Torsion points of small order on cyclic covers of $\mathbb P^1$

TL;DR

This paper generalizes previous results on torsion points of small order on hyperelliptic curves to cyclic covers of the projective line, establishing bounds on the order of torsion points on these curves.

Contribution

It extends known bounds for torsion points from hyperelliptic curves to arbitrary cyclic covers of the projective line, identifying when torsion points of certain orders can occur.

Findings

Points of order m on the curve are either m=d or m≥n.

Curves with points of order n are explicitly characterized.

No torsion points of order between 2 and d-1 exist on these curves.

Abstract

Let $d\geq 2$ be a positive integer, $K$ an algebraically closed field of characteristic not dividing $d$, $n\geq d+1$ a positive integer that is prime to $d$, $f(x)\in K[x]$ a degree $n$ monic polynomial without multiple roots, $C_{f,d}: y^d=f(x)$ the corresponding smooth plane affine curve over $K$, $\mathcal{C}_{f,d}$ a smooth projective model of $C_{f,d}$ and $J(\mathcal{C}_{f,d})$ the Jacobian of $\mathcal{C}_{f,d} $. We identify $\mathcal{C}_{f,d}$ with the image of its canonical embedding into $J(\mathcal{C}_{f,d})$ (such that the infinite point of $\mathcal{C}_{f,d}$ goes to the zero of the group law on $J(\mathcal{C}_{f,d})$). Earlier the second named author proved that if $d=2$ and $n=2g+1 \ge 5$ then the genus $g$ hyperelliptic curve $\mathcal{C}_{f,2}$ contains no points of orders lying between $3$ and $n-1=2g$. In the present paper we generalize this result to the case of arbitrary $d$. Namely, we prove that if $P$ is a point of order $m>1$ on $\mathcal{C}_{f,d}$, then either $m=d$ or $m\geq n$. We also describe all curves $\mathcal{C}_{f,d}$ having a point of order $n$.