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Wild Galois representations: a family of hyperelliptic curves with large inertia image
- Part of
-
- Journal:
- Mathematical Proceedings of the Cambridge Philosophical Society / Volume 173 / Issue 3 / November 2022
- Published online by Cambridge University Press:
- 26 January 2022, pp. 619-633
- Print publication:
- November 2022
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- Article
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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.
