# Complementary modules of Weierstrass canonical forms

J. Komeda, S. Matsutani and E. Previato

###### 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,
$y^{r}+A_{1}(x)y^{r-1}+A_{2}(x)y^{r-2}+\cdots+A_{r-1}(x)y+A_{r}(x)=0$ where
each $A_{j}$ is a polynomial in $x$ of degree $\leq js/r$ for certain coprime
positive integers $r$ and $s$ ($r<s$) such that the generators of the
Weierstrass non-gap sequence $H_{X}$ at $\infty$ include $r$ and $s$. It
provides the projection $\varpi_{r}:X\to{\mathbb{P}}$ as a covering space. Let
$R_{X}:={\mathbf{H}}^{0}(X,{\mathcal{O}}_{X}(*\infty))$ and
$R_{\mathbb{P}}:={\mathbf{H}}^{0}({\mathbb{P}},{\mathcal{O}}_{\mathbb{P}}(*\infty))$.
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 (E. Kunz, Introduction to plane algebraic curves,
trans. by R. G. Belshoff, Birkhäuser, Boston, 2004). 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 and phrases:

Keywords: Weierstrass canonical form, Complementary modules, plane and space
curves with higher genera, sigma function

###### 2010 Mathematics Subject Classification:

Primary, 14H55, 14H50. Secondary, 16S36, 13H10

##  1\. Introduction

The Weierstrass $\sigma$ function is defined for an elliptic curve of
Weierstrass’ equation $y^{2}=4x^{3}-g_{2}x-g_{3}$ in Weierstrass’ elliptic
function theory [W3, WW]. Since
$\Bigr{(}\wp(u)=-{\displaystyle\frac{d^{2}}{du^{2}}}\log\sigma(u),$
${\displaystyle\frac{d\wp(u)}{du}}\Bigr{)}$ is identical to a point $(x,y)$ of
the curve, we can use the equivalence between the algebraic objects of the
curve and the transcendental objects on its Jacobi variety. The equivalence
leads to the crucial relations among them and their algebraic and analytic
properties associated with the elliptic curves [W3]. These relations and
properties affect the various fields of science and mathematics.

We have studied the generalization of this picture to algebraic curves with
higher genera in the series of the studies [KM, KMP1, KMP2, KMP3] following
Mumford’s excellent studies for the hyperelliptic curves [Mu1, Mu2].

As the elliptic theta function was generalized by Riemann for an Abelian
variety, its equivalent function Al was defined for any hyperelliptic curve by
Weierstrass and was refined by Klein as a generalization of the elliptic sigma
function. Since Klein defined his hyperelliptic sigma function by using only
the data of the hyperelliptic Riemann surface and Jacobian transcendentally,
Baker re-constructed Klein’s sigma function by using only the data of the
hyperelliptic curve itself from an algebraic viewpoint [Ba1]. Buchstaber,
Enolskii, and Leykin extend the sigma functions to certain plane curves, so-
called $(n,s)$ curves, based on Baker’s construction which we call EEL
construction due to work by Eilbeck, Enolskii, and Leykin [EEL] [[BEL] and its
references.]. For the $(n,s)$ curves with the cyclic symmetry, the direct
relations between the affine rings and the sigma functions were obtained as
the Jacobi inversion formulae [MP1, MP2]. Further, we generalized the sigma
functions and the formulae to a particular class of the space curves using the
EEL-construction [MK, KMP1, KMP3].

Recently as a generalization of Klein’s sigma functions, D. Korotkin and V.
Shramchenko defined the sigma function of every compact Riemann surface
transcendentally [KS]. Every compact Riemann surface can be characterized by
the Weierstrass non-gap sequence, which is described by a numerical semigroup
$H$; we call the numerical semigroup _Weierstrass semigroup_. In [N2],
Nakayashiki refined the sigma function for every compact Riemann surface with
Weierstrass semigroup $H$ based on Sato’s theory on the universal Grassmannian
manifolds (UGM) [SN, SW].

On the other hand, it is well-known that for every compact Riemann surface $Y$
with the Weierstrass semigroup $H_{Y}$ at a point $P\in Y$, we find an
algebraic curve $X$ which is bi-rational to the surface $Y$ and is obtained by
the normalization of the curve satisfying the Weierstrass canonical form,
$y^{r}+A_{1}(x)y^{r-1}+A_{2}(x)y^{r-2}+\cdots+A_{r-1}(x)y+A_{r}(x)$ where each
$A_{j}$ is a polynomial in $x$ of degree $\leq js/r$ for certain coprime
positive integers $r$ and $s$ ($r<s$); the point $P\in Y$ corresponds to
$\infty\in X$ and the Weierstrass semigroup $H_{X}$ at $\infty\in X$ is equal
to $H_{Y}$ whose generators include $r$ and $s$. In this paper, we call such a
curve _Weierstrass curve_ or _W-curve_. The set of W-curves represents the set
of compact Riemann surfaces. It provides the projection
$\varpi_{r}:X\to{\mathbb{P}}$ $(e.g.,(x,y)\mapsto x)$ as a covering space. For
$R_{X}={\mathbf{H}}^{0}(X,{\mathcal{O}}_{X}(*\infty))$ and
$R_{\mathbb{P}}={\mathbf{H}}^{0}({\mathbb{P}},{\mathcal{O}}_{\mathbb{P}}(*\infty))$,
let these quotient fields be ${\mathcal{Q}}(R_{X})$ and
${\mathcal{Q}}(R_{\mathbb{P}})$. Then ${\mathcal{Q}}(R_{X})$ is a Galois
extension of ${\mathcal{Q}}({\mathbb{P}})$, or $X$ is the Galois covering on
${\mathbb{P}}$, and $R_{X}$ is regarded as an $R_{\mathbb{P}}$-module.

The $(n,s)$ curves are identical to the plane W-curves, whereas the space
curves studied in [MK, KMP1, KMP3] are particular classes of the space
W-curves. In [KMP3], we implicitly showed that these studies on the sigma
functions in [BEL, MP1, MP2, KMP1, O] are based on the algebraic structures of
$R_{X}$ as an $R_{\mathbb{P}}$-module for such particular W-curves. In [KM],
we show that if we have the data of the holomorphic one-forms over $X$ except
$\infty$, ${\mathbf{H}}^{0}(X,{\mathcal{A}}_{X}(*\infty))$, we have the
partial correspondence between $R_{X}$ and the Riemann theta function for the
subvarieties of its Jacobi variety, where ${\mathcal{A}}_{X}$ is the sheaf of
the holomorphic one-forms on $X$.

Our purpose of the studies series is to connect the algebra $R_{X}$ and the
sigma function for every W-curve $X$ and show the equivalence between the
algebraic objects of the W-curve $X$ and the transcendental objects on its
Jacobi variety using the sigma functions as Baker did for the hyperelliptic
sigma functions following Weierstrass’ elliptic function theory [Ba1]. The
equivalence also leads to the crucial relations among them and their algebraic
and analytic properties associated with the W-curve $X$, like Weierstrass’
elliptic function theory. In other words, we purpose to extend Weierstrass’
elliptic function theory [W3] and Mumford’s study [Mu1, Mu2] to every W-curve.

It is a critical perspective that Weierstrass himself and related researchers
had already obtained some, but not perfect, results [Ba1, W2]. Due to the
difficulty, these attempts were given up and forgotten for a century. However,
the development of mathematics enables us to revive their approaches. We also
emphasize that the progress in the studies on the sigma function has provided
several non-trivial results as a generalization of those in elliptic curves,
which had been regarded as impossible for the century, e.g., [[BEL, EEL,
EEMOP1, EEMOP2, FKMPA, MP1, MP2, N1, O], and reference therein] as Mumford did
for the hyperelliptic curves using $\theta$ functions in [Mu1, Mu2].

In the general theory of the algebraic curves [DW, Ku2, St], the algebraic
curves with the Galois covering are studied well, and it turns out that the
complementary module plays important roles and is connected with the Kähler
differentials. The origin of the complementary modules is in the study by
Dedekind and Weber [DW]; Weierstrass already stated some results in [W2]. Kunz
showed the explicit expression of the complementary module for every plane
W-curve [Ku2, Theorem 15.1] though he did not call it W-curve.

To connect Nakayashiki’s sigma functions in [N2] 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
$R_{X}$ as an $R_{\mathbb{P}}$-module and the complementary module
$R_{X}^{\mathfrak{c}}$. This paper aims to show the explicit expression of
$R_{X}^{\mathfrak{c}}$ since the expression of $R_{X}^{\mathfrak{c}}$ enables
us to apply the algebraic construction of the sigma function following the EEL
construction to every W-curve. 1) $R_{X}^{\mathfrak{c}}$ is directly related
to the meromorphic one forms ${\mathbf{H}}^{0}(X,{\mathcal{A}}_{X}(*\infty))$,
i.e., ${\mathbf{H}}^{0}(X,{\mathcal{A}}_{X}(*\infty))=R_{X}^{\mathfrak{c}}dx$;
we can find the Jaocobi inversion formula as mentioned in [MK]. 2) As we show
in a follow-up paper [KMP4], by using the structure of $R_{X}^{\mathfrak{c}}$,
we can define the fundamental differential of the second kind $\Omega$
following the EEL-construction to obtain the Jacobi inversion formula of
Nakayashiki’s sigma function.

We remark that in Weierstrass’ elliptic function theory, if we regard
$\displaystyle{x(u)=-\frac{d^{2}}{du^{2}}\log\sigma}$ with $du=dx/y$ as a
differential equation, it is known that it can be integrated $x$ twice in $u$
to obtain the elliptic sigma function without theta functions, at least,
formally. Similarly, we regard that the Jacobi inversion formula, i.e., the
relation between $R_{X}$ and the sigma function for every W-curve, also means
an algebraic construction of the sigma function as Baker considered [Ba1]. In
the construction, the most critical and most complicated point is to obtain
the explicit expression of $R_{X}^{\mathfrak{c}}$. Thus this paper is devoted
to the expression.

By using the general theory of the algebraic curves, in this paper, we obtain
the explicit expression of the complementary module $R_{X}^{\mathfrak{c}}$ for
every W-curve $X$, including a space curve as an extension of Kunz’s results
on the complementary modules of the plane W-curves [Ku2]. Though the
extension, especially to non-symmetric $H_{X}$ cases, is complicated, we have
already obtained the expressions of particular space curves in [MK, KMP1,
KMP3] heuristically. By investigating them, we found the essentials of the
general constructions of $R_{X}^{\mathfrak{c}}$ in terms of data of $R_{X}$
for general W-curves as follows. Since the W-curves $X$ are characterized by
the Weierstrass semigroups, $H_{X}$ [W1], the monomial curves $X_{H}$ and the
monomial rings $R_{H}^{Z}$ associated with $H_{X}$ determine the behavior of
$R_{X}$ at $\infty\in X$. Following such studies on the monomial curves by
Kunz [Ku1], Herzog [H], and Pinkham [P], we use the multiple groups
${\mathbf{G}}_{\mathfrak{m}}$ action on the monomial curve $X_{H}$ and the
monomial algebra $R_{H}^{Z}$. As we assume that the generators of the
Weierstrass semigroup $H_{X}$ include the minimal number $r$, the cyclic group
${\mathfrak{C}}_{r}$ of order $r$ gives the standard basis of $H_{X}$ with
respect to $r$. The standard basis of $H_{X}$ governs the monomial algebra and
induces the standard basis of $R_{X}$ as the $R_{\mathbb{P}}$-module globally.
By using the standard basis of $R_{X}$, we investigate the
$R_{\mathbb{P}}$-module structure of $R_{X}$. As a key propositions, we find
the identifies in $R_{X}$ in Proposition 3.14 which induce an embedding of
$m_{X}$-dimensional space curve $X$ to ${\mathbb{P}}^{2(m_{X}-1)}$ with
projection to each singular plane curve in ${\mathbb{P}}^{2}$ in Proposition
3.15. This embedding leads the trace structure $R_{H}^{Z}$ in Lemma 4.6 and
using the structure determines the dual basis with the trace of $R_{X}$ in
Proposition 4.17. Finally we find the explicit expression of
$R_{X}^{\mathfrak{c}}$ in Theorem 4.26. Using it, we describe
${\mathbf{H}}^{0}(X,{\mathcal{A}}_{X}(*\infty))=R_{X}^{\mathfrak{c}}dx$ in
terms of $R_{X}$ in Theorem 5.3.

Furthermore it is known that
${\overline{H}}_{X}^{\mathfrak{c}}:={\mathbb{Z}}\setminus H_{X}$ provides the
embedding of the algebraic systems associated with $X$ into the UGM due to
Segal and Wilson [SW, p.46]. Since due to Riemann-Roch theorem,
$R_{X}^{\mathfrak{c}}dx$ has the weight sequence which is equal to
${\overline{H}}_{X}^{\mathfrak{c}}-1$, $R_{X}^{\mathfrak{c}}dx$ might lead the
embedding as Nakayashiki did for $(n,s)$ curve in [N1] rather than the spin
connection in [N2]. Our results in this paper can be naturally applied to the
generalization of the EEL-constructions [KMP3] as we mentioned above. Our
previous report on the Riemann constant on the theta functions [KMP2] enables
us to construct the sigma functions of every W-curve algebraically and connect
them with the sigma functions, which is defined as a modified Nakayashiki’s
sigma function, as we show in a follow-up paper [KMP4].

Contents are as follows: Section 2 consists of the two subsections: Subsection
2.1 reviews Dedekind’s trace, and its related topics based on Kunz’s book
[Ku2]. Subsection 2.2 gives the summary of the numerical and Weierstrass
semigroups. We show the Weierstrass canonical forms and Weierstrass curves
(W-curves) and their properties in Section 3. Using their properties, we find
the identities in $R_{X}$ in Proposition 3.14, as the first key proposition,
to show the $R_{\mathbb{P}}$-module structure of $R_{X}$ and an embedding of
$X$ into ${\mathbb{P}}^{2(m_{X}-1)}$, where $m_{X}$ is the minimal number of
the generators of $H_{X}$. Section 4 is devoted to the construction of the
complementary module $R_{X}^{\mathfrak{c}}$. After rewriting the tools in
Subsection 2.1 for W-curves $X$ shortly, we start to review the explicit
expression of the complementary module for every plane W-curve of $(m_{X}=2)$
[Ku2, Theorem 15.1] in Proposition 4.3. In order to extend it to a general
W-curve $X$ including a non-symmetric case, we consider the trace structure of
the monomial ring $R_{H}^{Z}$ in Lemma 4.6 as the second key proposition.
Using it, we investigate the global structure of $R_{X}$ as the
$R_{\mathbb{P}}$-module in Proposition 4.17 and Lemma 4.21 as the third key
proposition. Finally, we construct the complementary module in Theorem 4.26 as
the first main theorem in this paper. In Section 5, we consider the
W-normalized Abelian differentials
${\mathbf{H}}^{0}(X,{\mathcal{A}}_{X}(*\infty))$ and show the second main
theorem in Theorem 5.3. In Section 6, we provide some examples of our results.

Acknowledgment: The second author is grateful to Professors Yohei Komori for
valuable discussions in his seminar and to Professor Takao Kato for letting
him know about the paper [CK]. He has been supported by the Grant-in-Aid for
Scientific Research (C) of Japan Society for the Promotion of Science Grant,
No.21K03289.

##  2\. Preliminary

###  2.1. Trace and complementary modules

We review Dedekind’s trace and its related topics based on Kunz’s book [Ku2]
whose origin appeared in the paper by Dedekind and Weber [DW].

Let $R_{P}$ be an algebra over ${\mathbb{C}}$ and $R_{Y}$ an $R_{P}$ algebra
such that $\displaystyle{R_{Y}=\bigoplus_{i=0}^{r-1}R_{P}{\mathbf{y}}_{i}}$.

The dual of $R_{Y}$ is defined by

$\omega_{R_{Y}/R_{P}}=\mathrm{Hom}_{R_{P}}(R_{Y},R_{P})$

with the basis
$\\{{\mathbf{y}}_{i}^{*}\\}_{i\in{\mathbb{Z}}_{r}}\subset\omega_{R_{Y}/R_{P}}$
satisfying ${\mathbf{y}}_{i}^{*}{\mathbf{y}}_{j}=\delta_{ij}$.

Let us introduce _standard trace_ ,
$\displaystyle{\tau_{R_{Y}/R_{P}}:=\sum_{i=0}^{r-1}{\mathbf{y}}_{i}{\mathbf{y}}_{i}^{*}}$
and its complementary module $R_{Y}^{\mathfrak{c}}$ given by

$R_{Y}^{\mathfrak{c}}:=\\{z\in{\mathcal{Q}}(R_{Y})\ |\
\tau_{R_{Y}/R_{P}}(zR_{Y})\subset R_{P}\\}.$

We construct the complementary module $R_{Y}^{\mathfrak{c}}$ as follows
because $R_{Y}^{\mathfrak{c}}$ is directly connected with the Kähler
differentials.

######  Lemma 2.1.

There are structure coefficients $a_{ijk}$ in $R_{P}$ satisfying
$\displaystyle{{\mathbf{y}}_{i}{\mathbf{y}}_{j}=\sum_{k}a_{ijk}{\mathbf{y}}_{k}}$,
which determines the structure of the $R_{P}$-module $R_{Y}$. Then the
standard trace shows $\tau_{R_{Y}/R_{P}}({\mathbf{y}}_{j})=\sum_{i}a_{iji}$.

###### Proof.

The former statement is obvious and due to the definition of the standard
trace, we have
$\tau_{R_{Y}/R_{P}}({\mathbf{y}}_{j})=\sum_{i}{\mathbf{y}}_{i}{\mathbf{y}}_{i}^{*}({\mathbf{y}}_{j})$
$=\sum_{i}{\mathbf{y}}_{i}^{*}({\mathbf{y}}_{i}{\mathbf{y}}_{j})$
$=\sum_{i}{\mathbf{y}}_{i}^{*}(\sum_{k}a_{ijk}{\mathbf{y}}_{k})=\sum_{i}a_{iji}$.

######  Definition 2.2.

If there is an element $\tau$ in $\omega_{R_{Y}/R_{P}}$ such that
$\omega_{R_{Y}/R_{P}}=R_{Y}\tau$ as an $R_{P}$-module, i.e.,
$\omega_{R_{Y}/R_{P}}$ is a free $R_{P}$-module with the basis $\\{\tau\\}$,
we say that $R_{Y}$ has the trace $\tau$.

We note that if $R_{Y}/R_{P}$ is separable, the standard trace
$\tau_{R_{Y}/R_{P}}$ is a trace in this definition.

######  Lemma 2.3.

If $R_{Y}$ has a trace $\tau$, the following hold.

  1. (1)

If an element $a$ in $R_{Y}$ satisfying $a\tau=0$, $a=0$. It means that
$\omega_{R_{Y}/R_{P}}\cong R_{Y}\tau$ as an $R_{Y}$-module.

  2. (2)

For a basis $\\{{\mathbf{y}}_{i}\\}$ of $R_{Y}$ as an $R_{P}$-module, there
exists a subset $\\{\widehat{{\mathbf{y}}}_{i}\\}\subset R_{Y}$ satisfying
$\tau(\widehat{{\mathbf{y}}}_{i}{\mathbf{y}}_{j})=\delta_{ij}$. (We call
$\\{\widehat{{\mathbf{y}}}_{i}\\}$ the dual basis with respect to the trace
$\tau$.) Then
$\displaystyle{\tau_{R_{Y}/R_{P}}=\left(\sum_{i=0}^{r-1}\widehat{{\mathbf{y}}}_{i}{\mathbf{y}}_{i}\right)\tau}$.

###### Proof.

(1). For any $x\in R_{Y}$, $0=a\tau(x)=\tau(ax)=x\tau(a)$. For every
$b\in\omega_{R_{Y}/R_{P}}$, $b(a)=0$ implies that $a=0$. The $R_{P}$-morphism
$\omega_{R_{Y}/R_{P}}\to R_{Y}\tau$ is injective. (2). For the dual basis
$\\{{\mathbf{y}}_{i}^{*}\\}$ of $\omega_{R_{Y}/R_{P}}$ such that
${\mathbf{y}}_{i}^{*}({\mathbf{y}}_{j})=\delta_{ij}$, we can find an element
$\widehat{{\mathbf{y}}}_{i}\in R_{Y}$ satisfying
${\mathbf{y}}^{*}_{i}=\widehat{{\mathbf{y}}}_{i}\tau$. Then
$\tau(\widehat{{\mathbf{y}}}_{i}{\mathbf{y}}_{j})={\mathbf{y}}_{i}^{*}({\mathbf{y}}_{j})=\delta_{ij}$,
and
$\displaystyle{\tau_{R_{Y}/R_{P}}=\sum_{i=0}^{r-1}{\mathbf{y}}_{i}{\mathbf{y}}_{i}^{*}=\left(\sum_{i=0}^{r-1}\widehat{{\mathbf{y}}}_{i}{\mathbf{y}}_{i}\right)\tau}$.

We construct $\tau$ and $\tau_{R_{Y}/R_{P}}$ in terms of the enveloping
algebra $R_{Y}^{e}:=R_{Y}\otimes_{R_{P}}R_{Y}$.

For $R_{Y}^{e}:=R_{Y}\otimes_{R_{P}}R_{Y}$, the standard multiplication
$\mu:R_{Y}^{e}\to R_{Y}$ is defined by $\mu(a\otimes b)=ab$.

We consider the annihilator of the kernel of $\mu$,

$\mathrm{Ann}_{R_{Y}^{e}}(\mathrm{Ker}\mu):=\\{z\in R_{Y}^{e}\ |\
z\cdot\mathrm{Ker}\mu=0\\}.$

For an element $\sum a_{i}\otimes
b_{i}\in\mathrm{Ann}_{R_{Y}^{e}}(\mathrm{Ker}\mu)$, $(c\otimes 1-1\otimes
c)\cdot\sum a_{i}\otimes b_{i}=0$ or $\sum ca_{i}\otimes b_{i}=\sum
a_{i}\otimes cb_{i}$.

######  Lemma 2.4.

There is a natural embedding
$\varphi:\mathrm{Ann}_{R_{Y}^{e}}(\mathrm{Ker}\mu)\to\mathrm{Hom}_{R_{Y}}(\omega_{R_{Y}/R_{P}},R_{Y})$.

###### Proof.

Since $\sum ca_{i}\otimes b_{i}=\sum a_{i}\otimes cb_{i}$ for an element $\sum
a_{i}\otimes b_{i}\in\mathrm{Ann}_{R_{Y}^{e}}(\mathrm{Ker}\mu)$ and $c\in
R_{P}$, it should be defined $\varphi(\sum_{i}a_{i}\otimes b_{i})(\rho)$
$=\sum_{i}\rho(a_{i})b_{i}\in R_{Y}$ for $\rho\in\omega_{R_{Y}/R_{P}}$; for
every $s\in R_{Y}$, we have $s\rho(a)b=\rho(a)sb=\rho(sa)b$.

