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% To appear in: BULLETIN (NEW SERIES) OF THE AMERICAN MATH. SOCIETY
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% Reference Number: BULL128
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% Letter of April 3 1990 from C. Venedettuoli
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% 5/10/90 small changes in top matter made for version 2.0 of AMSTeX
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% 6/8/90 eliminate Revisions 1985 duplication in printed version
% 6/22/90 changed zeta(12) to zeta(10) (3 places)
%         added bibliographic info for Rubin's paper, cleaned up extra ..'s
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\documentstyle{amsppt}


\magnification 1200
\vsize= 8.5 true in
\hsize= 5.5 true in
\hoffset= 0.5 true in 
\NoBlackBoxes

\topmatter

\title
The Behavior of the Mordell-Weil Group of Elliptic Curves
\endtitle

\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
{
BITNET\%"brumer\@fordmurh", BITNET\%"mcguiness\@fordmurh"
}
\endemail

\date
{October 30, 1989, revised March 1, 1990}
\enddate

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

\keywords
{Elliptic curve, Mordell-Weil group, rank,
Birch-Swinnerton-Dyer
conjecture, Hasse-Weil $L$-series, discriminant, 
period}
\endkeywords
%
% if genuine Cyrillic available, use that instead!
\def\Sha{\hbox{$\amalg\kern-.39em\amalg$}}
%
\def\F{\bold F}    % use for finite fields
\def\Q{\bold Q}
\def\C{\bold C}
\def\R{\bold R}
\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}
\def\Li{\mathop{\roman{Li}}\nolimits}
\define\Ls{L(E,s)}

% total number found
%
\def\totalfound{311243}
%
% total analysed so far
%
\def\totalnumber{310716}
\def\totalpositive{113969}
\def\totalnegative{196747}
\def\rationegpos{1.726}
\def\totalodd{155658}
\def\totaleven{155058}

\def\rankzero{93337}
\def\rankzeropos{31748}
\def\rankzeroneg{61589}
\def\rankzeropct{30.04} 

\def\ranktwo{61517}
\def\ranktwopos{24706}
\def\ranktwoneg{36811}
\def\ranktwopct{19.80}

\def\rankfour{804}
\def\rankfourpos{377}
\def\rankfourneg{427}
\def\rankfourpct{0.26}

\def\rankevenGtZeroPct{20.06}

\def\rankone{143192}
\def\rankonepos{51871}
\def\rankoneneg{91321}
\def\rankonepct{46.08}

\def\rankthree{11861}
\def\rankthreepos{5267}
\def\rankthreeneg{6594}
\def\rankthreepct{3.82}

\def\rankfive{5}
\def\rankfivepos{0}
\def\rankfiveneg{5}
\def\rankfivepct{}

\def\rankoddGtOnePct{3.83}
%
\endtopmatter

\document
\heading
\S1 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 set  $E(\Q)$ of solutions $(x,y)$ with $x$, $y\in\Q$, together with
the point at infinity, forms a finitely-generated abelian group, the
{\it Mordell-Weil group\/} of $E$. 
It is isomorphic to $\Z^r\oplus F$ where $F$ is
finite and where $r$ is the {\it rank\/} of $E$. 
The possibilities for the
finite group $F$ are completely known~\cite{9}.
The important question then is to understand the behavior of the rank
as $E$ varies over elliptic curves. 
It is still unknown whether $r$ is unbounded or not.
In fact, the opinion had been expressed that, in general, an elliptic
curve might tend to have the smallest possible rank, namely $0$ or
$1$, compatible with the rank parity predictions of \BSwD\ \cite{8}. 
We present evidence that this may not be
the case.

