Complementary Modules of Weierstrass Canonical Forms

Abstract

The Weierstrass curve is pointed (𝛸, ∞) with a numerical semigroup 𝐻𝛸, which is a normalization of the curve given by the Weierstrass canonical form, 𝑦ʳ + 𝐴₁(𝑥)𝑦ʳ⁻¹ + 𝐴₂(𝑥)𝑦ʳ⁻² +⋯+ 𝐴ᵣ₋₁(𝑥)𝑦 + 𝐴ᵣ(𝑥) = 0 where each 𝐴ⱼ is a polynomial in 𝑥 of degree ≤ 𝑗𝑠/𝑟 for certain coprime positive integers 𝑟 and 𝑠, 𝑟 < 𝑠, such that the generators of the Weierstrass non-gap sequence 𝐻𝛸 at ∞ include 𝑟 and 𝑠. The Weierstrass curve has the projection ϖᵣ: 𝛸 → ℙ, (𝑥, 𝑦) ↦ 𝑥, as a covering space. Let 𝑅𝛸 := 𝐇⁰(𝛸, 𝒪𝛸(∗∞)) and 𝑅ℙ := 𝐇⁰(ℙ, 𝒪ℙ(∗∞)) whose affine part is ℂ[𝑥]. In this paper, for every Weierstrass curve 𝛸, we show the explicit expression of the complementary module 𝑅ᶜ𝛸 of the 𝑅ℙ-module 𝑅X 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 𝐇⁰(ℙ, 𝒜ℙ(∗∞)) in terms of 𝑅𝛸. 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 𝑅ᶜ𝛸 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.