On 7-division fields of CM elliptic curves

TL;DR

This paper classifies the 7-division fields of CM elliptic curves over number fields, providing explicit generators, Galois groups, and applications to divisibility problems and modular curves.

Contribution

It offers a complete classification of the 7-division fields for CM elliptic curves, including explicit generators and Galois groups, advancing understanding of their arithmetic properties.

Findings

Explicit generators for K_7/K are given.

All Galois groups of K_7/K are determined.

Applications to Local-Global Divisibility and modular curves are demonstrated.

Abstract

Let $\mathcal{E}$ be a CM elliptic curve defined over a number field $K$, with Weiestrass form $y^3=x^3+bx$ or $y^2=x^3+c$. For every positive integer $m$, we denote by ${\mathcal{E}}[m]$ the $m$-torsion subgroup of ${\mathcal{E}}$ and by $K_m:=K({\mathcal{E}}[m])$ the $m$-th division field, i.e. the extension of $K$ generated by the coordinates of the points in ${\mathcal{E}}[m]$. We classify all fields $K_7$. In particular we give explicit generators for $K_7/K$ and produce all Galois groups ${\textrm{Gal}}(K_7/K)$. We also show some applications to the Local-Global Divisibility Problem and to modular curves.