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Author: William A. Stein
1%
2% To appear in: BULLETIN (NEW SERIES) OF THE AMERICAN MATH. SOCIETY
3%
4% Reference Number: BULL128
5%
6% Letter of April 3 1990 from C. Venedettuoli
7%
8%
9% 5/10/90 small changes in top matter made for version 2.0 of AMSTeX
10%
11% 6/8/90 eliminate Revisions 1985 duplication in printed version
12% 6/22/90 changed zeta(12) to zeta(10) (3 places)
13%         added bibliographic info for Rubin's paper, cleaned up extra ..'s
14%
15\documentstyle{amsppt}
16
17
18\magnification 1200
19\vsize= 8.5 true in
20\hsize= 5.5 true in
21\hoffset= 0.5 true in
22\NoBlackBoxes
23
24\topmatter
25
26\title
27The Behavior of the Mordell-Weil Group of Elliptic Curves
28\endtitle
29
30\author
31Armand Brumer and Ois\'{\i}n McGuinness
32\endauthor
33
34\affil
35Fordham University
36\endaffil
37
39% next line is a kludge, since don't have AMSFonts 2.0
40{\rm
41Mathematics Department, Fordham University, Bronx NY 10458,
42USA
43}
45
46% next added 6/8/90
47\email
48{
49BITNET\%"brumer\@fordmurh", BITNET\%"mcguiness\@fordmurh"
50}
51\endemail
52
53\date
54{October 30, 1989, revised March 1, 1990}
55\enddate
56
57% simplified subjclass June 8 1990
58\subjclass
59Primary 11G40, Secondary 11D25, 11G05, 11-04, 14K15
60\endsubjclass
61
62\keywords
63{Elliptic curve, Mordell-Weil group, rank,
64Birch-Swinnerton-Dyer
65conjecture, Hasse-Weil $L$-series, discriminant,
66period}
67\endkeywords
68%
69% if genuine Cyrillic available, use that instead!
70\def\Sha{\hbox{$\amalg\kern-.39em\amalg$}}
71%
72\def\F{\bold F}    % use for finite fields
73\def\Q{\bold Q}
74\def\C{\bold C}
75\def\R{\bold R}
76\def\Z{\bold Z}
77\def\divides{\,\vert\,}
78\def\notdivides{\,\not\vert\,}
79\define\SwD{Swinnerton-Dyer}
80\define\BSwD{Birch and \SwD}
81\define\heightpair#1#2{\langle #1, #2 \rangle}
82\def\Li{\mathop{\roman{Li}}\nolimits}
83\define\Ls{L(E,s)}
84
85% total number found
86%
87\def\totalfound{311243}
88%
89% total analysed so far
90%
91\def\totalnumber{310716}
92\def\totalpositive{113969}
93\def\totalnegative{196747}
94\def\rationegpos{1.726}
95\def\totalodd{155658}
96\def\totaleven{155058}
97
98\def\rankzero{93337}
99\def\rankzeropos{31748}
100\def\rankzeroneg{61589}
101\def\rankzeropct{30.04}
102
103\def\ranktwo{61517}
104\def\ranktwopos{24706}
105\def\ranktwoneg{36811}
106\def\ranktwopct{19.80}
107
108\def\rankfour{804}
109\def\rankfourpos{377}
110\def\rankfourneg{427}
111\def\rankfourpct{0.26}
112
113\def\rankevenGtZeroPct{20.06}
114
115\def\rankone{143192}
116\def\rankonepos{51871}
117\def\rankoneneg{91321}
118\def\rankonepct{46.08}
119
120\def\rankthree{11861}
121\def\rankthreepos{5267}
122\def\rankthreeneg{6594}
123\def\rankthreepct{3.82}
124
125\def\rankfive{5}
126\def\rankfivepos{0}
127\def\rankfiveneg{5}
128\def\rankfivepct{}
129
130\def\rankoddGtOnePct{3.83}
131%
132\endtopmatter
133
134\document
136\S1 Introduction
138Suppose that $E$ is an elliptic curve defined over $\Q$ given by the
139equation
140$$141y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6, 142\tag1$$
143where we assume that $a_i\in \Z$.
144The set  $E(\Q)$ of solutions $(x,y)$ with $x$, $y\in\Q$, together with
145the point at infinity, forms a finitely-generated abelian group, the
146{\it Mordell-Weil group\/} of $E$.
