2012/12/30 by Daeyeol Jeon, Soon-Yi Kang, Jeon, Daeyeol +3
Mathematics · #11F12 #11F30 #11F37 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1212.6695
openalex publication_date 2012/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Borcherds-Zagier bases of the spaces of weakly holomorphic modular forms of weights 1/2 and 3/2 share the Fourier coefficients which are traces of singular moduli. Recently, Duke, Imamoglu, and Tóth have constructed a basis of the space of weight 1/2 mock modular forms, each member in which has Zagier's generating series of traces of singular moduli as its shadow. They also showed that Fourier coefficients of their mock modular forms are sums of cycle integrals of the j-function which are real quadratic analogues of singular moduli. In this paper, we prove the Fourier coefficients of a basis of the space of weight 3/2 mock modular forms are sums of cycle integrals of a sesqui-harmonic Maass form of weight zero whose image under hyperbolic Laplacian is the j-function. Furthermore, we express these sums as regularized inner products of weakly holomorphic modular forms of weight 1/2.