Torsion points of small order on cyclic covers of $\mathbb P^1$
Boris M. Bekker, Yuri G. Zarhin
TL;DR
The paper advances the understanding of torsion on cyclic covers $C_{f,d}: y^d=f(x)$ by proving that any torsion point on the smooth projective model $\mathcal{C}_{f,d}$ has order either $d$ or at least $n$, thereby generalizing known results from the hyperelliptic $d=2$ case. It gives a precise algebraic characterization: order-$d$ points come from the $n$ ramification points, while order $n$ points arise exactly when $f(x)=(x-a)^n+v(x)^d$ with $\deg v\le (n-1)/d$, and it shows that, under several natural hypotheses, there is at most one $n$-packet (hence at most $d$ order-$n$ points). The work also investigates the existence and structure of order-$n$ points, using linear fractional transformations and cross-ratio arguments to rule out multiple $n$-packets in many cases, and it supplies explicit constructions in the Picard curve scenario ($n=4$, $d=3$) yielding two quartics with order-$4$ points at $(0,c)$ and $(-1,c')$. Overall, the results clarify the fine arithmetic of torsion on a broad family of superelliptic curves and extend prior bounds to a wide range of $(n,d)$ pairs.
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 $n$.