# Modularity of generating series of winding numbers

@article{Bruinier2017ModularityOG, title={Modularity of generating series of winding numbers}, author={J. Bruinier and Jens Funke and {\"O}. Imamoḡlu and Yingkun Li}, journal={Research in the Mathematical Sciences}, year={2017}, volume={5}, pages={1-23} }

The Shimura correspondence connects modular forms of integral weights and half-integral weights. One of the directions is realized by the Shintani lift, where the inputs are holomorphic differentials and the outputs are holomorphic modular forms of half-integral weight. In this article, we generalize this lift to differentials of the third kind. As an application, we obtain a modularity result concerning the generating series of winding numbers of closed geodesics on the modular curve.

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