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

TL;DR

This paper fully characterizes the degree of the $p$-torsion field extension of elliptic curves over $Q_ell$ for distinct primes $ell$ and $p$, providing a practical method to compute the discriminant ideal.

Contribution

It offers a complete description of the degree of the $p$-torsion field extension for elliptic curves over local fields, including a recipe for the discriminant ideal.

Findings

Explicit degree formulas for $Q_ell(E_p)/Q_ell$

A method to compute the discriminant ideal of the extension

Complete classification for all elliptic curves over $Q_ell$

Abstract

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 $E$.