Overconvergent modular forms and perfectoid Shimura curves

TL;DR

This paper introduces a new approach to overconvergent modular forms using affinoid subsets of infinite-level Shimura curves, and constructs an overconvergent Eichler-Shimura map with stronger results.

Contribution

It provides a novel construction of overconvergent modular forms via affinoid subsets and canonical coordinates, extending classical theory to the p-adic setting.

Findings

New construction of overconvergent modular forms on affinoid subsets

Definition of an overconvergent Eichler-Shimura map for Shimura curves

Stronger analogues of classical results by Andreatta-Iovita-Stevens

Abstract

We give a new construction of overconvergent modular forms of arbitrary weights, defining them in terms of functions on certain affinoid subsets of Scholze's infinite-level modular curve. These affinoid subsets, and a certain canonical coordinate on them, play a role in our construction which is strongly analogous with the role of the upper half-plane and its coordinate `z' in the classical analytic theory of modular forms. As one application of these ideas, we define and study an overconvergent Eichler-Shimura map in the context of compact Shimura curves over $\mathbb{Q}$, proving stronger analogues of results of Andreatta-Iovita-Stevens.