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.
Jones, N., Pappalardi, F., Stevenhagen, P. (2026). Locally imprimitive points on elliptic curves. MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 180(3), 663-683 [10.1017/S0305004125101795].
Locally imprimitive points on elliptic curves
Jones N.
Membro del Collaboration Group
;Pappalardi F.
Membro del Collaboration Group
;Stevenhagen P.
Membro del Collaboration Group
2026-01-01
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
