Supersingular prime

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If E is an elliptic curve defined over the rational numbers, then a prime p is supersingular for E if the reduction of E modulo p is a supersingular elliptic curve over the residue field $\mathbb{F}_p$ . More generally, if K is any global field -- i.e., a finite extension either of $\mathbb{Q}$ or of $\mathbb{F}_p(t)$ -- and A is an abelian variety defined over K, then a supersingular prime $\mathfrak{p}$ for A is a finite place of K such that the reduction of A modulo $\mathfrak{p}$ is a supersingular abelian variety.

Alternately, in some contexts the term supersingular prime is used for a prime divisor of the order of the Monster group M, the largest of the sporadic simple groups. 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 $X_0^+(p)$ has genus zero.

(Let H denote the upper half-plane. For a natural number n, let Γ₀(n) denote the modular group Γ₀, and let w_n be the Fricke involution defined by the block matrix [[0, −1], [n, 0]]. Furthermore, let the modular curve X₀(n) be the compactification (with added cusps) of

$Y_0(n) = \Gamma_0(n)\setminus H,$

and for any prime p, define

$X_0^+(p) = X_0(p)/w_p.$ )

(ii) Every supersingular elliptic curve of characteristic p can be defined over the prime subfield $\mathbb{F}_p$ .

(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.

[edit] See also

Hasse-Witt matrix

[edit] References

Eric W. Weisstein, Supersingular Prime at MathWorld.

Joseph H. Silverman (1986). The Arithmetic of Elliptic Curves. 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.