SUPERSINGULAR PRIME

In mathematics, there are two notions of 'supersingular primes'.
If $scriptstyle\; E/\; mathbb\; Q$ is an elliptic curve, then one says that a prime ''p'' is 'supersingular' for ''E'' if the number of points of ''E'' defined over $scriptstyle\; mathbb\; F\_p$ is congruent to 1 modulo ''p''. Equivalently, if the only ''p''-torsion point of ''E'' defined over $scriptstyle\; mathbb\; F\_p$ is the identity, then ''p'' is a supersingular prime. Note that a supersingular prime is not a singular prime (in that ''E'' modulo ''p'' is an elliptic curve, i.e. ''E'' has good reduction at a supersingular prime ''p'').
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 $scriptstylemathbb\; Q$, 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 . Furthermore, let the modular curve ''X''₀(''n'') be the compactification (with added cusps) of
: $Y\_0(n)\; =\; Gamma\_0(n)setminus\; H,,$
and for any prime ''p'', define
: $X\_0^+(p)\; =\; X\_0(p)/w\_p.$
Then ''p'' is supersingular means by definition that the genus of ''X''₀ ⁺ (''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 . It can also be shown that the supersingular primes are exactly the prime factors of the group order of the Monster group ''M''.
The supersingular primes are all Chen primes.

Contents

See also

References

See also

★ Hasse-Witt matrix

References

★ The Arithmetic of Elliptic Curves, Joseph H. Silverman, , , Springer, 1986,