  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
$p_X(P, Q)=\delta_{P,Q}$ 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 $\Omega$ and a connection to the sigma functions for $X$.
