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
, using heuristics involving the distribution of eigenvalues of the Frobenius endomorphism.
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