Wild Galois representations: elliptic curves with wild cyclic reduction

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.