Overconvergence and classicality: the case of curves

TL;DR

This paper establishes a classicality criterion for overconvergent sections of line bundles over curves, with applications to overconvergent modular forms on Shimura curves, advancing understanding of p-adic modular forms.

Contribution

It introduces a new classicality criterion applicable to overconvergent modular forms on Shimura curves, generalizing previous results to higher levels.

Findings

Proves a classicality criterion for overconvergent sections on curves.

Applies the criterion to overconvergent modular forms on Shimura curves.

Extends classicality results to higher-level modular forms.

Abstract

Given our set-up of a system of curves and maps between them satisfying certain assumptions, we prove a classicality criterion for overconvergent sections of line bundles over these curves. As a result, we prove such criteria for overconvergent modular forms over various Shimura curves. In particular, we provide a classicality criterion for overconvergent modular forms studied in [Kassaei: P-adic modular forms over Shimura curves over totally real fields, Compositio Math. 140 (2004), no 2, 359-395] and their higher-level generalizations.