######  Proposition 2.5.

[Ku2, Corollary H.20] Suppose $R_{Y}/R_{P}$ has a trace. Then $\varphi$
induces a bijection between the set of all traces of $R_{Y}/R_{P}$ and the set
of all generators of the $R_{Y}$-module
$\mathrm{Ann}_{R_{Y}^{e}}(\mathrm{Ker}\mu)$: Each trace
$\tau\in\omega_{R_{Y}/R_{P}}$ is mapped to the unique element
$\displaystyle{\Delta_{\tau}:=\sum_{i=0}^{r-1}\widehat{{\mathbf{y}}}_{i}\otimes{\mathbf{y}}_{i}\in}$$\displaystyle{\mathrm{Ann}_{R_{Y}^{e}}(\mathrm{Ker}\mu)}$
associated with $\tau$ such that
$\sum_{i=0}^{r-1}\tau(\widehat{{\mathbf{y}}}_{i}){\mathbf{y}}_{i}=1$.
Furthermore

  1. (1)

$\Delta_{\tau}$ generates the $R_{Y}$-module
$\mathrm{Ann}_{R_{Y}^{e}}(\mathrm{Ker}\mu)$, and
$\\{\widehat{{\mathbf{y}}}_{1},\ldots,\widehat{{\mathbf{y}}}_{r}\\}$ is the
dual basis of the basis $\\{{\mathbf{y}}_{1},\ldots,{\mathbf{y}}_{r}\\}$ of
$R_{X}$ with respect to $\tau$; i.e.,

$\tau(\widehat{{\mathbf{y}}}_{i}{\mathbf{y}}_{j})=\delta_{i,j},\quad(i,j=1,\ldots,r).$

  2. (2)

If
$\displaystyle{\sum_{i=0}^{r-1}\widehat{{\mathbf{y}}}^{\prime}_{i}\otimes{\mathbf{y}}_{i}}$
generates the $R_{Y}$-module $\mathrm{Ann}_{R_{Y}^{e}}(\mathrm{Ker}\mu)$, and
if $\tau^{\prime}\in\omega_{R_{Y}/R_{P}}$ is a linear form with
$\sum_{i=0}^{r-1}\tau^{\prime}(\widehat{{\mathbf{y}}}_{i}^{\prime}){\mathbf{y}}_{i}=1$,
then $\tau^{\prime}$ is a trace of $R_{Y}/R_{P}$ and
$\displaystyle{\Delta_{\tau^{\prime}}=\sum_{i=0}^{r-1}\widehat{{\mathbf{y}}}^{\prime}_{i}\otimes{\mathbf{y}}_{i}}$
is associated with the trace $\tau^{\prime}$; hence
$\\{\widehat{{\mathbf{y}}}^{\prime}_{1},\ldots,\widehat{{\mathbf{y}}}^{\prime}_{r}\\}$
is the dual basis of the basis
$\\{{\mathbf{y}}_{1},\ldots,{\mathbf{y}}_{r}\\}$ of $R_{X}$ with respect to
$\tau^{\prime}$.

  3. (3)

For each trace $\tau$ of $R_{Y}/R_{P}$,

$\tau_{R_{Y}/R_{P}}=\mu(\Delta_{\tau})\tau.$

###### Proof.

Let us consider $1\in\mathrm{Hom}_{R_{Y}}(\omega_{R_{Y}/R_{P}},R_{Y})$, and
consider the inverse image of $\\{1\\}$,
$\varphi^{-1}(1)\in\mathrm{Ann}_{R_{Y}^{e}}(\mathrm{Ker}\mu)$, which is
denoted by $\Delta_{\tau}$. By using the basis $\\{{\mathbf{y}}_{i}\\}\subset
R_{Y}$, we can express it as
$\Delta_{\tau}=\sum_{i}\widehat{{\mathbf{y}}}_{i}\otimes{\mathbf{y}}_{i}$ for
a certain $\\{\widehat{{\mathbf{y}}}_{i}\\}$. Thus
$\varphi(\sum_{i}\widehat{{\mathbf{y}}}_{i}\otimes{\mathbf{y}}_{i})(\tau)=\sum_{i}\tau(\widehat{{\mathbf{y}}}_{i}){\mathbf{y}}_{i}=1$.
The fact that $\sum_{i}\tau(\widehat{{\mathbf{y}}}_{i}){\mathbf{y}}_{i}=1$
implies that
${\mathbf{y}}_{j}={\mathbf{y}}_{j}\varphi(\sum_{i}\widehat{{\mathbf{y}}}_{i}\otimes{\mathbf{y}}_{i})(\tau)=\varphi(\sum_{i}\widehat{{\mathbf{y}}}_{i}\otimes{\mathbf{y}}_{i})({\mathbf{y}}_{j}\tau)=\sum_{i}\tau(\widehat{{\mathbf{y}}}_{i}{\mathbf{y}}_{j}){\mathbf{y}}_{i}$.
Thus $\tau(\widehat{{\mathbf{y}}}_{i}{\mathbf{y}}_{j})=\delta_{ij}$. Hence the
set $\\{\widehat{{\mathbf{y}}}_{i}\\}$ is a dual basis of $R_{Y}$ with respect
to $\tau$.

Let us consider the case
$\mathrm{Ann}_{R_{Y}^{e}}(\mathrm{Ker}\mu)=R_{Y}(\sum_{i}\widehat{{\mathbf{y}}}_{i}^{\prime}\otimes{\mathbf{y}}_{i})$
with
$\sum_{i=0}^{r-1}\tau^{\prime}(\widehat{{\mathbf{y}}}_{i}^{\prime}){\mathbf{y}}_{i}=1$
for $\tau^{\prime}\in\omega_{R_{Y}/R_{P}}$. Then by the above arguments,
obviously
$\varphi(\sum_{i}\widehat{{\mathbf{y}}}_{i}^{\prime}\otimes{\mathbf{y}}_{i})=1$.
we obtain (2).

Lemma 2.3 shows that
$\tau_{R_{Y}/R_{P}}=\sum_{i}{\mathbf{y}}_{i}{\mathbf{y}}_{i}^{*}=\sum_{i}\widehat{{\mathbf{y}}}_{i}{\mathbf{y}}_{i}\tau$
$=\mu(\Delta_{\tau})\tau$.

###  2.2. Numerical and Weierstrass semigroup

This subsection is on numerical and Weierstrass semigroups based on [ADGS,
KMP1]. An additive sub-monoid of the monoid, non-negative integers
${\mathbb{N}}_{0}$ is called _numerical semigroup_ if its compliment in
${\mathbb{N}}_{0}$ is a finite set. In this subsection, we review the
numerical semigroups associated with algebraic curves.

Let $X$ be a smooth (complex projective) curve of genus g. For a point $P\in
X$,

$H(X,P):=\\{n\in{\mathbb{N}}_{0}\ |\ \mbox{there exists
}f\in{\mathcal{M}}(X)\mbox{ such that }(f)_{\infty}=nP\ \\}$

forms a numerical semigroup by the Riemann-Roch theorem, which is called the
Weierstrass semigroup of the point $P$, If the Weierstrass gap sequence
$H^{\mathfrak{c}}(X,P):={\mathbb{N}}_{0}\setminus H(X,P)$ differs from the set
$\\{1,2,\cdots,g\\}$, we say that $P$ is a Weierstrass point of $X$ [FK].

In this paper, we consider a _pointed curve_ , a pair $(X,P)$ with $P$ a point
of the curve $X$ with the Weierstrass semigroup $H(X,P)$.

In general, a numerical semigroup $H$ has a unique (finite) minimal set of
generators, $M=M(H)$, ($H=\langle M\rangle$) and the finite cardinality $g$ of
$H^{\mathfrak{c}}={\mathbb{N}}_{0}\setminus H$; $g$ is the genus of $H$ or
$H^{\mathfrak{c}}$ and $H^{\mathfrak{c}}$ is called gap-sequence. For example,

(2.1) $\displaystyle\begin{split}&H^{\mathfrak{c}}=\\{1,2,4,5\\},\ \mbox{for}\
H=\langle 3,7,8\rangle.\\\ &H^{\mathfrak{c}}=\\{1,2,3,4,6,8,9,13\\},\
\mbox{for}\ H=\langle 5,7,11\rangle,\ \mbox{ and }\\\
&H^{\mathfrak{c}}=\\{1,2,3,4,5,7,8,9,10,17,23\\},\ \mbox{for}\ H=\langle
6,13,14,15,16\rangle.\\\ \end{split}$

We let $r_{\mathrm{min}}(H)$ be the smallest positive integer of $M(H)$. We
call the semigroup $H$ _an $r_{\mathrm{min}}(H)$-semigroup_, so that $\langle
3,7,8\rangle$ is a $3$-semigroup and $\langle 6,13,14,15,16\rangle$ is a
$6$-semigroup. Let $N(i)$ and $N^{\mathfrak{c}}(i)$ be the $i$-th ordered
element of $H=\\{N(i)\ |\ i\in{\mathbb{N}}_{0}\\}$ and
$H^{\mathfrak{c}}=\\{N^{\mathfrak{c}}(i)\ |\ i=0,1,\ldots,g-1\\}$ satisfying
$N(i)<N(i+1)$, and $N^{\mathfrak{c}}(i)<N^{\mathfrak{c}}(i+1)$ respectively.
The Schubert index of the set $H^{\mathfrak{c}}$ is

(2.2) $\alpha(H):=\\{\alpha_{0}(H),\alpha_{1}(H),\ldots,\alpha_{g-1}(H)\\},$

where $\alpha_{i}(H):=N^{\mathfrak{c}}(i)-i-1$. Moreover,

$\alpha(\langle
3,7,8\rangle)=\\{0^{2},1^{2}\\}\quad\mbox{and}\quad\alpha(\langle
6,13,14,15,16\rangle)=\\{0^{5},1^{5},6,11\\}.$

Further the conductor $c_{H}$ of $H$ is defined by the minimal natural number
satisfying $c_{H}+{\mathbb{N}}_{0}\subset H$. The number $c_{H}-1$ is known as
the Frobenius number, which is the largest element of $H^{\mathfrak{c}}$.

By letting the row lengths be $\Lambda_{i}=\alpha_{g-i}(H)+1$, $i<g$, we have
the Young diagram of the semigroup,
$\Lambda:=(\Lambda_{1},\ldots,\Lambda_{g})$, ($\Lambda_{i}\leq\Lambda_{i+1}$).
The Young diagram $\Lambda$ is a partition of
$\displaystyle{\sum_{i}\Lambda_{i}}$. We say that such a Young diagram is
associated with the numerical semigroup. If for a given Young diagram
$\Lambda$, we cannot find any numerical semigroups $H$ such that
$\Lambda_{i}=\alpha_{g-i}(H)+1$, we say that $\Lambda$ is not associated with
the numerical semigroup. It is obvious that in general, the Young diagrams are
not associated with the numerical semigroups.

(2.3)
$\displaystyle\begin{matrix}\yng(2,2,1,1),&\yng(6,3,3,2,1,1,1,1),&\yng(12,8,2,2,2,2,2,1,1,1,1,1).\\\
\langle 3,7,8\rangle&\langle 5,7,11\rangle&\langle 6,13,14,15,16\rangle\\\
\end{matrix}$

The Young diagram and the associated numerical semigroup are called symmetric
if the Young diagram is invariant under reflection across the main diagonal.
It is known that the numerical semigroup is symmetric if and only if $2g-1$
occurs in the gap sequence. It means that if $c_{H}=2g$, $H$ is symmetric.

We obviously have the following proposition:

######  Proposition 2.6.

The following hold:

  1. (1)

$N(n)-n\leq g$ for every $n\in{\mathbb{N}}_{0}$,

  2. (2)

$N(n)-n=g$ for $N(n)\geq c_{X}=N(g)$,

  3. (3)

$N(n)-n<g$ for $0\leq N(n)<c_{X}$,

  4. (4)

$\\#\\{n\ |\ N^{\mathfrak{c}}(n)\geq g\\}=\\#\\{n\ |\ N(n)<g\\}$

  5. (5)

for $N(i)<N^{\mathfrak{c}}(j)$, $N^{\mathfrak{c}}(j)-N(i)\in
H^{\mathfrak{c}}$, and

  6. (6)

when $H$ is symmetric, $c_{H}=N(g)=2g$ and
$c_{H}-N(i)-1=N^{\mathfrak{c}}(g-i-1)$ for $0\leq i\leq g-1$.

###### Proof.

(1)-(3) and (5) are obvious. Noting $\\#H^{\mathfrak{c}}=g$, (4) means that
what is missing must be filled later for $H^{\mathfrak{c}}$. (6) is left to
[ADGS].

In this paper, we mainly consider the $r$-numerical semigroup, $H$. We
introduce the tools as follows:

######  Definition 2.7.

  1. (1)

Let ${\mathbb{Z}}_{r}:=\\{0,1,2,\ldots,r-1\\}$ and
${\mathbb{Z}}_{r}^{\times}:={\mathbb{Z}}_{r}\setminus\\{0\\}$.

  2. (2)

Let ${\widetilde{\mathfrak{e}}}_{i}:=\min\\{h\in H\ |\ i\equiv h\ \mbox{ mod
}\ r\\}$, $i\in{\mathbb{Z}}_{r}$.

  3. (3)

Let ${\widetilde{\mathfrak{E}}}_{H}:=\\{{\widetilde{\mathfrak{e}}}_{i}\ |\
i\in{\mathbb{Z}}_{r}\\}$ be the standard basis of $H$. Further we define the
ordered set
${\mathfrak{E}}_{H}:=\\{{\mathfrak{e}}_{i}\in{\widetilde{\mathfrak{E}}}_{H}\
|\ {\mathfrak{e}}_{i}<{\mathfrak{e}}_{i+1}\\}$, and
${\mathfrak{E}}_{H}^{\times}:={\mathfrak{E}}_{H}\setminus\\{0\\}$, e.g.,
${\mathfrak{e}}_{0}={\widetilde{\mathfrak{e}}}_{0}=0$.

  4. (4)

Let ${\mathfrak{e}}_{\ell,i}^{*}$ be the element in ${\mathfrak{E}}_{H}$ such
that ${\mathfrak{e}}_{\ell,i}^{*}={\mathfrak{e}}_{\ell}-{\mathfrak{e}}_{i}$
modulo $r$.

  5. (5)

Let ${\overline{H}}^{\mathfrak{c}}:=H^{\mathfrak{c}}\bigcup(-{\mathbb{N}})$,
where ${\mathbb{N}}:={\mathbb{N}}_{0}\setminus\\{0\\}$.

  6. (6)

The Apéry set $\mathrm{Ap}(H,n)$ for a positive integer $n$ is defined by

$\mathrm{Ap}(H,n):=\\{s\in H\ |\ s-n\not\in H\\}.$

Since it is obvious that
$\mathrm{Ap}(H,r)={\mathfrak{E}}_{H}={\widetilde{\mathfrak{E}}}_{H}$ as a set,
the standard basis is sometimes defined by the Apéry set $\mathrm{Ap}(H,r)$.

We have the following elementary but essential results:

######  Lemma 2.8.

For $a\in{\mathbb{N}}_{0}$, we define

$[a]_{r}:=\\{a+kr\ |\
k\in{\mathbb{N}}_{0}\\},\quad\overline{[a]}_{r}^{\mathfrak{c}}:=\\{a-kr\ |\
k\in{\mathbb{N}}\\},\quad[a]_{r}^{\mathfrak{c}}:=\overline{[a]}_{r}^{\mathfrak{c}}\cap{\mathbb{N}}.$

  1. (1)

We have the following decomposition;

    1. (a)

$\displaystyle{{\mathbb{N}}_{0}=\bigoplus_{i\in{\mathbb{Z}}_{r}}[i]_{r},}$

    2. (b)

$\displaystyle{H=\bigoplus_{i\in{\mathbb{Z}}_{r}}[{\mathfrak{e}}_{i}]_{r},}$

    3. (c)

$\displaystyle{{\overline{H}}^{\mathfrak{c}}=\bigoplus_{i\in{\mathbb{Z}}_{r}}\overline{[{\mathfrak{e}}_{i}]}_{r}^{\mathfrak{c}}}$,
$H\bigcup{\overline{H}}^{\mathfrak{c}}={\mathbb{Z}}$,

    4. (d)

$\displaystyle{H^{\mathfrak{c}}=\bigoplus_{i\in{\mathbb{Z}}_{r}}[{\mathfrak{e}}_{i}]_{r}^{\mathfrak{c}}=\bigoplus_{i\in{\mathbb{Z}}_{r}^{\times}}[{\mathfrak{e}}_{i}]_{r}^{\mathfrak{c}}}$,
$H\bigcup H^{\mathfrak{c}}={\mathbb{N}}_{0}$,

  2. (2)

for every $x_{i}\in[{\mathfrak{e}}_{i}]_{r}$ $(i\in{\mathbb{Z}}_{r})$,

$\\{x_{i}\mbox{ modulo }r\ |\
i\in{\mathbb{Z}}_{r}\\}={\mathbb{Z}}/r{\mathbb{Z}},$

especially for $x\in[{\widetilde{\mathfrak{e}}}_{i}]_{r}$, $x=i$ modulo $r$,
and

  3. (3)

${\mathfrak{e}}_{\ell,\ell}^{*}={\mathfrak{e}}_{0}=0$, and
${\mathfrak{e}}_{\ell,0}^{*}={\mathfrak{e}}_{\ell}$.

###### Proof.

(1)(a), (2) and (3) are apparent. From the definition of ${\mathfrak{E}}_{H}$,
$H=\\{{\mathfrak{e}}_{i}+kr\ |\ i\in{\mathbb{Z}}_{r},\
k\in{\mathbb{N}}_{0}\\}$. For $i\neq j$,
$[{\mathfrak{e}}_{i}]_{r}\cap[{\mathfrak{e}}_{j}]_{r}=\emptyset$ and thus we
have the decomposition in (1)(b). Since
$H^{\mathfrak{c}}={\mathbb{N}}_{0}\setminus H$, we have (1)(d) and (c) noting
(1)(a).

The following is obvious:

######  Lemma 2.9.

For the generators $r$ and $s$ in the numerical semigroup $H$, there are
positive integers $i_{s}$ and $i_{r}$ such that $i_{s}s-i_{r}r=1.$

We remark that ${\overline{H}}^{\mathfrak{c}}$ determines the structure of the
differentials on a certain curve $X$ in Lemma 5.3 and the embedding of the
curve into the universal Grassmannian manifold as in [SW, p.46].

A numerical semigroup $H$ is said to be Weierstrass if there exists a pointed
curve $(X,P)$ such that $H=H(X,P)$. Hurwitz posed the problem of whether any
numerical semigroup $H$ is Weierstrass. The question was revived in the 1980s,
viewed as the question of deformations of a reduced complex curve singularity
$(X_{0},\infty)$. Buchweitz and Greuel showed a counterexample.

##  3\. Weierstrass canonical form and Weierstrass curves (W-curves)

###  3.1. Weierstrass canonical form

In this subsection, we now review the “Weierstrass canonical form”
(“Weierstrass normal form”) based on [KMP2, KMP3, KM], which is a
generalization of Weierstrass’ standard form for elliptic curves. This form
originated from Abel’s insight, and Weierstrass investigated its primitive
property [W1, W2]. Baker [Ba1, Ch. V, §§60-79] gives a complete review, proof,
and examples of the theory. Here we refer to Kato [Ka, CK], who also produces
this representation with proof.

######  Proposition 3.1.

[Ka, CK] For a pointed curve $(X,\infty)$ with Weierstrass semigroup
$H_{X}:=H(X,\infty)$ for which $r_{\mathrm{min}}(H_{X})=r$, and
${\mathfrak{e}}_{i}\in{\mathfrak{E}}_{H_{X}}$,
$(i\in{\mathbb{Z}}_{r}^{\times})$ in Definition 2.7, and we let
$\displaystyle{s:=\min_{i\in{\mathbb{Z}}_{r}^{\times}}\\{{\mathfrak{e}}_{i}\
|\ ({\mathfrak{e}}_{i},r)=1\\}}$ and $s={\mathfrak{e}}_{\ell_{s}}$.
$(X,\infty)$ is defined by an irreducible equation,

(3.1) $f_{X}(x,y)=0,$

for a polynomial $f_{X}\in{\mathbb{C}}[x,y]$ of type,

(3.2)
$f_{X}(x,y):=y^{r}+A_{1}(x)y^{r-1}+A_{2}(x)y^{r-2}+\cdots+A_{r-1}(x)y+A_{r}(x),$

where the $A_{i}(x)$’s are polynomials in $x$, $A_{0}=1$,

(3.3) $A_{i}=\sum_{j=0}^{\lfloor
is/r\rfloor}\lambda_{i,j}x^{j}=:\prod_{j=1}^{\lfloor
is/r\rfloor}a_{i,0}(x-a_{i,j}),$

and $\lambda_{i,j}\in{\mathbb{C}}$, $\lambda_{r,s}=-1$.

###### Proof.

We let
${\overline{\mathfrak{E}}}_{H_{X}}:=\\{{\mathfrak{e}}_{1},{\mathfrak{e}}_{2},\ldots,{\mathfrak{e}}_{r-1}\\}\setminus\\{s\\}$,
${\overline{\mathfrak{E}}}_{H_{X}}=:\\{{\overline{\mathfrak{e}}}_{2},\ldots,{\overline{\mathfrak{e}}}_{r-1}\\}$,
(${\overline{\mathfrak{e}}}_{i}<{\overline{\mathfrak{e}}}_{j}$ for $i<j$), and
${\overline{\mathfrak{e}}}_{1}=s$. Let
${\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{i}}$ be a meromorphic
function on $X$ whose only pole is $\infty$ with order
${\overline{\mathfrak{e}}}_{i}$, $(i\in{\mathbb{Z}}_{r}^{\times}$), taking
$y_{s}:={\widetilde{\mathfrak{y}}}_{s}={\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{1}}$.
From the definition of $X$, we have, as ${\mathbb{C}}$-vector spaces,

(3.4)
${\mathbf{H}}^{0}(X,{\mathcal{O}}_{X}(*\infty))={\mathbb{C}}\oplus\sum_{i=0}^{r-1}\sum_{j=0}{\mathbb{C}}x^{j}{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{i}}.$

Let the affine ring $R_{X}$ of $X$ be defined by
${\mathbf{H}}^{0}(X,{\mathcal{O}}_{X}(*\infty))$, i.e.,
$R_{X}:={\mathbf{H}}^{0}(X,{\mathcal{O}}_{X}(*\infty))$. Thus for every
${\overline{\mathfrak{e}}}_{j}\in{\overline{\mathfrak{E}}}_{H_{X}}$
$(j=2,3,\ldots,r-1)$, we obtain the following equations,

(3.5)
$\displaystyle\begin{split}\left\\{\begin{array}[]{rcl}y_{s}{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{2}}&=&A_{2,0}+A_{2,1}y_{s}+A_{2,2}{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{2}}+\cdots+A_{2,r-2}{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{r-2}}+A_{2,r-1}{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{r-1}},\\\
y_{s}{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{3}}&=&A_{3,0}+A_{3,1}y_{s}+A_{3,2}{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{2}}+\cdots+A_{3,r-2}{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{r-2}}+A_{3,r-1}{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{r-1}},\\\
&&\cdots\\\
y_{s}{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{r-1}}&=&A_{{r-1},0}+A_{r-1,1}y_{s}+A_{r-1,2}{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{2}}+\cdots+A_{r-1,r-2}{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{r-2}}+A_{r-1,r-1}{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{r-1}},\\\
\end{array}\right.\end{split}$ (3.6) $\displaystyle
y_{s}^{2}=A_{1,0}+A_{1,1}y_{s}+A_{1,2}{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{2}}+\cdots+A_{1,r-2}{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{r-2}}+A_{1,r-1}{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{r-1}},$

where $A_{i,j}\in{\mathbb{C}}[x]$.

