Wild Galois representations: elliptic curves with wild cyclic reduction

TL;DR

This paper completes the classification of Galois representations of elliptic curves with wild cyclic reduction over local fields, focusing on cases where the residue characteristic is 2 or 3 and the reduction becomes good over highly ramified extensions.

Contribution

It provides a comprehensive computation of Galois representations for elliptic curves with wild cyclic reduction, extending previous work to all remaining cases with highly ramified extensions.

Findings

Explicit Galois representations for wild cases with residue characteristic 2 or 3

Extension of classification to all wild cyclic reduction scenarios

Based on detailed analysis of ramified extensions and reduction types

Abstract

In 1990, Kraus classified all possible inertia images of the $\ell$-adic Galois representation attached to an elliptic curve over a non-archimedean local field. In previous work, the author computed explicitly the Galois representation of elliptic curves having non-abelian inertia image, a phenomenon which only occurs when the residue characteristic of the field of definition is $2$ or $3$ and the curve attains good reduction over some non-abelian ramified extension. In this work, the computation of the Galois representation in all the remaining wild cases, i.e. when the residue characteristic is $p=2$ or $3$ and the curve attains good reduction over an extension whose ramification degree is divisible by $p$ (without assuming the condition on the image of inertia being non-abelian), is completed. This is based on Chapter V of the author's PhD thesis.