On primitive points of elliptic curves with complex multiplication

Abstract

Let E be an elliptic curve defined over Q and P∈E(Q) a rational point of infinite order. Suppose that E has complex multiplication by an order in the imaginary quadratic field k. Denote by ME,P the set of rational primes ℓ such that ℓ splits in k, E has good reduction at ℓ, and P is a primitive point modulo ℓ. Under the generalized Riemann hypothesis, we can determine the positivity of the density of the set ME,P explicitly.