When $r=2$, (3.1) equals (3.6). We assume that $r>2$ and then (3.5) is reduced
to $(r-2)$ linear equations,

(3.7) $\displaystyle\begin{pmatrix}A_{2,1}-y_{s}&A_{2,2}&\cdots&A_{1,r-1}\\\
A_{3,1}&A_{3,2}-y_{s}&\cdots&A_{3,r-1}\\\ \vdots&\vdots&\ddots&\vdots\\\
A_{r-1,1}&A_{r-1,2}&\cdots&A_{r-1,r-1}-y_{s}\\\
\end{pmatrix}\begin{pmatrix}{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{2}}\\\
{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{3}}\\\ \vdots\\\
{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{r-1}}\\\
\end{pmatrix}=-\begin{pmatrix}A_{2,0}+A_{2,1}y_{s}\\\ A_{3,0}+A_{3,1}y_{s}\\\
\vdots\\\ A_{r-1,0}+A_{r-1,1}y_{s}\\\ \end{pmatrix}.$

One can check that the determinant of the matrix on the left-hand side of
(3.7) is not equal to zero by computing the order of pole at $\infty$ of the
monomials $B_{i}y_{s}^{r-2-i}$ in the expression,

$P(x,y_{s}):=\left|\begin{matrix}A_{2,1}-y_{s}&A_{2,2}&\cdots&A_{2,r-1}\\\
A_{3,1}&A_{3,2}-y_{s}&\cdots&A_{3,r-1}\\\ \vdots&\vdots&\ddots&\vdots\\\
A_{r-1,1}&A_{r-1,2}&\cdots&A_{r-1,r-1}-y_{s}\\\
\end{matrix}\right|=y_{s}^{r-2}+B_{1}y_{s}^{r-3}+\cdots+B_{r-3}y_{s}+B_{r-4},$

which is $s(r-2-i)+r\cdot\mathrm{deg}_{x}B_{i}$ by letting
$\mathrm{deg}_{x}h(x)$ be the degree of $h$ with respect to $x$. The fact that
$(r,s)=1$ shows that $s(r-2-i)+r\cdot\mathrm{deg}_{x}B_{i}\neq
s(r-2-j)+r\cdot\mathrm{deg}_{x}B_{j}$ for $i\neq j$.

Hence by solving equation (3.7), we have

(3.8)
${\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{i}}=\frac{Q_{i}(x,y_{s})}{P(x,y_{s})},$

where $Q_{i}(x,y_{s})\in{\mathbb{C}}[x,y_{s}]$ and a polynomial of order at
most $r-2$ in $y_{s}$. Note that the equations (3.8) are not algebraically
independent in general but in any cases the function field of the curve can be
generated by some of these
${\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{j}}$’s, and its affine
ring $R_{X}$ can be given by a quotient ring of
${\mathbb{C}}[x,y_{r_{2}},y_{r_{3}},\ldots,y_{r_{m_{X}}}]$ for $i_{j}\in
M_{X}:=M(H_{X})$, where
$M_{X}=\\{r_{1},r_{2},\ldots,r_{m_{X}}\\}\subset{\mathbb{N}}^{{m_{X}}}$ with
the conditions that mutually coprime, $(r_{1},\ldots,r_{m_{X}})=1$, $r_{1}=r$,
$r_{2}=s$, and $r_{i}<r_{j}$ for $2<i<j$, is a minimal set of generators for
$H_{X}$. Here the cardinality of the generator $M_{X}$ of $H_{X}$ is
$m_{X}(<r)$.

By putting (3.8) into (3.6), we obtain (3.2) if it is irreducible. If it is
reducible, $f_{X}(x,y_{s})$ is decomposed to polynomials whose degree is less
than $r$ with respect to $y_{s}$. However the relation $(r,s)=1$ shows that
there does not exist such monic polynomials due to the order of the
singularity at $\infty$.

Further the order of the singularity of $A_{i}y_{s}^{r-i}$ is given by
$s(r-i)+\mathrm{deg}_{x}A_{i}$. The cases satisfying that
$s(r-i)+\mathrm{deg}_{x}A_{i}=s(r-j)+\mathrm{deg}_{x}A_{j}$ mean that $i=j$ or
$(i,j)=(0,r),(r,0)$ due to $(r,s)=1$. Hence $\mathrm{deg}_{x}A_{r}=s$. For
$i=1,\ldots,r-1$, $r(m-i)+\mathrm{deg}_{x}A_{i}<rs$ leads that
$+\mathrm{deg}_{x}A_{i}<si/r$.

######  Remark 3.2.

Let us call the curve in Proposition 3.1 a _Weierstrass curve_ or a _W-curve_
in this paper. The Weierstrass canonical form characterizes the W-curve, which
has only one infinity point $\infty$. The infinity point $\infty$ is a
Weierstrass point if
$H_{X}^{\mathfrak{c}}=H^{\mathfrak{c}}(X,\infty)=\\{N^{\mathfrak{c}}(i)\\}$
differs from $\\{1,2,\cdots,g\\}$. Since every compact Riemann surface of the
genus, $g(>1)$, has a Weierstrass point whose Weierstrass gap sequence with
genus $g$ [ACGH, FK], it characterizes the behavior of the meromorphic
functions at the point, and thus there is a W-curve which is bi-rationally
equivalent to the compact Riemann surface.

Further Proposition 3.1 is also applicable to a pointed compact Riemann
surface $(Y,P)$ of genus $g$ whose point $P$ is an ordinary point rather than
the Weierstrass point; its Weierstrass gap sequence at $P$ is
$H^{\mathfrak{c}}(Y,P)=\\{1,2,\ldots,g\\}$. Even for the case, we find the
Weierstrass canonical form $f_{X}$ and the W-curve $X$ with
$H_{X}^{\mathfrak{c}}=\\{1,2,\ldots,g\\}$ which is bi-rational to $Y$.

######  Remark 3.3.

Let $R_{X^{\circ}}^{\circ}:={\mathbb{C}}[x,y]/(f_{X}(x,y))$ for (3.1) and its
normalized ring be $R_{X}^{\circ}$ if
$X^{\circ}:=\mathrm{Spec}\,R_{X^{\circ}}^{\circ}$ is singular. $R_{X}^{\circ}$
is the coordinate ring of the affine part of $X\setminus\\{\infty\\}$ and we
identify $R_{X}^{\circ}$ with
$R_{X}={\mathbf{H}}^{0}(X,{\mathcal{O}}_{X}(*\infty))$. Then the quotient
field ${\mathbb{C}}(X):={\mathcal{Q}}(R_{X})$ of $R_{X}$ is considered as an
algebraic function field on $X$ over ${\mathbb{C}}$.

By introducing
$R_{\mathbb{P}}:={\mathbf{H}}^{0}({\mathbb{P}},{\mathcal{O}}_{\mathbb{P}}(*\infty))={\mathbb{C}}[x]$
and its quotient field ${\mathbb{C}}(x):={\mathcal{Q}}(R_{\mathbb{P}})$,
${\mathcal{Q}}(R_{X})$ is considered a finite extension of
${\mathcal{Q}}(R_{\mathbb{P}})$. We regard $R_{X}$ as a finite extended ring
of $R_{\mathbb{P}}$ of rank $r$, e.g.,
$R_{X^{\circ}}^{\circ}=R_{\mathbb{P}}[y]/(f_{X}(x,y))$ as mentioned in
Subsection 3.4 [Ku2]. Further as we will mention in Subsection 3.3,
${\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{i}}$ in the proof of
Proposition 3.1 is the standard basis of $R_{X}$ as an
$R_{\mathbb{P}}$-module, and the matrix in (3.7) are naturally interpreted as
the $R_{\mathbb{P}}$-module.

For the local ring $R_{X,P}$ of $R_{X}$ at $P\in X$, we have the ring
homomorphism, $\varphi_{P}:R_{X}\to R_{X,P}$. We note that $R_{X,\infty}$
plays crucial roles in the Weierstrass canonical form. We let the minimal
generator $M_{X}=\\{r_{1},r_{2},\ldots,r_{m_{X}}\\}$ of the numerical
semigroup $H_{X}=H(X,\infty)$ appearing in the proof of Proposition 3.1. The
Weierstrass curve admits a local cyclic
${\mathfrak{C}}_{r}={\mathbb{Z}}/r{\mathbb{Z}}$-action at $\infty$ c.f.,
Subsection 3.2. The genus of $X$ is denoted by $g_{X}$, briefly $g$ and the
conductor of $H_{X}$ is denoted by $c_{X}:=c_{H_{X}}$; the Frobenius number
$c_{X}-1$ is the maximal gap in $H_{X}$. We let
${\overline{H}}_{X}^{\mathfrak{c}}:={\mathbb{Z}}\setminus H_{X}$.

####  3.1.1. Projection from $X$ to ${\mathbb{P}}$

There is the natural projection,

(3.9)
$\varpi_{r}:X\to{\mathbb{P}},\qquad(\varpi_{r}(x,y_{r_{2}},\ldots,y_{r_{m_{X}}})=x=y_{r})$

such that $\varpi_{r}(\infty)=\infty\in{\mathbb{P}}$.

Let $\\{y_{\bullet}\\}:=\\{y_{s}=y_{r_{2}},y_{r_{3}},\ldots,y_{r_{m_{X}}}\\}$
and
${\mathbb{C}}[x,y_{\bullet}]:={\mathbb{C}}[x,y_{s}=y_{r_{2}},y_{r_{3}},\ldots,y_{r_{m_{X}}}]$.

####  3.1.2. Symmetric and non-symmetric Weierstrass curves (W-curves)

We also investigate the W-curves whose Weierstrass semigroups $H_{X}$ are
symmetric and non-symmetric, which are called _symmetric Weierstrass curve_ or
_symmetric W-curve_ , and _non-symmetric Weierstrass curve_ or _non-symmetric
W-curve_ respectively in this paper.

###  3.2. The monomial curves and W-curves

This subsection shows the monomial curves and their relation to W-curves based
on [KMP2, KMP3, KM].

For a given W-curve $X$ with the Weierstrass semigroup $H=H_{X}$, and its
generator $M_{X}=\\{r=r_{1},r_{2},\ldots,r_{m_{X}}\\}$, the behavior of
singularities of the elements in $R_{X}$ at $\infty$ is described by a
monomial curve $X_{H}^{Z}$. For the numerical semigroup $H=\langle
M_{X}\rangle$, the _numerical semigroup ring_ $R_{H}$ is defined as
$R_{H}:={\mathbb{C}}[z^{r_{1}},z^{r_{2}},\cdots,z^{r_{m_{X}}}]$, and a
polynomial ring
${\mathbb{C}}[Z]:={\mathbb{C}}[Z_{r_{1}},Z_{r_{2}},\cdots,Z_{r_{m_{X}}}]$.

######  Proposition 3.4.

For the ${\mathbb{C}}$-algebra homomorphism
${\widetilde{\varphi}}^{Z}_{H}:{\mathbb{C}}[Z]\to R_{H}$, the kernel of
${\widetilde{\varphi}}^{Z}_{H}$ is generated by certain binomials
$f^{H}_{i}\in{\mathbb{C}}[Z]$ $(i=1,2,\ldots,k_{X})$ for a positive integer
$k_{X}$, $\displaystyle{\left(m_{X}-1\leq k_{X}<\infty\right)}$, i.e.,
$\ker{\widetilde{\varphi}}^{Z}_{H}=(f^{H}_{1},f^{H}_{2},\ldots,f^{H}_{k_{X}})$,
and

$R_{H}\simeq{\mathbb{C}}[Z]/\ker{\widetilde{\varphi}}^{Z}_{H}=:R^{Z}_{H}.$

###### Proof.

This follows from a result of Herzog’s [H]. There are the multiplicative group
actions (${\mathbb{G}}_{\mathfrak{m}}$-actions) on $Z_{r_{i}}$’s, whereas
$R_{H}$ is invariant for the action. It means that the number of generators of
$\ker{\widetilde{\varphi}}^{Z}_{H}$ is determined, i.e., $k_{X}$. The relation
in the $\ker{\widetilde{\varphi}}^{Z}_{H}$ is reduced to a binary one.

We call $R^{Z}_{H}={\mathbb{C}}[Z]/\ker{\widetilde{\varphi}}^{Z}_{H}$ a
_monomial ring_. Sending $Z_{r}$ to $1/x$ and $Z_{r_{i}}$ to $1/y_{r_{i}}$,
the monomial ring $R^{Z}_{H}$ determines the structure of gap sequence of $X$
[H, P]. Bresinsky showed that $k_{X}$ can be any finitely large number if
$m_{X}>3$ [Br].

Let $X_{H}:=\mathrm{Spec}\,R_{H}^{Z}$,which we call a _monomial curve_. We
also define the ring isomorphism on $R_{H}^{Z}$ induced from
${\widetilde{\varphi}}^{Z}_{H}$, which is denoted by $\varphi^{Z}_{H}$,

(3.10)
$\varphi^{Z}_{H}:{\mathbb{C}}[Z]/\ker{\widetilde{\varphi}}^{Z}_{H}=R^{Z}_{H}\to
R_{H}.$

Further, we let
$\\{Z_{\bullet}\\}:=\\{Z_{r_{2}},Z_{r_{3}},\ldots,Z_{r_{m_{X}}}\\}$, and
${\mathbb{C}}[Z_{r},Z_{\bullet}]:={\mathbb{C}}[Z_{r},Z_{r_{2}},Z_{r_{3}},\ldots,Z_{r_{m_{X}}}]$.
A monomial curve is an irreducible affine curve with
${\mathbb{G}}_{\mathfrak{m}}$-action, where ${\mathbb{G}}_{\mathfrak{m}}$ is
the multiplicative group of the complex numbers; $Z_{a}\mapsto g^{a}Z_{a}$ for
$g\in{\mathbb{G}}_{\mathfrak{m}}$, and it induces the action on the monomial
ring $R_{H}^{Z}$.

The following action of the cyclic group of order $r$ plays a crucial role in
this paper.

######  Lemma 3.5.

The cyclic group ${\mathfrak{C}}_{r}$ of order $r$ acts on the monomial ring
$R_{H}^{Z}$; the action of the generator
${\widehat{\zeta}}_{r}\in{\mathfrak{C}}_{r}$ on $Z_{a}$ is defined by sending
$Z_{a}$ to $\zeta_{r}^{a}Z_{a}$, where $\zeta_{r}$ is a primitive $r$-th root
of unity. By letting ${\mathfrak{r}}_{i}^{*}:=(r,r_{i})$,
${\mathfrak{r}}_{i}:=r/{\mathfrak{r}}_{i}*$, and
${\overline{\mathfrak{r}}}_{i}:=r_{i}/{\mathfrak{r}}_{i}^{*}$, the orbit of
$Z_{r_{i}}$ forms ${\mathfrak{C}}_{{\mathfrak{r}}_{i}}$; especially for the
case that $(r,r_{i})=1$, it recovers ${\mathfrak{C}}_{r}$.

Thus in $R_{H}^{Z}$, we have the following identities:

(3.11) $f_{H}^{(j)}(Z_{r},Z_{r_{j}})=0,\quad
f_{H}^{(j)}:=Z_{r_{j}}^{{\mathfrak{r}}_{j}}-Z_{r}^{{\overline{\mathfrak{r}}}_{j}},\quad(j=2,\ldots,m_{X}).$

For example, the case $M_{X}=\\{3,7,8\\}$ provides these elements
$\\{f^{H}_{1},f^{H}_{2},f^{H}_{3}\\}$ are given by the $2\times 2$ minors of
$\displaystyle{\left|\begin{matrix}Z_{3}^{2}&Z_{7}&Z_{8}\\\
Z_{7}&Z_{8}&Z_{3}^{3}\\\ \end{matrix}\right|}$. There is a cyclic group
${\mathfrak{C}}_{3}:=\\{\zeta_{3}^{a}\ |\ a=0,1,2\\}$ action on $R_{H}$ as a
${\mathbb{G}}_{\mathfrak{m}}$ action. Due to the relation, there are other
possibilities which are given by the $2\times 2$ minors of
$\displaystyle{\left|\begin{matrix}Z_{3}^{2}&\zeta_{3}^{a}Z_{7}&\zeta_{3}^{2a}Z_{8}\\\
\zeta_{3}^{a}Z_{7}&\zeta_{3}^{2a}Z_{8}&Z_{3}^{3}\\\ \end{matrix}\right|}$ for
$a=0,1,2$. It means that $f^{H}_{i}$ is unique up to the
${\mathbb{G}}_{\mathfrak{m}}$ action. On the other hand,
$f_{H}^{(2)}=Z_{7}^{3}-Z_{3}^{7}$ and $f_{H}^{(3)}=Z_{8}^{3}-Z_{3}^{8}$ for
(3.11).

There are non-negative integers $h_{j}^{(i\pm)}$ such that

(3.12)
$f^{H}_{i}=\left(\prod_{j=2}Z_{r_{j}}^{h_{j}^{(i+)}}\right)-\left(\prod_{j=1}Z_{r_{j}}^{h_{j}^{(i-)}}\right),$

where, in the first term, $Z_{r_{1}}$ does not exist because
$(r_{1},r_{2},\ldots,r_{m_{X}})=1$.

Corresponding to the standard basis of $H_{X}$ in Definition 2.7, we find the
monic monomial ${\mathfrak{Z}}_{{\mathfrak{e}}_{i}}\in R_{H}^{Z}$ such that
$\varphi^{Z}_{H}({\mathfrak{Z}}_{{\mathfrak{e}}_{i}})=z^{{\mathfrak{e}}_{i}}$,
and the standard basis $\\{{\mathfrak{Z}}_{{\mathfrak{e}}_{i}}\
|i\in{\mathbb{Z}}_{r}\\}$; ${\mathfrak{Z}}_{{\mathfrak{e}}_{0}}=1$.

######  Lemma 3.6.

The ${\mathbb{C}}[Z_{r}]$-module $R_{H}^{Z}$ is given by

$R_{H}^{Z}={\mathbb{C}}[Z_{r}]\oplus{\mathbb{C}}[Z_{r}]{\mathfrak{Z}}_{{\mathfrak{e}}_{1}}\oplus\cdots\oplus{\mathbb{C}}[Z_{r}]{\mathfrak{Z}}_{{\mathfrak{e}}_{r-1}},$

and thus
${\mathfrak{Z}}_{H}:=\\{{\mathfrak{Z}}_{{\mathfrak{e}}_{0}}=1,{\mathfrak{Z}}_{{\mathfrak{e}}_{1}},\ldots,{\mathfrak{Z}}_{{\mathfrak{e}}_{r-1}}\\}$
is the basis of the ${\mathbb{C}}[Z_{r}]$-module $R_{H}^{Z}$. Then there is a
monomial $b_{ijk}\in{\mathbb{C}}[Z_{r}]$ such that

${\mathfrak{Z}}_{{\mathfrak{e}}_{i}}{\mathfrak{Z}}_{{\mathfrak{e}}_{j}}=\sum_{k\in{\mathbb{Z}}_{r}}b_{ijk}{\mathfrak{Z}}_{{\mathfrak{e}}_{k}}.$

Further, there are elements ${\mathfrak{Z}}\in R_{H}^{Z}$ and
${\mathfrak{Z}}_{{\mathfrak{e}}_{i}}^{*}\in R_{H}^{Z}$ satisfying

${\mathfrak{Z}}_{{\mathfrak{e}}_{i}}^{*}{\mathfrak{Z}}_{{\mathfrak{e}}_{i}}={\mathfrak{Z}},$

and $\\{Z_{r_{j}}\ |j=2,\ldots,m_{X}\\}\subset{\mathfrak{Z}}_{H}$.

Moreover, the cyclic group of order $r$ acts on these elements; the action of
the generator ${\widehat{\zeta}}_{r}\in{\mathfrak{C}}_{r}$ on
${\mathfrak{Z}}_{{\mathfrak{e}}_{i}}$ $(i=0,\ldots,r-1)$ is defined by sending
${\mathfrak{Z}}_{{\mathfrak{e}}_{i}}$ to
$\zeta_{r}^{{\mathfrak{e}}_{i}}{\mathfrak{Z}}_{{\mathfrak{e}}_{i}}$, where
$\zeta_{r}$ is a primitive $r$-th root of unity; For $f,g\in R_{H}^{Z}$, by
letting
${\widehat{\zeta}}_{r}(fg)={\widehat{\zeta}}_{r}(f){\widehat{\zeta}}_{r}(g)$,
${\widehat{\zeta}}_{r}({\mathfrak{Z}}_{{\mathfrak{e}}_{1}}{\mathfrak{Z}}_{{\mathfrak{e}}_{2}}\cdots{\mathfrak{Z}}_{{\mathfrak{e}}_{r-1}})={\mathfrak{Z}}_{{\mathfrak{e}}_{1}}{\mathfrak{Z}}_{{\mathfrak{e}}_{2}}\cdots{\mathfrak{Z}}_{{\mathfrak{e}}_{r-1}}$.

###### Proof.

They are obtained from Definition 2.7 and Lemma 2.8. From the definition of
$M_{X}$, every $Z_{r_{j}}$ $(j=2,\ldots,m_{X})$ belongs to
${\mathfrak{Z}}_{H}$. The group action is obvious from Lemma 3.5.

To construct our curve $X$ from $R_{H}$ or $\mathrm{Spec}\,R_{H}$, we could
follow Pinkham’s strategy [P] with an irreducible curve singularity with
${\mathbb{G}}_{\mathfrak{m}}$ action, though we will not mention it in this
paper. Pinkham’s investigations provide the following proposition [P]:

######  Proposition 3.7.

For a given W-curve $X$ and its associated monomial ring,
$R_{H}^{Z}={\mathbb{C}}[Z]/(f^{H}_{1},\break f^{H}_{2},\ldots,f^{H}_{k_{X}})$,
there are a surjective ring-homomorphism [ADGS, p.80]

$\varphi^{X}_{H}:R_{X}\to R_{H}^{Z}$

such that $R_{X}/{\mathfrak{m}}_{{\mathbb{A}}}$ is isomorphic to $R_{H}^{Z}$,
where ${\mathfrak{m}}_{{\mathbb{A}}}$ is the maximal ideal $(\lambda_{ij})$ in
coefficient ring ${\mathbb{C}}[\lambda_{ij}]$, and
$\varphi^{X}_{H}(y_{r_{i}})=Z_{r_{i}}$, and a polynomial
$\\{f^{X}_{i}\\}_{i=1,\ldots,k_{X}}\in{\mathbb{C}}[x,y_{\bullet}]$ satisfying

  1. (1)

$\varphi^{X}_{H}(f^{X}_{i})(=\varphi^{X}_{H}(f^{X}_{i}$ modulo
${\mathfrak{m}}_{{\mathbb{A}}}))=f^{H}_{i}$ for $(i=1,\ldots,k_{X})$,

  2. (2)

the affine part of $R_{X}$ is given by
$R_{X}={\mathbb{C}}[x,y_{\bullet}]/(f^{X}_{1},f^{X}_{2},\ldots,f^{X}_{k_{X}})$,
and

  3. (3)

the rank of the matrix $\displaystyle{\left(\frac{\partial f^{X}_{i}}{\partial
y_{r_{j}}}\right)_{i=1,2,\ldots,k_{X},j=1,2,\ldots,m_{X}}}$ is $m_{X}-1$ for
every point $P$ in $X$.

