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Author: William A. Stein
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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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\documentstyle{amsppt}
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\magnification 1200
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\vsize= 8.5 true in
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\hsize= 5.5 true in
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\hoffset= 0.5 true in
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\NoBlackBoxes
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\topmatter
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\title
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The Behavior of the Mordell-Weil Group of Elliptic Curves
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\endtitle
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\author
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Armand Brumer and Ois\'{\i}n McGuinness
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\endauthor
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\affil
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Fordham University
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\endaffil
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\address
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% next line is a kludge, since don't have AMSFonts 2.0
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{\rm
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Mathematics Department, Fordham University, Bronx NY 10458,
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USA
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}
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\endaddress
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% next added 6/8/90
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\email
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{
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BITNET\%"brumer\@fordmurh", BITNET\%"mcguiness\@fordmurh"
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}
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\endemail
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\date
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{October 30, 1989, revised March 1, 1990}
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\enddate
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% simplified subjclass June 8 1990
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\subjclass
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Primary 11G40, Secondary 11D25, 11G05, 11-04, 14K15
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\endsubjclass
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\keywords
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{Elliptic curve, Mordell-Weil group, rank,
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Birch-Swinnerton-Dyer
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conjecture, Hasse-Weil $L$-series, discriminant,
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period}
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\endkeywords
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%
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% if genuine Cyrillic available, use that instead!
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\def\Sha{\hbox{$\amalg\kern-.39em\amalg$}}
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%
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\def\F{\bold F} % use for finite fields
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\def\Q{\bold Q}
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\def\C{\bold C}
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\def\R{\bold R}
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\def\Z{\bold Z}
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\def\divides{\,\vert\,}
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\def\notdivides{\,\not\vert\,}
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\define\SwD{Swinnerton-Dyer}
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\define\BSwD{Birch and \SwD}
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\define\heightpair#1#2{\langle #1, #2 \rangle}
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\def\Li{\mathop{\roman{Li}}\nolimits}
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\define\Ls{L(E,s)}
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% total number found
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%
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\def\totalfound{311243}
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%
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% total analysed so far
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%
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\def\totalnumber{310716}
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\def\totalpositive{113969}
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\def\totalnegative{196747}
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\def\rationegpos{1.726}
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\def\totalodd{155658}
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\def\totaleven{155058}
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\def\rankzero{93337}
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\def\rankzeropos{31748}
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\def\rankzeroneg{61589}
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\def\rankzeropct{30.04}
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\def\ranktwo{61517}
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\def\ranktwopos{24706}
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\def\ranktwoneg{36811}
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\def\ranktwopct{19.80}
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\def\rankfour{804}
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\def\rankfourpos{377}
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\def\rankfourneg{427}
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\def\rankfourpct{0.26}
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\def\rankevenGtZeroPct{20.06}
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\def\rankone{143192}
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\def\rankonepos{51871}
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\def\rankoneneg{91321}
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\def\rankonepct{46.08}
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\def\rankthree{11861}
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\def\rankthreepos{5267}
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\def\rankthreeneg{6594}
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\def\rankthreepct{3.82}
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\def\rankfive{5}
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\def\rankfivepos{0}
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\def\rankfiveneg{5}
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\def\rankfivepct{}
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\def\rankoddGtOnePct{3.83}
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%
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\endtopmatter
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\document
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\heading
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\S1 Introduction
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\endheading
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Suppose that $E$ is an elliptic curve defined over $\Q$ given by the
139
equation
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$$
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y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6,
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\tag1$$
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where we assume that $a_i\in \Z$.
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The set $E(\Q)$ of solutions $(x,y)$ with $x$, $y\in\Q$, together with
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the point at infinity, forms a finitely-generated abelian group, the
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{\it Mordell-Weil group\/} of $E$.
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It is isomorphic to $\Z^r\oplus F$ where $F$ is
148
finite and where $r$ is the {\it rank\/} of $E$.
