On endomorphism algebras of $\textup{GL}_{2}$-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 \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}+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} .