###### Proof.

As we showed in the proof of Proposition 3.1, $R_{X}$ is a quotient ring of
${\mathbb{C}}[x,y_{\bullet}]$. Since $\varphi_{H}^{X}$ must be a surjective,
there is a prime ideal
$(f^{X}_{1},f^{X}_{2},\ldots,f^{X}_{k_{X}})\subset{\mathbb{C}}[x,y_{\bullet}]$
generated by
$\\{f^{X}_{i}\\}_{i=1,\ldots,k_{X}}\subset{\mathbb{C}}[x,y_{\bullet}]$ such
that satisfying (1). Thus we prove (2). Noting $k_{X}\geq m_{X}$, Nagata’s
Jacobi criterion [M, Theorem 30.10],shows (3).

######  Definition 3.8.

  1. (1)

Recalling Lemma 2.9, we define _arithmetic local parameter_ at $\infty$ by [O]

$t=\frac{x^{i_{r}}}{y^{i_{s}}}.$

  2. (2)

The degree at ${\mathcal{Q}}(R_{X,\infty})$ as the order of the singularity
with respect to $t$ is naturally defined by

(3.13)
${\mathrm{wt}}=\mathrm{deg}_{\infty}:{\mathcal{Q}}(R_{X})\to{\mathbb{Z}},$

which is called _Sato-Weierstrass weight_ [W1].

  3. (3)

In the ring of the formal power series
${\mathbb{C}}[[t_{1},\ldots,t_{\ell}]]$, we define the symbols
$d_{>n}(t_{1},\ldots,t_{\ell})$ and $d_{\geq n}(t_{1},\ldots,t_{\ell})$, which
express that they belong to the ideals

$d_{>n}(t_{1},\ldots,t_{\ell})\in\\{\sum
a_{i_{1},\ldots,i_{\ell}}t_{1}^{i_{1}}\cdots t_{\ell}^{i_{\ell}}\ |\
a_{i_{1},\ldots,i_{\ell}}=0\mbox{ for }i_{1}+\cdots+i_{\ell}\leq n\\},$
$d_{\geq n}(t_{1},\ldots,t_{\ell})\in\\{\sum
a_{i_{1},\ldots,i_{\ell}}t_{1}^{i_{1}}\cdots t_{\ell}^{i_{\ell}}\ |\
a_{i_{1},\ldots,i_{\ell}}=0\mbox{ for }i_{1}+\cdots+i_{\ell}<n\\}.$

The weight of $y_{r_{i}}$ is given by

${\mathrm{wt}}(y_{r_{i}})=-r_{i},\quad(i=1,2,\ldots,{m_{X}}),\qquad
y_{r_{i}}=\frac{1}{t^{r_{i}}}(1+d_{>0}(t)).$

Since $R_{X}$ or $R_{X}^{\circ}$ is given by a quotient ring of
${\mathbb{C}}[x,y_{r_{2}},y_{r_{3}},\ldots,y_{r_{m_{X}}}]$ divided by certain
relations, we have the decomposition of $R_{X}$ as a ${\mathbb{C}}$-vector
space,

(3.14) $R_{X}=\bigoplus_{i=0}{\mathbb{C}}\phi_{i},$

where $\phi_{i}$ is a monomial in $R_{X}$ satisfying the inequalities
$-{\mathrm{wt}}\,\phi_{i}<-{\mathrm{wt}}\,\phi_{j}$ for $i<j$, i.e.,
$\phi_{0}=1$, $\phi_{1}=x$, $\ldots$.

Further by assigning a certain weight on each coefficient $\lambda_{i,j}$ in
(3.1) so that (3.1) is a homogeneous equation of weight $rs$, we also define
another weight,

(3.15) ${\mathrm{wt}}_{\lambda}:R_{X}\to{\mathbb{Z}}.$

######  Definition 3.9.

We define $S_{X}:=\\{\phi_{i}\ |\ i=0,1,2\ldots\\}$ by the basis of $R_{X}$ as
in (3.14).

Then $N(i)=-{\mathrm{wt}}(\phi_{i})$, for $\\{N(i)\ |\
i\in{\mathbb{N}}_{0}\\}=H_{X}$.

######  Lemma 3.10.

Let $t$ be the arithmetic local parameter at $\infty$ of $R_{X}$.

  1. (1)

By the isomorphism
$\displaystyle{\varphi_{\mathrm{inv}}:z\mapsto\frac{1}{t}}$,
$\varphi_{\mathrm{inv}}(R_{H})(\cong R_{H}^{Z})$ is a subring of
$\displaystyle{{\mathbb{C}}\left[\frac{1}{t}\right]}$; for $g(z)\in R_{H}$,
$\displaystyle{g\left(\frac{1}{t}\right)\in{\mathbb{C}}\left[\frac{1}{t}\right]}$.

  2. (2)

There is a surjection of ring $\varphi_{\infty}:R_{X}\to R_{H}(\cong
R_{H}^{Z})$; for $f\in R_{X}$, there is $g(z)\in R_{H}$ such that

$(f)_{\infty}=g\left(\frac{1}{t}\right)(1+d_{>0}(t))\in R_{X,\infty},$

where $(f)_{\infty}$ means the germ at $\infty$ or $(f)_{\infty}\in
R_{X,\infty}$ via $\varphi_{H}^{X}$ in Proposition 3.7. It induces the
surjection $R_{X,\infty}\to R_{H}(\cong R_{H}^{Z})$.

###### Proof.

By letting $g=\varphi_{H}^{Z}\circ\varphi_{H}^{X}(f)$, the existence of $g$ is
obvious.

###  3.3. $R_{\mathbb{P}}$-module $R_{X}$

$R_{X}$ is an $R_{\mathbb{P}}$-module, and its affine part is given by the
quotient ring of $R_{\mathbb{P}}[y_{\bullet}]$.

We recall Definition 2.7 and Lemma 3.6, and apply them to W-curves:

######  Proposition 3.11.

For ${\mathfrak{e}}_{i}\in{\mathfrak{E}}_{H_{X}}$, we let
${\mathfrak{y}}_{{\mathfrak{e}}_{i}}$ be the monic monomial in $R_{X}$ whose
weight is $-{\mathfrak{e}}_{i}$, $({\mathfrak{y}}_{{\mathfrak{e}}_{0}}=1)$
satisfying

$R_{X}=R_{\mathbb{P}}\oplus\bigoplus_{i=1}^{r-1}R_{\mathbb{P}}{\mathfrak{y}}_{{\mathfrak{e}}_{i}}=\bigoplus_{i=0}^{r-1}R_{\mathbb{P}}{\mathfrak{y}}_{{\mathfrak{e}}_{i}}=\langle{\mathfrak{y}}_{{\mathfrak{e}}_{0}},{\mathfrak{y}}_{{\mathfrak{e}}_{1}},\ldots,{\mathfrak{y}}_{{\mathfrak{e}}_{r-1}}\rangle_{R_{\mathbb{P}}}$

with the relations,

(3.16)
${\mathfrak{y}}_{{\mathfrak{e}}_{i}}{\mathfrak{y}}_{{\mathfrak{e}}_{j}}=\sum_{k=0}^{r-1}{\mathfrak{a}}_{ijk}{\mathfrak{y}}_{{\mathfrak{e}}_{k}},$

where ${\mathfrak{a}}_{ijk}\in R_{\mathbb{P}}$,
${\mathfrak{a}}_{ijk}={\mathfrak{a}}_{jik}$, especially
${\mathfrak{a}}_{0jk}={\mathfrak{a}}_{j0k}=\delta_{jk}$.

Further we let
${\widehat{H}}_{{\mathfrak{e}}}:=\displaystyle{\left\\{-{\mathrm{wt}}(f)\ |\
f\in\sum_{{\mathfrak{e}}_{i}\in{\mathfrak{E}}_{H}^{\times}}R_{\mathbb{P}}{\mathfrak{y}}_{{\mathfrak{e}}_{i}}\right\\}}$
and then ${\widehat{H}}_{{\mathfrak{e}}}\subset H_{X}$.

###### Proof.

The generating formula is directly obtained from (3.4) noting
${\mathfrak{y}}_{{\mathfrak{e}}_{0}}=1$ and Lemma 3.6.
${\widehat{H}}_{{\mathfrak{e}}}\subset H_{X}$ is obvious.

The set
${\mathfrak{Y}}_{X}:=\\{{\mathfrak{y}}_{0},{\mathfrak{y}}_{{\mathfrak{e}}_{1}},\ldots,{\mathfrak{y}}_{{\mathfrak{e}}_{r-1}}\\}$
is called the standard basis of $R_{X}$ as an $R_{\mathbb{P}}$-module, which
is essentially the same as
${\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{i}}$ in the proof of
Proposition 3.1;
${\mathfrak{Y}}_{X}=\\{{\widetilde{\mathfrak{y}}}_{{\overline{\mathfrak{e}}}_{i}}\\}\cup\\{1,y_{s}\\}$,
and thus we let $\ell_{s}$ be
$y_{s}={\mathfrak{y}}_{{\mathfrak{e}}_{\ell_{s}}}$.
${\mathfrak{a}}_{\ell_{s},k,\ell}$ in (3.16) corresponds to $A_{k\ell}$ in the
proof of Proposition 3.1.

The following corollary is obvious:

######  Corollary 3.12.

In ${\mathbb{C}}[x,y_{\bullet}]$, the ideal generated by (3.16) is a sub-ideal
of ${\mathcal{I}}_{R_{X}}:=(f_{1}^{X},f_{2}^{X},\ldots,$ $f_{k_{X}}^{X})$ in
Proposition 3.7.

As we regard an element in $R_{X}$ as an element in $R_{\mathbb{P}}$-module,
we introduce a polynomial in $R_{X}$ as an $R_{\mathbb{P}}$-monomial such that
it is given by $\delta(x)\prod_{i=2}^{m_{X}}y_{r_{i}}^{h_{i}}$ for certain
non-negative integers $h_{i}$ and $\delta(x)\in R_{\mathbb{P}}$, e.g.,
$(x^{2}-ax-b)y_{r_{2}}^{2}y_{r_{3}}$. Since we also define the weight,
${\mathrm{wt}}_{\lambda}$, on the $R_{\mathbb{P}}$-monomials, we can consider
homogeneous polynomials as elements in the $R_{\mathbb{P}}$-module.

######  Remark 3.13.

Corresponding to (3.12), the relation $f^{X}_{i}$ in Proposition 3.7 is
decomposed as

(3.17)
$f^{X}_{i}=\left(\prod_{j=2}y_{r_{j}}^{h_{j}^{(i+)}}\right)-\delta_{f^{X}_{i}}(x)\left(\prod_{j=2}y_{r_{j}}^{h_{j}^{(i-)}}\right)+\mbox{
lower weight terms with respect to }-{\mathrm{wt}}$

as an $R_{\mathbb{P}}$-module, which is relevant to (3.16). Here
$\delta_{f^{X}_{i}}(x)$ is an element of $R_{\mathbb{P}}$ whose degree is
$h_{1}^{(i+)}$ in (3.12). The first and the second terms are homogeneous
polynomials as the Sato-Weierstrass weight ${\mathrm{wt}}$.

####  3.3.1. Embedding of $X$ into ${\mathbb{P}}^{2(m_{X}-1)}$

The projection from $X$ to ${\mathbb{P}}$ in Subsubsection 3.1.1 with the
$R_{\mathbb{P}}$-module structure induces an embedding of $X$ into
${\mathbb{P}}^{2(m_{X}-1)}$ as follows. Besides $\\{f^{X}_{i}\ |\
i=1,\ldots,k_{X}\\}$, we introduce a subset of polynomials $\\{f_{X}^{(i)}\ |\
i=2,\cdots,m_{X}\\}$ of ${\mathbb{C}}[x,y_{\bullet}]$, which are
$R_{X}$-analog of $f_{H}^{(i)}$ in Lemma 3.5 such that $f_{X}^{(i)}=0$ an
identity in $R_{X}$.

######  Proposition 3.14.

Let $X$ be the W-curve in Proposition 3.1 whose the affine ring
$R_{X}={\mathbb{C}}[x,y_{\bullet}]/(f^{X}_{1},f^{X}_{2},\ldots,f^{X}_{k_{X}})$.
There are polynomials $A^{(j)}_{i}\in{\mathbb{A}}_{\mathbb{C}}[x]$
$(i=2,3,\ldots,{\mathfrak{r}}_{j}=r/(r,r_{j}),j=2,\cdots,m_{X})$ satisfying
$\displaystyle{A^{(j)}_{i}=\sum_{k=0}^{\lfloor
i{\overline{\mathfrak{r}}}_{j}/{\mathfrak{r}}_{j}\rfloor}\lambda^{(j)}_{i,k}x^{k}}$,
where ${\overline{\mathfrak{r}}}_{j}=r_{j}/(r,r_{j})$,
$\lambda^{(j)}_{i,k}\in{\mathbb{A}}_{\mathbb{C}}$,
$\lambda^{(j)}_{r,r_{j}}=1$, and an irreducible polynomial,

(3.18)
$f^{(j)}_{X}(x,y_{r_{j}}):=y_{r_{j}}^{{\mathfrak{r}}_{j}}+A^{(j)}_{1}y_{r_{j}}^{{\mathfrak{r}}_{j}-1}+A^{(j)}_{2}y_{r_{j}}^{{\mathfrak{r}}_{j}-2}+\cdots+A^{(j)}_{{\mathfrak{r}}_{j}-1}y_{r_{j}}+A^{(j)}_{{\mathfrak{r}}_{j}},$

in ${\mathbb{C}}[x,y_{r_{j}}]$, especially $f_{X}^{(2)}=f_{X}$ in (3.2), so
that $f^{(j)}_{X}$ satisfies the identity $f^{(j)}_{X}(x,y_{r_{j}})=0$ in
$R_{X}$.

###### Proof.

The $m_{X}=2$ case is trivial. Let us consider $m_{X}>2$ case. Then
$\mathrm{Spec}\,({\mathbb{C}}[x,y]$ $/(f_{X}(x,y)))$ is singular and the
commutative ring ${\mathbb{C}}[x,y]/(f_{X}(x,y))$ is not normal. We normalize
it to obtain $R_{X}$, in which every element in ${\mathcal{Q}}(R_{X})$ is
expressed by a monic equation with the coefficients in $R_{X}$. It means that
$y_{r_{j}}$ satisfies a certain relation
$y_{r_{j}}^{n}+b_{1}y_{r_{j}}^{n-1}+\cdots+b_{n-1}y_{r_{j}}+b_{n}=0$ for
certain positive integer $n$ and $b_{i}\in R_{X}$. We show that when
$n=r_{j}$, it is irreducible and $b_{j}\in R_{\mathbb{P}}$ as follows.

We remark that $y_{r_{j}}$ is equal to an element of the standard basis
${\mathfrak{Y}}_{X}$ from the definition of $y_{r_{j}}$ in the proof of
Proposition 3.1; we let ${\widetilde{j}}$ such that
$y_{r_{j}}={\mathfrak{y}}_{{\mathfrak{e}}_{{\widetilde{j}}}}$. We apply the
investigation of $A_{k\ell}$ for
${\mathfrak{y}}_{{\mathfrak{e}}_{\ell_{s}}}=y_{s}$ in the proof of Proposition
3.1 and ${\mathfrak{a}}_{\ell_{s}ik}$ in (3.16) to this
${\mathfrak{y}}_{{\mathfrak{e}}_{{\widetilde{j}}}}=y_{r_{j}}$ case. We
introduce

(3.19)
${\widetilde{A}}^{(j)}_{i,k}:={\mathfrak{a}}_{{\widetilde{j}},i,k}-{\mathfrak{y}}_{{\mathfrak{e}}_{\widetilde{j}}}\delta_{i,k},\quad{\widetilde{b}}_{i}:=-{\mathfrak{a}}_{{\widetilde{j}},i,0}-{\mathfrak{a}}_{{\widetilde{j}},i,\ell_{\widetilde{j}}}{\mathfrak{y}}_{{\mathfrak{e}}_{{\widetilde{j}}}},\quad(i,k\in{\mathbb{Z}}_{r}^{\times}),$

and then we consider the
${\mathfrak{y}}_{{\mathfrak{e}}_{{\widetilde{j}}}}=y_{r_{j}}$ action on
${\mathfrak{Y}}_{X}$ in (3.16), which is described by

(3.20)
$\sum_{\ell\in{\mathbb{Z}}_{r}\setminus\\{{\widetilde{j}}\\}}{\widetilde{A}}^{(j)}_{k\ell}{\mathfrak{y}}_{{\mathfrak{e}}_{\ell}}={\widetilde{b}}_{k},\quad(k\in{\mathbb{Z}}_{r}^{\times}\setminus\\{{\widetilde{j}}\\}),$
(3.21)
$\displaystyle{y_{r_{j}}^{2}={\overline{A}}_{{\widetilde{j}},0}+{\overline{A}}_{{\widetilde{j}}{\widetilde{j}}}y_{r_{j}}+\sum_{k\in{\mathbb{Z}}_{r}\setminus\\{{\widetilde{j}}\\}}{\overline{A}}_{{\widetilde{j}},k}{\mathfrak{y}}_{e_{k}}}.$

If $(r_{j},r)=1$, the matrix ${\widetilde{A}}^{(j)}_{k\ell}$ is regular and we
obtain (3.18) as in the proof of Proposition 3.1.

Thus we assume that $r_{j}$ and $r$ are not coprime. We recall
${\mathfrak{r}}_{j}^{*}=(r,r_{j})$,
${\mathfrak{r}}_{j}=r/{\mathfrak{r}}_{j}^{*}$ and
${\overline{\mathfrak{r}}}_{j}=r_{j}/{\mathfrak{r}}_{j}^{*}$.

If the determinant of the $(r-2)\times(r-2)$ matrix ${\widetilde{A}}^{(j)}$ is
not equal to zero, it is reduced to the above case. Then we obtain the formula
$f$ which is given by $y_{r_{j}}^{r}-x^{r_{j}}+\cdots$ and thus its image of
$\varphi_{H}^{X}$ is reduced to $Z_{r_{j}}^{r}-Z_{r}^{r_{j}}$, which is
decomposed
$(Z_{r_{j}}^{{\mathfrak{r}}_{j}}-Z_{r}^{{\overline{\mathfrak{r}}}_{j}})(Z_{r_{j}}^{{\mathfrak{r}}_{j}({\mathfrak{r}}_{j}^{*}-1)}+\cdots+Z_{r}^{{\overline{\mathfrak{r}}}_{j}({\mathfrak{r}}_{j}^{*}-1)})$.
If the formula $f$ is irreducible, the infinity point $\infty$ in $X$ must not
be unique, which contradicts the uniqueness of $\infty$ in W-curve as
mentioned in Remark 3.2. Thus the formula must be reduced to two formulae; one
of them must be the Sato-Weierstrass weight, $-rr_{j}/{\mathfrak{r}}_{j}^{*}$
and contain the terms
$y_{{\mathfrak{r}}_{j}}^{r}-x^{{\overline{\mathfrak{r}}}_{j}}$, which is equal
to $f_{X}^{(j)}$ (3.18). If the other formula $h_{X}^{(j)}$ is algebraic
independent to $f_{X}^{(j)}$, there are two proejctions,
$\pi_{r,r_{j}}:X\to\mathrm{Spec}\,({\mathbb{C}}[x,y_{r_{j}}]/(f_{X}^{(j)}))$
and
$\pi_{r,r_{j}}^{\prime}:X\to\mathrm{Spec}\,({\mathbb{C}}[x,y_{r_{j}}]/(h_{X}^{(j)}))$.
It is contradict with the uniqueness of the infinity $\infty$ at $X$ in
W-curves. Hence $f=(f_{X}^{(j)})^{{\mathfrak{r}}_{j}^{*}}$ up to constant
factor. Hence we have (3.18) and the identity $f_{X}^{(j)}=0$.

Hence we further assume that the matrix ${\widetilde{A}}^{(j)}$ is singular,
and its rank is $q(<r-2)$.

Now we consider the case that $q=0$. Then on the relation (3.21), we show that
${\overline{A}}_{{\widetilde{j}},k}$
($k\in{\mathbb{Z}}_{r}\setminus\\{{\widetilde{j}}\\}$) must vanish. If some of
${\overline{A}}_{{\widetilde{j}},k}\neq 0$, there is an element
${\widetilde{y}}\in\displaystyle{\bigoplus_{i\in{\mathbb{Z}}_{r}\setminus\\{{\widetilde{j}}\\}}R_{\mathbb{P}}{\mathfrak{y}}_{{\mathfrak{e}}_{i}}}$
which is expressed by a meromorphic function of $x$ and $y_{r_{j}}$ from the
relation. However, it means that there is a non-trivial relation in (3.20) and
thus the rank $q$ must not be zero. It contradicts the assumption. Hence
$y_{r_{j}}^{2}$ is expressed as
${\overline{A}}_{{\widetilde{j}},0}+{\overline{A}}_{{\widetilde{j}}{\widetilde{j}}}y_{r_{j}}$;
the order of ${\overline{A}}_{{\widetilde{j}},0}$ in $x$ is
${\overline{\mathfrak{r}}}_{j}$. Accordingly, if $q=0$, there exists a
$R_{\mathbb{P}}$-submodule
$\displaystyle{{\widetilde{R}}^{\prime}=R_{\mathbb{P}}\oplus
R_{\mathbb{P}}y_{r_{j}}}$, which forms a subring of $R_{X}$,
${\mathbb{C}}[x,y_{r_{j}}]/$
$(y_{r_{j}}^{2}-{\overline{A}}_{{\widetilde{j}},0}-{\overline{A}}_{{\widetilde{j}}{\widetilde{j}}}y_{r_{j}})\subset
R_{X}$. $X$ is a covering of a hyperelliptic (or elliptic) curve, and
${\mathrm{wt}}(x)$ is divisible by two and $2|r$. It is obvious that
$(y_{r_{j}}^{2}-{\overline{A}}_{{\widetilde{j}},0}-{\overline{A}}_{{\widetilde{j}}{\widetilde{j}}}y_{r_{j}})$
is irreducible as an $R_{\mathbb{P}}$-module since $X$ is not a covering of
decomposed curve, and we obtain (3.18).

Hence we let $q\neq 0$ or $0<q<r-2$. We introduce a subset
$I:=\\{n_{1},n_{2},\ldots,n_{q}\\}$ of ${\mathbb{Z}}_{r}^{\times}$ and a
submodule ${\widetilde{R}}_{X}$ of $R_{\mathbb{P}}$-module $R_{X}$ defined by

${\widetilde{R}}_{X}=R_{\mathbb{P}}\oplus
R_{\mathbb{P}}y_{r_{j}}\oplus\bigoplus_{i\in
I}R_{\mathbb{P}}{\mathfrak{y}}_{{\mathfrak{e}}_{i}}.$

We assume that ${\widetilde{R}}_{X}$ is closed for the $y_{r_{j}}$ action on
${\widetilde{R}}_{X}$, i.e.,

(3.22) $y_{r_{j}}{\widetilde{R}}_{X}\subset{\widetilde{R}}_{X}.$

The assumption enables us to find the regular submatrix
${\widetilde{A}}^{I}:=({\widetilde{A}}^{(j)}_{i,k})_{i,k\in I}$ of
${\widetilde{A}}^{(j)}$ satisfying

(3.23) $\sum_{k\in
I}{\widetilde{A}}^{(j)}_{ik}{\mathfrak{y}}_{{\mathfrak{e}}_{k}}={\widetilde{b}}_{i},\quad
i\in I.$

We let ${\overline{I}}:=I\cup\\{0,r_{j}\\}$, and
${\mathfrak{Y}}_{X}^{\overline{I}}:=\\{{\mathfrak{y}}_{{\mathfrak{e}}_{i}}\ |\
i\in{\overline{I}}\\}$, of ${\mathfrak{Y}}_{X}$.

