Windings of prime geodesics

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.

Accepted version, 39p., to appear in IMRN

References in corpus (2)

• The Manin-Drinfeld theorem and the rationality of Rademacher symbols
• Modular knots, automorphic forms, and the Rademacher symbols for triangle groups

Cited by in corpus (1)

• A Central Limit Theorem for the Winding Number of Low-Lying Closed Geodesics