Jacobi inversion formulae for a curve in Weierstrass normal form

Abstract

We consider a pointed curve (X, P ) which is given by the Weierstrass normal form,where x is an affine coordinate on P 1 , the point ∞ on X is mapped to x = ∞, and each A j is a polynomial in x of degree ≤ js/r for a certain coprime positive integers r and s (r < s) so that its Weierstrass non-gap sequence at ∞ is a numerical semigroup. It is a natural generalization of Weierstrass' equation in the Weierstrass elliptic function theory. We investigate such a curve and show the Jacobi inversion formulae of the strata of its Jacobian using the result of Jorgenson [Jo].

Complementary Modules of Weierstrass Canonical Forms

“…In this subsection, we now review the "Weierstrass canonical form" ("Weierstrass normal form") based on [17,19,20], which is a generalization of Weierstrass' standard form for elliptic curves. This form originated from Abel's insight, and Weierstrass investigated its primitive property [38,39].…”

“…This subsection shows the monomial curves and their relation to W-curves based on [17,19,20]. For a given W-curve X with the Weierstrass semigroup H = H X , and its generator M X = {r = r 1 , r 2 , .…”

“…In [20], we implicitly showed that these studies on the sigma functions in [5,18,27,28,33] are based on the algebraic structures of R X as an R P -module for such particular W-curves. In [17], we show that if we have the data of the holomorphic one-forms over X except ∞, H 0 (X, A X ( * ∞)), we have the correspondence between R X and the Riemann theta function for the subvarieties of its Jacobi variety associated with S k X, 0 ≤ k < g, where A X is the sheaf of the holomorphic one-forms on X.…”

“…We have studied the generalization of this picture to algebraic curves with higher genera in the series of the studies [17,18,19,20] following Mumford's excellent studies for the hyperelliptic curves [29,30].…”

“…To connect Nakayashiki's sigma functions in [32] for pointed compact Riemann surfaces with the Weierstrass semigroup H X and the algebraic properties of the W-curves X, in this paper, we study the algebraic properties of R X and the complementary module R c X as R P -modules. This paper aims to show the explicit expression of R c X since the expression of R c X enables us to apply the algebraic construction of the sigma function following the EEL construction to every W-curve: 1) R c X is directly related to the meromorphic one forms H 0 (X, A X ( * ∞)), i.e., H 0 (X, A X ( * ∞)) = R c X dx; we can find the Jaocobi inversion formula as mentioned in [17]. 2) As we show in a follow-up paper [21], by using the structure of R c X , we can define the fundamental differential of the second kind Ω following the EEL-construction to obtain the Jacobi inversion formula of Nakayashiki's sigma function.…”