Constructing Superelliptic Curves with non-trivial rational Torsion on their Jacobians

TL;DR

This paper presents a method to construct superelliptic curves with rational points of specified order on their Jacobians, using Hensel's Lemma, and demonstrates the approach with various examples.

Contribution

It introduces a novel construction technique for superelliptic curves with prescribed rational torsion points on their Jacobians.

Findings

Constructed superelliptic curves with rational N-division points

Method applicable for any integer N, producing curves of genus linear in N

Provided multiple explicit examples of such curves

Abstract

In this paper, we describe the construction of superelliptic curves with a rational point of prescribed order on their jacobians. The construction is based on Hensel's Lemma and produces for a given integer $N$ a superelliptic curve of genus linear in $N$ with a rational $N$-division point on the jacobian. The method is illustrated with multiple examples.