Integral differential forms for superelliptic curves
Sabrina Kunzweiler, Stefan Wewers
TL;DR
The paper develops a valuation-theoretic framework to compute the $\\mathcal{O}_K$-lattice $M$ of integral differential forms on a superelliptic curve $Y_K$ by constructing a regular model $X$ of $X_K = \mathbb{P}^1_K$ with a normal crossing divisor $D$ and passing to the normalization of $X$ in the function field of $Y_K$, which has only rational singularities. Central to the method are MacLane inductive valuations, residue-class discoids, and valuation trees that encode the vertical components and their intersections; these inputs drive the lattice computation $M = \{\\omega \\in M_K : v(\\omega) \\ge 0 \\forall v \\in V(Y)\\}$, where $V(Y)$ are the extensions of $V(X)$ to $F_Y$. The authors provide a concrete algorithm (with three main stages) to produce a regular model, compute a basis for $M_K$, extend valuations to $F_Y$, and intersect resulting lattices to obtain $M_Y$, with full implementations in Sage/MCLF and a dedicated module for superelliptic curves. The approach is demonstrated on explicit $p$-adic examples, yielding explicit lattice bases such as $\{5, x\}$ for certain genus-2 cases and $\{5^2, 5x, x^2-5, y\}$ for higher genus, illustrating the practical impact for arithmetic applications like BSD verifications and differential-form computations on nonregular models.
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.