Minimal Weierstrass models and regular models of hyperelliptic curves

Abstract

Let $C$ be a hyperelliptic curve of genus $g\ge 2$ over a discrete valuation field $K$ with perfect residue field. We study the minimal Weierstrass models of $C$. When there is more than one such model, we find interesting properties on the minimal regular model and the canonical model of $C$. For curves of genus $2$, we characterize the existence of the stable reduction in terms of the minimal Weierstrass models. When there is more than one such model, we can compute the Euler factor of $\mathrm{Jac}(C)$ and a volume form of the Néron model of $\mathrm{Jac}(C)$, using two specific minimal Weierstrass models.

arXiv commentAdd §1.7 on stably minimal Weierstrass models; more details for the extended Tate's algorithm §2.2; Add §3.5, §3.6 on Tamagawa number and conductor of Jac(C). Minor changes in the rest of the paper