# Differential Forms on Hyperelliptic Curves with Semistable Reduction

## TL;DR

This paper provides a method to determine a basis for the space of regular differentials on the minimal regular model of a hyperelliptic curve with semistable reduction, using the cluster picture of roots.

## Contribution

It introduces a direct way to read off a basis for differentials from the cluster picture and gives a valuation formula for a key generator.

## Key findings

- Basis for differentials can be obtained from cluster picture
- Valuation formula for the generator of the determinant of cohomology
- Applicable to 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 $\mathcal{X} \rightarrow \textrm{Spec}\, \mathcal{O}_K$ 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 $\lambda \cdot \frac{dx}{y} \land \dots \land x^{g-1}\frac{dx}{y}$ is a generator for $\det H^0(\mathcal{X},\omega_{\mathcal{X}/\mathcal{O}_K})$.