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arXiv · 2304.03964

Locally imprimitive points on elliptic curves

Nathan Jones·Francesco Pappalardi·Peter Stevenhagen
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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 $K^*$) 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.

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Nathan Jones, Francesco Pappalardi, Peter Stevenhagen. 2023-04-08. Locally imprimitive points on elliptic curves. https://doi.org/10.1017/s0305004125101795

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