# Wild Galois Representations: Elliptic curves over a $3$-adic field

## TL;DR

This paper explicitly describes the Galois representation associated with elliptic curves over a 3-adic field, especially in cases with complex inertia image structures, advancing understanding of local Galois actions.

## Contribution

It provides an explicit description of the Galois representation for elliptic curves with non-cyclic inertia image over a 3-adic field, addressing a previously unresolved case.

## Key findings

- Explicit description of Galois representation for complex inertia images
- Clarification of the case with inertia image isomorphic to C_3 ⋉ C_4
- Advancement in understanding Galois actions on elliptic curves over local fields

## Abstract

Given an elliptic curve $E$ over a local field $K$ with residue characteristic $3$, we investigate the action of the absolute Galois group of $K$ in the case of potentially good reduction. In particular the only not completely known case is that of the $\ell$-adic Galois representation attached to an elliptic curve such that the image of inertia is non-cyclic, and isomorphic to $C_3 \rtimes C_4$. In this work we describe such a representation explicitly.