# On the modularity of endomorphism algebras

## TL;DR

This paper demonstrates that all homomorphisms between Jacobians of modular curves can be expressed through Hecke correspondences, using adelic methods and group actions on cohomology, with explicit applications to Ribet's operators.

## Contribution

It provides a new adelic approach to characterize homomorphisms between Jacobians of modular curves and explicitly describes Ribet's twisting operators.

## Key findings

- Homomorphisms are generated by Hecke correspondences.
- Adelic methods effectively analyze Jacobian morphisms.
- Explicit description of Ribet's twisting operators.

## Abstract

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.