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Minimal regular models of quadratic twists of genus two curves

Volume 183 / 2018

Mohammad Sadek Acta Arithmetica 183 (2018), 317-337 MSC: Primary 11G20; Secondary 14G40. DOI: 10.4064/aa161222-15-12 Published online: 8 June 2018

Abstract

Let $K$ be a complete discrete valuation field with ring of integers $R$ and residue field $k$ of characteristic $p \gt 2$. We assume 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  $C^{\chi}$. This is accomplished by obtaining the stable model $\operatorname{\mathcal C}^{\chi}$ of $C^{\chi}$ from the stable model $\operatorname{\mathcal C}$ 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 $\operatorname{\mathcal C}^{\chi}$.

Authors

  • Mohammad SadekMathematics and Actuarial Science Department
    American University in Cairo
    AUC Avenue
    New Cairo, Egypt
    e-mail

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