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arXiv 1902.07784math.AGmath.NT
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半稳定约化的超椭圆曲线上的Differential Forms

Differential Forms on Hyperelliptic Curves with Semistable Reduction

Sabrina Kunzweiler

发表信息：
Research in Number Theory 6, 25 (2020): 1-17
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摘要：

设$C$是局部域$K$上的一条超椭圆曲线，该局部域具有奇剩余特征，由仿射Weierstrass方程$y^2=f(x)$定义.我们假设$C$具有半稳定约化，用$\mathcal{X} \rightarrow \textrm{Spec}\, \mathcal{O}_K$表示它的具有相对对偶层$ω_{\mathcal{X}/ \mathcal{O}_K}$的极小正则模型.我们展示了如何从$f$的根的聚类图中直接读出$H^0(\mathcal{X},ω_{\mathcal{X}/\mathcal{O}_K})$的基。此外，我们还给出了$λ$的赋值公式，使得$λ\cdot \frac{dx}{y} \land \dots \land x^{g-1}\frac{dx}{y}$是$\det H^0(\mathcal{X},ω_{\mathcal{X}/\mathcal{O}_K})$的生成元.

英文摘要：

Let $C$ be a hyperelliptic curve over a local field $K$ with odd residue characteristic, defined by some affine Weierstrass equation $y^2=f(x)$. We assume that $C$ has semistable reduction and denote by $\mathcal{X} \rightarrow \textrm{Spec}\, \mathcal{O}_K$ its minimal regular model with relative dualizing sheaf $ω_{\mathcal{X}/ \mathcal{O}_K}$. We show how to directly read off a basis for $H^0(\mathcal{X},ω_{\mathcal{X}/\mathcal{O}_K})$ from the cluster picture of the roots of $f$. Furthermore we give a formula for the valuation of $λ$ such that $λ\cdot \frac{dx}{y} \land \dots \land x^{g-1}\frac{dx}{y}$ is a generator for $\det H^0(\mathcal{X},ω_{\mathcal{X}/\mathcal{O}_K})$.

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