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\begin{center}
\vskip 1cm{\LARGE\bf Arithmetic Progressions on Edwards Curves
}
\vskip 1cm
\large
Andrew Bremner \\
School of Mathematical and Statistical Sciences \\
Arizona State University \\
Tempe AZ 85287-1804 \\
USA \\
\href{mailto:bremner@asu.edu}{\tt bremner@asu.edu} \\
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\begin{abstract}
Several authors have investigated the problem of finding elliptic curves
over $\Q$ that contain rational points whose $x$-coordinates are in arithmetic progression.
Traditionally, the elliptic curve has been taken in the form of an elliptic cubic
or elliptic quartic. Moody studied this question for elliptic curves
in Edwards form, and showed that there are infinitely many such curves upon which
there exist arithmetic progressions of length 9, namely, with $x=0,\pm 1,\pm 2,\pm 3,\pm 4$.
He asked whether any such curve will allow an extension to a progression of 11 points.
This note shows that such curves do not exist. A certain amount of luck comes into play,
in that we need only work over a quadratic extension field of $\Q$.
\end{abstract}
\section{Introduction}
A sequence of points on an elliptic curve is said to be an {\it arithmetic progression} on
the curve if
the $x$-coordinates of the points lie in arithmetic progression. The model chosen for
the elliptic curve will clearly affect the consequent theory. Bremner~\cite{Bre}, Campbell~\cite{Cam}, and
Garcia-Selfa and Tornero~\cite{GST} have investigated arithmetic progressions
on curves in Weierstrass cubic form; and Campbell~\cite{Cam}, Macleod~\cite{Mac}, and
Ulas~\cite{Ula} have investigated progressions on quartic models. Recntly, Moody~\cite{Moo}
investigates progressions on elliptic curves in the Edwards form
\[ E_d: \; x^2+y^2 =1 +d x^2 y^2, \qquad d \neq \pm 1, \]
which automatically contain the length three progression consisting of the points
$(-1,0)$, $(0,0)$, $(1,0)$. He shows that there exist infinitely many $d$ such that this
length 3 progression extends to a sequence of nine points whose $x$-coordinates
lie in arithmetic progression. He asks whether such progressions of length 9 can ever
be extended to progressions of length 11. By investigating a curve of genus 5 and using
elliptic Chabauty techniques, we prove here that such an extension is not possible.
\section{Progressions of length 11}
The condition that there be rational points on $E_d$ at $\pm x=2,3,4,5$ is precisely that
there exist rationals $p,q,r,s$ satisfying
\[ \frac{24 p^2+1}{25} = \frac{15 q^2+1}{16} = \frac{8 r^2+1}{9} = \frac{3 s^2+1}{4} (= d). \]
Homogenizing by setting $(p,q,r,s)=(P/T,Q/T,R/T,S/T)$, we get
\begin{equation}
\label{intersection}
32 P^2=25 S^2+7 T^2, \qquad 5 Q^2=4 S^2+T^2, \qquad 32 R^2=27 S^2+5 T^2.
\end{equation}
As an intersection of three quadrics in projective four-dimensional space, these equations
define a (non-singular) curve $C$ of genus 5. The trick in the following analysis is
finding an amenable way of writing down the equations for $C$. We choose the following formulation:
\[ 125Q^2=128 P^2-3 T^2, \quad 25R^2=27P^2-2T^2, \quad 25S^2=32P^2-7T^2. \]
From (\ref{intersection}), if both $P,T$ are divisible by $5$, then $S$, $R$, $Q$ are each
divisible by $5$; and so without loss of generality, $P$ (and hence $T$) is not divisible
by $5$. We may certainly suppose that $P>0$; and the sign of $T$ may be chosen so that
$P \equiv T \bmod{5}$. \\
Let $K=\Q(\theta)$, $\theta^2=6$, a field with class number 1. A fundamental unit is $\epsilon=5-2\theta>0$.
