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