$p$-adic Properties of Hauptmoduln with Applications to Moonshine

TL;DR

This paper explores the $p$-adic properties of Hauptmoduln related to monstrous moonshine, establishing new congruence relations and linking them to the monster group's structure.

Contribution

It develops a unified theory of prime power congruences for Hauptmoduln in moonshine, revealing connections to the monster group's subgroup structure.

Findings

Many Hauptmoduln satisfy prime power congruences

Established links between congruences and monster group structure

Identified subgroups with characters satisfying these congruences

Abstract

The theory of monstrous moonshine asserts that the coefficients of Hauptmoduln, including the $j$-function, coincide precisely with the graded characters of the monster module, an infinite-dimensional graded representation of the monster group. On the other hand, Lehner and Atkin proved that the coefficients of the $j$-function satisfy congruences modulo $p^n$ for $p \in \{2, 3, 5, 7, 11\}$, which led to the theory of $p$-adic modular forms. We combine these two aspects of the $j$-function to give a general theory of congruences modulo powers of primes satisfied by the Hauptmoduln appearing in monstrous moonshine. We prove that many of these Hauptmoduln satisfy such congruences, and we exhibit a relationship between these congruences and the group structure of the monster. We also find a distinguished class of subgroups of the monster with graded characters satisfying such congruences.