Modularity of generating series of winding numbers
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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.
Most of this work was done during visits by J.F. to TU Darmstadt, by J.F. and Y.L. to FIM in Zürich in 2017. We thank these organizations for their supports. We also thank the referees for careful readings and helpful comments and corrections. J.B and Y.L. are supported by the DFG Grant BR-2163/4-2 and the LOEWE research unit USAG. Y.L. is also partially supported by an NSF postdoctoral fellowship.
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