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, yr+A1(x)yr−1+A2(x)yr−2+⋯+Ar−1(x)y+Ar(x)=0, where r is a positive integer, and each Aj 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 ϖr:X→P as a covering space. Let RX:=H0(X,OX(∗∞)) and RP:=H0(P,OP(∗∞)). Recently, we obtained the explicit description of the complementary module RXc of RP-module RX, which leads to explicit expressions of the holomorphic form except ∞, H0(P,AP(∗∞)) and the trace operator pX such that pX(P,Q)=δP,Q for ϖr(P)=ϖr(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.