By considering the image of $\varphi_{H}^{Z}\circ\varphi_{H}^{X}$ of (3.23),
the weight $-{\mathrm{wt}}$ of each component in ${\widetilde{A}}^{I}$
obviously leads the fact that there is a sub-monoid $H^{\prime}:=\langle
r,r_{j},{\mathfrak{e}}_{n_{1}},\ldots,{\mathfrak{e}}_{n_{q}}\rangle$ such that
$\langle r,r_{j}\rangle\subset H^{\prime}\subset H_{X}$, and the set
${\overline{I}}$ is characterized by

(3.24) ${\overline{I}}=\\{\ell\in{\mathbb{Z}}_{r}^{\times}\ |\
{\mathfrak{e}}_{\ell}\in
H^{\prime}\\}\neq\varnothing,\quad\mbox{and}\quad\sum_{k\in
I}[{\mathfrak{e}}_{k}]_{r}=H^{\prime}.$

Then we find an expression ${\mathfrak{y}}_{e_{k}}$ of $k\in I$ as a
meromorphic function of $x$ and $y_{r_{j}}$,

(3.25)
${\mathfrak{y}}_{{\mathfrak{e}}_{k}}=\frac{{\widetilde{Q}}_{k}(x,y_{r_{j}})}{|{\widetilde{A}}^{I}|(x,y_{r_{j}})},$

where $|{\widetilde{A}}^{I}|$ is a monic degree $q$ polynomial of $y_{r_{j}}$
with $R_{\mathbb{P}}$ coefficients, whereas the degree of
${\widetilde{Q}}_{k}$ in $y_{r_{j}}$ is $q-1$. (3.25) means that
${\mathfrak{y}}_{{\mathfrak{e}}_{\ell}}$ satisfies the relation
$|{\widetilde{A}}^{I}|{\mathfrak{y}}_{{\mathfrak{e}}_{k}}={\widetilde{Q}}_{k}(x,y_{r_{j}})$.

Then by substituting (3.25) into ${\mathfrak{y}}_{{\mathfrak{e}}_{k}}$ in the
relation
$\displaystyle{y_{r_{j}}^{2}={\overline{A}}_{{\widetilde{j}},0}+{\overline{A}}_{{\widetilde{j}}{\widetilde{j}}}y_{r_{j}}+\sum_{k\in
I}{\overline{A}}_{{\widetilde{j}},k}{\mathfrak{y}}_{e_{k}}}$, we obtain

$y_{r_{j}}^{q+2}+{\overline{A}}_{1}y_{r_{j}}^{q+1}+\cdots+{\overline{A}}_{q+1}y_{r_{j}}+{\overline{A}}_{q+2}=0,$

where ${\overline{A}}_{i}$ is a certain element in $R_{\mathbb{P}}$. We state
$(q+2)|r$ because
$\varphi_{H}^{X}(y_{r_{j}}^{{\mathfrak{r}}_{j}}-x^{{\overline{\mathfrak{r}}}_{j}})=0$
and it must belong to $\varphi_{H}^{X-1}(\\{0\\})$. Thus we let
$(q+2)=n{\mathfrak{r}}_{j}$ for an integer $n(\geq 1)$. However due to
$\varphi_{H}^{X}(y_{r_{j}}^{{\mathfrak{r}}_{j}}-x^{{\overline{\mathfrak{r}}}_{j}})=0$
again, $y_{r_{j}}^{{\mathfrak{r}}_{j}}-x^{{\overline{\mathfrak{r}}}_{j}}$ is
equal to lower weight terms with respect to $-{\mathrm{wt}}$ because of the
uniqueness of the $\infty$ in $X$. Thus the equation is reduced to multiply
the same equation with ${\mathfrak{r}}_{j}$ order in $y_{r_{j}}$, due to the
above arguments. It means that $H^{\prime}\subset\langle r,r_{j}\rangle$, and
thus $H^{\prime}=\langle r,r_{j}\rangle$, i.e., $q=n{\mathfrak{r}}_{j}-2$.
Hence we obtain (3.18) as in the proof of Proposition 3.1, which is
irreducible.

We now show (3.22). Assume that
$y_{r_{j}}{\widetilde{R}}_{X}\setminus{\widetilde{R}}_{X}\neq\varnothing$ and
let
${\overline{I}}^{\mathfrak{c}}:={\mathbb{Z}}_{r}^{\times}\setminus({\overline{I}})$.
Obviously $y_{r_{j}}{\mathfrak{y}}_{{\mathfrak{e}}_{i}}$ belongs to
${\widetilde{R}}_{X}$ for $i\in I$ because of (3.23). Hence the assumption
means

$y_{r_{j}}^{2}={\overline{A}}_{{\widetilde{j}},0}+{\overline{A}}_{{\widetilde{j}}{\widetilde{j}}}y_{r_{j}}+\sum_{i\in
I}{\overline{A}}_{{\widetilde{j}},i}{\mathfrak{y}}_{{\mathfrak{e}}_{i}}+\sum_{k\in{\overline{I}}^{\mathfrak{c}}}{\overline{A}}_{{\widetilde{j}},k}{\mathfrak{y}}_{{\mathfrak{e}}_{k}}$

where there exists, at least, a non-vanishing
${\overline{A}}_{{\widetilde{j}},\ell}\in R_{\mathbb{P}}$ for a certain
$\ell\in{\overline{I}}^{\mathfrak{c}}$, and
${\overline{A}}_{{\widetilde{j}},i}\in R_{\mathbb{P}}$. By considering the
Sato-Weierstrass weight of the both hand sides, we have
$-{\mathrm{wt}}({\overline{A}}_{{\widetilde{j}},\ell}{\mathfrak{y}}_{{\mathfrak{e}}_{\ell}})=2r_{j}\in
H^{\prime}$ and ${\mathfrak{e}}_{\ell}$ belongs to $H^{\prime}$. It means
$\ell\in{\overline{I}}$ and thus is contradict with (3.24). Hence we show
$y_{r_{j}}{\widetilde{R}}_{X}\subset{\widetilde{R}}_{X}$ or (3.22).

Proposition 3.14 obviously leads the following observations. Using (3.18) in
Proposition 3.14, we introduce
$R_{X^{(i)}}:={\mathbb{C}}[x,y_{r_{i}}]/(f_{X}^{(i)})$, $(i=2,\cdots,m_{X}-1)$
which are $R_{\mathbb{P}}$-modules and the unnormalized rings for $m_{X}>2$,
and their associated singular curves $X^{(i)}$ $(i=2,\cdots,m_{X}-1)$ with the
projection $\varpi_{X^{(i)}}:X^{(i)}\to{\mathbb{P}}$,
($\varpi_{X^{(i)}}(x,y_{i})=x$), $(i=2,\ldots,m_{X})$. Since $f_{X}^{(i)}$ is
irreducible, $R_{X^{(i)}}$ is a subring of $R_{X}$ which is its normalized
ring of $R_{X^{(i)}}$. There are injective ring-homomorphisms
$R_{\mathbb{P}}\xrightarrow{\iota_{r}^{(i)}}R_{X^{(i)}}\xrightarrow{\iota_{r,r_{i}}}R_{X}$;
thus it induces the projections $\varpi_{r_{i},r}:X\to X^{(i)}$
$((x,y_{\bullet})\mapsto(x,y_{r_{i}}))$ and
$\varpi_{r}^{(i)}:X^{(i)}\to{\mathbb{P}}$ $((x,y_{r_{i}})\mapsto x)$; They
satisfy the commutative diagrams,

(3.26)
$\textstyle{R_{X}}$$\textstyle{R_{X^{(j)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{r_{j},r}}$$\textstyle{R_{X^{(i)}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{r_{i},r}}$$\textstyle{R_{\mathbb{P}},\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{r}^{(i)}}$$\scriptstyle{\iota_{r}}$$\scriptstyle{\iota_{r}^{(j)}}$
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varpi_{r}}$$\scriptstyle{\varpi_{r,r_{j}}}$$\scriptstyle{\varpi_{r,r_{i}}}$$\textstyle{X^{(j)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varpi_{r}^{(j)}}$$\textstyle{X^{(i)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varpi_{r}^{(i)}}$$\textstyle{{\mathbb{P}}.}$

Further we also define the tensor product of these rings
$R_{X^{(2)}}\otimes_{R_{\mathbb{P}}}R_{X^{(3)}}\otimes_{R_{\mathbb{P}}}\cdots\otimes_{R_{\mathbb{P}}}R_{X^{(m_{X})}}$,
and its geometrical picture
$X^{[m_{X}-1]}_{\mathbb{P}}:=X^{(2)}\times_{\mathbb{P}}X^{(3)}\times_{\mathbb{P}}\cdots\times_{\mathbb{P}}X^{(m_{X})}$.
By identifying ${\mathbb{C}}[x,y_{\bullet}]$ $/(f_{X}^{(2)},$
$f_{X}^{(3)},\ldots,f_{X}^{(m_{X})})=R_{X}^{\otimes[m_{X}-1]}$ with a ring
$R_{X^{(2)}}\otimes_{R_{\mathbb{P}}}R_{X^{(3)}}\otimes_{R_{\mathbb{P}}}\cdots\otimes_{R_{\mathbb{P}}}R_{X^{(m_{X})}}$,
we have the natural projection
$\varphi_{R_{X}^{\otimes[m_{X}-1]}}:R_{X}^{\otimes[m_{X}-1]}\to R_{X}$, i.e.,
$R_{X}=R_{X}^{\otimes[m_{X}-1]}/(f_{1}^{X},\ldots,f_{k_{X}}^{X})$, and the
injection $\iota_{R_{X}^{\otimes[m_{X}-1]}}:R_{\mathbb{P}}\hookrightarrow
R_{X}^{\otimes[m_{X}-1]}$. It induces the injection
$\iota_{X^{[m_{X}-1]}_{\mathbb{P}}}:X\to X^{[m_{X}-1]}_{\mathbb{P}}$ and the
projection
$\prod_{i}\varpi_{X^{(i)}}:X^{[m_{X}-1]}_{\mathbb{P}}\to{\mathbb{P}}$.

Moreover we also define the direct product of these rings
$R_{X}^{[m_{X}-1]}:=R_{X^{(2)}}\times R_{X^{(3)}}\times\cdots\times
R_{X^{(m_{X})}}$, and its geometrical picture $X^{[m_{X}-1]}:=\prod
X^{(i)}\subset{\mathbb{P}}^{2(m_{X}-1)}$. Then we have the following
proposition.

######  Proposition 3.15.

The (3.18) in Proposition 3.14 provides surjective and injective ring
homomorphisms,

$\varphi_{R_{X}^{\otimes[m_{X}-1]}}:R_{X}^{\otimes[m_{X}-1]}\to
R_{X},\quad\iota_{R_{X}^{\otimes[m_{X}-1]}}:R_{\mathbb{P}}\hookrightarrow
R_{X}^{\otimes[m_{X}-1]},$

so that they satisfy the commutative diagrams,

$\textstyle{R_{X}}$$\textstyle{R_{X}^{\otimes[m_{X}-1]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{R_{X}^{\otimes[m_{X}-1]}}}$$\textstyle{R_{\mathbb{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{R_{X}^{\otimes[m_{X}-1]}}}$
$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{X^{[m_{X}-1]}_{\mathbb{P}}}}$$\textstyle{X^{[m_{X}-1]}_{\mathbb{P}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\prod\varpi_{X^{(i)}}}$$\textstyle{\mathbb{P}}$

which is consistent with the projection
$\varpi_{X^{[m_{X}-1]}_{\mathbb{P}}}:X^{[m_{X}-1]}_{\mathbb{P}}\to X$. They
induce the homomorphism by the ideal $(x-x_{2},x-x_{3},\ldots,x-x_{m_{X}})$
and an embedding,

$\varphi_{R_{X}^{[m_{X}-1]}}:R_{X}^{[m_{X}-1]}\to
R_{X},\quad\iota_{X^{[m_{X}-1]}}:X\hookrightarrow
X^{[m_{X}-1]}(\subset{\mathbb{P}}^{2(m_{X}-1)}).$

######  Remark 3.16.

Since the normalization of a ring is not unique in general, the surjective
ring homomorphism $\varphi_{R_{X}^{\otimes[m_{X}-1]}}$ is not injective except
$m_{X}=2$. For example, $(3,4,5)$ case, there is an surjective ring
homomorphisms whose $\varphi_{H}$ image is given by

$\displaystyle\begin{split}&{\mathbb{C}}[Z_{3},Z_{4},Z_{5}]/(Z_{4}^{3}-Z_{3}^{4},Z_{5}^{2}-Z_{3}^{5})\\\
&\to{\mathbb{C}}[Z_{3},Z_{3},Z_{4}]/(Z_{4}^{2}-\zeta_{3}^{a}Z_{3}Z_{5},Z_{4}Z_{5}-Z_{3}^{3},Z_{5}^{2}-\zeta_{3}^{2a}Z_{3}^{2}Z_{4}).\end{split}$

Thus $y$ and $y_{r_{j}}$ are related via $f^{X}_{i}$’s whose image of
$\varphi_{H}^{X}$ are binomial relations.

Direct computation gives the following relation based on (3.18).

######  Lemma 3.17.

$\displaystyle{(\delta_{Y,y}f_{X}^{(j)})(x,Y,y):=\frac{f_{X}^{(j)}(x,Y)-f_{X}^{(j)}(x,y)}{Y-y}}$
$\displaystyle{=\sum_{\ell=0}^{{\overline{\mathfrak{r}}}_{j}-1}A_{\ell}^{(j)}(x)\sum_{i=0}^{{\mathfrak{r}}_{j}-\ell-1}Y^{i}y^{{\mathfrak{r}}_{j}-\ell-i-1}}$

$\in R_{X}\otimes_{R_{\mathbb{P}}}R_{X}$.

###  3.4. The covering structures in W-curves

We follow [Ku2, St] to investigate the covering structure in W-curves.

####  3.4.1. Galois covering

As mentioned in Remark 3.3, let us consider the Riemann sphere ${\mathbb{P}}$
and
$R_{\mathbb{P}}={\mathbf{H}}^{0}({\mathbb{P}},{\mathcal{O}}_{{\mathbb{P}}}(*\infty))$.
We identify $R_{\mathbb{P}}$ with its affine part
$R_{\mathbb{P}}^{\circ}={\mathbb{C}}[x]$ and its quotient field is denoted by
${\mathbb{C}}(x)={\mathcal{Q}}(R_{\mathbb{P}})$. The quotient field
${\mathcal{Q}}(R_{X})={\mathbb{C}}(X)$ of $R_{X}$ is an extension of the field
${\mathbb{C}}(x)$.

Following the above description, we consider the W-curve $X$. The covering
$\varpi_{r}:X\to{\mathbb{P}}$ $((x,y_{\bullet})\mapsto x)$ is obviously the
Galois covering. Further we have the Galois group
$\mathrm{Gal}({\mathcal{Q}}(R_{X})/$
${\mathcal{Q}}(R_{\mathbb{P}}))=\mathrm{Aut}(X/{\mathbb{P}})=\mathrm{Aut}(\varpi_{r})$,
which is denoted by $G_{X}$. The $\varpi_{r}$ is a finite branched covering. A
ramification point of $\varpi_{r}$ is defined as a point of such that is not
biholomorphic at the point. The image $\varpi_{r}$ of the ramification point
is called the branch point of $\varpi_{r}$. The number of the finite
ramification points is denoted by $\ell_{{\mathfrak{B}}}$.

We basically focus on the Galois covering $\varpi_{r}:X\to{\mathbb{P}}$:

######  Definition 3.18.

Let ${\mathfrak{B}}_{X}:={\mathfrak{B}}_{X,r}$
$=\\{B_{i}\\}_{i=0,\cdots,\ell_{{\mathfrak{B}}}}$ and
${\mathfrak{B}}_{{\mathbb{P}}}:=\varpi_{r}({\mathfrak{B}}_{X})=\\{b_{i}\\}_{i=0,\cdots,\ell_{{\mathfrak{B}}}}$,
where $\ell_{{\mathfrak{B}}}:=\\#{\mathfrak{B}}_{X}-1$, $B_{0}=\infty\in X$
and $b_{i}:=\varpi_{r}(B_{i})$.

We have the following results.

######  Lemma 3.19.

Every element $B_{i}$ in ${\mathfrak{B}}_{X}\setminus\\{\infty\\}$ is given by
the point $(x,y_{\bullet})$ at which there exists, at least, a certain $j$ in
$\\{2\leq j\leq m_{X}\\}$ such that $\displaystyle{\frac{\partial
f_{X}^{(j)}(x,y_{r_{j}})}{\partial y_{r_{j}}}=0}$ and
$\displaystyle{\frac{\partial f_{X}^{(j)}(x,y_{r_{j}})}{\partial x}\neq 0}$.

###### Proof.

When $\displaystyle{\frac{\partial f_{X}^{(i)}(x,y_{r_{i}})}{\partial x}=0}$
and $\displaystyle{\frac{\partial f_{X}^{(i)}(x,y_{r_{i}})}{\partial
y_{r_{j}}}=0}$ at a point $P\in X$, the point means the singular point as the
plane curve given by $f_{X}^{(i)}(x,y_{r_{i}})=0$ at $\varpi_{r,r_{i}}(P)$.
However, since $X$ is not singular, there exists $j$ satisfying the condition.
Then at the point, $dx$ is identically zero, and thus at the point, $x$ is not
a local parameter of $X$. Thus $P$ must be an element in
${\mathfrak{B}}_{X}\setminus\\{\infty\\}$, $P=B_{i}$. It means
$dx=d(t^{e_{i}})$ for a positive integer $e_{i}>1$ in terms of the local
parameter $t$ such that $t(B_{i})=0$ such that there exists $j$ satisfying
$dy_{r_{j}}=dt$ and $dy_{r_{i}}=t^{f_{i}}dt$ $(i\neq j,f_{i}\geq 1)$.

The $e_{i}$ appearing the proof for the ramification point $B_{i}$ in Lemma
3.19 is called _the ramification index_ , and denoted by $e_{B_{i}}$.

####  3.4.2. Riemann-Hurwitz theorem

Let us consider the behaviors of the covering $\varpi_{r}:X\to{\mathbb{P}}$,
including the ramification points. The Riemann-Hurwitz theorem [Ku2],

(3.27)
$2g-2=-2r+\sum_{i=1}^{\ell_{{\mathfrak{B}}}}(e_{B_{i}}-1)+(r-1)=\sum_{i=1}^{\ell_{{\mathfrak{B}}}}(e_{B_{i}}-1)-(r+1),$

shows the following:

######  Corollary 3.20.

The divisor of $dx$ is given by,

$\mathrm{div}(dx)=\sum_{i=1}^{\ell_{{\mathfrak{B}}}}(e_{B_{i}}-1)B_{i}-(r+1)\infty.$

##  4\. Complementary module $R_{X}^{\mathfrak{c}}$ of $R_{X}$

###  4.1. Trace in the covering structure

It is known that the Riemann-Hurwitz relation and the divisor of $dx$ are
obtained via the Dedekind different [Ku2, Theorem 15.11]. In this subsection,
we consider the properties of $R_{\mathbb{P}}$-module $R_{X}$, which are
related to the trace, complementary module, trace, and Dedekind different
[Ku2, Chapter 15].

####  4.1.1. Trace in $R_{X}/R_{\mathbb{P}}$

We review the general results of $R_{X}$ as a ring extension of
$R_{\mathbb{P}}$ following [Ku2] (see Subsection 2.1). Let us consider the
covering structures of $\varpi_{r}:X\to{\mathbb{P}}$ to discriminate its
lifted points, and the enveloping field
${\mathcal{Q}}(R_{X})^{e}={\mathbb{C}}(X)^{e}:={\mathbb{C}}(X)\otimes_{{\mathbb{C}}(x)}{\mathbb{C}}(X)={\mathcal{Q}}(R_{X})\otimes_{{\mathcal{Q}}(R_{\mathbb{P}})}{\mathcal{Q}}(R_{X})$.

The field extension ${\mathbb{C}}(X)/{\mathbb{C}}(x)$ induces the extension
ring $R_{X}$ of $R_{\mathbb{P}}$. As mentioned in Subsection 2.1, we consider
the dual of $R_{X}$ with respect to $R_{\mathbb{P}}$,

$\omega_{R_{X}/R_{\mathbb{P}}}:=\mathrm{Hom}_{R_{\mathbb{P}}}(R_{X},R_{\mathbb{P}}),$

which is a free $R_{\mathbb{P}}$-module. For
$\displaystyle{R_{X}=\oplus_{i=0}^{r-1}R_{\mathbb{P}}{\mathbf{y}}_{i}}$ (i.e.,
${\mathbf{y}}_{i}={\mathfrak{y}}_{{\mathfrak{e}}_{i}}$), it is expressed by
the dual basis $\\{{\mathbf{y}}_{i}^{*}\\}_{i=0,\ldots,r-1}$ such that

(4.1)
$\omega_{R_{X}/R_{\mathbb{P}}}=\bigoplus_{i=0}^{r-1}R_{\mathbb{P}}{\mathbf{y}_{i}^{*}},\quad\mbox{and}\quad{\mathbf{y}}_{i}^{*}{\mathbf{y}}_{j}=\delta_{i,j}.$

Here ${\mathbf{y}}_{i}^{*}$ and ${\mathbf{y}}_{j}$ correspond to the point in
fiber which lies over $x\in R_{\mathbb{P}}$. For the standard trace
$\displaystyle{\tau_{R_{X}/R_{\mathbb{P}}}:=\sum_{i=0}^{r-1}{\mathbf{y}}_{i}{\mathbf{y}}_{i}^{*}}$
$\in\omega_{R_{X}/R_{\mathbb{P}}}$, we define the complementary module
$R_{X}^{{\mathfrak{c}}}$ over $R_{X}/R_{\mathbb{P}}$ with respect to the
standard trace $\tau_{R_{X}/R_{\mathbb{P}}}$ by

(4.2) $R_{X}^{{\mathfrak{c}}}:=\\{z\in{\mathcal{Q}}(R_{X})\ |\
\tau_{R_{X}/R_{\mathbb{P}}}(za)\in R_{\mathbb{P}},\ \ {}^{\forall}a\in
R_{X}\\}.$

It is obviously that the localization $(R_{X}^{{\mathfrak{c}}})_{P}$ at $P\in
X$ is equal to $R_{X,P}^{{\mathfrak{c}}}$ from the definition. Since for a
point $P\in X$, $R_{X,P}$ is a principal ideal domain, every ideal is
generated by a certain element in $R_{X,P}$. There is an element
${\mathfrak{h}}_{X,P}\in R_{X,P}$ such that
$R_{X,P}^{{\mathfrak{c}}}={\mathfrak{h}}_{X,P}R_{X,P}$ [St, Proposition
3.4.2]. Following [St, Definition 3.4.3], we define the different of
$R_{X}/R_{\mathbb{P}}$.

