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% A Note on Arithmetic Progressions on Elliptic Curves
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\begin{document}
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\title{A note on arithmetic progressions on elliptic curves}
\maketitle
\centerline{Garikai Campbell}
\begin{center}
Department of Mathematics and Statistics \\
Swarthmore College\\
Swarthmore, PA 19081 \\
USA
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% Information for first author
%\author{Garikai Campbell}
% Address of record for the research reported here
%\address{Department of Mathematics and Statistics, Swarthmore College,
%Swarthmore, PA 19081}
% Current address
%\curraddr{Department of Mathematics and Statistics, Swarthmore College,
%Swarthmore, PA 19081}
%\email{kai@swarthmore.edu}
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% \keywords{Elliptic Curves, Arithmetic Progression}
\begin{abstract}
Andrew Bremner
({\it Experiment.\ Math.} {\bf 8} (1999), 409--413)
%\cite{BR}
has described a technique for producing infinite
families of elliptic curves containing length 7 and length 8 arithmetic
progressions. This note describes another way to produce infinite families
of elliptic curves containing length 7 and length 8 arithmetic progressions. We
illustrate how the technique articulated here gives an easy way to produce an
elliptic curve containing a length 12 progression and an infinite family of
elliptic curves containing a length 9 progression, with the caveat that these
curves are not in Weierstrass form.
\end{abstract}
% ------------ SECTION 1: INTRODUCTION -----------------------------
\section{Introduction.}
There are two (affine) models of elliptic curve that are very common.
They are $y^2=f(x)$ where $f(x)$ is either a cubic or a quartic.
We will say that {\em points on a particular model of an elliptic curve
are in arithmetic progression} if their $x$-coordinates form an arithmetic
progression. For example, Buhler, Gross and Zagier \cite{BU} found that the points
$(-3,0),$ $(-2,3),$ $(-1,3),$ $(0,2),$ $(1,0),$ $(2,0),$ $(3,3),$ and $(4,6)$
form an arithmetic progression of length 8 on the curve $y^2+y=(x-1)(x-2)(x+3)$.
Moreover, Bremner \cite{BR} proves:
\begin{theorem}
Each point on the elliptic curve $$C: y^2 = x^3 - x^2 - 36x + 36$$
corresponds to an elliptic curve in Weierstrass form containing at least 8
points in arithmetic progression.
\end{theorem}
Before proving this theorem, Bremner considers the following strategy.
First he remarks that any monic degree 8 polynomial, $P(x)$, can be written as
$Q(x)^2-R(x)$ where the degree of $R(x)$ is less than or equal to 3. If $R(x)$
has degree precisely 3 and no repeated zeros, then $y^2=R(x)$ is an elliptic
curve and for each zero, $\alpha$, of $P(x)$, this elliptic curve contains a pair of points
with $x$-coordinate $\alpha$. So one possible strategy for producing an elliptic curve with
an arithmetic progression of length 8 might be to let $P(x)=x(x+1)(x+2) \cdots(x+7)$
and compute the corresponding $R(x)$ so that $P(x)=Q(x)^2-R(x)$. Unfortunately,
in this case, $R(x)$ is linear and so this strategy fails for {\em any} degree 8 polynomial
whose zeros form an arithmetic progression. The goal of this note is to illustrate
how to turn this strategy into a successful one.
\section{Arithmetic Progressions of Length 8}
The statement that a degree 8 polynomial can be written as $Q(x)^2-P(x)$
is a special case of the following:
\begin{prop} \label{mprop}
If $P(x)$ is a monic polynomial of degree $2n$ defined over a field $k$,
then there are unique polynomials $Q(x)$ and $R(x)$ defined over $k$ such that
\begin{enumerate}
\item $P(x)= Q(x)^2 - R(x)$ and
\item the degree of $R(x)$ is strictly less than $n$.
