# The 3-adic representations arising from elliptic curves over $\mathbb{Q}_3$ with potential good reduction

## TL;DR

This paper classifies all potentially crystalline 3-adic Galois representations from elliptic curves over Q_3, especially focusing on cases with wild potential good reduction, using filtered module descriptions.

## Contribution

It provides a complete classification of these representations, detailing their structure via filtered (phi, Gal) modules, including wild reduction cases.

## Key findings

- Complete classification of 3-adic Galois representations from elliptic curves over Q_3
- Description of representations in terms of filtered (phi, Gal) modules
- Analysis of wild potential good reduction cases

## Abstract

We give a complete classification of all the potentially crystalline 3-adic representations of the absolute Galois group of $\mathbb{Q}_3$ that are isomorphic to the Tate module of an elliptic curve defined over $\mathbb{Q}_3$. These representations are described in terms of their associated filtered $(\varphi, \mathrm{Gal}(K/\mathbb{Q}_3))$-modules. The most interesting cases occur when the potential good reduction is wild.