Boubacar Sidy Balde, Mohamadou Mor Diogou Diallo, Oumar Sall
Mathematics and Applications Laboratory (MAL), U.F.R of Sciences and Technologies, Université Assane Seck of Ziguinchor, Ziguinchor, Senegal.
DOI: 10.4236/apm.2022.129039   PDF    HTML   XML   286 Downloads   1,031 Views   Citations

Abstract

In this work, we use the finiteness of the Mordell-weil group and the Riemann Roch spaces to give a geometric parametrization of the set of algebraic points of any given degree over the field of rational numbers Q on curve C₃ (11): y¹¹ = x³ (x-1)³. This result is a special case of quotients of Fermat curves C_r,s (p) : y^p = x^r(x-1)^s, 1 ≤ r, s, r + s ≤ p-1 for p = 11 and r = s = 3. The results obtained extend the work of Gross and Rohrlich who determined the set of algebraic points on C₁(11)(K) of degree at most 2 on Q.

Keywords

Mordell-Weil Group, Jacobian, Galois Conjugates, Algebraic Extensions, the Abel-Jacobi Theorem, Linear Systems

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1. Introduction

Let $\mathcal{C}$ be an algebraic curve defined on number field $\mathbb{K}$. We note $\mathcal{C}\left(\mathbb{K}\right)$ be the set of algebraic points on $\mathcal{C}$ defined on $\mathbb{K}$ and ${\displaystyle {\cup }_{\left[\mathbb{K}\mathrm{<mo>\; :\; </mo>}\mathbb{Q}\right]\le l}}\mathcal{C}\left(\mathbb{K}\right)$ the set of algebraic points on $\mathcal{C}$ to be coordinated in $\mathbb{K}$ of degree at most l over $\mathbb{Q}$. The degree of an algebraic point R is the degree of its defining field on $\mathbb{Q}$ ; $\mathrm{deg}\left(R\right)=\left[\mathbb{Q}\left(R\right)\mathrm{<mo>\; :\; </mo>}\mathbb{Q}\right]$. A famous theorem of Faltings states that if $g\ge 2$ then the set $\mathcal{C}\left(\mathbb{K}\right)$ of algebraic points on $\mathcal{C}$ defined on $\mathbb{K}$ is finite. A generalization to subvarieties of an abelian variety allows a qualitative study of the set ${\displaystyle {\cup }_{\left[\mathbb{K}\mathrm{<mo>\; :\; </mo>}\mathbb{Q}\right]\le l}}\mathcal{C}\left(\mathbb{K}\right)$ of algebraic points on $\mathcal{C}$ of degree at most l over $\mathbb{Q}$.

We propose to study in detail the set of algebraic points of any degree given on $\mathbb{Q}$ on the curve ${\mathcal{C}}_3\left(11\right)$ of affine equation $y^{11}=x^3{\left(x-1\right)}^3$.

Our affine equation curve ${\mathcal{C}}_3\left(11\right):y^{11}=x^3{\left(x-1\right)}^3$ is a special case of quotients of Fermat curves of equations ${\mathcal{C}}_{r,s}\left(p\right):y^p=x^r{\left(x-1\right)}^s$, $1\le r\mathrm{,}s\mathrm{,}r+s\le p-1$ studied in [1].

Let $P_0=\left(\mathrm{0\; <mo>\; :\; </mo>0\; <mo>\; :\; </mo>0}\right)$, $P_1=\left(1:0:1\right)$ and $P_{\infty }=\left(\mathrm{1\; <mo>\; :\; </mo>0\; <mo>\; :\; </mo>0}\right)$ denote the point at infinity of ${\mathcal{C}}_3\left(11\right)$. Consider the Jacobian folding defined by

$\begin{array}{ccc}j\mathrm{<mo>\; :\; </mo>}{\mathcal{C}}_3\left(11\right)\left(\mathbb{Q}\right)& \to & J\left(\mathbb{Q}\right)\\ P& \mapsto & \left[P-P_{\infty }\right]\end{array}$

We will designate J the Jacobian of ${\mathcal{C}}_3\left(11\right)$ and by $j\left(P\right)$ the class denoted $\left[P-P_{\infty }\right]$ of $P-P_{\infty }$.

