On 7 division fields of CM elliptic curves

Jessica Alessandrì, Laura Paladino

Published 2021-06-08Version 1

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 $K$ obtained by adding to it the coordinates of the points in ${\mathcal{E}}[m]$. We classify all the fields $K_7$ in terms of generators, degrees and Galois groups. We also show some applications to the Local-Global Divisibility Problem and to modular curves and Shimura curves.