Complementary Modules of Weierstrass Canonical Forms - Komeda, Jiryo; Matsutani, Shigeki; Previato, Emma (Symmetry, Integrability and Geometry: Methods and Applications, 2022)
The Weierstrass curve is pointed (, ∞) with a numerical semigroup , which is a normalization of the curve given by the Weierstrass canonical form, ʳ + ₁()ʳ⁻¹ + ₂()ʳ⁻² +⋯+ ᵣ₋₁() + ᵣ() = 0 where each ⱼ is a polynomial in of degree ≤ / for certain coprime positive integers and , < , such that the generators of the Weierstrass non-gap sequence at ∞ include and . The Weierstrass curve has the projection ϖᵣ: → ℙ, (, ) ↦ , as a covering space. Let := ⁰(, (∗∞)) and ℙ := ⁰(ℙ, ℙ(∗∞)) whose affine part is ℂ[]. In this paper, for every Weierstrass curve , we show the explicit expression of the complementary module ᶜ of the ℙ-module X as an extension of the expression of the plane Weierstrass curves by Kunz. The extension naturally leads to the explicit expressions of the holomorphic one form ∞, except ⁰(ℙ, ℙ(∗∞)) in terms of . Since for every compact Riemann surface, we find a Weierstrass curve that is bi-rational to the surface, we also comment that the explicit expression of ᶜ naturally leads to the algebraic construction of generalized Weierstrass' sigma functions for every compact Riemann surface and is also connected with the data on how the Riemann surface is embedded into the universal Grassmannian manifolds.