Intersections of Closed Geodesics on the Modular Curve.

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. In this paper, we give an alternative interpretation of this linking number by relating it to the intersection number of modular geodesics on the modular curve. We demonstrate a connection to the results of Gross and Zagier on the factorization of differences of singular moduli by finding a real quadratic analogue of one of their results. By relating the intersection number to rivers of Conway topographs, an efficient algorithm for computing intersection numbers is produced. The paper ends with a survey of future projects.