Generalized modular equations and the CM values of Hauptmoduln

TL;DR

This paper extends the algebraic integer property of CM values from specific Hauptmoduln in monstrous moonshine to a broader class with cyclotomic integer coefficients, using generalized modular equations.

Contribution

It generalizes the algebraic integer property of CM values to Hauptmoduln with cyclotomic coefficients via generalized modular equations.

Findings

CM values of certain Hauptmoduln are algebraic integers.

Generalized modular equations are effective in studying CM values.

Complete replicability implies algebraic integer CM values.

Abstract

Monstrous moonshine relates the representation of the Monster finite sporadic simple group to the distinguished modular functions, called Hauptmoduln. Chen-Yui~\cite{Chen-Yui} showed that the CM values of Hauptmoduln which appeare in monstrous moonshine (but not all) are algebraic integers, which is similar to the singular moduli of the $j$-function. In this paper, we generalize this result to Hauptmoduln whose $q$-coefficients are cyclotomic integers. A main idea for our proof is the use of generalized modular equations for Hauptmoduln, which was introduced by Cummins-Gannon~\cite{Cummins-Gannon} in the study of monstrous moonshine. As an application, we show that if a formal $q$-series satisfies the special combinatoric property called complete replicability, its CM values are algebraic integers, without assuming the modular invariance.