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

DOI10.1017/9781108773355.005zbMATH Open1473.14056arXiv1807.07008OpenAlexW4243663434MaRDI QIDQ5164746

Yuri G. Zarhin

Publication date: 12 November 2021

Abstract: Let $f(x)$ be a degree $(2g+1)$ monic polynomial with coefficients in an algebraically closed field $K$ with $char(K)e2$ and without repeated roots. Let $mathfrakRsubsetK$ be the $(2g+1)$-element set of roots of $f(x)$. Let $mathcalC:y^2=f(x)$ be an odd degree genus $g$ hyperelliptic curve over $K$. Let $J$ be the jacobian of $mathcalC$ and $J[2]subsetJ(K)$ the (sub)group of its points of order dividing $2$. We identify $mathcalC$ with the image of its canonical embedding into $J$ (the infinite point of $mathcalC$ goes to the identity element of $J$). Let $P=(a,b)inmathcalC(K)subsetJ(K)$ and $M_{1/2,P}subsetJ(K)$ 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 ${\displaystyle }$mathfrak{R}_{1/2,P}:={mathfrak{r}: mathfrak{R} o Kmid mathfrak{r}(alpha)^2=a-alpha forall alphainmathfrak{R}; prod_{alphainmathfrak{R}} mathfrak{r}(alpha)=-b}.${\displaystyle }$ 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 $mathfrakr(alpha)=sqrta-alpha$ should change).

Full work available at URL: https://arxiv.org/abs/1807.07008

Mathematics Subject Classification ID

Special algebraic curves and curves of low genus (14H45) Jacobians, Prym varieties (14H40) Theta functions and abelian varieties (14K25)

Recommendations

• Unnamed Item 👍 👎
• Integral points on hyperelliptic curves 👍 👎
• On the Jacobian of a family of hyperelliptic curves 👍 👎
• On Curves with Split Jacobians 👍 👎
• Points entiers sur les courbes hyperelliptiques 👍 👎
• Equations for the Jacobian of a hyperelliptic curve 👍 👎
• Division by on odd-degree hyperelliptic curves and their Jacobians 👍 👎
• Rational and Integral Points on Quadratic Twists of a Given Hyperelliptic Curve 👍 👎
• Rational points on Jacobians of hyperelliptic curves. 👍 👎
• On the number of points of given order on odd-degree hyperelliptic curves 👍 👎

This page was built for publication: Halves of Points of an Odd Degree Hyperelliptic Curve in its Jacobian