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Genus Zero Modular Functions

Bong H. Lian·Joshua L. Wiczer
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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.

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Bong H. Lian, Joshua L. Wiczer. 2006-11-09. Genus Zero Modular Functions. https://arxiv.org/abs/math/0611291

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