Our approach relies on the knowledge of the Mordell-Weil group of the Jacobian J-variety of ${\mathcal{C}}_3\left(11\right)$ and the condition that it is finite: it consists in using the Abel-Jacobi theorem to plunge the curve into its Jacobian and to study linear systems on the curve ${\mathcal{C}}_3\left(11\right)$.

The Mordell-Weil group $J\left(\mathbb{Q}\right)$ of rational points of the Jacobian J of ${\mathcal{C}}_3\left(11\right)$ is finite and given by $J\left(\mathbb{Q}\right)\cong \left(\mathbb{Z}/11\mathbb{Z}\right)$ ( [2], p. 219 and [3]).

Our study results from the work of Gross-Rohrlich who determined ${\displaystyle {\cup }_{\left[\mathbb{K}\mathrm{<mo>\; :\; </mo>}\mathbb{Q}\right]\le 2}}{\mathcal{C}}_1\left(11\right)\left(\mathbb{K}\right)$ the set of algebraic points on ${\mathcal{C}}_1\left(11\right)\left(\mathbb{K}\right)$ of degree at most 2 on $\mathbb{Q}$ and given by the following proposition:

Proposition 1.

The set of algebraic points on ${\mathcal{C}}_1\left(11\right)\left(\mathbb{K}\right)$ of degree at most 2 on $\mathbb{Q}$ is given by

${\displaystyle \underset{\left[\mathbb{K}\mathrm{<mo>\; :\; </mo>}\mathbb{Q}\right]\le 2}{\cup }}{\mathcal{C}}_1\left(11\right)\left(\mathbb{K}\right)=\left\{\left(\frac{1}{2}\pm \sqrt{y^{11}+\frac{1}{4}}\mathrm{,}y\right)\right\}\cup \left\{P_{\infty }\right\}$ (1)

We extend these results by giving a geometric parametrization of algebraic points of any given degree on $\mathbb{Q}$ on the curve ${\mathcal{C}}_3\left(11\right)$ of affine equation $y^{11}=x^3{\left(x-1\right)}^3$.

Our essential tools are:

1) The Mordell-Weil group $J\left(\mathbb{Q}\right)$ of the Jacobian of $\mathcal{C}$.

2) The Abel-Jacobi theorem (see in [4] page 156).

3) The study of linear systems on the curve ${\mathcal{C}}_3\left(11\right)$.

4) The theory of intersection.

Our main result is as follows:

Theorem

The set of algebraic points of degree $l\ge 9$ on ${\mathcal{C}}_3\left(11\right)$ is:

${\displaystyle \underset{\left[\mathbb{K}\mathrm{<mo>\; :\; </mo>}\mathbb{Q}\right]\le l}{\cup }}{\mathcal{C}}_3\left(11\right)\left(\mathbb{K}\right)={\mathcal{F}}_0\cup \left({\displaystyle \underset{k=1}{\overset{10}{\cup }}}{\mathcal{F}}_k\right)$ (2)

with

$\begin{array}{l}{\mathcal{F}}_0=\{\left(-\frac{{\displaystyle \underset{i\le \frac{l}{2}}{\sum }}{a}_i{y}^{\frac{i}{3}}}{{\displaystyle \underset{j\le \frac{l-11}{2}}{\sum }}{b}_j{y}^{\frac{j}{3}}},y\right)|a_0\ne 0,a_{\frac{l}{2}}\ne 0\text{if}l\text{is}\text{even},\\ b_{\frac{l-11}{2}}\ne 0\text{if}l\text{is}\text{odd}\text{and}y\text{root}\text{of}\text{the}\text{equation}\\ \begin{array}{c}\stackrel{}{}\\ \\ \\ \end{array}{y}^{11}{\left({\displaystyle \underset{j\le \frac{l-11}{2}}{\sum }}{b}_j{y}^j\right)}^2=\left({\displaystyle \underset{i\le \frac{l}{2}}{\sum }}{a}_i{y}^i\right){\left({\displaystyle \underset{i\le \frac{l}{2}}{\sum }}{a}_i{y}^{\frac{i}{3}}+{\displaystyle \underset{j\le \frac{l-11}{2}}{\sum }}{b}_j{y}^{\frac{j}{3}}\right)}^3\}\end{array}$ (3)

