Rational sequences on different models of elliptic curves

TL;DR

This paper investigates the existence of algebraic curves over number fields with rational points having specified x-coordinates, providing infinite families of elliptic curves for certain set sizes, thus extending previous results.

Contribution

It introduces new constructions of elliptic curves with prescribed rational points, generalizing earlier work on algebraic progressions on curves.

Findings

Infinite families of Edwards and Huff curves with specified rational x-coordinates

Existence results for sets of size 4, 5, or 6

Extension of previous progressions on algebraic curves

Abstract

Given a set $S$ of elements in a number field $k$, we discuss the existence of planar algebraic curves over $k$ which possess rational points whose $x$-coordinates are exactly the elements of $S$. If the size $|S|$ of $S$ is either $4,5$, or $6$, we exhibit infinite families of (twisted) Edwards curves and (general) Huff curves for which the elements of $S$ are realized as the $x$-coordinates of rational points on these curves. This generalizes earlier work on progressions of certain types on some algebraic curves.

Equations

$$\{(x,y)\in k^2:f(x,y)=0\}$$

$$a_1{x}^3+a_2{x}^{2}y+a_3{x}^2+a_{4}x{y}^2+a_{5}xy+a_{6}x+a_7{y}^3+a_8{y}^2+a_{9}y+a_{10}=0$$

$$E_d:x^2+y^2=1+d{x}^2{y}^2,$$

$$S=\{s_{-1}=-1,s_0=0,s_1=1,s_2,s_3,s_4\}\subset k,$$

$$d=\frac{(s_2^2-1)p^2+1}{s_2^2}=\frac{(s_3^2-1)q^2+1}{s_3^2}.$$

$$s_3^2\left[(s_2^2-1)p^2+1\right]-s_2^2\left[(s_3^2-1)q^2+1\right]=0$$

$$p$$

$$q$$

$$h(s_2,s_3)=-3+4s_3^2+s_2^4{s}_3^4+s_2^2(4-6s_3^2)\ne 0$$

$$E_d:x^2+y^2=1+d{x}^2{y}^2,d\in \mathbb{Q}$$

$$(-t^2{s}_2^2+s_3^2-s_2^2{s}_3^2+t^2{s}_2^2{s}_3^2{)}^{2}d$$

$$-4s_3^4+8s_2^2{s}_3^4-4s_2^4{s}_3^4)t+(-4s_2^2+4s_2^4-4s_3^2$$

$$+14s_2^2{s}_3^2-10s_2^4{s}_3^2+4s_3^4-10s_2^2{s}_3^4+6s_2^4{s}_3^4)t^2$$

$$+(4s_2^2-4s_2^4-8s_2^2{s}_3^2+8s_2^4{s}_3^2+4s_2^2{s}_3^4$$

$$-4s_2^4{s}_3^4)t^3+(s_2^4-2s_2^4{s}_3^2+s_2^4{s}_3^4)t^4.$$

$$r^2=\frac{s_4^2-1}{s_4^{2}d-1}=(A_0+A_{1}t+A_2{t}^2+A_3{t}^3+A_4{t}^4)/B(t{)}^2,$$

$$z^2=A_0+A_{1}t+A_2{t}^2+A_3{t}^3+A_4{t}^4,$$

$$(t,z)=\left(0,s_3^2(s_2^2-1)\right).$$

$$I$$

$$J$$

$$P=\left(-12(-1+s_2^2)(-1+s_3^2)(-3+s_2^2+s_3^2),-216(-1+s_2^2{)}^2(-1+s_3^2{)}^2\right).$$

$$3P=\left(\frac{g_1(s_2,s_3)}{h(s_2,s_3{)}^2},\frac{g_2(s_2,s_3)}{h(s_2,s_3{)}^3}\right),\text{ where }{g}_1,g_2\in \mathbb{Q}[s_2,s_3]$$

