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Wild Galois representations: a family of hyperelliptic curves with large inertia image
Published online by Cambridge University Press: 26 January 2022
Abstract
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.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 173 , Issue 3 , November 2022 , pp. 619 - 633
- Copyright
- © The Author(s), 2022. Published by Cambridge University Press on behalf of Cambridge Philosophical Society
Footnotes
Supported by EPSRC.
References
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