The Fourier coefficients of the McKay-Thompson series and the traces of CM values

TL;DR

This paper explores the Fourier coefficients of McKay-Thompson series at square-free levels, establishing formulas relating them to traces of CM values, extending known results about the modular j function.

Contribution

It generalizes Kaneko's arithmetic formula for Fourier coefficients to McKay-Thompson series of square-free level, linking them to traces of CM values.

Findings

Derived explicit formulas for Fourier coefficients of McKay-Thompson series.

Extended the connection between Fourier coefficients and CM value traces to new series.

Provided new insights into the arithmetic properties of modular functions.

Abstract

The elliptic modular j function enjoys many beautiful properties. Its Fourier coefficients are related to the Monster group, and its CM values generate abelian extensions over imaginary quadratic fields. Kaneko gave an arithmetic formula for the Fourier coefficients expressed in terms of the traces of the CM values. In this article, we are concerned with analogues of Kaneko's result for the McKay-Thompson series of square-free level.