Supersingular prime (moonshine theory)

In the mathematical branch of moonshine theory, a supersingular prime is a prime number that divides the order of the Monster group ${\displaystyle M}$, which is the largest sporadic simple group. There are precisely fifteen supersingular prime numbers: the first eleven primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31; as well as 41, 47, 59, and 71 (sequence A002267 in the OEIS).

The non-supersingular primes are 37, 43, 53, 61, 67, and any prime number greater than or equal to 73.

Supersingular primes are related to the notion of supersingular elliptic curves as follows. For a prime number ${\displaystyle p}$, the following are equivalent:

1. The modular curve ${\displaystyle X_{0}^{+}(p)=X_{0}(p)/w_{p}}$, where ${\displaystyle w_{p}}$ is the Fricke involution of ${\displaystyle X_{0}(p)}$, has genus zero.
2. Every supersingular elliptic curve in characteristic ${\displaystyle p}$ can be defined over the prime subfield ${\displaystyle \mathbb {F} _{p}}$.
3. The order of the Monster group is divisible by ${\displaystyle p}$.

The equivalence is due to Andrew Ogg. More precisely, in 1975 Ogg showed that the primes satisfying the first condition are exactly the 15 supersingular primes listed above and shortly thereafter learned of the (then conjectural) existence of a sporadic simple group having exactly these primes as prime divisors. This strange coincidence was the beginning of the theory of monstrous moonshine.

All supersingular primes are Chen primes, but 37, 53, and 67 are also Chen primes, and there are infinitely many Chen primes greater than 73.

• Supersingular prime (algebraic number theory)

• Weisstein, Eric W. "Supersingular Prime". MathWorld.
• Weisstein, Eric W. "Sporadic group". MathWorld.
• Ogg, A. P. (1980). "Modular Functions". In Cooperstein, Bruce; Mason, Geoffrey (eds.). The Santa Cruz Conference on Finite Groups. Held at the University of California, Santa Cruz, Calif., June 25–July 20, 1979. Providence, RI: Amer. Math. Soc. pp. 521–532. ISBN 0-8218-1440-0. MR 0604631.