## Current Developments in Mathematics

- Current Developments in Mathematics
- Volume 2008 (2009), 1-454

### Unearthing the visions of a master: harmonic Maass forms and number theory

#### Abstract

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.

#### Article information

**Source**

Current Developments in Mathematics Volume 2008 (2009), 347-454.

**Dates**

First available: 5 October 2009

**Permanent link to this document**

http://projecteuclid.org/euclid.cdm/1254748659

**Zentralblatt MATH identifier**

05635202

**Mathematical Reviews number (MathSciNet)**

MR2555930

#### Citation

Ono, Ken. Unearthing the visions of a master: harmonic Maass forms and number theory. Current Developments in Mathematics 2008 (2009), 347--454. http://projecteuclid.org/euclid.cdm/1254748659.