Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Nguyen, Chi, Yagci, Arman, Zhou, Yunchuan
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7125808541
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