Differential Forms on Hyperelliptic Curves with Semistable Reduction

Abstract

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 $\mathcal{X}\to \text{Spec}{\mathcal{O}}_K$ its minimal regular model with relative dualizing sheaf ${\omega }_{\mathcal{X}/{\mathcal{O}}_K}$. We show how to directly read off a basis for $H^0(\mathcal{X},{\omega }_{\mathcal{X}/{\mathcal{O}}_K})$ from the cluster picture of the roots of $f$. Furthermore we give a formula for the valuation of $\lambda$ such that $\lambda \cdot \frac{dx}{y}\wedge \cdots \wedge x^{g-1}\frac{dx}{y}$ is a generator for $\det H^0(\mathcal{X},{\omega }_{\mathcal{X}/{\mathcal{O}}_K})$.