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

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Zarhin,  Yuri G.
Max Planck Institute for Mathematics, Max Planck Society;

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arXiv:1807.07008.pdf
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Citation

Zarhin, Y. G. (2020). Halves of points of an odd degree hyperelliptic curve in its jacobian. In R. Donagi, & T. Shaska (Eds.), Integrable Systems and Algebraic Geometry: a celebration of Emma Previato's 65th birthday. Volume 2 (pp. 102-118). Cambridge: Cambridge University Press.


Cite as: https://hdl.handle.net/21.11116/0000-0006-BB4E-1
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).