Computing intersections of closed geodesics on the modular curve

TL;DR

This paper develops an efficient algorithm to compute intersection numbers of closed geodesics on the modular curve, linking geometric, combinatorial, and number-theoretic methods, and explores their distribution and total intersection properties.

Contribution

It introduces a novel combinatorial approach using Conway topographs to compute intersection numbers and derives a new formula for total intersections of certain discriminants.

Findings

Efficient algorithm for intersection number computation

Formula for total intersection of coprime fundamental discriminants

Numerical analysis and distribution questions addressed

Abstract

In a recent work of Duke, Imamo\={g}lu, and T\'{o}th, 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.