Local Transitivity and Entanglement Obstructions for Primitive Points

TL;DR

This work bounds the number of primitive points $|\mathcal{P}(E)|$ attached to a non-CM elliptic curve $E/\mathbb{Q}$ by adelic invariants, proving $|\mathcal{P}(E)| \le \min\{ m_0^2, 1+\frac{I(E)\,\sigma_0(m_0)}{2} \}$. It then develops a concrete uniqueness criterion: $|\mathcal{P}(E)|=1$ iff the action of $H(n)$ on $V_n$ is transitive for all $n|m_0$, and provides local transitivity (LT$_G$) and entanglement-stabilizer (EF$_{m_0}$) hypotheses plus an algorithm to verify them. A detailed fiber-product analysis yields a sufficient criterion for uniqueness, and the EF$_{m_0}$ framework isolates when entanglement affects primitive-point counts. Finally, the Serre-curve case is settled: every Serre curve satisfies $|\mathcal{P}(E)|=1$, and combined with a density-one result for Serre curves, almost all non-CM elliptic curves do not contribute isolated $j$-invariants. Collectively, these results provide effective, computable tools for diagnosing whether a given $j$-invariant can be isolated via modular curves, with practical implications for uniformity questions in Galois representations.

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.

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.2: Uniqueness criterion
• Theorem 1.3: Sufficient criterion for uniqueness
• Theorem 1.4
• Definition 2.1
• Definition 2.2
• Example 2.3
• Lemma 3.1
• proof
• Lemma 3.2