%1% To appear in: BULLETIN (NEW SERIES) OF THE AMERICAN MATH. SOCIETY2%3% Reference Number: BULL1284%5% Letter of April 3 1990 from C. Venedettuoli6%7%8% 5/10/90 small changes in top matter made for version 2.0 of AMSTeX9%10% 6/8/90 eliminate Revisions 1985 duplication in printed version11% 6/22/90 changed zeta(12) to zeta(10) (3 places)12% added bibliographic info for Rubin's paper, cleaned up extra ..'s13%14\documentstyle{amsppt}151617\magnification 120018\vsize= 8.5 true in19\hsize= 5.5 true in20\hoffset= 0.5 true in21\NoBlackBoxes2223\topmatter2425\title26The Behavior of the Mordell-Weil Group of Elliptic Curves27\endtitle2829\author30Armand Brumer and Ois\'{\i}n McGuinness31\endauthor3233\affil34Fordham University35\endaffil3637\address38% next line is a kludge, since don't have AMSFonts 2.039{\rm40Mathematics Department, Fordham University, Bronx NY 10458,41USA42}43\endaddress4445% next added 6/8/904647{48BITNET\%"brumer\@fordmurh", BITNET\%"mcguiness\@fordmurh"49}50\endemail5152\date53{October 30, 1989, revised March 1, 1990}54\enddate5556% simplified subjclass June 8 199057\subjclass58Primary 11G40, Secondary 11D25, 11G05, 11-04, 14K1559\endsubjclass6061\keywords62{Elliptic curve, Mordell-Weil group, rank,63Birch-Swinnerton-Dyer64conjecture, Hasse-Weil $L$-series, discriminant,65period}66\endkeywords67%68% if genuine Cyrillic available, use that instead!69\def\Sha{\hbox{$\amalg\kern-.39em\amalg$}}70%71\def\F{\bold F} % use for finite fields72\def\Q{\bold Q}73\def\C{\bold C}74\def\R{\bold R}75\def\Z{\bold Z}76\def\divides{\,\vert\,}77\def\notdivides{\,\not\vert\,}78\define\SwD{Swinnerton-Dyer}79\define\BSwD{Birch and \SwD}80\define\heightpair#1#2{\langle #1, #2 \rangle}81\def\Li{\mathop{\roman{Li}}\nolimits}82\define\Ls{L(E,s)}8384% total number found85%86\def\totalfound{311243}87%88% total analysed so far89%90\def\totalnumber{310716}91\def\totalpositive{113969}92\def\totalnegative{196747}93\def\rationegpos{1.726}94\def\totalodd{155658}95\def\totaleven{155058}9697\def\rankzero{93337}98\def\rankzeropos{31748}99\def\rankzeroneg{61589}100\def\rankzeropct{30.04}101102\def\ranktwo{61517}103\def\ranktwopos{24706}104\def\ranktwoneg{36811}105\def\ranktwopct{19.80}106107\def\rankfour{804}108\def\rankfourpos{377}109\def\rankfourneg{427}110\def\rankfourpct{0.26}111112\def\rankevenGtZeroPct{20.06}113114\def\rankone{143192}115\def\rankonepos{51871}116\def\rankoneneg{91321}117\def\rankonepct{46.08}118119\def\rankthree{11861}120\def\rankthreepos{5267}121\def\rankthreeneg{6594}122\def\rankthreepct{3.82}123124\def\rankfive{5}125\def\rankfivepos{0}126\def\rankfiveneg{5}127\def\rankfivepct{}128129\def\rankoddGtOnePct{3.83}130%131\endtopmatter132133\document134\heading135\S1 Introduction136\endheading137Suppose that $E$ is an elliptic curve defined over $\Q$ given by the138equation139$$140y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6,141\tag1$$142where we assume that $a_i\in \Z$.143The set $E(\Q)$ of solutions $(x,y)$ with $x$, $y\in\Q$, together with144the point at infinity, forms a finitely-generated abelian group, the145{\it Mordell-Weil group\/} of $E$.146It is isomorphic to $\Z^r\oplus F$ where $F$ is147finite and where $r$ is the {\it rank\/} of $E$.148The possibilities for the149finite group $F$ are completely known~\cite{9}.150The important question then is to understand the behavior of the rank151as $E$ varies over elliptic curves.152It is still unknown whether $r$ is unbounded or not.153In fact, the opinion had been expressed that, in general, an elliptic154curve might tend to have the smallest possible rank, namely $0$ or155$1$, compatible with the rank parity predictions of \BSwD\ \cite{8}.156We present evidence that this may not be157the case.158159Published examples \cite{2}, \cite{10} of curves of rank~$\geq 2$160might suggest that such curves are sparsely161distributed.