Supersingular Elliptic Curves and Moonshine

TL;DR

This paper extends Ogg's theorem on supersingular elliptic curves to higher levels, linking primes dividing sporadic simple groups' orders with supersingular invariants and exploring connections to moonshine phenomena.

Contribution

It generalizes Ogg's characterization of primes to supersingular elliptic curves with level structure and relates these to sporadic groups and moonshine.

Findings

Characterization of primes dividing sporadic groups via supersingular elliptic curves.

Connection established between supersingular elliptic curves and umbral moonshine.

Procedure developed for computing invariants of supersingular elliptic curves with level.

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.