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\begin{center}
\vskip 1cm{\LARGE\bf On Geometric Progressions \\
\vskip .1in
on Hyperelliptic Curves}
\vskip 1cm
\large
Mohamed Alaa\\
Department of Mathematics\\
 Faculty of Science\\
  Cairo University\\
   Giza\\
    Egypt\\
\href{mailto:malaa@sci.cu.edu.eg}{\tt malaa@sci.cu.edu.eg} \\
\ \\
Mohammad Sadek\\
Mathematics and Actuarial Science Department\\
American University in Cairo\\
  AUC Avenue\\
   New Cairo\\
    Egypt\\
\href{mailto:mmsadek@aucegypt.edu}{\tt mmsadek@aucegypt.edu}
\end{center}

\vskip .2in

\begin{abstract}
Let $C$ be a hyperelliptic curve over $\Q$ described by $y^2=a_0x^n+a_1x^{n-1}+\cdots+a_n$, $a_i\in\Q$. The points $P_{i}=(x_{i},y_{i})\in C(\mathbb{Q})$, $i=1,2,\ldots,k$, are said to be in a geometric progression of length $k$ if the rational numbers $x_{i}$, $i=1,2,\ldots,k$, form a geometric progression sequence in $\Q$, i.e., $x_i=pt^{i}$ for some $p,t\in\Q$. In this paper we prove the existence of an infinite family of hyperelliptic curves on which there is a sequence of rational points in a geometric progression of length at least eight.
\end{abstract}


\section{Introduction}
Let $C$ be a hyperelliptic curve defined over the rational field $\Q$ by a hyperelliptic equation of the form $y^2=f(x)$, $\deg f(x)\ge3$. One may construct a sequence of $\Q$-rational points in $C(\Q)$ such that the $x$-coordinates of these rational points form a sequence of rational numbers which enjoys a certain arithmetic pattern. For instance, an {\em arithmetic progression} sequence on $C$ is a sequence $(x_i,y_i)\in C(\Q)$, $i=1,2,\ldots$, where $x_i=a+ib$ for some $a,b\in\Q$. In a similar fashion one may define a {\em geometric progression} sequence on $C$.

 Bremner \cite{Bremner} discussed arithmetic progression sequences on elliptic curves over $\Q$. He investigated the size of these sequences and produced elliptic curves with long arithmetic progression sequences. Campbell, Macleod, and Ulas \cite{Campbell, Macleod, Ulas1} improved Bremner's techniques and used them to generate infinitely many elliptic curves with long arithmetic progression sequences of rational points. Furthermore, Ulas \cite{Ulas2} considered arithmetic progression sequences on genus $2$ curves.


   Bremner and Ulas \cite{BremnerUlas} studied a certain family of algebraic curves. They proved the existence of an infinite family of algebraic curves defined by $y^2 = ax^n + b,\,n\ge 1,\, a, b \in \mathbb Q$, with geometric progression sequences of rational points of length at least $4$. They remarked that their method can be exploited in order to increase the length of these sequences to be $5$.

  In this note we examine geometric progression sequences on hyperelliptic curves. We start with proving that unlike geometric progressions on the rational line, geometric progression sequences on hyperelliptic curves are finite. A certain family of hyperelliptic curves defined by an equation of the form  $y^{2}=ax^{2n}+bx^{n}+a$, $n \in \mathbb{N},\; a,b \in \mathbb{Q}$, is displayed. Each hyperelliptic curve in this family possesses a geometric progression sequence of rational points whose length is at least $8$. In fact, those hyperelliptic curves are parametrized by an elliptic surface $\mathcal H$ with positive rank. In particular, to each point of infinite order on $\mathcal H$ one can associates a hyperelliptic curve with a geometric progression sequence of length at least $8$.

