Torsion points of small order on cyclic covers of $\mathbb{P}^1$. III

TL;DR

This paper investigates the conditions under which certain torsion points of small order exist on the Jacobians of cyclic covers of the projective line, extending previous results to specific cases where $n-m_0+ ext{l}_0$ equals 0 or 1.

Contribution

It extends the characterization of torsion points on Jacobians of cyclic covers to cases where $n-m_0+ ext{l}_0$ is 0 or 1, providing new insights into their reachability.

Findings

Characterization of $(n,d)$-reachability for $m_0$ when $n-m_0+ ext{l}_0=0$ or $1$.

Conditions under which small order torsion points exist on Jacobians of cyclic covers.

Extension of previous reachability results to specific boundary cases.

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 $1$.