\documentclass[12pt,reqno]{article}
\usepackage[usenames]{color}
\usepackage{graphicx}
\usepackage{amscd}
\usepackage[colorlinks=true,
linkcolor=webgreen,
filecolor=webbrown,
citecolor=webgreen]{hyperref}
\definecolor{webgreen}{rgb}{0,.5,0}
\definecolor{webbrown}{rgb}{.6,0,0}
\usepackage{color}
\usepackage{fullpage}
\usepackage{float}
\usepackage{psfig}
\usepackage{graphics,amsmath,amssymb}
\usepackage{amsfonts}
\usepackage{amsthm}
\usepackage{latexsym}
\usepackage{epsf}
\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{.1in}
\setlength{\evensidemargin}{.1in}
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{8.9in}
\newcommand{\seqnum}[1]{\href{http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=#1}{\underline{#1}}}
\begin{document}
\begin{center}
\epsfxsize=4in
\leavevmode\epsffile{logo129.eps}
\end{center}
\begin{center}
\vskip 1cm{\LARGE\bf
Curves of Genus~2, Continued Fractions, and Somos
Sequences
}
\vskip 1cm
\large
Alfred J. van der Poorten\footnote{
The author's only support was a grant from the Australian
Research Council.}\\
Centre for Number Theory Research \\
1 Bimbil Place \\
Killara, Sydney \\
NSW 2071 \\
Australia \\
% Current address
%\curraddr{School of MPCE, Macquarie University, Sydney, NSW 2109, Australia}
\href{mailto:alf@@math.mq.edu.au}{\tt alf@math.mq.edu.au} \\
\end{center}
\vskip .2in
\begin{abstract} We detail the continued fraction expansion of the square
root of monic sextic polynomials. We note in passing that each line
of the expansion corresponds to addition of the divisor at infinity,
and interpret the data yielded by the general expansion. In
particular we obtain an associated Somos sequence defined by a
three-term recurrence relation of width~$6$.
\end{abstract}
\vskip .2in
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{lemma}{Lemma}[section]
%\documentclass{amsart}
%\usepackage{verbatim}
%\usepackage{syntonly}
%\usepackage{url} \urlstyle{sffont}
%\usepackage{showkeys}
%\syntaxonly
% Comment out preceding line to produce output.
\mathsurround=1pt
\def\Z{{\mathbb Z}} \def\Q{{\mathbb Q}}
\def\F{{\mathbb F}} %% means, with subscript, finite field to me
%\def\\{\cr} %This may be a bad thing; check that it does no harm
\def\lt{<}
\def\gt{>}
\def\conj{\overline}
\def\ov{\overline}
\def\id#1{\langle #1\rangle} %%% ideal, in other words
%-------------Abbreviations
\def\poly {polynomial}
\def\cf {continued fraction}
\def\pq {partial quotient}
\def\cq {complete quotient}
\def\ap {approximation}
\def\ex{expansion}
\def\cv {convergent}
\def\cfe{continued fraction expansion}
\def\Jac{{\rm Jac}}
\def\ch{{\rm char} }
\renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}}
%\def\ds{\displaystyle}
%\theoremstyle{plain}
%\newtheorem{theorem}{Theorem}
%\theoremstyle{definition}
%\newtheorem{definition}[theorem]{Definition}
%\newtheorem{Remark}{Remark}
%\maketitle
\section{Introduction}
In the present note I study the \cfe\ of the square root of a sextic
\poly, inter alia allowing the identification of sequences generated by
recursions
$$A_{h-3}A_{h+3}=aA_{h-2}A_{h+2}+bA_{h}^2\,.
$$
Specifically, see \S\ref{s:example} at page~\pageref{s:example} for the case
$(T_h)=(\ldots$, $2$, $1$, $1$, $1$, $1$, $1$, $1$, $2$, $3$, $4$, $8$, $17$,
$50$, $107$, $239$,
$1103$, $\ldots)$, where I illustrate how the
\cfe\ data readily allows one to identify the genus~$2$ curve $\mathcal C:
Y^2=(X^3-4X+1)^2+4(X-2)$ as giving rise to the sequence.