  It is worth mentioning that other types of sequences of rational points on algebraic curves are being studied. For example, Kamel and Sadek \cite{SadekKamel} construct an infinite family of elliptic curves such that every elliptic curve in the family has a sequence of rational points whose $x$-coordinates form a sequence of consecutive rational squares. The length of the latter sequence is at least $5$.

\section{Geometric progression sequences on hyperelliptic curves}
Let $C$ be a hyperelliptic curve defined over a number field $K$ by the equation $y^2=f(x)$ where $f(x)\in K[x]$ is of degree $n\ge 3$, and $f(x)$ has no double zeros. The set $C(K)$ of $K$-rational points on $C$ is defined by
$C(K)=\{(x,y):y^2=f(x),\,x,y\in K\}$.
\begin{definition}
Let $C:y^2=f(x)$ be a hyperelliptic curve over a number field $K$. The sequence $P_i=(x_i,y_i)\in C(K),\,i=1,2,\ldots$, is said to be a {\em geometric progression sequence} in $C(K)$ if there are $p,t\in K^{\times}$ such that $x_i=pt^i$. In other words, the $x$-coordinates of the rational points $P_i$ form a geometric progression sequence in $K$.
\end{definition}

We assume throughout that our geometric progression sequences contain distinct rational points, in particular $t\not\in\{\pm1\}$.

We will show that unlike geometric progressions in $K$, geometric progression sequences in $C(K)$ are finite.

\begin{theorem}
\label{thm:length}
Let $C:y^2=f(x)$ be a hyperelliptic curve over a number field $K$ with $\deg f(x)\ge 3$. Let $(x_i,y_i)$ form a geometric progression sequence in $C(K)$. Then the sequence $(x_i,y_i)$ is finite.
\end{theorem}
\begin{proof}
If $\deg f(x)\ge 5$, then it follows that the genus $g$ of $C$
satisfies $g\ge 2$. In view of Faltings' theorem, it is known that
$C(K)$ is finite; see \cite{Falting}.

If $\deg f(x)=3$ or $4$, then $C$ is an elliptic curve. Assume that $f(x)=a_0x^4+a_1x^3+a_2x^2+a_3x+a_4$. Assume on the contrary that there is an infinite sequence $(x_i,y_i)\in C(K)$, $x_i=pt^i,\,i=1,2,\ldots$, for some $p,t\in K^{\times}$. Considering the subsequence $(x_{2i},y_{2i})$, $i=1,2,\ldots$, one obtains
\[y_{2i}^2=a_0p^4t^{8i}+a_1p^3t^{6i}+a_2p^2t^{4i}+a_3p t^{2i}+a_4,\,i=1,2,\ldots.\]
In particular, the rational points $(t^i,y_i),i=1,2,\ldots$, form an infinite sequence of rational points on the new hyperelliptic curve
\[C':y^2=a_0p^4x^8+a_1p^3x^6+a_2p^2x^4+a_3px^2+a_4.\]
This contradicts Faltings' theorem, since the genus of $C'$ is $2$ if $a_0=0$; $3$ if $a_0\ne 0$.
\end{proof}

The theorem above motivates the following definition. Given a geometric progression sequence $(x_i,y_i)$, $i=1,2,\ldots,k$, in $C(K)$, the positive integer $k$ will be called the {\em length} of the sequence.

\section{Hyperelliptic curves with long geometric progressions}

In this note, we consider the family of hyperelliptic curves over $\Q$ described by the equation $y^2=ax^{2n}+bx^n+a$ where $a,b\in\Q$, and $n\ge 2$. We introduce an infinite family of these hyperelliptic curves with geometric progression sequences of length at least $8$. We remark that the existence of a sequence of rational points $(t^i,y_i)$, $i=1,2,\ldots,k$, in geometric progression on one of these hyperelliptic curves is equivalent to the existence of the following geometric progression sequence of rational points $(t^{ni},y_i)$ on the conic $y^2=ax^2+bx+a$. In fact, we will establish the existence of such an infinite family of conics on which there exist geometric progression sequences of rational points whose $x$-coordinates are $t^{-7},t^{-5},t^{-3},t^{-1},t,t^3,t^5,t^7$ for some $t\in\Q\setminus\{-1,0,1\}$.

