Locally imprimitive points on elliptic curves

Abstract

Under GRH, any element in the multiplicative group of a number field $K$ that is globally primitive (i.e., not a perfect power in $K^{\ast }$) 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.

20 pages