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% June 15 1990, 6/18/90
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% typed version of old stuff on archimedean height
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\documentstyle{amsppt}

\magnification 1200
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\topmatter

\title
The Archimedean Component \\
of the Canonical Height \\
on Elliptic Curves
\endtitle
\rightheadtext{Archimedean Component of the Canonical Height}

\author
{Armand Brumer and Ois\'{\i}n McGuinness}
\endauthor

\affil
{Fordham University}
\endaffil

\address
% next line is a kludge, since don't have AMSFonts 2.0
{\rm
Mathematics Department, Fordham University, Bronx NY 10458,
USA
}
\endaddress
% next added 6/8/90
\email
{\tt 
BITNET\%"brumer\@fordmurh", BITNET\%"mcguiness\@fordmurh"
}
\endemail

\date
{June 15, 1990}
\enddate

% simplified subjclass June 8 1990
\subjclass
{Primary 11G40, Secondary 11D25, 11G05, 11-04, 14K15} 
\endsubjclass

\keywords{Elliptic curve, Canonical Height}
\endkeywords

\def\F{\bold F}    % use for finite fields
\def\Q{\bold Q}
\def\C{\bold C}
\def\R{\bold R}
\define\UH{{\bold H}} % upper half plane
\def\Z{\bold Z}
\def\divides{\,\vert\,}       
\def\notdivides{\,\not\vert\,}
\define\SwD{Swinnerton-Dyer}
\define\BSwD{Birch and \SwD}
\define\heightpair#1#2{\langle #1, #2 \rangle}
\define\height{\hat h}
\define\archht{\hat\lambda_{\infty}}
\define\locht#1{\hat\lambda_{#1}}

\define\Intro{1}
\define\Definitions{2}
\define\Deltapositive{3}
\define\Deltanegative{4}

\define\half{{1\over 2}}
%
\endtopmatter

\document
\heading
\S\Intro\ Introduction
\endheading
Suppose that $E$ is an elliptic curve defined over $\Q$ given by the
equation
$$
y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6,
\tag1$$
where we assume that $a_i\in \Z$. 
The {\it canonical height\/}
$$
\height : E(\Q)\to\R
$$
is a non-degenerate quadratic form on $E(\Q)$ which may be defined globally by the limit formula
$$
\height(P) =\lim_{n\to\infty}4^{-n}h(x(2^nP))
$$
where for $x=m/n \in \Q$, $h(x)=\log\max(|m|, |n|)$, and for $P\in E(\Q)$, $x(P)$ denotes the
$x$-coordinate of $P$ in the model~(1) above. It is clear from this formula that $\height(P)=0$ if
and only if $P$ is a torsion point. Recall 
(see \cite{SI2}) that the most efficient way to compute the height function is to use the
decomposition of the canonical height as a sum of local heights
$$
\height(P) =\sum_{v}\locht{v}(P)
\tag2$$
where the sum is over all places $v$ of $\Q$,
and $\locht{v}: E(\Q_{v})-\{O\}\to\R$ is the local height function at $v$.
For example, suppose
$E$  has type $I_1$ reduction at all places of bad reduction. Then if $d$ is
the denominator of $x=x(P)$, for $P\in E(\Q)$,
$$
\height(P)=\archht(P) + {1\over 2}\log|d|,
\tag3$$
so that computing the height of $P$ reduces to computing the archimedean component.
This formula is easily seen from the explicit formulas for the local height functions in \cite{SI2}.
The assumption on $E$ is satisfied if the discriminant $\Delta$ of $E$ is square-free, as is true of
the curves studied in \cite{BrMcG}.
 It should be remarked that the local height functions are unique up to
addition of a constant, and we follow the conventions of \cite{SI2}, as recalled below 
in section~\Definitions\ for $\archht$.

In the work reported on briefly in \cite{BrMcG}, many interesting problems to do with the values of
the height function arose. One question that became of interest to us is the following:
{\it where does the archimedean height function $\archht$ achieve its minimum on $E(\R)$\/}?
If one can predict this from the equation of $E$, then perhaps one will attain some understanding of
the much more difficult problem of locating integral points on elliptic curves. (Compare~\cite{HS}.)
In this paper we shed some partial light on this problem, determining the exact location of the
minimum for the elliptic curves over~$\R$ with {\it two\/} components ($\Delta >0$), and reporting
some experimental evidence on the location of the minimum in the other case.

