Supersingular prime (for an elliptic curve)

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

Elkies (1987) showed that any elliptic curve over the rational numbers has infinitely many supersingular primes. However, the set of supersingular primes has asymptotic density zero. Lang & Trotter (1976) conjectured that the number of supersingular primes less than a bound X is within a constant multiple of $\frac{\sqrt{X}}{\ln X}$, using heuristics involving the distribution of eigenvalues of the Frobenius endomorphism. As of 2012, this conjecture is open.

More generally, if K is any global field—i.e., a finite extension either of Q or of 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.