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The possibilities for the
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finite group $F$ are completely known~\cite{9}.
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The important question then is to understand the behavior of the rank
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as $E$ varies over elliptic curves.
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It is still unknown whether $r$ is unbounded or not.
154
In fact, the opinion had been expressed that, in general, an elliptic
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curve might tend to have the smallest possible rank, namely $0$ or
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$1$, compatible with the rank parity predictions of \BSwD\ \cite{8}.
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We present evidence that this may not be
158
the case.
159
160
Published examples \cite{2}, \cite{10} of curves of rank~$\geq 2$
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might suggest that such curves are sparsely
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distributed.\footnote{This situation is not unrelated to the ranks of
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ideal class groups of quadratic fields, where similar phenomena
164
occur~\cite{13}.}
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Mestre and Oesterl\'{e}
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found the 436 modular elliptic curves of prime conductor up to 13100,
167
using~\cite{11}.
168
There were~80 rank~2 curves among the 233~curves of even rank.
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This proportion of rank~2 curves seemed too large
170
to conform to the conventional wisdom stated above (see also~\cite{18},
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pages~254--255).
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We decided to investigate the ranks of elliptic curves
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in a systematic way, over a significantly larger range. Curves
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of {\it prime\/} conductor only were considered for
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practical and theoretical reasons. This collection of curves appears to
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be a typical sample of the set of {\it all} curves (see section~5 for
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some evidence).
178
179
We have studied $\totalnumber$ elliptic curves of prime conductor
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less than $10^8$. There were $\totalodd$ curves with odd rank, and
181
$\totaleven$ curves with even rank.
182
We found $\rankevenGtZeroPct\%$ of {\it all\/} our curves have even
183
rank at least~2, or about $40\%$ of all the even rank curves.
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Even more striking is the behavior of the average rank, as discussed in
185
section~3.
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An incidental aspect of our computations is a massive corroboration of
187
the standard conjectures on elliptic curves, recalled in section~2.
188
189
Recent related work is described in \cite{6} and \cite{8}.
190
Contrasts with our results are given in section~3.
191
192
We expect to publish a fuller account,
193
including the behavior of other invariants of interest.
194
This announcement reports mainly on ranks.
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The computations have been carried out on
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Macintosh~II computers at Fordham University, with the
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partial support of an NSF SCREMS grant.
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We would like to thank our colleagues R.~Lewis, I.~Morrison, and
199
W.~Singer for the use of their machines.
200
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\heading
203
\S2 Definitions
204
\endheading
205
We recall standard notations and definitions~\cite{15}.
206
Associated with
207
equation~(1) is the {\it discriminant\/} $\Delta$, which we
208
will assume to be {\it minimal\/} among all models~(1) of $E$.
209
The fundamental property of the discriminant is that $p\divides\Delta$
210
if and only if the equation~(1) is singular modulo $p$, the
211
{\it conductor\/} $N$ of $E$ is a subtler invariant that has the same
212
property.
213
214
The Hasse-Weil $L$-series of $E$ is defined for $\Re(s)>3/2$ by
215
$$
216
\Ls = \prod_{p\divides N}\left(1-a_pp^{-s}\right)^{-1}
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\prod_{p\notdivides N}\left(1-a_pp^{-s}+p^{1-2s}\right)^{-1}
218
$$
219
where for $p\divides N$, $a_p\in\{-1,0,1\}$, and for
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$p\notdivides N$, $a_p=p+1-|E(\F_p)|$.
221
We shall assume that $E$ is a {\it modular\/} curve,
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so $E$ is a
223
factor\footnote{Note that Mestre-Oesterl\'{e} found their curves
224
by determining the $1$-dimensional factors of $J_0(N)$.
225
Needless to say, we have the same curves in their range.}
226
of the Jacobian $J_0(N)$ of the modular curve $X_0(N)$ of level~$N$.
227
(That is, the Taniyama-Weil
228
conjecture for $E$ is true.)