Published examples \cite{2}, \cite{10} of curves of rank~$\geq 2$ 
 might suggest that such curves are sparsely 
distributed.\footnote{This situation is not unrelated to the ranks of
ideal class groups of quadratic fields, where similar phenomena 
occur~\cite{13}.}
Mestre and Oesterl\'{e} 
found the 436 modular elliptic curves  of prime conductor up to 13100,
using~\cite{11}.
There were~80 rank~2 curves among the 233~curves of even rank.  
This  proportion of rank~2 curves seemed too large
to conform to the conventional wisdom stated above (see also~\cite{18},
pages~254--255).
We decided to investigate  the ranks of elliptic curves
in a systematic way, over a significantly larger range. Curves
of {\it prime\/} conductor only were considered for
practical and theoretical reasons. This collection of curves appears to
be a typical sample of the set of {\it all} curves  (see section~5 for
some evidence).

We have  studied  $\totalnumber$ elliptic curves of prime conductor
less than $10^8$. There were $\totalodd$ curves with odd rank, and
$\totaleven$ curves with even rank.
We found $\rankevenGtZeroPct\%$ of {\it all\/} our curves have even
rank at least~2, or about $40\%$ of all the even rank curves.
Even more striking is the behavior of the average rank, as discussed in
section~3.
An incidental aspect of our computations is a massive corroboration of
the standard conjectures on elliptic curves, recalled in section~2.

Recent related work is described in \cite{6} and \cite{8}.
Contrasts with our results are given in section~3. 

We expect to publish  a fuller  account,
including the behavior of other invariants of interest. 
This announcement  reports mainly on ranks. 
The computations have been carried out on
Macintosh~II computers at Fordham University, with the
partial support of  an NSF SCREMS grant.
We would like to thank our colleagues R.~Lewis, I.~Morrison, and
W.~Singer for the use of their machines. 


\heading
\S2 Definitions
\endheading
We recall standard notations and definitions~\cite{15}.
Associated with
equation~(1) is the {\it discriminant\/} $\Delta$, which we
will assume to be {\it minimal\/} among all models~(1) of $E$.
The fundamental property of the discriminant is that $p\divides\Delta$
if and only if the equation~(1) is singular modulo $p$, the
{\it conductor\/} $N$ of $E$ is  a subtler invariant that has the same
property. 

The Hasse-Weil $L$-series of $E$ is defined for $\Re(s)>3/2$ by
$$
\Ls = \prod_{p\divides N}\left(1-a_pp^{-s}\right)^{-1}
\prod_{p\notdivides N}\left(1-a_pp^{-s}+p^{1-2s}\right)^{-1}
$$
where for $p\divides N$, $a_p\in\{-1,0,1\}$, and for
$p\notdivides N$, $a_p=p+1-|E(\F_p)|$.
We shall assume that $E$ is a {\it modular\/} curve, 
so $E$ is  a 
factor\footnote{Note that Mestre-Oesterl\'{e} found their curves 
by determining the $1$-dimensional factors of $J_0(N)$.
Needless to say, we have the same curves in their range.}
of the Jacobian $J_0(N)$ of  the modular curve $X_0(N)$ of level~$N$.
(That is, the Taniyama-Weil
conjecture for $E$ is true.)
Hence $\Ls$ can be continued to an
entire function on $\C$,  satisfying a functional equation when $s
\mapsto 2-s$,
with a zero at $s=1$ of
order $\rho$, the {\it analytic\/} rank of $E$. 
For square-free conductor, 
the sign in the functional equation may be easily calculated from an
equation of the curve,  allowing a conjectural prediction
of the parity of the analytic rank~\cite{1}.
We also assume that the conjecture of 
\BSwD\ holds, so  that the analytic rank equals the rank, $\rho=r$,
and the leading term 
of $\Ls$ at $s=1$ is 
given by:
$$ 
\lim_{s\to1}\frac{\Ls}{(s-1)^r} =  \Omega
\frac{|\Sha|\det\left(\heightpair{P_i}{P_j}\right)} {[E(\Q):E']^2}
\prod_{p\divides\Delta}c_p. 
\tag2 
$$ 
Here $\Omega$ is the period $\int_{E(\R)}|\omega|$, for $\omega$ 
a N\'eron differential on $E$,
$\Sha$ denotes the conjecturally finite Tate-Shafarevich group of $E$, 
the $P_i$ for
$1\leq i\leq r$ are an independent set of points in $E(\Q)$
generating the subgroup $E'$, and $\heightpair{P_i}{P_j}$
denotes the height pairing. The fudge factors $c_p$
are all~1 for the curves we consider.
Recent work of Rubin and Kolyvagin 
\cite{14} confirms the
conjecture of Birch and Swinnerton-Dyer in many cases of rank $r\le 1$.