$$h(s_2,s_3)=-3+4s_3^2+s_2^4{s}_3^4+s_2^2(4-6s_3^2).$$

$$E_{a,d}:a{x}^2+y^2=1+d{x}^2{y}^2,$$

$$d=\frac{(a{u}_1^2-1)i^2+1}{u_1^2}=\frac{(a{u}_2^2-1)j^2+1}{u_2^2}.$$

$$u_2^2\left[(a{u}_1^2-1)i^2+1\right]-u_1^2\left[(a{u}_2^2-1)j^2+1\right]=0,$$

$$i$$

$$j$$

$$E_{a,d}:a{x}^2+y^2=1+d{x}^2{y}^2,d\in \mathbb{Q},a\in {\mathbb{Q}}^{\times }\text{ is arbitrary}$$

$$(a{u}_1^2{u}_2^2-u_2^2-a{t}^2{u}_1^2{u}_2^2+u_1^2{t}^2{)}^{2}d$$

$$+4u_1^2+4u_1^2{a}^2{u}_2^4-4u_1^{4}a-4u_1^4{a}^3{u}_2^4$$

$$+8u_1^4{a}^2{u}_2^2)t^3+(-4u_1^2-10u_1^2{a}^2{u}_2^4+14a{u}_1^2{u}_2^2$$

$$+6u_1^4{a}^3{u}_2^4-4u_2^2-10u_1^4{a}^2{u}_2^2+4u_1^{4}a+4a{u}_2^4)t^2$$

$$+(4u_2^2+8u_1^2{a}^2{u}_2^4-8a{u}_1^2{u}_2^2+4u_1^4{a}^2{u}_2^2-4a{u}_2^4$$

$$-4u_1^4{a}^3{u}_2^4)t+u_1^4{a}^3{u}_2^4-2u_1^2{a}^2{u}_2^4+a{u}_2^4.$$

$${\ell }^2=\frac{a{u}_3^2-1}{d{u}_3^2-1}=(C_0+C_{1}t+C_2{t}^2+C_3{t}^3+C_4{t}^4)/D(t{)}^2,$$

Full text

1. Introduction

An algebraic (affine) plane curve $C$ of degree $d$ over some field $k$ is defined by an equation of the form

[TABLE]

where $f$ is a polynomial of degree $d$. The algebraic affine plane curve $C$ can also be extended to the projective plane by homogenising the polynomial $f$. If $P=(x,y)$, then we write $x=x(P)$ and $y=y(P)$.

Studying the set of $k$-rational points on $C$, $C(k)$, has been subject to extensive research in arithmetic geometry and number theory, especially when $k$ is a number field. For example, if $f$ is a polynomial of degree $2$, then one knows that $C$ is of genus [math], and so if $C$ possesses one rational point then it contains infinitely many such points. If $f$ is of degree $3$, then $C$ is a genus $1$ curve if it is smooth. In this case, if $C(k)$ contains one rational point, then it is an elliptic curve, and according to Mordell-Weil Theorem, $C(k)$ is a finitely generated abelian group. In particular, $C(k)$ can be written as $T\times\mathbb{Z}^{r}$ where $T$ is the subgroup of points of finite order, and $r\geq 0$ is the rank of $C$ over $k$.

In enumerative geometry, one may pose the following question. Given a set of points $S$ in $k^{2}$, how many algebraic plane curve $C$ of degree $d$ satisfy that $S\subseteq C(k)$? It turns out that sometimes the answer is straightforward. For example, given $10$ points in $k^{2}$, in order for a cubic curve to pass through these points, a system of $10$ linear equations will be obtained by substituting the points of $S$ in

[TABLE]

and solving for $a_{1},\cdots,a_{10}$. Therefore, there exists a unique nontrivial solution to the system if the determinant of the corresponding matrix of coefficients is zero, hence a unique cubic curve through the points of $S$. Thus, one needs linear algebra to check the existence of algebraic curves of a certain degree through various specified points in $k^{2}$.

In this article, we address the following, relatively harder, question. Given $S\subset k$, are there algebraic curves $C$ of degree $d$ such that for every $x\in S$, $x=x(P)$ for some $P\in C(k)$? In other words, $S$ constitutes the $x$-coordinates of a subset of $C(k)$. The latter question can be reformulated to involve $y$-coordinates instead of $x$-coordinates. It is obvious that linear algebra cannot be utilized to attack the problem as substituting with the $x$-values of $S$ will not yield linear equations.

