Mock modular forms as $p$-adic modular forms

TL;DR

This paper investigates the relationship between mock modular forms and $p$-adic modular forms, demonstrating conditions under which mock modular forms can be corrected to become $p$-adic and linking corrections to their shadows.

Contribution

It establishes when mock modular forms are $p$-adic and connects their corrections to their shadows, advancing understanding of their structure in $p$-adic contexts.

Findings

Mock modular forms can be corrected to $p$-adic modular forms in certain cases.

The unique correction of a mock modular form relates to its shadow.

Conditions are identified under which mock modular forms are already $p$-adic.

Abstract

In this paper, we consider the question of correcting mock modular forms in order to obtain $p$-adic modular forms. In certain cases we show that a mock modular form $M^+$ is a $p$-adic modular form. Furthermore, we prove that otherwise the unique correction of $M^+$ is intimately related to the shadow of $M^+$.