Local Transitivity and Entanglement Obstructions for Primitive Points

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 $|\mathcal{P}(E)|$ 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 $|\mathcal{P}(E)|=1$; hence Serre curves do not contribute isolated $j$-invariants.

22 pages including references