Supersingular prime (algebraic number theory)
# 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 $F_p.$

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 (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 2019, 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.

