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

Let $\ell$ and $p \geq 3$ 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 $E$.

Categories: math.NT