229
Hence $\Ls$ can be continued to an
230
entire function on $\C$, satisfying a functional equation when $s
231
\mapsto 2-s$,
232
with a zero at $s=1$ of
233
order $\rho$, the {\it analytic\/} rank of $E$.
234
For square-free conductor,
235
the sign in the functional equation may be easily calculated from an
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equation of the curve, allowing a conjectural prediction
237
of the parity of the analytic rank~\cite{1}.
238
We also assume that the conjecture of
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\BSwD\ holds, so that the analytic rank equals the rank, $\rho=r$,
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and the leading term
241
of $\Ls$ at $s=1$ is
242
given by:
243
$$
244
\lim_{s\to1}\frac{\Ls}{(s-1)^r} = \Omega
245
\frac{|\Sha|\det\left(\heightpair{P_i}{P_j}\right)} {[E(\Q):E']^2}
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\prod_{p\divides\Delta}c_p.
247
\tag2
248
$$
249
Here $\Omega$ is the period $\int_{E(\R)}|\omega|$, for $\omega$
250
a N\'eron differential on $E$,
251
$\Sha$ denotes the conjecturally finite Tate-Shafarevich group of $E$,
252
the $P_i$ for
253
$1\leq i\leq r$ are an independent set of points in $E(\Q)$
254
generating the subgroup $E'$, and $\heightpair{P_i}{P_j}$
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denotes the height pairing. The fudge factors $c_p$
256
are all~1 for the curves we consider.
257
Recent work of Rubin and Kolyvagin
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\cite{14} confirms the
259
conjecture of Birch and Swinnerton-Dyer in many cases of rank $r\le 1$.
260
261
Examples illustrating (2) for the curves of ranks~4 and~5 of least
262
known conductor
263
are given in section~4.
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\heading
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\S3 Rank Results
267
\endheading
268
Elliptic curves of prime conductor $N$ were conjectured to have prime
269
discriminant, except for the Setzer-Neumann curves and for 5~other
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small conductor curves, see \cite{2, appendix}.
271
This is now known for {\it modular} curves by Theorem~2 of \cite{12}.
272
We therefore searched for curves of prime discriminant,
273
by looking for integral solutions to the equation
274
$$
275
c_4^3-c_6^2 = 1728\Delta,
276
\tag3
277
$$
278
where $c_4$ and $c_6$ are the usual invariants attached to
279
equation~(1), or more precisely, by fixing $a_4$, and searching for
280
$a_6$
281
for which~(3) has a solution with $\Delta$ prime and less than $10^8$.
282
This produced $\totalfound$ curves
283
including the $869$ expected curves with non-trivial torsion and
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rank~0.
285
The set of $\totalnumber$
286
curves that we studied is most simply described by
287
$$
288
\{E :\left|\Delta\right| \le 10^8, |a_6|\le 2^{31}-1\},
289
$$
290
with $|\Delta|$ {\it prime\/}.
291
292
We will not describe here all the details of the several thousand hours
293
of computations, but just say that,
294
imitating Mazur's description \cite{17} of ``infinite descent''
295
we searched for points by night, and calculated $L$-series derivatives
296
or regulators by day.
297
We may use the
298
infinite series formulas of~\cite{3} for the
299
derivatives of the $L$-series at $s=1$, since $E$ is assumed modular.
300
Then an upper bound for the analytic rank is found by
301
estimating the order of vanishing of $\Ls$ at $s=1$.
302
Using 2000 or 4000 terms of these series provided sufficient accuracy
303
for our purposes,
304
since the values are either~0 to several places or else are
305
far from~0 in most cases.
306
The period $\Omega$ is easily calculated using the
307
Arithmetic-Geometric mean algorithm of Gauss \cite{7},
308
and the height regulator $R=\det\left(\heightpair{P_i}{P_j}\right)$
309
is computed by using the method of Tate, as modified
310
by Silverman~\cite{16}, once points have been found by a search.
