On 7-division fields of CM elliptic curves

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 $\text{Gal}(K_7/K)$. We also show some applications to the Local-Global Divisibility Problem and to modular curves.

arXiv commentarXiv admin note: text overlap with arXiv:1808.00029