Windings of prime geodesics

TL;DR

This paper introduces a new method to compute winding numbers of closed geodesics around cusps in hyperbolic orbifolds, linking them to Rademacher symbols and automorphic forms for statistical analysis.

Contribution

It develops a novel construction of winding numbers for general cusped hyperbolic orbifolds and relates these to Rademacher symbols, enabling spectral analysis of geodesic distributions.

Findings

Winding numbers can be expressed via Rademacher symbols for various surfaces.

Spectral theory provides statistical insights into geodesic winding distributions.

New constructions extend previous methods to more general orbifolds.

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.