Algebraic Construction of the Sigma Function for General Weierstrass Curves

Abstract

The Weierstrass curve X is a smooth algebraic curve determined by the Weierstrass canonical form, yr+A1(x)yr−1+A2(x)yr−2+⋯+Ar−1(x)y+Ar(x)=0, where r is a positive integer, and each Aj is a polynomial in x with a certain degree. It is known that every compact Riemann surface has a Weierstrass curve X, which is birational to the surface. The form provides the projection ϖr:X→ as a covering space. Let RX:=0(X,X(*∞)) and R:=0(,(*∞)). Recently, we obtained the explicit description of the complementary module RX of R-module RX, which leads to explicit expressions of the holomorphic form except ∞, 0(,(*∞)) and the trace operator pX such that pX(P,Q)=δP,Q for ϖr(P)=ϖr(Q) for P,Q∈X∖{∞}. In terms of these, we express the fundamental two-form of the second kind Ω and its connection to the sigma function for X.

Complementary Modules of Weierstrass Canonical Forms

“…In [12], the explicit description of the Abelian functions in terms of the sigma function demonstrates the degenerating behavior of the sigma function for the degenerating family of curves given by the Weierstrass canonical form f X (x, y) = y 3 − x(x − s)(x − b 1 )(x − b 2 ) for s → 0 for disjoint non-zero complex numbers b 1 and b 2 recently, which is much more precise than the known results [14]. The results in this paper with this follow-up paper [21] mean that 1) as we handle the elliptic functions of an elliptic curve, we can handle the algebraic functions of any Wcurve X using the explicit connection between the sigma function for X and the affine ring R X , e.g., their additive structure, Jacobi inversion formula, and differential relations, 2) as we did in [12], we can basically express the degenerating behavior of sigma function (theta function) for any degenerating family of W-curves, and 3) in terms of them, we could have explicit expressions of the algebraic solutions of KP hierarchy more precisely: Though it has not been a concern in the study of the integrable system, even for soliton solutions of KP hierarchy, there is no study on explicit description associated with the space curves except the recent interesting work by Kodama and Xie [16]. Our results in this paper provide the bases.…”

“…Our results in this paper can be naturally applied to the generalization of the EEL-constructions [20] as we mentioned above. Our previous report on the Riemann constant on the theta function [19] enables us to construct the sigma function of every W-curve algebraically and connect R X with the sigma function as we show in the follow-up paper [21]. We mention it shortly in Remark 5.10.…”

“…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.…”

“…Theorems 4.27 and 5.3 enable us to define the fundamental 2-form of the second kind algebraically and to construct the sigma functions of every W-curve as we show in a follow-up paper [21]. Using the results [19], we connect them with the sigma functions, which is defined as a modified Nakayashiki's sigma function [21]. We find the explicit relations between R X and the meromorphic functions on the Jacobi variety J X associated with the sigma functions like Weierstrass' elliptic function theory.…”

“…This expression can be applied to the shifted Riemann constant for the non-symmetric Wcurves [19]. Theorems 4.27 and 5.3 enable us to define the fundamental 2-form of the second kind algebraically and to construct the sigma functions of every W-curve as we show in a follow-up paper [21]. Using the results [19], we connect them with the sigma functions, which is defined as a modified Nakayashiki's sigma function [21].…”