We have the prime ideal factorizations:
\[ (2)=\mathfrak{p}_2^2=(2+\theta)^2, \qquad (3)=\mathfrak{p}_3^2=(3-\theta)^2, \qquad (5)=\mathfrak{p}_{5}\mathfrak{p}'_{5}=(1+\theta)(1-\theta). \]
Now $\mbox{Norm}_{K/\Q}(16P-\theta T)=10(5Q)^2$ and thus
\[ 16P-\theta T = \pm (2+\theta)(1-\theta)\epsilon^i G^2= \pm (-4-\theta)\epsilon^i G^2 = (4+\theta) \epsilon^i G^2, \]
where $G$ is an integer of $K$, $i=0,1$, and the sign is determined by the fact that the left hand side is positive.
Similarly, $\mbox{Norm}_{K/\Q}(9P-\theta T)=3(5R)^2$, so that
\[ 9P-\theta T=\pm (3-\theta )\epsilon^j H^2 = (3-\theta)\epsilon^j H^2, \]
for an integer $H$ of $K$, $j=0,1$, and the sign determined by positivity. We can now deduce
\begin{align}
\label{ellquartic}
(16P-\theta T) (9P-\theta T) = (4+\theta)(3-\theta)\epsilon^k \square = (6-\theta) \epsilon^k \square, & \nonumber \\
(16P+\theta T) (9P+\theta T) (32P^2-7T^2) = (6+\theta) \epsilon^{-k} \square, &
\end{align}
where $k=0,1$. \\ \\
Case $k=0$.
The second equation at (\ref{ellquartic}) on setting $x=P/T$ gives the equation of an elliptic
quartic curve over $K$:
\begin{equation}
\label{quartic1}
(16x+\theta)(9x+\theta)(32x^2-7)=(6+\theta)y^2,
\end{equation}
with cubic form
\[ Y^2 = X(X^2 + 68(6+\theta) X + 6250(7+2\theta)). \]
Computations with Magma~\cite{Mag} show that this curve has rank 1 over the field $K$ of degree
2 over $\Q$, and since the rank is less than the degree of the field extension, elliptic Chabauty
methods apply. Magma returns that the only $K$-points of (\ref{quartic1}) with $x \in \Q$ are
$(x,\pm y)=(1,25)$. \\ \\
Case $k=1$.
The second curve at (\ref{ellquartic}) becomes on setting $x=P/T$
\begin{equation}
\label{quartic2}
(16x+\theta)(9x+\theta)(32x^2-7)= (42+17\theta)y^2,
\end{equation}
with cubic model
\[ Y^2=X(X^2 + 68(18-7\theta) X + 6250(103-42\theta)). \]
This curve has rank 0, and the $\Z/2\Z$ torsion points correspond to the points
$x=-\theta/9, -\theta/16$ on (\ref{quartic2}), with non-rational $x$. \\ \\
Consequently, the only rational points on the curve $C$ at (\ref{intersection})
are given by $P^2=Q^2=R^2=S^2=T^2$, and it follows there are no length $11$ arithmetic
progressions of the desired type.
\begin{remark}
There are ten ways to write the intersection at (\ref{intersection}), with the three formulations
\[ 5p^2 = 7r^2 - 2s^2, \qquad 25q^2 = 32r^2 - 7s^2, \qquad 5t^2 = 32r^2 - 27s^2 \]
\[ 7p^2 = 10q^2 - 3r^2, \qquad 7s^2 = -25q^2 + 32r^2, \qquad 7t^2 = 135q^2 - 128r^2 \]
\[ 3r^2 = -7p^2 + 10q^2, \qquad 3s^2 = -32p^2 + 35q^2, \qquad 3t^2 = 128p^2 - 125q^2 \]
also affording a similar treatment to the above involving factorization over a quadratic field,
namely $\Q(\sqrt{14})$, $\Q(\sqrt{30})$, and $\Q(\sqrt{70})$, respectively. An
argument over $\Q(\sqrt{30})$ is successful, with both elliptic curves that arise having rank 1;
but over $\Q(\sqrt{14})$ and $\Q(\sqrt{70})$ a corresponding elliptic curve has rank at least 2
over $K$, and elliptic Chabauty methods are inapplicable. In the remaining six formulations,
it would be necessary to work over a quartic extension of $\Q$.
