In 1990, Kraus [12] classified all possible inertia images of the \(\ell \) -adic Galois representation attached to an elliptic curve over a non-archimedean local field. In [1, 2], 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 [4].
