Perspectives on mock modular forms

Abstract

Mock modular forms have played many prominent roles in number theory and other areas of mathematics over the course of the last 15 years. While the term "mock modular form" was not formally defined in the literature until 2007, we now know in hindsight that evidence of this young subject appears much earlier, and that mock modular forms are intimately related to ordinary modular and Maass forms, and Ramanujan's mock theta functions. In this expository article, we offer several different perspectives on mock modular forms-some of which are number theoretic and some of which are not-which together exhibit the strength and scope of their developing theory. They are: combinatorics, q-series and mock theta functions, mathematical physics, number theory, and Moonshine. We also describe some essential results of Bruinier and Funke, and Zwegers, both of which have made tremendous impacts on the development of the theory of mock modular forms. We hope that this article is of interest to both number theorists and enthusiasts-to any reader who is interested in or curious about the history, development, and applications of the subject of mock modular forms, as well as some amount of the mathematical details that go along with them. 1 Preamble Modular forms have played central and far-reaching roles in number theory and mathematics over the last two centuries. Their footprints can be seen in the elliptic functions of the early 1800s, and their prominence has only risen since then. The theory and importance of their younger descendants, mock modular forms, has evolved in many analogous ways. Among the many papers written in the subject of mock modular forms are three excellent articles which are accessible to the non-specialist, and which we recommend to the readers of this article: Duke's 2014 article [54] in the the Notices of the AMS, Ono's 2010 article [120] in the Notices of the AMS, and Zagier's 2007 Séminaire Bourbaki article [138]. These articles highlight different aspects of the history, theory, and applications of mock modular forms, including discussions of Ramanujan's mock theta functions from 1920. The general term mock modular form was not defined in the literature until 2007 [138], and we now know in hindsight that Ramanujan's mock theta functions are among the oldest examples. In this expository article, we begin by offering a few concrete examples of mock modular forms in Section 2, which can be viewed as companions to some familiar ordinary modular forms. We offer these examples as previews to the more formal definitions and basic properties given in Section 3. In all of the following sections, we offer several perspectives on mock modular forms from the standpoint of different areas in which mock modular forms 1

Mock theta functions and related combinatorics

“…This context has also allowed us to make more sense of the notion of the order of a mock theta function. For more background and information on these aspects of the mock theta functions, see, e.g., [16,18,20,30].…”

Zagier's weight $3/2$ mock modular form

“…Zwegers showed that all of Ramanujan's mock theta functions are mock modular forms of weight 1/2. Over the last several years, mock modular forms have become a central topic in number theory with applications ranging from geometry, topology to black hole physics (see [9] for a nice review).…”

Module constructions for certain subgroups of the largest Mathieu group

“…whose trace functions, denoted H g (τ ), are certain mock modular forms. We refer to [5,23] for background on mock modular forms. In analogy with the work of Conway-Norton, work of Cheng, Eguchi-Hikami, and Gaberdiel-Hohenegger-Volpato [6,21,29,30] determined the mock modular forms H g (τ ) and then in 2012 Gannon [31] proved the existence of the associated module K ♮ .…”

“…Cheng, Duncan, and Harvey [9] conjectured that this relationship generalizes. More precisely, they formulated the umbral moonshine conjecture, stating that M 24 moonshine belongs to a class of 23 moonshines, each corresponding to one of the 23 Niemeier lattices with root systems of full rank [35]. The existence of these umbral moonshine modules was proven in 2015 by Duncan, Griffin, and Ono [16].…”