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 ${\mathrm{G}\mathrm{L}}_2$ 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.