Torsion points of small order on cyclic covers of $\mathbb{P}^1$. III
Boris M. Bekker, Yuri G. Zarhin
TL;DR
This work investigates which small-order torsion points can appear on the Jacobians of cyclic covers $\mathcal{C}_{f,d}$ of $\mathbb{P}^1$, where $y^d=f(x)$ with $f$ of degree $n$ and $(n,d)=1$. Building on prior results that restrict torsion orders to $m=d$, $m=n$, or $m\ge m_0$ with $m_0=d\ell_0$, the paper focuses on the edge case for $m_0$ when $n-m_0+\ell_0\in\{0,1\}$ and provides explicit, computable criteria for the existence of $K_0$-rational points of order $m_0$ via canonical normal forms and isomorphisms to standard models. It develops versal families of curves with $K_0$-rational torsion of order $m_0$, analyzes elliptic curves with a rational point of order $4$ and connects to Kubert’s universal family, and studies curves with two torsion packets, deriving a strict bound $d\le 5$ and detailed constraints on $(n,d)$ configurations. In the hyperelliptic case $d=2$, the authors show a two-packet structure forces a specific representation of $f$ and prove a finiteness result for parameter choices, highlighting the arithmetic rigidity of torsion in these cyclic covers. Overall, the paper provides explicit constructions, normal forms, and finiteness results that illuminate edge-case torsion phenomena on the Jacobians of cyclic covers of $\mathbb{P}^1$.
Abstract
Let $d>1$ be an integer and $K_0$ a perfect field such that $char(K_0)$ does not divide $d$. Let $n>d$ be an integer that is prime to $d$. Let $f(x)\in K_0[x]$ be a degree $n$ monic polynomial without repeated roots, and $\mathcal{C}_{f,d}$ a smooth projective model of the affine curve $y^d=f(x)$. Let $J(\mathcal{C}_{f,d})$ be the Jacobian of the $K_0$-curve $\mathcal{C}_{f,d} $. As usual, we identify $\mathcal{C}_{f,d}$ with its canonical image in $J(\mathcal{C}_{f,d})$ (such that the only ``infinite point'' of $\mathcal{C}_{f,d}$ goes to the zero of the group law on $J(\mathcal{C}_{f,d})$). We say that an integer $m>1$ is $(n,d)$-reachable over $K_0$ if there exists a polynomial $f(x)$ as above such that $\mathcal{C}_{f,d}(K_0)$ contains a torsion point of order $m$. Let us put $\ell_0:=[(n+d)/d], \ m_0:=\ell_0 d$. Earlier we proved that if $m$ is $(n,d)$-reachable, then either $m=d$ or $m = n$ or $m \ge m_0$ (in addition, both $d$ and $n$ are $(n,d)$-reachable over every $K_0$). We also proved that if $m_0$ is $(n,d)$-reachable over some $K_0$ then $n-m_0+\ell_0\ge 0$. In the present paper we discuss the $(n,d)$-reachability of $m_0$ when $n-m_0+\ell_0=0$ or $1$.