p-adic Eichler-Shimura maps for the modular curve

We give a new proof of Faltings's p-adic Eichler-Shimura decomposition of the modular curves via Bernstein-Gelfand-Gelfand (BGG) methods and the Hodge-Tate period map. The key property is the relation between the Tate module and the Faltings extension, which was used in the original proof. Then we construct overconvergent Eichler-Shimura maps for the modular curves providing 'the second half' of the overconvergent Eichler-Shimura map of Andreatta, Iovita and Stevens. We use higher Coleman theory on the modular curve developed by Boxer and Pilloni to show that the small-slope part of the Eichler-Shimura maps interpolates the classical p-adic Eichler-Shimura decompositions. Finally, we prove that overconvergent Eichler-Shimura maps are compatible with Poincare and Serre pairings.