- Supersingular prime
If "E" is an
elliptic curve defined over therational number s, then a prime "p" is supersingular for "E" if the reduction of "E" modulo "p" is asupersingular elliptic curve over theresidue field Fp. More generally, if "K" is anyglobal field — i.e., afinite extension either of Q or of Fp("t") — and "A" is anabelian variety defined over "K", then a supersingular prime for "A" is afinite place of "K" such that the reduction of "A" modulo is a supersingularabelian variety .Alternately, in some contexts, the term supersingular prime (without qualifier) is used for a prime
divisor of the order of theMonster group "M", the largest of thesporadic simple group s. In this sense, there are precisely 15 supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71.Although these two usages are certainly distinct (the first is relative to a particular elliptic curve, whereas the second is not), they are related. Indeed, for a prime number "p", the following are equivalent:
(i) The
modular curve "X"0+("p") = "X"0("p") / "w"p has genus zero.(ii) Every supersingular elliptic curve in characteristic "p" can be defined over the
prime subfield Fp.(iii) 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 (i) are exactly the 15 primes 2,...,71 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.ee also
*
Hasse-Witt matrix References
*MathWorld|title=Supersingular Prime|urlname=SupersingularPrime
* cite book
author =Joseph H. Silverman
year = 1986
title = The Arithmetic of Elliptic Curves
publisher = Springer*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.
Wikimedia Foundation. 2010.