\footnote{This situation is not unrelated to the ranks of162ideal class groups of quadratic fields, where similar phenomena163occur~\cite{13}.}164Mestre and Oesterl\'{e}165found the 436 modular elliptic curves of prime conductor up to 13100,166using~\cite{11}.167There were~80 rank~2 curves among the 233~curves of even rank.168This proportion of rank~2 curves seemed too large169to conform to the conventional wisdom stated above (see also~\cite{18},170pages~254--255).171We decided to investigate the ranks of elliptic curves172in a systematic way, over a significantly larger range. Curves173of {\it prime\/} conductor only were considered for174practical and theoretical reasons. This collection of curves appears to175be a typical sample of the set of {\it all} curves (see section~5 for176some evidence).177178We have studied $\totalnumber$ elliptic curves of prime conductor179less than $10^8$. There were $\totalodd$ curves with odd rank, and180$\totaleven$ curves with even rank.181We found $\rankevenGtZeroPct\%$ of {\it all\/} our curves have even182rank at least~2, or about $40\%$ of all the even rank curves.183Even more striking is the behavior of the average rank, as discussed in184section~3.185An incidental aspect of our computations is a massive corroboration of186the standard conjectures on elliptic curves, recalled in section~2.187188Recent related work is described in \cite{6} and \cite{8}.189Contrasts with our results are given in section~3.190191We expect to publish a fuller account,192including the behavior of other invariants of interest.193This announcement reports mainly on ranks.194The computations have been carried out on195Macintosh~II computers at Fordham University, with the196partial support of an NSF SCREMS grant.197We would like to thank our colleagues R.~Lewis, I.~Morrison, and198W.~Singer for the use of their machines.199200201\heading202\S2 Definitions203\endheading204We recall standard notations and definitions~\cite{15}.205Associated with206equation~(1) is the {\it discriminant\/} $\Delta$, which we207will assume to be {\it minimal\/} among all models~(1) of $E$.208The fundamental property of the discriminant is that $p\divides\Delta$209if and only if the equation~(1) is singular modulo $p$, the210{\it conductor\/} $N$ of $E$ is a subtler invariant that has the same211property.212213The Hasse-Weil $L$-series of $E$ is defined for $\Re(s)>3/2$ by214$$215\Ls = \prod_{p\divides N}\left(1-a_pp^{-s}\right)^{-1}216\prod_{p\notdivides N}\left(1-a_pp^{-s}+p^{1-2s}\right)^{-1}217$$218where for $p\divides N$, $a_p\in\{-1,0,1\}$, and for219$p\notdivides N$, $a_p=p+1-|E(\F_p)|$.220We shall assume that $E$ is a {\it modular\/} curve,221so $E$ is a222factor\footnote{Note that Mestre-Oesterl\'{e} found their curves223by determining the $1$-dimensional factors of $J_0(N)$.224Needless to say, we have the same curves in their range.}225of the Jacobian $J_0(N)$ of the modular curve $X_0(N)$ of level~$N$.226(That is, the Taniyama-Weil227conjecture for $E$ is true.)228Hence $\Ls$ can be continued to an229entire function on $\C$, satisfying a functional equation when $s230\mapsto 2-s$,231with a zero at $s=1$ of232order $\rho$, the {\it analytic\/} rank of $E$.233For square-free conductor,234the sign in the functional equation may be easily calculated from an235equation of the curve, allowing a conjectural prediction236of the parity of the analytic rank~\cite{1}.237We also assume that the conjecture of238\BSwD\ holds, so that the analytic rank equals the rank, $\rho=r$,239and the leading term240of $\Ls$ at $s=1$ is241given by:242$$243\lim_{s\to1}\frac{\Ls}{(s-1)^r} = \Omega244\frac{|\Sha|\det\left(\heightpair{P_i}{P_j}\right)} {[E(\Q):E']^2}245\prod_{p\divides\Delta}c_p.246\tag2247$$248Here $\Omega$ is the period $\int_{E(\R)}|\omega|$, for $\omega$249a N\'eron differential on $E$,250$\Sha$ denotes the conjecturally finite Tate-Shafarevich group