The paper is organised as follows: In section~\Definitions\  we recall several definitions of
$\archht$, and compute some special values. In section~\Deltapositive\ we determine the location of
the minimum in the case of two components, while in section~\Deltanegative\ we try to say something
useful about the other case. 

We would like to thank Ken Kramer for useful conversations. The work reported here was not
supported by the NSF, except in that an NSF SCREMS grant paid for the Macintosh~II computers
on which the numerical computations were carried out. 

\heading
\S\Definitions\ Definitions
\endheading
We use the usual notation for the invariants attached to elliptic curves, see the
formulaire in \cite{Mod4} or \cite{SI1}. The paper \cite{SI2} is our reference for the basic
properties of height functions and our notation generally is as used there. The field of definition
of the elliptic curves considered in this paper is taken to be~$\R$ unless otherwise stated.

According to N\'eron and Tate, the local height function $\archht:E(\R)-\{O\}\to\R$ is uniquely
determined by the three properties: 
$$
\archht(2P) = 4\archht(P)-\log|2y+a_1x+a_3|, \qquad P=(x,y),\quad 2P\neq O;
\tag1$$
$$
\lim_{P\to O}\left(\archht(P)-\half\log|x(P)|\right) \quad \text{exists};
\tag2$$
and lastly
$$
\archht\quad\text{is bounded on any open subset of $E(\R)-\{O\}$.}
\tag3$$
The duplication formula
$$
x(2P)={x^4-b_4x^2-2b_6x-b_8\over(2y+a_1x+a_3)^2}
\tag4$$
can be used to rewrite~(1) as
$$
\archht(2P)-\half\log|x(2P)| = 4\archht(P)-\half\log|x^4-b_4x^2-2b_6x-b_8|. 
$$
Define
$$
\mu(P) =\archht(P)-\half\log|x(P)|
\tag5$$
(this is 8~times the $\mu(P)$ in \cite{SI2}), so that
$$
\mu(2P) = 4\mu(P) -\half\log\left|1-{b_4\over x^2}-{2b_6\over x^3} - {b_8\over x^4}\right|.
\tag6$$
From this  it is clear that $\mu(P)\to 0$ as $P\to O$ in $E(\R)$.

These defining properties can be used to calculate some special values of $\archht(P)$.
Suppose first that $P_2$ is a two-division point of $E$, with $e=x(P_2)$ in either
$\R$ or in $\C$. Then in~(6) let $P\to P_2$, and deduce, using the definition
of $\mu$, that
$$
\archht(P_2)={1\over8}\log\left|e^4-b_4e^2-2b_6e-b_8\right|.
\tag7$$
If $b_2=0$ this simplifies to
$$
\archht(P_2)={1\over 4}\log\left|3e^2+b_4/2\right|,
\tag8$$
as in this case we have the identity 
$e^4-b_4e^2-2b_6e-b_8= e^4-b_4e^2-2b_6e-b_8+2e(4e^3+2b_4e+b_6)=(3e^2+b_4/2)^2$. Note that 
$b_2=0$ can always be achieved by a suitable translation.

If now $P_3$ is a 3-division point, then $2P_3=-P_3$, and we have $x(2P_3)=x(-P_3)=x(P_3)$, so that
$$
\mu(P_3)={1\over 6}\log\left|1-{b_4\over x^2}-{2b_6\over x^3} - {b_8\over x^4}\right|,
$$
and then
$$
\archht(P_3)={1\over 6}\log\left|{x^4-b_4x^2-2b_6 x - b_8\over x}\right|.
\tag9$$
If the model is such that $x=0$ gives a 3-division point, so that $b_8=0$ from the
explicit formula for multiplication by~3, (see \cite{SI2, equation~(29)}, this formula is 
modified: $\archht(P_3)=(1/6)\log|2b_6|$. 

One can similarly derive explicit formulas for the local height of any torsion point. 
Although our
focus is on the archimedean component of the height, the exact analogues of the properties above are
true for the other places of $\Q$, with the archimedean absolute value $|\ |$ replaced by the
$v$-adic absolute value. The resulting formulas for $\locht{v}(P_2)$ and $\locht{v}(P_3)$ are
the same formally as those we have derived. The product formula then shows that the canonical
height of these torsion points is~0, as we remarked in the introduction.