311
312
The rank predictions are based on a combination of three calculations:
313
the rank parity, the analytic rank or order of vanishing of the
314
$L$-series,
315
and the number of independent points found which is a lower bound for
316
the
317
algebraic rank~$r$.
318
When the ranks coincide, as they should, we get a prediction from~(2)
319
of
320
$|\Sha|$, which should be an integer square.
321
The largest $|\Sha|$ we found was~$289$, for a curve of rank~0.
322
323
For each curve, we keep its discriminant, parity, period, rank,
324
the appropriate $L$-derivative value, a list of $x$-coordinates of
325
the independent points found, and the regulator of these points.
326
327
Of the curves analysed, $\totalpositive$ had
328
positive discriminant, and $\totalnegative$ had negative
329
discriminant.\footnote{The quotient is about $\rationegpos$, near
330
$\sqrt{3}$.
331
See section~5 for an explanation.}
332
An interesting phenomenon was the systematic influence of the
333
discriminant sign on all aspects of the arithmetic of the curve.
334
The rank distribution is given in the following table:
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$$\vbox{\offinterlineskip
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\hrule
337
\halign{\vrule#&\strut\quad#\hfil\quad&#\vrule\,\vrule&%
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&\quad\hfil#\hfil\quad\vrule\cr
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%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr
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height2pt&\omit&&&&&&&\cr
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&Rank && 0 & 1 & 2 & 3 & 4 & 5\cr
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\noalign{\hrule}
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\noalign{\vskip 2pt}
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\noalign{\hrule}
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%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr
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height2pt&\omit&&&&&&&\cr
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&$\Delta>0$&&\rankzeropos&\rankonepos&\ranktwopos&\rankthreepos%
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&\rankfourpos&\rankfivepos\cr
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&$\Delta<0$&&\rankzeroneg&\rankoneneg&\ranktwoneg&\rankthreeneg%
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&\rankfourneg&\rankfiveneg\cr
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height2pt&\omit&&&&&&&\cr
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%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr
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\noalign{\hrule}
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%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr
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height2pt&\omit&&&&&&&\cr
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&Totals&&\rankzero&\rankone&\ranktwo&\rankthree%
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&\rankfour&\rankfive\cr
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&Percents&&\rankzeropct&\rankonepct&\ranktwopct&\rankthreepct%
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&\rankfourpct& \rankfivepct\cr
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%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr}
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height2pt&\omit&&&&&&&\cr}
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\hrule}
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$$
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Thus $\rankevenGtZeroPct\%$ of the curves have even rank at
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least~2. Note that the {\it
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positive\/} discriminant curves give an even higher percentage!
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368
Define $N(r, X)$ to be the proportion of curves with conductor at
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most $X$, and with rank at least $r$. Our data
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shows that these functions are {\it increasing\/} functions of
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$X$ for $r\geq 3$ and $X\le 10^8$.
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In contrast \cite{6}, dealing with quadratic twists of elliptic curves,
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suggests a decrease for the analogous functions.
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Denote the {\it average rank\/} among the curves
376
with discriminant sign
377
$\epsilon$ and conductor at most $X$ by $r_\epsilon(X)$.
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In our data, the functions $r_\epsilon(X)$ for $X<10^8$ are
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quite steadily climbing to the numbers
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$1.04$ for $\Delta >0$, and to
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$0.94$ for $\Delta <0$.
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In particular, the average rank of our curves is {\it
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not\/} $0.5$, as is expected to be the case for
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twists~\cite{5},\cite{8}.
385
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In most cases, the predicted analytic rank
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matches with the rank of points found, and the predicted $\Sha$ is
388
close to a non-zero integral square.
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More precisely, this is so for all the curves of rank at least~3,
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and for 95\% of the rank~2 curves.
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There is a very small number of rank~2
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curves for which the prediction of rank~2 is based solely on the
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vanishing of $L$-series.
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Note that in \cite{8}, a numerical study of one family of cubic twists,
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only the analytic rank is estimated.