of $E$,251the $P_i$ for252$1\leq i\leq r$ are an independent set of points in $E(\Q)$253generating the subgroup $E'$, and $\heightpair{P_i}{P_j}$254denotes the height pairing. The fudge factors $c_p$255are all~1 for the curves we consider.256Recent work of Rubin and Kolyvagin257\cite{14} confirms the258conjecture of Birch and Swinnerton-Dyer in many cases of rank $r\le 1$.259260Examples illustrating (2) for the curves of ranks~4 and~5 of least261known conductor262are given in section~4.263264\heading265\S3 Rank Results266\endheading267Elliptic curves of prime conductor $N$ were conjectured to have prime268discriminant, except for the Setzer-Neumann curves and for 5~other269small conductor curves, see \cite{2, appendix}.270This is now known for {\it modular} curves by Theorem~2 of \cite{12}.271We therefore searched for curves of prime discriminant,272by looking for integral solutions to the equation273$$274c_4^3-c_6^2 = 1728\Delta,275\tag3276$$277where $c_4$ and $c_6$ are the usual invariants attached to278equation~(1), or more precisely, by fixing $a_4$, and searching for279$a_6$280for which~(3) has a solution with $\Delta$ prime and less than $10^8$.281This produced $\totalfound$ curves282including the $869$ expected curves with non-trivial torsion and283rank~0.284The set of $\totalnumber$285curves that we studied is most simply described by286$$287\{E :\left|\Delta\right| \le 10^8, |a_6|\le 2^{31}-1\},288$$289with $|\Delta|$ {\it prime\/}.290291We will not describe here all the details of the several thousand hours292of computations, but just say that,293imitating Mazur's description \cite{17} of ``infinite descent''294we searched for points by night, and calculated $L$-series derivatives295or regulators by day.296We may use the297infinite series formulas of~\cite{3} for the298derivatives of the $L$-series at $s=1$, since $E$ is assumed modular.299Then an upper bound for the analytic rank is found by300estimating the order of vanishing of $\Ls$ at $s=1$.301Using 2000 or 4000 terms of these series provided sufficient accuracy302for our purposes,303since the values are either~0 to several places or else are304far from~0 in most cases.305The period $\Omega$ is easily calculated using the306Arithmetic-Geometric mean algorithm of Gauss \cite{7},307and the height regulator $R=\det\left(\heightpair{P_i}{P_j}\right)$308is computed by using the method of Tate, as modified309by Silverman~\cite{16}, once points have been found by a search.310311The rank predictions are based on a combination of three calculations:312the rank parity, the analytic rank or order of vanishing of the313$L$-series,314and the number of independent points found which is a lower bound for315the316algebraic rank~$r$.317When the ranks coincide, as they should, we get a prediction from~(2)318of319$|\Sha|$, which should be an integer square.320The largest $|\Sha|$ we found was~$289$, for a curve of rank~0.321322For each curve, we keep its discriminant, parity, period, rank,323the appropriate $L$-derivative value, a list of $x$-coordinates of324the independent points found, and the regulator of these points.325326Of the curves analysed, $\totalpositive$ had327positive discriminant, and $\totalnegative$ had negative328discriminant.\footnote{The quotient is about $\rationegpos$, near329$\sqrt{3}$.330See section~5 for an explanation.