Modularity of generating series of winding numbers

2017/11/24 by Jan H. Bruinier, Bruinier, Jan H., Jens Funke +5
Mathematics · #11F27 #11F37 #30F30 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11F27 #msc:11F37 #msc:30F30

paper · pdf · doi:10.48550/arxiv.1711.08907

arxiv created 2017/11/24 · arxiv updated 2017/11/27

Abstract

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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