On the Density of Coprime Reductions of Elliptic Curves
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866913105813962752 |
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| author | Hamakiotes, Asimina S. Lee, Sung Min Mayle, Jacob Wang, Tian |
| author_facet | Hamakiotes, Asimina S. Lee, Sung Min Mayle, Jacob Wang, Tian |
| contents | Given non-CM elliptic curves $E_1$ and $E_2$ over $\mathbb{Q}$, we study the natural density of primes $p$ of good reduction for which the orders of the groups $E_1(\mathbb{F}_p)$ and $E_2(\mathbb{F}_p)$ are coprime. This problem may be viewed as an elliptic curve analogue of the classical question concerning the density of coprime integer pairs. Motivated by Zywina's refinement of the Koblitz conjecture, we formulate a conjecture for the density of such primes. We prove that the series defining this constant converges and admits an almost Euler product expansion. In the case of Serre pairs, we give a closed formula for the constant and use it to prove a moments result describing the distribution of these constants as $(E_1, E_2)$ varies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24915 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Density of Coprime Reductions of Elliptic Curves Hamakiotes, Asimina S. Lee, Sung Min Mayle, Jacob Wang, Tian Number Theory Primary 11G05, Secondary 11F80, 11G10, 11N05 Given non-CM elliptic curves $E_1$ and $E_2$ over $\mathbb{Q}$, we study the natural density of primes $p$ of good reduction for which the orders of the groups $E_1(\mathbb{F}_p)$ and $E_2(\mathbb{F}_p)$ are coprime. This problem may be viewed as an elliptic curve analogue of the classical question concerning the density of coprime integer pairs. Motivated by Zywina's refinement of the Koblitz conjecture, we formulate a conjecture for the density of such primes. We prove that the series defining this constant converges and admits an almost Euler product expansion. In the case of Serre pairs, we give a closed formula for the constant and use it to prove a moments result describing the distribution of these constants as $(E_1, E_2)$ varies. |
| title | On the Density of Coprime Reductions of Elliptic Curves |
| topic | Number Theory Primary 11G05, Secondary 11F80, 11G10, 11N05 |
| url | https://arxiv.org/abs/2603.24915 |