  Under 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 $\mathfrak p$ of cyclic
reduction, possibly under GRH, a globally primitive point $P\in E(K)$ may fail
to generate any of the point groups $E(k_{\mathfrak p})$. We describe this
phenomenon in terms of an associated Galois representation $\rho_{E/K,
P}:G_K\to\mathrm{GL}_3(\hat{\mathbf Z})$, and use it to construct non-trivial
examples of global points on elliptic curves that are locally imprimitive.
