Genus Zero Modular Functions

Bong H. Lian Brandeis University Joshua L. Wiczer Brandeis University

Number Theory mathscidoc:1608.24002

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.
Borcherd recursion formuli, Schwarzian equation, Genus Zero Modular Functions
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  title={Genus Zero Modular Functions},
  author={Bong H. Lian, and Joshua L. Wiczer},
Bong H. Lian, and Joshua L. Wiczer. Genus Zero Modular Functions. 2006.
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