147It is isomorphic to $\Z^r\oplus F$ where $F$ is
148finite and where $r$ is the {\it rank\/} of $E$.
149The possibilities for the
150finite group $F$ are completely known~\cite{9}.
151The important question then is to understand the behavior of the rank
152as $E$ varies over elliptic curves.
153It is still unknown whether $r$ is unbounded or not.
154In fact, the opinion had been expressed that, in general, an elliptic
155curve might tend to have the smallest possible rank, namely $0$ or
156$1$, compatible with the rank parity predictions of \BSwD\ \cite{8}.
157We present evidence that this may not be
158the case.
159
160Published examples \cite{2}, \cite{10} of curves of rank~$\geq 2$
161 might suggest that such curves are sparsely
162distributed.\footnote{This situation is not unrelated to the ranks of
163ideal class groups of quadratic fields, where similar phenomena
164occur~\cite{13}.}
165Mestre and Oesterl\'{e}
166found the 436 modular elliptic curves  of prime conductor up to 13100,
167using~\cite{11}.
168There were~80 rank~2 curves among the 233~curves of even rank.
169This  proportion of rank~2 curves seemed too large
170to conform to the conventional wisdom stated above (see also~\cite{18},
171pages~254--255).
172We decided to investigate  the ranks of elliptic curves
173in a systematic way, over a significantly larger range. Curves
174of {\it prime\/} conductor only were considered for
175practical and theoretical reasons. This collection of curves appears to
176be a typical sample of the set of {\it all} curves  (see section~5 for
177some evidence).
178
179We have  studied  $\totalnumber$ elliptic curves of prime conductor
180less than $10^8$. There were $\totalodd$ curves with odd rank, and
181$\totaleven$ curves with even rank.
182We found $\rankevenGtZeroPct\%$ of {\it all\/} our curves have even
183rank at least~2, or about $40\%$ of all the even rank curves.
184Even more striking is the behavior of the average rank, as discussed in
185section~3.
186An incidental aspect of our computations is a massive corroboration of
187the standard conjectures on elliptic curves, recalled in section~2.
188
189Recent related work is described in \cite{6} and \cite{8}.
190Contrasts with our results are given in section~3.
191
192We expect to publish  a fuller  account,
193including the behavior of other invariants of interest.
194This announcement  reports mainly on ranks.
195The computations have been carried out on
196Macintosh~II computers at Fordham University, with the
197partial support of  an NSF SCREMS grant.
198We would like to thank our colleagues R.~Lewis, I.~Morrison, and
199W.~Singer for the use of their machines.
200
201
203\S2 Definitions
205We recall standard notations and definitions~\cite{15}.
206Associated with
207equation~(1) is the {\it discriminant\/} $\Delta$, which we
208will assume to be {\it minimal\/} among all models~(1) of $E$.
209The fundamental property of the discriminant is that $p\divides\Delta$
210if 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
212property.
213
214The Hasse-Weil $L$-series of $E$ is defined for $\Re(s)>3/2$ by
215$$216\Ls = \prod_{p\divides N}\left(1-a_pp^{-s}\right)^{-1} 217\prod_{p\notdivides N}\left(1-a_pp^{-s}+p^{1-2s}\right)^{-1} 218$$
219where for $p\divides N$, $a_p\in\{-1,0,1\}$, and for
220$p\notdivides N$, $a_p=p+1-|E(\F_p)|$.
221We shall assume that $E$ is a {\it modular\/} curve,
222so $E$ is  a
223factor\footnote{Note that Mestre-Oesterl\'{e} found their curves
224by determining the $1$-dimensional factors of $J_0(N)$.
225Needless to say, we have the same curves in their range.}
226of the Jacobian $J_0(N)$ of  the modular curve $X_0(N)$ of level~$N$.
227(That is, the Taniyama-Weil
228conjecture for $E$ is true.)
229Hence $\Ls$ can be continued to an
230entire function on $\C$,  satisfying a functional equation when $s 231\mapsto 2-s$,
232with a zero at $s=1$ of
233order $\rho$, the {\it analytic\/} rank of $E$.
234For square-free conductor,
235the sign in the functional equation may be easily calculated from an
236equation of the curve,  allowing a conjectural prediction
237of the parity of the analytic rank~\cite{1}.
