The Classification of the Stable Marked Reduction of Genus 2 Curves in Residue Characteristic 2

TL;DR

This paper classifies the stable reduction structures of genus 2 hyperelliptic curves over fields with residue characteristic 2, providing a computational implementation and analysis for curves over the rationals with small conductors.

Contribution

It offers a detailed classification of stable reductions in residue characteristic 2 and implements this classification in a computer algebra system for practical computations.

Findings

Classification of stable reduction structures in characteristic 2

Implementation of the classification in a computer algebra system

Computed reductions for curves over $\\mathbb{Q}$ with conductor up to $2^{20}$

Abstract

Consider a hyperelliptic curve of genus $2$ over a field $K$ of characteristic zero. After extending $K$ we can view it as a marked curve with its $6$ Weierstrass points. We classify the structure of the potentially stable reduction of such curves for a valuation of residue characteristic $2$. We implement this classification into a computer algebra system and compute it for a list of curves defined over $\mathbb{Q}$ with conductor at most $2^{20}$.