论文标题
伸缩曲线的Sigma功能的功率系列扩展的Hurwitz完整性
Hurwitz integrality of the power series expansion of the sigma function for a telescopic curve
论文作者
论文摘要
伸缩曲线是$ M-1 $方程在尺寸$ m $的仿射空间中定义的一定代数曲线，这可以是一种特殊情况和$（n，s）$曲线。 Sigma函数$σ（u）$与属于$ g $的望远镜曲线相关的是$ \ mathbb {c}^g $上的全态函数。对于一个$ \ mathbb {c} $和变量的$ r $ $ u = {}^t（u_1，\ dots，u_g）$ \rangle\rangle=\left\{\sum_{i_1,\dots,i_g\ge0}κ_{i_1,\dots,i_g}\frac{u_1^{i_1}\cdots u_g^{i_g}}{i_1!\cdots i_g！} \; \中间| \;κ_{i_1，\ dots，i_g} \ in r \ right \}。 $ f（u）$据说超过$ r $是Hurwitz的积分。在本文中，我们表明，与望远镜曲线相关的Sigma函数$σ（U）$在曲线定义方程的系数和$ \ frac {1} {2} {2} $上产生的环上是Hurwitz的积分。此外，我们表明，$σ（u）^2 $是由曲线定义方程式在$ \ mathbb {z} $上产生的环上不可或缺的。我们的结果是Y.ôisnishi的结果以$（n，s）$曲线为望远镜曲线的结果。
A telescopic curve is a certain algebraic curve defined by $m-1$ equations in the affine space of dimension $m$, which can be a hyperelliptic curve and an $(n,s)$ curve as a special case. The sigma function $σ(u)$ associated with the telescopic curve of genus $g$ is a holomorphic function on $\mathbb{C}^g$. For a subring $R$ of $\mathbb{C}$ and variables $u={}^t(u_1,\dots, u_g)$, let \[R\langle\langle u \rangle\rangle=\left\{\sum_{i_1,\dots,i_g\ge0}κ_{i_1,\dots,i_g}\frac{u_1^{i_1}\cdots u_g^{i_g}}{i_1!\cdots i_g!}\;\middle|\;κ_{i_1,\dots,i_g}\in R\right\}.\] If the power series expansion of a holomorphic function $f(u)$ on $\mathbb{C}^g$ around the origin belongs to $R\langle\langle u \rangle\rangle$, then $f(u)$ is said to be Hurwitz integral over $R$. In this paper, we show that the sigma function $σ(u)$ associated with the telescopic curve is Hurwitz integral over the ring generated by the coefficients of the defining equations of the curve and $\frac{1}{2}$ over $\mathbb{Z}$. Further, we show that $σ(u)^2$ is Hurwitz integral over the ring generated by the coefficients of the defining equations of the curve over $\mathbb{Z}$. Our results are a generalization of the results of Y. Ônishi for $(n,s)$ curves to telescopic curves.