238We also assume that the conjecture of
239\BSwD\ holds, so  that the analytic rank equals the rank, $\rho=r$,
240and the leading term
241of $\Ls$ at $s=1$ is
242given by:
243$$244\lim_{s\to1}\frac{\Ls}{(s-1)^r} = \Omega 245\frac{|\Sha|\det\left(\heightpair{P_i}{P_j}\right)} {[E(\Q):E']^2} 246\prod_{p\divides\Delta}c_p. 247\tag2 248$$
249Here $\Omega$ is the period $\int_{E(\R)}|\omega|$, for $\omega$
250a N\'eron differential on $E$,
251$\Sha$ denotes the conjecturally finite Tate-Shafarevich group of $E$,
252the $P_i$ for
253$1\leq i\leq r$ are an independent set of points in $E(\Q)$
254generating the subgroup $E'$, and $\heightpair{P_i}{P_j}$
255denotes the height pairing. The fudge factors $c_p$
256are all~1 for the curves we consider.
257Recent work of Rubin and Kolyvagin
258\cite{14} confirms the
259conjecture of Birch and Swinnerton-Dyer in many cases of rank $r\le 1$.
260
261Examples illustrating (2) for the curves of ranks~4 and~5 of least
262known conductor
263are given in section~4.
264
266\S3 Rank Results
268Elliptic curves of prime conductor $N$ were conjectured to have prime
269discriminant, except for the Setzer-Neumann curves and for 5~other
270small conductor curves, see \cite{2, appendix}.
271This is now known for {\it modular} curves by Theorem~2 of \cite{12}.
272We therefore searched for curves of prime discriminant,
273by looking for integral solutions to the equation
274$$275c_4^3-c_6^2 = 1728\Delta, 276\tag3 277$$
278where $c_4$ and $c_6$ are the usual invariants attached to
279equation~(1), or more precisely, by fixing $a_4$, and searching for
280$a_6$
281for which~(3) has a solution with $\Delta$ prime and less than $10^8$.
282This produced  $\totalfound$ curves
283including the $869$ expected curves with non-trivial torsion and
284rank~0.
285The set of $\totalnumber$
286curves that we studied is most simply described by
287$$288\{E :\left|\Delta\right| \le 10^8, |a_6|\le 2^{31}-1\}, 289$$
290with $|\Delta|$  {\it prime\/}.
291
292We will not describe here all the details of the several thousand hours
293of computations, but just say that,
294imitating Mazur's description \cite{17} of infinite descent''
295we searched for points by night, and calculated $L$-series derivatives
296or regulators by day.
297We may use the
298infinite series formulas of~\cite{3} for the
299derivatives of the $L$-series at $s=1$, since $E$ is assumed modular.
300Then an upper bound for the analytic rank is found by
301estimating the order of vanishing of $\Ls$ at $s=1$.
302Using 2000 or 4000 terms of these series provided sufficient accuracy
303for our purposes,
304since the values are either~0 to several places or else are
305far from~0 in most cases.
306The  period $\Omega$ is easily calculated using the
307Arithmetic-Geometric mean algorithm of Gauss \cite{7},
308and the height regulator $R=\det\left(\heightpair{P_i}{P_j}\right)$
309is computed  by using the method of Tate, as modified
310by Silverman~\cite{16}, once points have been found by a search.
311
312The rank predictions  are based on a combination of three calculations:
313the rank parity, the analytic rank or order of vanishing of the
314$L$-series,
315and the number of independent points found which is a lower bound for
316the
317algebraic rank~$r$.
318When the ranks coincide, as they should, we get a prediction from~(2)
319of
320$|\Sha|$, which should be an integer square.
321The largest $|\Sha|$ we found was~$289$, for a curve of rank~0.
322
323For each curve, we keep its discriminant, parity, period, rank,
324the appropriate $L$-derivative value, a list of $x$-coordinates of
325the independent points found, and the regulator of these points.
326
327Of the curves analysed,  $\totalpositive$ had
328positive discriminant, and $\totalnegative$ had negative
329discriminant.\footnote{The quotient is about $\rationegpos$, near
330$\sqrt{3}$.
331See section~5 for an explanation.}
332An interesting  phenomenon was the systematic influence of the
333discriminant sign on all aspects of the arithmetic of the curve.
