2025/05/08 by Tomiyama, Kazuki
Computer Science · Mathematics · #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2505.05135
openalex publication_date 2025/05/08 · openalex created_date 2025/10/19 · openalex updated_date 2026/08/03
Monstrous moonshine relates the representation of the Monster finite sporadic simple group to the distinguished modular functions, called Hauptmoduln. Chen-Yui~\citeChen-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~\citeCummins-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.