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

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 $1$.

24 pages