Geodesic
Cycle integrals of the $j$-function and mock modular forms
William Duke1 ; Özlem Imamoglu 2 ; Á. Tóth3
1 University of California<br/>Los Angeles, CA
2 ETH<br/> Zürich<br/> Switzerland
3 Eotvos Lorand University<br/> Department of Analysis<br/> Budapest<br/>Hungary
Annals of mathematics, Tome 173 (2011) no. 2, pp. 947-981.

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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.
DOI : 10.4007/annals.2011.173.2.8

William Duke 1 ; Özlem Imamoglu  2 ; Á. Tóth 3

1 University of California<br/>Los Angeles, CA
2 ETH<br/> Zürich<br/> Switzerland
3 Eotvos Lorand University<br/> Department of Analysis<br/> Budapest<br/>Hungary 
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William Duke; Özlem Imamoglu ; Á. Tóth. Cycle integrals of the $j$-function and mock modular forms. Annals of mathematics, Tome 173 (2011) no. 2, pp. 947-981. doi : 10.4007/annals.2011.173.2.8. http://geodesic.mathdoc.fr/articles/10.4007/annals.2011.173.2.8/

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