We start with assuming that the points $(t,U)$ and $(t^3,V)$ are two rational points in $C(\Q)$ where $C$ is given by $y^2=f(x)=ax^2+bx+a$. This implies that
\begin{eqnarray*}
U^{2}&amp;=&amp;at^{2}+bt+a,\\
V^{2}&amp;=&amp;at^{6}+bt^{3}+a,
\end{eqnarray*}
hence
\begin{eqnarray}
\label{eq1}
a&amp;=&amp;\frac{t^{2}U^{2}-V^{2}}{(t^{2}-1)^{2}(t^{2}+1)},\nonumber\\
b&amp;=&amp;\frac{(t^{4}-t^{2}+1)U^{2}-V^{2}}{t(t^{2}-1)^{2}}.
\end{eqnarray}

The symmetry in the polynomial $f(x)$ implies that if the points $(t,U)$ and $(t^{3},V)$ are in $C(\Q)$, then so are the points $(t^{-1},Ut^{-1})$ and $(t^{-3},Vt^{-3})$. So we already have four points in geometric progression in $C(\Q)$.

In order to increase the length of the progression, we assume that $(t^{5},R)$ is in $C(\Q)$, hence $(t^{-5},Rt^{-5})$ is in $C(\Q)$ as well. Given the description of $a$ and $b$, (\ref{eq1}), one obtains
$$R^{2}=-t^{2}(t^{4}+1)U^{2}+(1+t^{2}+t^{4})V^{2}.$$
\begin{theorem}
\label{Thm2}
The conic $\mathcal{C}:R^{2}=-t^{2}(t^{4}+1)U^{2}+(1+t^{2}+t^{4})V^{2}$ defined over $\Q(t)$ has infinitely many rational points given by the following parametrization
\begin{eqnarray}
\label{eq2}
U&amp;=&amp;t^{2}(1+t^{4})p^{2}+(1+t^{2}+t^{4})q^{2}-2t(1+t^{2}+t^{4})pq,\nonumber\\
V&amp;=&amp;t^2(1+t^4)p^2+(1+t^{2}+t^{4})q^{2}-2t(1+t^{4})pq,\nonumber\\
R&amp;=&amp;t^{3}(1+t^{4})p^{2}-t(1+t^{2}+t^{4})q^{2}.
\end{eqnarray}
\end{theorem}
\begin{proof}
The point $(U:V:R)=(1:t:t^2)$ lies in $\mathcal{C}(\Q(t))$. This implies the existence of infinitely many rational points on the conic $\mathcal C$. For the parametric description of these rational points; see \cite[p.\ 69]{Mordell}.
\end{proof}
\begin{corollary}
There exists an infinite family of conics $y^2=ax^2+bx+a$, $a,b\in\Q$, containing $6$ rational points in geometric progression. In particular, there exist infinitely many hyperelliptic curves described by the equation $y^2=ax^{2n}+bx^n+a$ with $6$ rational points in geometric progression.
\end{corollary}

