Integral differential forms for superelliptic curves

Abstract

Given a superelliptic curve $Y_K : y^n = f(x)$ over a local field $K$, we describe the theoretical background and an implementation of a new algorithm for computing the $\mathcal{O}_K$-lattice of integral differential forms on $Y_K$. We build on the results of Obus and the second author, which describe arbitrary regular models of the projective line using only valuations. One novelty of our approach is that we construct an $\mathcal{O}_K$-model of $Y_K$ with only rational singularities, but which may not be regular.