REDUCTION modp OF SUBGROUPS OF THE MORDELL–WEIL GROUP OF AN ELLIPTIC CURVE

Abstract

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, [Formula: see text] 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 [Formula: see text] has density zero.