\section{Some Brief Reminders}
\noindent A reminder exposition on \cf s in quadratic function fields appears as
\S4 of \cite{169}. However, the na\"{\i}ve reader needs little more than that a
\cfe\ of a
quadratic irrational integer function $Z$ is a two-sided sequence of lines, $h$
in $\Z$,
\begin{equation*}
\frac{Z+P_h}{Q_h}=a_h-\frac{\conj Z+P_{h+1}}{Q_h}\,;\quad\text{in brief
$Z_h=a_h-\conj R_h$}\,,
\end{equation*}
with $(Z+P_{h+1})(\conj Z+P_{h+1})=-Q_hQ_{h+1}$ defining the sequences $(P_h)$
and $(Q_h)$ of \poly s. Necessarily, one must have $Q_0$ divides
$(Z+P_{0})(\conj Z+P_{0})$ in which case the sequence $(a_h)$
consisting of \poly s
guarantees that always $Q_h$ divides $(Z+P_{h})(\conj Z+P_{h})$.
If each \emph{\pq} $a_h$ is always chosen as the \poly\ part of
$Z_h$ then
$Z_0$ \emph{reduced} (that is, $\deg Z>0$ and $\deg \conj Z<0$) implies that all
the
$Z_h$ and $R_h$ are reduced; and always $a_h$ also is the \poly\ part of
$R_h$. Then conjugation --- in brief: studying the lines $R_h=a_h-\conj Z_h$ ---
retrieves the
left hand half of the
\ex\ of
$Z_0$ from the right hand half of the \ex\ of $R_0$.
\section{Curves of Genus}
\noindent Set $A(X)=X^3+fX+g$ and $R(X)=u(X^2-vX+w)$. Then
\begin{equation}\label{eq:Z}
\mathcal C: Z^2-AZ-R=0\end{equation}
defines a quadratic irrational integer function $Z$ as a Laurent series
$\sum_{h=-3}^\infty
z_hX^{-h}$ in $X^{-1}$. Here $\deg Z=3$ refers to the degree in $X$ of $Z$. Note
that because the
other zero $\conj Z$ of equation \eqref{eq:Z} satisfies $Z\conj Z=-R$, and $\deg
R\le2$, we must
have $\deg \conj Z\lt0$, so $Z$ is reduced. We also note that the discriminant
$D(X)$ of \eqref{eq:Z} is given by $D=A^2+4R$ and is thus a general sextic
\poly. Evidently, if
$Y^2=D$ we may think of $Z$ as $Z=\frac12(Y+A)$.
However, in defining $Z$ by \eqref{eq:Z} we allow the base field $\F$ to be of
arbitrary
characteristic, whereas any talk of $Y$ of course requires that $\ch \F\ne2$.
Now set $Z_0=(Z+P_0)/Q_0$ with $P_0=d_0(X+e_0)$. Suppose that
$Q_0(X)=X^2-v_0X+w_0$ divides the norm
$$Z_0\conj Z_0=-R+d_0(X+e_0)\bigl(A+d_0(X+e_0)\bigr)\,,$$
and that $Z_0$ has been so chosen that its \pq s are of degree~$1$. Such a
choice is `generic' if
the base field is infinite. It follows from recursion formul\ae\ immediately
below that our
requirement on $Z_0$ is the same as insisting that the $d_h$ alll\footnote{Of
course
it suffices that just those $d_h$ actually participating in our discussion not
vanish.} be
nonzero.
