Entanglement of elliptic curves upon base extension

TL;DR

The paper investigates how the $(p,q)$-entanglement type of an elliptic curve $E$ over a number field $K$ changes under base extension, focusing on the finite set of entanglement types $\operatorname{Ent}_{p,q}(E/K)$. It proves that for every prime $\ell$ dividing $d(E,p,q)=\gcd(|\operatorname{im}\rho_{E,p}|,|\operatorname{im}\rho_{E,q}|)$ there exists a base field $L/K$ with $T_{p,q}(E/L)\cong \mathbb{Z}/\ell\mathbb{Z}$, i.e., cyclic entanglement of order $\ell$ can be forced. The authors develop a group-theoretic framework, relate entanglement to subfield structure via $K_{p,q}(E)=K(E[p])\cap K(E[q])$, and introduce entangleable fields to study how entanglement can be glued or shrunk under base change, with explicit results for $(2,q)$-entanglements over $\mathbb{Q}$. They also provide a detailed classification of $(2,q)$-entanglement types, including constructions ensuring the presence of $S_3$-type entanglement and identifying exceptional primes and small primes where the behavior deviates. Together, these results give constructive tools to control and predict how torsion-field intersections influence the full mod $pq$ Galois image under base extension, with concrete implications for the arithmetic of elliptic curves.

Abstract

Fix distinct primes $p$ and $q$ and let $E$ be an elliptic curve defined over a number field $K$. The $(p,q)$-entanglement type of $E$ over $K$ is the isomorphism class of the group $\operatorname{Gal}(K(E[p])\cap K(E[q])/K)$. The size of this group measures the extent to which the image of the mod $pq$ Galois representation attached to $E$ fails to be a direct product of the mod $p$ and mod $q$ images. In this article, we study how the $(p,q)$-entanglement group varies over different base fields. We prove that for each prime $\ell$ dividing the greatest common divisor of the size of the mod $p$ and $q$ images, there are infinitely many fields $L/K$ such that the entanglement over $L$ is cyclic of order $\ell$. We also classify all possible $(2,q)$-entanglement types that can occur as the base field $L$ varies.

Figures (1)

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Theorems & Definitions (63)

• Theorem 1.2
• Corollary 1.3
• Example 1.4
• Remark 1.5
• Corollary 1.6
• Definition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof