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arXiv · 2209.06233

Windings of prime geodesics

Claire Burrin·Flemming von Essen
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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.

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Claire Burrin, Flemming von Essen. 2024-09-27. Windings of prime geodesics. https://doi.org/10.1093/imrn%2Frnae225

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