Title: Endomorphism algebras of {\rm GL}_2-type abelian varieties and Diophantine applications.

Abstract: Let f and g be two newforms with equal coefficient fields. In this talk we will see how a congruence between the Galois representations of f and g for a sufficiently large prime p implies strong conditions between the algebras of endomorphisms of the abelian varieties associated to the newforms f and g. Then, we will see one of the possible applications of such a result for the proof of the non-existence of solutions of Diophantine equations.