}331An interesting phenomenon was the systematic influence of the332discriminant sign on all aspects of the arithmetic of the curve.333The rank distribution is given in the following table:334$$\vbox{\offinterlineskip335\hrule336\halign{\vrule#&\strut\quad#\hfil\quad&#\vrule\,\vrule&%337&\quad\hfil#\hfil\quad\vrule\cr338%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr339height2pt&\omit&&&&&&&\cr340&Rank && 0 & 1 & 2 & 3 & 4 & 5\cr341\noalign{\hrule}342\noalign{\vskip 2pt}343\noalign{\hrule}344%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr345height2pt&\omit&&&&&&&\cr346&$\Delta>0$&&\rankzeropos&\rankonepos&\ranktwopos&\rankthreepos%347&\rankfourpos&\rankfivepos\cr348&$\Delta<0$&&\rankzeroneg&\rankoneneg&\ranktwoneg&\rankthreeneg%349&\rankfourneg&\rankfiveneg\cr350height2pt&\omit&&&&&&&\cr351%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr352\noalign{\hrule}353%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr354height2pt&\omit&&&&&&&\cr355&Totals&&\rankzero&\rankone&\ranktwo&\rankthree%356&\rankfour&\rankfive\cr357&Percents&&\rankzeropct&\rankonepct&\ranktwopct&\rankthreepct%358&\rankfourpct& \rankfivepct\cr359%height2pt&\omit&&\omit&\omit&\omit&\omit&\omit&\cr}360height2pt&\omit&&&&&&&\cr}361\hrule}362$$363Thus $\rankevenGtZeroPct\%$ of the curves have even rank at364least~2. Note that the {\it365positive\/} discriminant curves give an even higher percentage!366367Define $N(r, X)$ to be the proportion of curves with conductor at368most $X$, and with rank at least $r$. Our data369shows that these functions are {\it increasing\/} functions of370$X$ for $r\geq 3$ and $X\le 10^8$.371In contrast \cite{6}, dealing with quadratic twists of elliptic curves,372suggests a decrease for the analogous functions.373374Denote the {\it average rank\/} among the curves375with discriminant sign376$\epsilon$ and conductor at most $X$ by $r_\epsilon(X)$.377In our data, the functions $r_\epsilon(X)$ for $X<10^8$ are378quite steadily climbing to the numbers379$1.04$ for $\Delta >0$, and to380$0.94$ for $\Delta <0$.381In particular, the average rank of our curves is {\it382not\/} $0.5$, as is expected to be the case for383twists~\cite{5},\cite{8}.384385In most cases, the predicted analytic rank386matches with the rank of points found, and the predicted $\Sha$ is387close to a non-zero integral square.388More precisely, this is so for all the curves of rank at least~3,389and for 95\% of the rank~2 curves.390There is a very small number of rank~2391curves for which the prediction of rank~2 is based solely on the392vanishing of $L$-series.393Note that in \cite{8}, a numerical study of one family of cubic twists,394only the analytic rank is estimated.395For many rank~1 curves (i.e., curves whose rank is predicted to be odd396for which $L'(E,1)$ does not vanish),397we have found no points by point searches398over moderate ranges, and do not expect to find any small points.399400\heading401\S4 Examples402\endheading403In the literature, one finds very few404verifications of the conjecture of Birch and Swinnerton-Dyer405for ranks $\geq 2$.406The paper \cite{3} on the curve of conductor~5077, found in \cite{2},407works out408the only rank~3 case that we know of.409It is perhaps not without interest to report the details for the410curves\footnote{These curves are new.411The curves of rank~4 reported in \cite{2} and \cite{10}412are respectively the 5th, 7th and 2nd curves of rank~4 in our list.}413of smallest known conductor of rank 4 and414of rank 5.415416The first rank~4 curve is:417$$418y^2+y=x^3+x^2-72x+210, \qquad \Delta =501029.419$$420For this curve, the predicted parity is even, the period421$\Omega=2.952580$,422the value and 2nd derivative of the $L$-series vanish to several423places, and $L^{(4)}(E,1)/4!=9.357978$.424The points with $x$-coordinates $5$, $4$, $3$, $6$ in order425of increasing height, form a basis. The regulator is $R=3.169424$,426which matches the427quotient of $L^{(4)}(E,1)/4!