m \ge 5$ and $p \ge 3$.To aputative 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 helpof level lowering and our modularity result. We deduce the non-existence ofnon-trivial primitive solutions if $p \le 11$, or if $p=13$ and $\ell$, $m \ne7$.

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