On Galois representations of superelliptic curves

TL;DR

This paper describes the Galois representations associated with superelliptic curves over a DVR, using a cluster picture approach under specific conditions on the polynomial and residual characteristic.

Contribution

It introduces a new description of Galois representations for superelliptic curves using cluster pictures, extending prior algorithms to a broader context.

Findings

Provides a cluster picture-based description of Galois representations

Extends previous algorithms to cases where roots are in the fraction field

Applicable when p does not divide n

Abstract

A superelliptic curve over a DVR ${\mathcal O}$ of residual characteristic $p$ is a curve given by an equation $C:y^n=f(x)$. The purpose of the present article is to describe the Galois representation attached to such a curve under the hypothesis that $f(x)$ has all its roots in the fraction field of ${\mathcal O}$ and that $p \nmid n$. Our results are inspired on the algorithm given in [BW17] but our description is given in terms of a cluster picture as defined in [DDMM18].