For
$h=0$,
$1$,
$2$,
$\ldots\,$ we denote the \cq s of $Z_0$ by
\begin{equation}
\label{eq:notation}Z_h=\bigl(Z+d_h(X+e_h)\bigr)/u_h(X^2-v_hX+w_h)\,,
\end{equation}
noting that the $Z_h$ all are reduced, namely $\deg Z_h\gt0$ but $\deg
\conj Z_h\lt0$. The upshot is that the $h$-th line of the
\cfe\ of
$Z_0$ is
\begin{equation}\label{eq:P}
Z_h=\frac{Z+d_h(X+e_h)}{u_h(X^2-v_hX+w_h)}=\frac{X+v_h}{u_h} -
\frac{\conj Z+d_{h+1}(X+e_{h+1})}{u_h(X^2-v_hX+w_h)}\,.
\end{equation}
\begin{theorem}\label{th:d} In the special case $u=0$, that is: $R=-v(X-w)$, the
sequence of
parameters $(d_h)$ given by the \cfe\ of $Z_0$ satisfies
\begin{equation}\label{eq:more}
d_{h-2}d_{h-1}^2d_h^3d_{h+1}^2d_{h+2}=v^2d_{h-1}d_h^2d_{h+1}-v^3(g+wf+w^3)\,.
\end{equation}
\end{theorem}
\noindent Note here that $g+wf+w^3=A(w)$ and, I add this \emph{en passant}, the
result, with
$g+wf+w^3$ replaced by $A(w)$, does not in fact depend on the convenient
assumption throughout
that $A(X)$ has no term in $X^2$. However, the main feature is that the
recursion
\eqref{eq:more} depends only on the given curve $\mathcal C$ and not on the
`initial' complete
quotient $Z_0$.
Now define a sequence $(T_h)$ of elements of $\F$ by the recursive relation
\begin{equation}\label{eq:relation}
T_{h-1}T_{h+1}=d_hT_h^2\,, \quad\text{$h\in\Z$.}\quad
\end{equation}
One then sees fairly readily that
$$
T_{h-2}T_{h+2}=d_{h-1}d_h^2d_{h+1}T_h^2 \quad\text{and}\quad
T_{h-3}T_{h+3}=d_{h-2}d_{h-1}^2d_h^3d_{h+1}^2d_{h+2}T_h^2\,;
$$
and thus that multiplying \eqref{eq:more} by $T_h^2$ provides the principal
result of this note.
\begin{theorem}\label{th:T} A curve
$$
\mathcal C: Z^2-(X^3+fX+g)Z+v(X-w)=0
$$
gives rise to sequences
$(T_h)$ of Somos type defined by suitable initial values and the recursive
relation
\begin{equation}\label{eq:recurrence}
T_{h-3}T_{h+3}=v^2T_{h-2}T_{h+2}-v^3(g+wf+w^3)T_h^2\,.
\end{equation}
\end{theorem}
{\noindent \bf Remarks.}
All that is well and good of course but the real point
%\begin{proof}[Remarks]
here is this. The
$d_h$ are generically, so to speak, random rationals growing in height with $h$
so as to have
logarithmic height $O(h^2)$; thus they become complicated indeed. Very
differently, however,
recursions such as \eqref{eq:recurrence} of `Somos type' and width, or `gap'
(the maximum
difference of the indices), at most seven --- $6$ in the present case --- are
now well known to
`want to' consist of integers. Specifically, results of Fomin and Zelevinsky
summarised
in \cite{FZ} guarantee that the $T_h$ are Laurent polynomials in the `initial
values' $T_{-3}$,
$\ldots\,$, $T_{2}$, say, with coefficients in the ring $\Z[a,b]$ --- where in
the present case
$a=v^2$ and $b=-v^3(g+wf+w^3)$. This explains why the example sequence (at
page~\pageref{s:example}) with
$a=b=1$ and the six initial values all $1$ takes only integer values --- mind
you, integers whose
logarithm grows at rate $O(h^2)$.