334The rank distribution is given in the following table:
335\vbox{\offinterlineskip 336\hrule 337\halign{\vrule#&\strut\quad#\hfil\quad&#\vrule\,\vrule&% 338&\quad\hfil#\hfil\quad\vrule\cr 339%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr 340height2pt&\omit&&&&&&&\cr 341&Rank && 0 & 1 & 2 & 3 & 4 & 5\cr 342\noalign{\hrule} 343\noalign{\vskip 2pt} 344\noalign{\hrule} 345%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr 346height2pt&\omit&&&&&&&\cr 347&\Delta>0&&\rankzeropos&\rankonepos&\ranktwopos&\rankthreepos% 348 &\rankfourpos&\rankfivepos\cr 349&\Delta<0&&\rankzeroneg&\rankoneneg&\ranktwoneg&\rankthreeneg% 350 &\rankfourneg&\rankfiveneg\cr 351height2pt&\omit&&&&&&&\cr 352%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr 353\noalign{\hrule} 354%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr 355height2pt&\omit&&&&&&&\cr 356&Totals&&\rankzero&\rankone&\ranktwo&\rankthree% 357 &\rankfour&\rankfive\cr 358&Percents&&\rankzeropct&\rankonepct&\ranktwopct&\rankthreepct% 359 &\rankfourpct& \rankfivepct\cr 360%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr} 361height2pt&\omit&&&&&&&\cr} 362\hrule} 363
364Thus $\rankevenGtZeroPct\%$ of the curves have even rank at
365least~2. Note that the {\it
366positive\/} discriminant curves give an even higher percentage!
367
368Define $N(r, X)$ to be the proportion of curves with conductor at
369most $X$, and with rank at least $r$.  Our data
370shows that these functions are {\it increasing\/} functions of
371$X$ for $r\geq 3$ and $X\le 10^8$.
372In contrast \cite{6}, dealing with quadratic twists of elliptic curves,
373suggests a decrease for the analogous functions.
374
375Denote the {\it average rank\/} among the curves
376with discriminant sign
377$\epsilon$ and conductor at most $X$ by $r_\epsilon(X)$.
378In our data, the functions $r_\epsilon(X)$ for $X<10^8$ are
379quite steadily climbing to the numbers
380$1.04$ for $\Delta >0$, and to
381$0.94$ for $\Delta <0$.
382In particular, the average rank of our curves is {\it
383not\/} $0.5$, as  is expected to be the case for
384twists~\cite{5},\cite{8}.
385
386In most cases, the predicted  analytic rank
387matches with the rank of points found, and the predicted $\Sha$ is
388close to a non-zero integral square.
389More precisely, this is so for all the curves of rank at least~3,
390and for 95\% of the rank~2 curves.
391There is a very small number of rank~2
392curves for which the prediction of rank~2 is based solely on the
393vanishing of $L$-series.
394Note that in \cite{8}, a numerical study of one family of cubic twists,
395only the analytic rank is estimated.
396For many rank~1 curves (i.e., curves whose rank is predicted to be odd
397for which $L'(E,1)$ does not vanish),
398we have found no points by  point searches
399over moderate ranges, and do not expect to find any small points.
400
402\S4 Examples
404In the literature, one finds very few
405verifications of the conjecture of Birch and Swinnerton-Dyer
406for ranks $\geq 2$.
407The paper \cite{3} on the curve of conductor~5077, found in \cite{2},
408works out
409the only rank~3 case that we know of.
410It is perhaps not without interest to report the details for the
411curves\footnote{These curves are new.
412The curves of rank~4 reported in \cite{2} and \cite{10}
413are respectively the 5th, 7th and 2nd curves of rank~4 in our list.}
414of smallest known conductor of rank 4 and
415of rank 5.
416
417The first rank~4 curve  is:
418$$419y^2+y=x^3+x^2-72x+210, \qquad \Delta =501029. 420$$
421For this curve, the predicted parity is even, the period
422$\Omega=2.952580$,
423the value and 2nd derivative of the $L$-series vanish to several
424places, and  $L^{(4)}(E,1)/4!=9.357978$.
425The points with $x$-coordinates $5$, $4$, $3$, $6$ in  order
426of increasing height, form a basis. The regulator is $R=3.169424$,
427which matches the
428quotient of $L^{(4)}(E,1)/4!$ and $\Omega$, so $|\Sha|$ appears to be
4291.
430There are $21\times2$ integral points with $|x|<10^6$, while the second
431curve of rank~4 has $28\times2$.