Examples illustrating (2) for the curves of ranks~4 and~5 of least
known conductor
are given in section~4.

\heading
\S3 Rank Results
\endheading
Elliptic curves of prime conductor $N$ were conjectured to have prime
discriminant, except for the Setzer-Neumann curves and for 5~other
small conductor curves, see \cite{2, appendix}. 
This is now known for {\it modular} curves by Theorem~2 of \cite{12}.
We therefore searched for curves of prime discriminant, 
by looking for integral solutions to the equation
$$
c_4^3-c_6^2 = 1728\Delta,
\tag3
$$
where $c_4$ and $c_6$ are the usual invariants attached to
equation~(1), or more precisely, by fixing $a_4$, and searching for
$a_6$
for which~(3) has a solution with $\Delta$ prime and less than $10^8$.
This produced  $\totalfound$ curves  
including the $869$ expected curves with non-trivial torsion and
rank~0.
The set of $\totalnumber$ 
curves that we studied is most simply described by 
$$ 
\{E :\left|\Delta\right| \le 10^8, |a_6|\le 2^{31}-1\}, 
$$
with $|\Delta|$  {\it prime\/}. 

We will not describe here all the details of the several thousand hours
of computations, but just say that, 
imitating Mazur's description \cite{17} of ``infinite descent''
we searched for points by night, and calculated $L$-series derivatives
or regulators by day. 
We may use the 
infinite series formulas of~\cite{3} for the 
derivatives of the $L$-series at $s=1$, since $E$ is assumed modular.
Then an upper bound for the analytic rank is found by
estimating the order of vanishing of $\Ls$ at $s=1$.
Using 2000 or 4000 terms of these series provided sufficient accuracy
for our purposes, 
since the values are either~0 to several places or else are
far from~0 in most cases.
The  period $\Omega$ is easily calculated using the
Arithmetic-Geometric mean algorithm of Gauss \cite{7},
and the height regulator $R=\det\left(\heightpair{P_i}{P_j}\right)$ 
is computed  by using the method of Tate, as modified 
by Silverman~\cite{16}, once points have been found by a search.

The rank predictions  are based on a combination of three calculations:
the rank parity, the analytic rank or order of vanishing of the
$L$-series,
and the number of independent points found which is a lower bound for
the
algebraic rank~$r$.
When the ranks coincide, as they should, we get a prediction from~(2)
of
$|\Sha|$, which should be an integer square. 
The largest $|\Sha|$ we found was~$289$, for a curve of rank~0.

For each curve, we keep its discriminant, parity, period, rank,
the appropriate $L$-derivative value, a list of $x$-coordinates of
the independent points found, and the regulator of these points.

