On endomorphism algebras of GL2 -type abelian varieties and Diophantine applications

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 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 $Q(\sqrt{-d})$ 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 $x^2+dy^6 =z^p$.