Complex Multiplication Formulae for Hyperelliptic Curves of Genus Three
Complex Multiplication Formulae for Hyperelliptic Curves of Genus Three
复制标题
三格超椭圆曲线的复乘法公式
DOI：
10.3836/tjm/1270041822
复制
发表时间：
1998
影响因子：
0.6
通讯作者：
Y. Ônishi
中科院分区：
文献类型：
--
作者：
Y. Ônishi
所有的系数都属于Z[\zeta]。（爱森斯坦似乎已经知道了这些事实。）因此根$\{\wp(u)\}$除分子$0$外的乘积等于$\pm b$，而根$\{\wp(u)\}$的倒数乘积等于$b^{2}$。因此，我们在Z[\zeta]$的扩展整数环中分解了$b$或$b^{2}$。类似的事实已知有一个函数$\wp(u)$满足$\wp^{\素数}(u)^{2}=4\wp(u)^{3}-\wp(u)$。利用这些事实，我们深入研究了三次和四次高斯和（见[12]和[13]）。因此，我们似乎很自然地期望存在与（0.1）类似的公式，用于高属曲线。D. Grant对定义为$y^{2}=x^{5}+1/4$（[9]）的2属曲线发现了一个显著的公式。本文的目的是推广他的公式。设C是一条属曲线
and all the coefficients belong to $Z[\zeta]$ . (These facts seem to be already known to Eisenstein [6]). Therefore the product of the roots $\{\wp(u)\}$ except for $0$ of the numerator is equal to $\pm b$ , and the product of reciprocals of the roots $\{\wp(u)\}$ of the denominator is equal to $b^{2}$ . So we have factorization of $b$ or $b^{2}$ in an extended integer ring of $Z[\zeta]$ . Analogous fact is known for a function $\wp(u)$ satisfying $\wp^{\prime}(u)^{2}=4\wp(u)^{3}-\wp(u)$ . By using these facts essentially, the cubic and quartic Gauss sums were deeply investigated (see [12] and [13]). So it seems natural for us to expect the existence of formulae analogous to (0.1) for curves of higher genus. A remarkable formula was discovered by D. Grant for the curve of genus two defined by $y^{2}=x^{5}+1/4$ ([9]). The purpose of this paper is to generalize his formula. Let $C$ be a curve of genus