On 5-torsion of CM elliptic curves

TL;DR

This paper classifies the fields generated by 5-torsion points of CM elliptic curves over number fields, detailing their structure, degrees, and Galois groups, with applications to divisibility problems and modular/ Shimura curves.

Contribution

It provides a complete classification of the fields $K_5$ for CM elliptic curves with specific Weierstrass forms, including generators, degrees, and Galois groups, and explores their applications.

Findings

Classified $K_5$ fields for CM elliptic curves with given Weierstrass forms.

Determined degrees and Galois groups of these fields.

Applied results to divisibility problems and modular/Shimura curves.

Abstract

Let $\mathcal{E}$ be an elliptic curve defined over a number field $K$. Let $m$ be a positive integer. We denote by ${\mathcal{E}}[m]$ the $m$-torsion subgroup of $\mathcal{E}$ and by $K_m:=K({\mathcal{E}}[m])$ the number field obtained by adding to $K$ the coordinates of the points of ${\mathcal{E}}[m]$. We describe the fields $K_5$, when $\mathcal{E}$ is a CM elliptic curve defined over $K$, with Weiestrass form either $y^2=x^3+bx$ or $y^2=x^3+c$. In particular we classify the fields $K_5$ in terms of generators, degrees and Galois groups. Furthermore we show some applications of those results to the Local-Global Divisibility Problem, to modular curves and to Shimura curves.