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

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## Abstract

In this paper we present a description of the $\ell$-adic 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 $\SL_2(\F_3)$, and in each case we need to distinguish whether the inertia degree of $K$ over $\Q_2$ is even or odd. The result presented here are being implemented in an algorithm to compute explicitly the Galois representation in these four cases.
Original language English 10 International Journal of Number Theory 10 Feb 2020 https://doi.org/10.1142/S179304212050061X E-pub ahead of print - 10 Feb 2020

## Keywords

• elliptic curves
• local fields
• wild ramification
• Galois representations