Reduction mod p of Subgroups of the Mordell-Weil Group of an Elliptic Curve

Let E be an elliptic curve defined over Q. Let Γ be a free subgroup of rank r of E(Q). For any prime p of good reduction, let Γp be the reduction of Γ modulo p and Ep be the reduction of E modulo p. We prove that if E has CM then for all but o(x/ log x) of primes p ≤ x, |Γp| ≥ p r r+2 + , where (p) is any function of p such that (p)→ 0 as p→∞. This is a consequence of two other results. Denote by Np the cardinality of Ep(Fp), where Fp is a finite field of p elements. Then for any δ > 0, the set of primes p for which Np has a divisor in the range (pδ− (p), pδ+ (p)) has density zero. Moreover, the set of primes p for which |Γp| < p r r+2 − (p) has density zero.