Division by 2 on odd degree hyperelliptic curves and their jacobians

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 $2g$.