p-adic Eichler-Shimura maps for the modular curve

TL;DR

This paper presents a new proof of the p-adic Eichler-Shimura decomposition for modular curves using BGG methods and constructs overconvergent Eichler-Shimura maps, demonstrating their interpolation properties and compatibility with pairings.

Contribution

It provides a novel proof of the p-adic Eichler-Shimura decomposition and constructs overconvergent ES maps that interpolate classical decompositions.

Findings

New proof of p-adic Eichler-Shimura decomposition

Construction of overconvergent Eichler-Shimura maps

Compatibility with Poincaré and Serre pairings

Abstract

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.