Of the curves analysed,  $\totalpositive$ had
positive discriminant, and $\totalnegative$ had negative
discriminant.\footnote{The quotient is about $\rationegpos$, near
$\sqrt{3}$.
See section~5 for an explanation.}
An interesting  phenomenon was the systematic influence of the
discriminant sign on all aspects of the arithmetic of the curve.
The rank distribution is given in the following table:
$$\vbox{\offinterlineskip
\hrule
\halign{\vrule#&\strut\quad#\hfil\quad&#\vrule\,\vrule&%
&\quad\hfil#\hfil\quad\vrule\cr
%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr
height2pt&\omit&&&&&&&\cr
&Rank && 0 & 1 & 2 & 3 & 4 & 5\cr
\noalign{\hrule}
\noalign{\vskip 2pt}
\noalign{\hrule}
%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr
height2pt&\omit&&&&&&&\cr
&$\Delta>0$&&\rankzeropos&\rankonepos&\ranktwopos&\rankthreepos%
 &\rankfourpos&\rankfivepos\cr
&$\Delta<0$&&\rankzeroneg&\rankoneneg&\ranktwoneg&\rankthreeneg%
 &\rankfourneg&\rankfiveneg\cr
height2pt&\omit&&&&&&&\cr
%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr
\noalign{\hrule}
%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr
height2pt&\omit&&&&&&&\cr
&Totals&&\rankzero&\rankone&\ranktwo&\rankthree%
 &\rankfour&\rankfive\cr
&Percents&&\rankzeropct&\rankonepct&\ranktwopct&\rankthreepct%
 &\rankfourpct& \rankfivepct\cr
%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr}
height2pt&\omit&&&&&&&\cr}
\hrule}
$$
Thus $\rankevenGtZeroPct\%$ of the curves have even rank at
least~2. Note that the {\it
positive\/} discriminant curves give an even higher percentage!

Define $N(r, X)$ to be the proportion of curves with conductor at
most $X$, and with rank at least $r$.  Our data
shows that these functions are {\it increasing\/} functions of
$X$ for $r\geq 3$ and $X\le 10^8$. 
In contrast \cite{6}, dealing with quadratic twists of elliptic curves,
suggests a decrease for the analogous functions.

Denote the {\it average rank\/} among the curves 
with discriminant sign
$\epsilon$ and conductor at most $X$ by $r_\epsilon(X)$.
In our data, the functions $r_\epsilon(X)$ for $X<10^8$ are
quite steadily climbing to the numbers
$1.04$ for $\Delta >0$, and to
$0.94$ for $\Delta <0$. 
In particular, the average rank of our curves is {\it
not\/} $0.5$, as  is expected to be the case for
twists~\cite{5},\cite{8}.

In most cases, the predicted  analytic rank
matches with the rank of points found, and the predicted $\Sha$ is
close to a non-zero integral square.
More precisely, this is so for all the curves of rank at least~3, 
and for 95\% of the rank~2 curves.
There is a very small number of rank~2
curves for which the prediction of rank~2 is based solely on the
vanishing of $L$-series. 
Note that in \cite{8}, a numerical study of one family of cubic twists,
only the analytic rank is estimated.
For many rank~1 curves (i.e., curves whose rank is predicted to be odd
for which $L'(E,1)$ does not vanish),
we have found no points by  point searches
over moderate ranges, and do not expect to find any small points. 

\heading
\S4 Examples
\endheading
In the literature, one finds very few 
verifications of the conjecture of Birch and Swinnerton-Dyer
for ranks $\geq 2$. 
The paper \cite{3} on the curve of conductor~5077, found in \cite{2},
works out
the only rank~3 case that we know of. 
It is perhaps not without interest to report the details for the
curves\footnote{These curves are new.
The curves of rank~4 reported in \cite{2} and \cite{10}
are respectively the 5th, 7th and 2nd curves of rank~4 in our list.} 
of smallest known conductor of rank 4 and
of rank 5.

The first rank~4 curve  is:  
$$
y^2+y=x^3+x^2-72x+210, \qquad \Delta =501029.
$$
For this curve, the predicted parity is even, the period
$\Omega=2.952580$,
the value and 2nd derivative of the $L$-series vanish to several
places, and  $L^{(4)}(E,1)/4!=9.357978$. 
The points with $x$-coordinates $5$, $4$, $3$, $6$ in  order
of increasing height, form a basis. The regulator is $R=3.169424$, 
which matches the
quotient of $L^{(4)}(E,1)/4!$ and $\Omega$, so $|\Sha|$ appears to be
1.
There are $21\times2$ integral points with $|x|<10^6$, while the second
curve of rank~4 has $28\times2$.

The first rank~5 curve is: 
$$
y^2+y=x^3-79x + 342, \qquad \Delta = -19047851.
$$
The previously found  curves of rank~5
have conductors about ten times larger~\cite{2}, \cite{10}.  

