Lifting cusp forms to Maass forms with an application to partitions
Abstract
For 2 < k ∈ 1/2 &Z; , we define lifts of cuspidal Poincaré series in S_{k}(Γ_{0}(N)) to weight 2  k harmonic weak Maass forms. This construction answers a question of Dyson by providing the general framework "explaining" Ramanujan's mock theta functions. As an application, we show that the number of partitions of a positive integer n is the "trace" of singular moduli of a Maass form arising from the lift of a weight 4 cusp form corresponding to a CalabiYau threefold.
 Publication:

Proceedings of the National Academy of Science
 Pub Date:
 March 2007
 DOI:
 10.1073/pnas.0611414104
 Bibcode:
 2007PNAS..104.3725B