Algebraic Construction of the Sigma Function for General Weierstrass Curves

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 + ⋯ + 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 ϖ r : X → P as a covering space. Let R X : = H 0 ( X , O X ( ∗ ∞ ) ) and R P : = H 0 ( P , O P ( ∗ ∞ ) ) . Recently, we obtained the explicit description of the complementary module R X c of R P -module R X , which leads to explicit expressions of the holomorphic form except ∞ , H 0 ( P , A P ( ∗ ∞ ) ) and the trace operator p X such that p X ( P , Q ) = δ P , Q for ϖ r ( P ) = ϖ r ( Q ) for P , Q ∈ X \ { ∞ } . In terms of these, we express the fundamental two-form of the second kind Ω and its connection to the sigma function for X .