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For many rank~1 curves (i.e., curves whose rank is predicted to be odd
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for which $L'(E,1)$ does not vanish),
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we have found no points by point searches
399
over moderate ranges, and do not expect to find any small points.
400
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\heading
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\S4 Examples
403
\endheading
404
In the literature, one finds very few
405
verifications of the conjecture of Birch and Swinnerton-Dyer
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for ranks $\geq 2$.
407
The paper \cite{3} on the curve of conductor~5077, found in \cite{2},
408
works out
409
the only rank~3 case that we know of.
410
It is perhaps not without interest to report the details for the
411
curves\footnote{These curves are new.
412
The curves of rank~4 reported in \cite{2} and \cite{10}
413
are respectively the 5th, 7th and 2nd curves of rank~4 in our list.}
414
of smallest known conductor of rank 4 and
415
of rank 5.
416
417
The first rank~4 curve is:
418
$$
419
y^2+y=x^3+x^2-72x+210, \qquad \Delta =501029.
420
$$
421
For this curve, the predicted parity is even, the period
422
$\Omega=2.952580$,
423
the value and 2nd derivative of the $L$-series vanish to several
424
places, and $L^{(4)}(E,1)/4!=9.357978$.
425
The points with $x$-coordinates $5$, $4$, $3$, $6$ in order
426
of increasing height, form a basis. The regulator is $R=3.169424$,
427
which matches the
428
quotient of $L^{(4)}(E,1)/4!$ and $\Omega$, so $|\Sha|$ appears to be
429
1.
430
There are $21\times2$ integral points with $|x|<10^6$, while the second
431
curve of rank~4 has $28\times2$.
432
433
The first rank~5 curve is:
434
$$
435
y^2+y=x^3-79x + 342, \qquad \Delta = -19047851.
436
$$
437
The previously found curves of rank~5
438
have conductors about ten times larger~\cite{2}, \cite{10}.
439
440
The predicted parity is odd,
441
the period $\Omega=2.047641$, and
442
then the 1st and 3rd
443
$L$-series derivatives are 0 to several places, and
444
$L^{(5)}(E,1)/5! = 30.285711$.
445
Dividing by $\Omega$ gives $14.790539$.
446
There are $38\times 2$ integral points with $|x| <10^6$.
447
Those with $x=5$, 4, 3, 7, 0 form a basis for $E(\Q)$ with height
448
regulator $R= 14.790528$.
449
So $\Sha$ is predicted to be trivial.
450
There are four other curves of rank~5 with conductor less than
451
100~million.
452
The next one has discriminant~$-64921931$.
453
All turn out to have trivial $\Sha$, if we believe the conjectures.
454
455
Some of the curves found by the search give rise to rather spectacular
456
cancellations.
457
A particular example is the curve of rank~0
458
$$
459
y^2+xy+y=x^3+x^2-12632622x-17287039382, \qquad \Delta=38593.
460
$$
461
Here $c_4 = 606365857$, and $c_6 = 14931454281967$,
462
and the equation $1728\Delta = c_4^3-c_6^2$
463
involves the difference of two 27~digit numbers!
464
465
466
\heading
467
\S 5 Heuristics
468
\endheading
469
While its connection with the Taniyama-Weil conjecture makes the
470
conductor a natural invariant,
471
the discriminant has turned out to be more useful in our heuristics.
472
The idea is to arrange the curves in
473
order by discriminant size, and then replace lattice-point counting
474
problems by area computations.
475
This works well, for instance, to count curves.
476
477
% \zeta(12) changed to \zeta(10) 3 times
478
\proclaim{Heuristic Estimate}
479
We have the following estimates for the number of positive
480
and negative discriminants of absolute value at most $N$,
481
$$\align
482
A_{+}(N)&\sim \frac{\alpha_{+} }{\zeta(10)}N^{5/6},\\
483
A_{-}(N)&\sim \frac{\alpha_{-} }{\zeta(10)}N^{5/6}.
