# Supersingular prime (algebraic number theory)

In algebraic number theory, a supersingular prime for a given elliptic curve is a prime number with a certain relationship to that curve. If the curve E is 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 Fp.

Noam Elkies showed that every elliptic curve over the rational numbers has infinitely many supersingular primes. However, the set of supersingular primes has asymptotic density zero (if E does not have complex multiplication). (Lang Trotter) conjectured that the number of supersingular primes less than a bound X is within a constant multiple of $\displaystyle{ \frac{\sqrt{X}}{\ln X} }$, using heuristics involving the distribution of eigenvalues of the Frobenius endomorphism. As of 2019, this conjecture is open.

More generally, if K is any global field—i.e., a finite extension either of Q or of Fp(t)—and A is an abelian variety defined over K, then a supersingular prime $\displaystyle{ \mathfrak{p} }$ for A is a finite place of K such that the reduction of A modulo $\displaystyle{ \mathfrak{p} }$ is a supersingular abelian variety.