Given a set $S=\{x_{1},x_{2},\cdots,x_{n}\}\subset k$, if $(x_{i},y_{i})$, $i=1,\cdots,n$, are $k$-rational points on an algebraic curve $C$, then these rational points are said to be an $S$-sequence of length $n$. In what follows, we summarize the current state of knowledge for different types of $S$.

We first describe the state-of-art when the elements of $S\subset\mathbb{Q}$ are chosen to form an arithmetic progression, Lee and Vélez [10] found infinitely many curves described by $y^{2}=x^{3}+a$ containing $S$-sequences of length $4$. Bremner [2] showed that there are infinitely many elliptic curves with $S$-sequences of length $7$ and $8$. Campbell [5] gave a different method to produce infinite families of elliptic curves with $S$-sequences of length $7$ and $8$. In addition, he described a method for obtaining infinite families of quartic elliptic curves with $S$-sequences of length $9$, and gave an example of a quartic elliptic curve with an $S$-sequence of length $12$. Ulas [17] first described a construction method for an infinite family of quartic elliptic curves on which there exists an $S$-sequence of length $10$. Secondly he showed that there is an infinite family of quartics containing $S$-sequences of length $12$. Macleod [11] showed that simplifying Ulas’ approach may provide a few examples of quartics with $S$-sequences of length $14$. Ulas [18] found an infinite family of genus two curves described by $y^{2}=f(x)$ where $\deg(f(x))=5$ possessing $S$-sequences of length $11$. Alvarado [1] showed the existence of an infinite family of such curves with $S$-sequences of length $12$. Moody [12] found an infinite number of Edwards curves with an $S$-sequence of length $9$. He also asked whether any such curve will allow an extension to an $S$-sequence of length $11$. Bremner [3] showed that such curves do not exist. Also, Moody [14] found an infinite number of Huff curves with $S$-sequences of length $9$, and Choudhry [6] extended Moody’s result to find several Huff curves with $S$-sequences of length $11$.

Now we consider the case when the elements of $S$ form a geometric progression, Bremner and Ulas [4] obtained an infinite family of elliptic curves with $S$-sequences of length $4$, and they also pointed out infinitely many elliptic curves with $S$-sequences of length $5$. Ciss and Moody [13] found infinite families of twisted Edwards curves with $S$-sequences of length $5$ and Edwards curves with $S$-sequences of length $4$. When the elements of $S\subset\mathbb{Q}$ are consecutive squares, Kamel and Sadek [9] constructed infinitely many elliptic curves given by the equation $y^{2}=ax^{3}+bx+c$ with $S$-sequences of length $5$. When the elements of $S\subset\mathbb{Q}$ are consecutive cubes, Çelik and Soydan [7] found infinitely many elliptic curves of the form $y^{2}=ax^{3}+bx+c$ with $S$-sequences of length $5$.

In the present work, we consider the following families of elliptic curves due to the symmetry enjoyed by the equations defining them: (twisted) Edwards curves and (general) Huff curves. Given an arbitrary subset $S$ of a number field $k$, we tackle the general question of the existence of infinitely many such curves with an $S$-sequence when there is no restriction on the elements of $S$. We provide explicit examples when the length of the $S$-sequence is $4,5$, or $6$. This is achieved by studying the existence of rational points on certain quadratic and elliptic surfaces.

2. Edwards curves with $S$-sequences of length $6$

Throughout this work, $k$ will be a number field unless otherwise stated.

An Edwards curve over $k$ is defined by

[TABLE]

where $d$ is a non-zero element in $k$. It is clear that the points $(x,y)=(-1,0),(0,\pm 1),(1,0)\in E_{d}(k)$. We show that given any set

[TABLE]

$s_{i}\neq s_{j}$ if $i\neq j$, there are infinitely many Edwards curves $E_{d}$ that possess rational points whose $x$-coordinates are $s_{i}$, $-1\leq i\leq 4$, i.e., the set $S$ is realized as $x$-coordinates in $E_{d}(k)$. In other words, there are infinitely many Edwards curves that possess an $S$-sequence.

