Halves of points of an odd degree hyperelliptic curve in its jacobian

TL;DR

This paper explores the structure of halves of points on odd degree hyperelliptic curves within their Jacobians, explicitly describing how 2-torsion points act on associated square root collections.

Contribution

It provides an explicit description of the action of 2-torsion points on the set of square root collections related to points on hyperelliptic curves.

Findings

Explicit bijection between halves of points and square root collections

Description of the action of Jacobian 2-torsion on these collections

Clarification of sign changes in square roots under this action

Abstract

Let $f(x)$ be a degree $(2g+1)$ monic polynomial with coefficients in an algebraically closed field $K$ with $char(K)\ne 2$ and without repeated roots. Let $\mathfrak{R}\subset K$ be the $(2g+1)$-element set of roots of $f(x)$. Let $\mathcal{C}: y^2=f(x)$ be an odd degree genus $g$ hyperelliptic curve over $K$. Let $J$ be the jacobian of $\mathcal{C}$ and $J[2]\subset J(K)$ the (sub)group of its points of order dividing $2$. We identify $\mathcal{C}$ with the image of its canonical embedding into $J$ (the infinite point of $\mathcal{C}$ goes to the identity element of $J$). Let $P=(a,b)\in \mathcal{C}(K)\subset J(K)$ and $M_{1/2,P}\subset J(K)$ the set of halves of $P$ in $J(K)$, which is $J[2]$-torsor. In a previous work we established an explicit bijection between $M_{1/2,P}$ and the set of collections of square roots $$\mathfrak{R}_{1/2,P}:=\{\mathfrak{r}: \mathfrak{R} \to K\mid \mathfrak{r}(\alpha)^2=a-\alpha \ \forall \alpha\in\mathfrak{R}; \ \prod_{\alpha\in\mathfrak{R}} \mathfrak{r}(\alpha)=-b\}.$$ The aim of this paper is to describe the induced action of $J[2]$ on $\mathfrak{R}_{1/2,P}$ (i.e., how signs of square roots $\mathfrak{r}(\alpha)=\sqrt{a-\alpha}$ should change).