On the modularity of endomorphism algebras
Volume 71 / 2023
Bulletin Polish Acad. Sci. Math. 71 (2023), 23-33
MSC: Primary 11F41; Secondary 11F25, 11F70, 11F80, 14G32.
DOI: 10.4064/ba230310-29-3
Published online: 14 April 2023
Abstract
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.