We start with assuming that $s_{2}$ is the $x$-coordinate of a point in $E_{d}(k)$, then one must have $\displaystyle y^{2}=\frac{s_{2}^{2}-1}{s_{2}^{2}d-1}$, or $s_{2}^{2}d-1=(s_{2}^{2}-1)p^{2}$ for some $p=1/y\in k$.

Similarly, if $s_{3}$ is the $x$-coordinate of a point in $E_{d}(k)$, then $\displaystyle y^{2}=\frac{s_{3}^{2}-1}{s_{3}^{2}d-1}$, or $s_{3}^{2}d-1=(s_{3}^{2}-1)q^{2}$. So

[TABLE]

Thus we have the following quadratic curve

[TABLE]

on which we have the rational point $(p,q)=(1,1)$. Parametrizing the rational points on the latter quadratic curve yields

[TABLE]

Therefore, fixing $s_{2}$ and $s_{3}$ in $k$, one sees that $p$ and $q$ lie in $k(t)$. Now we obtain the following result.

Theorem 2.1**.**

Let $s_{-1}=-1$, $s_{0}=0$, $s_{1}=1$, $s_{2}$, $s_{3}$ and $s_{4}$, $s_{i}\neq s_{j}$ if $i\neq j$, be a sequence in $\mathbb{Z}$ such that

[TABLE]

where either $g_{1}(s_{2},s_{3})/h(s_{2},s_{3})^{2}$ or $g_{2}(s_{2},s_{3})/h(s_{2},s_{3})^{3}$ are not integers, $g_{1}$ and $g_{2}$ are defined in (2.3). There are infinitely many Edwards curves described by

[TABLE]

on which $s_{i}$, $-1\leq i\leq 4$, are the $x$-coordinates of rational points in $E_{d}(\mathbb{Q})$. In other words, there are infinitely many Edwards curves that possess an $S$-sequence where $S=\{s_{i}:-1\leq i\leq 4\}.$

Proof.

Substituting the value for $p$ in $\displaystyle d=\frac{(s_{2}^{2}-1)p^{2}+1}{s_{2}^{2}}$ yields that

[TABLE]

Thus, for fixed values of $s_{2}$ and $s_{3}$, we have $d\in\mathbb{Q}(t)$.

Now we show the existence of infinitely many values of $t$ such that $s_{4}$ is the $x$-coordinate of a rational point on $E_{d}$. In fact, we will show that $t$ can be chosen to be the $x$-coordinate of a rational point on an elliptic curve with positive Mordell-Weil rank, hence the existence of infinitely many such possible values for $t$. Forcing $(s_{4},r)$ to be a point in $E_{d}(\mathbb{Q})$ for some rational $r$ yields that

[TABLE]

where $A_{i}\in\mathbb{Z}$ and $B(t)=-t^{2}s_{2}^{2}+t^{2}s_{2}^{2}s_{3}^{2}+s_{3}^{2}-s_{2}^{2}s_{3}^{2}.$ This implies that $A_{0}+A_{1}t+A_{2}t^{2}+A_{3}t^{3}+A_{4}t^{4}$ must be a rational square. This yields the elliptic curve $C$ defined by

[TABLE]

with the following rational point

[TABLE]

The latter elliptic curve is isomorphic to the elliptic curve described by the Weierstrass equation $E_{I,J}:y^{2}=x^{3}-27Ix-27J$ where

[TABLE]

see for example [16, §2]. The latter elliptic curve has the following rational point

[TABLE]

One notices that the coordinates of $3P$ are rational functions. Indeed,

[TABLE]

and

[TABLE]

