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.