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

## TL;DR

This paper characterizes the Galois representations of elliptic curves over 2-adic fields with non-abelian inertia images, providing a basis for explicit computational algorithms in four distinct cases.

## Contribution

It classifies possible non-abelian inertia images for elliptic curves over 2-adic fields and details how to explicitly compute the associated Galois representations.

## Key findings

- Inertia image can be $Q_8$ or $SL_2(\mathbb{F}_3)$.
- Distinction based on whether inertia degree over $\mathbb{Q}_2$ is even or odd.
- Provides an algorithmic framework for explicit Galois representation computation.

## 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 $SL_2(\mathbb{F}_3)$, 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.