Hence, as long as $h(s_{2},s_{3})\neq 0$, and $g_{1}/h^{2}\not\in\mathbb{Z}$ or $g_{2}/h^{3}\not\in\mathbb{Z}$, one sees that $3P$ is a point of infinite order by virtue of Lutz-Nagell Theorem. Thus, $P$ itself is a point of infinite order. It follows that $E_{I,J}$ is of positive Mordell-Weil rank. Since $C$ is isomorphic to $E_{I,J}$, it follows that $C$ is also of positive Mordell-Weil rank. Therefore, there are infinitely many rational points $(t,z)\in C(\mathbb{Q})$, each giving rise to a value for $d$, by substituting in (2.2), hence an Edwards curve $E_{d}$ possessing the aforementioned rational points. That infinitely many of these curves are pairwise non-isomorphic over $\mathbb{Q}$ follows, for instance, from Proposition 6.1 in [8]. ∎

3. Twisted Edwards curves with $S$-sequences of length $4$

A Twisted Edwards curve over $k$ is given by

[TABLE]

where $a$ and $d$ are nonzero elements in $k.$ Note that the point $(x,y)=(0,\pm 1)\in E_{a,d}(k)$. Given a set $\{u_{0}=0,u_{1},u_{2},u_{3}\}\subset k$, $u_{i}\neq u_{j}$ if $i\neq j$, we prove that there are infinitely many twisted Edwards curves $E_{a,d}$ for which $S$ is realized as the $x$-coordinates of rational points on $E_{a,d}$.

We begin by assuming that $u_{1}$ is the $x$-coordinate of a point in $E_{a,d}(k)$, then one must get $\displaystyle y^{2}=\frac{au_{1}^{2}-1}{u_{1}^{2}d-1}$, or $u_{1}^{2}d-1=(au_{1}^{2}-1)i^{2}$ for some $i\in k$.

Now, if $u_{2}$ is the $x$-coordinate of a point in $E_{a,d}(k)$, then $\displaystyle y^{2}=\frac{au_{2}^{2}-1}{u_{2}^{2}d-1}$ or $u_{2}^{2}d-1=(au_{2}^{2}-1)j^{2}$. So

[TABLE]

Hence we obtain the following quadratic surface

[TABLE]

on which we have the rational point $(i,j)=(1,1)$. Solving the above quadratic surface gives the following

[TABLE]

Now we get the following result.

Theorem 3.1**.**

Let $u_{0}=0$, $u_{1}$, $u_{2}$ and $u_{3}$ , $u_{i}\neq u_{j}$ if $i\neq j$, be a sequence in $\mathbb{Z}$ such that $h(u_{1},u_{2})\neq 0$, and either $g_{1}(s_{2},s_{3})/h(s_{2},s_{3})^{2}$ or $g_{2}(s_{2},s_{3})/h(s_{2},s_{3})^{3}$ are not integers, where $h,g_{1},g_{2}$ are defined in (3). There are infinitely many twisted Edwards curves described by

[TABLE]

on which $u_{i}$, $0\leq i\leq 3$, are the $x$-coordinates of rational points in $E(\mathbb{Q})$. In other words, there are infinitely many twisted Edwards curves that possess an $S$-sequence where $S=\{u_{i}:0\leq i\leq 3\}.$

Proof.

Substituting the expression for $i$ in $\displaystyle d=\frac{(au_{1}^{2}-1)i^{2}+1}{u_{1}^{2}}$ gives that

[TABLE]

Then, assuming $(u_{3},\ell)\in E(\mathbb{Q})$ yields

[TABLE]

where $C_{i}\in\mathbb{Q}$ and $D(t)=au_{1}^{2}u_{2}^{2}-u_{2}^{2}-at^{2}u_{1}^{2}u_{2}^{2}+u_{1}^{2}t^{2}.$

For the latter equation to be satisfied, one needs to find rational points on the elliptic curve $C^{\prime}$ defined by

[TABLE]

that possesses the rational point

[TABLE]

The latter elliptic curve is isomorphic to the elliptic curve described by the Weierstrass equation $E_{I,J}:y^{2}=x^{3}-27Ix-27J$ where

[TABLE]

see for example [16, §2]. The latter elliptic curve has the following rational point

