An application of sieve methods to elliptic curves

Abstract

Let E be an elliptic curve defined over the rationals. Koblitz conjectured that the number of primes p ≤ x such that the number of points |E(Fp)| on the curve over the finite field of p elements has prime order is asymptotic to CE x/(log x)2 for some constant CE. We consider curves without complex multiplication. Assuming the GRH (that is, the Riemann Hypothesis for Dedekind zeta functions) we prove that for ≫ x/(log x)2 primes p ≤ x, the group order |E(Fp)| has at most 16 prime divisors. We also show (again, assuming the GRH) that for a random prime p, the group order |E(Fp)| has log log p prime divisors.