Recovering Kodaira types from $\mathrm{\ell }$-torsion on elliptic curves

The classical Néron-Ogg-Shafarevich criterion characterises good reduction of an elliptic curve $E$ over a $p$-adic field via the action of inertia on the $\mathrm{\ell }$-adic Tate module. However, the action of inertia on $E[\mathrm{\ell }]$ is not sufficient to distinguish between potentially good and multiplicative reduction, and the action on $T_{\mathrm{\ell }}(E)$ is not sufficient to determine the Kodaira type. We remedy this situation by endowing $E[\mathrm{\ell }]$ with a distance function that records the $p$-adic distances between the $x$-coordinates of the points. We show that, equipped with this additional structure, $E[\mathrm{\ell }]$ determines the Kodaira type of the elliptic curve. In the case of residue characteristic $2$, we assume that $E$ does not have potentially good reduction of type $I_n^{\ast }$.