On the modularity of endomorphism algebras

We show that any homomorphism between Jacobians of modular curves arises from a linear combination of Hecke modular correspondences. The proof uses the adelic language and is based on a study of the actions of GL$_2$ and Galois on the étale cohomology of the tower of modular curves. We also make this result explicit for Ribet’s twisting operators on modular abelian varieties.