484
\endalign$$
485
\endproclaim
486
487
The $\zeta(10)$ factor arises from taking into account the non-minimal
488
discriminants. Here
489
$\alpha_{+} = 0.4206$ and
490
$\alpha_{-} = 0.7285$ are
491
given by the elliptic integrals
492
$$
493
\alpha_{\pm} = \frac{\sqrt{3}}{10 }
494
\int_{\pm 1}^\infty \frac{du}{\sqrt{u^{3}\mp 1}},
495
$$
496
which arise from parametrising suitably the integrals that replace the
497
lattice-point counts. A well known identity of
498
Legendre, related to complex multiplication, is
499
$\alpha_{-} =\sqrt{3}\alpha_{+}$.
500
501
\noindent{\smc Remark.} One should compare this estimate with the
502
results of \cite{4}.
503
While the number of elliptic curves grows like $N^{5/6}$, the
504
number of cubic fields grows like~$N$.
505
506
Assuming the distribution of {\it prime\/} discriminants among
507
discriminants is that of prime numbers among all integers,
508
the number of prime discriminants of sign $\epsilon$ and of
509
size less than $N$ is then~$\alpha_\epsilon \Li(N^{5/6})$, where
510
$\Li(x)$ is the logarithmic integral.
511
The expected
512
number of curves with prime $|\Delta|<10^8$ is then~$311586$, comparing
513
rather well with the number $\totalfound$ found.
514
515
Similar heuristic arguments have been applied to other invariants.
516
For instance, the {\it average period of a curve with
517
positive discriminant is $\sqrt{3/2}$ times the average period of a
518
curve with negative discriminant}.
519
This is also confirmed by the data.
520
The fits with the experimental data are so good that one
521
could hope for proofs in the near future.
522
523
We have not as yet been able to provide heuristics for the
524
growth of the functions $N(r,X)$.
525
While our data may seem massive, $N= 10^8$ is not sufficient to
526
distinguish growth laws of $\log\log N$, $N^{1/12}$ or $N^{1/24}$
527
from constants.
528
So we have to be cautious in formulating conjectures
529
based on the numerical evidence.
530
531
532
\Refs
533
534
\ref\no 1
535
\by B.~Birch and W.~Kuyk, editors
536
\book Modular Functions of One Variable IV
537
\bookinfo volume 476 of Lecture Notes in Mathematics
538
%\vol 476
539
\publ Springer-Verlag
540
\publaddr New York
541
\yr1975
542
\endref
543
544
\ref\no 2
545
\by A.~Brumer and K.~Kramer
546
\paper The rank of elliptic curves
547
\jour Duke Math. J.
548
\vol 44
549
\pages 715--743
550
\yr1977
551
\endref
552
553
\ref\no 3
554
\by J.~Buhler, B.~H. Gross, and D.~B. Zagier
555
\paper On the conjecture of {B}irch and {S}winnerton-{D}yer for an
556
elliptic curve of rank 3
557
\jour Math. Comp.
558
\vol 44
559
\pages 473--481
560
\yr 1985
561
\endref
562
563
\ref\no 4
564
\by H.~Davenport and H.~Heilbronn
565
\paper On the density of discriminants of cubic fields {(II)}
566
\jour Proc. Royal Soc., (A)
567
\vol 322
568
\pages 405--420
569
\yr 1971
570
\endref
571
572
\ref\no 5
573
\by D.~M. Goldfeld
574
\paper Conjectures on elliptic curves over quadratic fields
575
\inbook Number Theory Carbondale 1979
576
%\vol 751
577
\bookinfo Lecture Notes in Mathematics 751
578
\pages 108--118
579
\publ Springer Verlag
580
\publaddr New York
581
\yr 1979
582
\endref
583
584
\ref\no 6
585
\by F.~Gouvea and B.~Mazur
586
\paper The square-free sieve and the rank of {M}ordell-{W}eil
587
\paperinfo (Preprint, April 1989)
588
%\yr 1989
589
\endref
590
591
\ref\no 7
592
\by D.~Grayson
593
\paper The arithogeometric mean
594
\jour Arch. Math.
