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arXiv 1809.03061math.AGmath.NT
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奇次超椭圆曲线的2整除及其Jacobian

Division by 2 on odd degree hyperelliptic curves and their jacobians

Yuri G. Zarhin

备注：
18 pages. The paper overlaps with arXiv:1606.05252 and arXiv:1807.07008
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摘要：

设$K$是特征不等于2的代数闭域，$g$是正整数，$f(x)$是系数在$K$中且无重根的$(2g+1)$次多项式，$C:y^2=f(x)$是K上相应亏格的$g$超椭圆曲线，$J$是$C$的雅可比数.我们识别$C$与其规范嵌入到$J$的图像（$C$的无穷远点到$J$的单位元）。众所周知，对于每个$\mathfrak{b} \in J(K)$，恰好存在$2^{2g}$元素$\mathfrak{a} \in J(K)$，使得$2\mathfrak{a}=\mathfrak{b}$。M. Stoll构造了一个算法，该算法根据$\mathfrak{b}$的芒福德表示，提供了所有此类$\mathfrak{a}$的芒福德表示。当$\mathfrak{b}\in J(K)$由$P=(a,b) \in C(K)\subset J(K)$以坐标$a,b$给出时，本文的目的是给出所有这类$\mathfrak{a}$的Mumford表示的显式公式.我们还证明，如果$g>1$，那么$C(K)$不包含扭转点之间的$3$和$2g$的顺序。

英文摘要：

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 identity element of $J$). It is well known that for each $\mathfrak{b} \in J(K)$ there are exactly $2^{2g}$ elements $\mathfrak{a} \in J(K)$ such that $2\mathfrak{a}=\mathfrak{b}$. M. Stoll constructed an algorithm that provides Mumford representations of all such $\mathfrak{a}$, in terms of the Mumford representation of $\mathfrak{b}$. The aim of this paper is to give explicit formulas for Mumford representations of all such $\mathfrak{a}$, when $\mathfrak{b}\in J(K)$ is given by $P=(a,b) \in C(K)\subset J(K)$ in terms of coordinates $a,b$. We also prove that if $g>1$ then $C(K)$ does not contain torsion points with order between $3$ and $2g$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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