Algebraic Construction of the Sigma Function for General Weierstrass Curves (vol 10, 3010, 2022)

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 omega(r) : X -> P as a covering space. Let R-X := H-0 (X, O-X (*infinity)) and R-P := H-0 (P, O-P (*infinity)). 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 infinity, H-0(P, A(P) (*infinity)) and the trace operator p(X) such that p(X) (P, Q) = delta(P,Q) for omega(r) (P) = omega(r) (Q) for P, Q is an element of X \ {infinity}. In terms of these, we express the fundamental two-form of the second kind Omega and its connection to the sigma function for X.