For a different emphasis, notice that the pair of zeros of each $Q_h(X)$
produced by the \cfe\ defines a
\emph{divisor} on $\mathcal C$; I talk loosely immediately below of the
`divisor class $Q_h$', meaning the
class of the divisor given by the pair of points on
$\mathcal C$ defined over some quadratic extension of the base field $\F$ with
$X$ co-ordinates the two zeros
of the \poly\ $Q_h(X)$. Viewed as points on the additive group $\Jac\mathcal C$
it is well understood that the
sequence of divisor classes $(Q_h)$ is an arithmetic progression with common
difference $S$ the class of the
divisor at infinity. In brief, exactly as in the elliptic case \cite{169}. each
step of the \cfe\ adds the
divisor at imfinity to the divisor belonging to the complete quotient. Concerned
readers might contemplate the
introduction to Cantor's paper \cite{Ca} and the instructive discussion by
Kristin Lauter in \cite{La}. A
central theme of the paper \cite{BCZ} is a generalisation of the phenomenon to
Pad\'e
approximation in arbitrary algebraic function fields.
More of course, Theorem~\ref{th:T} is in tight analogy with the
corresponding result for quartic \poly s detailed in \cite{169}. In that case,
however,
\emph{singular} cases are incorporated. Here, cases when one or more of the
$d_h$ vanish and
therefore one or more of the $T_h$ vanish are more problematic and will have to
be the subject of
further analysis elsewhere. I do not yet know whether my assumption that the
\cfe\ is generic,
thus that none of the
$d_h$ vanish, is or is not essential to the validity of the present results.
%\end{proof}
\section{Continued Fraction Expansion of the Square Root of a Sextic}
\label{s:cfe}
\noindent Given that the $h$-th line of the \cfe\ of $Z_0$ is given by
\begin{equation}
Z_h=\frac{Z+d_h(X+e_h)}{u_h(X^2-v_hX+w_h)}=\frac{X+v_h}{u_h} -
\frac{\conj Z+d_{h+1}(X+e_{h+1})}{u_h(X^2-v_hX+w_h)}\,,\tag{\ref{eq:P}}
\end{equation}
evident recursion formulas yield
\begin{equation} \label{eq:d}
f+d_h+d_{h+1}=-v_h^2+w_h
\end{equation}
\begin{equation} \label{eq:e}
g+d_he_h+d_{h+1}e_{h+1}=v_hw_h
\end{equation}
and
\begin{multline}\label{eq:Q}
-u_hu_{h+1}(X^2-v_hX+w_h)(X^2-v_{h+1}X+w_{h+1})\\
=\bigl(Z+d_{h+1}(X+e_{h+1})\bigr)\bigl(\conj Z+d_{h+1}(X+e_{h+1})\bigr).
\end{multline}
Hence, noting that $Z\conj Z=-u(X^2-vX+w)$ and $Z+\conj Z=A=X^3+fX+g$, we may
equate coefficients in \eqref{eq:Q} to see that
\begin{equation} \label{eq:X4}
d_{h+1}=-u_hu_{h+1}\,.
\tag{$\ref{eq:Q}:X^4$} \end{equation}
Given that, we obtain, after in each case dividing by
$-u_hu_{h+1}$,
\begin{equation} \label{eq:X3}
e_{h+1}=-v_h-v_{h+1}\,;
\tag{$\ref{eq:Q}:X^3$} \end{equation}
\begin{equation} \label{eq:X2}
(f+d_{h+1})=v_hv_{h+1}+(w_h+w_{h+1})+u/d_{h+1}\,;
\tag{$\ref{eq:Q}:X^2$} \end{equation}
\begin{equation} \label{eq:X1}
(f+d_{h+1})e_{h+1}+(g+d_{h+1}e_{h+1})
=-v_hw_{h+1}-v_{h+1}w_h-uv/d_{h+1}\,;
\tag{$\ref{eq:Q}:X^1$} \end{equation}
\begin{equation} \label{eq:X0}
(g+d_{h+1}e_{h+1})e_{h+1}
=w_hw_{h+1}+uw/d_{h+1}\,.
