Algebraic Construction of the Sigma Function for General Weierstrass Curves

The Weierstrass curve X is a smooth algebraic determined by the canonical form, yr+A1(x)yr−1+A2(x)yr−2+⋯+Ar−1(x)y+Ar(x)=0, where r positive integer, and each Aj polynomial in x with certain degree. It known that every compact Riemann surface has X, which birational to surface. form provides projection ϖr:X→P as covering space. Let RX:=H0(X,OX(∗∞)) RP:=H0(P,OP(∗∞)). Recently, we obtained explicit description of complementary module RXc RP-module RX, leads expressions holomorphic except ∞, H0(P,AP(∗∞)) trace operator pX such pX(P,Q)=δP,Q for ϖr(P)=ϖr(Q) P,Q∈X\{∞}. In terms these, express fundamental two-form second kind Ω its connection sigma function X.