A geometric perspective on p-adic properties of mock modular forms

TL;DR

This paper provides a geometric proof that certain mock modular forms can be corrected to p-adic modular forms, using harmonic Maass forms and overconvergent modular forms theory.

Contribution

It offers a new geometric proof for the p-adic properties of mock modular forms, extending previous results by employing harmonic Maass forms and overconvergent modular forms.

Findings

New proof of p-adic properties for mock modular forms

Application of harmonic Maass forms and overconvergent modular forms

Validation for good primes p

Abstract

Bringmann, Guerzhoy and Kane have shown how to correct mock modular forms by a certain linear combination of the Eichler integral of their shadows in order to obtain p-adic modular forms in the sense of Serre. In this paper, we give a new proof of their results (for good primes p) by employing the geometric theory of harmonic Maass forms developed by the first author and the theory of overconvergent modular forms due to Katz and Coleman.