Algebraic construction of the sigma function for general Weierstrass curves

Abstract

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 $X$.