Genus Zero Modular Functions

TL;DR

This paper constructs Schwarzian differential equations for genus zero modular functions using power series methods, enabling the recovery of these functions from their associated equations, advancing understanding of their structure.

Contribution

It introduces a method to derive Schwarzian equations for genus zero modular functions using power series and Borcherd recursion, providing a new approach to analyze these functions.

Findings

Successfully derived Schwarzian equations for genus zero modular functions.

Demonstrated how to recover modular functions from their Schwarzian equations.

Extended the application of power series methods to modular function analysis.

Abstract

This project was sponsored through the Schiff Fellowship program of Brandeis University. This project involved using the power series method to construct a third order nonlinear ordinary differential equation, a Schwarzian equation, for each of the "genus zero" modular functions, described in the Conway-Norton paper. We first use the Borcherd recursion formuli to generate, in each case, a modular function up to whatever degree we desire, and then use the fact that there is a Schwarzian equation, determined by a single rational function we call a Q-value. By similar power series methods, we compute the coefficients of our rational function, and hence have all the necessary data to create a Schwarzian differential equation for each modular function. This equation can, in turn, be used to recover the modular function itself.