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

Abstract

The classical N\'eron-Ogg-Shafarevich criterion characterises good reduction of an elliptic curve $E$ over a $p$-adic field via the action of inertia on the $\ell$-adic Tate module. However, the action of inertia on $E[\ell]$ is not sufficient to distinguish between potentially good and multiplicative reduction, and the action on $T_{\ell}(E)$ is not sufficient to determine the Kodaira type. We remedy this situation by endowing $E[\ell]$ 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, $E[\ell]$ 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 $I_n^*$.