On the Greatest Prime Divisor of Np

Let E be an elliptic curve defined over Q. For any prime p of good reduction, let E_p be the reduction of E mod p. Denote by N_p the cardinality of E_p(F_p), where F_p is the finite field of p elements. Let P(N_p) be the greatest prime divisor of N_p. We prove that if E has CM then for all but o(x/ log x) of primes

                                      p ≤ x,P(Np) > p^ϑ(p),

where ϑ is any number less than ϑ₀ = 1 − 1/2 e^−1/4 = 0.6105 · · ·. As an application of this result we prove the following. Let Γ be a free subgroup of rank r ≥ 2 of the group of rational points E(Q), and Γp be the reduction of Γ mod p, then for a positive proportion of primes p ≤ x, we have

                                            |Γ_p| > p^ϑ₀−ϵ,

where ϵ > 0.