In what follows we parametrize the family of conics $C:y^2=ax^2+bx+a$ containing a seventh rational point $(t^7,S)$. We recall that the existence of this seventh rational point implies the existence of an eighth point $(t^{-7},St^{-7})$ on the conic $C$. The point $(t^7,S)$ satisfies the equation of the conic where $a,b$ are described as in (\ref{eq1}) and (\ref{eq2}). This gives rise to the following curve over $\Q(t)$
\begin{multline}
\label{eq3}
\mathcal{H}:S^2=H_t(p,q):=t^8(1+t^4)^2p^4+4t^5(1+2t^4+2t^8+t^{12})p^3q \\
-2t^4(4+3t^2+9t^4+4t^6+9t^8+3t^{10}+4t^{12})p^2q^2\\
        +4t^3(1-t^2+t^4)(1+t^2+t^4)^2pq^3+t^4(1+t^2+t^4)^2q^4.
        \end{multline}
        \begin{theorem}
        \label{thm3}
        The curve $\mathcal H$ defined over $\Q(t)$ is an elliptic curve with $\rank \mathcal{H}(\Q(t))\ge 1$.
        \end{theorem}
        \begin{proof}
        The following point lies in $\mathcal H(\Q(t))$:
\scriptsize
\begin{displaymath}
(p:q:S)=\left(\frac{-t}{1-t^2+t^4}:1-\frac{3+2t^2+4t^4+2t^6+3t^8}{2(1+t^4+t^8)}:\frac{t^2(3+4t^2+8t^4+8t^6+10t^8+8t^{10}+8t^{12}+4t^{14}+3 t^{16})}{4(1-t^2+t^4)^2(1+t^2+t^4)}\right).
\end{displaymath}
\normalsize
The existence of the latter rational point in $\mathcal H(\Q(t))$ implies that the curve $\mathcal H$ is birationally equivalent over $\Q(t)$ to its Jacobain $\mathcal E$ described by $Y^2=4X^3-g_2X-g_3$ where
\begin{eqnarray*}
g_2&amp;=&amp;\frac{4}{3}t^8(1+t^2+t^4)^2(1+t^2+4t^4+t^6+7t^8+t^{10}+4t^{12}+t^{14}+t^{16}),\\
g_3&amp;=&amp;-\frac{4}{27}t^{12}(1+t^2+t^4)^4(2+t^2+3t^4+15t^6-9t^8+30t^{10}-9t^{12}+15t^{14}+3t^{16}+t^{18}+2t^{20});
\end{eqnarray*}
see \cite[Thm.\ 2, p.\ 77]{Mordell}. The point $P=(X_P,Y_P)$ where
\begin{eqnarray*}
X_P&amp;=&amp;-\frac{t^4(1+t^2+t^4)^2(2-5t^2-2t^4-2t^6-2t^8-5t^{10}+2t^{12})}{3(1+t^2)^4},\\
Y_P&amp;=&amp;\frac{4t^7(1+t^2+t^4)^2}{(1+t^2)^6}(1+t^2+2t^4+2t^6+3t^8+2t^{10}+3t^{12}+2t^{14}+2t^{16}+t^{18}+t^{20})
\end{eqnarray*}
is a point in $\mathcal E(\Q(t))$. In fact, specializing $t=2$ and using \Magma  \cite{MAGMA}, we find that the specialization of the point $P$ is a point of infinite order on the specialization of $\EE$ when $t=2$. It follows that the point $P$ itself is a point of infinite order in $\EE(\Q(t))$.
\end{proof}
\begin{corollary}
\label{cor:hyperelliptic}
Fix $t_0\in\Q$. For any nontrivial geometric progression sequence of the form $t_0^{\pm1},t_0^{\pm3},t_0^{\pm5},t_0^{\pm 7}$, there exist infinitely many hyperelliptic curves $C_m:y^2=a_mx^{2n}+b_mx^n+a_m,\;m\in\Z\setminus\{0\},n\ge 2$, such that the numbers $t_0^{\pm i},i=1,3,5,7$, are the $x$-coordinates of rational points on $C_m$.
\end{corollary}
\begin{proof}
The point $P=(p:q:S)$ described by 
\scriptsize
$$\left(\frac{-t_0}{1-t_0^2+t_0^4}:1-\frac{3+2t_0^2+4t_0^4+2t_0^6+3t_0^8}{2(1+t_0^4+t_0^8)}:\frac{t_0^2(3+4t_0^2+8t_0^4+8t_0^6+10t_0^8+8t_0^{10}+8t_0^{12}+4t_0^{14}+3 t_0^{16})}{4(1-t_0^2+t_0^4)^2(1+t_0^2+t_0^4)}\right)$$
\normalsize
is a point of infinite order on the curve $\mathcal H$ over $\Q(t_0)$; see Theorem \ref{thm3}. For any nonzero $m$, we write $mP=(p_m:q_m:S_m)$ for the $m$-th multiple of $P$.

