Modularity of generating series of winding numbers

TL;DR

This paper generalizes the Shintani lift to differentials of the third kind and applies it to prove modularity of generating series of winding numbers of closed geodesics on modular curves.

Contribution

It extends the Shintani lift to a broader class of differentials and establishes a new modularity result for winding numbers of geodesics.

Findings

Generalization of the Shintani lift to third-kind differentials

Proof of modularity for generating series of winding numbers

New connections between modular forms and geometric invariants

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.