Local Galois representations for higher genus curves

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The aim of this research project is to describe the Galois representation attached to a hyperelliptic curve over a local field. Partial results on this problem exist: in particular, for elliptic curves it is known how to compute the inertia image and, in some cases, even the whole Galois representation. The first step of this project is to complete all the remaining cases, giving an explicit solution which can be implemented in a computer program. For higher genus curves, less is known, so the final objective is to understand the Galois representation for special families of curves, for which recent developments in the theory of models of curves can be applied, and again provide an explicit description of it.