Harmonic Maass Forms, Mock Modular Forms, and Quantum Modular Forms

This short course is an introduction to the theory of harmonic Maass forms, mock modular forms, and quantum modular forms. These objects have many applications: black holes, Donaldson invariants, partitions and q-series, modular forms, probability theory, singular moduli, Borcherds products, central values and derivatives of modular L-functions, generalized Gross-Zagier formulae, to name a few. Here we discuss the essential facts in the theory, and consider some applications in number theory. This mathematics has an unlikely beginning: the mystery of Ramanujan’s enigmatic “last letter” to Hardy written three months before his untimely death. Section 15 gives examples of projects which arise naturally from the mathematics described here. Modular forms are key objects in modern mathematics. Indeed, modular forms play crucial roles in algebraic number theory, algebraic topology, arithmetic geometry, combinatorics, number theory, representation theory, and mathematical physics. The recent history of the subject includes (to name a few) great successes on the Birch and Swinnerton-Dyer Conjecture, Mirror Symmetry, Monstrous Moonshine, and the proof of Fermat’s Last Theorem. These celebrated works are dramatic examples of the evolution of mathematics. Here we tell a different story, the mathematics of harmonic Maass forms, mock modular forms, and quantum modular forms. This story has an unlikely beginning, the mysterious last letter of the Ramanujan. 1. Ramanujan The story begins with the legend of Ramanujan, the amateur Indian genius who discovered formulas and identities without any rigorous training in mathematics. He recorded his findings in mysterious notebooks without providing any indication of proofs. Some say he did not see the need to provide proofs; indeed, it is said that his findings came to him as visions from Goddess Namagiri. As is well known, in 1913 Ramanujan wrote G. H. Hardy, a celebrated English analytic number theorist, a famous letter which included pages of formulas and claims (without proof). Some of his claims were well known, some were false, but many were so strange that Hardy (and his colleague Littlewood) were unable to evaluate them. Impressed by the creativity exhibited in this letter, Hardy invited Ramanujan to Cambridge. Ramanujan accepted the invitation, and he spent much of 1914-1919 in Cambridge where he wrote about 30 papers on an exceptionally wide variety of subjects in analytic number These notes are an edited and expanded version of the author’s expository paper for the 2008 HarvardMIT Current Developments in Mathematics [194] Conference Proceedings. The author thanks the National Science Foundation and the Asa Griggs Candler Trust for their generous support. Ramanujan flunked out of two different colleges. 1