📝 SummarXiv

Abstract

Monstrous moonshine reveals the connection between the representation theory of the Monster finite sporadic group and distinguished modular functions, called Hauptmoduln. In this paper, we study the CM values of Hauptmoduln whose Fourier coefficients are cyclotomic integers, and prove that these special values are algebraic integers, analogous to the singular moduli of the elliptic modular $j$-function. Especially, the CM values of all of the McKay-Thompson series are algebraic integers. A main tool for our proof is the use of generalized modular equations for Hauptmoduln, introduced by Cummins-Gannon in the study of monstrous moonshine.