Locally imprimitive points on elliptic curves

Mathematical Proceedings of the Cambridge Philosophical Society, vol. 180, pp. 663–683

Abstract

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 ${\mathfrak{p}}$ 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 $\rho_{E/K, P}\,:\,G_K\to\mathrm{GL}_3({\widehat {{\mathbf{Z}}}})$ , and use it to construct non-trivial examples of global points on elliptic curves that are locally imprimitive.

Authors 2

1. Francesco Pappalardi
   Affiliation as printed

   Largo Giovanni Battista Marzi

2. Peter Stevenhagen Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, Universiteit Leiden