\tag{$\ref{eq:Q}:X^0$} \end{equation}
The :$X^2$ equation readily becomes
\begin{equation*}
-d_h=f-w_h+v_h^2+d_{h+1}=v_h(v_h+v_{h+1})+w_{h+1}+u/d_{h+1}\,,
\end{equation*}
so $d_{h+1}(v_he_{h+1}-w_{h+1})=d_hd_{h+1}+u$. With similar manipulation of the
next two equations we felicitously obtain
\begin{subequations}\label{eq:helpful}
\begin{align}
d_{h+1}(v_he_{h+1}-w_{h+1})&=d_hd_{h+1}+u\,;\\
-v_hd_{h+1}(v_he_{h+1}-w_{h+1})&=d_hd_{h+1}(e_h+e_{h+1})-uv\,;\\
w_{h}d_{h+1}(v_he_{h+1}-w_{h+1})&=d_hd_{h+1}e_he_{h+1}+uw\,.
\end{align}
\end{subequations}
That immediately yields
\begin{subequations}\label{eq:useful}
\begin{align}
d_hd_{h+1}(e_h+e_{h+1}+v_h)&=u(v-v_h)\,;\\
d_hd_{h+1}(e_he_{h+1}-w_h)&=-u(w-w_h)\,.
\end{align}
\end{subequations}
Incidentally, by
\begin{equation*}
-d_{h+1}=f-w_h+v_h^2+d_{h}=v_h(v_{h-1}+v_{h})+w_{h-1}+u/d_{h}\,,
\end{equation*}
we also discover that, mildly surprisingly,
\begin{equation}\label{eq:surprise}
d_hd_{h+1}+u=d_{h+1}(v_he_{h+1}-w_{h+1})=d_{h}(v_he_{h}-w_{h-1}).
\end{equation}
\section{A Ridiculous Computation}\label{s:ridiculous}
\noindent It is straightforward to notice that the three final equations
\eqref{eq:Q} yield
$$
e_h^2(v_{h-1}v_h+w_{h-1}+w_h)+e_h(v_{h-1}w_h+v_hw_{h-1})+w_{h-1}w_h=
-u(e_h^2+ve_h+w)/d_h\,.
$$
Remarkably, by \eqref{eq:surprise}
\begin{multline*}
(d_{h-1}d_h+u)(d_hd_{h+1}+u)=d_h^2(v_{h-1}e_{h}-w_{h})(v_he_{h}-w_{h-1})\\
=e_h^2v_{h-1}v_h-e_h(v_{h-1}w_{h-1}+v_hw_h)+w_{h-1}w_h
\end{multline*}
and so, because
\begin{multline*}
-(v_{h-1}w_{h-1}+v_hw_h)=v_{h-1}w_h+v_hw_{h-1}-(w_{h-1}+w_h)(v_{h-1}+v_h)\\
=v_{h-1}w_h+v_hw_{h-1}+e_h(w_{h-1}+w_h)\,,
\end{multline*}
we obtain the surely useful identity
\begin{equation}\label{eq:identity}
(d_{h-1}d_h+u)(d_hd_{h+1}+u)=-ud_h(e_h^2+ve_h+w)\,.
\end{equation}
This is just one of the nine such identities provided by the
equations
\eqref{eq:helpful}, and
\eqref{eq:surprise}.
