Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 033, 35 pages      arXiv:1809.02913      https://doi.org/10.3842/SIGMA.2019.033
Contribution to the Special Issue on Moonshine and String Theory

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

Ryan C. Chen, Samuel Marks and Matthew Tyler
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA

Received September 19, 2018, in final form April 10, 2019; Published online April 29, 2019

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.

Key words: modular forms congruences; $p$-adic modular forms; moonshine.

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