595
\vol 52
596
\pages 507--512
597
\yr 1989
598
\endref
599
600
\ref\no 8
601
\by G.~Kramarz and D.~B. Zagier
602
\paper Numerical investigations related to the {$L$}-series of certain
603
elliptic curves
604
\jour J. Indian Math. Soc.
605
\vol 52
606
\pages 51--60
607
\yr 1987
608
\finalinfo (Ramanujan Centenary volume)
609
\endref
610
611
\ref\no 9
612
\by B.~Mazur
613
\paper Modular curves and the {E}isenstein ideal
614
\jour Publ. Math. IHES
615
\vol 47
616
\pages 33--186
617
\yr 1977
618
\endref
619
620
\ref\no 10
621
\by J.-F.~Mestre
622
\paper {F}ormules explicites et minorations de conducteurs de
623
vari\'et\'es alg\'ebriques
624
\jour Comp. Math.
625
\vol 58
626
\pages 209--232
627
\yr 1986
628
\endref
629
630
\ref\no 11
631
\by J.-F.~Mestre
632
\paper La m{\'e}thode des graphes. {E}xemples et applications
633
\inbook Class Numbers and Units of Number Fields
634
\pages 217--242
635
\publ Nagoya University
636
\publaddr Nagoya
637
\yr 1986
638
\bookinfo Katata conference
639
\finalinfo Unpublished tables
640
\endref
641
642
\ref\no 12
643
\by J.-F.~Mestre and J.~Oesterl{\'{e}}
644
\paper Courbes de {W}eil semi-stables de discriminant une puissance
645
$m$-i{\`e}me
646
\jour J. Reine Angew. {M}ath.
647
\vol 400
648
\pages 173--184
649
\yr 1989
650
\endref
651
652
\ref\no 13
653
\by J.~Quer
654
\paper Corps quadratiques de 3-rang~6 et courbes elliptiques de rang~12
655
\jour C.~R. Acad. Sc. Paris
656
\vol 305
657
\pages 215--218
658
\yr 1987
659
\endref
660
661
% updated this entry 6/22/90
662
\ref\no 14
663
\by K.~Rubin
664
\paper The work of {K}olyvagin on the arithmetic of elliptic curves
665
\inbook Arithmetic of Complex Manifolds
666
\eds W.P.~Barth and H.~Lange
667
\pages 128--136
668
\bookinfo Lecture Notes in Mathematics 1399
669
%\paperinfo (Preprint, 1989)
670
\publ Springer Verlag
671
\publaddr New York
672
\yr 1989
673
\endref
674
675
\ref\no 15
676
\by J.~H.~Silverman
677
\book The arithmetic of elliptic curves
678
%\vol 106
679
\bookinfo volume 106, Graduate Texts in Mathematics
680
\publ Springer Verlag
681
\publaddr New York
682
\yr 1986
683
\endref
684
685
\ref\no 16
686
\by J.~H.~Silverman
687
\paper Computing heights on elliptic curves
688
\jour Math. Comp.
689
\vol 51
690
\pages 339--358
691
\yr 1988
692
\endref
693
694
695
\ref\no 17
696
\by J.~Tate
697
\paper The arithmetic of elliptic curves.
698
\jour Invent. Math.
699
\vol 23
700
\pages 179--206
701
\yr 1974
702
\endref
703
704
\ref\no 18
705
\by L.~C.~Washington
706
\paper Number fields and elliptic curves
707
\pages 245--278
708
\inbook {NATO} {A}dvanced {S}tudy {I}nstitute on {N}umber {T}heory,
709
{B}anff 1988
710
\publ Kluwer
711
\publaddr Netherlands
712
\yr 1989
713
\endref
714
715
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\endRefs
717
\enddocument
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