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In this work we generalise the main result of [1] to the family of hyperelliptic curves with potentially good reduction over a p-adic field which have genus $g=({p-1})/{2}$ and the largest possible image of inertia under the $\ell$ -adic Galois representation associated to its Jacobian. We will prove that this Galois representation factors as the tensor product of an unramified character and an irreducible representation of a finite group, which can be either equal to the inertia image (in which case the representation is easily determined) or a $C_2$ -extension of it. In this second case, there are two suitable representations and we will describe the Galois action explicitly in order to determine the correct one.

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We study a 3-dimensional stratum M 3,V of the moduli space M 3 of curves of genus 3 parameterizing curves Y that admit a certain action of V C 2 × C 2 . We determine the possible types of the stable reduction of these curves to characteristic different from 2. We define invariants for M 3,V and characterize the occurrence of each of the reduction types in terms of them. We also calculate the j-invariant (resp. the Igusa invariants) of the irreducible components of positive genus of the stable reduction Y in terms of the invariants.

In this paper we present a description of the Galois representation attached to an elliptic curve defined over a 2-adic field K, in the case where the image of inertia is non-abelian. There are two possibilities for the image of inertia, namely Q 8 and SL 2 (F 3 ), and in each case we need to distinguish whether the inertia degree of K over Q 2 is even or odd. The result presented here can be implemented in an algorithm to compute explicitly the Galois representation in these four cases.