p-Adic Properties of Hauptmoduln with Applications to Moonshine

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 ∈ {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. and the right-hand sides are simple sums involving the dimensions of the irreducible representations of the monster group M: