Current Developments in Mathematics

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

Ken Ono

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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 in Project Euclid: 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.


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