\end{remark}
\section{Concluding remarks}
The result of this note shows that Moody's centrally symmetric $9$-term progression
of points on $E_d$ cannot be extended to a centrally symmetric $11$-term progression.
The question as to whether $E_d$ may possess arithmetic progressions that are not
centrally symmetric has been addressed in Gonz\'alez-Jim\'enez~\cite{Gon},
where he expresses the belief that {\it no} arithmetic progression on $E_d$ can have
more than $9$ elements.
An upper bound for the length of arithmetic progressions on elliptic curves
under any presentation, remains elusive. In the cases of Weierstrass cubics or elliptic
quartics, examples are known (Bremner~\cite{Bre}, Macleod~\cite{Mac}) of progressions of length $8$
and length $14$, respectively (the latter examples being centrally symmetric).
To show, for example, that there are no progressions of length $9$ on a cubic model
of an elliptic curve is equivalent to finding all the rational points on the
(non-singular) intersection of five quadrics in $\mathbb{P}^8$, a variety of
dimension $3$, and which at present seems an intractable problem.
\begin{thebibliography}{99}
\bibitem{Bre} A. Bremner, On arithmetic progressions on elliptic
curves, {\it Experiment. Math.} {\bf 8} (1999), 409--413.
\bibitem{Cam} G. Campbell, A note on arithmetic progressions on
elliptic curves, {\it J. Integer Seq.} {\bf 6} (2003),
\href{https://cs.uwaterloo.ca/journals/JIS/VOL6/Campbell/campbell4.html}{Article 03.1.3}.
\bibitem{GST} I. Garc\'ia-Selfa and J. Tornero, Searching for
simultaneous arithmetic progressions on elliptic curves, {\it Bull.
Austral. Math. Soc.} {\bf 71} (2005), 417--424.
\bibitem{Gon} E. Gonz\'alez-Jim\'enez, On arithmetic progressions on
Edwards curves, preprint, \url{http://arxiv.org/pdf/1304.4361.pdf}.
\bibitem{Mac} A. Macleod, 14-term arithmetic progressions on quartic
elliptic curves, {\it J. Integer Seq.} {\bf 9} (2006),
\href{https://cs.uwaterloo.ca/journals/JIS/VOL9/MacLeod/macleod11.html}{Article 06.1.2}.
\bibitem{Mag} Wieb Bosma, John Cannon, and Catherine Playoust, The
Magma algebra system. I. The user language, {\it J. Symbolic Comput.},
{\bf 24} (1997), 235--265.
\bibitem{Moo} D. Moody, Arithmetic Progressions on Edwards Curves, {\it
J. Integer Seq.} {\bf 14} (2011),
\href{https://cs.uwaterloo.ca/journals/JIS/VOL14/Moody/moody2.html}{Article 11.1.7}.
\bibitem{Ula} M. Ulas, A note on arithmetic progressions on quartic
elliptic curves, {\it J. Integer Seq.} {\bf 8} (2005),
\href{https://cs.uwaterloo.ca/journals/JIS/VOL8/Ulas/ulas7.html}{Article 05.3.1}.
\end{thebibliography}
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\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11G05; Secondary 11B25, 11G30.
\noindent \emph{Keywords: }
arithmetic progression, elliptic curve, Edwards curve, Chabauty.
\bigskip
\hrule
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\noindent
Received August 6 2013;
revised version received September 9 2013.
Published in {\it Journal of Integer Sequences}, October 12 2013.
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