Supersingular prime (moonshine theory)

In the mathematical branch of moonshine theory, a supersingular prime is a certain type of prime number. Namely, a supersingular prime is a prime divisor of the order of the Monster group M, the largest of the sporadic simple groups. There are precisely 15 supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71 — all 15 are Chen primes.

This definition is related to the notion of supersingular elliptic curves as follows. For a prime number p, the following are equivalent:

1. The modular curve X₀⁺(p) = X₀(p) / w_p, where w_p is the Fricke involution of X₀(p), has genus zero.
2. Every supersingular elliptic curve in characteristic p can be defined over the prime subfield F_p.
3. The order of the Monster group is divisible by 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.

References

• Weisstein, Eric W. "Supersingular Prime". 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.