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}}rightarrow text {Spec}, {mathcal {O}}_K its minimal regular model with relative dualising 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}{2y} wedge ldots wedge x^{g-1}frac{dx}{2y} is a generator for det H^0({mathcal {X}},omega _{{mathcal {X}}/{mathcal {O}}_K}).