On inertial types of elliptic curves

Let $F/\mathbb Q_p$  be a finite extension and $E/F$ an elliptic curve. The action of inertia on the $\ell$-adic Tate module of $E (\ell \neq p)$ defines a Weil-Deligne inertial type $\tau_E: I_F \rightarrow GL_2(\mathbb C)$. In this talk I will present a classification of such inertial types, extending the work of Dembele, Freitas and Voight for the case $F = \mathbb Q_p$. The main difficulty arises over 2-adic fields where, in particular, there are triply imprimitive types, which are not present over $\mathbb Q_2$, and in proving that each type admits a realization as the inertial type of an elliptic curve $E$. We will also discuss an algorithm based on this characterization that determines all inertia types over a given $F$. This characterization, done purely in terms of Weil-Deligne representations, can be used outside of the realm of elliptic curves. For instance, it has been used to solve diophantine equations using Jacobians of hyperelliptic curves by Pacetti and Villagra. This is joint work with Enric Florit and Nuno Freitas.