Local Transitivity and Entanglement Obstructions for Primitive Points

TL;DR

This paper investigates primitive points on modular curves to understand isolated points related to elliptic curves, providing bounds, criteria, and algorithms, and showing Serre curves have a unique primitive point, thus not contributing isolated $j$-invariants.

Contribution

It introduces bounds and criteria for primitive points on modular curves and presents an algorithm for their uniqueness, linking primitive points to elliptic curve properties.

Findings

Bound on the number of primitive points in terms of adelic index

Criteria and algorithm for primitive point uniqueness

Serre curves have exactly one primitive point, excluding isolated $j$-invariants

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.