Differential forms on hyperelliptic curves with semistable reduction

Abstract

AbstractLet C be a hyperelliptic curve over a local field K with odd residue characteristic, defined by some affine Weierstraß equation $$y^2=f(x)$$y2=f(x). We assume that C has semistable reduction and denote by $${\mathcal {X}}\rightarrow \text {Spec}\, {\mathcal {O}}_K$$X→SpecOK its minimal regular model with relative dualising sheaf $$\omega _{{\mathcal {X}}/ {\mathcal {O}}_K}$$ωX/OK. We show how to directly read off a basis for $$H^0({\mathcal {X}},\omega _{{\mathcal {X}}/{\mathcal {O}}_K})$$H0(X,ωX/OK) from the cluster picture of the roots of f. Furthermore we give a formula for the valuation of $$\lambda $$λ such that $$\lambda \cdot \frac{dx}{2y} \wedge \ldots \wedge x^{g-1}\frac{dx}{2y}$$λ·dx2y∧…∧xg-1dx2y is a generator for $$\det H^0({\mathcal {X}},\omega _{{\mathcal {X}}/{\mathcal {O}}_K})$$detH0(X,ωX/OK).