$\begin{array}{l}{\mathcal{F}}_k=\{\left(-\frac{{\displaystyle \underset{11-k\le i\le \frac{l+11-k}{2}}{\sum }}{a}_i{y}^{\frac{i}{3}}}{{\displaystyle \underset{j\le \frac{l-k}{2}}{\sum }}{b}_j{y}^{\frac{j}{3}}},y\right)|b_0\ne 0,a_{\frac{l+11-k}{2}}\ne 0\text{if}l\text{is}\text{even},\\ b_{\frac{l-k}{2}}\ne 0\text{if}l\text{is}\text{odd}\text{and}y\text{root}\text{of}\text{the}\text{equation}\\ \begin{array}{c}\stackrel{}{}\\ \\ \\ \end{array}{y}^k{\left({\displaystyle \underset{j\le \frac{l-k}{2}}{\sum }}{b}_j{y}^j\right)}^2=\left({\displaystyle \underset{11-k\le i\le \frac{l+11-k}{2}}{\sum }}{a}_i{y}^{i-\left(11-k\right)}\right){\left({\displaystyle \underset{11-k\le i\le \frac{l+11-k}{2}}{\sum }}{a}_i{y}^{\frac{i}{3}}+{\displaystyle \underset{j\le \frac{l-k}{2}}{\sum }}{b}_j{y}^{\frac{j}{3}}\right)}^3\}\end{array}$ (4)

2. Auxiliary Results

Let x and y be the rational functions defined on ${\mathcal{C}}_3\left(11\right)$ by: $x\left(X\mathrm{,}Y\mathrm{,}Z\right)=\frac{X}{Z}$ and $y\left(X\mathrm{,}Y\mathrm{,}Z\right)=\frac{Y}{Z}$.

For a divisor D on ${\mathcal{C}}_3\left(11\right)$, let $\mathcal{L}\left(D\right)$ be the $\bar{\mathbb{Q}}$ -vector space of the rational functions f defined by

$\mathcal{L}\left(D\right)=\left\{f\in \bar{\mathbb{Q}}{\left({\mathcal{C}}_3\left(11\right)\right)}^{*}|div\left(f\right)\ge -D\right\}\cup \left\{0\right\}$ (5)

The projective equation of the curve ${\mathcal{C}}_3\left(11\right)$ is: $Y^{11}=X^3{Z}^5{\left(X-Z\right)}^3$.

We have the following Lemma:

Lemma 1

${\mathcal{C}}_3\left(11\right):y^{11}=x^3{\left(x-1\right)}^3$

$div\left(x\right)=11P_0-11P_{\infty }$ ;

$div\left(y\right)=3P_0+3P_1-6P_{\infty }$ ;

$div\left(x-1\right)=11P_1-11P_{\infty }$.

Proof 1 It is a calculation of type

$div\left(x-i\right)=\left(\left(X-iZ\right)=0\right).{\mathcal{C}}_3\left(11\right)-\left(Z=0\right).{\mathcal{C}}_3\left(11\right)$ (6)

From (6), we have $div\left(x\right)=\left(X=0\right).\mathcal{C}-\left(Z=0\right).\mathcal{C}$.

For $X=0$, the projective equation gives $Y^{11}=0$ ; and for $Z=1$, we obtain the point $P_0=\left(0:0:1\right)$ of multiplicity equal to 11.

For $Z=0$, the projective equation gives $Y^{11}=0$ ; and for $X=1$, we obtain the point $P_{\infty }=\left(1:0:0\right)$ of multiplicity equal to 11. Thus $div\left(x\right)=11P_0-11P_{\infty }$.

In the same way we show that $div\left(x-1\right)=11P_1-11P_{\infty }$ and $div\left(y\right)=3P_0+3P_1-6P_{\infty }$.

