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

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 $n$.