Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalized modular equations and the CM values of Hauptmoduln

Kazuki Tomiyama

TL;DR

The paper addresses the arithmetic of CM values of Hauptmoduln with cyclotomic $q$-coefficients by developing generalized modular equations in the spirit of Cummins–Gannon. It proves that if a genus-zero group $\Gamma$ contains a $\Gamma_0(N)$ and its Hauptmodul $h$ has cyclotomic integer coefficients, then $h(\tau)$ is an algebraic integer for CM points $\tau$ with $a\tau^2+b\tau+c=0$, $(a,N)=1$; the argument hinges on diagonal generalized modular polynomials $F_n^h(h,h)$ and their unit-leading coefficients. The work extends Chen–Yui’s CM-integrality results to a broader class of Hauptmoduln, and shows that completely replicable formal $q$-series with cyclotomic integer coefficients also yield CM-values that are algebraic integers without invoking modular invariance. These results illuminate the algebraic structure behind monstrous moonshine, reinforce the connection between replicability and Hauptmodul properties, and suggest further links to ring class fields and the representation theory of sporadic groups. Overall, the paper advances our understanding of the arithmetic of CM values in the moonshine landscape and provides tools for exploring deeper connections between modular objects and finite group representations.

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.

Generalized modular equations and the CM values of Hauptmoduln

TL;DR

The paper addresses the arithmetic of CM values of Hauptmoduln with cyclotomic -coefficients by developing generalized modular equations in the spirit of Cummins–Gannon. It proves that if a genus-zero group contains a and its Hauptmodul has cyclotomic integer coefficients, then is an algebraic integer for CM points with , ; the argument hinges on diagonal generalized modular polynomials and their unit-leading coefficients. The work extends Chen–Yui’s CM-integrality results to a broader class of Hauptmoduln, and shows that completely replicable formal -series with cyclotomic integer coefficients also yield CM-values that are algebraic integers without invoking modular invariance. These results illuminate the algebraic structure behind monstrous moonshine, reinforce the connection between replicability and Hauptmodul properties, and suggest further links to ring class fields and the representation theory of sporadic groups. Overall, the paper advances our understanding of the arithmetic of CM values in the moonshine landscape and provides tools for exploring deeper connections between modular objects and finite group representations.

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 -function. In this paper, we generalize this result to Hauptmoduln whose -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 -series satisfies the special combinatoric property called complete replicability, its CM values are algebraic integers, without assuming the modular invariance.
Paper Structure (18 sections, 19 theorems, 56 equations)

This paper contains 18 sections, 19 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. The $j$-function and Hauptmoduln
  3. Monstrous moonshine
  4. Main results
  5. Modular equations
  6. Modular equations for the $j$-function
  7. Generalized modular equations
  8. Proof of the main results
  9. Construction of generalized modular equations
  10. Coefficients of the generalized modular polynomials
  11. Proof of results
  12. Replicable functions
  13. Definition of replicability
  14. Replicability and Hauptmodul property
  15. Complete replicability and modular equations
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $T_g$ be a fundamental McKay-Thompson series, and suppose that the invariance group $\Gamma_g$ contains $\Gamma_0(N)$. Let $\tau\in\mathbb{H}$ be a root of $az^2+bz+c=0~(a,b,c\in\mathbb{Z}, (a,N)=1)$. Then the CM value $T_g(\tau)$ is an algebraic integer.

Theorems & Definitions (33)

  • Theorem 1.1: Chen-Yui
  • Theorem 1.2: Theorem \ref{['main thm']}
  • Definition 2.1
  • Theorem 2.2: Kozlov
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Cummins-Gannon
  • Proposition 3.1: Cummins-Gannon
  • Lemma 3.2: Cummins-Gannon
  • Proposition 3.3
  • ...and 23 more