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 an isomorphism between the residual Galois representations attached to $f$ and to $g$ for a large prime $p$ (depending only on $g$) implies that the endomorphism algebra of the abelian variety $A_f$, attached to $f$ by the Eichler–Shimura construction (after tensoring with $\mathbb{Q}$), is a subalgebra of the endomorphism algebra of the abelian variety $A_g$ attached to $g$. 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 $x^4+d{y}^2=z^p$ has no non-trivial primitive solutions when $p$ is large enough. We prove a similar result for the equation $x^2+d{y}^6=z^p$.