Computing the space of differential forms of a plane curve and its Cartier-Manin matrix

TL;DR

This paper presents an algorithm to explicitly compute a basis for the space of regular differential forms on plane algebraic curves and applies it to determine the Cartier-Manin matrix, with implementation in Magma.

Contribution

It introduces a feasible algorithm for computing differential form bases on plane curves and applies it to determine the Cartier-Manin matrix, enhancing computational algebraic geometry methods.

Findings

Algorithm successfully computes bases for differential forms.

Implementation verified on concrete examples in Magma.

Provides explicit Cartier-Manin matrices for studied curves.

Abstract

In this paper, we propose a feasible algorithm to give an explicit basis of the space of regular differential forms on the nonsingular projective model of any given plane algebraic curve. The algorithm is demonstrated for concrete examples, with our implementation over the computer algebra system Magma. As an application, we also describe the Cartier-Manin matrix of the nonsingular projective curve with respect to the basis computed by the algorithm.