Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 098, 39 pages      arXiv:2207.01905      https://doi.org/10.3842/SIGMA.2022.098

Complementary Modules of Weierstrass Canonical Forms

Jiryo Komeda a, Shigeki Matsutani b and Emma Previato c
a) Department of Mathematics, Center for Basic Education and Integrated Learning, Kanagawa Institute of Technology, 1030 Shimo-Ogino, Atsugi, Kanagawa 243-0292, Japan
b) Faculty of Electrical, Information and Communication Engineering, Kanazawa University, Kakuma Kanazawa, 920-1192, Japan
c) Department of Mathematics and Statistics, Boston University, Boston, MA 02215-2411, USA

Received July 12, 2022, in final form December 07, 2022; Published online December 18, 2022; Misprints corrected April 12, 2023

Abstract
The Weierstrass curve is a pointed curve $(X,\infty)$ with a numerical semigroup $H_X$, which is a normalization of the curve given by the Weierstrass canonical form, $y^r + A_{1}(x) y^{r-1} + A_{2}(x) y^{r-2} +\dots + A_{r-1}(x) y + A_{r}(x)=0$ where each $A_j$ is a polynomial in $x$ of degree $\leq j s/r$ for certain coprime positive integers $r$ and $s$, $r$<$s$, such that the generators of the Weierstrass non-gap sequence $H_X$ at $\infty$ include $r$ and $s$. The Weierstrass curve has the projection $\varpi_r\colon X \to {\mathbb P}$, $(x,y)\mapsto x$, as a covering space. Let $R_X := {\mathbf H}^0(X, {\mathcal O}_X(*\infty))$ and $R_{\mathbb P} := {\mathbf H}^0({\mathbb P}, {\mathcal O}_{\mathbb P}(*\infty))$ whose affine part is ${\mathbb C}[x]$. In this paper, for every Weierstrass curve $X$, we show the explicit expression of the complementary module $R_X^{\mathfrak c}$ of $R_{\mathbb P}$-module $R_X$ as an extension of the expression of the plane Weierstrass curves by Kunz. The extension naturally leads the explicit expressions of the holomorphic one form except $\infty$, ${\mathbf H}^0({\mathbb P}, {\mathcal A}_{\mathbb P}(*\infty))$ in terms of $R_X$. 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 $R_X^{\mathfrak c}$ naturally leads 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.

Key words: Weierstrass canonical form; complementary modules; plane and space curves with higher genera; sigma function.

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