######  Definition 4.1.

The different $\mathrm{diff}(R_{X}/R_{\mathbb{P}})$ is a divisor defined by

$\mathrm{diff}(R_{X}/R_{\mathbb{P}}):=\sum_{P\in
X\setminus\\{\infty\\}}d_{P}\,P,$

where the different divisor $d_{P}:=-\mathrm{deg}_{P}({\mathfrak{h}}_{X,P})$
for $R_{X,P}^{\mathfrak{c}}={\mathfrak{h}}_{X,P}R_{X,P}$.

By Dedekind’s different theorem, we have the following:

######  Proposition 4.2.

$d_{B_{i}}=e_{B_{i}}-1$ for $P\in{\mathfrak{B}}_{X}\setminus\\{\infty\\}$, and
the support of $\mathrm{diff}(R_{X}/R_{\mathbb{P}})$ equals
${\mathfrak{B}}_{X}\setminus\\{\infty\\}$.

###### Proof.

See [Ku2, Theorem 15.11] and [St, Theorem 3.5.1].

###  4.2. Trace operator for plane W-curves: $R_{X}=R_{X^{\circ}}$ ($m_{X}=2$)
case

Following Kunz [Ku2, Theorem 15.1], we we review $m_{X}=2$ case:

######  Proposition 4.3.

[Ku2, Theorem 15.1] For the plane W-curve ($m_{X}=2$), we have the relation,

(4.3) $R_{X}^{\mathfrak{c}}=\frac{1}{f_{X,y}}\cdot R_{X}.$

###### Proof.

Let us show the proof by Kunz. We note that the extension of field
${\mathcal{Q}}(R_{X})$ of ${\mathcal{Q}}(R_{\mathbb{P}})$ is separable and
$f_{X}$ is monic,

${\mathcal{Q}}(R_{X})={\mathcal{Q}}(R_{\mathbb{P}})[Y]/(f_{X})=\bigoplus_{i=0}^{r-1}{\mathcal{Q}}(R_{\mathbb{P}})y^{i},$

and thus we note $y^{i}={\mathbf{y}}_{i}={\mathfrak{y}}_{{\mathfrak{e}}_{i}}$
in Proposition 3.11 and in Subsection 2.1, and

${\mathcal{Q}}(R_{X})^{e}={\mathcal{Q}}(R_{X})\otimes_{{\mathcal{Q}}(R_{\mathbb{P}})}{\mathcal{Q}}(R_{X})\cong{\mathcal{Q}}(R_{X})[Y]/(f_{X}).$

The ring-homomorphism $\mu:{\mathcal{Q}}(R_{X})^{e}\to{\mathcal{Q}}(R_{X})$
$(a\otimes b\mapsto ab)$, the standard trace
$\displaystyle{\tau_{R_{X}/R_{\mathbb{P}}}:=\sum_{i=0}^{r-1}y^{i}{\mathbf{y}}_{i}^{*}}$
is obtained by the extension of a trace $\tau$. Let us find the basis
$\\{\widehat{{\mathbf{y}}}_{i}\\}$ of $R_{X}$ with respect to $\tau$, and
$\Delta_{\tau}:=\displaystyle{\bigoplus_{i=0}^{r-1}\widehat{{\mathbf{y}}}_{i}\otimes
y^{i}}$ as an element of $R_{\mathbb{P}}$-module
$\mathrm{Ann}_{R_{X}^{e}}(\mathrm{Ker}\mu)$. If we find
$\\{\widehat{{\mathbf{y}}}_{i}\\}$ and $\tau$, using them, we obtain the
standard trace $\tau_{R_{X}/R_{\mathbb{P}}}=\mu(\Delta_{\tau})\tau$.
Accordingly, we construct the $\tau$ and $\\{\widehat{{\mathbf{y}}}_{i}\\}$ as
follows.

For an element ${\widetilde{h}}\in{\mathcal{Q}}(R_{X})[Y]$ such that
$f_{X}(x,Y)=(Y-y)\cdot{\widetilde{h}}$, we can identify
$\mathrm{Ann}_{R_{X}^{e}}(\mathrm{Ker}\mu)$ with the principal ideal
$({\widetilde{h}})/(f_{X}(x,Y))$. Indeed, in
${\mathcal{Q}}(R_{X})[Y]/(f_{X}(x,Y))$,

$f_{X}(x,Y)=f_{X}(x,Y)-f_{X}(x,y)=(Y-y)\frac{f_{X}(x,Y)-f_{X}(x,y)}{Y-y}=(Y-y){\widetilde{h}}=0.$

It means that ${\widetilde{h}}(x,Y,y):=\delta_{Y,y}f_{X}(x,Y,y)$ belongs to
$R_{X}\otimes_{R_{\mathbb{P}}}R_{X}$ noting Lemma 3.17. Thus we have

$\Delta_{{\widetilde{h}}}=\sum_{\ell=0}^{r-1}A_{\ell}(x)\sum_{j=0}^{r-\ell-1}y^{j}\otimes
y^{r-\ell-j-1}\in R_{X}^{e}=R_{X}\otimes_{R_{\mathbb{P}}}R_{X},$

which generates the ideal $\mathrm{Ann}_{R_{X}^{e}}(\mathrm{Ker}\mu)$ in
$R_{X}^{e}$, i.e., $(a\otimes 1-1\otimes a)\Delta_{{\widetilde{h}}}=0$ for
every $a\in R_{X}$. Further from Proposition 2.5, it corresponds to the trace
$\tau_{\widetilde{h}}\in\omega_{R_{X}/R_{\mathbb{P}}}$ by

$1=\sum_{\ell=0}^{r-1}A_{\ell}(x)\sum_{j=0}^{r-\ell-1}\tau_{\widetilde{h}}(y^{j})y^{r-\ell-j-1}.$

For example, $r=4$ case, it is

$1=\tau_{\widetilde{h}}(y^{3}+A_{1}y^{2}+A_{2}y+A_{3})+\tau_{\widetilde{h}}(y^{2}+A_{1}y+A_{2})y+\tau_{\widetilde{h}}(y+A_{1})y^{2}+\tau_{\widetilde{h}}(1)y^{3},$

which should be interpreted as $\tau_{\widetilde{h}}(1)=0$,
$\tau_{\widetilde{h}}(y)=0$, $\tau_{\widetilde{h}}(y^{2})=0$, and thus
$\tau(y^{3})=1$. Similarly since $A_{0}=1$ and $y^{r-\ell-j-1}=1$ when
$\ell=0$ and $j=r-1$, we compare the both sides in the equation and obtain

$\tau_{\widetilde{h}}(y^{i})=\left\\{\begin{matrix}1&\mbox{ for }i=r-1,\\\
0&\mbox{ otherwise. }\end{matrix}\right.$

In other words, for ${\mathbf{y}}_{i}=y^{i}$, $(i\in{\mathbb{Z}}_{r})$, we
have

$\widehat{{\mathbf{y}}}_{i}=\sum_{\ell=0}^{r-1-i}A_{\ell}(x)y^{\ell}$

as the dual basis of the basis
$\\{y^{i}={\mathbf{y}}_{i}={\mathfrak{y}}_{{\mathfrak{e}}_{i}}\\}_{i\in{\mathbb{Z}}_{r}}$
with respect to $\tau_{\widetilde{h}}$ in Subsection 2.1. Then
$\tau_{\widetilde{h}}(R_{X})\subset R_{\mathbb{P}}$. It is obvious that
$\mu(\Delta_{{\widetilde{h}}})=f_{X,y}$. Thus the standard trace
$\tau_{R_{X}/R_{\mathbb{P}}}$ of $R_{X}/R_{\mathbb{P}}$ is given by

$\tau_{R_{X}/R_{\mathbb{P}}}=\mu(\Delta_{\widetilde{h}})\cdot\tau_{\widetilde{h}}=f_{X,y}\cdot\tau_{\widetilde{h}}.$

We have

$\tau_{R_{X}/R_{\mathbb{P}}}\left(\frac{y^{i}}{f_{X,y}(x,y)}\right)=\left\\{\begin{matrix}1&\mbox{
for }i=r-1,\\\ 0&\mbox{ otherwise }.\end{matrix}\right.$

Let us consider an element in $R_{X}^{{\mathfrak{c}}}$ as in (4.2). We
consider $u\in{\mathcal{Q}}(R_{X})$ which is expressed by

$u=\frac{1}{f_{X,y}(x,y)}\sum_{i=0}^{r-1}a_{i}y^{i}\in{\mathcal{Q}}(R_{X}),$

and its conditions in (4.2). For every element
$\displaystyle{v=\sum_{j=0}^{r-1}b_{j}y^{j}}\in R_{X}$, ($b_{j}\in
R_{\mathbb{P}}$), $u$ satisfies $\tau_{R_{X}/R_{\mathbb{P}}}(uv)$ $\in
R_{\mathbb{P}}$, i.e.,

$\tau_{R_{X}/R_{\mathbb{P}}}(uv)=\sum_{i,j}\tau_{\widetilde{h}}(a_{i}b_{j}y^{i}y^{j})=\sum_{i=0}^{r-1}a_{i}b_{r-1-i}\in
R_{\mathbb{P}}.$

It implies that $a_{i}\in R_{\mathbb{P}}$ for every $i=0,1,\ldots,r-1$. Thus
we obtain the relation (4.3).

######  Remark 4.4.

Here we note that instead of $\\{\widehat{{\mathbf{y}}}_{i}\\}$ in the proof,
we define a simpler dual basis of
$\\{y^{i}={\mathbf{y}}_{i}={\mathfrak{y}}_{{\mathfrak{e}}_{i}}\\}_{i\in{\mathbb{Z}}_{r}}$
with respect to $\tau_{\widetilde{h}}$ by

$\\{{\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}:=y^{r-i-1}\\}_{i\in{\mathbb{Z}}_{r}},$

because it is obviously
$\tau_{\widetilde{h}}({\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}{\mathfrak{y}}_{{\mathfrak{e}}_{j}})=\delta_{ij}$
from the definition of $\tau_{\widetilde{h}}$.

From (4.3), the Dedekind different of the plane W-curve is given by

$\mathrm{diff}(R_{X}/R_{\mathbb{P}})=-\sum_{P\in
X\setminus\\{\infty\\}}\mathrm{deg}_{P}(f_{X,y})P.$

By Dedekind’s different theorem (Proposition 4.2), we have the following [Ku2,
Chapter 15]:

######  Proposition 4.5.

$\mathrm{deg}_{B_{i}}(f_{X,y})=e_{B_{i}}-1$.

####  4.2.1. Trace in the Galois covering at $R_{X}$ I

We generalize the result of the plane curves to the space curves. The
surjective ring homomorphism in Proposition 3.15 can be interpreted as
follows. We implicitly introduce the dual modules
$\omega_{R_{X^{(i)}}/R_{\mathbb{P}}}$ of $R_{X^{(i)}}$ and their tensor
product
$\omega_{R_{X}/R_{\mathbb{P}}}^{\otimes[m_{X}-1]}:=\omega_{R_{X^{(2)}}/R_{\mathbb{P}}}\otimes_{R_{\mathbb{P}}}\cdots\otimes_{R_{\mathbb{P}}}\omega_{R_{X^{(m_{X})}}/R_{\mathbb{P}}}$
to find $R_{X}^{\mathfrak{c}}$. More precisely, we implicitly construct the
trace $\tau$ in $R_{X}$ by using the data of $(\tau_{2},\cdots,\tau_{m_{X}})$
of $R_{X}^{\otimes[m_{X}-1]}$ by regarding $R_{X}$ as
$R_{X}=R_{X}^{\otimes[m_{X}-1]}/(f^{X}_{1},f^{X}_{2},\ldots,f^{X}_{k_{X}})$.
By investigating them, we obtain $R_{X}^{\mathfrak{c}}$ for the $m_{X}>2$
case.

We first investigate the similar relations in $R_{H}$ following Lemma 3.5.

####  4.2.2. Trace in the Galois covering at $R_{H}$

In order to generalize the investigation of the complementary module for the
plane curves $m_{X}=2$ to general W-curves, we investigate the trace structure
at $R_{H}$ since the monomial curve in Subsection 3.2 is crucial in W-curves.
We assume $m_{X}\geq 2$.

We use the surjection $\varphi_{\infty}:R_{X}\to R_{H}$ in Lemma 3.10, and
consider the behavior of the trace in $R_{H}^{Z}$. We investigate a “covering”
structure in
$\varpi_{H}:X_{H}=\mathrm{Spec}\,R_{H}^{Z}\to{\mathbb{P}}=\mathrm{Spec}\,{\mathbb{C}}[Z_{r}]$.
The cyclic group ${\mathfrak{C}}_{r}$ of order $r$ acts on $R_{H}^{Z}$ and
$X_{H}$ as the ${\mathbb{G}}_{\mathfrak{m}}$ action. We regard it as the
Galois covering and consider ${\mathbb{C}}[Z_{r}]$-module $R_{H}^{Z}$ and
$R_{H}$.

We introduce a meromorphic function on
$\mathrm{Spec}\,R_{H}^{Z}\times_{\mathrm{Spec}\,{\mathbb{C}}[Z_{r}]}\mathrm{Spec}\,R_{H}^{Z}$
or an element $p$ in the enveloping field
${\mathcal{Q}}(R_{H}^{Z})^{e}:={\mathcal{Q}}(R_{H}^{Z}\otimes_{{\mathbb{C}}[Z_{r}]}R_{H}^{Z})$,
and an element $h$ in its associated enveloping ring
${R_{H}^{Z}}^{e}:=R_{H}^{Z}\otimes_{{\mathbb{C}}[Z_{r}]}R_{H}^{Z}$, i.e.,
$Z_{r_{i}}^{r}=Z_{r}^{r_{i}}=Z_{r_{i}}^{\prime r}$ for $i=2,\ldots,m_{X}$. We
extend the group action of ${\mathfrak{C}}_{r}$ to that on
${\mathcal{Q}}(R_{H}^{Z}\otimes_{{\mathbb{C}}[Z_{r}]}R_{H}^{Z})$ such as
$\zeta h(Z_{r},Z_{\bullet},Z^{\prime}_{\bullet})=h(Z_{r},\zeta
Z_{\bullet},\zeta Z^{\prime}_{\bullet})$ for $\zeta\in{\mathfrak{C}}_{r}$. We
define an element $p_{H}$ in
${\mathcal{Q}}(R_{H}^{Z}\otimes_{{\mathbb{C}}[Z_{r}]}R_{H}^{Z})$ by

(4.4)
$\begin{split}p_{H}(Z_{r},Z_{\bullet},Z^{\prime}_{\bullet})&:=\prod_{i=2}^{m_{X}}p_{H,r_{i}}(Z_{r},Z_{r_{i}},Z^{\prime}_{r_{i}}),\\\
p_{H,r_{i}}(Z_{r},Z_{r_{i}},Z^{\prime}_{r_{i}})&:=\frac{Z^{{\mathfrak{r}}_{i}-1}_{r_{i}}+Z^{{\mathfrak{r}}_{i}-2}_{r_{i}}Z^{\prime}_{r_{i}}+Z^{{\mathfrak{r}}_{i}-3}_{r_{i}}Z^{\prime
2}_{r_{i}}+\cdots+Z^{\prime{\mathfrak{r}}_{i}-1}_{r_{i}}}{rZ_{r_{i}}^{{\mathfrak{r}}_{i}-1}}.\end{split}$

Noting
$Z_{r_{i}}^{{\mathfrak{r}}_{i}}=Z_{r}^{{\mathfrak{r}}_{i}}=Z_{r_{i}}^{\prime{\overline{\mathfrak{r}}}_{i}}$
in Lemma 3.5, the each factor behaves like

$p_{H,r_{i}}(Z_{r},Z_{r_{i}},Z^{\prime}_{r_{i}})=\left\\{\begin{matrix}1&\mbox{
for }Z^{\prime}_{r_{i}}=Z_{r_{i}},\\\ 0&\mbox{ otherwise.}\end{matrix}\right.$

We have the trace property,

(4.5)
$p_{H}(Z_{r},Z_{\bullet},Z^{\prime}_{\bullet})=\left\\{\begin{matrix}1&\mbox{
for }Z^{\prime}_{\bullet}=Z_{\bullet},\\\ 0&\mbox{
otherwise.}\end{matrix}\right.$

######  Lemma 4.6.

There are polynomials
${\widetilde{h}}_{H}(Z_{r},Z_{\bullet},Z^{\prime}_{\bullet})\in
R_{H}^{Z}\otimes_{{\mathbb{C}}[Z_{r}]}R_{H}^{Z}$ and $h_{H}(Z)=$
$h_{H}(Z_{r},Z_{\bullet})$
$:={\widetilde{h}}_{H}(Z_{r},Z_{\bullet},Z_{\bullet})\in R_{H}^{Z}$ such that

(4.6)
$\varphi_{H}^{Z}({\widetilde{h}}_{H}(Z_{r},Z_{\bullet},Z^{\prime}_{\bullet}))=z^{d_{h}}\sum_{i=0}^{r-1}z^{\prime\,{\mathfrak{e}}_{i}}z^{-{\mathfrak{e}}_{i}}=\sum_{i=0}^{r-1}z^{\prime\,{\mathfrak{e}}_{i}}z^{{\widehat{\mathfrak{e}}}_{i}}\in
R_{H}\otimes_{{\mathbb{C}}[z^{r}]}R_{H},$

where 1) $z$ and $z^{\prime}$ are given by
$\varphi_{H}^{Z}(Z_{r_{i}})=z^{r_{i}}$,
$\varphi_{H}^{Z}(Z_{r_{i}}^{\prime})=z^{\prime\,r_{i}}$, for
$i\in{\mathbb{Z}}_{r}$, and $z^{r}=z^{\prime\,r}$, 2) $d_{h}$,
${\widehat{\mathfrak{e}}}_{i}$, $\delta_{i}$ $(i\in{\mathbb{Z}}_{r})$ and
$\ell$ are non-negative integers satisfying the conditions that a)
$d_{h}\geq{\mathfrak{e}}_{i}$ for every $i\in{\mathbb{Z}}_{r}$, b) $d_{h}$ is
determined by Lemma 4.15, c) $d_{h}={\mathfrak{e}}_{\ell}+\delta_{0}r$ such
that
$\displaystyle{{\mathfrak{e}}_{\ell}=\sum_{i=2}^{m_{X}}({\mathfrak{r}}_{i}-1)r_{i}}$
modulo $r$, and d)

(4.7)
${\widehat{\mathfrak{e}}}_{i}:=d_{h}-{\mathfrak{e}}_{i}={\mathfrak{e}}_{\ell,i}^{*}+\delta_{i}r,\quad
z^{{\mathfrak{e}}_{i}}z^{{\widehat{\mathfrak{e}}}_{i}}=z^{{\mathfrak{e}}_{0}}z^{{\widehat{\mathfrak{e}}}_{0}}=z^{d_{h}},$

for every $i\in{\mathbb{Z}}_{r}$, especially

(4.8)
${\widehat{\mathfrak{e}}}_{0}=d_{h}={\mathfrak{e}}_{\ell}+\delta_{0}r,\quad{\widehat{\mathfrak{e}}}_{\ell}=\delta_{0}r,\quad(\delta_{\ell}=\delta_{0}),$

and 3) ${\widetilde{h}}_{H}(Z_{r},Z_{\bullet},Z^{\prime}_{\bullet})$ consists
of $r$ monomials corresponding to each term in (4.6) and satisfies

(4.9)
$p_{H}(Z_{r},Z_{\bullet},Z^{\prime}_{\bullet})=\frac{{\widetilde{h}}_{H}(Z_{r},Z_{\bullet},Z^{\prime}_{\bullet})}{h_{H}(Z)}.$

###### Proof.

The cyclic group ${\mathfrak{C}}_{r}$ acts on
$p_{H}(Z_{r},Z_{\bullet},Z^{\prime}_{\bullet})$ so that it is invariant. We
consider $R_{H}$ rather than $R_{H}^{Z}$. We note the following: 1) For
$j=2,\ldots,m_{X}$, $r_{i}$ and $r_{j}\in M_{X}$, $\\{r_{j}i\mbox{ modulo }r\
|\ i\in{\mathbb{Z}}_{r}\\}={\mathbb{Z}}_{r}$, 2) the numerator in each
$p_{H,r_{i}}$ in (4.4) is homogeneous, and 3) their product is also
homogeneous. Therefore it is obvious that there are ${\widetilde{d}}_{\Delta}$
and $\\{\Delta_{i}\ |\ \Delta_{i}<\Delta_{i+1}\\}$ such that

$\frac{z^{{\widetilde{d}}_{\Delta}}\sum_{i=0}^{r-1}z^{\prime\,\Delta_{i}}z^{-\Delta_{i}}}{rz^{{\widetilde{d}}_{\Delta}}}$

has the property of the right hand side of (4.5) after acting
$\varphi_{H}^{Z}$ both sides in (4.5), and $\\{\Delta_{i}\mbox{ modulo }r\ |\
i\in{\mathbb{Z}}_{r}\\}={\mathbb{Z}}_{r}$. It means that $\\{\Delta_{i}\ |\
i\in{\mathbb{Z}}_{r}\\}$ equals $\\{{\mathfrak{e}}_{i}+n_{i}r\ |\
i\in{\mathbb{Z}}_{r}\\}$ for a certain non-negative number
$n_{i}\in{\mathbb{N}}_{0}$ from Lemma 2.8, and $\Delta_{0}=0$.

Due to the isomorphism $\varphi_{H}^{Z}$, for sufficiently large $n$ and $m$,
we find an element in $R_{H}^{Z}\otimes_{{\mathbb{C}}[Z_{r}]}R_{H}^{Z}$ whose
image of $\varphi_{H}^{Z}$ is $z^{n}z^{\prime\,m}$. For $\ell_{i}$ satisfying
$\sum_{i}r_{i}\ell_{i}\equiv 0$ modulo $r$, $\prod Z_{r_{i}}^{\ell_{i}}=\prod
Z_{r_{i}}^{\prime\ell_{i}}$, and thus we can find $d_{h}$ such that
$z^{{\widetilde{d}}_{\Delta}}\sum_{i=0}^{r-1}z^{\prime\,\Delta_{i}}z^{-\Delta_{i}}=z^{d_{h}}\sum_{i=0}^{r-1}z^{\prime\,{\mathfrak{e}}_{i}}z^{-{\mathfrak{e}}_{i}}$
noting $z^{r}=z^{\prime\,r}$. There is a preimage
${\widetilde{h}}_{H}(Z_{r},Z_{\bullet},Z^{\prime}_{\bullet})$ as an element in
$R_{H}^{Z}\otimes_{{\mathbb{C}}[Z_{r}]}R_{H}^{Z}$. It is obvious that
${\widetilde{h}}_{H}(Z_{r},Z_{\bullet},Z^{\prime}_{\bullet})$ consists of $r$
monomials and satisfies (4.6) and (4.9) for
$p_{H}(Z_{r},Z_{\bullet},Z^{\prime}_{\bullet})$.

