Locally imprimitive points on elliptic curves
Fuente:
arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866913051481997312 |
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| author | Jones, Nathan Pappalardi, Francesco Stevenhagen, Peter |
| author_facet | Jones, Nathan Pappalardi, Francesco Stevenhagen, Peter |
| contents | 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 $ρ_{E/K, P}:G_K\to\mathrm{GL}_3(\hat{\mathbf Z})$, and use it to construct non-trivial examples of global points on elliptic curves that are locally imprimitive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_03964 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Locally imprimitive points on elliptic curves Jones, Nathan Pappalardi, Francesco Stevenhagen, Peter Number Theory Algebraic Geometry 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 $ρ_{E/K, P}:G_K\to\mathrm{GL}_3(\hat{\mathbf Z})$, and use it to construct non-trivial examples of global points on elliptic curves that are locally imprimitive. |
| title | Locally imprimitive points on elliptic curves |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2304.03964 |