Torsion points on Fermat quotients of the form $y^n = x^d + 1$

TL;DR

This paper classifies all geometric torsion points on certain Fermat quotient curves and extends previous results to more general superelliptic curves, providing a comprehensive understanding of torsion points in these settings.

Contribution

It provides a complete classification of geometric torsion points on Fermat quotients and extends existing classifications to generic superelliptic curves.

Findings

Classified all geometric torsion points on Fermat quotients $y^n = x^d + 1$.

Extended classification of torsion points to generic superelliptic curves.

Generalized previous results from hyperelliptic to superelliptic cases.

Abstract

We classify all geometric torsion points on the Fermat quotients $y^n = x^d + 1$ where $n, d \ge 2$ are coprime. In addition, we classify all geometric torsion points on the generic superelliptic curve $y^n = (x - a_1) \cdots (x - a_d)$, extending a result of Poonen and Stoll, who considered the hyperelliptic $n = 2$ case.