Consequence 1

$11j\left(P_0\right)=11j\left(P_1\right)=0$ ;

$3j\left(P_0\right)+3j\left(P_1\right)=0$

so $j\left(P_0\right)$ and $j\left(P_1\right)$ generate the same subgroup $J\left(\mathbb{Q}\right)$.

Lemma 2 A $\mathbb{Q}$ -base of $\mathcal{L}\left(l{P}_{\infty }\right)$ is given by :

$\mathcal{B}=\left\{{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^i|i\in \mathbb{N}\mathrm{,}i\le \frac{l}{2}\right\}\cup \left\{x{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^j|j\in \mathbb{N}\mathrm{,}j\le \frac{l-11}{2}\right\}$ (7)

Proof 2. It is clear that $\mathcal{B}$ is free. It remains to show that

$dim\left(\mathcal{B}\right)=dim\left(\mathcal{L}\left(l{P}_{\infty }\right)\right)$.

By the Riemann-Roch theorem, we have $dim\left(\mathcal{L}\left(l{P}_{\infty }\right)\right)=l-g+1$ as soon as $l\ge 2g-1$ with $g=\frac{11-1}{2}$

Let us consider the following cases:

Case 1: Suppose that l is even, and let $l=2h$. Then we have

$i\le \frac{l}{2}=h$

and

$j\le \frac{l-11}{2}\iff j\le \frac{2h-11}{2}\iff j\le \frac{2h-11-1}{2}=h-6=h-g-1$.

So we obtain

$\mathcal{B}=\left\{1,\frac{x^2{\left(x-1\right)}^2}{y^7},\cdots ,{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^h\right\}\cup \left\{x,x\frac{x^2{\left(x-1\right)}^2}{y^7},\cdots ,x{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^{h-g-1}\right\}$,

and therefore $dim\left(\mathcal{B}\right)=\left(h+1\right)+\left(h-g\right)=2h-g+1=l-g+1=dim\left(\mathcal{L}\left(l{P}_{\infty }\right)\right)$.

Case 2: Suppose that l is odd, and let $l=2h+1$.

$i\le \frac{l}{2}\iff i\le \frac{2h+1}{2}\iff i\le h$

and

$j\le \frac{l-11}{2}\iff j\le \frac{2h-10}{2}=h-g$

So we obtain

$\mathcal{B}=\left\{\mathrm{1,}\frac{x^2{\left(x-1\right)}^2}{y^7}\mathrm{,}\cdots \mathrm{,}{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^h\right\}\cup \left\{x\mathrm{,}x\frac{x^2{\left(x-1\right)}^2}{y^7}\mathrm{,}\cdots \mathrm{,}x{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^{h-g}\right\}$,

and therefore

$dim\left(\mathcal{B}\right)=\left(h+1\right)+\left(h-g+1\right)=2h+1-g+1=l-g+1=dim\left(\mathcal{L}\left(l{P}_{\infty }\right)\right)$.

3. Demonstration of the Theorem

Let $R\in {\mathcal{C}}_3\left(11\right)\left(\bar{\mathbb{Q}}\right)$ with $\left[\mathbb{Q}\left(R\right)\mathrm{<mo>\; :\; </mo>}\mathbb{Q}\right]=l$. Let $R_1\mathrm{,}\cdots \mathrm{,}{R}_l$ be the Galois conjugates of R, and let $t=\left[R_1+\cdots +R_l-l{P}_{\infty }\right]$ which is a point of $J\left(\mathbb{Q}\right)=\left\{mj\left(P_0\right)\mathrm{,0}\le m\le 10\right\}$ ; so $t=mj\left(P_0\right)$ with $0\le m\le 10$. This gives the relation

$\left[R_1+\cdots +R_l-l{P}_{\infty }\right]=mj\left(P_0\right)\mathrm{.}$ (8)

We note that $R\notin \left\{P_0\mathrm{,}{P}_1\mathrm{,}{P}_{\infty }\right\}$.

