Algebraic Construction of the Sigma Function for General Weierstrass Curves - Jiryo Komeda; Shigeki Matsutani; Emma Previato (Mathematics, 2022)
The Weierstrass curve X is a smooth algebraic curve determined by the Weierstrass canonical form, <semantics>yr+A1(x)yr−1+A2(x)yr−2+⋯+Ar−1(x)y+Ar(x)=0</semantics>, where r is a positive integer, and each <semantics>Aj</semantics> 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 <semantics>ϖr:X→P</semantics> as a covering space. Let <semantics>RX:=H0(X,OX(∗∞))</semantics> and <semantics>RP:=H0(P,OP(∗∞))</semantics>. Recently, we obtained the explicit description of the complementary module <semantics><msubsup>RXc</msubsup></semantics> of <semantics>RP</semantics>-module <semantics>RX</semantics>, which leads to explicit expressions of the holomorphic form except ∞, <semantics>H0(P,AP(∗∞))</semantics> and the trace operator <semantics>pX</semantics> such that <semantics>pX(P,Q)=δP,Q</semantics> for <semantics>ϖr(P)=ϖr(Q)</semantics> for <semantics>P,Q∈X\{∞}</semantics>. In terms of these, we express the fundamental two-form of the second kind <semantics>Ω</semantics> and its connection to the sigma function for X.