Division by 2 on odd degree hyperelliptic curves and their jacobians

TL;DR

This paper provides explicit formulas for halving points on the Jacobian of odd degree hyperelliptic curves and proves the absence of certain torsion points on the curve over algebraically closed fields.

Contribution

It offers explicit Mumford representations for points that are halves of given Jacobian points and establishes torsion point restrictions for hyperelliptic curves of genus greater than one.

Findings

Explicit formulas for halving points on Jacobians

Algorithmic approach to compute square roots in Jacobians

Proof that certain torsion points do not exist on the curve

Abstract

Let $K$ be an algebraically closed field of characteristic different from 2, $g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficients in $K$ and without multiple roots, $C:y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over K, and $J$ the jacobian of $C$. We identify $C$ with the image of its canonical embedding into $J$ (the infinite point of $C$ goes to the identity element of $J$). It is well known that for each $\mathfrak{b} \in J(K)$ there are exactly $2^{2g}$ elements $\mathfrak{a} \in J(K)$ such that $2\mathfrak{a}=\mathfrak{b}$. M. Stoll constructed an algorithm that provides Mumford representations of all such $\mathfrak{a}$, in terms of the Mumford representation of $\mathfrak{b}$. The aim of this paper is to give explicit formulas for Mumford representations of all such $\mathfrak{a}$, when $\mathfrak{b}\in J(K)$ is given by $P=(a,b) \in C(K)\subset J(K)$ in terms of coordinates $a,b$. We also prove that if $g>1$ then $C(K)$ does not contain torsion points with order between $3$ and $2g$.