The computation of $\archht(P)$ for non-torsion points is best done using 
the method of~\cite{SI2}.
A special case is that given by Tate in his letter to Serre of
October~1, 1979. 
Suppose that there is an $\epsilon>0$ so that $|x(P)|>\epsilon$
for all $P\in E(\R)-\{O\}$. Then we have
$$
\archht(P) = \half\log|x(P)| + {1\over 8}\sum_{n=0}^{\infty}4^{-n}\log|z(2^nP)|,
\tag10
$$
where $t=t(P)=1/x$, $z=z(P)=1-b_4t^2-2b_6t^3-b_8t^4$, $w=4t+b_2t+2b_4t^3+b_6t^4$, so that
$x(2P)=z/w$ and $t(2P)=w/z$. 
This series converges extremely rapidly; we refer to \cite{SI2,
Theorem~4.2} for explicit estimates. 
(Note that if $P$ is torsion, then the sequence $\{2^nP\}$
repeats, and the infinite series collapses into a finite sum of explicitly summable geometric
series. 
The formulas~(7) and~(9) for  $\archht(P_2)$ and $\archht(P_3)$ are also obtainable in
this way.) 
It is easy to check that the series on the right side of~(10) satisfies the axioms for
$\archht$ given at the beginning of this section, so one could regard~(10) as an existence proof for
the local height. 
If the condition that $|x(P)|$ be bounded away from~0 is not satisfied, then the
series needs to be modified. 
We refer to \cite{SI2,  section~2} for the details. 
This is the case if the base field is $\C$, for example. 
Over~$\R$ one can always perform a suitable translation to obtain the condition on $|x|$.

We have implemented this series in an MPW tool, {\tt ShowHeightArch}, that given an elliptic curve
and a list of $x$-coordinates of points on the curve, calculates the corresponding values of
$\archht$. 
This was used in the experimental calculations described later.

There are  two other viewpoints on the local height function that are of interest, the
second of which we will use later. 
Firstly, following \cite{GR}, the height function can be
described using the idea of {\it Green's Functions on Riemann Surfaces\/}. 
(This is also the context in which the local height function appears in a corner of {\it Superstring
theory\/}; some references are given below.)
Suppose that $X$ is a curve over $\C$. Regard $X(\C)$ as a Riemann surface. Then for each $x\in
X(\C)$ there exists a Green's function $g_{(x)} : X(\C)-\{x\}\to\R$ characterised by proposition~6.1
in \cite{GR}. One of its properties is that $g_{(x)}-\log|z|$ is real analytic near~$x$ where $z$ is
a uniformizing parameter at~$x$. Then one defines $G(x,y)=g_{(x)}(y)=g_{(y)}(x)$ for $x\neq y$,
and the {\it height pairing\/} is defined on relatively prime divisors of degree~0 on $X$
by $\langle \sum m_x (x), \sum m_y (y)\rangle = \sum m_x m_yG(x,y)$. 
The height pairing for an elliptic curve may then be constructed explicitly by constructing a
Green's function, see section~8 of \cite{GR}. 
We omit the details of passing from the height pairing to the height function.

Without using this general theory, the local height function on $E(\R)$ can be described as
follows. 
By the theory of elliptic functions (see \cite{LAN} for example) there is a lattice
$L=\Z\omega_1+\Z\omega_2$ with $\omega_1>0$ and $\tau=\omega_2/\omega_1$ in the upper half plane
$\UH$, so that over $\C$ there is a complex analytic isomorphism
$$
\phi:\C/L \to E(\C).
$$
Furthermore $\phi(0)=O$, and $\phi(z)=(x, y)$ where the connection with the Weierstra\ss\ 
$\wp$-function is: $x+b_2/12 = \wp(z;L)$, $2y+a_1x+a_3=\wp'(z;L)$.
There are functions $\zeta(z;L)$ and $\sigma(z;L)$ such that $-\zeta'=\wp$, and
$\zeta=\sigma'/\sigma$. 
The Weierstra\ss\ sigma function is entire with simple zeros at the
lattice points:
$$
\sigma(z;L) = z\prod_{0\neq\omega\in L}
\left(1-{z\over\omega}\right)e^{{z\over\omega}+{1\over2}\left({z\over\omega}\right)^2}
$$
so then 
$$
\zeta(z;L) ={1\over z} +\sum_{0\neq\omega\in L}\left({1\over
z-\omega}+{1\over\omega}+{z\over\omega^2}\right)
$$
These functions are not $L$-periodic but almost are: there are constants $\eta_1\in\R$ and
$\eta_2$ satisfying the Legendre relation $\eta_1\omega_2-\eta_2\omega_1=2\pi i$, and so that
$$ 
\zeta(z+\omega_1;L)= \zeta(z;L)+\eta_1, \qquad
\zeta(z+\omega_2;L)= \zeta(z;L)+\eta_2.
$$
Define  the real-linear function $\eta(z)$ so that $\eta(\omega_1)=\eta_1$ and
$\eta(\omega_2)=\eta_2$.
Finally we define the {\it Klein form\/} $k(z;L)=e^{-z\eta(z)/2}\sigma(z)\Delta^{1/12}$,
and define for $z\in\C-L$:
$$
\lambda(z;L) = -\log|k(z;L)|.
\tag11
$$
Near $z=0$ we have $\lambda(z;L)=-\log|z|+O(1)$, and since $\log|x|=-2\log|z|+O(1)$
from $x =\wp(z;L)+b_2/12$, we have $\lambda(z;L)-\half\log|x|$ bounded near~0.
This is axiom~2 for the local height function. It turns out that $\lambda$ and $\archht$
differ by a constant:
$$
\lambda(z;L)=\archht(\phi(z)) -{1\over 12}\log\Delta.
\tag12
$$
In the next section we will use this expression for the height function.