\end{enumerate}
\end{prop}
Since $R(x)$ is a square at every zero of $P(x)$, if $R(x)$ is a cubic
or a quartic with no repeated zeros, then we can produce elliptic curves $y^2=R(x)$
with great control over many of the $x$-coordinates.
\begin{remark} We note that Mestre \cite{ME1} was first to observe that this relatively simple proposition
could be used to produce elliptic curves of large rank. Since Mestre's
first paper exploiting this idea, many others (\cite{CA1}, \cite{FE}, \cite{KI1}, \cite{KU}, \cite{NA1})
have used the proposition in clever ways to produce elliptic curves and infinite
families of elliptic curves with the largest known rank (often with some condition on
the torsion subgroup).
\end{remark}
Now consider the polynomial
$$p_t(x)=(x-t)^2 \prod_{j=0}^5 (x-j) \ \in \Q(t)[x].$$
In this case, we can write
$$p_t(x) = q_t(x)^2 - f_t(x),$$
where $f_t(x)$ is a polynomial of degree 3 in $\Q(t)[x]$ such that
\begin{enumerate}
\item the discriminant of $f_t(x)$ is an irreducible polynomial in $\Q[t]$
\item the coefficient of $x^3$ is $c (2t-5)$, where $c\in \Q$.
\end{enumerate}
Therefore, we have that
\begin{theorem}
The curve $E_t$ defined by $y^2 = f_t(x)$
is an elliptic curve defined over $\Q(t)$, containing at least six points in
arithmetic progression and for each $t_0\in \Q$, $t_0\neq 5/2$, the specialization of
$E_t$ at $t=t_0$ gives an elliptic curve defined over $\Q$ containing at least six points in
arithmetic progression.
\end{theorem}
We next observe that $f_t(6)$ is a conic in $\Q[t]$ which is a rational square when
$t=6$. Therefore, we can parameterize all rational solutions
to $y^2 = f_t(6)$ by letting
\begin{eqnarray}
t & = & \frac{6m^2 - 126m - 285360}{m^2 - 72256}. \label{t}
\end{eqnarray}
Since no rational value of $m$ gives $t=5/2$, we
have:
\begin{cor}
Let $g_m(x)$ be the polynomial $f_t(x)$ with $t$ given by (\ref{t}).
The curve $E_m$ defined by $y^2=g_m(x)$ is an elliptic curve defined over $\Q(m)$ containing at least
seven points in arithmetic progression and for each $m_0\in \Q$, the specialization
of $E_m$ at $m=m_0$ gives an elliptic curve defined over $\Q$ containing at least seven
points in arithmetic progression.
\end{cor}
If we continue in this vein and explore the conditions imposed by $y^2=g_m(7)$,
we find the following.
\begin{theorem}
Let $D$ be the elliptic curve defined by
\begin{eqnarray*}
D: y^2= -264815m^4 - 19343520m^3 + 62846856064m^2 \\
-2906312951808m - 495507443511296.
\end{eqnarray*}
Let
\begin{eqnarray*}
g_3 & = & -18816m^4 + 677376m^3 + 1922543616m^2 \\
& & \ \ \ \ \ \ - 48944480256m - 40678301368320, \\
g_2& = & 236896m^4 - 9821952m^3 - 22598349824m^2 \\
& & \ \ \ \ \ \ + 508953231360m + 520252184657920,\\
g_1& = & -958800m^4 + 40985280m^3 + 89932669440m^2 \\
& & \ \ \ \ \ \ - 1957723729920m - 2113363439616000,\mbox{ and}\\
g_0& = & 1292769m^4 - 57304800m^3 - 118795148928m^2 \\
& & \ \ \ \ \ \ + 2647001548800m + 2758336954896384.
\label{e}
\end{eqnarray*}
Then
$$E_m^\prime: y^2 = g_3\ x^3 + g_2\ x^2 + g_1\ x + g_0,$$
is an elliptic curve defined over $\Q(D)$ containing the 8 points in arithmetic
progression with $x$-coordinates 0, 1, 2, $\ldots$ , 7.
\end{theorem}
\begin{proof}
$E_m^\prime$ is isomorphic to $E_m$ via the change of variables
$y\mapsto y/(m^2-72256)$. Substituting $x=7$ into $E_m^\prime$, we get the curve $D$.
