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Wild Galois representations: a family of hyperelliptic curves with large inertia image

Part of: Arithmetic algebraic geometry Number theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  26 January 2022

NIRVANA COPPOLA
NIRVANA COPPOLA*
Affiliation:
Vrije Universiteit Amsterdam, De Boelelaan 1111, 1081 HV Amsterdam, The Netherlands. e-mail: n.coppola@vu.nl
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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.

Keywords

11G2011G20

MSC classification

Primary: 11G20: Curves over finite and local fields
Secondary: 11F80: Galois representations 11F85: $p$-adic theory, local fields 11-04: Explicit machine computation and programs (not the theory of computation or programming)
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

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Footnotes

†

Supported by EPSRC.

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