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

Let $K$ be an algebraically closed field of characteristic differentfrom $2$, $g$ 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 ofgenus $g$ over $K$, and $J$ its Jacobian. We identify $\mathcal{C}$ withthe image of its canonical embedding in $J$ (the infinite point of$\mathcal{C}$ goes to the identity element of $J$). It is well known thatfor every $\mathfrak{b} \in J(K)$ there are exactly $2^{2g}$ elements$\mathfrak{a}\in J(K)$ such that $2\mathfrak{a}=\mathfrak{b}$. Stollconstructed an algorithm that provides the Mumford representationsof all such $\mathfrak{a}$ in terms of the Mumford representation of$\mathfrak{b}$. The aim of this paper is to give explicit formulaefor the Mumford representations of all such $\mathfrak{a}$ in terms ofthe 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 ordersbetween $3$ and $2g$.

hyperelliptic curves, Weierstrass points, Jacobians, torsion points