\subsection{The special case $u=0$} Consider now the case in which $R$, the
remainder term
$u(X^2-vX+w)$, is replaced by $-v(X-w)$. In effect $u\gets0$ except that
$uv\gets
v$, $uw\gets vw$. For instance, \eqref{eq:identity} becomes
\begin{equation}\label{eq:identity'}
d_{h-1}d_hd_{h+1}=-v(e_h+w)\,,\tag{\ref{eq:identity}$'$}
\end{equation}
and, we'll need this, we now have
\begin{align}
\label{eq:v} e_h+e_{h+1}+v_h&=v/d_hd_{h+1}\,;\tag{\ref{eq:useful}$'$a}\\
\label{eq:w} e_he_{h+1}-w_h&=-vw/d_hd_{h+1}\,.\tag{\ref{eq:useful}$'$b}
\end{align}
Indeed, we find that
\begin{equation}
d_{h-1}d_h^2d_{h+1}^2d_{h+2}=v^2(e_he_{h+1}+w(e_h+e_{h+1})+w^2)=
v^2(w_h-wv_h+w^2)
\end{equation}
and therefore that
\begin{multline}
d_{h-2}d_{h-1}^3d_h^4d_{h+1}^3d_{h+2}=\\
v^4\bigl(w_{h-1}w_h+w^2\bigl(v_{h-1}v_h+(w_{h-1}+w_h)\bigr)-w(v_{h-1}w_h+w_{h-1}
v_{h})-
w^3(v_{h-1}+v_h)+w^4\bigr)\,.
\end{multline}
This last expression is transformed by the equations \eqref{eq:Q} to become
\begin{multline}
v^4\bigl((g+d_he_h)e_h-vw/d_h+w^2(f+d_h)+
\\+w\bigl((f+d_h)e_h+(g+d_he_h)+v/d_h\bigr)+w^3e_h+w^4\bigr)\\
=v^4(e_h+w)\bigl((g+d_he_h)+w(f+d_h)+w^3\bigr)\,.
\end{multline}
Thus
\begin{equation}
d_{h-2}d_{h-1}^2d_h^3d_{h+1}^2d_{h+2}=-v^3\bigl((g+d_he_h)+w(f+d_h)+w^3\bigr)\,.
\end{equation}
But wait, there's more! By \eqref{eq:identity'} we know that
$-ve_h=d_{h-1}d_hd_{h+1}+vw$, so
\begin{equation}
d_{h-2}d_{h-1}^2d_h^3d_{h+1}^2d_{h+2}=v^2d_{h-1}d_h^2d_{h+1}-v^3(g+wf+w^3)\,,\
tag{\ref{eq:recurrence}}
\end{equation}
already announced as Theoren~\ref{th:d}.
\section{A Cute Example}\label{s:example}
\noindent The example
\begin{equation}\label{eq:example}
T_{h-3}T_{h+3}=T_{h-2}T_{h+2}+T_h^2\,,
\end{equation}
with $T_0=T_1=T_2=T_3=T_4=T_5=1$ is readily found to derive from the genus~$2$
curve
\begin{equation}\label{eq:examplecurve}
\mathcal C: Z^2-(X^3-4X+1)Z+(X-2)=0\,.
\end{equation}
To indeed see this, we first note that of course we need $d_1=d_2=d_3=d_4=1$ to
produce the
initial values from
$T_0=T_1=1$. Since, plainly, $T_{-1}=T_6=2$, clearly $d_0=2$. By the Theorem, we
expect to
require
$v^2=1$ and
$-v^3(g+wf+w^3)=1$. Without loss of generality, we may take $v=-1$. From
\eqref{eq:identity'} we then read off that
\begin{equation*}
e_1=2-w \quad\text{and}\quad e_2=1-w\,.
\end{equation*}
Thus, by \eqref{eq:d} and \eqref{eq:e} we have
$$
f+2=-v_1^2+w_1 \quad\text{and}\quad g +3-2w=v_1w_1\,.
$$
But from \eqref{eq:v} and \eqref{eq:w} we evaluate $v_1$ and $w_1$ in terms of
$w$ as
$$3-2w+v_1=-1 \quad\text{and}\quad (2-w)(1-w)-w_1= w\,.
$$
Substituting appropriately we find that $1=g+fw+w^3=6w-11$ so, as already
announced,
$v=-1$, $w=2$,
$g=1$, and
$f=-4$.