We note that the determination of $d_{h}$ has the ambiguity up to $r$, and
thus we set it such that it satisfies Lemma 4.15. Here
$d_{h}={\widetilde{d}}_{h}+n_{h}r$ so that ${\widetilde{d}}_{h}$ is the
minimal element satisfying the above relations.

From the definition and Lemma 2.8 3, we obtain the relations (4.7) of $i=0$
and $i=\ell$, and $\delta_{0}=\delta_{\ell}$.

We consider the cases $d_{h}$ modulo $r$.

  1. (1)

$d_{h}={\mathfrak{e}}_{0}=0$ modulo $r$ case: We note that
$\\{{\mathfrak{e}}_{i}$ modulo $r\ |\
i\in{\mathbb{Z}}_{r}^{\times}\\}={\mathbb{Z}}_{r}^{\times}$, and
${\mathfrak{e}}_{\ell}<{\mathfrak{e}}_{r-1}$, whereas
${\widehat{\mathfrak{e}}}_{\ell}$ must be non-negative. Each
${\mathfrak{e}}_{i}$ ($i\in{\mathbb{Z}}_{r}^{\times}$) cannot be divided by
$r$ and thus ${\mathfrak{e}}_{i}$, especially, ${\mathfrak{e}}_{r-1}$ is not
equal to $d_{h}$. From Definition 2.7. we find a non-negative integer
$\delta_{i}$ such that
${\widehat{\mathfrak{e}}}_{i}={\mathfrak{e}}_{0,i}^{*}+\delta_{i}r$,
$i\in{\mathbb{Z}}_{r}^{\times}$. It is obvious that
$d_{h}={\widehat{\mathfrak{e}}}_{0}=\delta_{0}r>{\mathfrak{e}}_{r-1}>0$. We
also note ${\widehat{\mathfrak{e}}}_{r-1}>0$ because of
${\mathfrak{e}}_{0,r-1}^{*}>0$.

  2. (2)

$d_{h}={\mathfrak{e}}_{\ell}$ modulo $r$ case ($\ell=1,\cdots,r-2$):
Similarly, since $d_{h}={\mathfrak{e}}_{\ell}+\delta_{0}r$ satisfies
$d_{h}-{\mathfrak{e}}_{i}\geq 0$ for $i\in{\mathbb{Z}}_{r}$, (especially
$d_{h}\geq{\mathfrak{e}}_{r-1}$) we find non-negative integers $\delta_{i}$
such that
${\widehat{\mathfrak{e}}}_{i}={\mathfrak{e}}_{\ell,i}^{*}+\delta_{i}r$ for
$i\in{\mathbb{Z}}_{r}$, and
${\widehat{\mathfrak{e}}}_{\ell}=\delta_{\ell}r=\delta_{0}r>0$, noting
${\mathfrak{e}}_{\ell,\ell}^{*}=0$. Then ${\widehat{\mathfrak{e}}}_{r-1}>0$
because of ${\mathfrak{e}}_{\ell,r-1}^{*}>0$.

  3. (3)

$d_{h}={\mathfrak{e}}_{r-1}$ modulo $r$ case: Similarly since
$d_{h}={\mathfrak{e}}_{r-1}+\delta_{0}r$ satisfies
$d_{h}-{\mathfrak{e}}_{i}\geq 0$, especially $d_{h}\geq{\mathfrak{e}}_{r-1}$,
$\delta_{0}$ is non-negative. We find non-negative integers $\delta_{i}$ such
that ${\widehat{\mathfrak{e}}}_{i}={\mathfrak{e}}_{r-1,i}^{*}+\delta_{i}r$ for
$i\in{\mathbb{Z}}_{r}^{\times}$, and
${\widehat{\mathfrak{e}}}_{r-1}=\delta_{r-1}r\geq 0$ or
$\delta_{r-1}=\delta_{0}$ because of ${\mathfrak{e}}_{r-1,r-1}^{*}=0$.

These show the statements in the proposition.

######  Proposition 4.7.

The $\delta_{0}$ in (4.8) equals zero if and only if
$d_{h}={\mathfrak{e}}_{r-1}$.

The case $d_{h}={\mathfrak{e}}_{r-1}$ or $\delta_{0}=0$, occurs if and only if
$H$ is symmetric whereas the case $\delta_{0}\neq 0$ if and only if $H_{X}$ is
not symmetric.

Thus we say that if $\delta_{0}=0$, $d_{h}$ is symmetric and otherwise,
$d_{h}$ is not symmetric.

###### Proof.

They are proved in Lemma 5.4.

######  Remark 4.8.

We remark that $R_{H}$ and $R_{H}^{Z}$ are characterized by these parameters
$(M_{X}=\\{r_{i}\\},m_{X},k_{X},\\{{\mathfrak{e}}_{i}\\},\\{{\widehat{\mathfrak{e}}}_{i}\\},d_{h},\\{\delta_{i}\\},\ell)$.
Especially $\ell$ is a fixed number for a given $X$ in this paper.

######  Example 4.8.1.

  1. (1)

$H=\langle 4,6,7,9\rangle$ (non-symmetric) case $(d_{h}=\delta_{0}4+9,\
\ell=3)$ $\delta_{0}=1$:

$H=\\{0,4,6,7,8,9,10,11,\cdots\\}$, $H^{\mathfrak{c}}=\\{1,2,3,5\\}$.

$R_{H}={\mathbb{C}}[Z_{4},Z_{6},Z_{7},Z_{9}]/\sim$ and then

$\displaystyle
h_{{R_{Z}^{H}}}(Z,Z_{\bullet},Z_{\bullet}^{\prime})=Z_{4}Z_{9}+Z_{7}Z_{6}^{\prime}+Z_{6}Z_{7}^{\prime}+Z_{4}Z_{9}^{\prime},\quad
h_{H}(Z)=4Z_{4}Z_{9}.$

$i$ | 0 | 1 | 2 | 3
---|---|---|---|---
${\mathfrak{e}}_{i}$ | 0 | 6 | 7 | 9
${\widehat{\mathfrak{e}}}_{i}$ | $4+9$ | $7$ | $6$ | 4
  2. (2)

$H=\langle 5,7,11,13\rangle$ (non-symmetric) case $(d_{h}=\delta_{0}r,\
\ell=0)$:

$H=\\{0,5,7,10,11,12,13,14,\cdots\\}$, $H^{\mathfrak{c}}=\\{1,2,3,4,6,8,9\\}$.

$R_{H}={\mathbb{C}}[Z_{5},Z_{7},Z_{11},Z_{13}]/\sim$ and then

$\displaystyle
h_{{R_{Z}^{H}}}(Z,Z_{\bullet},Z_{\bullet}^{\prime})=Z_{5}^{5}+Z_{5}Z_{13}Z_{7}^{\prime}+Z_{14}Z_{11}^{\prime}+Z_{5}Z_{7}Z_{13}^{\prime}+Z_{11}Z_{14}^{\prime},\quad
h_{H}(Z)=5Z_{5}^{5}.$

$i$ | 0 | 1 | 2 | 3 | 4
---|---|---|---|---|---
${\mathfrak{e}}_{i}$ | 0 | 7 | 11 | 13 | 14
${\widehat{\mathfrak{e}}}_{i}$ | 25 | 13+5 | 14 | 7+5 | 11
  3. (3)

$H=\langle 6,13,14,15,16\rangle$ (symmetric) case
$(d_{h}={\mathfrak{e}}_{r-1},\ \delta_{0}=0,\ \ell=r-1)$:

$H=\\{0,6,12,13,14,15,16,18,19,20,21,22,24,\cdots\\}$,

$H^{\mathfrak{c}}=\\{1,2,3,4,5,7,8,9,10,11,17,23\\}$.

$R_{H}={\mathbb{C}}[Z_{6},Z_{13},Z_{14},Z_{15},Z_{16}]/\sim$ and noting
$Z_{13}Z_{16}=Z_{14}Z_{15}$

$\displaystyle
h_{{R_{Z}^{H}}}(Z,Z_{\bullet},Z_{\bullet}^{\prime})=Z_{13}Z_{16}+Z_{13}Z_{16}^{\prime}+Z_{14}Z_{15}^{\prime}+Z_{15}Z_{14}^{\prime}+Z_{16}Z_{13}^{\prime}+Z_{13}^{\prime}Z_{14}^{\prime}\quad
h_{H}(Z)=6Z_{13}Z_{16}.$

$i$ | 0 | 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---|---
${\mathfrak{e}}_{i}$ | 0 | 13 | 14 | 15 | 16 | 29
${\widehat{\mathfrak{e}}}_{i}$ | 29 | 16 | 15 | 14 | 13 | 0

By comparing the semigroup $H_{X}$ with $\\{{\mathfrak{e}}_{i}\\}$, we have
the following corollaries:

######  Corollary 4.9.

  1. (1)

${\widetilde{h}}_{H}(Z_{r},Z_{\bullet},Z^{\prime}_{\bullet})$ is a homogeneous
polynomial whose degree is $d_{h}={\mathfrak{e}}_{\ell}+\delta_{0}r$ in (4.8),

  2. (2)

$\displaystyle{{\widetilde{h}}_{H}(Z_{r},Z_{\bullet},Z^{\prime}_{\bullet})=\sum_{i=0}^{r-1}Z_{r}^{\delta_{i}}{\mathfrak{Z}}_{{\mathfrak{e}}_{\ell,i}^{*}}{\mathfrak{Z}}_{{\mathfrak{e}}_{i}}^{\prime}}$,

  3. (3)

$Z_{r}^{\delta_{0}}{\mathfrak{Z}}_{{\mathfrak{e}}_{\ell}}=Z_{r}^{\delta_{i}}{\mathfrak{Z}}_{{\mathfrak{e}}_{\ell,i}^{*}}{\mathfrak{Z}}_{{\mathfrak{e}}_{i}}$,

  4. (4)

$h_{H}(Z)=rZ_{r}^{\delta_{i}}{\mathfrak{Z}}_{{\mathfrak{e}}_{i}^{*}}{\mathfrak{Z}}_{{\mathfrak{e}}_{i}}=rZ_{r}^{\delta_{0}}{\mathfrak{Z}}_{{\mathfrak{e}}_{\ell}}$
for $i\in{\mathbb{Z}}_{r}$.

###### Proof.

They are obvious from Definition 2.7, Lemma 2.8 3, Lemma 3.6 and Lemma 4.6.

######  Corollary 4.10.

${\mathfrak{e}}_{i}-r-1\geq 0$, ${\widehat{\mathfrak{e}}}_{i}-r-1\geq 0$ and
$d_{h}-r-1\geq 0$.

###### Proof.

$\displaystyle{{\mathfrak{e}}_{i}\geq\min_{j=2}^{m_{X}}r_{j}=r_{2}\geq r+1}$
because of $r+1\leq r_{2}$.

######  Remark 4.11.

Corollary 4.9 determines the $R_{\mathbb{P}}$-module structure of $R_{X}$ in
Proposition 3.11.

####  4.2.3. Trace structure in Weierstrass curves (W-curves)

We use the $R_{\mathbb{P}}$-module structure of $R_{X}$ in the previous
Subsubsection and the properties in $R_{H}$ and $R_{H}^{Z}$ noting the
surjection $\varphi_{\infty}:R_{X}\to R_{H}$.

We define $p_{X}^{(j)}$ in the quotient field
${\mathcal{Q}}(R_{X}\otimes_{R_{\mathbb{P}}}R_{X})$ for every affine ring
${\mathbb{C}}[x,y]/(f_{X}^{(j)})$:

######  Definition 4.12.

For $f_{X}^{(j)}\in{\mathbb{C}}[x,y]$, we define

$p_{X}^{(j)}(x,y,y^{\prime}):=\frac{(\delta_{y,y^{\prime}}f_{X}^{(j)})(x,y,y^{\prime})}{(f_{X,y}^{(j)})(x,y)}.$

We regard $p_{X}^{(j)}(x,y,y^{\prime})$ as an element in
${\mathcal{Q}}({\mathbb{C}}[x,y]/(f_{X}^{(j)})\otimes_{{\mathbb{C}}[x]}{\mathbb{C}}[x,y]/(f_{X}^{(j)}))$
associated with ${\mathcal{Q}}(R_{X}\otimes_{R_{\mathbb{P}}}R_{X})$. We extend
the Galois group $G_{X}$ to the action on
$R_{X}\otimes_{R_{\mathbb{P}}}R_{X}$, such that
${\widehat{\zeta}}(x,y_{\bullet},y_{\bullet}^{\prime})=(x,{\widehat{\zeta}}y_{\bullet},{\widehat{\zeta}}y_{\bullet}^{\prime})$.

######  Lemma 4.13.

$\displaystyle{p_{X}^{(j)}(x,y,y^{\prime})=\left\\{\begin{array}[]{rcl}1&\mbox{
for }&y=y^{\prime}\\\ 0&\mbox{ for }&y\neq y^{\prime}\end{array}\right.}$ and
for a Galois action ${\widehat{\zeta}}\in G_{X}$,

$p_{X}^{(j)}(x,{\widehat{\zeta}}y,{\widehat{\zeta}}y^{\prime})=p_{X}^{(j)}(x,y,y^{\prime}).$

Let us consider $\displaystyle{p_{R_{X}}:=\prod_{j=2}^{m_{X}}p_{X}^{(j)}}$ as
an element of ${\mathcal{Q}}(R_{X}\otimes_{R_{\mathbb{P}}}R_{X})$. The
following is obvious:

######  Proposition 4.14.

For $(P,Q)\in X\times_{\mathbb{P}}X$,

$p_{R_{X}}(P,Q)=\left\\{\begin{array}[]{rcl}1&\mbox{ for }&P=Q,\\\ 0&\mbox{
for }&P\neq Q.\end{array}\right.$

However, some parts in its numerator and denominator are canceled because they
belong to $R_{\mathbb{P}}$. Thus we introduce an element
$h(x,y_{\bullet},y_{\bullet}^{\prime})\in R_{X}\otimes_{R_{\mathbb{P}}}R_{X}$
such that $h(x,y_{\bullet},y_{\bullet}^{\prime})/h(x,y_{\bullet},y_{\bullet})$
reproduces the product.

The ring homomorphism $\varphi^{X}_{R_{X}}$ in Proposition 3.7 is extended to
the surjective ring homomorphism from $R_{X}\otimes_{R_{\mathbb{P}}}R_{X}$ to
$R_{H}^{Z}\otimes_{{\mathbb{C}}[Z_{r}]}R_{H}^{Z}$.

######  Lemma 4.15.

For a point $(P=(x,y_{\bullet}),P^{\prime}=(x,y_{\bullet}^{\prime}))\in
X\times_{\mathbb{P}}X$, there is a polynomial
${\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime})\in
R_{X}\otimes_{R_{\mathbb{P}}}R_{X}$ such that

  1. (1)

by regarding the element $a$ in $R_{X}$ as $a\otimes 1$ in
$R_{X}\otimes_{R_{\mathbb{P}}}R_{X}$,
${\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet})$ and
${\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime})$ are coprime as
elements in $R_{X}\otimes_{R_{\mathbb{P}}}R_{X}$,

  2. (2)

for a Galois action ${\widehat{\zeta}}\in G_{X}$,
$\displaystyle{\frac{{\widetilde{h}}_{R_{X}}(x,{\widehat{\zeta}}y_{\bullet},{\widehat{\zeta}}y_{\bullet}^{\prime})}{{\widetilde{h}}_{R_{X}}(x,{\widehat{\zeta}}y_{\bullet},{\widehat{\zeta}}y_{\bullet})}=\frac{{\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime})}{{\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet})}}$,

  3. (3)

it satisfies
$\displaystyle{\frac{{\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime})}{{\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet})}=p_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime})}$,
and

  4. (4)

$\displaystyle{\varphi_{H}^{X}({\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime}))={\widetilde{h}}_{H}(Z_{r},Z_{\bullet},Z^{\prime}_{\bullet})}$
(${\mathrm{wt}}({\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime}))=d_{h}$
with respect to $(x,y)$) for ${\widetilde{h}}_{H}$ in Lemma 4.6 by letting
$\varphi_{H}^{X}(y_{i})=Z_{i}$,
$\varphi_{H}^{X}(y_{i}^{\prime})=Z_{i}^{\prime}$.

###### Proof.

By considering the numerator and denominator in
$\displaystyle{\prod_{j=2}^{m_{X}}p_{X}^{(j)}\left(\varpi_{r}(P),\varpi_{r_{j}}(P),\varpi_{r_{j}}(P^{\prime}).\right)}$
modulo $(f_{i}^{X})_{i=1,\ldots,k_{X}}$, they are reduced. Let the numerator
be denoted by ${\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime})\in
R_{X}\otimes_{R_{\mathbb{P}}}R_{X}$. It is obvious that
$\displaystyle{\frac{{\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime})}{{\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet})}}$
must be invariant for the Galois action $G_{X}$ on
$R_{X}\otimes_{R_{\mathbb{P}}}R_{X}$. Due to the condition (1), we have a
unique ${{\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet})}$. Then we can
find $d_{h}$ in Lemma 4.6 such that the image
$\varphi^{X}_{H}({\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime}))$
is equal to ${\widetilde{h}}_{H}$ in Lemma 4.6 because it is invariant for the
${\mathbb{G}}_{\mathfrak{m}}$ action; the reduction in (3) correspond to the
reduction in $R_{H}^{Z}\otimes_{{\mathbb{C}}[Z_{r}]}R_{H}^{Z}$ as in (4). Then
it is clearly that (1), (2), and (3) are satisfied.

######  Definition 4.16.

Let
$h_{X}(x,y_{\bullet}):={\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet})$.

Noting Corollary 4.9, we have the expression of
${\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime})$:

######  Proposition 4.17.

${\widetilde{h}}_{R_{X}}\in R_{X}\otimes_{R_{\mathbb{P}}}R_{X}$ is expressed
by

$\displaystyle\begin{split}{\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime})&={\widehat{\Upsilon}}_{0}\cdot
1+{\widehat{\Upsilon}}_{1}{\mathfrak{y}}^{\prime}_{{\mathfrak{e}}_{1}}+\cdots+{\widehat{\Upsilon}}_{r-1}{\mathfrak{y}}^{\prime}_{{\mathfrak{e}}_{r-1}}\\\
&=1\cdot{\widehat{\Upsilon}}_{0}^{\prime}+{\mathfrak{y}}_{{\mathfrak{e}}_{1}}{\widehat{\Upsilon}}_{1}^{\prime}+\cdots+{\mathfrak{y}}_{{\mathfrak{e}}_{r-1}}{\widehat{\Upsilon}}_{r-1}^{\prime}\\\
&={\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{0}}{\mathfrak{y}}^{\prime}_{{\mathfrak{e}}_{0}}+{\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{1}}{\mathfrak{y}}^{\prime}_{{\mathfrak{e}}_{1}}+\cdots+{\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{r-1}}{\mathfrak{y}}^{\prime}_{{\mathfrak{e}}_{r-1}}\\\
&\qquad\qquad+\mbox{ lower weight terms with respect to
$-{\mathrm{wt}}$}\end{split}$

as an $R_{\mathbb{P}}$-module. Here
${\mathfrak{y}}^{\prime}_{{\mathfrak{e}}_{0}}=1$, and each
${\widehat{\Upsilon}}_{i}$ holds the following properties:

  1. (1)

$\displaystyle{{\widehat{\Upsilon}}_{i}=\sum_{j=0}^{r-1}{\mathfrak{b}}_{i,j}{\mathfrak{y}}_{{\mathfrak{e}}_{j}},}$
with certain ${\mathfrak{b}}_{i,j}\in{\mathbb{C}}[x]$, and

  2. (2)

$\displaystyle{{\widehat{\Upsilon}}_{i}={\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}+\mbox{
lower weight terms with respect to $-{\mathrm{wt}}$}}$, where
$\displaystyle{{\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}={\widetilde{\delta}}_{i}(x){\mathfrak{y}}_{{\mathfrak{e}}_{\ell,i}^{*}}}$
with a monic polynomial ${\widetilde{\delta}}_{i}(x)\in{\mathbb{C}}[x]$ whose
weight is $-\delta_{i}r$, (especially,
${\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{0}}={\widetilde{\delta}}_{0}(x){\mathfrak{y}}_{{\mathfrak{e}}_{\ell,0}^{*}}={\widetilde{\delta}}_{0}(x){\mathfrak{y}}_{{\mathfrak{e}}_{\ell}}$)
such that

$\displaystyle{\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{0}}$ $\displaystyle=$
$\displaystyle{\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}{\mathfrak{y}}_{{\mathfrak{e}}_{i}}+\mbox{
lower weight terms with respect to $-{\mathrm{wt}}$}$

for $i\in{\mathbb{Z}}_{r}$,
${\mathrm{wt}}({\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}})=-{\widehat{\mathfrak{e}}}_{i}=-(d_{h}-{\mathfrak{e}}_{i})$
in Lemma 4.6, where $b_{i,j}$ is a certain element in $R_{\mathbb{P}}$ for
$(i,j)$.

###### Proof.

From the construction,
${\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime})={\widetilde{h}}_{R_{X}}(x,y_{\bullet}^{\prime},y_{\bullet})$;
${\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime})$ is invariant for
the exchanging between $y_{\bullet}$ and $y_{\bullet}^{\prime}$. From
Proposition 3.11, ${\widehat{\Upsilon}}_{i}$’s are uniquely determined. It is
obvious that ${\mathfrak{b}}_{ij}$ belongs to
${\mathbb{C}}[x]=R_{\mathbb{P}}$. However by letting $\ell_{i}$ satisfy
${\mathfrak{e}}_{\ell_{i}}={\mathfrak{e}}_{\ell,i}^{*}$, due to the Sato-
Weierstrass weight of ${\widehat{\Upsilon}}_{i}$,
$-{\mathrm{wt}}({\mathfrak{b}}_{i,j}{\mathfrak{y}}_{{\mathfrak{e}}_{j}})<-{\mathrm{wt}}({\mathfrak{b}}_{i,\ell_{i}}{\mathfrak{y}}_{{\mathfrak{e}}_{\ell_{i}}})$
and
${\mathrm{wt}}({\widehat{\Upsilon}}_{i})={\mathrm{wt}}({\mathfrak{b}}_{i,\ell_{i}}{\mathfrak{y}}_{{\mathfrak{e}}_{\ell_{i}}})$.
Thus we let ${\widetilde{\delta}}_{i}:={\mathfrak{b}}_{i,\ell_{i}}$, and then
we obtain
${\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}={\widetilde{\delta}}_{i}{\mathfrak{y}}_{{\mathfrak{e}}_{\ell,i}^{*}}$,
especially
${\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{0}}={\widetilde{\delta}}_{0}{\mathfrak{y}}_{{\mathfrak{e}}_{\ell}}$.
${\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}$ is monic from Corollary 4.9.
Hence ${\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime})$ must have
the form mentioned above.

Since ${\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{0}}$ and
${\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}{\mathfrak{y}}_{{\mathfrak{e}}_{i}}$
are homogeneous with respect to $-{\mathrm{wt}}_{\lambda}$ and correspond to
the relations in Corollary 4.9,
${\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{0}}-{\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}{\mathfrak{y}}_{{\mathfrak{e}}_{i}}$
is given by the lower weight terms with respect to $-{\mathrm{wt}}$. Further
(3.16) shows ${\widetilde{\delta}}_{0}{\mathfrak{y}}_{{\mathfrak{e}}_{\ell}}$.

