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

Symmetry Integrability and Geometry-Methods and Applications

article

Ryan C. Chen¹  Samuel Marks¹  Matthew Tyler¹ 

[1] Department of Mathematics, Princeton University

关键词: modular forms congruences;    p-adic modular forms;    moonshine;   

DOI  :  10.3842/SIGMA.2019.033

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown