Computing the Stable Reduction of Hyperelliptic Curves in Residue Characteristic 2

TL;DR

This paper studies the stable reduction of hyperelliptic curves in residue characteristic 2, providing theoretical insights, refining an algorithm, and demonstrating explicit examples up to genus 30.

Contribution

It offers new theoretical properties and an improved algorithm for computing stable reduction of hyperelliptic curves in characteristic 2.

Findings

Proved general properties of stable reduction in residue characteristic 2.

Refined an existing algorithm for computing stable reduction.

Explicitly computed examples up to genus 30.

Abstract

Consider a hyperelliptic curve of genus $g$ over a field $K$ of characteristic zero. After extending $K$ we can view it as a marked curve with its $2g+2$ Weierstrass points. We prove some general properties of the stable reduction of this marked curve for a valuation of residue characteristic $2$ and refine an existing algorithm for its computation. We work out explicit examples up to genus $g=30$.