Substituting these values of $p_m,q_m\in\Q(t_0)$ into (\ref{eq2}), one obtains a parametric solution $U_m,V_m,R_m$ of the quadratic $R^2=-t_0^2(t_0^4+1)U^2+(1+t_0^2+t_0^4)V^4$. Hence, one obtains $a_m$ and $b_m$ by substituting $U_m$ and $V_m$ into the formulas of $a,b$ in (\ref{eq1}).

   We get an infinite family of hyperelliptic curves $C_m:y^2=a_mx^{2n}+b_mx^n+a_m$, where $m$ is nonzero. This family satisfies the property that the points $(t_0^i,u_i),(t_0^{-i},u_it_0^{-i})$, $i=1,3,5,7$, are lying in $C_m(\Q)$ for some $u_i\in\Q$. Thus, one obtains an infinite family of hyperelliptic curves with an $8$-term geometric progression sequence of rational points.
  \end{proof}
\section{A numerical example}
  The curve $C: y^{2}=a(T)x^{2n}+b(T)x^{n}+a(T),n \in \mathbb N$, where  $a(T)$ is given by \\
$$\frac{T^{4n}(1+T^{2n})(1+T^{8n})}{2(1+T^{4n})(-1+T^{2n}-T^{4n}+T^{6n}-T^{8n}+T^{10n})^2}$$
and $b(T)$ is defined by
\scriptsize
$$
\frac{1-2T^{2n}-T^{4n}-12T^{6n}-3T^{8n}-14T^{10n}-13T^{12n}-40T^{14n}-13T^{16n}-14T^{18n}-3T^{20n}-12T^{22n}-T^{24n}-2T^{26n}+T^{28n}}{16T^{3n}
(-1+T^{2n})^2(1+T^{4n})^2(1-T^{2n}+T^{4n})^2(1+T^{2n}+T^{4n})^2}$$
\normalsize
has the following $8$-term geometric progression sequence
\scriptsize
\begin{gather*}
\left(T^{-7},\frac{3+4T^{2n}+5T^{4n}+4T^{6n}+5T^{8n}+4T^{10n}+3T^{12n}}{4T^{5n}(1+2T^{4n}+2T^{8n}+T^{12n})}\right),\\
\left(T^{-5},\frac{1+4T^{2n}+3T^{4n}+4T^{6n}+3T^{8n}+4T^{10n}+T^{12n}}{4T^{4n}(1+2T^{4n}+2T^{8n}+T^{12n})}\right),
\left(T^{-3},\frac{1+3T^{4n}+4T^{6n}+3T^{8n}+T^{12n}}{4T^{3n}(1+2T^{4n}+2T^{8n}+T^{12n})}\right),\\
\left(T^{-1},\frac{-1+T^{4n}+4T^{6n}+T^{8n}-T^{12n}}{4T^{2n}(1+2T^{4n}+2T^{8n}+T^{12n})}\right),
\left(T,\frac{-1+T^{4n}+4T^{6n}+T^{8n}-T^{12n}}{4T^n(1+2T^{4n}+2T^{8n}+T^{12n})}\right),\\
\left(T^3,\frac{1+3T^{4n}+4T^{6n}+3T^{8n}+T^{12n}}{4(1+2T^{4n}+2T^{8n}+T^{12n})}\right),
\left(T^5,\frac{T^n(1+4T^{2n}+3T^{4n}+4T^{6n}+3T^{8n}+4T^{10n}+T^{12n})}{4(1+2T^{4n}+2T^{8n}+T^{12n})}\right),\\
\left(T^7,\frac{T^{2n}(3+4T^{2n}+5T^{4n}+4T^{6n}+5T^{8n}+4T^{10n}+3T^{12n})}{4(1+2T^{4n}+2T^{8n}+T^{12n})}\right).
\end{gather*}
\normalsize

