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

SIGMA 15 (2019), 007, 17 pages      arXiv:1809.07421      https://doi.org/10.3842/SIGMA.2019.007
Contribution to the Special Issue on Moonshine and String Theory

Supersingular Elliptic Curves and Moonshine

Victor Manuel Aricheta ab
a) Department of Mathematics, Emory University, Atlanta, GA 30322, USA
b) Institute of Mathematics, University of the Philippines, Diliman 1101, Quezon City, Philippines

Received September 30, 2018, in final form January 19, 2019; Published online January 29, 2019

Abstract
We generalize a theorem of Ogg on supersingular $j$-invariants to supersingular elliptic curves with level. Ogg observed that the level one case yields a characterization of the primes dividing the order of the monster. We show that the corresponding analyses for higher levels give analogous characterizations of the primes dividing the orders of other sporadic simple groups (e.g., baby monster, Fischer's largest group). This situates Ogg's theorem in a broader setting. More generally, we characterize, in terms of supersingular elliptic curves with level, the primes arising as orders of Fricke elements in centralizer subgroups of the monster. We also present a connection between supersingular elliptic curves and umbral moonshine. Finally, we present a procedure for explicitly computing invariants of supersingular elliptic curves with level structure.

Key words: moonshine; modular curves; supersingular elliptic curves; supersingular polynomials.

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