The predicted parity is odd,
the period $\Omega=2.047641$, and
then the 1st and 3rd
$L$-series derivatives are 0 to several places, and 
$L^{(5)}(E,1)/5! = 30.285711$. 
Dividing by $\Omega$ gives  $14.790539$. 
There are $38\times 2$ integral points with  $|x| <10^6$. 
Those with $x=5$, 4, 3, 7, 0 form a basis for $E(\Q)$ with height
regulator $R= 14.790528$. 
So $\Sha$ is predicted to be trivial. 
There are four other curves of rank~5 with conductor less than
100~million.
The next one has discriminant~$-64921931$.
All turn out to have trivial $\Sha$, if we believe the conjectures.

Some of the  curves found by the search give rise to rather spectacular
cancellations. 
A particular example is the curve of rank~0
$$
y^2+xy+y=x^3+x^2-12632622x-17287039382, \qquad \Delta=38593.
$$
Here $c_4 =  606365857$, and $c_6 =  14931454281967$,
and the equation $1728\Delta = c_4^3-c_6^2$
involves the difference of two 27~digit numbers!


\heading
\S 5 Heuristics
\endheading
While its connection with  the Taniyama-Weil conjecture makes the 
conductor a natural invariant,
the discriminant  has turned out to be more useful in our heuristics. 
The idea is to arrange the curves in
order by discriminant size, and then replace lattice-point counting
problems by area computations. 
This works well, for instance, to count curves.

% \zeta(12) changed to \zeta(10) 3 times
\proclaim{Heuristic Estimate}
We have the following estimates for the number of positive 
and negative discriminants of absolute value at most $N$,
$$\align
A_{+}(N)&\sim \frac{\alpha_{+} }{\zeta(10)}N^{5/6},\\
A_{-}(N)&\sim \frac{\alpha_{-} }{\zeta(10)}N^{5/6}. 
\endalign$$
\endproclaim

The $\zeta(10)$ factor  arises from taking into account the non-minimal
discriminants. Here 
$\alpha_{+} = 0.4206$ and 
$\alpha_{-} = 0.7285$ are
given by the elliptic integrals
$$
\alpha_{\pm} = \frac{\sqrt{3}}{10 }
\int_{\pm 1}^\infty \frac{du}{\sqrt{u^{3}\mp 1}},  
$$
which arise from parametrising suitably the integrals  that replace the
lattice-point counts. A well known identity of
Legendre, related to complex multiplication, is
$\alpha_{-} =\sqrt{3}\alpha_{+}$.

\noindent{\smc Remark.} One should compare this estimate with the
results of \cite{4}. 
While the number of elliptic curves grows like $N^{5/6}$, the
number of cubic fields grows like~$N$.

Assuming  the distribution of {\it prime\/} discriminants  among
discriminants is that of prime numbers among all integers, 
the number of prime discriminants of sign $\epsilon$ and of
size less than $N$ is then~$\alpha_\epsilon \Li(N^{5/6})$, where
$\Li(x)$ is the logarithmic integral. 
The expected
number of curves with prime $|\Delta|<10^8$ is then~$311586$, comparing
rather well with the number $\totalfound$ found.
 
Similar heuristic arguments have been applied to other invariants. 
For instance, the {\it average period of a curve with
positive discriminant is $\sqrt{3/2}$ times the average period of a
curve with negative discriminant}. 
This is also confirmed by the data.
The fits with the experimental data are so good that one 
could hope for proofs in the near future. 

We have not as yet been able to provide heuristics for the
growth of the functions $N(r,X)$.
While our data may seem massive, $N= 10^8$ is not sufficient to
distinguish growth laws of $\log\log N$, $N^{1/12}$ or $N^{1/24}$
from constants.
So we have to be cautious in formulating conjectures
based on the numerical evidence.


