Abstract Under the Generalised Riemann Hypothesis (GRH), any element in the multiplicative group of a number field K that is globally primitive (i. e. , not a perfect power in K^*) is a primitive root modulo a set of primes of K of positive density. For elliptic curves E/K that are known to have infinitely many primes p of cyclic reduction, possibly under GRH, a globally primitive point P E (K) may fail to generate any of the point groups E (k). We describe this phenomenon in terms of an associated Galois representation ₄/₊, \,: \, GK₃ ({{Z}}), and use it to construct non-trivial examples of global points on elliptic curves that are locally imprimitive.
Jones et al. (Thu,) studied this question.