Geodesic
$p$-Adic Properties of Hauptmoduln with Applications to Moonshine
Symmetry, integrability and geometry: methods and applications, Tome 15 (2019) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: modular forms congruences; $p$-adic modular forms; moonshine. 
@article{SIGMA_2019_15_a32,
     author = {Ryan C. Chen and Samuel Marks and Matthew Tyler},
     title = {$p${-Adic} {Properties} of {Hauptmoduln} with {Applications} to {Moonshine}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2019},
     volume = {15},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a32/}
}
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Ryan C. Chen; Samuel Marks; Matthew Tyler. $p$-Adic Properties of Hauptmoduln with Applications to Moonshine. Symmetry, integrability and geometry: methods and applications, Tome 15 (2019). http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a32/

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