Cycle integrals of the j-function and mock modular forms

2011/03/01 by W. Duke, William Duke, Özlem Imamoḡlu +3 · 14 citations
Mathematics · #Advanced Mathematical Identities #Advanced Algebra and Geometry #Analytic Number Theory Research

paper · pdf · doi:10.4007/annals.2011.173.2.8

Abstract

In this paper we construct certain mock modular forms of weight 1/2 whose Fourier coefficients are given in terms of cycle integrals of the modular j-function. Their shadows are weakly holomorphic forms of weight 3/2. These new mock modular forms occur as holomorphic parts of weakly harmonic Maass forms. We also construct a generalized mock modular form of weight 1/2 having a real quadratic class number times a regulator as a Fourier coefficient. As an application of these forms we study holomorphic modular integrals of weight 2 whose rational period functions have poles at certain real quadratic integers. The Fourier coefficients of these modular integrals are given in terms of cycle integrals of modular functions. Such a modular integral can be interpreted in terms of a Shimura-type lift of a mock modular form of weight 1/2 and yields a real quadratic analogue of a Borcherds product.

Citations

• On Modular Forms of Half Integral Weight

Cited by

• Geometric invariants for real quadratic fields
• Shintani lifts and fractional derivatives for harmonic weak Maass forms
• Periods of the j -function along infinite geodesics and mock modular forms
• The first moment of Salie sums
• Extended-cycle integrals of modular functions for badly approximable numbers
• The basis functions of Fourier interpolation
• Recurrence and congruences for the smallest parts function
• Locally harmonic Maass forms and the kernel of the Shintani lift
• A polyharmonic Maass form of depth 3/2 for SL₂(Z)
• Singular invariants and coefficients of weak harmonic Maass forms of weight 5/2
• Harmonic Maass forms associated to real quadratic fields
• On p-adic haromonic Maass functions
• On cycle integrals of weakly holomorphic modular forms
• The Fourier coefficients and singular moduli of the elliptic modular function j(τ), revisited

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