Case $m=0$

Then there exists a rational function f such that $div\left(f\right)=R_1+\cdots +R_l-l{P}_{\infty }$, so $f\in \mathcal{L}\left(l{P}_{\infty }\right)$. According to Lemma 2, we have

$f={\displaystyle \underset{i\le \frac{l}{2}}{\sum }}{a}_i{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^i+x{\displaystyle \underset{j\le \frac{l-11}{2}}{\sum }}{b}_j{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^j$

with $a_{\frac{l}{2}}\ne 0$ if l is even (otherwise the $R_i$ would be equal to $P_{\infty }$ ) and $b_{\frac{l-11}{2}}\ne 0$ if l is odd (otherwise the $R_i$ would be equal to $P_{\infty }$ ). At the points $R_i$ we have

${\displaystyle \underset{i\le \frac{l}{2}}{\sum }}{a}_i{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^i+x{\displaystyle \underset{j\le \frac{l-11}{2}}{\sum }}{b}_j{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^j=0$

hense

$x=-\frac{{\displaystyle \underset{i\le \frac{l}{2}}{\sum }}{a}_i{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^i}{{\displaystyle \underset{j\le \frac{l-11}{2}}{\sum }}{b}_j{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^j}$

and therefore

$y^{11}=x^3{\left(x-1\right)}^3\iff y^{\frac{1}{3}}=\frac{x^2{\left(x-1\right)}^2}{y^7}$,

so

$x=-\frac{{\displaystyle \underset{i\le \frac{l}{2}}{\sum }}{a}_i{y}^{\frac{i}{3}}}{{\displaystyle \underset{j\le \frac{l-11}{2}}{\sum }}{b}_j{y}^{\frac{j}{3}}}$.

So the equation $y^{11}=x^3{\left(x-1\right)}^3$ becomes

$y^{11}{\left({\displaystyle \underset{j\le \frac{l-11}{2}}{\sum }}{b}_j{y}^j\right)}^2=\left({\displaystyle \underset{i\le \frac{l}{2}}{\sum }}{a}_i{y}^i\right){\left({\displaystyle \underset{i\le \frac{l}{2}}{\sum }}{a}_i{y}^{\frac{i}{3}}+{\displaystyle \underset{j\le \frac{l-11}{2}}{\sum }}{b}_j{y}^{\frac{j}{3}}\right)}^3$

which is an equation of degree l in y. We thus find a family of points of degree l

$\begin{array}{l}{\mathcal{F}}_0=\{\left(-\frac{{\displaystyle \underset{i\le \frac{l}{2}}{\sum }}{a}_i{y}^{\frac{i}{3}}}{{\displaystyle \underset{j\le \frac{l-11}{2}}{\sum }}{b}_j{y}^{\frac{j}{3}}},y\right)|a_0\ne 0,a_{\frac{l}{2}}\ne 0\text{if}l\text{is}\text{even},\\ b_{\frac{l-11}{2}}\ne 0\text{if}l\text{is}\text{odd}\text{and}y\text{root}\text{of}\text{the}\text{equation}\\ \begin{array}{c}\stackrel{}{}\\ \\ \\ \end{array}{y}^{11}{\left({\displaystyle \underset{j\le \frac{l-11}{2}}{\sum }}{b}_j{y}^j\right)}^2=\left({\displaystyle \underset{i\le \frac{l}{2}}{\sum }}{a}_i{y}^i\right){\left({\displaystyle \underset{i\le \frac{l}{2}}{\sum }}{a}_i{y}^{\frac{i}{3}}+{\displaystyle \underset{j\le \frac{l-11}{2}}{\sum }}{b}_j{y}^{\frac{j}{3}}\right)}^3\}\end{array}$

In the same way we show that for $m=k$ with $k\in \left\{\mathrm{1,}\cdots \mathrm{,10}\right\}$, the relation (8) gives $\left[R_1+\cdots +R_l-l{P}_{\infty }\right]=kj\left(P_0\right)=\left(k-11\right)j\left(P_0\right)$. Then there exists a rational function f such that $div\left(f\right)=R_1+\cdots +R_l+\left(11-k\right)P_0-\left(l+11-k\right)P_{\infty }$, so $f\in \mathcal{L}\left(l+11-k\right)P_{\infty }$. According to the Lemma 2, we have $f={\displaystyle \underset{i\le \frac{l+11-k}{2}}{\sum }}{a}_i{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^i+x{\displaystyle \underset{j\le \frac{l-k}{2}}{\sum }}{b}_j{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^j$ ; and as $ord{f}_{P_0}=11-k$,

