Towards a classification of entanglements of Galois representations attached to elliptic curves

TL;DR

This paper classifies how Galois representations attached to elliptic curves can have smaller-than-expected images due to entanglements between division fields, providing a group-theoretic framework for understanding these phenomena.

Contribution

It introduces a group-theoretic classification of entanglements in Galois representations of elliptic curves, especially over abelian extensions, advancing understanding of their image structures.

Findings

Categorization of entanglement types in Galois representations

Results on elliptic curves with entanglements over abelian extensions

Framework applicable to principally polarized abelian varieties

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve, let $\overline{\mathbb{Q}}$ be a fixed algebraic closure of $\mathbb{Q}$, and let $G_{\mathbb{Q}}=\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group of $\mathbb{Q}$. The action of $G_{\mathbb{Q}}$ on the adelic Tate module of $E$ induces the adelic Galois representation $\rho_E\colon G_{\mathbb{Q}} \to \text{GL}(2,\widehat{\mathbb{Z}}).$ The goal of this paper is to explain how the image of $\rho_E$ can be smaller than expected. To this end, we offer a group theoretic categorization of different ways in which an entanglement between division fields can be explained and prove several results on elliptic curves (and more generally, principally polarized abelian varieties) over $\mathbb{Q}$ where the entanglement occurs over an abelian extension.