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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.17559 |
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| _version_ | 1866912845865680896 |
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| author | Nguyen, Chi Yagci, Arman Zhou, Yunchuan |
| author_facet | Nguyen, Chi Yagci, Arman Zhou, Yunchuan |
| contents | Primitive points on the tower of modular curves $X_1(n)$ provide a finite "certificate set" for detecting isolated points above a fixed $j$-invariant: for a non-CM elliptic curve $E/\mathbb{Q}$, $j(E)$ arises from an isolated point on some $X_1(N)$ if and only if one of the associated primitive point is isolated. We bound the number $\lvert \mathcal{P}(E)\rvert$ of primitive points in terms of the adelic index $I(E)$ and give criteria as well as an algorithm for uniqueness of primitive point. As an application, every Serre curve has $\lvert \mathcal{P}(E)\rvert =1$; hence Serre curves do not contribute isolated $j$-invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_17559 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local Transitivity and Entanglement Obstructions for Primitive Points Nguyen, Chi Yagci, Arman Zhou, Yunchuan Number Theory 14H52, 11F80 Primitive points on the tower of modular curves $X_1(n)$ provide a finite "certificate set" for detecting isolated points above a fixed $j$-invariant: for a non-CM elliptic curve $E/\mathbb{Q}$, $j(E)$ arises from an isolated point on some $X_1(N)$ if and only if one of the associated primitive point is isolated. We bound the number $\lvert \mathcal{P}(E)\rvert$ of primitive points in terms of the adelic index $I(E)$ and give criteria as well as an algorithm for uniqueness of primitive point. As an application, every Serre curve has $\lvert \mathcal{P}(E)\rvert =1$; hence Serre curves do not contribute isolated $j$-invariants. |
| title | Local Transitivity and Entanglement Obstructions for Primitive Points |
| topic | Number Theory 14H52, 11F80 |
| url | https://arxiv.org/abs/2601.17559 |