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arXiv · 2601.17559

Local Transitivity and Entanglement Obstructions for Primitive Points

Chi Nguyen·Arman Yagci·Yunchuan Zhou
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Abstract

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.

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Chi Nguyen, Arman Yagci, Yunchuan Zhou. 2026-01-24. Local Transitivity and Entanglement Obstructions for Primitive Points. https://arxiv.org/abs/2601.17559

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