\end{proof}
Moreover, if we let $D(\Q)$ be the group of rational points on $D$,
then we have that $D(\Q)$ is infinite. More specifically, we have:
\begin{prop}
$D$ has rank 2 and torsion subgroup $\Z/2\Z$.
\end{prop}
\begin{proof}
A short computer search reveals that $O=(-88, 15628032)$ is a point
in $D(\Q)$. Taking $O$ taken to be the identity, $D(\Q)$
is generated by
\begin{eqnarray*}
P_0 & = & (10984/79, -80015523840/6241) \mbox{ and } \\
P_1 & = & (-1363640/2531, 31969540657152/6405961),
\end{eqnarray*}
and contains the point of order two:
\begin{eqnarray*}
P_2 & = & (10984/79, 80015523840/6241).
\end{eqnarray*}
\end{proof}
(The calculations above were performed with the help of
{\tt mwrank} \cite{CR} and GP \cite{GP}.)
An immediate consequence of the proposition above is the following:
\begin{cor}
Each point on the elliptic curve $D$
corresponds to an elliptic curve in Weierstrass form containing at least 8
points in arithmetic progression.
\end{cor}
\begin{remark}
This condition is very similar to the
condition found in Bremner's construction-- namely, that points on the curve
$C$ give rise to elliptic curves with 8 points in
arithmetic progression. The differences are that $C$ has rank 1 and
torsion subgroup $\Z/2\Z \times \Z/2\Z$,
while $D$ has rank 2 and torsion subgroup $\Z/2\Z$.
\end{remark}
\section{Longer Progressions}
This construction can also be used to produce progressions of length greater than
8 on elliptic curves of the form $y^2=f(x)$ where $f(x)$ is a quartic. More specifically,
we have:
\begin{theorem}
There exists an elliptic curve in the form $y^2=w(x)$, with $w(x)$ a quartic,
containing 12 points in arithmetic progression.
\end{theorem}
\begin{proof}
Let
$$g_0(x)=\prod_{j=0}^{11} (x-j).$$
Then $g_0(x)=u_0(x)^2 - (81/4)\cdot v_0(x)$, with
\begin{eqnarray*}
u_0(x)& = & x^6 - 33 x^5 + 418 x^4 - 2541 x^3 + (14993 /2) x^2 \\
& & \ \ \ \ \ - (18513 /2) x + (4851 /2), \mbox{ and } \\
v_0(x)& = & 429 x^4 - 9438 x^3 + 74295 x^2 - 246246 x + 290521.
\end{eqnarray*}
Since the discriminant of $v_0(x)$ is nonzero, the curve $E: y^2 = v_0(x)$ is an
elliptic curve. This elliptic curve then contains a length 12 arithmetic
progression.
\end{proof}
(Note that by using {\tt mwrank}, we computed the rank of
this curve to be 4 with torsion subgroup $\Z/2\Z$.)
The construction above produces a single curve and it is unclear how to produce
an infinite family of curves containing a length 12 progression using this idea.
The problem is that, in general, if the $P(x)$ of proposition \ref{mprop} is
taken to have degree 12, then the $R(x)$ is only guaranteed to have degree less
than or equal to $5$, not $4$. Therefore, the curve $y^2 = R(x)$ need not be
an elliptic curve. We can, however, prove the following.
\begin{theorem}
There are infinitely many elliptic curves of the form $y^2=w(x)$, with
$w(x)$ a quartic, containing 9 points in arithmetic progression.
\end{theorem}
\begin{proof}
Let
$$g(x)=(x-a) \cdot \prod_{j=0}^8 (x-j),$$
and write $g(x)$ as $u(x)^2 - v(x)$. $v(x)$ is a degree four polynomial in
$\Q(a)[x]$ with discriminant zero only for $a\in \{0,4,8\}$.
\end{proof}
The work here (and that of Bremner) leaves open the following questions:
\begin{question}
Is there an elliptic curve of the form $y^2=f(x), f(x)$ a cubic, containing
a length 9 arithmetic progression? Are there infinitely many?