therefore

$f={\displaystyle \underset{11-k\le i\le \frac{l+11-k}{2}}{\sum }}{a}_i{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^i+x{\displaystyle \underset{j\le \frac{l-k}{2}}{\sum }}{b}_j{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^j$

with $a_{\frac{l+11-k}{2}}\ne 0$ if l is even (otherwise the $R_i$ would be equal to $P_{\infty }$ ) and $b_{\frac{l-k}{2}}\ne 0$ if l is odd (otherwise the $R_i$ would be equal to $P_{\infty }$ ). At the points $R_i$ we have

${\displaystyle \underset{11-k\le i\le \frac{l+11-k}{2}}{\sum }}{a}_i{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^i+x{\displaystyle \underset{j\le \frac{l-k}{2}}{\sum }}{b}_j{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^j=0$

hense $x=-\frac{{\displaystyle \underset{11-k\le i\le \frac{l+11-k}{2}}{\sum }}{a}_i{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^i}{{\displaystyle \underset{j\le \frac{l-k}{2}}{\sum }}{b}_j{\left(\frac{x^2{\left(x-1\right)}^2}{y^7}\right)}^j}$ and therefore $x=-\frac{{\displaystyle \underset{11-k\le i\le \frac{l+11-k}{2}}{\sum }}{a}_i{y}^{\frac{i}{3}}}{{\displaystyle \underset{j\le \frac{l-k}{2}}{\sum }}{b}_j{y}^{\frac{j}{3}}}$.

So the equation $y^{11}=x^3{\left(x-1\right)}^3$ becomes

$y^k{\left({\displaystyle \underset{j\le \frac{l-k}{2}}{\sum }}{b}_j{y}^j\right)}^2=\left({\displaystyle \underset{11-k\le i\le \frac{l+11-k}{2}}{\sum }}{a}_i{y}^{i-\left(11-k\right)}\right){\left({\displaystyle \underset{11-k\le i\le \frac{l+11-k}{2}}{\sum }}{a}_i{y}^{\frac{i}{3}}+{\displaystyle \underset{j\le \frac{l-k}{2}}{\sum }}{b}_j{y}^{\frac{j}{3}}\right)}^3$

which is an equation of degree l in y. We thus find a family of points of degree l

$\begin{array}{l}{\mathcal{F}}_k=\{\left(-\frac{{\displaystyle \underset{11-k\le i\le \frac{l+11-k}{2}}{\sum }}{a}_i{y}^{\frac{i}{3}}}{{\displaystyle \underset{j\le \frac{l-k}{2}}{\sum }}{b}_j{y}^{\frac{j}{3}}},y\right)|b_0\ne 0,a_{\frac{l+11-k}{2}}\ne 0\text{if}l\text{is}\text{even},\\ b_{\frac{l-k}{2}}\ne 0\text{if}l\text{is}\text{odd}\text{and}y\text{root}\text{of}\text{the}\text{equation}\\ \begin{array}{c}\stackrel{}{}\\ \\ \\ \end{array}{y}^k{\left({\displaystyle \underset{j\le \frac{l-k}{2}}{\sum }}{b}_j{y}^j\right)}^2=\left({\displaystyle \underset{11-k\le i\le \frac{l+11-k}{2}}{\sum }}{a}_i{y}^{i-\left(11-k\right)}\right){\left({\displaystyle \underset{11-k\le i\le \frac{l+11-k}{2}}{\sum }}{a}_i{y}^{\frac{i}{3}}+{\displaystyle \underset{j\le \frac{l-k}{2}}{\sum }}{b}_j{y}^{\frac{j}{3}}\right)}^3\}\end{array}$

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

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[3]	Faddeev, D. (1961) On the Divisor Class Groups of Some Algebraic Curves. Soviet Mathematics, 2, 67-69.
[4]	Griffiths, P.A. (1989) Introduction to Algebraic Curves. Translations of Mathematical Monographs, Providence. [CrossRef]