On 5-torsion of CM elliptic 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.