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 $X$.
Categories: math.AG nlin.SI