Minimal regular models of quadratic twists of genus two curves

TL;DR

This paper determines the minimal regular models of quadratic twists of genus two curves over local fields by analyzing invariants and singularities, extending the understanding of their arithmetic and geometric properties.

Contribution

It provides a method to explicitly compute the minimal regular model of quadratic twists of genus two curves from the original curve's stable model and invariants.

Findings

Explicit description of minimal regular models for quadratic twists

Analysis of Igusa and affine invariants for curve comparison

Calculation of singularity degrees of twisted curve models

Abstract

Let $K$ be a complete discrete valuation field with ring of integers $R$ and residue field $k$ of characteristic $p>2$. We assume moreover that $k$ is algebraically closed. Let $C$ be a smooth projective geometrically connected curve of genus $2$. If $K(\sqrt{D})/K$ is a quadratic field extension of $K$ with associated character $\chi$, then $C^{\chi}$ will denote the quadratic twist of $C$ by $\chi$. Given the minimal regular model $\mathcal X$ of $C$ over $R$, we determine the minimal regular model of the quadratic twist $C^{\chi}$. This is accomplished by obtaining the stable model $\mathcal{C}^{\chi}$ of $C^{\chi}$ from the stable model $\mathcal{C}$ of $C$ via analyzing the Igusa and affine invariants of the curves $C$ and $C^{\chi}$, and calculating the degrees of singularity of the singular points of $\mathcal{C}^{\chi}$.