In themathematical branch ofmoonshine theory, asupersingular prime is aprime number thatdivides theorder of theMonster group${\displaystyle M}$, which is the largestsporadic simple group. There are precisely fifteen supersingular prime numbers: the first eleven primes2,3,5,7,11,13,17,19,23,29, and31; as well as41,47,59, and71 (sequenceA002267 in theOEIS).

The non-supersingular primes are37,43,53,61,67, and any prime number greater than or equal to73.

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

1. Themodular curve${\displaystyle X_0^{+}(p)=X_0(p)/w_p}$, where${\displaystyle w_p}$ is theFricke involution of${\displaystyle X_0(p)}$, hasgenus zero.
2. Every supersingular elliptic curve in characteristic${\displaystyle p}$ can be defined over theprime subfield${\displaystyle {\mathbb{F}}_p}$.
3. The order of the Monster group is divisible by${\displaystyle p}$.

The equivalence is due toAndrew 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 (thenconjectural) existence of a sporadic simple group having exactly these primes as prime divisors. This strange coincidence was the beginning of the theory ofmonstrous moonshine.

All supersingular primes areChen 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.

• Classes of prime numbers
• Moonshine theory
• Sporadic groups

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• Short description matches Wikidata