For example, when $n=2$ and $t=2$, one has the elliptic curve
 \[y^2=\frac{142608512}{250308167443425}x^4+\frac{62553486161362657 }{65873099809751270400}x^2+\frac{142608512}{250308167443425}\] which contains the following $8$-term geometric progression sequence
$$\left(2^{-7},\frac{54871363}{69258448896}\right),\left(2^{-5},\frac{21185345}{17314612224}\right),\left(2^{-3},\frac{5663659}{1442884352}\right),\left(2^{-1},\frac{16695041}{1082163264}\right),$$
$$\left(2,\frac{16695041}{270540816}\right),\left(2^3,\frac{5663659}{22545068}\right),\left(2^5,\frac{21185345}{16908801}\right),\left(2^7,\frac{219485452}{16908801}\right).$$


\section{A remark on geometric progressions of length 10}
In order to extend the length of the $8$-term geometric progression sequence we constructed in Corollary \ref{cor:hyperelliptic} to a geometric progression of length $10$, one assumes that a point of the form $(t^9,S')$, and consequently the point $(t^{-9},S't^{-9})$, exists on the hyperelliptic curve $y^2=a(t)x^{2n}+b(t)x^n+a(t)$. This yields the existence of a rational point $(p:q:S')$ on the elliptic curve $\mathcal L$ defined by
\begin{multline*}
S'^2=H_t'(p,q):=t^{10}(1+t^4)^2p^4+4t^5(1+t^2+2t^4+2t^6+2t^8+2t^{10}+2t^{12}+t^{14}+t^{16})p^3q\\
-2t^4(4+6t^2+11t^4+11t^6+12t^8+11t^{10}+11t^{12}+6t^{14}+4t^{16})p^2q^2\\
+4t^3(1+2t^2+3t^4+3t^6+3t^8+3t^{10}+3t^{12}+2t^{14}+t^{16})pq^3
+t^6(1+t^2+t^4)^2q^4.
\end{multline*}
One recalls that the pair $(p,q)$ makes up the first two coordinates of a point $(p:q:S)$ on the elliptic curve $\mathcal H:S^2=H_t(p,q)$ defined over $\Q(t)$. This implies that one needs to find a solution $(p,S,S')$ on the genus $5$ curve $\mathcal C$ defined by the affine equation
\[S^2=H_t(p,1),\;S'^2=H'_t(p,1).\] In view of Faltings' theorem, a genus five curve possesses finitely many rational points. Therefore, one reaches the following result.
\begin{proposition}
 Fix $t_0\in\Q$. For any nontrivial $10$-term geometric progression sequence of the form $t_0^{\pm1},t_0^{\pm3},t_0^{\pm5},t_0^{\pm 7},t_0^{\pm 9}$, there exist finitely many hyperelliptic curves of the form $C:y^2=ax^{2n}+bx^n+a,\,a,b\in\Q$, such that the numbers $t_0^{\pm i},i=1,3,5,7,9$, are the $x$-coordinates of rational points in $C(\Q)$.
\end{proposition}

\section{Acknowledgment}
We would like to thank Professor Nabil Youssef, Cairo University, for
his support, careful reading of an earlier draft of the paper, and
several useful suggestions that improved the manuscript.

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\bibitem{Ulas2} M. Ulas, On arithmetic progressions on genus two
curves, {\it Rocky Mountain J. Math.} \textbf{39} (2009), 971--980.

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\noindent 2010 {\it Mathematics Subject Classification:} Primary 14G05; Secondary 11B83.

\noindent \emph{Keywords:} geometric progression, hyperelliptic curve, rational point.

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\noindent
Received February 18 2016;
revised versions received  June 14 2016; June 17 2016.
Published in {\it Journal of Integer Sequences}, June 29 2016.

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