Division by 2 on hyperelliptic curves and jacobians

TL;DR

This paper explicitly describes the points on the Jacobian of hyperelliptic curves that are halved from points on the curve, providing concrete Mumford representations and discussing rationality issues.

Contribution

It provides explicit formulas for the halving of points on hyperelliptic Jacobians and analyzes rationality questions for these points.

Findings

Explicit Mumford representations for 1/2 P points

Description of rationality conditions for halved points

Generalization to hyperelliptic curves of arbitrary genus

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 zero point of $J$). For each point $P=(a,b)\in C(K)$ there are $2^{2g}$ points $\frac{1}{2}P \in J(K)$. We describe explicitly the Mumford represesentations of all $\frac{1}{2}P$. The rationality questions for $\frac{1}{2}P$ are also discussed.