Reduction of Hyperelliptic Curves in Residue Characteristic 2

TL;DR

This paper introduces an explicit algorithm for computing the stable reduction of hyperelliptic curves of any genus over fields with residue characteristic 2, with detailed cases for genus up to 2.

Contribution

It provides a new explicit algorithm for stable reduction of hyperelliptic curves in residue characteristic 2, including simplified conditions for low genus cases.

Findings

Algorithm successfully computes stable reduction in characteristic 2

Simplified conditions for genus 1 and 2 cases

Enhances understanding of hyperelliptic curve reductions

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 present an explicit algorithm to compute the stable reduction of this marked curve for a valuation of residue characteristic $2$ over a finite extension of $K$. In the cases $g\le2$ we work out relatively simple conditions for the structure of this reduction.