The half-integral weight eigencurve

TL;DR

This paper constructs a p-adic eigencurve for half-integral weight modular forms using Banach spaces and proves that low-slope overconvergent forms are classical, extending Coleman's theorem.

Contribution

It introduces Banach spaces and modules for overconvergent half-integral weight p-adic modular forms and constructs the associated eigencurve.

Findings

Construction of Banach spaces for overconvergent forms

Development of an eigencurve parameterizing eigenvalues

Proof that low-slope eigenforms are classical

Abstract

In this paper we define Banach spaces of overconvergent half-integral weight $p$-adic modular forms and Banach modules of families of overconvergent half-integral weight $p$-adic modular forms over admissible open subsets of weight space. Both spaces are equipped with a continuous Hecke action for which $U_{p^2}$ is moreover compact. The modules of families of forms are used to construct an eigencurve parameterizing all finite-slope systems of eigenvalues of Hecke operators acting on these spaces. We also prove an analog of Coleman's theorem stating that overconvergent eigenforms of suitably low slope are classical.