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Supersingular Prime


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

Group-theoretically, let Gamma_0(N) be the modular group Gamma0, and let X_0(N) be the compactification (by adding cusps) of Y_0(N)=Gamma_0(N)H, where H is the upper half-plane. Also define w_N to be the Fricke involution defined by the block matrix [[0,-1],[N,0]]. For p a prime, define X_0^+(p)=X_0(p)/(w_p). Then p is a supersingular prime if the genus of X_0^+(p)=0.

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

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.


See also

Modular Group Gamma0, Monster Group

This entry contributed by John McKay

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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."

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Supersingular Prime

Cite this as:

McKay, John. "Supersingular Prime." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SupersingularPrime.html

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