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 Galois representation attached to an elliptic curve defined over a $2$-adic field $K$, in the case where the image of inertia is non-abelian. There are two possibilities for the image of inertia, namely $Q_8$ and $S{L}_2({\mathbb{F}}_3)$, and in each case we need to distinguish whether the inertia degree of $K$ over ${\mathbb{Q}}_2$ is even or odd. The result presented here can be implemented in an algorithm to compute explicitly the Galois representation in these four cases.

Final accepted version, to appear in IJNT. No diagrams in this version. 10 pages