Semistable reduction of covers of degree $p$

TL;DR

This paper presents a method for computing semistable reduction of degree-$p$ covers of curves over local fields, including superelliptic and plane quartic curves, using non-archimedean geometry and the different function.

Contribution

It introduces a general approach for semistable reduction of degree-$p$ covers beyond Galois cases, with explicit implementation for plane quartics at $p=3$.

Findings

Method for semistable reduction of degree-$p$ covers

Implementation for plane quartics at $p=3$ in SageMath

Use of non-archimedean analytic geometry and the different function

Abstract

Let $K$ be a local field of residue characteristic $p>0$. We explain how to compute the semistable reduction of $K$-curves $Y$ equipped with a degree-$p$ morphism from $Y$ to the projective line. This includes the reduction at $p$ of superelliptic curves of degree $p$, but our approach is not limited to Galois covers. We give particular attention to the reduction of plane quartics at $p=3$, which case is implemented in SageMath. We use the language of non-archimedean analytic geometry in the sense of Berkovich. A key tool is the different function of Cohen, Temkin, and Trushin.