[TABLE]

One notices that the coordinates of $3Q$ are rational functions. In fact,

[TABLE]

and

[TABLE]

Therefore, as long as $h(u_{1},u_{2})\neq 0$ and $g_{1}/h^{2}\not\in\mathbb{Z}$ or $g_{2}/h^{3}\not\in\mathbb{Z}$, one sees that $E_{I,J}$ is of positive Mordell-Weil rank where the point $Q$ is of infinite order. Since $C^{\prime}$ is isomorphic to $E_{I,J}$, it follows that $C^{\prime}$ is also of positive Mordell-Weil rank. Hence, there are infinitely many rational points $(t,z)\in C^{\prime}(\mathbb{Q})$, each giving rise to a value for $d$, by substituting in (3.2), therefore a twisted Edwards curve $E_{a,d}$ possessing the aforementioned rational points. That infinitely many of these curves are pairwise non-isomorphic over $\mathbb{Q}$ again follows from Proposition 6.1 in [8].

∎

Remark 1**.**

Since $(0,-1),(0,1)$ are rational points on any twisted Edwards curve, one can show that if $u_{-1}=-1$, $u_{1}=1$, $u_{2}$, $u_{3}$ and $u_{4}$, $u_{i}\neq u_{j}$ if $i\neq j$, is a sequence in $\mathbb{Z}$, there are infinitely many Edwards curves on which $u_{i}$, $i\in\{-1,1,2,3,4\}$, are the $y$-coordinates of rational points in $E_{a,d}(\mathbb{Q})$.

4. Huff curves with $S$-sequences of length $5$

A Huff curve over a number field $k$ is defined by

[TABLE]

with $a^{2}\neq b^{2}$. Note that the points $(x,y)=(-1,\pm 1),(0,0),(1,\pm 1)$ are in $H_{a,b}(k)$. We prove that given $s_{-1}=-1$, $s_{0}=0$, $s_{1}=1$, $s_{2},$ $s_{3}$ $\in k$, $s_{i}\neq s_{j}$ if $i\neq j$, there are infinitely many Huff curves on which these numbers are realized as the $x$-coordinates of rational points.

Assuming $(s_{2},p)$ and $(s_{3},q)$ are two points on $H_{a,b}$ yields

[TABLE]

and

[TABLE]

respectively. Using (4.2) and (4.3), one obtains

[TABLE]

therefore, one needs to consider the curve

[TABLE]

where $A=s_{3}s_{2}^{2}-s_{2}$ and $B=s_{2}s_{3}^{2}-s_{2}$. Dividing both sides of the above equality by $q^{3}$ gives

[TABLE]

Substituting $x=\frac{p}{q}$ and $y=\frac{1}{q^{2}}$ in the above equation yields the following quadratic curve

[TABLE]

on which we have the rational point $(x,y)=(1,1)$. Parametrizing the rational points on the latter quadratic curve gives

[TABLE]

[TABLE]

Now we have the following result.

Theorem 4.1**.**

Let $s_{-1}=-1,s_{0}=0,s_{1}=1,s_{2},s_{3},$ $s_{m}\neq s_{n}$ if $m\neq n$, be a sequence in $\mathbb{Z}$ such that

[TABLE]

where $A$ and $B$ are defined as above, and either $g_{1}/h^{2}$ or $g_{2}/h^{3}$ are not integers, where $g_{1},g_{2}$ are defined in (4.6). There are infinitely many Huff curves described by

[TABLE]

on which $s_{m}$, $-1\leq m\leq 3$, are the $x$-coordinates of rational points in $H_{a,b}(\mathbb{Q})$. In other words, there are infinitely many Huff curves that possess an $S$-sequence where $S=\{s_{i}:-1\leq i\leq 3\}.$

Proof.

