On the degree of the $p$-torsion field of elliptic curves over ${\mathbb{Q}}_{\ell }$ for $\ell \ne p$

Abstract

Let $\ell$ and $p\ge 3$ be distinct prime numbers. Let $E/{\mathbb{Q}}_{\ell }$ be an elliptic curve with $p$-torsion module $E_p$. Let ${\mathbb{Q}}_{\ell }(E_p)$ be the $p$-torsion field of $E$. We provide a complete description of the degree of the extension ${\mathbb{Q}}_{\ell }(E_p)/{\mathbb{Q}}_{\ell }$. As a consequence, we obtain a recipe to determine the discriminant ideal of the extension ${\mathbb{Q}}_{\ell }(E_p)/{\mathbb{Q}}_{\ell }$ in terms of standard information on $E$.

36 pages