## 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 ℓ-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 C2-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.

Original language | English |
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Pages (from-to) | 619-633 |

Number of pages | 15 |

Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

Volume | 173 |

Issue number | 3 |

Early online date | 26 Jan 2022 |

DOIs | |

Publication status | Published - Nov 2022 |

### Bibliographical note

Funding Information:Supported by EPSRC.

Publisher Copyright:

© 2022 The Author(s).