Lifting cusp forms to Maass forms with an application to partitions

2007/03/01 by Kathrin Bringmann, Ken Ono · 2 citations
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Advanced Algebra and Geometry #Lift (data mining) #Cusp (singularity) #Mathematics #Moduli #Ramanujan's sum #Pure mathematics #Geometry #Computer science #Physics

paper · doi:10.1073/pnas.0611414104

openalex publication_date 2007/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/02

Abstract

For 2 < k [abstract: see text] we define lifts of cuspidal Poincaré series in S(k)(Gamma(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 Calabi-Yau threefold.

Citations

• The Selberg Trace Formula for PSL(2,ℝ)
• Distribution of the Partition Function Modulo m
• The f(q) mock theta function conjecture and partition ranks
• Arithmetic properties of coefficients of half-integral weight Maass–Poincaré series
• On two geometric theta lifts
• Congruence properties for the partition function
• On Modular Forms of Half Integral Weight

Cited by

• A modular framework for generalized Hurwitz class numbers III
• Mock modular form [wikipedia]