Torsion Points of order 2g+1 on odd degree hyperelliptic curves of genus g

TL;DR

This paper investigates torsion points of order 2g+1 on odd degree hyperelliptic curves of genus g, revealing bounds on their number over various fields and generalizing classical elliptic curve results.

Contribution

It extends known results about torsion points of order 3 on elliptic curves to higher genus hyperelliptic curves, providing bounds and existence results for points of order 2g+1.

Findings

At most two points of order 2g+1 over algebraically closed fields when p=2g+1 is prime

At most two real points of order 2g+1 if g is odd and f(x) has real coefficients

At most two rational points of order 2g+1 for g<52 with rational coefficients

Abstract

Let $K$ be an algebraically closed field of characteristic different from $2$, $g$ a positive integer, $f(x)\in K[x]$ a degree $2g+1$ monic polynomial without repeated roots, $C_f: y^2=f(x)$ the corresponding genus g hyperelliptic curve over $K$, and $J$ the jacobian of $C_f$. We identify $C_f$ with the image of its canonical embedding into $J$ (the infinite point of $C_f$ goes to the zero of group law on $J$). It is known (arXiv:1809.03061 [math.AG]) that if $g>1$ then $C_f(K)$ does not contain torsion points, whose order lies between $3$ and $2g$. In this paper we study torsion points of order $2g+1$ on $C_f(K)$. Despite the striking difference between the cases of $g=1$ and $g> 1$, some of our results may be viewed as a generalization of well-known results about points of order $3$ on elliptic curves. E.g., if $p=2g+1$ is a prime that coincides with $char(K)$, then every odd degree genus $g$ hyperelliptic curve contains, at most, two points of order $p$. If $g$ is odd and $f(x)$ has real coefficients, then there are, at most, two real points of order $2g+1$ on $C_f$. If $f(x)$ has rational coefficients and $g<52$, then there are, at most, two rational points of order $2g+1$ on $C_f$. (However, there are exist genus $52$ hyperelliptic curves over the field of rational numbers that have, at least, four rational points of order 105.)