Supersingular Prime

There are two definitions of the supersingular primes: one group-theoretic, and the other number-theoretic.

Group-theoretically, let  be the modular group Gamma0, and let  be the compactification (by adding cusps) of , where  is the upper half-plane. Also define  to be the Fricke involution defined by the block matrix . For  a prime, define . Then  is a supersingular prime if the genus of .

The number-theoretic definition involves supersingular elliptic curves defined over the algebraic closure of the finite field . They have their j-invariant in .

Supersingular curves were mentioned by Charlie the math genius in the Season 2 episode "In Plain Sight" of the television crime drama NUMB3RS.

There are exactly 15 supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71 (OEIS A002267). The supersingular primes are exactly the set of primes that divide the group order of the Monster group.

REFERENCES:

Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, 1985.

Conway, J. H. and Norton, S. P. "Monstrous Moonshine." Bull. London Math. Soc. 11, 308-339, 1979.

Ogg, A. P. "Modular Functions." In The Santa Cruz Conference on Finite Groups. Held at the University of California, Santa Cruz, Calif., June 25-July 20, 1979 (Ed. B. Cooperstein and G. Mason). Providence, RI: Amer. Math. Soc., pp. 521-532, 1980.

Silverman, J. H. The Arithmetic of Elliptic Curves. New York: Springer-Verlag, 1986.

Silverman, J. H. The Arithmetic of Elliptic Curves II. New York: Springer-Verlag, 1994.

Sloane, N. J. A. Sequence A002267 in "The On-Line Encyclopedia of Integer Sequences."