Galois Theory, discriminants and torsion subgroups of elliptic curves

TL;DR

This paper explores the connection between torsion subgroups of elliptic curves over rationals and their mod 2 Galois representations, utilizing discriminant properties to establish a precise relationship.

Contribution

It introduces a novel characterization linking the squareness of elliptic curve discriminants to the structure of their torsion subgroups via Galois representations.

Findings

Established a direct relationship between torsion subgroups and mod 2 Galois images.

Connected discriminant properties to torsion subgroup classifications.

Provided new criteria for analyzing elliptic curve torsion structures.

Abstract

We find a tight relationship between the torsion subgroup and the image of the mod 2 Galois representation associated to an elliptic curve defined over the rationals. This is shown using some characterizations for the squareness of the discriminant of the elliptic curve.