Cycle integrals of the j-function and mock modular forms

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 f...

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Bibliographic Details
Published in:Annals of mathematics 2011-03, Vol.173 (2), p.947-981
Main Authors: Duke, W., Imamoḡlu, Ö., Tóth, Á.
Format: Article
Language:English
Subjects:
Algebra
Discriminants
Exact sciences and technology
Fourier coefficients
General mathematics
General, history and biography
Integers
Mathematical analysis
Mathematical cusps
Mathematical functions
Mathematical integrals
Mathematics
Number theory
Real functions
Sciences and techniques of general use
Series convergence
Whittaker functions
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Summary: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.
ISSN:0003-486X
1939-8980
DOI:10.4007/annals.2011.173.2.8