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

Abstract: Let $f$ and $g$ be two different newforms without complex multiplication having the same coefficient field. The main result of the present article proves that a congruence between the Galois representations attached to $f$ and to $g$ for a large prime $p$ implies an isomorphism between the endomorphism algebras of the abelian varieties $A_f$ and $A_g$ attached to $f$ and $g$ by the Eichler-Shimura construction. This implies important relations between their building blocks. A non-trivial application of our result is that for all prime numbers $d$ congruent to $3$ modulo $8$ satisfying that the class number of $\mathbb{Q}(\sqrt{-d})$ is prime to $3$, the equation $x^4+dy^2 =z^p$ has no non-trivial primitive solutions when $p$ is large enough. We prove a similar result for the equation $x^2+dy^6=z^p$.