ON THE DISTRIBUTION OF SUPERSINGULAR PRIMES

Let E be a fixed elliptic curve defined over the rational numbers. We prove that the number of primes p less than or equal to x such that E has supersingular reduction mod p is greater than log(3) x/(log(4) x)( 1+delta) for any positive delta and x sufficiently large. Here log(k) x is defined recursively as log(log(k-1) x) and log(1) x = log x. We a lso establish several results related to the Lang-Trotter conjecture.