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arXiv · 2106.04579

On 7-division fields of CM elliptic curves

Jessica Alessandrì·Laura Paladino
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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.

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Jessica Alessandrì, Laura Paladino. 2021-06-08. On 7-division fields of CM elliptic curves. https://arxiv.org/abs/2106.04579

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