$\displaystyle{\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}{\mathfrak{y}}_{{\mathfrak{e}}_{i}}$
$\displaystyle=$
$\displaystyle{\widetilde{\delta}}_{i}{\mathfrak{y}}_{{\mathfrak{e}}_{\ell_{i}}}{\mathfrak{y}}_{{\mathfrak{e}}_{i}}=\sum_{j\in{\mathbb{Z}}_{r}}{\widetilde{\delta}}_{i}{\mathfrak{a}}_{i,\ell_{i},j}{\mathfrak{y}}_{{\mathfrak{e}}_{j}}$
$\displaystyle=$
$\displaystyle{\widetilde{\delta}}_{i}{\mathfrak{a}}_{i,\ell_{i},\ell}{\mathfrak{y}}_{{\mathfrak{e}}_{\ell}}+\mbox{
lower weight terms with respect to $-{\mathrm{wt}}$}.$

Unless
${\widetilde{\delta}}_{i}{\mathfrak{a}}_{i,\ell_{i},\ell}/{\widetilde{\delta}}_{0}$
belongs to $R_{\mathbb{P}}$, it must be expressed $\alpha/\beta$. where
$\alpha$ and $\beta$ are elements of $R_{\mathbb{P}}$ and their Sato-
Weierstrass weights are the same, and thus we redefine

${\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}:=\beta{\widetilde{\delta}}_{i}{\mathfrak{y}}_{{\mathfrak{e}}_{\ell,i}^{*}},\quad{\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{0}}:=\alpha{\widetilde{\delta}}_{0}{\mathfrak{y}}_{{\mathfrak{e}}_{\ell}}.$

We repeat such operation for each $i$ and then, for every $i$, we obtain

${\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}{\mathfrak{y}}_{{\mathfrak{e}}_{i}}={\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{0}}+\sum_{j\in{\mathbb{Z}}_{r},j\neq\ell}{\widetilde{\delta}}_{i}{\mathfrak{a}}_{i,\ell_{i},j}{\mathfrak{y}}_{{\mathfrak{e}}_{j}}.$

However if ${\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime})$ and
${\widetilde{h}}_{X}(x,y_{\bullet},y_{\bullet})=h_{X}(x,y_{\bullet})$ are not
coprime, it means that there is a cofactor in
${\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime})$ and we can
divide ${\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet})$ by the factor.

Recalling Proposition 4.7, ${\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}$
and $h_{X}(x,y_{\bullet})$ are expressed as follows.

######  Corollary 4.18.

  1. (1)

If $d_{h}$ is symmetric,
$\displaystyle{{\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}={\widetilde{\delta}}_{i}(x){\mathfrak{y}}_{{\mathfrak{e}}_{r-1,i}^{*}}}$
for $i\in{\mathbb{Z}}_{r}$,
${\widetilde{\delta}}_{0}={\widetilde{\delta}}_{r-1}=1$,
${\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{0}}={\mathfrak{y}}_{{\mathfrak{e}}_{r-1}}$,
and
${\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{r-1}}={\mathfrak{y}}_{{\mathfrak{e}}_{0}}=1$.

  2. (2)

If $d_{h}$ is not symmetric,
${\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}={\widetilde{\delta}}_{i}(x){\mathfrak{y}}_{{\mathfrak{e}}_{\ell,i}^{*}}$
for $i\in{\mathbb{Z}}_{r}^{\times}$, and ${\widetilde{\delta}}_{0}\neq 1$.

  3. (3)

$h_{X}(x,y_{\bullet})=r{\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{0}}+$ lower
weight terms with respect to $-{\mathrm{wt}}$.

We extend the arguments for $m_{X}=2$ case.

Obviously, due to Proposition 3.11, the dual $\omega_{R_{X}/R_{\mathbb{P}}}$
of $R_{X}$ as an $R_{\mathbb{P}}$-module has the standard basis as a trace.

######  Lemma 4.19.

We define

$\Delta_{{\widetilde{h}}}:={\widehat{\Upsilon}}_{0}\otimes
1+\sum_{k\in{\mathbb{Z}}_{r}^{\times}}{\widehat{\Upsilon}}_{k}\otimes{\mathfrak{y}}_{{\mathfrak{e}}_{k}}=\sum_{i,k\in{\mathbb{Z}}_{r}}{\mathfrak{y}}_{{\mathfrak{e}}_{i}}{\mathfrak{b}}_{i,k}\otimes{\mathfrak{y}}_{{\mathfrak{e}}_{k}}=\sum_{i,k\in{\mathbb{Z}}_{r}}{\mathfrak{y}}_{{\mathfrak{e}}_{i}}\otimes{\mathfrak{b}}_{i,k}{\mathfrak{y}}_{{\mathfrak{e}}_{k}},$

Then for every $a\in R_{X}$, $(a\otimes 1-1\otimes
a)\Delta_{{\widetilde{h}}}=0$ and $\Delta_{{\widetilde{h}}}R_{X}^{e}$ is equal
to $\mathrm{Ann}_{R_{X}^{e}}(\mathrm{Ker}\mu_{\mathcal{Q}})$.

###### Proof.

First we consider
${\mathfrak{y}}_{{\mathfrak{e}}_{j}}{\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime})$
$=\sum_{j,k}{\mathfrak{b}}_{j,k}{\mathfrak{y}}_{{\mathfrak{e}}_{i}}{\mathfrak{y}}_{{\mathfrak{e}}_{j}}{\mathfrak{y}}_{{\mathfrak{e}}_{k}}^{\prime}=\sum_{j,k,\ell}{\mathfrak{b}}_{j,k}{\mathfrak{a}}_{ij\ell}{\mathfrak{y}}_{{\mathfrak{e}}_{\ell}}{\mathfrak{y}}_{{\mathfrak{e}}_{k}}^{\prime}$
and
${\mathfrak{y}}_{{\mathfrak{e}}_{j}}^{\prime}{\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime})$
$=\sum_{j,k,\ell}{\mathfrak{b}}_{j,k}{\mathfrak{a}}_{ik\ell}{\mathfrak{y}}_{{\mathfrak{e}}_{j}}{\mathfrak{y}}_{{\mathfrak{e}}_{\ell}}^{\prime}$.
The latter can be expressed by
$\sum_{\ell,j,k}{\mathfrak{b}}_{\ell,j}{\mathfrak{a}}_{ijk}{\mathfrak{y}}_{{\mathfrak{e}}_{\ell}}{\mathfrak{y}}_{{\mathfrak{e}}_{k}}^{\prime}$.
Lemma 4.15 with Proposition 4.14 shows that both cases agree

${\mathfrak{y}}_{{\mathfrak{e}}_{j}}{\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime})={\mathfrak{y}}_{{\mathfrak{e}}_{j}}^{\prime}{\widetilde{h}}_{R_{X}}(x,y_{\bullet},y_{\bullet}^{\prime})=\left\\{\begin{array}[]{rl}{\mathfrak{y}}_{{\mathfrak{e}}_{j}}h_{X}&\mbox{
for }y_{\bullet}=y_{\bullet}^{\prime},\\\ 0&\mbox{ otherwise.
}\end{array}\right.$

It means that
$\sum_{j}{\mathfrak{b}}_{k,j}{\mathfrak{a}}_{ij\ell}=\sum_{j}{\mathfrak{b}}_{\ell,j}{\mathfrak{a}}_{ijk}$.
This relation shows
$\sum_{i,k}{\mathfrak{b}}_{j,k}{\mathfrak{y}}_{{\mathfrak{e}}_{i}}{\mathfrak{y}}_{{\mathfrak{e}}_{j}}\otimes{\mathfrak{y}}_{{\mathfrak{e}}_{k}}=\sum_{i,k}{\mathfrak{b}}_{j,k}{\mathfrak{y}}_{{\mathfrak{e}}_{j}}\otimes{\mathfrak{y}}_{{\mathfrak{e}}_{i}}{\mathfrak{y}}_{{\mathfrak{e}}_{k}}$
and we obtain the equality $(a\otimes 1-1\otimes
a)\Delta_{{\widetilde{h}}}=0$.

Further from Proposition 2.5, $\Delta_{{\widetilde{h}}}$ in Lemma 4.19
provides the trace $\tau_{\widetilde{h}}\in\omega_{R_{X}/R_{\mathbb{P}}}$ with
the dual basis $\\{{\widehat{\Upsilon}}_{i}\\}$ of $R_{X}$ with respect to the
trace $\tau_{\widetilde{h}}$.

######  Lemma 4.20.

$\\{{\widehat{\Upsilon}}_{i}\\}$ is the dual basis of $R_{X}$ with respect to
the trace $\tau_{\widetilde{h}}$, such that

$\tau_{\widetilde{h}}({\widehat{\Upsilon}}_{i})=\left\\{\begin{matrix}1&\mbox{
for }i=0,\\\ 0&\mbox{ otherwise. }\end{matrix}\right.$

###### Proof.

These $\\{{\widehat{\Upsilon}}_{i}\\}$ correspond to the dual basis
$\\{\widehat{{\mathbf{y}}}_{i}\\}$ of $R_{X}$ with respect to the trace
$\tau_{\widetilde{h}}$. By considering

$1=\tau_{\widetilde{h}}({\widehat{\Upsilon}}_{0})\cdot
1+\sum_{k\in{\mathbb{Z}}_{r}^{\times}}\tau_{\widetilde{h}}\left({\widehat{\Upsilon}}_{k}\right){\mathfrak{y}}^{\prime}_{{\mathfrak{e}}_{k}},$

we have the relation.

We introduce an $R_{\mathbb{P}}$-module,
$R_{X,\tau_{\widetilde{h}}}^{*}:=\langle{\widehat{\Upsilon}}_{0},{\widehat{\Upsilon}}_{1},\ldots,{\widehat{\Upsilon}}_{r-1}\rangle_{R_{\mathbb{P}}}$,
and consider its structure as an $R_{X}$-module. Due to Proposition 2.5,
$\omega_{R_{X}/R_{\mathbb{P}}}=R_{X}\tau_{\widetilde{h}}$, and thus
$\omega_{R_{X}/R_{\mathbb{P}}}\cong R_{X,\tau_{\widetilde{h}}}^{*}$ as an
$R_{X}$-module:

######  Lemma 4.21.

$R_{X,\tau_{\widetilde{h}}}^{*}$ is an ideal of $R_{X}$, especially
${\mathfrak{y}}_{{\mathfrak{e}}_{i}}R_{X,\tau_{\widetilde{h}}}^{*}\subset
R_{X,\tau_{\widetilde{h}}}^{*}$, and

$R_{X,\tau_{\widetilde{h}}}^{*}=\langle{\widehat{\Upsilon}}_{0},{\widehat{\Upsilon}}_{1},\ldots,{\widehat{\Upsilon}}_{r-1}\rangle_{R_{X}}=\langle{\widehat{\Upsilon}}_{1},\ldots,{\widehat{\Upsilon}}_{r-1}\rangle_{R_{X}}.$

It means that as $R_{\mathbb{P}}$-modules,

$\langle{\widehat{\Upsilon}}_{0},{\widehat{\Upsilon}}_{1},\ldots,{\widehat{\Upsilon}}_{r-1}\rangle_{R_{\mathbb{P}}}=\langle{\widehat{\Upsilon}}_{1},\ldots,{\widehat{\Upsilon}}_{r-1}\rangle_{R_{X}}.$

###### Proof.

Let us consider ${\mathfrak{y}}_{{\mathfrak{e}}_{i}}{\widehat{\Upsilon}}_{j}$,
which is equal to

${\mathfrak{y}}_{{\mathfrak{e}}_{i}}{\widehat{\Upsilon}}_{j}=\sum_{k=0}^{r-1}{\mathfrak{b}}_{jk}{\mathfrak{y}}_{{\mathfrak{e}}_{j}}{\mathfrak{y}}_{{\mathfrak{e}}_{k}}=\sum_{k,\ell=0}^{r-1}{\mathfrak{b}}_{jk}{\mathfrak{a}}_{ik\ell}{\mathfrak{y}}_{{\mathfrak{e}}_{\ell}}.$

However from the proof in Lemma 4.19,
$\displaystyle{\sum_{k}{\mathfrak{b}}_{jk}{\mathfrak{a}}_{ik\ell}=\sum_{k}{\mathfrak{b}}_{\ell
k}{\mathfrak{a}}_{ikj}}$, and thus

${\mathfrak{y}}_{{\mathfrak{e}}_{i}}{\widehat{\Upsilon}}_{j}=\sum_{k=0}^{r-1}{\mathfrak{b}}_{k\ell}{\mathfrak{a}}_{ikj}{\mathfrak{y}}_{{\mathfrak{e}}_{\ell}}=\sum_{k=0}^{r-1}{\mathfrak{a}}_{ikj}{\widehat{\Upsilon}}_{k}.$

Hence, $R_{X,\tau_{\widetilde{h}}}^{*}$ is closed for the action of
${\mathfrak{y}}_{{\mathfrak{e}}_{i}}$.

On the other hand, we take an integer $i$ ($i\neq\ell$) and then consider
${\mathfrak{y}}_{{\mathfrak{e}}_{i}}{\widehat{\Upsilon}}_{i}$ which is
decomposed by ${\widehat{\Upsilon}}_{i}$’s but has the form,

(4.10)
${\mathfrak{y}}_{{\mathfrak{e}}_{i}}{\widehat{\Upsilon}}_{i}={\widehat{\Upsilon}}_{0}+\sum_{j=1}^{r-1}{\mathfrak{d}}_{ij}{\widehat{\Upsilon}}_{j},$

where ${\mathfrak{d}}_{ij}$ belongs to $R_{\mathbb{P}}$ because the leading
terms of the both
${\mathfrak{y}}_{{\mathfrak{e}}_{i}}{\widehat{\Upsilon}}_{i}$ and
${\widehat{\Upsilon}}_{0}$ must agree in the both sides. It means that
${\widehat{\Upsilon}}_{0}\in\langle{\widehat{\Upsilon}}_{1},\ldots,{\widehat{\Upsilon}}_{r-1}\rangle_{R_{X}}$.

This proof shows the following lemma:

######  Lemma 4.22.

$\tau_{\widetilde{h}}({\mathfrak{y}}_{{\mathfrak{e}}_{i}}{\widehat{\Upsilon}}_{i})=1$
for every $i\in{\mathbb{Z}}_{r}^{\times}$, and thus
$\tau_{\widetilde{h}}({\mathfrak{y}}_{{\mathfrak{e}}_{i}}{\widehat{\Upsilon}}_{j})=\delta_{ij}$.

###### Proof.

(4.10) shows this.

Noting Remark 4.4 for the $m_{X}=2$ case, we also introduce more convenient
basis
$\\{{\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}\\}_{i\in{\mathbb{Z}}_{R}}$
with respect to $\tau_{{\widetilde{h}}}$ instead of
$\\{{\widehat{\Upsilon}}_{i}\\}_{i\in{\mathbb{Z}}_{R}}$:

######  Definition 4.23.

For $i\in{\mathbb{Z}}_{r}$, we define a truncated polynomial
${\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}$ of ${\widehat{\Upsilon}}_{i}$
such that the weight $-{\mathrm{wt}}$ of
${\widehat{\Upsilon}}_{i}-{\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}$ is
less than $-{\mathrm{wt}}({\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}})$,
i.e.,
${\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}={\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}+$
certain terms, and the number of the terms is minimal satisfying the relations
as $R_{\mathbb{P}}$-modules,

$\langle{\widehat{\Upsilon}}_{1},\ldots,{\widehat{\Upsilon}}_{r-1}\rangle_{R_{X}}=\langle{\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{1}},\ldots,{\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{r-1}}\rangle_{R_{X}}=\langle{\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{0}},\ldots,{\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{r-1}}\rangle_{R_{\mathbb{P}}},\quad\tau_{{\widetilde{h}}}({\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{i}})=\left\\{\begin{matrix}1&\mbox{
for }i=0,\\\ 0&\mbox{ otherwise. }\end{matrix}\right.$

######  Lemma 4.24.

  1. (1)

When $d_{h}$ is symmetric,
${\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}={\mathfrak{y}}_{{\mathfrak{e}}_{r-1-i}}$,
especially for $m_{X}=2$ case,
${\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}={\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}=y_{s}^{i}$.

  2. (2)

When $X$ has a cyclic symmetry of order $r$,
${\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}={\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}$.

###### Proof.

The $m_{X}=2$ case is described in Subsection 4.2 and the statement is
obvious. For symmetric $d_{h}$ case, under which the numerical semigroup
$H_{X}$ is symmetric due to Proposition 4.7,
${\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}={\mathfrak{y}}_{{\mathfrak{e}}_{r-1-i}}$,
(${\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{r-1}}={\widehat{\Upsilon}}_{r-1}=1$)
and thus it is obvious

$\langle{\widehat{\Upsilon}}_{1},{\widehat{\Upsilon}}_{2},\ldots,{\widehat{\Upsilon}}_{r-1}\rangle_{R_{X}}=\langle{\mathfrak{y}}_{{\mathfrak{e}}_{0}},{\mathfrak{y}}_{{\mathfrak{e}}_{1}},\ldots,{\mathfrak{y}}_{{\mathfrak{e}}_{r-2}}\rangle_{R_{X}}=R_{X}.$

Further, when $X$ has the cyclic symmetry of the order $r$,
${\widehat{\Upsilon}}_{i}={\mathring{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}$
because of the invariance for the cyclic action.

Since $\mu(\Delta_{{\widetilde{h}}})=h_{X}(x,y_{\bullet})$, the trace
$\tau_{R_{X}/R_{\mathbb{P}}}$ of $R_{X}/R_{\mathbb{P}}$ is given by

$\tau_{R_{X}/R_{\mathbb{P}}}=h_{X}(x,y_{\bullet})\cdot\tau_{\widetilde{h}}.$

Then we obviously have the following lemma:

######  Lemma 4.25.

For a monomial $\phi$ in $R_{X}$ as an $R_{\mathbb{P}}$-module and
${\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}$, we have the following:

  1. (1)

When $d_{h}$ is symmetric $(d_{h}={\mathfrak{e}}_{r-1},\delta_{0}=0)$ in
(4.8),

$\displaystyle{\tau_{R_{X}/R_{\mathbb{P}}}\left(\frac{{\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}}{h_{X}(x,y_{\bullet})}\right)=\left\\{\begin{array}[]{ll}1&\mbox{
for }i=0,\\\ 0&\mbox{ otherwise. }\end{array}\right.}$

  2. (2)

When $d_{h}$ is not symmetric
$(d_{h}={\mathfrak{e}}_{\ell}+\delta_{0}r,\delta_{0}\neq 0)$ in (4.8),

$\displaystyle{\tau_{R_{X}/R_{\mathbb{P}}}\left(\frac{\phi}{h_{X}(x,y_{\bullet})}\right)=\left\\{\begin{array}[]{ll}\phi/{\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{0}}&\mbox{
for }{\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{0}}|\phi,\\\ 0&\mbox{
otherwise. }\\\ \end{array}\right.\quad}$

Here we note that the case
${\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{0}}|\phi$ means
$({\mathfrak{y}}_{{\mathfrak{e}}_{\ell}}{\widetilde{\delta}}_{0})|\phi$
whereas the case ${\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{0}}\nmid\phi$
consists of two cases 1) $\phi=(f(x){\mathfrak{y}}_{{\mathfrak{e}}_{\ell}})$,
${\widetilde{\delta}}_{0}\nmid f$ and 2)
$\phi=(f(x){\mathfrak{y}}_{{\mathfrak{e}}_{i}})\
i\in{\mathbb{Z}}_{r}\setminus\\{\ell\\}$.

Let us consider elements in $R_{X}^{{\mathfrak{c}}}$ as in (4.2). We consider
$u\in{\mathcal{Q}}(R_{X})$ given by

(4.11)
$u=\frac{1}{h_{X}(x,y_{\bullet})}\sum_{i=0}^{r-1}a_{i}{\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}\in{\mathcal{Q}}(R_{X}),$

which satisfies the condition in (4.2). Indeed, for any element
$\displaystyle{v=\sum_{j=0}^{r-1}b_{j}{\mathfrak{y}}_{{\mathfrak{e}}_{j}}\in
R_{X}}$, ($b_{j}\in R_{\mathbb{P}}$), $u$ satisfies
$\tau_{R_{X}/R_{\mathbb{P}}}(uv)\in R_{\mathbb{P}}$, i.e.

$\tau_{R_{X}/R_{\mathbb{P}}}(uv)=\sum_{i,j}\tau_{\widetilde{h}}(a_{i}b_{j}{\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}{\mathfrak{y}}_{{\mathfrak{e}}_{j}})=\sum_{i=0}^{r-1}(a_{i}b_{i})\in
R_{\mathbb{P}}.$

The condition $\tau_{R_{X}/R_{\mathbb{P}}}(uv)\in R_{\mathbb{P}}$ means that
$a_{i}$ belongs to $R_{\mathbb{P}}$ for every $i=0,1,\ldots,r-1$, and thus we
obtain $R_{X}^{\mathfrak{c}}$.

We, now, state the first theorem in this paper.

######  Theorem 4.26.

The complementary module $R_{X}^{\mathfrak{c}}$ is given as a fractional ideal
of $R_{X}$,

$R_{X}^{\mathfrak{c}}=\frac{\langle{\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{1}},\ldots,{\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{r-1}}\rangle_{R_{X}}}{h_{X}(x,y_{\bullet})}.$

###### Proof.

It is obvious that due to the above arguments and the identity
$\langle{\widehat{\Upsilon}}_{0},{\widehat{\Upsilon}}_{1},\ldots,{\widehat{\Upsilon}}_{r-1}\rangle_{R_{X}}=\langle{\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{1}},\ldots,{\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{r-1}}\rangle_{R_{X}}$
due to Lemma 4.21 and Definition 4.23.

Let us identify the generator ${\mathfrak{h}}_{X,P}$ of the principal ideal
$R_{X,P}^{\mathfrak{c}}$ locally.

######  Proposition 4.27.

  1. (1)

Symmetric $d_{h}$ case ($\delta_{0}=0$ in (4.8) or
${\widetilde{\delta}}_{0}=1$ and $d_{h}={\mathfrak{e}}_{r-1}$): The
complementary module (as a fractional ideal) is given by

(4.12)
$R_{X}^{\mathfrak{c}}=\frac{1}{h_{X}(x,y_{\bullet})}R_{X}={\mathfrak{h}}_{X}R_{X},$

and we define
$\displaystyle{{\mathfrak{h}}_{X}:=\frac{1}{h_{X}(x,y_{\bullet})}}$.

  2. (2)

Non-symmetric $d_{h}$ case ($\delta_{0}\neq 0$ in (4.8) or
${\widetilde{\delta}}_{0}\neq 1$ and
$d_{h}={\mathfrak{e}}_{\ell}+\delta_{0}r$: For complex numbers $(a_{i}(\neq
0))_{i}$, e.g., $a_{i}=1$, we define

(4.13)
${\mathfrak{h}}_{X}:=\frac{\sum_{i=1}^{r-1}a_{i}{\widehat{\mathfrak{y}}}_{{\mathfrak{e}}_{i}}}{h_{X}(x,y_{\bullet})}.$