\Refs

\ref\no 1
\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\no 2
\by A.~Brumer and K.~Kramer
\paper The rank of elliptic curves
\jour Duke Math. J.
\vol 44
\pages 715--743 
\yr1977
\endref  

\ref\no 3
\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\no 4
\by H.~Davenport and H.~Heilbronn
\paper On the density of discriminants of cubic fields {(II)}
\jour Proc. Royal Soc., (A)
\vol 322 
\pages 405--420
\yr 1971
\endref  

\ref\no 5
\by D.~M. Goldfeld
\paper Conjectures on elliptic curves over quadratic fields
\inbook Number Theory Carbondale 1979
%\vol 751 
\bookinfo  Lecture Notes in Mathematics 751 
\pages 108--118 
\publ Springer Verlag
\publaddr New York
\yr 1979
\endref  

\ref\no 6
\by F.~Gouvea and B.~Mazur
\paper The square-free sieve and the rank of {M}ordell-{W}eil
\paperinfo (Preprint, April 1989)
%\yr 1989
\endref  

\ref\no 7
\by D.~Grayson
\paper The arithogeometric mean
\jour Arch. Math.
\vol 52 
\pages 507--512
\yr 1989
\endref  

\ref\no 8
\by G.~Kramarz and D.~B. Zagier
\paper Numerical investigations related to the {$L$}-series of certain
  elliptic curves
\jour J. Indian Math. Soc. 
\vol 52 
\pages 51--60 
\yr 1987
\finalinfo (Ramanujan Centenary volume)
\endref  

\ref\no 9
\by B.~Mazur
\paper Modular curves and the {E}isenstein ideal
\jour Publ. Math. IHES
\vol 47 
\pages 33--186
\yr 1977
\endref  

\ref\no 10
\by J.-F.~Mestre
\paper {F}ormules explicites et minorations de conducteurs de
vari\'et\'es alg\'ebriques
\jour Comp. Math.
\vol 58 
\pages 209--232
\yr 1986
\endref  

\ref\no 11
\by J.-F.~Mestre
\paper La m{\'e}thode des graphes. {E}xemples et applications
\inbook Class Numbers and Units of Number Fields
\pages 217--242
\publ  Nagoya University
\publaddr Nagoya
\yr 1986
\bookinfo Katata conference
\finalinfo Unpublished tables
\endref  

\ref\no 12
\by J.-F.~Mestre and J.~Oesterl{\'{e}}
\paper Courbes de {W}eil semi-stables de discriminant une puissance
  $m$-i{\`e}me
\jour J. Reine Angew. {M}ath.
\vol 400 
\pages 173--184
\yr 1989
\endref  

\ref\no 13
\by J.~Quer
\paper Corps quadratiques de 3-rang~6 et courbes elliptiques de rang~12
\jour C.~R. Acad. Sc. Paris 
\vol 305 
\pages 215--218
\yr 1987
\endref  

% updated this entry 6/22/90
\ref\no 14
\by K.~Rubin
\paper The work of {K}olyvagin on the arithmetic of elliptic curves
\inbook Arithmetic of Complex Manifolds
\eds W.P.~Barth and H.~Lange
\pages 128--136
\bookinfo  Lecture Notes in Mathematics 1399
%\paperinfo (Preprint, 1989)
\publ Springer Verlag
\publaddr New York
\yr 1989
\endref  

\ref\no 15
\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\no 16
\by J.~H.~Silverman
\paper Computing heights on elliptic curves
\jour Math. Comp.
\vol 51 
\pages 339--358
\yr 1988
\endref  


\ref\no 17
\by J.~Tate
\paper The arithmetic of elliptic curves.
\jour Invent. Math.
\vol 23 
\pages 179--206
\yr 1974
\endref  

\ref\no 18
\by L.~C.~Washington
\paper Number fields and elliptic curves
\pages 245--278
\inbook {NATO} {A}dvanced {S}tudy {I}nstitute on {N}umber {T}heory,
  {B}anff 1988 
\publ Kluwer
\publaddr Netherlands
\yr 1989
\endref

% added next line 6/22/90
\endRefs
\enddocument
%
% end of file
%