432
433The first rank~5 curve is:
434$$435y^2+y=x^3-79x + 342, \qquad \Delta = -19047851. 436$$
437The previously found  curves of rank~5
438have conductors about ten times larger~\cite{2}, \cite{10}.
439
440The predicted parity is odd,
441the period $\Omega=2.047641$, and
442then the 1st and 3rd
443$L$-series derivatives are 0 to several places, and
444$L^{(5)}(E,1)/5! = 30.285711$.
445Dividing by $\Omega$ gives  $14.790539$.
446There are $38\times 2$ integral points with  $|x| <10^6$.
447Those with $x=5$, 4, 3, 7, 0 form a basis for $E(\Q)$ with height
448regulator $R= 14.790528$.
449So $\Sha$ is predicted to be trivial.
450There are four other curves of rank~5 with conductor less than
451100~million.
452The next one has discriminant~$-64921931$.
453All turn out to have trivial $\Sha$, if we believe the conjectures.
454
455Some of the  curves found by the search give rise to rather spectacular
456cancellations.
457A particular example is the curve of rank~0
458$$459y^2+xy+y=x^3+x^2-12632622x-17287039382, \qquad \Delta=38593. 460$$
461Here $c_4 = 606365857$, and $c_6 = 14931454281967$,
462and the equation $1728\Delta = c_4^3-c_6^2$
463involves the difference of two 27~digit numbers!
464
465
467\S 5 Heuristics
469While its connection with  the Taniyama-Weil conjecture makes the
470conductor a natural invariant,
471the discriminant  has turned out to be more useful in our heuristics.
472The idea is to arrange the curves in
473order by discriminant size, and then replace lattice-point counting
474problems by area computations.
475This works well, for instance, to count curves.
476
477% \zeta(12) changed to \zeta(10) 3 times
478\proclaim{Heuristic Estimate}
479We have the following estimates for the number of positive
480and negative discriminants of absolute value at most $N$,
481\align 482A_{+}(N)&\sim \frac{\alpha_{+} }{\zeta(10)}N^{5/6},\\ 483A_{-}(N)&\sim \frac{\alpha_{-} }{\zeta(10)}N^{5/6}. 484\endalign
485\endproclaim
486
487The $\zeta(10)$ factor  arises from taking into account the non-minimal
488discriminants. Here
489$\alpha_{+} = 0.4206$ and
490$\alpha_{-} = 0.7285$ are
491given by the elliptic integrals
492$$493\alpha_{\pm} = \frac{\sqrt{3}}{10 } 494\int_{\pm 1}^\infty \frac{du}{\sqrt{u^{3}\mp 1}}, 495$$
496which arise from parametrising suitably the integrals  that replace the
497lattice-point counts. A well known identity of
498Legendre, related to complex multiplication, is
499$\alpha_{-} =\sqrt{3}\alpha_{+}$.
500
501\noindent{\smc Remark.} One should compare this estimate with the
502results of \cite{4}.
503While the number of elliptic curves grows like $N^{5/6}$, the
504number of cubic fields grows like~$N$.
505
506Assuming  the distribution of {\it prime\/} discriminants  among
507discriminants is that of prime numbers among all integers,
508the number of prime discriminants of sign $\epsilon$ and of
509size less than $N$ is then~$\alpha_\epsilon \Li(N^{5/6})$, where
510$\Li(x)$ is the logarithmic integral.
511The expected
512number of curves with prime $|\Delta|<10^8$ is then~$311586$, comparing
513rather well with the number $\totalfound$ found.
514
515Similar heuristic arguments have been applied to other invariants.
516For instance, the {\it average period of a curve with
517positive discriminant is $\sqrt{3/2}$ times the average period of a
518curve with negative discriminant}.
519This is also confirmed by the data.
520The fits with the experimental data are so good that one
521could hope for proofs in the near future.
522
523We have not as yet been able to provide heuristics for the
524growth of the functions $N(r,X)$.
525While our data may seem massive, $N= 10^8$ is not sufficient to
526distinguish growth laws of $\log\log N$, $N^{1/12}$ or $N^{1/24}$
527from constants.
528So we have to be cautious in formulating conjectures
529based 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
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
556elliptic 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
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
623vari\'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
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
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
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
712\yr 1989
713\endref
714
715% added next line 6/22/90
716\endRefs
717\enddocument
718%
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720%
721
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