Elliptic curves with maximally disjoint division fields

TL;DR

This paper constructs numerous examples of pairs of rational elliptic curves with maximally disjoint division fields, ensuring their combined Galois representation has the largest possible image, which is significant for understanding their algebraic properties.

Contribution

It provides explicit examples of elliptic curve pairs with maximally disjoint division fields, advancing the understanding of their Galois representations and division field intersections.

Findings

Constructed infinite examples of elliptic curve pairs with maximally disjoint division fields.

Analyzed the intersection properties of division fields for rational elliptic curves.

Showed that maximal disjointness leads to maximal Galois representation images.

Abstract

One of the many interesting algebraic objects associated to a given rational elliptic curve, $E$, is its full-torsion representation $\rho_E:\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})\to\mathrm{GL}_2(\hat{\mathbf{Z}})$. Generalizing this idea, one can create another full-torsion Galois representation, $\rho_{(E_1,E_2)}:\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})\to\left(\mathrm{GL}_2(\hat{\mathbf{Z}})\right)^2$ associated to a pair $(E_1,E_2)$ of rational elliptic curves. The goal of this paper is to provide an infinite number of concrete examples of pairs of elliptic curves whose associated full-torsion Galois representation $\rho_{(E_1,E_2)}$ has maximal image. The size of the image is inversely related to the size of the intersection of various division fields defined by $E_1$ and $E_2$. The representation $\rho_{(E_1,E_2)}$ has maximal image when these division fields are maximally disjoint, and most of the paper is devoted to studying these intersections.