Unearthing the visions of a master: harmonic Maass forms and number theory. (English) Zbl 1229.11074

Jerison, David (ed.) et al., Current developments in mathematics, 2008. Somerville, MA: International Press (ISBN 978-1-57146-139-1/pbk). 347-454 (2009).
Harmonic Maass forms were defined by Bruinier and Funke in 2004 and quickly gained fame for their connection with Ramanujan’s mock theta functions. Today these non-holomorphic modular forms play a role in subjects such as arithmetic geometry, combinatorics, number theory, and mathematical physics. Ken Ono is one of the principal players in this domain, and here he has written an excellent and essential survey. The opening sections contain historical background on Ramanujan’s life and work, his last letter to Hardy, his lost notebook and the influence of his mock theta functions on \(20\)th century mathematics. In Section 6 the author briefly describes Zwegers’ fundamental work on the modular completions of mock theta functions. In Sections 7-8, he defines harmonic Maass forms and gives some examples. Each of Sections 9–15 is then devoted to an application in number theory: Dyson-Ramanujan theory of partition congruences, Eulerian series as modular forms, exact formulas, applications to classical modular forms, generating functions for singular moduli, Borcherds products, and derivatives and values of \(L\)-functions.
For the entire collection see [Zbl 1173.00021].


11F37 Forms of half-integer weight; nonholomorphic modular forms
11F11 Holomorphic modular forms of integral weight
11P81 Elementary theory of partitions
11P82 Analytic theory of partitions
11P83 Partitions; congruences and congruential restrictions
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
Full Text: Euclid