Complementary Modules of Weierstrass Canonical Forms

Abstract

The Weierstrass curve is a pointed curve (X, ∞) with a numerical semigroup H X , which is a normalization of the curve given by the Weierstrass canonical form,where each A j 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 H X at ∞ include r and s. The Weierstrass curve has the projection ϖ r : X → P, (x, y) → x, as a covering space. Let R X := H 0 (X, O X ( * ∞)) andIn this paper, for every Weierstrass curve X, we show the explicit expression of the complementary module R c X of R 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 ∞, H 0 (P, A P ( * ∞)) 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 c X 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.

Algebraic Construction of the Sigma Function for General Weierstrass Curves

“…We have studied further generalization of the picture in terms of the Weierstrass canonical form [16,17]. The Weierstrass curve X is a normalized curve of the curve given by the Weierstrass canonical form, y r + A 1 (x)y r−1 + A 2 (x)y r−2 + • • • + A r−1 (x)y + A r (x) = 0, where r is a positive integer, and each A j is a polynomial in x of a certain degree (c.f.…”

“…Let R X := H 0 (X, O X ( * ∞)) and R P := H 0 (P, O P ( * ∞)). In [17], we have the explicit description of the complementary module R c X of R P -module R X , which leads the explicit expressions of the holomorphic one form except ∞, H 0 (P, A P ( * ∞)).…”

“…In this paper, we use our recent results on the complementary module of the W-curve [17] to define the trace operator p X such that p X (P, Q) = δ P,Q for r (P) = r (Q) for P, Q ∈ X \ {∞}. In terms of them, we express the fundamental two-form of the second kind Ω algebraically in Theorem 3, and finally obtain a connection to Nakayashiki's sigma function by modifying his definition in Theorem 4, which means an algebraic construction of the sigma function for every W-curve as Baker performed for Klein's sigma function following Weierstrass' elliptic function theory.…”

“…The contents are as follows: Section 2 reviews the Weierstrass curves (W-curves) based on [17] (1) the numerical semigroup in Section 2.1, (2) Weierstrass canonical form in Section 2.2, (3) their relations to the monomial curves in Section 2.3, (4) the properties of the R P -module R X in Section 2.4, (5) the covering structures in W-curves in Section 2.5, and (6) especially the complementary module R c X of R X in Section 2.6; the explicit description of R c X is the first main result in [17]. Section 3 provides the first and the second theorems in this paper on the W-normalized Abelian differentials on X.…”

“…Section 3 provides the first and the second theorems in this paper on the W-normalized Abelian differentials on X. In Section 3.1, we review the second main result in [17] on the W-normalized Abelian differentials H 0 (X, A X ( * ∞)), which contain the Abelian differentials of the first kind. Further, we extend the trace operator p ∈ R X ⊗ R P R X introduced in [17] to R X ⊗ C R X in Section 3.2.…”