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On the modularity of endomorphism algebras

Volume 71 / 2023

François Brunault 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


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.


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