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

TL;DR

The paper investigates which torsion orders can occur on cyclic d-fold covers of the projective line given by y^d=f(x). It develops a framework tying torsion orders to pole orders of rational functions with a single pole at the infinite point O, yielding congruence restrictions modulo d and divisibility by d for orders in the range between n and 2n. It provides constructive criteria for when specific orders m (notably m = d·[(n+d)/d], n+d, and n+ed) are reachable by selecting suitable polynomials f(x), including a detailed polynomial-algebra apparatus based on binomial truncations V_{r,E}(x). The results include precise reachability conditions in characteristic zero and, in the hyperelliptic case (d=2), a complete range for m. The methods combine divisor calculus on C_{f,d}, properties of rational functions with poles at O, and explicit polynomial constructions to realize torsion points on the Jacobians of these curves.

Abstract

Let $d\geq 2$ be an integer, $K_0$ a perfect field such that $char(K_0)$ does not divide $d$, $n > d$ an integer prime to $d$, $f(x)\in K_0[x]$ a degree $n$ monic polynomial without repeated roots, and $C_{f,d}$ a smooth projective model of the affine curve $y^d=f(x)$. Let $J(C_{f,d})$ be the Jacobian of the $K_0$-curve $C_{f,d} $. We identify $C_{f,d}$ with its canonical image in $J(C_{f,d})$ (such that the infinite point of $C_{f,d}$ goes to the zero of the group law on $J(C_{f,d})$). 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 $C_{f,d}(K_0)$ contains a torsion point of order $m$. Earlier we proved that if $m$ is $(n,d)$-reachable, then either $m=d$ or $m \geq n$ (in addition, both $d$ and $n$ are $(n,d)$-reachable). In the present paper we prove the following. If $n<m<2n$ and if $m$ is $(n,d)$-reachable over $K_0$, then either $d|m$ or $m \equiv n \bmod d$. If either $char(K_0)=0$ or $K_0$ in infinite and $char(K_0)>n$, then $d\cdot [(n+d)/d]$ is $(n,d)$-reachable if and only if $n-(d-1)\cdot [(n+d)/d]\ge 0$. If $char(K_0)=0$, then $n+d$ is $(n,d)$-reachable if and only if $d^2-2d<n$. If $d=2$ (the hyperelliptic case) and $char(K_0)=0$, then $m$ is $(n,d)$-reachable if $n+1 \le m \le 2n+1$. (The case when $n \le m \le 3(n-1)/2$ was done earlier by E.V. Flynn.)

Theorems & Definitions (35)

• Theorem 1
• Remark 1
• Corollary 1
• proof
• Definition 1
• Remark 2
• Theorem 2: Main Theorem
• Remark 3
• Corollary 2
• proof