2022/07/06 by Komeda, Jiryo, Matsutani, Shigeki, Previato, Emma · 1 citation
#14H05 #14H42 #14H50 #14H55 #Algebraic Geometry (math.AG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences
paper · doi:10.48550/arxiv.2207.02690
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 \varpir : X → ℙ as a covering space. Let RX := ℍ0(X, OX(*∞)) and Rℙ := ℍ0(ℙ, Oℙ(*∞)). Recently we have the explicit description of the complementary module RX^\mathfrakc of Rℙ-module RX, which leads the explicit expressions of the holomorphic one form except ∞, ℍ0(ℙ, Aℙ(*∞)) and the trace operator pX such that pX(P, Q)=δP,Q for \varpir(P)=\varpir(Q) for P, Q ∈ X∖\∞\. In terms of them, we express the fundamental 2-form of the second kind Ω and a connection to the sigma functions for X.