Supersingular prime

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In mathematics, there are two notions of supersingular primes.

If $E/ \mathbb Q$ is an elliptic curve, then one says that a prime $p$ is supersingular for $E$ if the number of points of $E$ defined over $\mathbb F_p$ is congruent to 1 modulo $p$ . Equivalently, if the only $p$ -torsion point of $E$ defined over $\mathbb F_p$ is the identity, then $p$ is a supersingular prime. Note that a supersingular prime need not be singular (in the sense that $E$ modulo $p$ is a singular curve.)

If $E$ has complex multiplication, then the set of primes which are supersingular has Dirichlet density 1/2. However, if $E$ does not have CM, then this density is 0. Nonetheless, Noam Elkies has proven that for any elliptic curve over $\mathbb Q$ , there are infinitely many supersingular primes.

An alternative definition of supersingular primes comes from the theory of modular curves. Formally, 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

Y₀(n) = Γ₀(n)\H,

and for any prime p, define

X₀ ⁺ (p) = X₀(p) / w_p.

Then p is supersingular means by definition that the genus of X₀ ⁺ (p) is zero.

It is also possible to define supersingular primes in a number-theoretic way using supersingular elliptic curves defined over the algebraic closure of the finite field GF(p) that have their j-invariant in GF(p ²). As it turns out, there are exactly fifteen supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71 (sequence A002267 in OEIS). It can also be shown that the supersingular primes are exactly the prime factors of the group order of the Monster group M.