Abstract
The winding of a closed oriented geodesic around the cusp of the modular orbifold is computed by the Rademacher symbol, a classical function from the theory of modular forms. In this article, we introduce a new construction of winding numbers to record the winding of closed oriented geodesics about a prescribed cusp of a general cusped hyperbolic orbifold. For various arithmetic families of surfaces, this winding number can again be expressed by a Rademacher symbol, and access to the spectral theory of automorphic forms yields statistical results on the distribution of closed (primitive) oriented geodesics with respect to their winding.
Originalsprog | Engelsk |
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Tidsskrift | International Mathematics Research Notices |
Vol/bind | 2024 |
Udgave nummer | 22 |
Sider (fra-til) | 13931-13963 |
Antal sider | 33 |
ISSN | 1073-7928 |
DOI | |
Status | Udgivet - 2024 |
Bibliografisk note
Funding Information:This work was supported by the Swiss National Science Foundation [201557 to C.B.]. The first named author would like to thank Marc Burger and Alessandra Iozzi for first mentioning winding numbers in relation to the work of Goldstein, Jay Jorgenson for discussions, and the Hausdorff Institute for Mathematics in Bonn, where most of the final draft of this manuscript was completed. The second author would like to thank Morten Risager for advice and help.
Funding Information:
This work was supported by the Swiss National Science Foundation [201557 to C.B.]. Acknowledgments
Publisher Copyright:
© The Author(s) 2024.