\heading
\S\Deltapositive\ Two components
\endheading
Suppose that $E$ has two components over $\R$. 
Then $\Delta>0$, and all of the 2-division points are defined over $\R$.
In terms of the complex analytic parametrisation, we have $\omega_2\in i\R$, the lattice~$L$
has a {\it rectangular\/} fundamental domain. The Legendre relation then shows $\eta_2\in i\R$.
The 2-division points are at $x=e_1=\phi(\omega_1/2)$, 
$e_2=\phi(\omega_2/2)$ and $e_3=\phi((\omega_1+\omega_2)/2)$. 
In the usual picture of $E(\R)$, we have $e_1>e_2>e_3$, and the line segment $[0, \omega_1]$ in
$\C$ is mapped by $\phi$ to the connected component of the identity in $E(\R)$, and the line
segment $[\omega_2/2, \omega_1+\omega_2/2]$ maps to the ``oval''.
In this situation, numerical experiments showed unequivocally that the minimum of $\archht$ on
$E(\R)$ was achieved at $e_1$. And the actual minimum value was calculated in section~1.

We prove this by showing that the minimum of $\lambda(z;L)$ on the line segments $(0,
\omega_1/2]$ and on $[\omega_2/2, (\omega_1+\omega_2)/2]$ occur at the 2-division points.
(Note that we need only look at half of the original segments.)
First a convenient normalization is to take $\omega_1=1$, $\omega_2=\tau$. 
This is possible since one easily verifies that for $\gamma >0$ one has
the homogeneity relation
$$
\lambda(\gamma z;\gamma L)=\lambda(z;L).
$$
(This was the reason for including the $\Delta$ factor in the definition of $k(z;L)$).
Now we set, as is usual in modular function theory, $q=e^{2\pi i\tau}$. The well-known
$q$-expansions of the elliptic functions defined above give:
$$
\lambda(z; 1, \tau)=
-\log\left|e^{-z\eta(z)/2}e^{\eta_1z^2/2}
     { \left(u^{1/2}-u^{-1/2}\right) \over 2\pi i }
   \prod_{n=1}^{\infty} 
     { (1-q^nu)(1-q^nu^{-1}) \over (1-q^n)^2 }\right|
\tag1
$$
where we have put $u=e^{2\pi iz}$.
(Compare equation~(15) in \cite{HS}, and also formulas on pages~16, 73, 224 and 234 of
\cite{GSW}.)
Since $\tau=it$ with $t>0$, we have $q=e^{-2\pi t}$ and so $0<q<1$.
Along the line segment $0<z=s\leq 1/2$, the function reduces to (throwing away
the contribution of $\Delta$, namely the factors $(1-q^n)$ that don't vary):
$$
-\log\left|{1\over\pi}\sin(\pi s)\prod_{n=1}^{\infty} 
\left(1-2\cos(2\pi s)q^n +q^{2n}\right)
\right|
\tag2
$$
Note that $\eta(s)=\eta_1s$ here, so the $\eta(z)$ terms cancelled.
We want to {\it maximize\/} the quantity inside the logarithm.
The maximum of $\sin(\pi s)$ certainly occurs at $s=1/2$. 
And since $0<q<1$, the maximum of each term in the infinite product occurs
when $s=1/2$ also. We have proved our claim for the first line segment.
The argument even gives, if we are careful, a formula for the minimum height.
A comparison of this formula with that in~(1.7) gives a formula
well known from $\theta$-function theory. 
(See the appendix to chapter~8 of \cite{GSW}, for
an expression of $\lambda$ in terms of $\theta$-functions.)

