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/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

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

Zendy Tools

Zendy Open

Let $f$ and $g$ be two different newforms without complex multiplicationhaving the same coefficient field. The main result of the present articleproves that a congruence between the Galois representations attached to $f$ andto $g$ for a large prime $p$ implies an isomorphism between the endomorphismalgebras of the abelian varieties $A_f$ and $A_g$ attached to $f$ and $g$ bythe Eichler-Shimura construction. This implies important relations betweentheir building blocks. A non-trivial application of our result is that for allprime numbers $d$ congruent to $3$ modulo $8$ satisfying that the class numberof $\mathbb{Q}(\sqrt{-d})$ is prime to $3$, the equation $x^4+dy^2 =z^p$ has nonon-trivial primitive solutions when $p$ is large enough. We prove a similarresult for the equation $x^2+dy^6=z^p$.

On endomorphism algebras of $\text{GL}_2$-type abelian varieties and Diophantine applications