Open Access
VOL. 2008 | 2009 Unearthing the visions of a master: Harmonic Maass forms and number theory
Ken Ono

Editor(s) David Jerison, Barry Mazur, Tomasz Mrowka, Wilfried Schmid, Richard P. Stanley, Shing-Tung Yau

Current Developments in Mathematics, 2009: 347-454 (2009)


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.


Published: 1 October 2009
First available in Project Euclid: 5 October 2009

zbMATH: 1229.11074
MathSciNet: MR2555930

Rights: Copyright © 2009 International Press of Boston

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