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(F_p)∣ on the curve over the finite field of p elements has prime order is asymptotic to $c_{\text{e}} \frac{x} {{\left( {\log x} \right)^2 }}$ for some constant C _E. We consider curves without complex multiplication. Assuming the GRH (that is, the Riemann Hypothesis for Dedekind zeta functions) we prove that for $\gg \frac{x} {{\left( {\log x} \right)^2 }}$ primes p ≤x, the group order ∣E(F_p)∣ has at most 16 prime divisors. We also show (again, assuming the GRH) that for a random prime p, the group order ∣E(F_p)∣ has log log p prime divisors.