Using the equalities (4.4) and (4.5), we obtain the following

[TABLE]

In both cases we need $(B+At)(-1+t)(B(-1+t)-At)$ to be a square or in other words we need $t$ to be the $x$-coordinate of a rational point on the elliptic curve $C^{\prime\prime}$ defined by

[TABLE]

with the following $k$-rational point $(t,z)=(0,B).$ The latter curve can be described by the following equation

[TABLE]

where $A(B-A)t=X$ and $A(B-A)z=Y$. This curve has the rational point

[TABLE]

Observing that

[TABLE]

where $h(A,B)=-4+A^{2}-3AB+B^{2}$, one concludes as in the proof of Theorem 2.1. ∎

5. General Huff curves with $S$-sequences of length $4$

A general Huff curve over a number field $k$ is defined by

[TABLE]

where $a,b\in k$ and $ab(a-b)\neq 0$. It is clear that the point $(x,y)=(0,0)\in G_{a,b}(k)$. We show that given $u_{0}=0,u_{1},u_{2},u_{3}$ in $k$, $u_{i}\neq u_{j}$ if $i\neq j$, there are infinitely many general Huff curves over which these points are realized as the $x$-coordinates of rational points.

We start by assuming that if $u_{1}$ is the $x$-coordinates of a point in $G_{a,b}(k)$, then one must have $\displaystyle\frac{ay^{2}-1}{y}=\frac{bu_{1}^{2}-1}{u_{1}}$ or $\displaystyle\frac{a-i^{2}}{i}=\frac{bu_{1}^{2}-1}{u_{1}}$ for some $i\in k$.

Similarly, if $u_{2}$ is the $x$-coordinate of a point in $G_{a,b}$, then $\displaystyle\frac{ay^{2}-1}{y}=\frac{bu_{2}^{2}-1}{u_{2}}$ or $\displaystyle\frac{a-j^{2}}{j}=\frac{bu_{2}^{2}-1}{u_{2}}$ for some $j\in k$. Thus, one obtains

[TABLE]

which gives the following quadratic curve

[TABLE]

where $A=-u_{1}u_{2}$, $B=u_{1}u_{2}$, $C=-u_{1}^{2}u_{2}b+u_{2}$, $D=bu_{1}u_{2}^{2}-u_{1}$. Then consider the line

[TABLE]

connecting the rational points $P=(i:j:z)=(0:0:1)$ and $Q=(p:q:r)$ lying on $S\subset\mathbb{P}^{2}$. The intersection of $S$ and $mP+nQ$ yields the quadratic equation

[TABLE]

Using $P$ and $Q$ lying on $S$, one solves this quadratic equation and obtains formulae for the solution $(i:j:z)$ with the following parametrization:

[TABLE]

Now we obtain the following result.

Theorem 5.1**.**

Let $u_{0}=0$, $u_{1}$, $u_{2}$ and $u_{3}$ , $u_{i}\neq u_{j}$ if $i\neq j$, be a sequence in $k$. There are infinitely many general Huff curves described by

[TABLE]

on which $u_{i}$, $0\leq i\leq 3$, are the $x$-coordinates of rational points in $G_{a,b}(k)$. In other words, there are infinitely many general Huff curves that possess an $S$-sequence where $S=\{u_{i}:0\leq i\leq 3\}.$

Proof.

Substituting the value for $i$ in $\displaystyle a=\frac{(bu_{1}^{2}-1)i+u_{1}i^{2}}{u_{1}}$ yields that

[TABLE]

Now we assume that $(u_{3},\ell)\in G_{a,b}(k)$. This yields that

[TABLE]

This can be rewritten as

[TABLE]

where $T=1/\ell.$ One sees that the rational point $P=(q:T:Z)=(1:0:u_{1}(-1+bu_{2}^{2})/pu_{2}(-1+bu_{1}^{2}))$ lies on the quadratic curve above, hence we may parametrize the rational points on the quadratic curve above. This is obtained by considering the intersection of the line $dP+eQ$ where $Q=(q_{1}:q_{2}:q_{3})$ is a point on the quadratic curve. In fact, this yields that

[TABLE]

[TABLE]

∎

Acknowledgements

We would like to thank the referees for carefully reading our manuscript and for giving such constructive comments which substantially helped improving the presentation of the paper.