$ and $\Omega$, so $|\Sha|$ appears to be4281.429There are $21\times2$ integral points with $|x|<10^6$, while the second430curve of rank~4 has $28\times2$.431432The first rank~5 curve is:433$$434y^2+y=x^3-79x + 342, \qquad \Delta = -19047851.435$$436The previously found curves of rank~5437have conductors about ten times larger~\cite{2}, \cite{10}.438439The predicted parity is odd,440the period $\Omega=2.047641$, and441then the 1st and 3rd442$L$-series derivatives are 0 to several places, and443$L^{(5)}(E,1)/5! = 30.285711$.444Dividing by $\Omega$ gives $14.790539$.445There are $38\times 2$ integral points with $|x| <10^6$.446Those with $x=5$, 4, 3, 7, 0 form a basis for $E(\Q)$ with height447regulator $R= 14.790528$.448So $\Sha$ is predicted to be trivial.449There are four other curves of rank~5 with conductor less than450100~million.451The next one has discriminant~$-64921931$.452All turn out to have trivial $\Sha$, if we believe the conjectures.453454Some of the curves found by the search give rise to rather spectacular455cancellations.456A particular example is the curve of rank~0457$$458y^2+xy+y=x^3+x^2-12632622x-17287039382, \qquad \Delta=38593.459$$460Here $c_4 = 606365857$, and $c_6 = 14931454281967$,461and the equation $1728\Delta = c_4^3-c_6^2$462involves the difference of two 27~digit numbers!463464465\heading466\S 5 Heuristics467\endheading468While its connection with the Taniyama-Weil conjecture makes the469conductor a natural invariant,470the discriminant has turned out to be more useful in our heuristics.471The idea is to arrange the curves in472order by discriminant size, and then replace lattice-point counting473problems by area computations.474This works well, for instance, to count curves.475476% \zeta(12) changed to \zeta(10) 3 times477\proclaim{Heuristic Estimate}478We have the following estimates for the number of positive479and negative discriminants of absolute value at most $N$,480$$\align481A_{+}(N)&\sim \frac{\alpha_{+} }{\zeta(10)}N^{5/6},\\482A_{-}(N)&\sim \frac{\alpha_{-} }{\zeta(10)}N^{5/6}.483\endalign$$484\endproclaim485486The $\zeta(10)$ factor arises from taking into account the non-minimal487discriminants. Here488$\alpha_{+} = 0.4206$ and489$\alpha_{-} = 0.7285$ are490given by the elliptic integrals491$$492\alpha_{\pm} = \frac{\sqrt{3}}{10 }493\int_{\pm 1}^\infty \frac{du}{\sqrt{u^{3}\mp 1}},494$$495which arise from parametrising suitably the integrals that replace the496lattice-point counts. A well known identity of497Legendre, related to complex multiplication, is498$\alpha_{-} =\sqrt{3}\alpha_{+}$.499500\noindent{\smc Remark.} One should compare this estimate with the501results of \cite{4}.502While the number of elliptic curves grows like $N^{5/6}$, the503number of cubic fields grows like~$N$.504505Assuming the distribution of {\it prime\/} discriminants among506discriminants is that of prime numbers among all integers,507the number of prime discriminants of sign $\epsilon$ and of508size less than $N$ is then~$\alpha_\epsilon \Li(N^{5/6})$, where509$\Li(x)$ is the logarithmic integral.510The expected511number of curves with prime $|\Delta|<10^8$ is then~$311586$, comparing512rather well with the number $\totalfound$ found.513514Similar heuristic arguments have been applied to other invariants.515For instance, the {\it average period of a curve with516positive discriminant is $\sqrt{3/2}$ times the average period of a517curve with negative discriminant}.518This is also confirmed by the data.519The fits with the experimental data are so good that one520could hope for proofs in the near future.521522We have not as yet been able to provide heuristics for the523growth of the functions $N(r,X)$.524While our data may seem massive, $N= 10^8$ is not sufficient to525distinguish growth laws of $\log\log N$, $N^{1/12}$ or $N^{1/24}$526from constants.527So we have to be cautious in formulating conjectures528based on the numerical evidence.529530531\Refs532533\ref\no 1534\by B.~Birch and W.~Kuyk, editors535\book Modular Functions of One Variable IV536\bookinfo volume 476 of Lecture Notes in Mathematics537%\vol 476538\publ Springer-Verlag539\publaddr New York540\yr1975541\endref542543\ref\no 2544\by A.~Brumer and K.~Kramer545\paper The rank of elliptic curves546\jour Duke Math. 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