Wild Galois representations: Elliptic curves over a 2-adic field with non-abelian inertia action

Abstract

In this paper, we present a description of the [Formula: see text]-adic Galois representation attached to an elliptic curve defined over a [Formula: see text]-adic field [Formula: see text], in the case where the image of inertia is non-abelian. There are two possibilities for the image of inertia, namely [Formula: see text] and [Formula: see text], and in each case, we need to distinguish whether the inertia degree of [Formula: see text] over [Formula: see text] is even or odd. The results presented here are being implemented in an algorithm to compute explicitly the Galois representation in these four cases.