Furthermore, we have $v_1=0$ and $v_0+v_1+e_1=0$, so $v_0=0$; then
$f+3=-v_0^2+w_0$ yields
$w_0=-1$. Noting that $g+2e_0+e_1=0$, we find that $e_0=-1/2$. Thus the relevant
\cfe\ commences
\begin{align*}
Z_0:=\frac{Z+2X-1}{X^2-1}&=X-\frac{\conj Z+X}{X^2-1}\\
\frac{Z+X}{-(X^2-2)}&=-X-\frac{\conj Z+X-1}{-(X^2-2)}\\
\frac{Z+X-1}{X^2-X-1}&=X+1-\frac{\conj Z+X-1}{X^2-X-1}\\
\frac{Z+X-1}{-(X^2-2)}&=-X-\frac{\conj Z+X}{-(X^2-2)}\\
\frac{Z+X}{X^2-1}&=X-\frac{\conj Z+2X-1}{X^2-1}\\
&\cdots\end{align*}
providing a useful check on our allegations and displaying an expected symmetry
(both the
defining recursion and the set of initial values are symmetric). Denote by $M$
the divisor class
defined by the pair of points
$(\varphi,0)$ and
$(\conj\varphi,0)$ --- here, $\varphi$ is the golden ratio, a
happenstance that will please adherents to the cult of Fibonacci ---
and by
$S$ the divisor class at infinity. Then the sequence
$(T_h)=(\ldots$, $2$, $1$, $1$, $1$, $1$, $1$, $1$, $2$, $3$, $4$, $8$, $17$,
$50$, $107$, $239$,
$1103$, $\ldots)$ may be thought of as arising from the points $\ldots$,
$M-S$, $M$, $M+S$, $M+2S$, $\ldots$ on the Jacobian of the curve
$\mathcal C$ displayed at \eqref{eq:examplecurve}. Evidently,
$M-S=-M$ so $2M=S$ on $\Jac(\mathcal C)$.\medskip
\noindent\textbf{Allegation.} Of course I do not do it here, but I suggest that
my remarks suffice to show
that one may readily prove that given a sequence $(A_h)$ satisfying a recusive
relation
$A_{h-3}A_{h+3}=aA_{h-2}A_{h+2}+bA_{h}^2$ and with given values $A_{h-3}$,
$A_{h-2}$, $\ldots\,$, $A_{h+2}$
one may identify both a genus~$2$ curve $\mathcal C:Z^2-AZ-R=0$, $\deg A=3$,
$\deg R=1$ and a divisor $M$
on $\mathcal C$ giving rise to the sequence.
\section{Comments}
\noindent I consider the argument given in \S\ref{s:ridiculous} above to be
quite absurd
and am ashamed to have spent a great deal of time in extracting it. Such are the
costs of
truly low lowbrow arguments; see \cite{CF} for heights of `brow'. The only
saving grace is
my mildly ingenious use of symmetry in the argument's later stages. I do not
know whether
there is an appealing result of the present genre if
$u\ne0$; but see my remarks below. I should admit that I realised, but only
after having
successfully selected
$u=0$, that Noam Elkies had suggested to me at ANTS, Sydney 2002, that an
identity of the
genre
\eqref{eq:recurrence} would exist, but had in fact specified just the special
case $\deg
R=1$.
Mind you, with some uninteresting effort one can show (say by counting free
parameters) that over an algebraic extension of the base field there is a
birational
transformation which transforms the given curve to one where $\deg R=1$. That
does not
truly better the present theorem.
