Singular invariants and coefficients of weak harmonic Maass forms of weight 5/2

2014/10/27 by Nickolas Andersen, Andersen, Nickolas
Mathematics · #11F37 #11P82 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1410.7349

openalex publication_date 2014/10/27 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We study the coefficients of a natural basis for the space of weak harmonic Maass forms of weight 5/2 on the full modular group. The non-holomorphic part of the first element of this infinite basis encodes the values of the partition function p(n). We show that the coefficients of these harmonic Maass forms are given by traces of singular invariants. These are values of non-holomorphic modular functions at CM points or their real quadratic analogues: cycle integrals of such functions along geodesics on the modular curve. The real quadratic case relates to recent work of Duke, Imamoglu, and Tóth on cycle integrals of the j-function, while the imaginary quadratic case recovers the algebraic formula of Bruinier and Ono for the partition function.

Citations

• The f(q) mock theta function conjecture and partition ranks
• Heegner points and derivatives ofL-series. II
• Dyson’s ranks and Maass forms
• On two geometric theta lifts
• Cycle integrals of the j-function and mock modular forms

Related

• Weak harmonic Maass forms of weight 5/2 and a mock modular form for the partition function
• Cycle integrals of a sesqui-harmonic Maass form of weight zero
• Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms
• Harmonic weak Maass forms and periods II
• Integrality Properties of the CM-values of Certain Weak Maass Forms