Computing intersections of closed geodesics on the modular curve

Abstract

In a recent work of Duke, Imamoglu, and Tóth, the linking number of certain links on the space $\text{SL}(2,\mathbb{Z})\backslash \text{SL}(2,\mathbb{R})$ is investigated. This linking number has an alternative interpretation as the intersection number of closed geodesics on the modular curve, which is the focus of this paper. By relating the intersection number to a combinatorial computation involving rivers of Conway topographs, an efficient algorithm for computing intersection numbers is produced. A formula for the total intersection of a pair of positive coprime fundamental discriminants is also derived, which can be thought of as a real quadratic analogue of a classical result of Gross and Zagier. The paper ends with numerical computations and distribution questions relating to intersection numbers.

28 pages. Updated formatting to be consistent with my other papers, and added a few errata

References in corpus (1)

• Counting intersection numbers of closed geodesics on Shimura curves

Cited by in corpus (5)

• A Hyperbolic Analogue of the Rademacher Symbol
• Modular knots, automorphic forms, and the Rademacher symbols for triangle groups
• Counting intersection numbers of closed geodesics on Shimura curves
• Hecke operators acting on optimal embeddings in indefinite quaternion algebras
• Intersecting geodesics on the modular surface