Supersingular Elliptic Curves and Moonshine Article Swipe
YOU?
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· 2019
· Open Access
·
· DOI: https://doi.org/10.3842/sigma.2019.007
· OA: W2892272683
We generalize a theorem of Ogg on supersingular $j$-invariants to\nsupersingular elliptic curves with level. Ogg observed that the level one case\nyields a characterization of the primes dividing the order of the monster. We\nshow that the corresponding analyses for higher levels give analogous\ncharacterizations of the primes dividing the orders of other sporadic simple\ngroups (e.g., baby monster, Fischer's largest group). This situates Ogg's\ntheorem in a broader setting. More generally, we characterize, in terms of\nsupersingular elliptic curves with level, the primes arising as orders of\nFricke elements in centralizer subgroups of the monster. We also present a\nconnection between supersingular elliptic curves and umbral moonshine. Finally,\nwe present a procedure for explicitly computing invariants of supersingular\nelliptic curves with level structure.\n
