Conductor exponents for families of hyperelliptic curves

TL;DR

This paper computes conductor exponents at odd places for specific hyperelliptic curve families using cluster pictures, extending previous results and providing alternative methods for these calculations.

Contribution

It introduces a new approach to compute conductor exponents for hyperelliptic curves, extending known results to broader signatures and offering an alternative to existing methods.

Findings

Computed conductor exponents at odd places for three hyperelliptic families.

Extended previous results to the signature (p,p,r) case.

Provided an alternative method using cluster pictures.

Abstract

We compute the conductor exponents at odd places using the machinery of cluster pictures of curves for three infinite families of hyperelliptic curves. These are families of Frey hyperelliptic curves constructed by Kraus and Darmon in the study of the generalised Fermat equations of signatures $(r,r,p)$ and $(p,p,r)$, respectively. Here, $r$ is a fixed prime number and $p$ is a prime that is allowed to vary. In the context of the modular method, Billerey-Chen-Dieulefait-Freitas computed all conductor exponents for the signature $(r,r,p)$. We recover their computations at odd places, providing an alternative approach. In a similar setup, Chen-Koutsianas computed all conductor exponents for the signature $(p,p,5)$. We extend their work to the general case of signature $(p,p,r)$ at odd places. Our work can also be used to compute local arithmetic data for the curves in these families.