Unearthing the Visions of a Master: Harmonic Maass Forms and Number Theory

Together with his collaborators, most notably Kathrin Bringmann and Jan Bruinier, the author has been researching harmonic Maass forms. These non-holomorphic modular forms play central roles in many subjects: arithmetic geometry, combinatorics, modular forms, and mathematical physics. Here we outline the general facets of the theory, and we give several applications to number theory: partitions and q-series, modular forms, singular moduli, Borcherds products, extensions of theorems of Kohnen-Zagier and Waldspurger on modular L-functions, and the work of Bruinier and Yang on Gross-Zagier formulae. What is surprising is that this story has an unlikely beginning: the pursuit of the solution to a great mathematical mystery. Modular forms are central in contemporary 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 SwinnertonDyer Conjecture, Mirror Symmetry, Monstrous Moonshine, and the proof of Fermat’s Last Theorem. These celebrated works are dramatic examples of the evolution of mathematics; indeed, it would have been impossible to prophesy them fifty years ago. Instead of travelling back in time to the 1950s, our story (also see [165]) begins in 1887, in a village in India. Our mathematics, which is about harmonic Maass forms, begins with the legend (see [6, 39, 40, 110, 111, 133, 165]) of the great mathematician Srinivasa Ramanujan, and the mathematics he conjured from his death bed. 1. The gift from Kumbakonam Ramanujan was born on December 22, 1887 in Erode, a town about 250 miles southwest of Chennai (formerly known as Madras). He was a Brahmin, a member of India’s priestly caste. He grew up in Kumbakonam, a small town roughly 150 miles south of Chennai, where his father Srinivasa was a cloth merchant’s clerk. Kumbakonam, which is situated on the banks of the sacred Cauvery River, was (and remains today) a cosmopolitan center of the rural Indian district of Tanjore in the state of Tamil Nadu. As a boy, Ramanujan was a stellar student. Indeed, he won many awards at Town High School in Kumbakonam. Thanks to his exemplary performance, Ramanujan won a scholarship to Government College. His life took a dramatic turn when a friend loaned The author thanks the support of the NSF, the Manasse family, and the Hilldale Foundation. This is an exposition of the author’s lectures at the ’08 Harvard-MIT CDM Conference. Present day visitors to Town High School can view copies of these treasured awards. 1