It remains to take care of the segment $z=s+it/2$, for $0\leq s\leq 1/2$.
Here $\eta(z)=s\eta_1+\eta_2/2$.
Then  $-z\eta(z)/2+\eta_1z^2/2$ has real part $-\eta_1 t^2/8$ which does not depend on~$s$.
Here $u=e^{2\pi is-\pi t}$, and 
removing constant factors as before we want to minimize:
$$
-\log\left|{1\over\pi}\left(e^{\pi is -\pi t/2}-e^{-\pi is +\pi t/2}\right)
\prod_{n=1}^{\infty} 
\left(1-q^{n+1/2}e^{2\pi is}\right)\left(1-q^{n-1/2}e^{-2\pi is}\right)
\right|
\tag3
$$
Removing a factor of $-e^{-\pi is}q^{-1/4}$ from the term in front of the infinite product,
and shifting indices by~1 in the second term in the infinite product we see this becomes:
$$
-\log\left|q^{-1/4}{1\over\pi}
\prod_{n=0}^{\infty} 
\left(1-q^{n+1/2}e^{2\pi is}\right)\left(1-q^{n+1/2}e^{-2\pi is}\right)
\right|
\tag4
$$
or
$$
-\log\left|q^{-1/4}{1\over\pi}
\prod_{n=0}^{\infty} 
\left(1-2\cos(2\pi s)q^{n+1/2}+q^{2n+1}\right)
\right|
\tag5
$$
which clearly has its minimum at $s=1/2$, as we needed to check. 
Now  we know the minimum of $\archht$ occurs at one of the three 2-division points.
From  equation~(1.8) we see that the smallest of the three values of $\archht$ is 
at $x=e_1$, as was claimed. (Oops, I think I need more details here!)

\heading
\S\Deltanegative\ One component
\endheading
By contrast with the last section, here we will see that the formula for the
archimedean height that comes from the complex analytic viewpoint of elliptic curves does {\it not\/}
appear to be useful for finding the location of the minimum height in the case where $E(\R)$ has one
component.

\Refs

\ref\key Mod4
\by B.~Birch and W.~Kuyk, editors
\book Modular Functions of One Variable IV
\bookinfo volume 476 of Lecture Notes in Mathematics
%\vol 476
\publ Springer-Verlag
\publaddr New York 
\yr1975
\endref  

\ref\key BrMcG
\by A.~Brumer and O.~McGuinness
\paper The behavior of the Mordell-Weil group of elliptic curves
\jour Bull. Amer. Math. Soc.
\vol ??
\pages ???
\yr1990
\endref  

\ref\key BGZ
\by J.~Buhler, B.~H. Gross, and D.~B. Zagier
\paper On the conjecture of {B}irch and {S}winnerton-{D}yer for an
elliptic curve of rank 3
\jour Math. Comp.
\vol 44
\pages 473--481
\yr 1985
\endref  


\ref\key GR
\by B.H.~Gross
\paper Local Heights on Curves
\inbook Arithmetic Geometry, edited by G.~Cornell and J.~Silverman
\pages 327--339
\yr 1986
\publ Springer-Verlag
\publaddr New York 
\endref

\ref\key GSW
\by M.~Green, J.~Schwarz, and E.~Witten
\book Superstring Theory: 2 Loop amplitudes, Anomalies and Phenomenology
\vol  2
\yr 1987
\publ Cambridge University Press
\publaddr Cambridge and New York 
\endref

\ref\key HS
\by M.~Hindry and J.~Silverman
\paper The canonical height and integral points on elliptic curves
\jour Invent. Math.
\vol 93
\pages 419--450
\yr 1988
\endref

\ref\key LAN
\by S.~Lang
\book Elliptic Functions
\yr 1987
\publ Springer Verlag
\publaddr  New York 
\endref

\ref\key SI1
\by J.~H.~Silverman
\book The arithmetic of elliptic curves
%\vol 106 
\bookinfo volume 106, Graduate Texts in Mathematics
\publ Springer Verlag
\publaddr New York
\yr 1986
\endref  

\ref\key SI2
\by J.~H.~Silverman
\paper Computing heights on elliptic curves
\jour Math. Comp.
\vol 51 
\pages 339--358
\yr 1988
\endref  


\ref\key TA
\by J.~Tate
\paper The arithmetic of elliptic curves.
\jour Invent. Math.
\vol 23 
\pages 179--206
\yr 1974
\endref  

\endRefs

\enddocument
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