Modular Functions and the Monstrous Exponents

TL;DR

The paper develops a modular-function framework to explain the exponents in the monster group's order for primes $p>3$ by linking $v_p(\#\mathbb{M})$ to differences of Hauptmoduls $J_1$, $J_p$, $J_{p^+}$, and $J_{p^2}$, and to a Deligne-type $p$-adic expansion involving supersingular data. It introduces level-lowering machinery ($U_N$, $X_p$, and Atkin–Lehner twists) and genus-zero Hauptmoduls $J_N$, $J_{N^+}$ to organize the arithmetic of $v_p$ via modular curves. The main contributions are two precise, compatible formulas for $v_p(\#\mathbb{M})$ for $p>3$, plus a detailed analysis using Deligne's theorem that explains the $p$-adic divisibility in terms of supersingular $J_1$-values and Faber polynomials, with exceptional handling for $p=2,3$. This work provides a geometric, modular-arithmetic perspective on the monster exponents, connecting moonshine phenomena, genus-zero modular curves, and supersingular elliptic curves to the prime factorization of the monster order.

Abstract

We present a modular function-based approach to explaining, for primes larger than 3, the exponents that appear in the prime decomposition of the order of the monster finite simple group.

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4