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

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 $2^{20}$.

28 pages, 16 figures, to appear in Mathematics of Computation