论文标题
一般Weierstrass曲线的Sigma函数的代数构建
Algebraic construction of the sigma function for general Weierstrass curves
论文作者
论文摘要
WeierStrass曲线$ x $是由Weierstrass规范形式确定的平滑代数曲线，$ y^r + a_ + a_ {1}（x）y^{r-1} + a_ {2}（x）y^{y^{r-2} + \ cdots + cdots + cdots + cdots + cd + a _ {r-1}并且每个$ a_j $都是$ x $的多项式，并具有一定程度的程度。众所周知，每个紧凑的Riemann表面都有一个Weierstrass曲线$ x $，对表面具有含糊。表单提供投影$ \ varpi_r：x \ to {\ mathbb {p}} $作为覆盖空间。令$ r_x：= {\ Mathbb {h}}^0（x，x，{\ Mathcal {o}} _ x（*\ infty））$和$ r _ {\ Mathbb {p {p {p}}：= = {\ MathBBB { {\ Mathcal {O}} _ {\ Mathbb {p}}}（*\ infty））$。 Recently we have the explicit description of the complementary module $R_X^{\mathfrak{c}}$ of $R_{\mathbb{P}}$-module $R_X$, which leads the explicit expressions of the holomorphic one form except $\infty$, $ {\ MATHBB {h}}^0（{{\ Mathbb {p}}，{\ Mathcal {a}} _ {\ Mathbb {p}}}（*\ infty）） $ \ varpi_r（p）= \ varpi_r（q）$ for $ p，q \ in x \ setMinus \ {\ infty \} $。就它们而言，我们表达了第二种$ω$的基本2形，以及与$ x $的Sigma函数的连接。
The Weierstrass curve $X$ is a smooth algebraic curve determined by the Weierstrass canonical form, $y^r + A_{1}(x) y^{r-1} + A_{2}(x) y^{r-2} +\cdots + A_{r-1}(x) y + A_{r}(x)=0$, where $r$ is a positive integer, and each $A_j$ is a polynomial in $x$ with a certain degree. It is known that every compact Riemann surface has a Weierstrass curve $X$ which is birational to the surface. The form provides the projection $\varpi_r : X \to {\mathbb{P}}$ as a covering space. Let $R_X := {\mathbb{H}}^0(X, {\mathcal{O}}_X(*\infty))$ and $R_{\mathbb{P}} := {\mathbb{H}}^0({\mathbb{P}}, {\mathcal{O}}_{\mathbb{P}}(*\infty))$. Recently we have the explicit description of the complementary module $R_X^{\mathfrak{c}}$ of $R_{\mathbb{P}}$-module $R_X$, which leads the explicit expressions of the holomorphic one form except $\infty$, ${\mathbb{H}}^0({\mathbb{P}}, {\mathcal{A}}_{\mathbb{P}}(*\infty))$ and the trace operator $p_X$ such that $p_X(P, Q)=δ_{P,Q}$ for $\varpi_r(P)=\varpi_r(Q)$ for $P, Q \in X\setminus\{\infty\}$. In terms of them, we express the fundamental 2-form of the second kind $Ω$ and a connection to the sigma functions for $X$.
