4 - Halves of Points of an Odd Degree Hyperelliptic Curve in its Jacobian

Summary

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