On the modularity of endomorphism algebras

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