\end{question}
\begin{question}
Is there an elliptic curve of the form $y^2=f(x), f(x)$ a quartic, containing
a length 13 arithmetic progression? Are there infinitely many curves in this
form containing a length 10 progression?
\end{question}
And finally,
\begin{question}
What is the longest arithmetic progression one can find on an elliptic curve
in the form $y^2=f(x)$, where $f(x)$ is a cubic? a quartic?
\end{question}
\section{Acknowledgments}
This work was completed with the support of the
Lindback Foundation Minority Junior Faculty Grant.
% \bibliographystyle{amsplain}
% \begin{thebibliography}{10}
% \end{thebibliography}
\nocite{*}
\begin{thebibliography}{10}
\bibitem{GP}
C.~Batut, K.~Belabas, D.~Bernardi, H.~Cohen, and M.~Olivier.
\newblock The Pari system.
\newblock {\tt ftp://megrez.math.u-bordeaux.fr/pub/pari/}, 2000.
\bibitem{BR}
Andrew Bremner.
\newblock On arithmetic progressions on elliptic curves.
\newblock {\em Experiment. Math.}, {\bf 8} (1999), 409 -- 413.
\bibitem{BU}
J.~P. Buhler, B.~H. Gross, and D.~B. Zagier.
\newblock On the conjecture of {Birch} and {Swinnerton-Dyer} for an elliptic
curve of rank 3.
\newblock {\em Math. Comp.}, {\bf 44} (1985), 473 -- 481.
\bibitem{CA1}
Garikai Campbell.
\newblock {\em Finding elliptic curves and infinite families of elliptic curves
defined over ${Q}$ of large rank}.
\newblock PhD thesis, Rutgers University, June 1999.
\newblock Available at {\tt http://math.swarthmore.edu/kai/thesis.html}.
\bibitem{CR}
John Cremona.
\newblock Home page.
\newblock {\tt http://www.maths.nottingham.ac.uk/personal/jec/ftp/progs/}.
\bibitem{FE}
Stefane F{\'{e}}rmigier.
\newblock Un exemple de courbe elliptique d{\'{e}}finie sur ${Q}$ de rang $\geq
19$.
\newblock {\em C. R. Acad. Sci. Paris S\'er. I}, {\bf 315} (1992), 719 -- 722.
\bibitem{KI1}
Shoichi Kihara.
\newblock On an infinite family of elliptic curves with rank $\geq 14$ over
${Q}$.
\newblock {\em Proc. Japan Acad. Ser. A.}, {\bf 73} (1997) 32.
\bibitem{KU}
L.~Kulesz.
\newblock {\em {Arithm\'etique} des courbes {alg\'ebriques} de genre au moins
deux}.
\newblock PhD thesis, {Universit\'e} Paris 7, 1998.
\bibitem{ME1}
Jean-Fran{\c{c}}ois Mestre.
\newblock Construction d'une courbe elliptique de rang $\geq 12$.
\newblock {\em C. R. Acad. Sci. Paris S\'er. I}, {\bf 295} (1982), 643 -- 644.
\bibitem{ME2}
Jean-Fran{\c{c}}ois Mestre.
\newblock Courbes elliptiques de rang $\geq 11$ sur ${Q}(t)$.
\newblock {\em C. R. Acad. Sci. Paris S\'er. I}, {\bf 313} (1991), 139 -- 142.
\bibitem{NA1}
Koh-Ichi Nagao.
\newblock Examples of elliptic curves over ${Q}$ with rank $\geq 17$.
\newblock {\em Proc. Japan Acad. Ser. A.}, {\bf 68} (1997), 287 -- 289.
\end{thebibliography}
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\bigskip
\noindent 2000 {\it Mathematics Subject Classification}:
11G05, 11B25 .\ \
\noindent \emph{Keywords: elliptic curves, arithmetic progression}
\bigskip
\hrule
\bigskip
\vspace*{+.1in}
\noindent
Received February 5, 2003;
revised version received February 7, 2003.
Published in {\it Journal of Integer Sequences} February 25, 2003.
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Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.math.uwaterloo.ca/JIS/}.
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