On the other hand, a dozen years ago\footnote{I have a revision of his
manuscript dated
November, 1992.}, David Cantor \cite{Ca} mentions that his results lead readily
to Somos
sequences both in genus~$1$ and $2$; the latter of width~$8$. That his results
provide Somos sequences in
genus~$2$ is not obvious; however, recently, Cantor has told me a rather
ingenious idea which clearly yields
the result for all hyperelliptic curves $Y^2=E(X)$, $E$ a quintic, say with
constant coefficient~$1$. In
brief, Cantor's result is more general than mine but does not deal with all
cases I handle
here; nor does it produce the expected recursion formul\ae\ of width~$6$.
Moreover, after this paper was
submitted I learned of the work \cite{BEH} which produces Cantor's width~$8$
recurrences from
addition formulas for the corresponding hyperelliptic functions.
The most serious disappointment is that the best argument I can produce here is
just a
much more complex version of that of \cite{169} for genus~$1$. Seemingly a new
view on the issues is needed if my methods are
to yield results in higher genus.
\bibliographystyle{amsalpha}
\begin{thebibliography}{ABC}
\bibitem{BCZ} {Enrico Bombieri and Paula B. Cohen, Siegel's
lemma, Pad\'e approximations and Jacobians (with an appendix by Umberto
Zannier, and dedicated to Enzio De Giorgi), {\it Ann. Scuola Norm.
Sup. Pisa\/} Cl. Sci. (4) {\bf 25\/} (1997),
155--178.}
\bibitem{BEH} Harry W. Braden, Victor Z. Enolskii, and Andrew N. W. Hone,
Bilinear
recurrences and addition formulae for hyperelliptic sigma functions, (2005),
15pp: at
\href{http://arxiv.org/abs/math.NT/0501162}{\tt http://arxiv.org/abs/math.NT/0501162}.
\bibitem{CF} J. W. S. Cassels and E. V. Flynn, \emph{Prolegomena to a
middlebrow arithmetic of curves of genus $2$}, London Mathematical
Society Lecture Note Series, 230. Cambridge University Press,
Cambridge, 1996. xiv+219 pp. %{ISBN: 0-521-48370-0 MR 97i:11071}
\bibitem{Ca1} David G. Cantor, Computing in the Jacobian of a hyperelliptic
curve,
{\it Math. Comp.\/} {\bf 48\/}.177 (1987), 95--101.
\bibitem{Ca} David G. Cantor, On the analogue of the division
polynomials for hyperelliptic curves, {\it J. Reine Angew. Math.\/}
{\bf 447\/} (1994), 91--145.
\bibitem{FZ} {Sergey Fomin and Andrei Zelevinsky}, The
Laurent phenomenon, \textit{Adv. in Appl. Math.}, \textbf{28},
(2000), {119--144}.
Also 21pp: at \href{http://www.arxiv.org/math.CO/0104241}{\tt http://www.arxiv.org/math.CO/0104241}.
\bibitem{La} Kristin E. Lauter, The equivalence of the geometric and
algebraic group laws for Jacobians of genus $2$ curves,
{\it Topics in algebraic and noncommutative geometry\/} (Luminy/Annapolis, MD,
2001), 165--171, {\it Contemp. Math.\/}, {\bf 324\/},
Amer.\ Math.\ Soc., Providence, RI, 2003.
\bibitem{169} Alfred J. van der Poorten,
\href{http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Poorten/vdp40.html}{Elliptic curves and continued fractions.} {\it J. Integer Sequences\/} {\bf
8\/}, (2005), article 05.2.5.
\end{thebibliography}
\end{document}
\bigskip
\hrule
\bigskip
\noindent 2000 {\it Mathematics Subject Classification}:
Primary: 11A55, 11B83; Secondary: 11G30, 14H05.
\noindent \emph{Keywords:}
continued fraction expansion, function field of characteristic zero,
hyperelliptic curve, Somos sequence.
\bigskip
\hrule
\bigskip
\vspace*{+.1in}
\noindent
Received November 12 2004;
revised version received July 21 2005.
Published in {\it Journal of Integer Sequences}, July 28 2005.
\bigskip
\hrule
\bigskip
\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in
\end{document}