Division by $1-\zeta$ on superelliptic curves and jacobians

TL;DR

This paper generalizes Zarhin's formulas for dividing points on hyperelliptic curves to superelliptic curves by a factor of $1 - $, providing explicit functions for divisors despite the absence of Mumford's representation.

Contribution

It extends division formulas from hyperelliptic to superelliptic curves by deriving explicit functions for divisors without relying on Mumford's representation.

Findings

Formulas for dividing points by $1 - $ on superelliptic curves.

Explicit functions describing divisors on superelliptic curves.

Generalization of Zarhin's hyperelliptic results to superelliptic case.

Abstract

In 2016, Yuri Zarhin gave formulas for "dividing a point on a hyperelliptic curve by 2." Given a point $P$ on a hyperelliptic curve $\mathcal{C}$, Zarhin gives the Mumford's representation of every degree $g$ divisor $D$ such that $2(D - g \infty) \sim P - \infty$. The aim of this paper is to generalize Zarhin's result to the superelliptic situation; instead of dividing by 2, we divide by $1 - \zeta$. Even though there is no Mumford's representation for superelliptic curves, we give a formula for functions which cut out $D$.