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

Nuno Freitas, Alain Kraus

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 f…