Division by 2 on odd-degree hyperelliptic curves and their Jacobians

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