• Eichler-Shimura

After a brief review of the necessary prerequisites - concerning modular forms and elliptic curves - we go through the study of modular curves.
Such spaces are crucial for our discussion since their Jacobians can be seen as quotients of spaces of modular forms, and they are, at the same time, moduli spaces of elliptic curves with some extra torsion datum.
Our first goal is to find an algebraic model for X_0(N) in which the curve is defined over Q.
Another fundamental tool in our study is the algebra of the Hecke operators. These objects, besides being defined analytically as operators on spaces of modular forms, act on the Jacobians of the modular curves in a way compatible with the Q-structure.
This remarkable fact allows us to consider the reduction modulo p of all the objects in play and prove the Eichler-Shimura Correspondence. The theorem, written in the language of correspondences, reveals a close connection between the correspondence induced by the Hecke operator T_p and one induced by the Frobenius of the reduced curve.